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Transcript
Topological phases and polaron physics
in ultra cold quantum gases
Dissertation
Fabian Grusdt
Vom Fachbereich Physik der Technischen Universität Kaiserslautern zur Erlangung
des akademischen Grades ”Doktor der Naturwissenschaften” genehmigte Dissertation
Betreuer: Prof. Dr. Michael Fleischhauer
Zweitgutachter: Prof. Dr. Eugene Demler
Datum der wissenschaftlichen Aussprache:
15. April 2015
D 386
Gewidmet den wichtigsten Lehrern
und Betreuern während meiner Schulzeit:
Anton Pöpperl
Matthias Schweinberger
Rudolf Lehn
Contents
Abstract
7
Kurzfassung
10
I
Topological States of Interacting Bosons
13
1 Introduction
1.1 Summary and Overview . . . . . . . . . . .
1.2 Fundamental Concepts . . . . . . . . . . . .
1.2.1 Topological order . . . . . . . . . . .
1.2.2 (Abelian) topological invariants . . .
1.2.3 Non-Abelian topological invariants .
1.2.4 The Hofstadter-Bose-Hubbard model
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2 Topology in the Superlattice Bose Hubbard Model
2.1 Outline and Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Superlattice Bose Hubbard model . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Hard-core bosons and chiral symmetry . . . . . . . . . . . . . . . . . .
2.2.2 Bulk phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Topological order in the superlattice Bose Hubbard model . . . . . . . . . . .
2.4 Topological edge states in the superlattice Bose Hubbard model . . . . . . . .
2.4.1 Failure of the bulk-boundary correspondence . . . . . . . . . . . . . .
2.4.2 Generalized bulk-boundary correspondence . . . . . . . . . . . . . . .
2.4.3 Experimental considerations . . . . . . . . . . . . . . . . . . . . . . . .
2.4.4 Relation to Majorana fermions . . . . . . . . . . . . . . . . . . . . . .
2.5 Extended superlattice Bose Hubbard model . . . . . . . . . . . . . . . . . . .
2.5.1 Model and bulk phases . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.2 Symmetry-protected topological classification of Mott insulators . . .
2.5.3 Topological excitations . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Thouless pump classification of inversion-symmetric models in one dimension
2.7 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 Realization of Fractional Chern Insulators in the Thin-torus Limit
3.1 Outline and Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Relation to the thin-torus-limit of the Hofstadter-Hubbard model
3.2.2 Possible experimental implementation . . . . . . . . . . . . . . .
3.3 Topology in the non-interacting system – Thouless pump . . . . . . . .
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CONTENTS
3.4
3.5
3.6
Interacting topological states . . . . . . . . . . . . . . .
3.4.1 Grand-canonical phase diagram . . . . . . . . . .
3.4.2 Harmonic trapping potential . . . . . . . . . . .
Topological classification and fractional Thouless pump
3.5.1 1 + 1D model and fractional Thouless pump . . .
3.5.2 1D model and SPT CDW . . . . . . . . . . . . .
Summary and Outlook . . . . . . . . . . . . . . . . . . .
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4 Fractional Quantum Hall E↵ect with Rydberg Interactions
4.1 Summary and Introduction . . . . . . . . . . . . . . . . . . . .
4.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Rydberg dressing . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Rydberg-interaction pseudopotentials in the LLL . . . .
4.3 Ground state for small blockade radii . . . . . . . . . . . . . . .
4.3.1 Ground states at ⌫ = 1/2 and ⌫ = 1/4 . . . . . . . . . .
4.3.2 Ground states at small fillings . . . . . . . . . . . . . . .
4.3.3 Correlated Wigner crystal of composite particles . . . .
4.4 E↵ects of finite blockade radius . . . . . . . . . . . . . . . . . .
4.4.1 Bubble crystal at small fillings . . . . . . . . . . . . . .
4.4.2 Large filling – indications for cluster liquids . . . . . . .
4.5 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . .
5 Topological Growing Scheme for Laughlin States
5.1 Outline and Introduction . . . . . . . . . . . . . .
5.2 Growing quantum states with topological order . .
5.3 Model . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Protocol – continuum . . . . . . . . . . . . . . . .
5.5 Performance . . . . . . . . . . . . . . . . . . . . . .
5.6 Protocol – lattice . . . . . . . . . . . . . . . . . . .
5.6.1 Buckyball-Hofstadter-Bose-Hubbard model
5.6.2 Numerical Simulation . . . . . . . . . . . .
5.6.3 Possible experimental realizations . . . . . .
5.7 Outlook – Beyond Laughlin states . . . . . . . . .
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Interferometry-based Detection of Topological Invariants
6 Introduction
6.1 Outline . . . . . . . . . . . . . . . . .
6.2 Fundamental Concepts . . . . . . . . .
6.2.1 Interferometric Measurement of
6.2.2 Z2 topological invariant . . . .
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Invariants
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7 Interferometric Measurement of Z2 Topological Invariants
7.1 Outline and Introduction . . . . . . . . . . . . . . . . . . . .
7.2 Interferometric measurement of the Z2 invariant . . . . . . .
7.2.1 Discontinuity of time-reversal polarization . . . . . . .
7.2.2 The twist scheme . . . . . . . . . . . . . . . . . . . . .
7.2.3 The Wilson loop scheme . . . . . . . . . . . . . . . . .
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Topological
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CONTENTS
7.3
7.4
7.5
7
7.2.4 Relation between Wilson loops and TRP
Twist scheme . . . . . . . . . . . . . . . . . . . .
7.3.1 Interferometric sequence . . . . . . . . . .
7.3.2 Dynamical-phase-free sequence . . . . . .
7.3.3 Experimental realization and limitations .
7.3.4 Formal definition and calculation of cTRP
7.3.5 Example: Kane-Mele model . . . . . . . .
Wilson loop scheme . . . . . . . . . . . . . . . .
7.4.1 TR Wilson loops and their phases . . . .
7.4.2 Zak phases . . . . . . . . . . . . . . . . .
7.4.3 Experimental realization . . . . . . . . . .
Summary and outlook . . . . . . . . . . . . . . .
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8 Interferometric Measurement of Many-Body Topological Invariants
8.1 Outline and Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Theoretical Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.2 Strong coupling approximation . . . . . . . . . . . . . . . . . . .
8.2.3 TP invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.4 Topological invariants: general considerations . . . . . . . . . . .
8.2.5 Strong coupling external TP invariant . . . . . . . . . . . . . . .
8.3 Integer Chern insulators and Integer Quantum Hall e↵ect . . . . . . . .
8.3.1 Topological invariants . . . . . . . . . . . . . . . . . . . . . . . .
8.3.2 TP in the Hofstadter Chern insulator - single hole approximation
8.3.3 Solution in strong coupling approximation . . . . . . . . . . . . .
8.3.4 Interacting fermions . . . . . . . . . . . . . . . . . . . . . . . . .
8.3.5 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3.6 TP in the integer quantum Hall e↵ect . . . . . . . . . . . . . . .
8.4 Fractional Quantum Hall e↵ect and Fractional Chern Insulators . . . . .
8.4.1 Topological invariants . . . . . . . . . . . . . . . . . . . . . . . .
8.4.2 Fractional Chern insulators . . . . . . . . . . . . . . . . . . . . .
8.4.3 Fractional quantum Hall e↵ect . . . . . . . . . . . . . . . . . . .
8.4.4 Experimental considerations . . . . . . . . . . . . . . . . . . . . .
8.5 Mott insulators and symmetry protected topological order . . . . . . . .
8.5.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.5.2 Polaron transformation . . . . . . . . . . . . . . . . . . . . . . .
8.5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.5.4 Approximate descriptions . . . . . . . . . . . . . . . . . . . . . .
8.6 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . .
III
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Polaron Physics with Ultra Cold Atoms
9 Introduction
9.1 Summary and Overview . . . . . . . . . .
9.2 Fundamental Concepts . . . . . . . . . . .
9.2.1 Polaron Hamiltonian for Impurities
9.2.2 Experimental Considerations . . .
9.2.3 The Lee-Low-Pines Transformation
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in a BEC
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8
CONTENTS
9.2.4
9.2.5
Weak-coupling or Mean-Field Polaron Theory . . . . . . . . . . . . . . . 200
Strong-coupling polaron theory . . . . . . . . . . . . . . . . . . . . . . . 206
10 RF Spectra of Fröhlich Polarons in a BEC
10.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2 RF spectra . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2.1 Model . . . . . . . . . . . . . . . . . . . . . . . . .
10.2.2 Formulation as a non-equilibrium problem . . . . .
10.3 Time-dependent MF theory . . . . . . . . . . . . . . . . .
10.3.1 Equations of motion – Dirac’s variational principle
10.4 Discussion of RF spectra . . . . . . . . . . . . . . . . . . .
10.4.1 Leading-order expansions . . . . . . . . . . . . . .
10.4.2 Universal high-energy RF tail . . . . . . . . . . . .
10.5 Non-equilibrium polaron dynamics . . . . . . . . . . . . .
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11 Weak-coupling theory of polaron Bloch oscillations in optical lattices
11.1 Summary and Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2.1 Derivation from microscopic model . . . . . . . . . . . . . . . . . .
11.2.2 Time-dependent Lee-Low-Pines transformation in the lattice . . .
11.3 Weak-coupling Theory of Lattice Polarons . . . . . . . . . . . . . . . . . .
11.3.1 Mean-field polaron wavefunction . . . . . . . . . . . . . . . . . . .
11.3.2 Results: equilibrium properties . . . . . . . . . . . . . . . . . . . .
11.4 Polaron Bloch Oscillations and Adiabatic Approximation . . . . . . . . .
11.4.1 Time-dependent variational wavefunctions . . . . . . . . . . . . . .
11.4.2 Adiabatic approximation . . . . . . . . . . . . . . . . . . . . . . .
11.4.3 Polaron trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.5 Non-Adiabatic Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.5.1 Impurity dynamics beyond the adiabatic approximation . . . . . .
11.5.2 Beyond wavepacket dynamics . . . . . . . . . . . . . . . . . . . . .
11.6 Polaron Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.6.1 General observations . . . . . . . . . . . . . . . . . . . . . . . . . .
11.6.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.6.3 Semi-analytical current-force relation . . . . . . . . . . . . . . . . .
11.6.4 Insufficiencies of the phenomenological Esaki-Tsu model . . . . . .
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219
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. 241
12 All-coupling Theory of the Fröhlich Polaron
12.1 Summary and Introduction . . . . . . . . . . . . . . . . .
12.2 Fröhlich Model and RG coupling constants . . . . . . . .
12.2.1 Towards the supersonic regime . . . . . . . . . . .
12.3 Renormalization Group Formalism for the Fröhlich model
12.3.1 Dimensional analysis . . . . . . . . . . . . . . . . .
12.3.2 Formulation of the RG . . . . . . . . . . . . . . . .
12.4 Polaron Groundstate Energy . . . . . . . . . . . . . . . .
12.4.1 Logarithmic UV Divergence of the polaron energy
12.4.2 Regularization of the Polaron Energy . . . . . . .
12.5 Other Groundstate Polaron Properties – Derivation . . . .
12.5.1 Polaron Mass . . . . . . . . . . . . . . . . . . . . .
12.5.2 Phonon Number . . . . . . . . . . . . . . . . . . .
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261
CONTENTS
12.5.3
12.6 Other
12.6.1
12.6.2
12.6.3
12.6.4
9
Quasiparticle weight . . . . . . .
Groundstate Polaron Properties –
Solutions of RG flow equations .
Polaron Mass . . . . . . . . . . .
Phonon Number . . . . . . . . .
Quasiparticle weight . . . . . . .
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Results
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13 Dynamical RG for Intermediate-coupling Fröhlich Polarons
13.1 Formulation of the dynamical RG . . . . . . . . . . . . . . . . .
13.1.1 Phonon number and momentum . . . . . . . . . . . . .
13.1.2 Time-dependent overlap . . . . . . . . . . . . . . . . . .
13.2 Results: Spectral Function of the Fröhlich Polaron . . . . . . .
13.3 Results: Dynamics of polaron formation . . . . . . . . . . . . .
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Appendices
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261
262
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266
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269
. 270
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. 275
. 281
. 283
287
A Quantization of the fractional part of the charge on the edge
289
B Exact diagonalization in the lowest Landau level
293
C Proof of the Wilson loop formula for the Z2 invariant
295
D Bloch oscillation’s equations of motion
297
E Non-universal Franck-Condon factor phases
299
F TR invariant non-adiabatic two-band dynamics
301
G Hofstadter TP in the polaron frame
305
H Measurement of TP invariant in the Hofstadter problem
309
H.0.1 E↵ect of driving terms on impurity . . . . . . . . . . . . . . . . . . . . . 309
H.0.2 Exact treatment of driving terms . . . . . . . . . . . . . . . . . . . . . . 310
I
Lowest Chern band projection
313
J Impurity-boson interactions in a lattice
315
K Static MF polarons in a lattice
317
L Impurity density in the lab frame
319
M Adiabatic wavepacket dynamics
321
N Discussion and extension of the analytical current-force relation
323
O Alternative derivation of polaron current
325
P Renormalized impurity mass
327
10
CONTENTS
Q Polaron Properties from RG
329
Q.1 Polaron phonon number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
Q.2 Polaron momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
Q.3 Quasiparticle weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
R Alternative Check of the RG – Kagan-Prokof ’ev theory
R.1 Simplified Model . . . . . . . . . . . . . . . . . . . . . . . .
R.2 Relation to Kagan and Prokof’ev theory . . . . . . . . . . .
R.2.1 Kagan-Prokof’ev theory . . . . . . . . . . . . . . . .
R.2.2 Polaron Hamiltonian . . . . . . . . . . . . . . . . . .
R.2.3 Application to polaron case . . . . . . . . . . . . . .
R.3 Comparison with RG . . . . . . . . . . . . . . . . . . . . . .
R.4 Results: Kagan-Prokof’ev versus RG . . . . . . . . . . . . .
R.4.1 Polaron mass term . . . . . . . . . . . . . . . . . . .
R.4.2 Polaron energy . . . . . . . . . . . . . . . . . . . . .
R.5 Asymptotic solutions of Kagan-Prokof’ev theory . . . . . .
R.5.1 UV asymptotics . . . . . . . . . . . . . . . . . . . .
R.5.2 IR asymptotics . . . . . . . . . . . . . . . . . . . . .
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333
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S Summary of the dRG
341
S.1 Time-dependent observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
S.2 Time-dependent overlap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
T Time-dependent overlap – MF versus dRG
345
Publications
347
Bibliography
350
Thanks to...
377
12
CONTENTS
Abstract
Since the advent of quantum mechanics, physicists believe to have at hand a microscopic
description of almost any phenomenon than can be observed at moderate energy scales. This
typically means that we can write down a Hamiltonian Ĥ (or a Lagrangian) which describes
the relevant microscopic degrees of freedom. However, this by far does not mean that all
the physics is well understood at moderate energies! The interplay of many indistinguishable
particles – of the order of 1023 – gives rise to rich physics, which some – including the author
– ultimately believe to include even as complex phenomena as human life.
One key challenge today is to understand what di↵erent phases of matter can exist in the
context of non-relativistic many-body quantum mechanics (at equilibrium). Until recently
there have been two main approaches to unravel the behavior of a many-body quantum system. One approach starts by considering non-interacting particles and treating their mutual
interactions in a perturbative manner. For example, Landau-Fermi-liquid theory is of this nature [1, 2]. Another approach is based on mean-field theory, describing quantum systems by
essentially classical order parameters. This approach was pioneered by Ginzburg and Landau
in their description of superconductivity [3].
This thesis deals with a class of systems which lie beyond the realm of perturbation theory,
and can neither be described by a mean-field ansatz. In the first two parts (I and II) phases
of matter are investigated which are characterized by their topological order. Topological
order is a true quantum phenomenon, related to the entanglement present in the many-body
wavefunction [4]. In the third part III of the thesis polarons are investigated, a system
involving impurities that, in some regimes of the coupling constant, can not be understood
from perturbation theory. In contrast to most impurity problems with known non-perturbative
descriptions, it involves mobile impurities.
Physics is the natural science which ultimately combines a sound mathematical description
of nature with factual experiments. The many-body systems considered in this thesis can be
realized in various experimentally accessible situations. Topological order was discovered in
the context of the quantum Hall e↵ect of interacting electrons, but it may as well play a role
in high-energy physics. Polarons, too, were discovered in solid-state systems, where electrons
(= impurities) interact with the surrounding phonons.
A shortcoming of electronic systems for a systematic comparison with theoretical models is
their limited tunability. The parameters entering model Hamiltonians can only be changed by
small amounts, or not at all, and often they stand in a non-universal relation to one another.
Ultra cold quantum gases, but also systems of photons with a wide range of frequencies, have
become promising alternatives. By tuning optical lattice potentials almost at will, paradigmatic many-body systems can now be quantum simulated [5] and individual quanta can be
controlled fully coherently. In this thesis theoretical aspects of the quantum simulation of
the non-perturbative phenomena of topological order and polaron formation at intermediate
coupling are explored in the context of ultra cold atoms and photons.
Topological order is a consequence of the constraint posed on symmetric quantum systems by demanding locality of the Hamiltonian [6]. What kind of topological order may exist
depends crucially on the dimensionality. Its most interesting intrinsic form is found in two
dimensions, when time-reversal symmetry is broken. In this case a particularly exciting direction is the possibility of excitations with anyonic statistics, which are neither bosons nor
fermions. When non-Abelian anyons are considered [7, 8] this allows to build a topological
quantum computer [9, 10, 11]. Because local modifications of the Hamiltonian are unable to
change the topological order of a state, topological quantum computers are completely robust
to disorder.
CONTENTS
13
An obvious goal of experiments with ultra cold atoms (or photons) is to carry over their
coherent control to individual anyons. Potentially, this will enable first the unambiguous
detection of anyons and second the implementation of a topological quantum computer. To
investigate topological order with an analogue quantum simulator the following steps have to
be taken. All of them will be addressed in this thesis.
(i) Implementation of a suitable Hamiltonian.
(ii) Preparation of the topological ground state.
(iii) Detection of its topological order.
In part I we start by suggesting an implementation of symmetry protected topological
order in a paradigmatic one-dimensional model of interacting bosons. To this end the SuSchrie↵er-Heeger model [12], which has recently been realized with non-interacting bosons
[13], is supplemented by boson-boson interactions. Indicators of its topological order are
discussed, with special emphasis on the bulk-boundary correspondence. Next, this model is
generalized to quasi one-dimensional ladder systems and we show how they can be employed
to quantum simulate the thin-torus limit of the two-dimensional Hofstadter Bose Hubbard
model [14] with current experiments [15]. We continue by investigating truly two-dimensional
systems and examine the fractional quantum Hall e↵ect of bosons with Rydberg interactions.
In particular we show that they give rise to exotic quantum states in the lowest Landau level,
including states with non-Abelian excitations and correlated Wigner crystals.
At the end of part I, we propose a novel scheme for the preparation of topologically ordered
ground states. We focus in particular on Laughlin states and show how they can be grown
step-by-step using a dynamical protocol.
In part II we examine how topological order can be detected in ultra cold quantum gases.
While local measurements are incapable to distinguish di↵erent topological orders, interferometric measurements are ideally suited for this purpose. This was recently demonstrated in
a direct measurement of the topological invariant of the Su-Schrie↵er-Heeger model [13]. In
this thesis we generalize the interferometric approach to non-Abelian topological invariants,
involving multiple bands. In particular we develop a measurement scheme for the Z2 topological invariant of two- and three-dimensional topological insulators. In addition, we generalize
the interferometric scheme to interacting topological invariants characterizing e.g. fractional
Chern insulators. To this end we introduce the concept of a topological polaron and couple a
mobile impurity to a topological excitation.
When a mobile impurity interacts with a surrounding bath, e.g. of phonons [16], it becomes
a dressed quasiparticle with an increased mass. This polaron state was introduced by Landau
and Pekar, who solved the problem in the strong-coupling regime [17, 18]. Shortly afterwards,
using perturbative and mean-field methods, the polaron problem was also solved in the weakcoupling regime by Lee, Low and Pines [19]. Later the intermediate-coupling regime received
much attention, where neither mean-field nor perturbative methods work. Nowadays it is
established that no phase transition occurs from weak- to strong-coupling [20], but until
recently an efficient description of intermediate coupling polarons was still lacking.
In part III of this thesis a novel theoretical description is developed for intermediate coupling polarons, based on a renormalization group approach. By comparing the predictions of
our method to numerically involved quantum Monte Carlo calculations [21, 22] we establish
its validity all the way from weak to intermediate couplings. We also compare our findings to
results obtained from Feynman’s variational ansatz [23], which was for a long time believed
to constitute a superior all-coupling polaron theory. Our method is superior to Feynman’s
approach, however, in that it sheds new light on shortcomings of Feynman’s ansatz in describing accurately acoustic polarons [24, 22], while being computationally cheap. In addition, we
14
CONTENTS
demonstrate that our new approach can be generalized to non-equilibrium situations, which
are beyond the scope of numerical Monte Carlo calculations.
We apply our newly developed polaron theory to Fröhlich polarons in a Bose Einstein
condensate (BEC), where impurity atoms interact with the Bogoliubov phonons. We start
by presenting a mean-field description of the BEC polaron, which we employ to calculate
the spectral function of the impurity in the weak-coupling regime. This furthermore enables
us to predict non-equilibrium dynamics of polarons. We proceed by generalizing our meanfield description to lattice polarons, where the impurities are confined to an optical lattice
potential. By applying an external force, polaron Bloch oscillations can be driven which we
examine in detail. In particular we predict a sub-Ohmic current-force relation, which strongly
depends on the dimensionality of the system.
Next we return to continuum polarons and derive our all-coupling renormalization groupbased polaron theory. We use it to predict the e↵ective mass of BEC polarons, which is
subject of much controversy and can be measured in experiments with ultra cold atoms that
are currently under construction in di↵erent laboratories. A particular result of our theory –
which we derive analytically – is that the polaron energy predicted by the Fröhlich Hamiltonian
is logarithmically divergent in the ultra-violet regime in three dimensions. We attribute it to
an insufficient treatment of the Lippmann-Schwinger equation relating the interaction strength
to the universal scattering length, which allows us to construct a regularization scheme for
the logarithmic divergence. Finally we generalize our approach to non-equilibrium situations,
and calculate in particular the spectral function of the polaron in the intermediate coupling
regime.
CONTENTS
15
Kurzfassung
Seit der Formulierung der Quantenmechanik glauben Physiker im Besitz einer mikroskopischen Beschreibung fast jeden Phänomens bei moderaten Energien zu sein. In der Praxis heißt
das typischerweise, dass wir einen Hamiltonoperator Ĥ (bzw. einen Lagrangian) aufschreiben
können, der die relevanten mikroskopischen Freiheitsgrade beschreibt. Trotzdem wäre es
weit gefehlt zu behaupten, dass die gesamte Physik bei moderaten Energieskalen gut verstanden ist! Das Zusammenspiel vieler ununterscheidbarer Teilchen – größenordnungsmäßig
1023 Stück – führt zu vielfältigen physikalischen E↵ekten, die, so der Glaube einiger – der
Autor eingeschlossen, schlussendlich selbst solch komplexe Phänomene wie das menschliche
Leben mit einschließen.
Eine der wichtigsten aktuellen Herausforderungen besteht darin, zu verstehen, welche
verschiedenen Phasen der Materie existieren können, die durch die nicht-relativistische Vielteilchen Quantenmechanik (im Gleichgewicht) beschrieben werden. Bis vor einiger Zeit wurden insbesondere zwei Ansätze verfolgt, um das Verhalten eines Vielteilchen Quantensystems
zu verstehen. Eine Methode beginnt damit, nicht-wechselwirkende Teilchen zu betrachten
und deren Wechselwirkung untereinander störungstheoretisch zu behandeln. Zum Beispiel
ist die Landau-Fermi-liquid Theorie von dieser Natur [1, 2]. Ein alternativer Ansatz basiert
auf der mean-field Theorie, wobei Quantensysteme durch essentiell klassische Ordnungsparameter beschrieben werden. Dieser Ansatz wurde zuerst von Ginzburg und Landau in deren
Beschreibung der Supraleitung erforscht [3].
Die vorliegende Arbeit handelt von einer Klasse von Systemen, die über die Grenzen der
Störungstheorie hinweg gehen und auch nicht mit einem mean-field Ansatz gelöst werden
können. In den ersten zwei Teilen der Arbeit (I und II) werden Phasen der Materie untersucht, die durch ihre topologische Ordnung charakterisiert sind. Topologische Ordnung
ist ein reines Quantenphänomen, das im Zusammenhang steht mit der Verschränkung in
der Vielteilchen Wellenfunktion [4]. Im dritten Teil (III) der Arbeit werden Polaronen untersucht, ein System bestehend aus Fremdkörpern, das in bestimmten Bereichen der Kopplungsstärke nicht allein störungstheoretisch verstanden werden kann. Im Gegensatz zu den
meisten Fremdkörperproblemen mit bekannten nicht-störungstheoretischen Beschreibungen
behandeln wir bewegliche Fremdkörper.
Physik ist die Naturwissenschaft, die ultimativ eine saubere mathematische Beschreibung der Natur mit faktischen Experimenten vereint. Die im Rahmen dieser Arbeit betrachteten Vielteilchensysteme können in verschiedenen experimentellen Situationen realisiert
werden. Topologische Ordnung wurde im Zusammenhang mit dem Quanten Hall E↵ekt
wechselwirkender Elektronen entdeckt, aber sie könnte auch in der Hochenergiephysik eine
Rolle spielen. Auch Polaronen wurden in Festkörpersystemen erforscht, wobei Elektronen (=
Fremdkörper) mit den umgebenden Phononen wechselwirken.
Ein Nachteil elektronischer Systeme für einen systematischen Vergleich mit theoretischen
Modellen ist ihre limitierte Verstellbarkeit. Die in Modellhamiltonoperatoren eingehenden
Parameter können nur um kleine Beträge verändert werden, wenn überhaupt, und oft stehen
sie in nicht-universellem Zusammenhang miteinander. Ultrakalte Quantengase, aber auch
Systeme mit Photonen aus einem weitreichenden Frequenzbereich, haben sich als vielversprechende Alternativen erwiesen. Durch die fast unbegrenzte Verstimmbarkeit optischer
Gitter können paradigmatische Vielteilchensysteme heute quantensimuliert werden [5], und
individuelle Quanten können vollkommen kohärent kontrolliert werden. In dieser Arbeit werden theoretische Aspekte der Quantensimulation der nicht-störungstheoretischen Phänomene
der topologischen Ordnung und der Polaronenbildung bei mittlerer Kopplung untersucht, im
Zusammenhang mit ultrakalten Atomen und Photonen.
16
CONTENTS
Topologische Ordnung ist eine Konsequenz aus den Einschränkungen von symmetrischen
Quantensystemen durch die Forderung der Lokalität [6]. Welche Arten von topologischer
Ordnung existieren können, hängt daher von der Dimensionalität ab. Die interessanteste
intrinsische Form der topologischen Ordnung tritt in zwei Dimensionen auf, wenn die Zeitumkehrinvarianz gebrochen ist. In diesem Fall beschäftigt sich eine besonders spannende
Forschungsrichtung mit der Möglichkeit von Elementaranregungen mit anyonischer Statistik,
die weder Bosonen noch Fermionen sind. Mit Hilfe nicht-Abelscher Anyonen [7, 8] kann
sogar ein Quantencomputer gebaut werden [9, 10, 11]. Nachdem lokale Veränderungen des
Hamiltonoperators nicht dazu in der Lage sind, die topologische Ordnung eines Zustandes zu
verändern, sind topologische Quantencomputer vollständig robust gegenüber Unordnung.
Ein o↵ensichtliches Ziel von Experimenten mit ultrakalten Atomen (wie auch Photonen)
ist es, die erlangte kohärente Kontrolle auf Anyonen zu übertragen. Das wird es potentiell
ermöglichen, zunächst eindeutig die Existenz von Anyonen nachzuweisen und dann einen
topologischen Quantencomputer zu implementieren. Um topologische Ordnung mit Hilfe eines
analogen Quantensimulators zu erforschen, müssen die folgenden Hürden überwunden werden.
Alle Punkte werden im Laufe dieser Arbeit behandelt.
(i) Konstruktion eines geeigneten Hamiltonoperators.
(ii) Präparation eines topologischen Grundzustandes.
(iii) Detektion seiner topologischen Ordnung.
Im Teil I starten wir damit, einen Vorschlag zur Implementierung von symmetriegeschützter
topologischer Ordnung in einem paradigmatischen eindimensionalen Modell wechselwirkender
Bosonen zu machen. Dazu erweitern wir das Su-Schrie↵er-Heeger Modell [12], das kürzlich mit
schwach wechselwirkenden Bosonen realisiert wurde [13], durch starke Boson-Boson Wechselwirkungen. Indikatoren für seine topologische Ordnung werden diskutiert, mit einem Schwerpunkt auf der bulk-boundary Korrespondenz. Danach wird dieses Modell verallgemeinert
zu quasi-ein-dimensionalen Leitermodellen, und wir zeigen, wie solche Systeme dazu verwendet werden können, den thin-torus Limes des zwei-dimensionalen Hofstadter Bose Hubbard
Modells [14] mit aktuellen Experimenten [15] quantenzusimulieren. Wir gehen dann zur Betrachtung zwei-dimensionaler Systeme über und untersuchen den fraktionalen Quanten Hall
E↵ekt von Bosonen mit Rydberg Wechselwirkungen. Insbesondere zeigen wir, dass diese zu
exotischen Quantenzuständen im untersten Landau Level führen, einschließlich Zuständen mit
nicht-Abelschen Anregungen und korrelierten Wigner Kristallen.
Am Ende des ersten Teils I schlagen wir ein neuartiges Präparationsschema für topologisch
geordnete Grundzustände vor. Wir betrachten insbesondere Laughlin Zustände und zeigen
wie sie Schritt für Schritt in einem dynamischen Protokoll gewachsen werden können.
Im zweiten Teil II untersuchen wir, wie topologische Ordnung in ultrakalten Quantengasen
detektiert werden kann. Während lokale Messungen unzulänglich sind, um verschiedene topologische Ordnungen voneinander zu unterscheiden, sind interferometrische Methoden hierzu
ideal geeignet. Das wurde kürzlich mit einer direkten Messung der topologischen Invarianten des Su-Schrie↵er-Heeger Modells demonstriert [13]. In dieser Arbeit verallgemeinern wir
den interferometrischen Ansatz für nicht-Abelsche topologische Invarianten, die Sammlungen
mehrerer Bänder charakterisieren. Insbesondere entwickeln wir eine Messmethode für die Z2
topologische Invariante von zwei- und drei-dimensionalen topologischen Isolatoren. Zusätzlich
verallgemeinern wir das interferometrische Schema für wechselwirkende topologische Invarianten, die beispielsweise fraktionale Chern-Isolatoren charakterisieren. Dazu führen wir das
Konzept eines topologischen Polarons ein und koppeln ein bewegliches Fremdkörperatom an
eine topologische Anregung.
CONTENTS
17
Wenn ein beweglicher Fremdkörper mit einem umgebenden Bad wechselwirkt, bestehend
z.B. aus Phononen [16], dann wird es zu einem dekorierten Quasiteilchen mit einer erhöhten
Masse. Dieser Polaron Zustand wurde eingeführt von Landau and Pekar, die das Problem im
stark wechselwirkenden Grenzfall gelöst haben [17, 18]. Kurz darauf wurde das Polaron Problem außerdem mit Hilfe störungstheoretischer und mean-field Methoden im schwach wechselwirkenden Grenzfall von Lee, Low und Pines gelöst [19]. Später erlangte insbesondere
der Bereich mittlerer Wechselwirkungsstärke viel Aufmerksamkeit, wo weder mean-field noch
störungstheoretische Methoden funktionieren. Heutzutage ist etabliert, dass es zwischen den
schwach- und stark wechselwirkenden Grenzfällen keinen Phasenübergang gibt [20]. Trotzdem existierte bis vor kurzem keine effiziente theoretische Beschreibung von Polaronen bei
mittleren Wechselwirkungen.
Im Teil III dieser Arbeit wird eine neuartige theoretische Beschreibung von Polaronen bei
mittleren Wechselwirkungen entwickelt, basierend auf einem Renormierungsgruppenansatz.
Durch Vergleich von Vorhersagen unserer Methode mit anspruchsvollen numerischen MonteCarlo Rechnungen [21, 22] etablieren wir die Gültigkeit unseres Ansatzes im gesamten Bereich
von schwachen zu mittleren Wechselwirkungen. Außerdem vergleichen wir unsere Ergebnisse mit Rechnungen nach Feynmans Variationsansatz [23], der bisher als übermächtige Polarontheorie für beliebige Kopplungsstärken galt. Unsere Methode macht dagegen bessere
Vorhersagen als Feynmans Ansatz, der für akustische Polaronen keine akkurate Beschreibung darstellt [24, 22]. Des weiteren zeigen wir, dass unsere Herangehensweise auf NichtGleichgewichtssituationen verallgemeinert werden kann, was über die Reichweite numerischer
Monte-Carlo Rechnungen hinausgeht.
Wir wenden unsere neu entwickelte Polaronentheorie bei Fröhlich Polaronen in einem Bose
Einstein Kondensat (BEC) an, wobei Fremdkörperatome mit Bogoliubov-Phononen wechselwirken. Wir beginnen damit, eine mean-field Behandlung des BEC Polarons vorzustellen,
womit wir die spektrale Funktion des Fremdkörperatoms im schwach wechselwirkenden Bereich ausrechnen. Des weiteren ermöglicht uns diese Methode, die Nicht-Gleichgewichtsdynamik
von Polaronen vorherzusagen. Wir fahren fort, indem wir unsere mean-field Beschreibung
zu Gitterpolaronen verallgemeinern, wobei die Fremdkörperatome in einem optischen Gitter
gefangen sind. Durch das Anlegen einer externen Kraft können Polaron-Blochoszillationen
getrieben werden, die wir im Detail untersuchen. Insbesondere sagen wir eine sub-Ohmsche
Strom-Kraft Beziehung vorher, die stark von der Dimensionalität des Systems abhängt.
Zuletzt wenden wir uns wieder den Polaronen im Kontinuum zu und leiten unsere Renormierungsgruppen Polarontheorie für beliebige Kopplungsstärken her. Wir verwenden sie, um
die e↵ektive Polaronenmasse im BEC zu berechnen, die Gegenstand kontroverser Diskussionen
ist und in Experimenten mit ultrakalten Atomen gemessen werden kann, welche sich derzeit
in verschiedenen Labors im Aufbau befinden. Ein spezielles Ergebnis unserer Theorie – das
wir analytisch ableiten – ist, dass die vom Fröhlich-Hamiltonoperator vorhergesagte Polaronenergie in drei Dimensionen im Ultraviolettbereich logarithmisch divergiert. Wir führen diese
Divergenz auf eine unzulängliche Behandlung der Lippmann-Schwinger Gleichung zurück,
welche die Wechselwirkungsstärke mit der universellen Streulänge verbindet. Dies erlaubt
es uns, ein Regularisierungsschema zu konstruieren. Schlussendlich verallgemeinern wir unseren Zugang auf Nicht-Gleichgewichtssituationen und berechnen insbesondere die spektrale
Funktion des Polarons bei mittleren Kopplungsstärken.
18
CONTENTS
Part I
Topological States of Interacting
Bosons
19
Chapter 1
Introduction
1.1
Summary and Overview
This first part of this thesis deals with interacting bosons, in phases of matter which are
characterized by their topological order. Roughly speaking, topology is a concept of classifying
objects by their global properties. Two objects belong to the same topological class if and
only if they can be continuously transformed into each other without violating certain sets
of rules. Topology as a classification scheme for phases of matter received attention among
condensed matter physicists for the first time when the quantum Hall e↵ect was discovered by
von Klitzing and co-workers in 1980 [25]. Completely unexpected they made two observations
of major importance: Firstly, for large magnetic fields the Hall conductivity in a quantum
Hall setup no longer increases monotonically but forms plateaus. Secondly, the value of the
Hall conductivity on these plateaus takes strictly quantized values, xy = Ce2 /h, where the
so-called Chern number C is an integer with a reproducibility to within (nowadays) 2 ⇥ 10 9
[26] despite the presence of residual disorder in experimental samples (e is the electron charge
and h the Planck constant). In a celebrated paper Thouless, Kohmoto, Nightingale and den
Nijs (TKNN) [27] shortly afterwards explained the quantization of the quantum Hall e↵ect by
relating the Hall response to the topology encoded in the quantum mechanical wavefunction.
From a modern point of view, the lesson learned by the community from the quantum
Hall e↵ect was that Ginzburg’s and Landau’s classification scheme for the phases of matter
is incomplete. Their approach consists of classifying phases by their broken symmetries. For
example, a superfluid spontaneously breaks the U (1) gauge symmetry and is hence distinct
from a Mott insulating phase which has no long-range order. The quantum Hall e↵ect, on the
other hand, does not break any symmetries but is nevertheless distinct from a trivial band
insulator by its topological order [4]. The Chern number C (or TKNN invariant) provides a
measure of this topological order. Recently this concept was generalized by Xiao-Gang Wen
and co-workes, who introduced a more general topological classification scheme [6].
After the discovery of the quantum Hall e↵ects, intense subsequent search for further
quantum phases with topological order revealed the existence of an entire class of topological
insulators and superconductors. In a groundbreaking work [28] Kane and Mele showed how
additional symmetries (in their case time-reversal invariance) can give rise to subclasses of
quantum phases with symmetry protected topological order. This not only led to the discovery
of the quantum spin Hall e↵ect [28, 29, 30, 31, 32, 33] but triggered a wave of renown interest
in topological phases of matter (see [34, 35, 36] for reviews). In a first step, all topological
insulators realizable with free fermions were classified [37], and extensions of the scheme to
arbitrary dimensions were developed [38, 39]. Along these lines also topological superconduc21
22
CHAPTER 1. INTRODUCTION
tors were classified, which – as pointed out previously by Kitaev [9] – are ideal candidates
for realizing a decoherence-free quantum memory with Majorana fermions. This is only one
among many possible applications of topological order and triggered an experimental search
for Majorana fermions [40, 41, 42, 43, 44].
Among the key applications of topological order is the possibility of implementing robust quantum gates with non-Abelian anyons. Anyons are quasiparticle excitations in twodimensional systems which, upon exchange, behave neither like bosons nor fermions. Instead
of acquiring a sign ±1 in the process of being exchanged, they can pick up an arbitrary
phase ei# or even a unitary U (N ) matrix Û acting on a degenerate ground state manifold.
In a seminal paper Kitaev pointed out that these matrices can be used to implement robust
quantum gates [10] and to realize a topological quantum computer [11]. Shortly after the
discovery of the fractional quantum Hall e↵ect of interacting electrons [45, 46], Laughlin suggested that its elementary excitations are fractionally charged (Abelian) anyons [47]. Later
the unexpected discovery of a quantized Hall plateau at the even filling fraction ⌫ = 5/2 [48]
was interpreted as a fractional quantum Hall state with non-Abelian anyonic excitations [49],
triggering additional theoretical interest in states of interacting particles in the presence of
strong magnetic fields [7, 50]. On the experimental side ongoing e↵orts are made to gain control over individual anyons in fractional quantum Hall samples and observe the theoretically
predicted non-Abelian braiding statistics.
A major limitation for solid state experiments are the small involved length scales, well
below an optical wavelength. This makes experiments aiming to gain coherent control over
individual anyons extremely challenging. Therefore alternative systems of ultra cold atoms
and photons are currently explored, where an unprecedented coherent control on the level of
single quanta has been achieved [51, 52]. While the ultimate goal of such experiments will
be to carry over this control to exotic topological excitations, the applications are manifold:
From a fundamental point of view, the realization of the fractional quantum Hall e↵ect and,
more generally, topological states with bosons is an outstanding challenge. The ability to
tune interactions and engineer Hamiltonians for ultra cold atoms and photons will allow to
investigate the intricate interplay of topology and interactions in detail, in a regime where the
computational capabilities even of the best conventional computers are insufficient by a huge
margin. Eventually this might even lead to the discovery of completely new quantum phases
of matter, going beyond the current understanding of theorists.
In this first part of the thesis possible routes are explored towards achieving the goals
described above. To this end we will mainly discuss models of interacting bosons, and show
how they can be realized in current experiments with ultra cold atoms and photons. We will
start by discussing a simple toy model of interacting bosons in a one-dimensional topologically
non-trivial band structure, and address the question how the topological order manifests itself
in a possible experiment [P12], [P2]. In particular we discuss the bulk-boundary correspondence. Then we move on by suggesting a realistic experimental setup for the realization of
Laughlin-type fractional quantum Hall states in quasi one-dimensional ladder systems using
ultra cold bosons [P7]. This is an important step towards finding a suitable setup for realizing the bosonic fractional quantum Hall e↵ect, because in a simplified experiment new
techniques can be tested and developed. Finally we demonstrate how the versatile quantum
optics toolbox allows to engineer long-range interactions using Rydberg excitations, and ask
the question how they modify the ”standard” fractional quantum Hall physics familiar from
electrons in the lowest Landau level [P1].
The first requirement to realize a state with topological order is to find a Hamiltonian
of which it is the ground state, and to engineer this Hamiltonian in an experiment. Then
1.2. FUNDAMENTAL CONCEPTS
23
a second fundamental question is how such a ground state can reliably be prepared. While
the cooling methods available for ultra cold quantum gases have reached remarkably small
values of the temperature on an absolute scale [53], the achievable temperatures are still
comparably large when it comes to relative scales. For example the experimental observation
of an antiferromagnet with ultra cold atoms in the Fermi-Hubbard model is an outstanding
challenge [54]. Similarly the preparation of a fractional quantum Hall state requires extremely
small temperatures, out of reach with current technology. In this thesis an alternative route
is suggested how to prepare states with topological order, based on a dynamical scheme [P6],
[P11]. In particular we show how the concept of crystal growth can be generalized to fractional
quantum Hall states, where in every step of the growing scheme the topological order has to
be preserved. More generally, dynamical phenomena in models with topological order can be
discussed, see e.g. [P3].
The first part of this thesis is organized as follows. In the remainder (Section 1.2) of this
introductory chapter the fundamental concepts of topological order will be presented. An
overview of the paradigmatic Hofstadter-Bose-Hubbard model is given. In Chap.2 we discuss
topological aspects of the one-dimensional super-lattice Bose Hubbard model. In particular
we explore its relation to the paradigmatic Su-Schrie↵er-Heeger model [12] and investigate
the breakdown of the bulk-boundary correspondence. In Chap.3 we consider a quasi onedimensional ladder system in a magnetic field, fractionally filled with interacting bosons. We
show how Laughlin-type states and a fractionally quantized Thouless pump can be realized
in this setup. Our proposal is motivated by – and can be implemented in – the experiment
described in Ref. [15]. In Chap. 4 we discuss fractional quantum Hall physics in the lowest
Landau level with bosons subject to Rydberg interactions. In particular we discuss Wigner
crystallization at small atomic densities and the emergence of non-Abelian quantum liquids at
large densities. This Chapter completes the discussion presented in the diploma thesis of the
author, Ref. [55]. In Chap. 5 we develop a dynamical growing scheme for correlated Laughlin
states and present exact numerical simulations for the Hofstadter-Bose-Hubbard model.
1.2
Fundamental Concepts
In this chapter we introduce the fundamental concepts underlying topological order. Instead
of presenting them in the chronological order of their invention, we adapt a modern point
of view in Section 1.2.1, put forward by Wen and co-workers [6]. Specifically we discuss a
general classification scheme for gapped topological phases of matter. In Section 1.2.2 a review
of topological invariants is given, which provide a quantitative measure how two topological
phases di↵er from one another. The material in this chapter has all been discussed at various
places in the existing literature and all relevant references will be provided. However the
selection of topics and the particular presentation reflects the author’s personal view on recent
developments in the field. Calculations without a reference were performed by the author.
1.2.1
Topological order
We start the discussion of topological order by summarizing results obtained in an important
paper by Chen, Gu and Wen [6]. The starting point for the analysis is the question what
di↵erent phases of matter can – in principle – be realized by local Hamiltonians. Note that at
this point we take a mathematical point of view where a Hamiltonian is merely a very large
Hermitian matrix. For such a matrix to describe any fundamental physical Hamiltonian, it
has to be local, given that all known fundamental physical process are local.
24
CHAPTER 1. INTRODUCTION
Before we can start discussing how topology comes into play, let us first ask more fundamentally how di↵erent phases of matter can be distinguished in the first place. An elegant
way how this can be achieved is by classifying phase transitions rather than the actual phases
themselves. To this end let us consider a Hamiltonian Ĥ(g) which depends continuously on
some parameter g. The physical phase is then described by the ground state wave function
| (g)i of this Hamiltonian. Next, consider observables Ô, represented by hermitian operators, Ô† = Ô. In the ground state the expectation value of such observables is given by
O(g) = h (g)|Ô| (g)i. Now we say that a quantum phase transition takes place at a critical
value of g = gc whenever an observable O(g) has a singularity at gc . This, in turn, allows to
define physical phases:
Two states | (g1 )i and | (g2 )i belong to the same phase if and only if they can be transformed
into one another without crossing a phase transition. I.e. if a set of local Hamiltonians Ĥ(g)
exists, which depend continuously on the parameter g and have groundstates | (g)i, such that
no phase transition takes place in the groundstate | (g)i between g1  g  g2 .
Some comments are in order about the above definitions. The first concerns the systemsize: it is always assumed that the thermodynamic limit is taken, i.e. both particle number
N and linear system size L are send to infinity, while the particle density ⇢ = N/Ld is kept
constant (where d is the dimensionality). Only in this limit a true singularity can develop at
gc , while for finite-size systems there will always be an upper limit to any physical observable
– typically given by the system size Ld .
The second remark concerns the energy gap E(g) separating the ground state | (g)i from
excited states. Here we will be mostly concerned with gapped phases, where E takes a finite
value in the thermodynamic limit. (In fact, for gapless phases the definition of topological
order is an outstanding problem.) In this case, a quantum phase transition can only occur
when the gap closes, i.e. E(gc ) = 0 [6, 56]. The reason, roughly speaking, is that a finite
value of the gap allows a perturbative treatment, such that no non-analyticity can occur.
This brings us to the last, and most important, comment. Both the Hamiltonian and the
observable have to be local quantities. I.e. they have to be of a form
X
X
Ĥ(g) =
ĥj (g),
Ô =
ôj ,
(1.1)
j
j
where the operators ĥj (g) (ôj ) only act on a finite-size patch labeled by j 1 . Let us imagine that
this was not the case. Then we can show that any two states are in the same physical phase –
i.e. there are no distinct physical phases at all. To this end we note that we can define unitary
transformations Û (g), such that | (g)i = Û (g)| (g1 )i for any state | (g)i. Starting from a
gapped Hamiltonian Ĥ(g1 ) with ground state | (g1 )i and defining Ĥ(g) = Û (g)Ĥ(g1 )Û † (g)
we furthermore found a family of gapped Hamiltonians with ground states | (g)i which can
be transformed into one another without closing the energy gap. Hence there can not be a
phase transition, and | (g1 )i and any | (g2 )i belong to the same phase. If, on the other hand,
the Hamiltonian Ĥ(g) has to remain a local one, this leads to additional constraints on the
allowed unitary transformations Û (g) from which new Hamiltonians may be constructed. In
fact, as we discuss below, these constraints are severe enough to give rise to the entire class
of topologically ordered phases of matter.
1
More generally, the operators ĥj (g) should at least be exponentially localized. We shall not discuss such
technical details here, however.
1.2. FUNDAMENTAL CONCEPTS
25
Now that we defined two ground states of local Hamiltonians to be in the same physical
phase if and only if they can be transformed into one another without crossing a phase
transition, let us ask what di↵erent phases there could be. Ginzburg and Landau suggested
a classification scheme, based on the (spontaneous) breaking of symmetries. They postulated
that any phase transition can be characterized by how certain symmetries are broken, which
they described by introducing local order parameters. For example the superfluid - Mott
insulator transition of bosons in more than two dimensions is characterized by the spontaneous
breaking of the U (1) gauge symmetry in the superfluid phase, which gives rise to the complex
local superfluid order parameter (r) 2 C obeying the Gross-Pitaevskii equation.
For a long time it was believed that Ginzburg and Landau’s scheme is sufficient for characterizing all phases of matter. However, the quantum Hall phase provides a counter example.
Although – like the band insulator – it does not break any symmetries, it can not be transformed into a trivial band insulating phase without closing the energy gap. In the remainder
of this introductory chapter we will formalize the definition of topological order, and give a
brief overview of its possible signatures.
Intrinsic topological order and local unitary transformations
Chen, Gu and Wen [6] argued that topological order is a pattern of long-range entanglement.
This becomes apparent from their definition of topological equivalence classes, which classify
di↵erent phases of matter. According to the definition given above, two states | (g1 )i and
| (g2 )i are topologically equivalent if they can be transformed into each other without crossing a phase transition. To make this definition more operational, Chen et al. showed that the
following definition is equivalent:
Two gapped states2 | 1 i and | 2 i are topologically equivalent if and only if they are connected
by a local unitary (LU) transformation, i.e. i↵ they are related by a finite time-evolution
g = 0...1 with a local bounded Hamiltonian H̃(g),
|
2 i = P exp

i
Z
1
0
dg H̃(g) |
1 i.
(1.2)
Here P denotes path-ordering. This equation defines an equivalence relation | 1 i ⇠ | 2 i, the
equivalence classes of which correspond to topologically distinct phases of matter. States in
the trivial class, generated from an unentangled product state by applications of LU transformations, are called short-range entangled (SRE). All other states are said to support intrinsic
topological order, and they are called long-range entangled (LRE).
The main advantage of this definition is that it is independent of the Hamiltonians of
which | 1,2 i are ground states. On the other hand, the definition is not constructive: Given
two states it is often complicated – if not impossible – to check whether they belong to the
same universality class. To check this more easily – in specific cases at least – we will discuss
topological invariants in the following subsection 1.2.2.
Before however, we briefly discuss the physical meaning of Eq.(1.2). To this end, let us
start from the unentangled product state | 1 i = |0i, which is SRE. By applying a timeevolution with some local Hamiltonian H̃(g), for g = 0...1, entanglement can be built into the
2
A state is called gapped here, when it can be written as the ground state of a gapped local Hamiltonian.
Because of the finite gap, such states have a finite correlation length ⇠, beyond which correlations are strongly
suppressed. This demonstrates that not all states can be gapped.
26
CHAPTER 1. INTRODUCTION
wavefunction | 2 i. However, because the initial state is gapped, such that there are no longrange correlations, and the evolution is for a finite period of time with a local and bounded
Hamiltonian only, no long-range entanglement can be built up. Therefore the resulting state
is called SRE, too. This demonstrates that LU transformations can only modify the local
structure of the entanglement pattern in a wavefunction3 .
Quite surprisingly, gapped states exist which can not be transformed into SRE product
states using LU transformations. Whether such LRE states can exist or not depends crucially
on the dimensionality d of the system under consideration. Moreover, as pointed out above,
it is intimately related to the concept of locality. In fact, to define dimensionality, we rely
on a concept of locality: Any local system in d dimensions can formally be mapped onto
a one-dimensional system, which however has non-local couplings in general. Now we will
demonstrate this intricate interplay by discussing the specific example of the LRE integer
quantum Hall state of non-interacting fermions, which we compare to a trivial free fermion
band insulator. Both states can be constructed by filling up all the Wannier functions wj (r)
of, say, the lowest energy band,
YZ
| i=
dd r wj (r) ˆ† (r)|0i.
(1.3)
j
Here j is a d-dimensional index labeling unit-cells (or, more generally, magnetic unit cells in
a continuum system), |0i is the vacuum state and ˆ† (r) creates a fermion at position r.
Let us start by discussing one-dimensional systems, d = 1, and ask whether it is possible
to transform | i into a trivial product state using LU transformations. In this case (d = 1)
it is well known that the Wannier function w(x)4 can be chosen such that it is exponentially
localized [57]. This result has important consequences, because it implies that all free fermion
band insulators in one dimension are topologically trivial. To see this, we can simply choose
LU transformations which locally modify the – already localized – Wannier function to become
completely localized. A more rigorous proof along the same lines goes as follows: We may
define the following local Hamiltonian,
Z
XZ
†
ˆ
Ĥ(g) =
dx (x)wj (x; g) dx0 ˆ(x0 )wj⇤ (x0 ; g),
(1.4)
j
where the Wannier functions w(x; g) depend on the parameter g 2 [0, 1]. For g = 0 we fix
them to be given by the band insulator Wannier function in Eq.(1.3), w(x; 0) = w(x). For
g = 1, on the other hand, we fix them to be completely localized, w(x, 1) = (x). In between
any continuous interpolation can be chosen, which has to be localized at any point, for example w(x, g) = (1 g)w(x) + g (x). Now that we found a continuous interpolation Ĥ(g)
between the band insulating state | i and the trivial product state (of fully localized Wannier
functions), without crossing a phase transition, it follows that every one-dimensional band
insulator is SRE. This is only one example of a more general result [58]:
All gapped states in one dimension are short range entangled. I.e., there exists no intrinsic
topological order in one dimension.
3
For readers familiar with the concepts of quantum circuits, we note that this becomes even more apparent
when formulating the classification above in terms of quantum circuits. In fact, Chen et al. showed that their
classification is equivalent to the following: Two states are topologically equivalent if and only if they can be
connected by a finite-depth quantum circuit.
4
With wj (r) = w(r Rj ) and Rj being the lattice vector corresponding to site j.
1.2. FUNDAMENTAL CONCEPTS
27
Note that this result is also closely related to the success of the matrix product state
formalism, which is widely used nowadays for simulating correlated one-dimensional quantum
systems using the density-matrix renormalization group (DMRG) method [59, 60]. At least
partly it is the absence of topological order which guarantees that any gapped state in one
dimension can efficiently be represented by a matrix product state. Indeed Chen et al. [58]
provided a direct proof that any one-dimensional matrix product state is SRE5 .
Now we return to two-dimensional systems, d = 2. First of all we note that, by the same
arguments as presented above, any band insulator with localized Wannier functions w(r) is
SRE also in two dimensions. For the integer quantum Hall e↵ect, however, it was shown by
Brouder et al. [61] that there is no way to choose exponentially (or even better) localized
Wannier functions w(r). At best they may fall o↵ polynomially. As a consequence, the
integer quantum Hall states are LRE. Although they are trivial product states, they can not
be transformed into a trivial product of localized states using only LUs 6 . The reason why this
scenario is possible only in two dimensions and not in one has to do with topology: To localize
Wannier functions, a continuous gauge for the Bloch wavefunctions has to be chosen. The
possibility of having vortices – topological objects – in two dimensions can lead to unavoidable
discontinuities in the gauge choice, giving rise non-localized Wannier functions, see [62, 61].
Symmetry protected topological order
So far we excluded symmetries from our discussion of topological order, and we observed
that intrinsic topological order is a concept that can be formulated independently of any
symmetries. On the other hand, it is well known that symmetries can enormously enrich the
properties of a physical system. For instance, the lattice-translational symmetry gives rise to
the band-theory of solids, which is one of the corner-stones of modern physics. Now we will
review how symmetries give rise also to new topological orders even for SRE states, so-called
symmetry-protected topological (SPT) order.
Let us consider classes of Hamiltonians Ĥ which all respect a certain symmetry. Important
examples are time-reversal symmetry (giving rise to topological insulators [28, 29, 32, 36]),
particle-hole symmetry (giving rise to topological superconductors [9, 36]) or crystal pointgroup symmetries (giving rise to topological crystalline insulators [63]). The concept of SPT
order can be introduced as a straightforward generalization of intrinsic topological order, by
restricting LU transformations to respect the symmetries of the Hamiltonian:
Two gapped states | 1 i and | 2 i, which are ground states of Hamiltonians that respect some
set of symmetries S, are topologically equivalent if and only if they are connected by a symmetric local unitary (LU) transformation, i.e. i↵ they are related by a finite time-evolution
5
In the authors opinion this insight is of particular importance if one attempts to generalize the matrix
product state concept to higher-dimensional systems. In that case one may hope to find an exhaustive class of
variational wavefunctions separately for each LRE topological sector, which – similar to matrix product states
– describes the local entanglement structure in the quantum state.
6
This is easy to understand in the case when Hamiltonians are assumed to be particle number conserving
and quadratic, such that LUs can merely modify the Wannier functions. In a completely general case, start
from an integer quantum Hall wavefunction of non-interacting fermions. According to Brouder et al.[61] the
many-body wavefunction can, at maximum, fall o↵ polynomially when the distance between two fermions
becomes large. By applying a LU transformation, i.e. finite-time evolution with a local, gapped and bounded
Hamiltonian, the many-body wavefunction at large inter-particle distances can not change. Therefore no LU
transformation can bring the integer quantum Hall wavefunction into a SRE product form. In such a case the
new wavefunction would vanish at least exponentially at large inter-particle separations.
28
CHAPTER 1. INTRODUCTION
symmetric
SPT2
SPT1
gapless region
Figure 1.1: The generic phase diagram in the vicinity of a SPT phase transition is shown.
As described in the text, a gapless region separates two gapped states (SPT1,2 ) with the
symmetry S from each other on the axis SB = 0 where the Hamiltonian is symmetric. Inside
the gapped region, S may be spontaneously broken.
g = 0...1 with a bounded local Hamiltonian H̃(g) which respects the symmetries S,
|
2 i = P exp

i
Z
1
0
dg H̃(g) |
1 i.
(1.5)
This equation defines an equivalence relation | 1 i ⇠ | 2 i, the equivalence classes of which
correspond to distinct symmetry protected topological (SPT) phases of matter.
From its definition above it is clear that SPT order classifies di↵erent patterns of entanglement
in a wavefunction, taking into account symmetry restrictions. It is also immediately obvious
that SPT order is a refinement of intrinsic topological order. In particular the class of SRE
states is classified further by SPT order and splits into di↵erent SPT equivalence classes. If
LRE states are distinguished further by the inclusion of a symmetry, one speaks of symmetry
enriched topological (SET) order.
Finally, let us discuss the generic phase diagram of SPT phases, see also FIG.1.1. For
concreteness we discuss only the one-dimensional case where all states are SRE. To this end
we consider a class of one-dimensional Hamiltonians Ĥ( SY , SB ) which are parametrized by
two real parameters SY and SB . Let us assume that Ĥ respects some symmetry S only if
SB = 0, but for all values of SY . In the simplest case the Hamiltonian is gapped everywhere
on the symmetric axis ( SY , SB = 0), and there is no SPT order. In general, because the
ground state is SRE everywhere, after properly choosing SB any two gapped states can
be adiabatically transformed into each other by following a loop L in the two-dimensional
parameter space SY ⇥ SB . If there is SPT order, we can find a situation where L can not
be deformed to lie entirely on the symmetric axis at SB = 0. In this case, somewhere on this
axis the gap of the Hamiltonian Ĥ( SY , 0) has to close and there is a SPT phase transition at
this critical point. The two gapped phases left and right of the critical point are topologically
distinct, while now local measurement can distinguish between them.
Indicators for topological order
The last question we would like to address during this general discussion of topological order
is how it can be witnessed. Over the last few years a number of unique indicators were found
for topological order. For many of them the question whether they are necessary, sufficient or
both is still an open problem. Here we will give a brief overview of such indicators, without
explaining them in depth however, before we proceed with a detailed discussion of topological
invariants in 1.2.2.
Topological invariants.– Historically the first indicator for topological order was the quantized value of the Hall conductivity observed in the quantum Hall e↵ect [25]. Thouless,
1.2. FUNDAMENTAL CONCEPTS
29
Kohmoto, Nightingale and denNijs [27] pointed out that this quantization can be understood
from a topological property of the wavefunction, which is characterized by an integer number (the Chern number) and does not depend on any details of the system. In mathematics
topological invariants are numbers which label topological equivalence classes. Often they are
defined as winding numbers, e.g. in the context of complex analysis or homotopy theory.
In quantum mechanics it is natural to view sets of wavefunctions, which are parametrized
by some continuous external parameters from a manifold, as vector bundles. The topology of
such a vector bundle can be characterized by a winding number. How such an invariant can
be expressed in terms of the original wavefunction, and what values it may take, depends on
the dimensionality of the system. In practice topological invariants are ofter defined in terms
of geometric phases, as will be discussed in the following section 1.2.2.
Gapless edge states.– Many topological phases are characterized by gapless modes localized on the edge of the system, or more generally at the interface to a trivial phase. They
can only disappear (or ”be gapped out”) when the bulk gap of the system closes during a
topological phase transition. For the integer quantum Hall e↵ect, for example, it was shown
that the number of chiral edge modes is in one-to-one correspondence to the integer topological invariants of the bulk [64, 65]. A similar result is true for SPT phases, e.g. for Z2
topological insulators. In one dimensional systems, on the other hand, the edge is a point
defect, such that no gapless mode can appear. Nevertheless robust edge states can survive, for
instance when the particle-hole symmetry provides special protection for zero-energy modes
in the context of topological superconductors [9].
More generally it is established that non-trivial topology in the bulk leads to unconventional physics at the edge. Because the edge states can not be gapped out, they provide a
means for perfect transport without any backscattering. For example the quantum Hall edge
states are chiral, i.e. they allow propagation only in one direction along the edge. Hence the
absence of any backscattering channels leads to robust transport, completely eliminating the
possibility of Anderson localization [66] in the presence of disorder. Interestingly, although the
chiral edge states are one-dimensional, no one-dimensional tight-binding model can capture
their chiral character. The same holds true for three-dimensional Z2 topological insulators,
which host single Dirac cones on their two-dimensional surface. This contradicts the general
fermion-doubling theorem [67], stating that in a lattice model such Dirac cones have to come
in pairs. From the examples discussed above we see that edge e↵ects can reflect the topological
order in the system. This idea was even used to classify di↵erent topological orders [39].
Ground state degeneracy.– A further indicator for topological order is a robust ground
state degeneracy if the system is placed on a topologically non-trivial manifold. For example
in the fractional quantum Hall e↵ect on a manifold of genus g, the ground state degeneracy
grows exponentially with g. The splitting between the low-energy levels decays exponentially
with the system size, and it is robust to disorder as long as the bulk gap does not close. This
shows that local order parameters can not distinguish between the ground states. Instead nonlocal topological order parameters (e.g. string order-parameters [68] or Wilson loops [69]) are
required. Note however, that a finite ground state degeneracy is not a necessary requirement
for topological order. E.g. the integer quantum Hall state does not show any ground state
degeneracy on a torus, while it certainly contains some form of topological order.
Fractionalization.– A characteristic feature of some topologically ordered states is the
fractionalization of their elementary quasiparticle excitations. For example, the quasiparticles
in the fractional quantum Hall e↵ect carry fractional charge [47, 70]. This has an important
consequence: Because only an integer number of particles can be added to or taken from a
system, such elementary quasiparticle excitations can not be locally created (or annihilated).
30
CHAPTER 1. INTRODUCTION
Again, however, this property is not necessary for topological order, as it is not present in the
integer quantum Hall e↵ect.
Anyonic statistics.– The elementary excitations of a gapped phase provide useful information about the underlying state. They can not only have fractional charge, but in two spatial
dimensions, they can even have fractional braiding statistics [71, 72, 73]. When two identical elementary excitations are adiabatically exchanged, the wavefunction picks up a phase ei#
where # can be any fraction of 2⇡ in principle. More generally, the braiding statistics can even
be non-Abelian. In this case the degenerate ground state manifold picks up a unitary matrix
Û when two identical excitations are adiabatically exchanged. Nowadays such topological
order is classified by the use of category theory [11], which is subject of active research.
Anyonic statistics is not limited to two-dimensional systems. In one-dimensional settings
the T-junction approach was shown to be sufficient for realizing non-Abelian braiding operations [74]. In higher dimensions braiding in the usual sense is not defined, but for loop-type
excitations similar concepts can be formulated. Anyonic statistics is not a necessary requirement for topological order.
Entanglement entropy/spectrum.– Recently a new indicator of topological order was introduced by Kitaev and Preskill [75], the topological entanglement entropy . Their idea was to
consider the ground state of a large system (compared to the correlation length ⇠) and divide
it into an interior disc of diameter L/2⇡ and trace out the surrounding exterior degrees of
freedom. This yields a reduced density-matrix ⇢ˆ, the von-Neumann entropy S = trˆ
⇢ log ⇢ˆ of
which measures the entanglement of the interior with the exterior. By cutting the system into
two along a smooth boundary of length L, a non-universal amount of entropy proportional to
L is generated originating from short range entanglement close to the boundary. In addition,
however, they could show that a universal constant amount of entropy survives,
S = ↵L +
+ O(1/L),
(1.6)
which characterizes the long range entanglement inp
the wavefunction. The universal entanP 2
glement entropy is given by = log D, where D =
a da is the total quantum dimension
of the system. Here a labels the super-selection sectors of the theory, and the quantum dimension da of quasiparticle of type a defines the dimension of the Hilbertspace. For example,
N
the dimension of the Hilbertspace defined by N Majorana fermions
p is given by dMF , and the
quantum dimension of the Majorana fermion is given by dMF = 2. More details about the
quantum dimension can be found e.g. in the review by Nayak et al. [11].
In view of the topological classification scheme by Chen et al. [6], based on patterns
of long range entanglement, it is an obvious goal to understand more about the entanglement pattern in a wavefunction. To this end Li and Haldane [76] generalized the concept of
entanglement entropy and introduced the concept of entanglement spectra. They identified
conserved quantum numbers in the system, and plotted the singular values resulting from a
partition of the system as a function of these conserved quantum numbers. Entanglement
spectra are routinely used nowadays to check whether a state is topologically ordered, but
also to gain detailed information about the specific topological order of a state.
Topological EM response.– Finally, let us discuss a less generic feature of topological order,
which has to do with the response of the (classical) electromagnetic (EM) field when coupled
to a topological state. When the matter component in the topological state is integrated
out, the e↵ective action for the EM field may contain additional topological terms besides the
Maxwell term. This is the case for example for the quantum Hall e↵ect, where a Chern-Simons
term appears, or for three-dimensional Z2 topological insulators where a topological ✓-term
appears in the EM Lagrangian [77]. In both cases, their coupling constants are quantized.
1.2. FUNDAMENTAL CONCEPTS
1.2.2
31
(Abelian) topological invariants
In this section we give a brief review of topological invariants, which are among the most useful
indicators for topological order. Specifically we will discuss geometric (Berry) phases, characterizing one-dimensional systems, and formulate higher-dimensional topological invariants in
terms of these Berry phases. Here we restrict ourselves to the Abelian case with only a single band. For a more rigorous review, including connections to the underlying mathematical
concepts, see e.g. Les Houches lecture notes by Prof. J. Moore7 .
We start this section by posing the following simple question: What is a topological
invariant? Constructed as a unique indicator of topological order, a topological invariant ⌫
is a number which labels topological equivalence classes. As such, it may only take discrete,
say, integer values ⌫ 2 Z, and it may only change its value at a topological quantum phase
transition. Note that this second point is crucial: It is in not useful to define some arbitrary
observable which is integer quantized, as long its change has no concrete physical meaning.
Geometric phases: Berry, Zak and King-Smith - Vanderbilt
Geometric Berry phases are at the heart of many topological invariants because they provide
a direct measure of the geometry of a complex vector bundle. We will now give an overview
what their origin is, where they appear and what sort of information they contain about a
quantum mechanical system.
Berry’s phase.– Geometric phases appeared in di↵erent physical contexts (mostly involving
Dirac cones) before Berry formulated the e↵ect in a general framework [78]. He considered an
adiabatic time-evolution of quantum state, described by a gapped Hamiltonian Ĥ( ) where
is a vector of parameters. If we start from an eigenstate | (0)i = | ( 0 )i and vary (t)
adiabatically in time, the quantum state | (t)i follows the instantaneous eigenbasis (energy
E( )) and picks up a additional phase '(t),
| (t)i = ei'(t) | ( (t))i.
(1.7)
When (t) follows a closed path L in parameter space which returns to itself, (T ) = 0 , the
wavefunction also returns to its initial state up the phase '(T ). Berry showed that this phase
contains a dynamical part, which depends on the total evolution time T ,
'(T ) = ⌦dyn T + 'geo ,
⌦dyn = T
1
Z
T
dt E( (t)),
(1.8)
0
and a geometrical contribution which is independent of T ,
I
'geo =
d · A( ),
A( ) = h ( )|ir | ( )i.
(1.9)
L
This phase is called geometric (or Berry) phase, because it only depends on the path L chosen
in parameter space. If, moreover, 'geo becomes independent of the details of the path L, the
geometric phase becomes a topological phase. Before turning our attention to examples of
this below, we discuss some properties of the Berry phase.
Above we introduced the Berry connection A, which has a structure similar to the vector
potential describing a gauge field. In particular, it is not gauge-invariant, as can easily be seen
by performing a gauge transformation, | ( )i ! ei#( ) | ( )i. After this re-definition of the
eigenstates, the Berry connection transforms as A( ) ! A( ) r #( ), which indeed has
7
http://topo-houches.pks.mpg.de/wp-content/uploads/2014/11/LH_merged.pdf
32
CHAPTER 1. INTRODUCTION
the from of a gauge transformation
for a vector potential. Meanwhile the geometric phase 'geo
H
is gauge invariant, because d · r # 2 2⇡Z. We thus conclude that the geometric phase
contains information about the gauge structure of the collection of wavefunctions | ( )i.
Although the Berry connection A is not an actual gauge-field, its close analogy to the latter
can indeed be employed to construct artificial gauge fields e.g. for neutral atoms [79].
From the explicit gauge-dependence of the Berry connection follows a problem for the
numerical calculation of the Berry phase. In order to use Eq.(1.9) directly to obtain 'geo from
a microscopic wavefunction | ( )i, a smooth gauge choice has to be made. This can lead to
problems for two reasons. Firstly, choosing a continuous gauge with just a discrete number
of points available is a non-trivial numerical task. Secondly, in some cases a smooth gauge
choice does not even exist [62]. Now we will present a workaround consisting of a discretized
expression for the Berry phase which, moreover, is fully gauge invariant, see e.g. [80]. To this
end, let us divide the loop L into discrete points j , for j = 0, 1, ..., M , such that 0 = M .
Then the Berry phase is given by
'geo =
lim Im log
M !1
M
Y1
j=0
h (
j )|
(
j+1 )i.
(1.10)
The expression on the right is well-defined up to integer multiples of 2⇡, and because every
bra h ( j )| and every ket | ( j )i appear once each, the expression is gauge invariant when
| ( j )i ! | ( j )iei#j and h ( j )| ! h ( j )|e i#j .
To proof Eq.(1.10), let us assume that locally smooth gauge choice can be found (we don’t
actually have to find one). This allows us to perform a Taylor expansion and write,
h
i
h ( j )| ( j+1 )i = h ( j )| | ( j )i i ( j+1
j ) · ir | ( j )i =
=1
i(
j+1
= exp [ i (
j+1
j)
· A(
j)
j)
· A(
=
j )] .
(1.11)
Higher orders can safely be neglected when M ! 1. Hence
lim
M !1
M
Y1
j=0
h (
j )|
(
j+1 )i
= exp

i
I
L
d · A( ) ,
(1.12)
from which Eq.(1.10) immediately follows.
Zak phase.– Now we discuss one particular kind of geometric phases, which appears in
the band theory of solids. There (single-particle) quantum states can be labeled by their
quasimomentum k 2 BZ within the first Brillouin zone (BZ). According to Bloch’s theorem
they can be written as k (r) = eik·r uk (r), where the Bloch wavefunction is lattice periodic,
uk (r + R) = uk (r) for R a lattice vector. Because of the periodicity in reciprocal space, with
reciprocal lattice vectors G, the BZ defines a d-dimensional torus T d . It serves as a parameter
space, where to each k a Bloch vector |uk i can be assigned.
As discussed in the last paragraphs, the geometry of the so-defined vector bundle over the
BZ is encoded in the corresponding Berry phases. This geometry was investigated by Zak
[81], who pointed out the importance of one particular set of loops L through the BZ: Loops
which cross the edge of the BZ. In the following we will focus on one-dimensional systems,
where such loops can be parametrized by
k( ) =
⇡/a + 2⇡/a,
2 [0, 1],
(1.13)
1.2. FUNDAMENTAL CONCEPTS
33
with G = 2⇡/a and a being the lattice constant. Quite remarkably, the geometric Zak
phases corresponding to such loops provide signatures for non-trivial geometry even in a onedimensional parameter space. What makes these loops special, from a mathematical point of
view, is that the corresponding geometric (Zak) phases are not completely gauge invariant.
To solve this problem, Zak generalized the concept of geometric phases to open paths in
parameter space which are connected by a gauge transformation [81]. We will now review his
approach and explain the subtleties which appear when using Bloch wavefunctions.
The trouble with Bloch wavefunctions |uk i comes from the fact that the Bloch Hamiltonian
ĥ(k)8 is not cyclic in the BZ in general: By making the standard gauge choice k+G (x) = k (x)
and defining the Bloch wavefunction as uk (x) = e ikx k (x), it automatically follows that
uk+G (x) = e
iGx
uk (x).
(1.14)
That is |uk+G i = ÛG |uk i with ÛG = e iGx̂ , such that the Bloch Hamiltonian is not cyclic in
general, ÛG† ĥ(k + G)ÛG = ĥ(k).
Importantly, however, Eq.(1.14) defines a one-to-one relation between the Bloch wavefunction |uk i and |uk+G i, which is obtained from the former by the gauge transformation ÛG .
Thus from the adiabatic theorem it follows for the time evolution of the full wavefunction
k (x, t) under a weak force F (or an electric field) that [81]
k (x, T )
= ei'Zak
i
RT
0
dt ✏k(t) ikx
e
ÛG uk (x),
k(t) =
⇡/a + F t,
(1.15)
where k (x, 0) = eikx uk (x) and T = 2⇡/F a corresponds to one full Bloch cycle9 . In a direct
calculation, along the lines of Berry’s derivation [78], Zak showed that the geometric phase
'Zak in the last equation (1.15) is given by [81]
'Zak =
Z
⇡/a
⇡/a
dk huk |[email protected] |uk i.
(1.16)
This is the definition of the Zak phase corresponding to the Bloch band |uk i.
We can draw several conclusions from Zak’s result Eq.(1.15). First of all, by making a
particular gauge choice, we find a universal number 'Zak which characterizes energy bands in
solids. We will discuss below how the Zak phase is a↵ected by this particular gauge choice
(1.14), but in any case the prescription above provides a unique fingerprint for any Bloch band.
As we will see in the following subsection, Zak phases contain all the information about the
topology of the Bloch band. Secondly, Zak phase di↵erences have a real physical meaning and
can be measured. For example an interference experiment between two Bloch bands can be
used to detect their relative Zak phase, see Ref. [13]. The required matrix elements in this case
are independent of the gauge transformation ÛG in the final wavefunctions Eq.(1.15). Finally
let us point out that, like in the case of standard Berry phases, the Zak phase is invariant
under gauge transformations where the Bloch wavefunction |uk i ! ei (k) |uk i acquires an
arbitrary phase. Because of the condition (1.14) we require (k + G) = (k) + 2⇡Z, such that
'Zak is invariant up to integer multiples of 2⇡.
Now we turn to the discussion of the absolute meaning of the Zak phase. To this end we
re-examine the gauge choices above, which were due to Zak [81], and investigate how the Zak
8
The Bloch Hamiltonian is defined such that the Schrödinger equation reads ĥ(k)|uk i = ✏k |uk i in terms of
the Bloch wavefunction.
9
Note that Zak’s result (1.15) is valid only for a particular gauge choice where the electric field is defined
as the time-derivative of the vector potential E = @t A. For a di↵erent choice, using e.g. a scalar potential
V (x) = qE · x, the corresponding result can be obtained by a gauge transformation.
34
CHAPTER 1. INTRODUCTION
phase changes when they are modified. We start with the choice k+G (x) = k (x), which
is convenient but may be replaced by the more general expression k+G (x) = ei#G k (x). In
this case Eq.(1.14) is generalized to u0k+G (x) = e iGx+i#G u0k (x). From the previous Bloch
wavefunction |uk i we can construct a new one by writing |u0k i := ei#G k/G |uk i, which fulfills
this condition. For the new Zak phase we obtain from Eq.(1.16) '0Zak = 'Zak #G . However
because ÛG0 = e+i#G , the form of the final wavefunction Eq.(1.15) remains invariant.
Next we turn to the gauge choice for the Bloch wavefunction, k (x) = eikx uk (x). This
definition assigns a special role to the origin x = 0, which is the unique single point in space
where the complex phases of k (x) and uk (x) coincide for all quasimomenta k. Now it is
natural to ask what happens if we displace the origin from 0 to some value x0 , i.e. if we define
a new Bloch wavefunction by k (x) = eik(x x0 ) u0k (x). In this case u0k (x) = eikx0 uk (x) and it
follows that u0k+G (x) = e iG(x x0 ) u0k (x), in agreement with k+G (x) = k (x). Now, however,
the Zak phase Eq.(1.16) changes,
'0Zak = 'Zak
x0
2⇡
.
a
(1.17)
This is an important result, because it shows that:
The Zak phase depends on the choice of the origin relative to the unit-cell. Shifts of the origin
by multiples of the lattice constant a leave the Zak phase invariant.
King-Smith and Vanderbilt.– Alternative to displacing the origin by x0 , we may as well
ask what happens if we displace the entire system. The new wavefunction is given by k0 (x) =
k (x + x0 ) in this case, which still fulfills the Schrödinger equation if we displace also the
coordinates in the Hamiltonian by x0 . Again a simple calculation shows that '0Zak = 'Zak
2⇡
a x0 . This result explains the physical meaning of the Zak phase: it essentially measures
the center of mass of the system. More precisely, because it is only well-defined up to 2⇡, it
measures the center of mass relative to the unit-cell – that is the polarization [80, 82].
The result that the Zak phase measures the polarization was obtained for the first time
by King-Smith and Vanderbilt [83]. More concretely they showed the following:
The polarization of an electronic band ↵ in a solid can be defined from the corresponding
Wannier function w↵ (x) as
Z
P↵ = dx w↵⇤ (x)xw↵ (x).
(1.18)
It is directly related to the Zak phase 'Zak of the corresponding Bloch function |u↵ (k)i,
P↵ =
a
'Zak .
2⇡
(1.19)
This can easily be proven in a few lines using the definition of Wannier functions,
Z 1
P =
dx w⇤ (x)xw(x) =
1
Z 1
Z
a2
ik0 x
=
dx
dk dk 0 e ikx u⇤k (x) xe
|
{z
} uk0 (x) =
2
(2⇡)
1
BZ
0
= [email protected] eik
x
1.2. FUNDAMENTAL CONCEPTS
35
Z
1 Z a
a2 X
0
0
=
dx
dk dk 0 e i(k k )ja e i(k k )x u⇤k (x)[email protected] uk0 (x) =
2
(2⇡)
BZ
j= 1 0
Z a
Z
Z
a
a
⇤
=
dx
dk uk (x)[email protected] uk0 (x) =
dk huk |[email protected] |uk i .
| {z }
2⇡ 0
2⇡ BZ
BZ
(1.20)
=A(k)
Similar relations were obtained for interacting systems, see e.g. [84, 85].
The result by King-Smith and Vanderbilt is of enormous importance, because it provides a
simple physical interpretation of the Zak phase. We will make use of it in the next subsection,
where we introduce the Chern number as a measure for the quantized current in the quantum
Hall e↵ect. There we will use that a current is equivalent to a change of polarization. Before
doing so, however, let us add an intuitive explanation why the Zak phase should be related
to the polarization.
To this end, imagine a wavepacket with momentum k in a Bloch band. We can apply a
weak homogeneous force F to the system, such that the wavepacket moves through the band
structure adiabatically, k̇ = F . During this process it picks up a dynamical and a geometric
phase. The dynamical phase is only due to the kinetic energy in the lattice, which we may
neglect if the bandwidth is sufficiently small. In that case,
R the wavepacket (x) is quasi-static
in position space, and picks up only a phase 'geo = F T dx | (x)|2 x. First of all, note that
although this contribution appears to be a dynamical phase, it does not vanish in the limit
T ! 0 because F T = 2⇡/a is required to complete a full Bloch cycle. Using this result we find
that 'geo = 2⇡hx̂i/a, i.e. the geometric phase measures the center of mass. Because 'geo can
be measured only up to multiples of 2⇡, the result of a measurement yields the polarization.
The Chern number
After our extensive discussion of one-dimensional geometric phases, we now turn our attention
to two-dimensional systems. Here we discuss the Chern number as a topological invariant
for free particles, before generalizing this concept to interacting many-body systems in the
following subsection. We will start by giving a definition as a winding number of the Zak phase
and explain its relation to the integer quantum Hall e↵ect. Then we explain the connection to
earlier expressions for the Chern number and discuss briefly the consequences of non-trivial
topology.
Let us consider an arbitrary two-dimensional band structure, with a gapped Bloch band
|u(k)i. To characterize its topology, we view it as a collection of one-dimensional Bloch
bands |uky (kx )i along kx , which we label by their quasimomentum
ky . To each of these
R
one-dimensional bands we may assign a Zak phase, 'Zak (ky ) = dkx huky (kx )|[email protected] |uky (kx )i.
Because of the periodicity in reciprocal space, uky +Gy (kx ; x) = e iGy y uky (kx ; x), it follows
that 'Zak (ky + Gy ) = 'Zak (ky ) mod 2⇡. Here Gy = 2⇡/ay . As a consequence the winding
of the Zak phase, when ky crosses one BZ, is quantized in units of 2⇡. This leads to the
definition of the Chern number as an integer characterizing the topology of the Bloch band:
The Chern number C is defined as the winding of the Zak phase across the BZ,
C :=
1
2⇡
Z
2⇡/ay
0
dky @ky 'Zak (ky ) 2 Z.
(1.21)
To calculate the derivative of the Zak phase @ky 'Zak (ky ) the gauge has to be chosen continuously only on a patch around ky .
36
CHAPTER 1. INTRODUCTION
As pointed out at the beginning of this section, in order to show that the Chern number
is a topological invariant we have to proof that it can only change its value when the band
gap closes at a quantum phase transition. To this end consider a small perturbation of order
, well below the band gap ⌧ . In this case the wavefunction depends continuously on
, and because the Berry connection is a continuous function of the wavefunction, also the
Zak phase changes continuously with , i.e. '0Zak (ky ) = 'Zak (ky ) + O( / ). Hence the Chern
number can only change by an order of / ⌧ 1, which means it can’t change at all because
it is integer quantized. An alternative proof can be found e.g. in [86].
Because of the King-Smith Vanderbilt relation (1.19) between Zak phase and polarization,
the Chern number has a simple graphical interpretation. To this end, let us consider the
quasimomentum ky as an external parameter, and define an e↵ective one-dimensional system
along x at each ky . In particular, we can define one-dimensional Wannier functions wky (x).
They can always be chosen to be exponentially localized [57], and their center of mass is
given by the polarization 'Zak (ky )ax /2⇡ = P (ky ) = hwky |x̂|wky i. In FIG. 1.2 (a) some
Wannier centers hwj (ky )|x̂|wj (ky )i = P (ky ) + ja are sketched as a function of ky . When two
quasimomenta ky and ky0 di↵er by 2⇡/ay , all Wannier centers have to coincide because of the
periodicity in reciprocal space. However, when ky is adiabatically changed by 2⇡/ay , Wannier
centers may reconnect to their neighbors. During this process, which is sketched in FIG.1.2
(a), the polarization changes by an integer amount given by the Chern number P = Cax .
Let us add a few remarks about this picture. Firstly, it seems rather unintuitive to plot
together (quasi)momentum along one, and position along the second axis. However, for a
topologically non-trivial case it is not surprising that it actually helps: As we saw in the
beginning, topological order is a restriction to express wavefunctions in a local basis. Because
no intrinsic topological order exists in one dimension, we can always treat one axis in a local
basis, forcing us to choose a non-local basis in the second direction to keep things simple.
The second remark concerns the question under which circumstances topological order,
i.e. in this context a non-vanishing Chern number, may appear. FIG. 1.2 (a) suggestes that,
quite generically, topological order can appear. The only way to prevent topological order
in general is to impose additional symmetries. The most important example is time-reversal
symmetry, which maps x ! x while ky ! ky . This implies C ! C and we conclude: For
systems with time reversal symmetry the Chern number vanishes, C = 0.
Next we clarify the relation of the Chern number to the integer quantum Hall e↵ect. To
(a)
(b)
Figure 1.2: The Chern number C (here C = 1) can be interpreted as the change of polarization
P (in units of the lattice constant ax ) of a Wannier function when the quasimomentum ky is
adiabatically changed by ky = 2⇡/ay . To this end e↵ective one-dimensional systems along x
are defined at each fixed value of ky , and the Wannier centers hwj (ky )|x̂|wj (ky )i are sketched
as a function of ky , for j = 0, 1, .... In (a) the Abelian case (one band) is shown, whereas in
the non-Abelian case (b) Wannier centers of two bands are sketched (red and blue).
1.2. FUNDAMENTAL CONCEPTS
37
this end a Bloch band is considered, characterized by a Chern number C, which is completely
filled with non-interacting electrons (here we introduce the electron charge e and the Planck
constant ~ which is set equal to one otherwise). We will now calculate the quantum Hall
response xy = Jx /Ey , where Jx is the current density along x when an electric field Ey is
applied along y. Instead of using the Kubo formula, as Thouless et al. did in their famous
paper [27], we will make use of the concept of polarization. Applying an electric field Ey
corresponds to driving Bloch oscillations, i.e. the momentum ky of every electron changes
in time according to eEy = [email protected] ky . At the same time the polarization P (ky ) of each onedimensional system at fixed ky changes accordingly, P (t) = P (ky (t)). If in time T , ky changes
by 2⇡/ay , we have ky (t) = ky0 + Tt 2⇡
ay . The change of the total polarization gives the current
density
✓
◆
Z
e X T
t 2⇡
0
Jx =
dt @t P ky +
,
(1.22)
T Ly 0 0
T ay
ky
where Ly is the length of the sample in y-direction. Since @ky P (ky ) is 2⇡/ay -periodic in ky
this simplifies and we obtain the relation to the Chern number:
Z
e2 ⇡
e2
dky @ky P (ky )/ax = C.
(1.23)
xy =
h
h
⇡
We note that along the same lines it can be shown, more generally, that adiabatic particle
transport is quantized in periodically driven one dimensional systems. This e↵ect was first
discussed by Thouless [87], who essentially suggested to replace the quasimomentum ky by
a more general external parameter which periodically changes in time. These systems are
therefore called adiabatic or Thouless pumps.
Related expressions.– We close our discussion of the Chern number by deriving equivalent
expressions which are more often found in the literature than Eq.(1.21). The most important
one makes use of Stokes theorem to write the Chern number as an integral over the Berry
curvature F(k). As result we obtain
Z
1
C=
d2 k F(k),
F(k) = (rk ⇥ A(k))z .
(1.24)
2⇡ BZ
From this expression it is immediately clear that the Chern number is fully gauge invariant,
because the Berry curvature does not change when A(k) ! A(k) rk (k) for an arbitrary
function (k). Moreover Eq.(1.24) is a good starting point to generalize the Chern number
to higher-dimensional systems (which is possible for all even spatial dimensions).
To derive Eq.(1.24) let us consider a patch P which covers only a portion of the BZ and
where the gauge of the Bloch
H wavefunctions can be continuously chosen. Then we can define
the Zak phase 'Zak (P) = @P dk · A(k) corresponding to a path around the edge @P of the
patch. The Chern number (1.21) is obtained from the Zak phase in the limit when the patch
covers the entire BZ, C = limP!BZ 'Zak (P)/2⇡. On the other hand, because of the continuous
gauge choice, we can use Stokes theorem on P to write
Z
1
C=
lim
d2 k F(k).
(1.25)
2⇡ P!BZ P
Because the Berry curvature is gauge-invariant we do not have to use explicitly the limit
P ! BZ, and the simpler equation (1.24) follows.
Finally we would like to mention two more equivalent formulations of Chern-type topo-
38
CHAPTER 1. INTRODUCTION
logical order10 . The first involves the possibility to chose a continuous gauge. It was proven
by Kohmoto [62] that the Chern number is an obstruction to making a continuous gauge
choice. This is possible if and only if C = 0. The second, related, point is concerned with
the localization properties of Wannier functions. As pointed out by Brouder et al. [61] exponentially localized Wannier functions can be found if and only if a continuous gauge choice
can be made. Together with the last result this implies that exponentially localized Wannier
functions exists only for topologically trivial systems.
The many-body Chern number
Now we turn our attention to generalizations of the Chern number to interacting many-body
systems, which may even contain disorder. There are at least three di↵erent approaches how
this can be achieved. Here we will mostly discuss the flux-insertion trick where twisted periodic
boundary conditions are introduced [88]. An alternative approach is based on Ward-Takahasi
identities in quantum field theory [89]. Moreover, Green’s functions have been invoked to
define topological invariants of interacting systems [90]. Note however that it is unclear at
present whether all approaches are equivalent.
If we want to generalize the Chern number, as defined e.g. in Eq.(1.24), to systems
with interactions and without any symmetries, we face the problem that the single particle
quasimomentum k is no longer a good quantum number. Without the structure of the BZ
torus T 2 however, we can no longer interpret the wavefunctions as a vector bundle, which is
the fundamental mathematical object classified by the Chern number. Before re-introducing
vector bundles in the general case, let us first take a physical point of view and calculate the
many-body quantum Hall response xy . In the following we will reproduce the result of Niu
et al. [88] using a simplified argument based on polarization. Afterwards we briefly review
their more formal derivation.
To avoid edge e↵ects for the moment, let us consider a system placed on a torus in position
space11 . To apply a circular electric field along, say, y on the torus, we can adiabatically
introduce magnetic flux ✓y through one of the two holes of the torus (embedded in three
dimensional space). Next we calculate the resulting current along x, i.e. the change of the
center-of-mass hX̂i of the many-body system. To this end we repeat the naive argument given
below Eq.(1.20) and apply a weak circular electric field Fx along x around the torus for some
time T . According to King-Smith and Vanderbilt the corresponding geometric phase is thus
given by 'mb
Zak (✓y ) = h (✓y )|X̂| (✓y )i2⇡/Lx , where the length of the system in x-direction, Lx ,
takes the role of the lattice constant (in a system with only one unit cell). For the geometric
phase we can also write
'mb
Zak (✓y ) =
Z
2⇡
0
d✓x h (✓x , ✓y )|[email protected]✓x | (✓x , ✓y )i,
(1.26)
which is the many-body generalization of the Zak phase. Here we adiabatically introduced
magnetic flux ✓x through the second hole of the torus, which realizes the circular electric field
Fx . Due to Faraday’s law ✓˙x = Fx Lx the flux ✓x has to change from 0 to 2⇡ to realize one
”Bloch cycle” in the large ”unit cell”, i.e. Fx T = 2⇡/Lx .
10
Note that the Chern number measures intrinsic topological order as defined in 1.2.1. Vice-versa it is not
obvious whether all intrinsic topological orders are characterized by a Chern number in two dimensions.
11
Note that this torus in position space is di↵erent from the BZ torus and can not be used to define a vector
bundle: In the most general case, when all symmetries are broken, there is only one many-body wavefunction
associated with the entire torus. To define a vector bundle, one wavefunction would be required for every point
on the torus.
1.2. FUNDAMENTAL CONCEPTS
39
Now we can easily obtain an expression for the quantum Hall response, xy = Jx /Fy .
Let us introduce magnetic flux ✓y in time T 0 . Then the current density is given by Jx =
hX̂i/Lx Ly T 0 , and the electric field is obtained from Faraday’s law, ✓y /T 0 = Fy Ly . Com'mb
1
Zak
bining these equations, we arrive at xy = 2⇡
✓y . I.e. the quantum Hall response xy is
given by the change of the many-body Zak phase per change of magnetic flux.
Next we note that 'mb
✓y . Furthermore, in
Zak ( ✓y ) is 2⇡-periodic as a function of
order to obtain a more handy expression for xy , we make use of the fact that the manybody Hamiltonian is only changed by a gauge transformation when ✓y changes by 2⇡ (i.e.
when one magnetic flux quantum is introduced into the system). The properties of a gapped
thermodynamically large system can not be modified by introducing a single unit of magnetic
flux. Therefore we may average over one period of flux insertion and obtain for the many-body
quantum Hall response:
xy
1
=
(2⇡)2
Z
2⇡
0
d✓y @✓y 'mb
Zak (✓y ) =
1
C,
2⇡
(1.27)
in complete analogy with the single-particle result (here e2 /~ = 1).
Above we introduced magnetic flux ✓x,y merely as a means to realize circular electric fields.
Niu et al. [88] realized, however, that this allows one to re-introduce a vector bundle structure
in the problem. For every configuration of the magnetic flux ✓x , ✓y they define a many-body
wavefunction | (✓x , ✓y )i. Because of the fundamental gauge symmetry, the parameter space
should be viewed as a torus, ✓x , ✓y 2 T 2 , and wavefunctions are recovered up to a gauge
transformation when ✓x,y ! ✓x,y + 2⇡ changes by one quantum of flux. The topological
invariant (1.27), which may alternatively be formulated by the many-body Berry curvature
cf. Eq.(1.24)
Z 2⇡
1
C=
d2 ✓ F(✓),
F(✓) = (r✓ ⇥ A(✓))z ,
(1.28)
2⇡ 0
is exactly the Chern number characterizing this vector bundle.
Let us close this section by adding a few remarks about Niu et al.’s calculations [88].
First, instead of talking about magnetic fluxes ✓x,y , they used an equivalent formulation in
terms of twisted boundary conditions. In fact, it is easy to show by a non-trivial gauge
transformation12 , that introducing magnetic flux ✓ is equivalent to imposing twisted periodic
boundary conditions,
(x1 , ..., xi + L, ..., xN ) = ei✓ (x1 , ..., xi , ..., xN ).
(1.29)
Secondly, to calculate the quantum Hall response, Niu et al. used the Kubo formula. They
realized that the e↵ect of twisted boundary conditions is to modify the current operator,
[email protected] ! [email protected] + ✓/L. Then, following TKNN’s original calculations [27], they arrived at
x,y = F(✓), which explicitly depends on the boundary conditions. After averaging over
di↵erent boundary conditions as above, Eq.(1.28) is obtained.
1.2.3
Non-Abelian topological invariants
So far we discussed exclusively Abelian topological invariants which characterize a gapped
band consisting of a single state. For example in the free-fermion case this was the Bloch
wavefunction |u(k)i, and for the many-body case this was a single many-body wavefunction
| (✓)i. Now we will generalize our discussion to non-Abelian cases, where the band consists of
12
Here a gauge transformation
(x) ! (x)ei#(x) is referred to as non-trivial i↵
H
dx @x #(x) 6= 0.
40
CHAPTER 1. INTRODUCTION
a collection of states | ↵ ( )i with ↵ = 1, ..., M . Between these states we allow for degeneracies,
but from any other state in the Hilbertspace they need to be separated by a gap.
We start by generalizing the concept of Berry phases to sets of bands, which leads to the
concept of Wilson loops. This idea goes back to Wilczek and Zee [91]. Then we generalize
the King-Smith - Vanderbilt relation as well as the Chern number. This allows us to relate
the Hall conductivity xy in the fractional quantum Hall e↵ect to a topological invariant [88].
The Z2 invariant of the quantum spin Hall e↵ect, which we discuss in Chapter 6.2.2, is an
example for a non-Abelian topological invariant which does not exist in the single-band case.
Wilson loops
Shortly after Berry introduced his geometric phase [78], the mathematical meaning of which
was further clarified by Simon [92], Wilczek and Zee [91] asked the question how Berry’s
scenario changes when it is generalized to multiple bands. Concretely they considered a
family of Hamiltonians Ĥ( ) consisting of M degenerate bands, which are protected by a
gap from further states. They showed that under an adiabatic change of the Hamiltonian,
= (t), the dynamics within the degenerate bands can take a non-trivial form.
Consider an initial state | (0)i = | ↵ ( 0 )i, which is an eigenstate of the initial Hamiltonian Ĥ( 0 ). According to the adiabatic theorem, when (t) is slowly varied in time the
wavefunction is restricted to the manifold of degenerate states. Therefore the most general exP
pression for the time-evolved state reads | (t)i =
| ( (t))iU ↵ (t). Plugging this ansatz
into the Schrödinger equation, it is easy to show that the propagator Û (t) is given by
 Z
Û (t) = T exp i
t
0
d⌧ ˙ (⌧ ) · Â( (⌧ )) .
(1.30)
Here T denotes time-ordering and the non-Abelian Berry connection is given by
A↵,
µ ( )=h
↵(
)|i
@
@
µ
|
( )i.
(1.31)
As in the Abelian case, the Berry connection transforms like a gauge-field when the basis
{| ↵ i}↵ of the degenerate manifold is changed. In this case, however, the most general
ˆ ). I.e. | ↵0 ( )i =
basis transformation is described by a U (M ) unitary transformation ⌦(
ˆ )| ↵ ( )i. Accordingly the Berry connection transforms like a U (M ) gauge potential,
⌦(
0
ˆ † ( )Â( )⌦(
ˆ ) + i⌦
ˆ † ( )r ⌦(
ˆ ).
 ( ) = ⌦
(1.32)
This emergence of an e↵ective non-Abelian gauge structure can be used, for example, to
engineer artificial non-Abelian gauge fields for neutral particles [93].
Now we generalize the concept of Berry phases to non-Abelian situations. To this end,
let us consider a closed loop L in parameter space, such that (T ) = 0 . In this case the
propagator (1.30) takes the form of a U (M ) Wilson loop,
 I
Ŵ = P exp i d · Â( ) ,
(1.33)
L
which only depends on the loop L and, hence, is a measure for the geometry of the degenerate
states. Here P denotes the path ordering operator. As in the Abelian case, a fully gaugeinvariant discretized formula for the Wilson loop can be obtained, see e.g. [94, 95]. This is of
particular importance for numerical calculations.
1.2. FUNDAMENTAL CONCEPTS
41
We close this paragraph by summarizing the most important properties of Wilson loops. A
more detailed pedagogical review can be found for example in [95]. First of all, in the Abelian
case the Wilson loop reduces to a U (1) phase factor, W = ei'geo . Similarly, the Wilson loop
is a U (M ) unitary matrix, Ŵ † Ŵ = 1. The invariance modulo 2⇡ of the Berry phase under
gauge transformations | ( )i ! ei ( ) | ( )i translates into U (M ) gauge-invariance of the
ˆ ), the
Wilson loop. When the basis is changed according to the unitary transformation ⌦(
Wilson loop becomes13
ˆ † ( 0 )Ŵ ⌦(
ˆ 0 ).
Ŵ ! ⌦
(1.34)
This merely represents the freedom to choose an arbitrary basis to represent the Wilson loop.
Meanwhile the trace trŴ , the determinant det Ŵ and the spectrum of the Wilson loop are
fully U (M )-gauge invariant.
Non-Abelian King-Smith - Vanderbilt relation
Now we generalize the King-Smith - Vanderbilt relation (1.19) to multi-band systems. To this
end we consider free fermions filling up M partly degenerate Bloch bands |u↵ (k)i, protected
by a gap from further states. We need to calculate the center-of-mass hX̂i from the Wannier
functions |w↵ i corresponding to the degenerate bands. In contrast to the Abelian case, there
can be non-vanishing matrix elements hw↵ |X̂|w i =
6 0 between Wannier functions of di↵erent
bands however.
To resolve this issue, we make use of the freedom to chose an arbitrary basis to represent
the M partly degenerate bands14 . To this end we first choose a basis {|u0↵ (k)i}↵=1...M where
the non-Abelian Berry connection A↵ (k) = hu0↵ (k)|[email protected] |u0 (k)i / ↵, is diagonal for every
value of k. Note that such a choice is always possible because † =  is Hermitian. In this
eigenbasis, generalizing Eq.(1.20) is straight-forward and we obtain the non-Abelian version
of the King-Smith - Vanderbilt relation in its diagonal form,
Z
ax
0
0
P↵ = hw↵ |X̂|w↵ i =
dk A↵↵
(1.35)
x (k).
2⇡ BZ
Second, we bring this equation into a basis-independent form.
⇣ To⌘ this end we exponentiate
Eq.(1.35) because – unlike k  itself – the expression exp i k  appearing in the Wilson
loop is gauge invariant. Hence we obtain the non-Abelian King-Smith - Vanderbilt relation:
The polarization matrix P↵, = hw↵ |X̂|w i of a degenerate set of bands ↵,
given by the logarithm of the Wilson loop, i.e.
✓
◆
✓ Z
◆
2⇡
exp i P̂ = P exp i
dk Âx (k) ⌘ Ŵ .
ax
BZ
= 1, ..., M is
(1.36)
This can
Eq.(1.32)
form of the Wilson
loop. Ŵ ! Ŵ 0 where
⇣ easily be 0shown
⌘ from
⇣ and using the discretized
⌘⇣
⌘
Q
Q
ˆ † ( j ) 1 + i · Â( j ) ⌦(
ˆ j)
ˆ j) = ⌦
ˆ † ( 0 )Ŵ ⌦(
ˆ N ).
Ŵ 0 = j 1 + i · Â ( j ) = j ⌦
· r⌦(
13
14
In general, this leads to a tight-binding Hamiltonian with hopping elements between di↵erent partly degenerate bands ↵, . Since we are only interested in the collection of these bands as a whole, however, this is
acceptable because no hoppings into gapped bands can be generated in this way.
42
CHAPTER 1. INTRODUCTION
Multi-band Chern number
Now we generalize also the Chern number to the multi-band case, which allows us in particular
to describe the fractional quantization of the quantum Hall e↵ect of interacting electrons
following Niu et al.[88]. In the simplest case the bands ↵ = 1, ..., M are non-degenerate,
allowing us to treat them independently and assign a Chern number C↵ to each one of them
as described above. This is no longer possible, however, when degeneracies are present at any
point in parameter space. In this case, the freedom to change the basis at the point of the
degeneracy makes the individual Chern numbers of the bands ill-defined. In order to calculate
the Hall current and relate it to a quantized topological invariant in this case, we will now
utilize the King-Smith - Vanderbilt relation (1.36).
Free fermions.– The Hall current Jx of a multi-band system consisting of M Bloch states
|u↵ (k)i filled by non-interacting fermions can easily be calculated in the basis where the KingSmith - Vanderbilt relation takes its diagonal form Eq.(1.35). Using the same arguments as
in the single-band case shows that the quantum Hall response is given by
Z
e2 1 X
e2
=
dky @ky P↵ (ky ) =: C,
(1.37)
xy
h ax ↵ BZ
h
cf. Eq.(1.23). Here, as before, we define e↵ective one-dimensional systems along x by fixing
the value of ky , for which we calculate the polarization P (ky ). C denotes the total Chern
number of the multi-band system.
Using the basis-independent expression (1.36) for the polarization P↵ (ky ) we arrive at the
non-Abelian generalization of the Chern number:
Z
1
C=
dky @ky Im log det Ŵ (ky ).
(1.38)
2⇡ BZ
As before, this expression can alternatively be formulated in terms of the Berry curvature,
Z
⇣
⌘
1
C=
d2 k trF̂(k),
F̂(k) = rk ⇥ Â(k) .
(1.39)
2⇡ BZ
z
Many-body Chern number and fractional quantum Hall e↵ect.– The generalization of the
non-Abelian Chern number to interacting systems is straight-forward when the flux-insertion
trick of Niu et al.[88] is used. In this case, for a given configuration ✓ of the magnetic fluxes,
we consider M (closely) degenerate gapped states | ↵ (✓)i, where ↵ = 1...M . The many-body
Chern number is thus defined as the winding number of the corresponding Wilson loop,
 Z 2⇡
Z
1
C=
d✓y @✓y Im log det Ŵ (✓y ),
Ŵ (✓y ) = P exp i
d✓x Âx (✓) .
(1.40)
2⇡ BZ
0
As mentioned above, ground state degeneracies appear in the fractional quantum Hall
e↵ect for a system placed on a torus (genus g = 1), which is also an indicator for topological
order. In this case only the total Chern number C of the collection of all ground states can
be defined using Eq.(1.40). To calculate the quantum Hall response xy in this case, a weak
circular electric field Ey is realized by adiabatically changing ✓y in time. The corresponding
Hall current Jx is proportional to the change of polarization P (✓y ) which is derived from
the generalized King-Smith - Vanderbilt relation (1.36). To this end the eigenvalues of the
Wilson loop, corresponding to the Wannier centers, have to be tracked when ✓y is changed.
The generic scenario is sketched in FIG.1.2 (b), and we observe that instead of returning
1.2. FUNDAMENTAL CONCEPTS
43
to themselves after one cycle of flux insertion, quantum states switch partners. Because
degenerate eigenvalues of the Wilson loop Ŵ (✓y ) are prevented by avoided crossings – unless
symmetries are present – we find that, in general, M cycles of flux insertion are necessary for
a quantum state to return to itself. Meanwhile the polarization changes by an amount given
by the total Chern number C. Therefore, as suggested by Niu et al.[88], we can average over
M flux insertions to obtain the fractional quantum Hall response,
xy
=
e2 C
e2
⌘ C̃,
hM
h
C̃ = C/M.
(1.41)
Here C̃ = C/M is the Chern number per state, which may take fractional values.
1.2.4
The Hofstadter-Bose-Hubbard model
Now we will discuss a specific model of interacting bosons which supports topological order.
It combines two paradigmatic models of modern physics, the Hofstadter model [96] describing
non-interacting charged particles hopping on a lattice which is placed in a homogeneous
magnetic field, and the Bose-Hubbard model [97, 98, 99] describing interacting bosons in a
lattice potential. Because in the continuum limit of the Hofstadter model the quantum Hall
problem is recovered, the first ingredient, a magnetic field, gives rise to topological order in this
case. Both the lattice structure and the interactions between the bosons enrich the resulting
topological phases dramatically. In the continuum limit of the Hofstadter-Bose-Hubbard
model the fractional quantum Hall problem of bosons is recovered, where fractionalization
can give rise to quantum states whose excitations obey anyonic statistics, which may even be
non-Abelian [100, 101]. Lattice e↵ects, on the other hand, cause the fractal structure of the
Hofstadter butterfly energy spectrum [96], and it is an interesting question how the lattice
a↵ects interacting states.
The Hofstadter-Bose-Hubbard model was discussed for the first time by Sørensen, Demler
and Lukin [14] in the context of ultra cold quantum gases. In this work, which was supplemented by a subsequent paper by Hafezi et al.[102], it was shown that fractional quantum
Hall states of bosons exist even in the presence of a lattice. Later this idea was generalized
to include lattices without a magnetic field, for which Haldane showed [103] that nevertheless topological order can emerge. In the presence of interactions this leads to fractional
quantum Hall states even in the absence of a magnetic field, so-called fractional Chern insulators [104, 105]. From this general point of view, the Hofstadter-Bose-Hubbard model is one
particular model realizing fractional Chern insulators.
This section is organized as follows. We start by introducing the model Hamiltonian and
discuss briefly how it can be implemented in various systems. We move on and consider the
continuum limit of the Hofstadter-Bose-Hubbard model, which brings us to a short review of
the bosonic fractional quantum Hall e↵ect. We close the section by summarizing the results
of [14, 102] how a lattice a↵ects the fractional quantum Hall states.
Model Hamiltonian
The Hofstadter-Bose-Hubbard model is described by the following Hamiltonian,
ĤHBH =
J
X⇣
hi,ji
⌘ UX
⇣
â†i âj ei'ij + h.c. +
â†j âj â†j âj
2
j
⌘
1 .
(1.42)
44
CHAPTER 1. INTRODUCTION
Here J > 0 is the hopping amplitude between the sites (labeled by i, j) of a two dimensional
square lattice. U is the standard on-site interaction strength familiar from the Bose-Hubbard
model. â†j (âj ) creates (annihilates) a boson at site j.
In contrast to the standard Bose-Hubbard model, the hopping elements are modified by
additional phase factors ei'ij picked up when particles hop between neighboring sites hi, ji.
These so-called Peierls phases mimic the Aharonov-Bohm phase 'AB [106] acquired by a
charged particle when it encircles an area A enclosing magnetic flux ; 'AB = 2⇡ / 0 ,
where 0 is the magnetic flux quantum. Note that 'ij itself is not gauge invariant, because
by re-defining âj ! âj ei#j we obtain 'ij ! 'ij +#j #i . However, because the magnetic flux
can not be gauged away, the sum of the phases 'ij around one plaquette p is a gauge-invariant
quantity, which we write as
X
'ij = 2⇡↵.
(1.43)
p
In summary, the Hofstadter-Bose-Hubbard model is characterized by two dimensionless
parameters: U/J and ↵. In addition, to determine its phase diagram the density ⇢ (i.e. the
number of particles per lattice site) or, equivalently, the chemical potential µ is required. The
density is related to the magnetic filling fraction ⌫ (i.e. the number of particles per magnetic
flux quantum) by
⇢ = ⌫↵.
(1.44)
Implementation
The Hofstadter-Hubbard model can be implemented in various physical systems. The weakly
interacting two-dimensional electron gas in the presence of a magnetic field is described by
a similar Hamiltonian, in a regime where interactions may be neglected. In such a setting
the flux per plaquette ↵ ⌧ 1 is small because of the microscopic atomic lattice spacing.
However, by the use of Moiré superlattice structures, larger values of ↵ can be realized. This
has recently lead to the first observation of the Hofstadter butterfly, in a system of bilayer
graphene coupled to hexagonal boron nitride [107].
The Hofstadter model of free bosons has also been realized experimentally with ultra cold
atoms [108, 109, 110] and optical photons [111]. The main challenge for these experiments was
to implement artificial gauge fields. Several di↵erent approaches how gauge fields for neutral
particles can be implemented were explored [112, 113, 79, 93, 14, 114, 115, 116, 117, 118],
as summarized also in a recent review article by Dalibard et al. [119]. More specifically, for
photonic systems Refs.[120, 121, 122, 123] explored alternative approaches.
In the atomic system strong local interactions are readily available [14, 5] and the HofstadterHubbard model can be implemented. Currently the biggest challenge for these experiments
is to prepare the corresponding ground states, see e.g. [110, 124]. Alternatively, dipolar spin
systems were suggested as candidates to realize the Hofstadter-Bose-Hubbard model [125]. In
photonic systems, on the other hand, achieving strong interactions on a single-photon level
is difficult. In this context Rydberg systems are currently explored [126, 127, 128, 129], or
quantum dots could be coupled to photonic cavities, but an implementation in an experiment
like [111] is not yet in sight.
Continuum limit – fractional quantum Hall e↵ect
The continuum limit of the Hofstadter-Bose-Hubbard model is obtained when the magnetic
area AB – i.e. the area by which one unit of magnetic flux 0 is enclosed – is large compared
1.2. FUNDAMENTAL CONCEPTS
45
to the size of one plaquette Ap . In a square lattice with period a this corresponds to
a2 /↵ = AB
Ap = a 2 ,
i.e.
↵ ⌧ 1.
In this limit the Hamiltonian (1.42) may be written (up to an overall energy shift)
"
#
Z
2
⇣
⌘
(
ir
A(r))
g
r
BB
ˆ(r) +
ˆ† (r) ˆ(r) ˆ† (r) ˆ(r) 1 ,
Ĥ = d2 r ˆ† (r)
2M
2
(1.45)
(1.46)
where M 1 = 2Ja2 and gBB = U/a2 . The vector potential A(r) is defined such that the
corresponding magnetic field is (rr ⇥ A(r))z = 0 /AB = ↵ 0 /a2 . Therefore the continuum
limit is defined by
↵ ! 0,
↵/a2 ! const.,
Ja2 ! const.,
U/a2 ! const.
(1.47)
The Hamiltonian (1.46) realizes the fractional quantum Hall e↵ect for bosons. At this
point, no attempt shall be made to review the rich physics of the quantum Hall e↵ect in any
detail. Instead the interested reader is referred to the classic book by Prange and Girvin [130]
and the books by Ezawa [131] and Jain [26]. The notations used in this thesis are the same
as in the diploma thesis of the author [55], where also a brief review of the quantum Hall
e↵ect can be found. In the following an overview is given of the theoretical and experimental
progress made towards a realization of the bosonic fractional quantum Hall e↵ect.
The single-particle eigenstates of the Hamiltonian (1.46) are largely degenerate Landau
levels, split by the so-called cyclotron energy !c . Depending on the symmetry of the gauge
choice for A(r), the degenerate states within the Landau levels can be labeled by their angular
momentum `z (symmetric gauge) or by their linear momentum kx in x-direction (Landau
gauge). For sufficiently small temperatures kB T ⌧ !c well below the cyclotron gap, bosons can
occupy only the lowest Landau level. Then, interactions between the bosons are switched on.
It is mostly assumed that the interaction strength is sufficiently small, such that occupations
of higher Landau levels (i.e. Landau level mixing) can be neglected (for a discussion see
[132]). In this limit the interaction Hamiltonian is characterized completely by Haldane’s
pseudo-potentials [133].
When the boson density is large, with many bosons per flux quantum ⌫
1, a condensate
is formed. Because of the artificial gauge field vortices appear in the condensate and form a
triangular Abrikosov lattice, in analogy with the behavior of a type-II superconductor. This
regime was investigated experimentally using rotating ultra cold Bose gases in the groups of
Ketterle [134] and Cornell [135]. A review of such rapidly rotating quantum gases was given by
Cooper [115]. The regime where ⌫ approaches unity was investigated theoretically by Cooper
et al.[100], where the authors predicted a quantum phase transition to a correlated quantum
liquid state around ⌫c ⇡ 6. Attempts were made to access this regime experimentally, but the
smallest achievable filling fraction was ⌫ & 500 [135], well within the condensed phase. For
even smaller filling fractions large heating rates as well as small atomic densities pose serious
problems for these kinds of experiments. A possible way out might be the use an array of
rapidly rotating micro traps with only a few atoms per well, an approach that was already
tried by Gemelke et al. [136].
In the regime of small fillings ⌫ . 1 the fractional quantum Hall e↵ect for bosons can
occur. This was first realized in the context of ultra cold quantum gases by Wilkin et al.[137].
In particular it was pointed out that the bosonic version of the Laughlin state [47] at filling
46
CHAPTER 1. INTRODUCTION
⌫ = 1/2,
LN (z1 , ..., zN )
=
Y
i<j
(zi
zj ) 2 e
P
j
|zj |2 /4
,
(1.48)
is an exact, incompressible eigenstate of the Hamiltonian (1.46) with zero interaction energy (see also [138]). In fact, from the form of the wavefunction (1.48) it is obvious to see
that the interaction energy vanishes. Because the Laughlin wavefunction is purely analytic,
@/@zj⇤ LN = 0, it follows that it lies entirely within the lowest Landau level.
Following this discovery, the analogy to the fractional quantum Hall problem was further
explored. Wilkin and Gunn [139] and Cooper and Wilkin [140] employed particle-flux composites (so-called composite bosons and fermions) to describe the bosonic fractional quantum
Hall states, following earlier ideas by Read [141], Zhang et al. [142] and Jain [143] for describing the electronic fractional quantum Hall e↵ect. In this context it was suggested by
Cooper et al. [100] that even more exotic quantum states, so-called parafermion states, may
appear. The parafermion states were introduced by Read and Rezayi [50] as a generalization
of the celebrated Moore-Read Pfaffian state [7, 8] which is believed to describe the ⌫ = 5/2
incompressible fractional quantum Hall state observed for interacting electrons [48]. Extensive exact numerical calculations by Regnault and Joliceur [101, 144] supported the existence
of parafermions in bosonic systems and estimated their excitation gaps.
The fractional quantum Hall Hamiltonian (1.46) can be generalized by considering more
general forms of interactions. For example dipolar [145] or Rydberg interactions [55, 146]
add a long-range character. In Chapter 4 of this thesis we will show how tunable Rydberg
interactions can give rise to correlated quantum states which go even beyond the realm of
parafermion states. In circuit QED systems, on the other hand, even three- (and more) body
interactions may be engineered [147] using superconducting qubits. This allows to implement
parent Hamiltonians for which e.g. the Moore-Read Pfaffian state is an exact eigenstate.
The experiments searching for the fractional quantum Hall e↵ect of bosons encountered
difficulties when trying to implement large artificial magnetic fields. This goal has recently
been achieved in optical lattices. In such systems, however, it is crucial to understand the
e↵ect of lattices, which we will briefly discuss now.
Lattice e↵ects
Sørensen et al. [14] and Hafezi et al. [102] investigated what e↵ect the lattice has on the
fractional quantum Hall states described above. To this end they used exact numerical diagonalization of systems at ⌫ = 1/2 with a few particles and considered the ground states of
(1.42) at finite values ↵. As a first indicator they calculated the overlap of the actual ground
state to the Laughlin wavefunction (1.48) evaluated at the lattice sites only. For values below
↵ . 0.3 good overlaps indicate that the ground state has topological order. Similarly, the
excitation gap takes finite values up to ↵ ⇡ 0.25 after which it drops, indicating a quantum
phase transition to a further state. Sørensen et al. [14] found an optimum value of ↵ where
the bulk excitation gap ⇡ 0.25J takes its maximum value.
To clarify further for which values of ↵ a phase transition away from the topologically ordered Laughlin state takes place, Hafezi et al. [102] calculated the many-body Chern number
of the ground state. To this end they considered the system on a torus, where a two-fold topological ground state degeneracy was reported. To calculate the Chern number they followed
an approach introduced by Kohmoto [62], which requires a gauge fixing. Here we will present
our own results for their system, based on the method described in Sec.1.2.3 which does not
require any gauge-fixing. To this end we calculate the Wilson loop Ŵ (✓y ) at di↵erent values
1.2. FUNDAMENTAL CONCEPTS
47
2
1.5
1
0.5
0
0
0.2
0.4
0.6
0.8
1
Figure 1.3: The U (2) Wilson loop phase 'W (✓y ) = Im log det Ŵ (✓y ) is shown (in units of ⇡),
calculated for the Hofstadter-Bose-Hubbard model on a torus (6 ⇥ 7 sites). Other parameters
were U/J = 5, ↵ = 1/7 and N = 3. The Chern number C = 1 of the two-fold degenerate
ground state manifold is obtained as the winding of 'W /2⇡, thus each states carries a Chern
number C̃ = 1/2. We also checked that there is always a finite energy gap
to further
states, and that the discretization yields a reasonable approximation for the Wilson loop.
(We need | det Ŵ | = 1 because Wilson loops are unitary. Finding | det W | ⇡ 0.5 is sufficient
when keeping in mind that errors lead to an exponential decay of | det W | with the number
of discretization points.)
of ✓y and plot the phase 'W (✓y ) = Im log det Ŵ (✓y ) in FIG.1.3. Its winding over ✓y yields
the total Chern number, C = 1 in this case. Because M = 2 topologically degenerate ground
states exist, the Chern number per particle is C̃ = 1/2, consistent with the results in [102].
For values of ↵ & ↵c ⇡ 1/4 Hafezi et al.[102] no longer found a two-fold degenerate ground
state manifold. Therefore they can no longer assign a Chern number per state C̃ = 1/2 to the
system, which they interpret as a quantum phase transition to a further state. Interestingly,
this transition takes place only around ↵c ⇡ 1/4, when the overlap to the Laughlin state
has already dropped to a value close to zero. This indicates the power of the Chern number
method to unambiguously identify topological order. The nature of the ground state at larger
values ↵ & 1/4 is not completely understood at present.
48
CHAPTER 1. INTRODUCTION
Chapter 2
Topology in the Superlattice Bose
Hubbard Model
2.1
Outline and Introduction
In this chapter we investigate a paradigmatic toy model for a symmetry-protected topological
(SPT) phase of interacting bosons in one dimension. To this end we consider repulsively interacting bosons hopping between the sites of a superlattice, with alternating hopping elements
t1 and t2 , for di↵erent fillings ⇢. This generalizes the non-interacting inversion-symmetric SuSchrie↵er-Heeger (SSH) type model [12], which is a minimal model for the realization of SPT
order. On the one hand, our model can be easily implemented in experiments with ultra cold
atoms. This allows to generalize previous experiments with weakly interacting Bose-Einstein
condensates [13] to a highly correlated regime. On the other hand, it serves as a toy model
to investigate the properties of interacting SPT phases of bosons in general.
Topological phases have become an intensely studied subject in many fields of physics.
A key signature of topological order is the emergence of edge states at a sharp boundary.
These edge states are not only robust and protected against perturbations, but moreover they
can have exotic properties. In particular edge states appear in the integer- and fractional
quantum Hall e↵ects [64, 148, 65] (where they are chiral), in topological insulators [28, 32, 33]
(where they are helical), in topological superconductors [9] (where they can support nonAbelian Majorana fermions) or in various one-dimensional models including spin-one chains
[149, 150, 151]. In fact the phenomenon appears to be so generic that the question arises
if topological order can exist at all in the absence of edge states. In this chapter we will
investigate this question in detail using our inversion-symmetric toy model. In this particular
case we can show that, although inversion symmetry is sufficient to protect the topological
order of the state, it is incapable of protecting the edge state (in the sense of a quantum
mechanical degree of freedom localized at the edge).
The topological properties of the non-interacting SSH model are classified by a Z2 topological invariant, defined by the Zak phase 'Zak . It is quantized, due to the inversion symmetry
of the model, an can only take two values 'Zak = 0, ⇡. In an experiment with weakly interacting ultra cold bosons the quantized Zak phase has recently been measured [13]. Here
we generalize the topological invariant to strongly interacting many-body systems, by using
twisted periodic boundary conditions as described in Sec.1.2.2. A similar quantized invariant,
which moreover classifies the topological Haldane spin-one phase [149, 152], was introduced
previously by Hatsugai [153]. We show that the ⇢ = 1/2 dimerized Mott insulator (MI) phase
of the superlattice Bose Hubbard model is characterized by such a many-body Zak phase. It
49
50
CHAPTER 2. TOPOLOGY IN THE SUPERLATTICE BOSE HUBBARD MODEL
has two distinct dimerizations which are separated by a topological phase transition.
The topological properties of non-interacting superlattice models have been studied previously by several authors, especially focusing on zero-energy edge modes, see [154, 155, 156,
157, 158]. In particular it was pointed out in Refs. [154, 155] that there is a strict relation
between the existence of a single-particle mid-gap edge state at an open boundary, and the
value of the bulk topological invariant. Here we show that in the case of moderately interacting bosons, a similar bulk-boundary correspondence does not hold in general due to the
absence of the chiral symmetry. Instead, as we establish using analytic approximations and
numerical density-matrix renormalization group (DMRG) simulations [59, 159, 60], a generalized relation holds. While either a state with half a particle or half a hole1 localized at the
edge remains stable until the MI melts due to tunneling, one of the two states hybridizes with
the bulk already substantially before the bulk phase transition takes place.
We suggest experiments with ultra cold atoms to probe for the existence of fractionally
charged edge states. Introducing a localized potential step allows to create an interface between gapped MI phases with di↵erent topological invariants. As a result many-body ground
states emerge that display density minima or maxima at the interface in analogy to an unoccupied or occupied single-particle edge state for free fermions. This can easily be observed
with techniques developed in recent years [160, 161, 162].
We extend our analysis of the super-lattice Bose Hubbard model and include repulsive
nearest neighbor interactions. This stabilizes incompressible MI phases also at fractional
fillings ⌫ := 2⇢ = 1/2 of the lowest Bloch band of the SSH model. Unlike in the ⌫ = 1
MI phase discussed previously, the ⌫ = 1/2 ground state spontaneously breaks an inversion
symmetry. This leads to an interesting interplay of SPT order and spontaneous symmetrybreaking, which we discuss in detail. This leads us to a precise definition of the topological
order in the extended superlattice Bose Hubbard model, using Chen et al.’s [6] classification
scheme. We show that, even when the topological order survives, many of the usually expected
indicators of topological order are not robust. Furthermore we investigate finite systems with
open boundary conditions and find interesting behavior of the fractionally charged edge states.
In particular they can be transformed into topological excitations in the bulk of the MI, which
are fractionally charged domain walls.
This chapter is based on the results from Refs. [P2] and [P12]. It is organized as follows.
In Sec.2.2 the superlattice Bose Hubbard model is introduced and its bulk phase diagram is
obtained perturbatively. In Sec.2.3 the topological order of the model is discussed. How this
relates to topological edge states is examined in Sec.2.4, where also possible experiments are
discussed. In Sec.2.5 long-range interactions are added and the rich topological properties
of the extended superlattice Bose Hubbard model are investigated. A general classification
scheme for inversion symmetric one-dimensional models is suggested in Sec.2.6. We close with
a summary and by giving an outlook in Sec.2.7.
2.2
Superlattice Bose Hubbard model
The one-dimensional superlattice Bose Hubbard model can be described by a grand-canonical
Hamiltonian, see FIG.2.1 (a),
Ĥ =
1
X⇣
j odd
t1 â†j âj+1 + h.c.
⌘
X ⇣
j even
⌘ UX
t2 â†j âj+1 + h.c. +
n̂j (n̂j
2
j
1) +
X
(✏j
µ)n̂j .
j
(2.1)
The charge is defined in relation to the density distribution of the topologically trivial dimerization.
2.2. SUPERLATTICE BOSE HUBBARD MODEL
(a)
(b)
51
0.8
0.6
0.4
0.2
0
−1
−0.5
0
0.5
1
Figure 2.1: (a) Schematic picture of the ⇢ = 1/2 Mott insulating phase in the one-dimensional
SSH-type superlattice Bose Hubbard model. It has two topologically distinct dimerizations
I and II, the second of which may support topologically protected edge states at an open
boundary. (b) Grand canonical phase-diagram of the superlattice Bose Hubbard model with
t2 = 0.2t1 obtained by DMRG. This figure was taken from Ref. [167]. One recognizes the
presence of MI phases with integer and half-integer filling ⇢, which are surrounded by a
superfluid region. The ⇢ = 1/2 MI phase, which will be examined in detail, is blue-shaded.
Here âj and â†j are the boson annihilation and creation operators at lattice site j, and n̂j =
â†j âj . Particles can tunnel between the sites j, with alternating hopping amplitudes t1 and t2 ,
and their mutual interaction is described by a Hubbard-type U term. In addition, ✏j describes
an external potential (e.g. a harmonic confinement or hard walls) and µ denotes the chemical
potential. In the case of non-interaction bosons (i.e. U ! 0) and for ✏j ⌘ 0, Eq. (2.1) reduces
to the inversion symmetric free SSH model [13]. The model (2.1) can be realized with ultra
cold atoms [5] or in superconducting networks [163, 164, 165].
The model (2.1), and generalizations thereof, have been studied intensely in the past.
Using a supercell mean-field approach it was shown that the superlattice can induce additional
loophole-shaped Mott lobes in the grand-canonical phase diagram [166]. This prediction was
confirmed using DMRG calculations [167], see FIG.2.1 (b). Now we will briefly review those
results which are of relevance for the following discussion of topological order in Sec.2.3. From
now on our main focus will be on the ⇢ = 1/2 MI phase.
2.2.1
Hard-core bosons and chiral symmetry
To relate our model to previously obtained results with non-interacting fermions, we consider
the hard-core limit where U ! 1. In this regime the system becomes incompressible at integer
and half-integer fillings. This can be understood from the fermionization of the underlying
particles and the corresponding Pauli blockade. Hence the hard-core boson model inherits
its topological properties from the free-fermion model, which are protected by the chiral
symmetry.
To map hard-core bosons onto non-interacting fermions in one dimension, the JordanWigner transformation [168] can be employed. One can easily check that the following operators,
P
P
†
†
ĉ†j = ei⇡ i<j âi âi â†j ,
ĉj = âj e i⇡ i<j âi âi ,
(2.2)
fulfill fermionic anti-commutation relations, {ĉi , ĉ†j } = i,j . Moreover the Hilbertspace of
hard-core bosons can directly be mapped to that of fermions.
If we apply the transformation (2.2) to the Hamiltonian (2.1) we obtain the free-fermion
52
CHAPTER 2. TOPOLOGY IN THE SUPERLATTICE BOSE HUBBARD MODEL
SSH model. When µ = ✏j = 0 it simply reads
⌘
X⇣
ĤU !1 =
t1 ĉ†j ĉj+1 + h.c.
X ⇣
j even
j odd
⌘
t2 ĉ†j ĉj+1 + h.c. ,
(2.3)
and it can easily be solved by Fourier transformation,
Z
ĤU !1 =
dk ĉ†k h(k)ĉk .
(2.4)
BZ
The corresponding Bloch Hamiltonian has two bands with energy ✏± (k),
h(k) =
✓
t1
0
t2 e
◆
t2 eika
,
0
t1
ika
✏± (k) = ±
q
t21 + t22 + 2t1 t2 cos (ak).
(2.5)
Unless t1 = t2 we obtain a band insulator, which corresponds to the incompressible MI phase
of the hard-core bosons at half-filling ⇢ = 1/2, see FIG.2.1 (b).
Chiral symmetry
Thus in the hard-core limit of our model, the free-fermion SSH case is recovered. In particular
its topological properties carry over. It was noted previously that the topological order in
the SSH model is protected by the chiral symmetry [154, 155]. A Hamiltonian Ĥ has a chiral
symmetry, if it anti-commutes with some unitary matrix ˆ , i.e. {Ĥ, ˆ } = 0. Indeed, from
Eq.(2.5) it is easy to read o↵ that {ĥ(k), ˆz } = 0, i.e. ˆz provides the chiral symmetry2 .
Let us briefly discuss how the chiral symmetry protects topological properties in the SSH
model. To this end we make use of the fact that ˆ can be used to generate a solution of energy
E from an eigenstate | (+) i with energy E. In fact it is easy to check that ˆ | (+) i is such
an eigenstate,
Ĥ ˆ |
(+)
ˆ Ĥ|
i=
(+)
i=
E ˆ|
(+)
i,
|
( )
i=e
i
ˆ|
(+)
i.
(2.6)
Here is an arbitrary phase factor. In particular, this property protects zero-energy edge
states which can not hybridize with the bulk but have to remain at zero energy E = 0 as long
as the chiral symmetry is fulfilled [155].
The second important consequence of the chiral symmetry for a band structure is that it
leads to a quantization of the Zak phase. Consider a chiral-symmetric system with 2N bands
and an energy gap around zero energy. Then the total Zak phase of the N negative-energy
( )
( )
bands |u↵ i is quantized to 'Zak = 0, ⇡. To show this, we first of all note that the total Zak
( )
(+)
phase is trivial, 'Zak + 'Zak = 0 mod 2⇡. Moreover, the Berry connections of positive and
negative energy bands are related by
A(↵
( )
)
⌘ hu(↵ ) (k)|[email protected] |u(↵ ) (k)i = @k
↵ (k)
+ A(+)
↵ .
(+)
(2.7)
( )
Hence we also obtain 'Zak 'Zak = 0 mod 2⇡, which only leaves the possibilities 'Zak = 0, ⇡.
Finally we discuss what physical meaning the chiral symmetry ˆz , which we identified in
the free-fermion model, has in the bosonic system. The action of ˆz translates into
â2n
2
ˆz
! â2n ,
â2n+1
ˆz
!
â2n+1 .
(2.8)
Note that, in order to find a chiral symmetry on the basis of the Bloch Hamiltonian it is important that
the unitary matrix ˆ is independent of the quasimomentum k.
2.3. TOPOLOGICAL ORDER IN THE SUPERLATTICE BOSE HUBBARD MODEL 53
From Eq.(2.1) we observe that the hopping-part of the Hamiltonian reverses sign when ˆz is
applied. The interaction part, in contrast, does not change its sign. Therefore the super-lattice
Bose Hubbard model (2.1) can only be chiral-symmetric if U = 1, where both U = ±1
prohibit double-occupancy of any site. This break-down of the chiral symmetry for finite
interactions U < 1 has important consequences for the edge states of the super-lattice Bose
Hubbard model.
2.2.2
Bulk phase diagram
To derive the bulk phase diagram of the super-lattice Bose Hubbard model (2.1), we apply
a simple perturbative treatment. This so-called cell-strong coupling perturbative expansion
(CSCPE) technique was developed in Ref.[169] and applied to the super-lattice model previously in [170]. For the ⇢ = 1/2 Mott phase we obtain for the critical chemical potentials,
µ
1/2
1/2
µ+
=
(t1
= (t1
t2 ),
U
t2 ) +
2
1
2
q
16t21 + U 2
(2.9)
4t1 t2
.
U
(2.10)
These expressions are valid when t1 > t2 , up to corrections of order O t22 /U, t2 t21 /U 2 . Note
1/2
1/2
that in the limit U ! 1 because of the developing chiral symmetry µ
⇡ µ+ . When
1/2
1/2
the chemical potential is between these bounds, µ < µ < µ+ , the Mott insulator is stable
against melting. As shown in FIG.2.2 below, these expression are in good agreement with the
phase boundaries determined from DRMG calculations.
A generic ground-state phase diagram of the super-lattice Bose Hubbard model, taken
from Ref.[167], is shown in Fig.2.1 (b) for ✏j ⌘ 0. Besides MI phases with integer filling it
shows loophole insulating regions with half-integer filling for t1 > t2 . These regions shrink
when t1 decreases, and vanish at t1 = t2 (simple Bose-Hubbard model). They reappear when
t1 < t2 and, as we will argue now, the point t1 = t2 marks a topological phase transition even
for finite interactions U < 1.
2.3
Topological order in the superlattice Bose Hubbard model
In the case of non-interacting fermions, the topology of the band structure is determined by
its Zak phase 'Zak . As we discussed above, due to the chiral symmetry the Zak phase is
Z2 quantized in the SSH model. For finite interactions U < 1 the chiral symmetry is broken,
so one might expect that also the quantization of 'Zak is lost.
However, the inversion symmetry Iˆ leads to a similar quantization of the Zak phase,
'Zak = 0, ⇡, as pointed out already by Zak [81]. In this case the Bloch wavefunction at
ˆ
|u( k)i = ei k I|u(k)i.
Hence it follows that A( k) = A(k) + @k k and thus
'Zak =
0
⇡/a
= i log ⇠0
i log ⇠⇡
mod 2⇡.
(2.11)
Here ⇠0,⇡ are the eigenvalues of the inversion-operator Iˆ at k = 0, ⇡/a. Because of Iˆ2 = 1 they
are restricted to ⇠ = ±1, which leads the quantization of the Zak phase.
Even in the presence of soft-core bosons, U < 1, the super-lattice Bose Hubbard model
(2.1) is inversion-symmetric. Therefore we expect that the symmetry-protected topological
order remains, even though the chiral symmetry is broken. To show that this is indeed the
case, we now introduce a many-body topological invariant which characterizes the soft-core
super-lattice Bose Hubbard model (2.1).
54
CHAPTER 2. TOPOLOGY IN THE SUPERLATTICE BOSE HUBBARD MODEL
For interacting systems there is no conserved lattice quasi-momentum k and we employ a
many-body generalization of the winding number. It can be defined by introducing twisted
boundary conditions [88, 84], see Sec.1.2.2. I.e. we assume (xj +L) = ei# (xj ) for all particle
coordinates j = 1, ..., N and with the system size L, which corresponds to a magnetic flux
# threading the system. When this flux is adiabatically varied # = 0 ! 2⇡, the many-body
wavefunction | (#)i picks up a Zak phase [81, 78]
⌫=
Z
2⇡
0
d# h (#)|[email protected]# | (#)i.
(2.12)
As is the case for all Zak phases, the Hamiltonians at the beginning, Ĥ(# = 0), and the
end of the flux insertion, Ĥ(# = 2⇡), di↵er only by a gauge transformation. In our lattice
model we will make a particularly simple gauge choice, assuming that the flux # only modifies
the hopping amplitude across a single bond, where the two ends of the system are connected.
In this case, Ĥ(# + 2⇡) = Ĥ(#) and the Zak phase ⌫ reduces to a simpler Berry phase.
Because flux insertion adiabatically modifies the lattice momentum k, the many-body
invariant ⌫ reduces to the Zak phase of the lowest SSH band in the limit of hard-core bosons.
This can also be checked by a direct calculation where MI states can be described using
direct-product Gutzwiller wavefunctions. In this way we find that MIs with integer filling
are topologically trivial with ⌫ = 0 and those with half-integer filling can take the values
⌫ I = 0 for dimerization I and ⌫ II = ⇡ for dimerization II. (We defined the dimerizations I,
II as in FIG.2.1 and added the complex hopping t2 ei# between the ends of the system when
introducing twisted periodic boundary conditions.)
Now we proof that the topological invariant ⌫ stays strictly quantized by inversion symmetry, even for finite U < 1, as long as the particle-hole gap is finite. To this end we first note
that our system Ĥ(#) is invariant under the combined action of the inversion symmetry Iˆ and
time-reversal K (i.e. complex conjugation), for every value of #. First, inversion symmetry
maps the twisted hopping element t(#) = t2 ei# ! t2 e i# = t0 (#). Then complex conjugation
maps it back to its initial value, t0 (#) = t2 e i# ! t2 ei# = t(#). All remaining hoppings t1,2
are assumed to be real and thus una↵ected by K. The resulting anti-unitary symmetry of the
super-lattice Bose Hubbard Hamiltonian,
ˆ
⌧ˆ = IK,
[Ĥ(#), ⌧ˆ] = 0,
(2.13)
leads to the quantization of the Zak phase ⌫ = 0, ⇡, as shown by Hatsugai [153].
Here we quickly repeat the proof of Hatsugai and point out the relation of the Zak phase ⌫
to the eigenvalues of the inversion operator. From the symmetry (2.13) it immediately follows
that | (#)i = ei # ⌧ˆ| (#)i, in the absence of degeneracies of the ground state | (#)i. Hence
A(#) = ihˆ
⌧ (#)|@# ⌧ˆ| (#)i
= ihK (#)|@# K| (#)i
= [email protected]# (#)| (#)i
@#
@#
#
@#
#
=
=
#
(2.14)
=
A(#)
(2.15)
@#
#.
(2.16)
Because the wavefunction | (#)i can be chosen real at # = 0, ⇡, we obtain the quantization
of ⌫ in direct analogy to Eq.(2.11),
⌫=
0
⇡
= i log ⇠0
i log ⇠⇡ mod2⇡ = 0, ⇡.
Again, ⇠0,⇡ = ±1 are the inversion-eigenvalues at # = 0, ⇡.
(2.17)
2.4. TOPOLOGICAL EDGE STATES IN THE SUPERLATTICE BOSE HUBBARD
MODEL
55
Figure 2.2: (a) Sketch of the super-lattice Bose Hubbard model in the topologically non-trivial
dimerization (II) with an open boundary. Plots in (b)-(d), calculated by Michael Höning using
DMRG, show density distributions below (blue triangles) and above (red squares) the critical
chemical potential of edge-state occupation µe for t1 /U = 0.1, 0.2, 0.3 respectively. While a
well localized hole state (empty edge) can be observed in all cases, the particle state (occupied
edge) becomes unstable already before the MI melts (e). Due to the absence of the chiral
1/2
symmetry µ+ (green up-pointing triangles) approaches µe (black circles) already at small
values of t1 /U . The solid curves show the analytic results from CSCPE, the dashed straight
lines correspond to hard-core bosons and symbols were obtained from DMRG by Michael
1/2
Höning. (f) When µe approaches µ+ the particle state becomes delocalized, as can be see
from the localization length ⇠p,h of particle and hole edge states. Systems of length L = 65
are considered in the DMRG simulations.
We confirmed the quantization of the Z2 invariant ⌫ in small systems by exact numerical
diagonalization. Alternatively it could be calculated using DMRG, or as was recently shown
(in related spin systems) using quantum Monte Carlo methods [171]. Thus we conclude that
the non-trivial topology of the SSH bands carries over to bosons with finite interactions. This
is our motivation to study edge states of topologically non-trivial MI phases in the super-lattice
Bose Hubbard model in the following section.
2.4
Topological edge states in the superlattice Bose Hubbard
model
For the SSH model of free fermions it has been proven that there are N = 'Zak /⇡ zeroenergy edge states localized at an open boundary of the system [155]. They may serve as
a direct indicator for the symmetry-protected topological order in the system. Now we will
show that there are no genuine topologically protected zero-energy edge states in the superlattice Bose Hubbard model at finite interaction strength U < 1. The reason is that only
a global symmetry (inversion) remains, whereas the local chiral symmetry is broken by the
finite interactions. Nevertheless, the charge localized at the boundaries of the system takes
fractional values and remains quantized everywhere in the incompressible MI phase.
56
2.4.1
CHAPTER 2. TOPOLOGY IN THE SUPERLATTICE BOSE HUBBARD MODEL
Failure of the bulk-boundary correspondence
Let us consider adding an open boundary to a super-lattice Bose Hubbard model which has a
non-trivial dimerization. This corresponds to the situation (II) in Fig.2.1(a) and is sketched
in FIG.2.2 (a). From the connection to the SSH model, we expect a mid-gap (i.e. zero-energy)
edge state in the limit U ! 1 in this case. Indeed, for small values of t1 /U = 0.1 and 0.2 (i.e.
large U ) the DMRG simulations performed by Michael Höning show both, a well-localized
hole state | h i and particle state | p i. The corresponding chemical potentials were chosen
below and above a critical chemical potential µe , see Fig.2.2(b) and (c). This critical chemical
potential corresponds to the energy cost of adding one particle at the edge of the system. It
divides the grand-canonical phase diagram in two well-separated incompressible parts. In the
limit U ! 1 considered so far, the system is chiral-symmetric and µe = 0, see FIG.2.2 (e).
However, as shown in Fig.2.2(d), already for slightly smaller U , e.g. t1 /U = 0.3, the
situation changes. When increasing the chemical potential, the density of the bulk increases
non-locally instead of filling up the hole which is localized at the edge. Note that this takes
place well before the MI melts due to increased tunneling. Within CSCPE this corresponds to
a situation where it is energetically favorable for the particle to tunnel into the bulk instead of
localizing at the edge. We find for the critical chemical potential µe where the grand canonical
ground state turns from | h i into | p i,
µe =
2t22
U 2t1
.
(U + t1 )(U 3t1 )
(2.18)
We have plotted this result for µe along with the values from DMRG simulations in
Fig.2.2(e). As the chiral symmetry is broken for finite values of the interaction, µe 6= 0 is
1/2
1/2
no longer exactly midway between µ
and µ+ . More dramatically, at a tunneling rate of
1/2
t1 /U ⇡ 0.25 the curve touches µ+ indicating that it becomes energetically favorable to add
a particle to the bulk rather than localizing it at the edge. Within CSCPE we find for the
critical value
✓ ◆
t1
1 ⌘
t2
⇡
, ⌘= ,
(2.19)
2
U c 4(1 + ⌘ ⌘ /2)
t1
which is slightly below the numerically obtained value.
In FIG.2.2 (e) we observe that the curve for µe (t1 /U ) almost remains a straight line, and
1/2
starts to bend only when it approaches µ+ . This is related to an increasing delocalization
of the particle edge state | p i. In Fig. 2.2(f) we have plotted the numerically determined
localization lengths ⇠p,h for the particle and hole states in units of the lattice constant. They
are defined following Ref. [172] through the participation ratio
⇣P
⇠= P
j
j
nj
nj
⌘2
2
,
nj = |nj
✓ ✓
1
| ⇥ ± nj
2
1
2
◆◆
,
(2.20)
where “+” (“ ”) has to be used for ⇠p (⇠h ) corresponding to the particle (hole) edge-state respectively; ⇥ is the Heaviside step function. While the hole remains localized, the localization
length of the particle-state diverges as t1 /U approaches the critical value (t1 /U )c .
From our findings we conclude that the bulk-boundary correspondence does not hold in
the sense of a protected and localized many-body mid-gap state. The reason for this is that
the chiral symmetry, which restricts edge states to have zero energy in the SSH model, is
broken. Nevertheless, we will now show that the density distributions at an open boundary
are markedly di↵erent in topologically trivial and non-trivial phases.
2.4. TOPOLOGICAL EDGE STATES IN THE SUPERLATTICE BOSE HUBBARD
MODEL
2.4.2
57
Generalized bulk-boundary correspondence
In the topologically non-trivial case the edge-state with a localized particle, | p i, disappears
in the compressible region of the phase diagram at (t1 /U )c . Meanwhile, the edge-state with
a localized hole remains stable. Both these states | h,p i are characterized by the fractional
charge q localized at the edge. Now we show that it is given by the bulk topological invariant,
q = ⌫/⇡
mod 1,
(2.21)
and constitutes a topological invariant which is protected by the inversion symmetry of the
bulk. Note that this is true even though the edge breaks the inversion symmetry locally.
We will furthermore argue that independent of the precise number of particles localized at
the edge, all ground states with an incompressible bulk carry a fractional part of a charge q
mod 1 which is localized at the edge.
Our argument is completely generic and requires nothing more but an incompressible onedimensional system which is inversion symmetric. The fractional quantization of the charge q
mod 1 localized at the edges is a direct consequence of topology. To show this, consider first a
topologically trivial configuration, corresponding to the dimerization I in the interacting SSH
model, see FIG.2.1 (a). We will define the (fractional part of the) charge of the corresponding
ground state |Ii to vanish, q mod 1 = 0. Note that the integer part of the charge is not a
universal number and may, for example, be changed by adding local disorder to the system.
The definition of q mod 1 = 0 is more or less arbitrary, and it only serves as a reference here.
Next, a state in the topologically non-trivial configuration can be prepared using Laughlin’s argument [173] by adiabatically introducing half a Thouless-pump flux quantum [87, 174].
Let us explain what is meant by this. In one spatial dimension all states are short-range entangled, such that any incompressible state can be adiabatically transformed into any other
incompressible state [6]. In particular, we can find a Hamiltonian Ĥg which connects the incompressible ground state |Ii, Ĥ0 |Ii = EI |Ii, in the trivial dimerization I to the incompressible
ground state |IIi, Ĥ1 |IIi = EII |IIi, in the non-trivial dimerization II. The required Hamiltonian Ĥg necessarily breaks the inversion-symmetry for some intermediate 0 < g < 1, because
|I, IIi have distinct symmetry-protected topological order. During the adiabatic deformation
the polarization of the ground state changes by
P =
d
⌫ II
2⇡
⌫I ,
(2.22)
see Sec.1.2.2. Here d = 2a is the size of the unit-cell in the SSH model3 . An alternative and
more detailed discussion of this argument can be found in Appendix A.
From the quantization of the bulk topological invariants ⌫ I,II it follows that the halfThouless pump cycle described above pumps a fractional charge q = P/d mod 1 = 1/2
across the bulk of the system (see FIG.1.1). Hence, in the resulting state |IIi a fractional
charge q = 1/2 is localized at the edge4 . This proofs Eq.(2.21).
3
Note that we assumed that the simplest-possible half-Thouless pump, with no complete winding around
the critical region, was chosen, see e.g. [174]. If it would wind around the critical region an integer number of
d
times n, the polarization change would be P = 2⇡
⌫ II ⌫ I + nd.
4
A comment is order about fractional charges. Constructing a wavefunction with fractional charge localized
somewhere is trivial, by simply choosing a suitable density distribution. A state is only called fractionally
charged, if it is also stable, i.e. if the fractional charge is not subject to quantum fluctuations. This explains
the importance of the Thouless pump in our construction above: The pumped charge P is related to the
topological invariant ⌫, which is not fluctuating. Therefore we obtain a genuine fractional charge. This is also
the essence of Laughlin’s argument for the quantization of the Hall e↵ect [173].
58
CHAPTER 2. TOPOLOGY IN THE SUPERLATTICE BOSE HUBBARD MODEL
Let us emphasize again, that the integer part of the charge q in the final state is nonuniversal. In fact, in the hard-core limit U ! 1, two edge states with a fractionally charged
hole and a fractionally charged particle are stabilized at the same time at zero energy by
the chiral symmetry5 . This illustrates that the fractional part of the charge is a genuine
topological order parameter which can only be changed by global modifications of the system.
In our analysis of the super-lattice Bose Hubbard model we found that the state with
a fractional hole, | h i, remains stable all the way until the MI melts into the bulk and the
symmetry-protected topological order is destroyed. Let us emphasize again that there is no
way how the fractional part of the charge can disappear without a closing of the bulk gap. In
the dynamical argument above this is apparent from the fact that, during the entire adiabatic
evolution from |Ii to |IIi under Ĥg , the system remains fully gapped (by construction of
Ĥg ). Hence the state is always short-range correlated and the fractional charge can not
melt into a bulk excitation. An alternative argument starts from a grand-canonical setting.
There the bulk phase is determined by the chemical potential µ alone. As long as the bulk
is incompressible6 , we can calculate the fractional charge localized at the edge simply by
comparing the static bulk polarizations. Its fractional part is universal because of the particlenumber super-selection rule.
An interesting exception of the last rule follows if the incompressible ground state in
the bulk is degenerate. This situation generically occurs when a symmetry is spontaneously
broken, and we will discuss it in the following section.
2.4.3
Experimental considerations
In the last section we showed that edge states contain information about the symmetryprotected topological order of the bulk. This motivates us to discuss possible experimental
realizations of topological edge states with ultra cold atoms now.
Although a sharp boundary as discussed above is difficult to realize, an interface between
two MI phases with di↵erent fillings can be created by increasing the potential energy ✏j by
✏ for a number of consecutive lattice sites. For example, if we chose ✏ such that
µ1 < µ < µ1+
µ
1/2
+
1/2
✏ < µ < µ+ +
(2.23)
✏,
(2.24)
1/2
we expect an interface between a ⇢ = 1 and a ⇢ = 1/2 MI. Here µ± and µ1± denote the upper
(+) and lower ( ) boundaries of the insulating regions in the phase diagram of Fig.2.1 (b).
As shown e.g. in Ref.[175] for the case of Bose-Fermi mixtures, a sharp potential step can
be created by an admixture of a second atomic species, e.g. fermions, with very small hopping
rates. Under appropriate conditions (see [175]) the fermions form a connected cluster at the
center of the trap with unity filling and sharp boundaries. Due to boson-fermion interactions
this results in an increase of the potential energy of the bosons by ✏ in the center of the
5
The same is true in the interacting case. As we saw in the previous subsection, the state with a fractional
particle is no longer stabilized in general in the particular super-lattice Bose Hubbard model. Nevertheless, we
can easily envision a well-designed trapping potential for the fractional particle at the edge, preventing it from
melting into the bulk. Such additional terms in the Hamiltonian would not break the inversion-symmetry in
the bulk which is required to protect the topological order.
6
Let us clarify the meaning of incompressibility in this context. Here we call a system incompressible,
if it is incompressible in its bulk. For example, if the boundary of the system is determined by a smooth
potential extending over a length Le , generically we expect the gap of the full many-body Hamiltonian to scale
exponentially with Le , i.e.
⇠ e Le . This should be contrasted to a situation where the system becomes
critical and the bulk gap closes. In this case ⇠ e L and we may assume L
Le .
2.4. TOPOLOGICAL EDGE STATES IN THE SUPERLATTICE BOSE HUBBARD
MODEL
59
Figure 2.3: Ground-state density distributions of the super-lattice Bose Hubbard model with
a potential step ✏ between sites j = 6 and j = 5 leading to interfaces between ⇢ = 1/2 (in
the center) and ⇢ = 1 MI regions. A harmonic trapping potential ✏trap
= !(j + 0.5)2 was also
j
added, with !/U = 0.001. Results were obtained by DMRG simulations by Michael Höning
for µ/U = 0.55. (a) Topological trivial ⇢ = 1/2 MI phase with t1 /U = 0.04, t2 /U = 0.2
and ✏/U = 0.6. (b) Topological non-trivial ⇢ = 1/2 phase with t1 /U = 0.2, t2 /U = 0.04
and ✏/U = 0.6(blue squares), 0.7(red stars). The right panel illustrates the interface in the
topologically trivial (c) and in the nontrivial case with occupied (d) and unoccupied edge (e).
trap. Depending on the location of the interfaces relative to the sub-lattices the interface is
I
II
either topologically trivial, ⌫ = ⌫1/2
⌫1 = 0, or non-trivial if ⌫ = ⌫1/2
⌫1 = ⇡. (Recall
that the topological invariant of the ⇢ = 1 MI is always zero, ⌫1 = 0.)
Fig.2.3 shows typical density distributions in a weak harmonic trap, with an additional
potential step ✏. One clearly recognizes interfaces between a central ⇢ = 1/2 MI and
surrounding ⇢ = 1 MI regions. Since the number of heavy particles was taken to be even,
both interfaces are characterized by the same change of the topological invariant ⌫. The
upper plot shows the case ⌫ = 0, the lower one ⌫ = ⇡. In the first case there is a simple
step in the density and no additional structure at the edge. In the second case, in contrast, one
sees pronounced dips (peaks) in the average density, indicating the localization of a fractional
charge at the interface. We checked that between any two MI phases with integer fillings no
fractional edge states can be observed, irrespective of the dimerization.
To get a better microscopic understanding of the physics at the interface between ⇢ = 1
and ⇢ = 1/2 MIs, we will now consider a single potential step ✏ inside a system with open
boundary conditions. We start by generalizing our CSCPE analysis to this case, which is valid
in the the thermodynamic limit and for large interactions U
t1 and small hopping t2 ⌧ t1 .
The following analysis can furthermore be generalized for an interface between a ⇢ = m+1
2 and
m
a ⇢ = 2 MI, with m being a positive integer. In the case when m > 1 the low-lying bands
can be treated as inert, and only the renormalization of the hopping amplitudes due to Bose
enhancement has to be taken into account.
To derive the grand-canonical ground state we consider states | 1 i and | 0 i with and
without an additional particle localized at the interface, see FIG.2.3 (d) and (e). The critical
chemical potentials µ1± defining the borders of the ⇢ = 1 MI phases in the bulk in FIG.2.1 (b)
60
CHAPTER 2. TOPOLOGY IN THE SUPERLATTICE BOSE HUBBARD MODEL
(1)
(0)
Figure 2.4: The red line, µc = Ee
Ee , separates parameter regions with an occupied
interface (upper part – state | 1 i) and an unoccupied interface (lower part – state | 0 i). When
µc lies within the shaded region, both states | 0 i and | 1 i are stable against hybridization
with the bulk. Parameters are t1 /U = 0.2 and t1 /t2 = 5 and the results are obtained from
our CSCPE analysis.
can easily be calculated using CSCPE. Up to terms of order O t22 /U, t2 t21 /U 2 we find
q
q
t22 3U
1
2t1 U + 4t21 + U 2
+
+
16t21 + U 2
t1
2
2
✓
◆
q
1
4t1 t2
t2
1
2
µ = t2 + t1 +
U
16t1 + U 2 +
+ 2 .
2
U
2t1
µ1+ =
2t2
t1
4t1 t2
, (2.25)
U
(2.26)
To understand whether the particle localized at the edge can be bound to the interface or
di↵uses into the bulk, we perform a stability analysis. To this end we investigate under what
conditions the energy of the many-body state | 1 i (with a particle localized at the edge) lies
within the gap to collective bulk excitations on the state | 0 i. If | 1 i lies within the gap,
both many-body states | 0 i and | 1 i are stable edge states. By tuning the chemical potential
µ they can both be realized as the grand-canonical ground state.
The additional particle in | 1 i can tunnel into the bulk by creating an excitation in either
of the two MIs. Reducing (increasing) the number of particles at the MI interface by one and
simultaneously creating a particle (hole) in the bulk costs a finite energy if
µ1 < µc < µ1+
µ
(1)
(0)
1/2
(0)
+
✏ < µc <
1/2
µ+
(2.27)
+
✏,
(2.28)
(1)
where µc = Ee
Ee . Here Ee and Ee denote the energies of the states | 0 i and | 1 i
respectively. The CSCPE yields
✓
◆
✓ 2
◆
q
1
t21
t22
t22
(4t21 + 5t22 ) 1
t2 1
2
2
µc = ✏
2 2
+
✏ + 4t1 + O
,
. (2.29)
2
U
2t1 2 ✏
U
2
U U3
In Fig.2.4 µc is plotted and we can read o↵ the potential heights ✏ for which non of the
states | 0 i, | 1 i hybridizes with the bulk. This is the case when the red line µc lies within
the shaded area. Moreover, | 0 i (| 1 i) is the grand-canonical ground state for a chemical
potential µ < µc (µ > µc ).
To verify our analytical results, Michael Höning performed DMRG simulations for a step
potential ✏j /U = ✏⇥(j jstep + 0.5). In Fig.2.5 (a) we show the density distribution for
2.4. TOPOLOGICAL EDGE STATES IN THE SUPERLATTICE BOSE HUBBARD
MODEL
61
Figure 2.5: (a) Density distributions around a potential step ✏/U = 0.6 at jstep = 6 are
shown. Parameters are t1 /U = 0.2 and t1 /t2 = 5 while the chemical potential µ/U increases
from curve to curve. (b) The local particle number hn̂edge
i at the right edge of the ⇢ = 1 MI
1
region is shown as function of µ/U . The interface is occupied when µ > µc = 0.47 (dashed
red line).
di↵erent values of µ, which are indicated in FIG.2.4 by red bullets. For µ/U = 0.45 and
µ/U = 0.50 the system is inside the stability region of both edge states | 0 i and | 1 i. One
clearly recognizes a well localized dip in the density at µ/U = 0.45, corresponding to | 0 i,
and a peak at µ/U = 0.50 corresponding to | 1 i. When µ is chosen such that the system
is outside the region of the ⇢ = 1/2 MI (µ/U = 0.37 and µ/U = 0.65) the density dip on
the ⇢ = 1/2 side starts to vanish, while interestingly the peak on the ⇢ = 1 MI side remains.
Fig.2.5(b) shows the local occupation number hn̂edge
i at the edge of the ⇢ = 1 MI side as
1
function of µ/U . As soon as µ exceeds µc , as calculated in CSCPE (red dashed line), there is
a clear jump indicating the transition from unoccupied to occupied edge state.
A particular feature of the edge state in the super-lattice Bose Hubbard model is an
increase of the local density above unity on the ⇢ = 1 MI side. This is possible due to virtual
tunneling of a fractional particle localized at the interface. Hence the e↵ect is not present in
the hard-core limit where tunneling is completely suppressed. Within CSCPE we can calculate
Figure 2.6: The density hn̂edge
i at the edge of the ⇢ = 1 MI region is shown as function of
1
t1 /U . It was extracted from DMRG calculations by Michael Höning (blue squares) and we
compare it to our analytic result, Eq.(2.30) (green dashed line). Somewhat better agreement
is obtained when higher orders of t2 are included the CSCPE analysis (red full line). Other
parameters are µ/U = 0.96 1.6t1 /U , ✏/U = 0.88 0.8t1 /U and t1 /t2 = 5.
62
CHAPTER 2. TOPOLOGY IN THE SUPERLATTICE BOSE HUBBARD MODEL
this density, for which we obtain in zeroth order of t2
hn̂edge
i=1+
1
4 ✏t21 6 ✏2 t21
+
+ O(1/U 5 ).
U3
U4
(2.30)
Fig.2.6 shows a comparison of hn̂edge
i obtained by DMRG with Eq.(2.30). We find very good
1
agreement even for rather large tunneling rates just before the ⇢ = 1 MI starts to melt.
For t1 /U ! 0, hn̂edge
i ! 1 approaches the hard-core value, while finite tunneling leads to
1
hn̂edge
i
>
1.
Beyond
some
optimum value of t1 /U the edge state starts to delocalize as the
1
particle-hole gap of the insulator becomes smaller and hn̂edge
i decreases.
1
2.4.4
Relation to Majorana fermions
It was predicted recently that one-dimensional semi-conducting nano-wires may host Majorana fermions [9, 41, 42], which are zero-energy edge excitations of Kitaev’s topological p-wave
superconductor [9]. Because of the enormous interest in Majorana fermions, and the experimental e↵orts for their realization and detection [43, 44], we would like to comment briefly
on their relation to this work.
P
Majorana fermions can be understood from a quadratic Hamiltonian ĤBdG = k ĉ†k h(k)ĉk
(where ĉk = (ĉk," , ĉk,# , ĉ† k," , ĉ† k,# )T are Nambu-spinors), which is obtained after proximitycoupling fermions to an s-wave superconductor. In many respects this quadratic BogoliubovDe-Gennes Hamiltonian is similar to the free-fermion SSH model. In particular it hosts zeroenergy topological edge states which are protected by the intrinsic particle-hole symmetry.
On the level of the Bloch Hamiltonian h(k) this is completely analogous to the SSH model.
The exotic properties of the zero-energy edge states become apparent, however, when
expressing them in terms of the original fermions. Because of the superconducting terms in the
Hamiltonian, the resulting edge states are zero-energy Majorana fermions. In particular, this
gives rise to non-Abelian braiding properties of the edge excitations [74]. This phenomenology
is completely missing in the SSH model, and we would like to stress that the edge excitations
of the SSH model are no anyons (although they are fractionally charged).
In the SSH model, realized with soft-core bosons, we found that the zero-energy edge
states are no longer energetically protected when U < 1. In semi-conducting nano-wires,
on the other hand, interactions among the fermions still respect the particle-hole symmetry.
Hence Majorana fermions are robust against interactions. In the soft-core boson SSH model
we showed that the fractional part of the charge q mod 1 of edge excitations remains a welldefined topological invariant. This has a direct analogue in semi-conducting nano-wires. In
that case the fermion parity Nf mod 2, where Nf is the total fermion number, is a well-defined
topological invariant.
2.5
Extended superlattice Bose Hubbard model
In the half-filling soft-core boson SSH model studied so far, interactions are needed merely to
stabilize the incompressible Mott insulating ground state. In the limit of infinite interactions,
on the other hand, the model can be described as a simple band-insulator of free fermions.
Thus all topological order in the ⇢ = 1/2 MI can ultimately be attributed to the properties of
the underlying SSH-type band structure. Now we will consider the e↵ect of nearest-neighbor
interactions, which can stabilize Mott insulating phases at even smaller fillings. In particular
we will study the ⇢ = 1/4 MI phase, corresponding to a fractional filling ⌫ = 1/2 of the lowest
SSH-band.
2.5. EXTENDED SUPERLATTICE BOSE HUBBARD MODEL
63
The goal of this section is two-fold. Firstly, we discuss the specific model system of
interacting bosons in the SSH super-lattice potential. We demonstrate that, at fractional
fillings ⌫ < 1 of the lowest SSH band, it shows features which have previously been attributed
to topological order. In particular we investigate topological excitations, both at the edge and
in the bulk, and classify them by their topological charges. This system can be implemented
in current experiments with ultra cold quantum gases, which enables observations of its rich
properties. Secondly, we derive a rigorous topological classification of the model by applying
the general scheme introduced by Chen et al.[6]. It relies on the inversion symmetry of the
model. We show how the aforementioned observable characteristics of the model relate to
this topological classification. Furthermore we discuss the interplay of symmetry breaking
and topological order in the system.
The section is organized as follows. In 2.5.1 we introduce the extended super-lattice Bose
Hubbard model and present its grand-canonical bulk phase diagram. In 2.5.2 we carry out
the topological classification of the model. Finally we discuss the topological excitations of
the model in a setting with open boundary conditions in Subsec.2.5.3.
2.5.1
Model and bulk phases
The super-lattice Bose Hubbard Hamiltonian (2.1) will now be complemented by nearest and
next-nearest neigbor interactions,
Ĥ =
X⇣
t1 â†j âj+1 + h.c.
j odd
⌘
X ⇣
t2 â†j âj+1 + h.c.
j even
+
UX
n̂j (n̂j
2
⌘
1) +
j
X
µn̂j
j
X
(V1 n̂j n̂j+1 + V2 n̂j n̂j+2 ) . (2.31)
j
The finite-range interactions V1,2 can be implemented in ultra cold quantum gases by various
means. One possibility is the use of magnetic dipole-dipole interactions. Alternatively ultra
cold molecules with permanent electric dipole moments can be employed, which have an order
of magnitude larger dipole moment. The interaction strength can be increased even further
by the use of long-range Rydberg-Rydberg interactions. See Ref. [176] for a recent review
of the topic. Finally we note that yet another option would be the use of phonon-mediated
interactions when the system is coupled to a superfluid reservoir [177].
For true long-range interactions which are convex, the conventional extended Bose Hubbard model supports MI states for all rational fractional fillings ⇢ = p/q[178, 179, 180], also
referred to as the devil’s staircase. Here we have only nearest and next-nearest neighbor interactions, which can stabilize a quarter-filling ⇢ = 1/4 MI phase in the super-lattice potential.
We will now discuss this phase by considering the integrable limit of the model (2.31),
|t1
t2 |
U, V1,2 ,
t1 + t2
|t1
t2 | ⌧ U, V1,2
(integrable limit).
(2.32)
The first condition justifies projection of the model on the lowest SSH band. The second
condition guarantees that its band-width is much smaller than all relevant interaction energies.
Integrable Limit
In the integrable limit (2.32) the extended super-lattice Bose Hubbard model is classical. We
can solve the quadratic SSH part by introducing independent dimers,
p
p
b̂j = (â2j 1 + â2j ) / 2,
ĉj = (â2j 1 â2j ) / 2,
(2.33)
64
CHAPTER 2. TOPOLOGY IN THE SUPERLATTICE BOSE HUBBARD MODEL
with energies t1 (for b̂-dimers) and +t1 (for ĉ-dimers). We assumed t1 > t2 for concreteness.
The resulting e↵ective low-energy Hamiltonian reads
Ĥe↵ =
t
X⇣
b̂†j+1 b̂j + h.c.
j
⌘
(µ + t1 )
X
j
b̂†j b̂j + Ĥedge +
UX † ⇣ †
+
b̂j b̂j b̂j b̂j
4
j
⌘
X †
1 +w
b̂j+1 b̂j+1 b̂†j b̂j , (2.34)
j
where the reduced hopping reads t = t2 /2 ⌧ U, V1,2 and the renormalized nearest-dimer
interaction is⇣ w = (V1 + 2V
⌘ 2 ) /4. The edge part Ĥedge may contain non-interacting terms
†
†
Ĥedge = µ âL âL + âR âR in the non-trivial dimerization, where âL,R denote the left- and
right-most lattice sites in a system with open boundary conditions.
Indeed, when t ! 0 the e↵ective model (2.34) is diagonal in the Fock basis, [Ĥe↵ , b̂†j b̂j ] = 0
for all j. At half dimer-filling ⌫ = 1/2, corresponding to ⇢ = 1/4 in the original model, there
are two unique incompressible ground states with zero interaction energy. They correspond
to charge density waves (CDWs) with di↵erent polarizations, see FIG.2.9 (a). Using CSCPE
techniques to treat the renormalized hopping t perturbatively, we obtain for the critical chemical potentials of the ⇢ = ⌫/2 = 1/4 MI phase to lowest order in t
µ
1/4
=
t1 + 4t,
By introducing ⌘ = t1 /t2 > 1 we obtain
✓
◆
2
1/4
µ = t1 1
,
⌘
1/4
µ+ =
1/4
µ+
=
t1 + 2w
t1
✓
2
1+
⌘
4t.
(2.35)
◆
(2.36)
+ 2w.
Grand-canonical bulk phase diagram
To check the range of validity of the e↵ective Hamiltonian, we calculate the grand-canonical
phase diagram of the extended super-lattice Bose Hubbard model (2.31) in the bulk. To this
end DMRG simulations were performed by Richard Jen. Our results are presented in FIG.2.7.
We find incompressible MI phases at fractional fillings ⇢ = 1/2, 1/3 and 1/4. The ⇢ = 1/2
phase was studied in detail in the previous sections. The existence of a loophole phase at filling
⇢ = 1/4 is expected from our integrable e↵ective model (2.34). Note, however, that the latter
provides a microscopic description only when t1 /U & 0.25 is sufficiently large (for parameters
as in the figure). In this regime the simple CSCPE analysis (2.35) predicts instability of the
MI phase. Indeed, the tip of the Mott lobe in FIG.2.7 has to be understood as a BerezinskyKosterlitz-Thouless transition [181, 182]. For smaller t1 /U . 0.25 on the other hand, the
⇢ = 1/4 MI phase extends all the way to t1 /U ! 0.
Finally we also find an incompressible charge density wave at ⇢ = 1/3, which is incommensurate with the underlying super-lattice. When t1 /U becomes larger, the system develops a
superfluid fraction. In addition we find indications for super-solidity, i.e. a finite structure factor, but we did not investigate its robustness carefully. In fact it was shown in the conventional
extended Bose Hubbard model that no super-solid phase exists [183].
Bulk signatures of topological order
Now we summarize bulk signatures for topological order in the incompressible ⇢ = 1/4 MI
phase and discuss their stability to perturbations. First we note that the ground state (in
2.5. EXTENDED SUPERLATTICE BOSE HUBBARD MODEL
(a)
65
(c)
(b)
15
10
5
0
30
32
34
Figure 2.7: (a) We show the grand-canonical bulk phase diagram of the extended super-lattice
Bose Hubbard model as it is expected in an infinite system. Simulations were performed in a
finite-size system with open boundary conditions (b), where edges pin the CDW configuration.
This illustrates the spontaneous symmetry breaking expected to occur in an infinite system:
the presence of point-like edges in a one-dimensional system is sufficient to pin the CDW
pattern. Parameters are t1 /t2 = 5, V1 /V2 = 2, and V2 /U = 0.1. This corresponds to
t/w = t1 /U in our e↵ective model, which is valid in a regime 1/4 . t1 /U . 1. The phase
boundaries were determined by finite size extrapolation to the infinite lattice from systems
of size L = 34, 66, and 130. In the shaded regions incompressible MI phases are found at
the indicated densities ⇢. They are surrounded by superfluid (SF) regions, where we also find
weak indications for a super-solid (SS). We did not examine the stability of a possible SS in
detail. The picture was prepared by Michael Höning. (b) The ⇢ = 1/4 MI can be understood
from an integrable e↵ective model. (c) The Schmidt values n of the reduced density matrix
for a cut of the system at site j are shown. The calculations in (a) and (c) were carried out
by Richard Jen.
the bulk) is two-fold degenerate. However, this degeneracy is not a topological degeneracy,
because it can be lifted by infinitesimal local perturbations, which shift the energy of only one
of the two CDWs, see FIG.2.9 (a). Note that such perturbations do not even have to break
the inversion-symmetry which is required to protect topological order of the model.
A second indicator for the topological order of the ⇢ = 1/4 MI is obtained from the
entanglement spectrum. We checked using DMRG simulations that the Schmidt-values corresponding to a cut of the system at a strong bond (tunneling t1 > t2 ) are two-fold degenerate,
see FIG.2.7 (c). From a simple Gutzwiller ansatz, which is valid in the integrable limit of
our model, we expect this degeneracy to be robust to inversion-symmetric perturbations. It
merely reflects the fact that the MI phases in the super-lattice model are composed of dimers.
It is instructive to go through the remaining points of the list with indicators for topological
order that we presented in the introductory chapter, see Sec.1.2.1. We will do this now,
anticipating the results derived in the remainder of this section.
(i) Topological invariants. We can define quantized topological invariants which distinguish
di↵erent ⇢ = 1/4 CDWs. See Sec.2.5.2.
(ii) Edge states. As in the ⇢ = 1/2 bosonic SSH model, the strict bulk-boundary correspondence (involving zero-energy edge states) is violated. Nevertheless the edge states carry
a quantized fractional part of their charge which is a finger-print of the topological order
66
CHAPTER 2. TOPOLOGY IN THE SUPERLATTICE BOSE HUBBARD MODEL
in the system. See Sec.2.5.3.
(iii) Fractionalization. In the bulk of the ⇢ = 1/4 MI phase, fractionally charged topological
excitations exist. They correspond to domain-walls between two CDW configurations.
See Sec.2.5.3.
(iv) Anyonic Statistics. There are no excitations with anyonic statistics in the model. See
also Sec.2.4.4.
2.5.2
Symmetry-protected topological classification of Mott insulators
Now we give a complete topological classification of the incompressible ⇢ = 1/4 MI phases. To
this end we apply the classification scheme introduced by Chen et al.[6], which we summarized
in the introductory Chapter 1.2.1. This requires us to keep track of the intricate relation
between symmetries and topology. In this entire section, for simplicity, we will mostly restrict
ourselves to the bulk classification and consider infinite systems. In the following Sec.2.6 a
more general classification scheme is presented, making use of Thouless pumps.
As a first step in the classification, we identify the symmetries of the system. Without
edges, the Hamiltonian (2.31) has two sets of inversion symmetries, around even and odd
bonds. Labeling the inversion operators around these bonds Iˆ2n , Iˆ2n+1 respectively, it holds
[Ĥ, Iˆ2n ] = 0,
[Ĥ, Iˆ2n+1 ] = 0,
(2.37)
and they are defined by Iˆj ân Iˆj = â2j+1 n . In the following, without loss of generality, we
will only consider inversions around odd bonds and write Iˆ := Iˆ2n+1 . In particular this
allows perturbations Ĥ in the Hamiltonian which break the even-bond inversion symmetry
ˆ = 0. This situation also applies to finite systems with edges,
[ Ĥ, Iˆ2n ] 6= 0, whereas [ Ĥ, I]
where the boundaries only break odd-bond inversion while one global inversion symmetry Iˆ
remains (in this case only for one fixed value of n). Now we investigate the properties of the
ˆ Apparently, in the bulk the choice of the symmetry
system in the presence of the symmetry I.
ˆ
ˆ
(I2n or I2n+1 ) is arbitrary.
We start by classifying symmetry-breaking phases. In FIG.2.8 (a) the possibilities are
shown how the inversion symmetry Iˆ can spontaneously be broken. (Note that the states
sketched in FIG.2.8 are exact eigenstates in the integrable limit of the model.) To distinguish
the two resulting phases we can introduce a local order parameter P̂n , which breaks the
(a)
...
...
...
...
(b)
...
...
...
...
Figure 2.8: (a) We show two quarter-filling MI phases which are distinguished by how they
break the odd-bond inversion symmetry. The parity Pn defined in the text is hP̂n i = 1 in the
upper and hP̂n i = +1 in the lower case. (b) In this case the two quarter-filling MI phases can
not be distinguished by how they break odd-bond inversion symmetry. Instead, a topological
classification is required. The topological invariant is ⌫ = 0 in the upper and ⌫ = ⇡ in the
lower case.
2.5. EXTENDED SUPERLATTICE BOSE HUBBARD MODEL
symmetry breaking
P = ±1
⌫ 2 [0, 2⇡)
67
symmetric
P =0
⌫ = 0, ⇡
Table 2.1: Classification of the infinite odd-bond-inversion symmetric super-lattice Bose Hubbard model (2.34), with additional even-bond-inversion symmetry breaking perturbations.
odd-bond inversion symmetries [Iˆ2n+1 , P̂n ] 6= 0. This parity operator
P̂n := (n̂2n + n̂2n+1
n̂2n+2
n̂2n+3 ) /2
(2.38)
detects the anti-ferromagnetic order of the CDWs. See FIG.2.8 for the definitions of site labels.
From this figure we can also read o↵ the parities hP̂n i = ±1 of the two CDWs. This shows
that the symmetry breaking states are characterized by long-range order, hP̂n P̂m i ⇠ |n m|0 .
For example in the integrable limit, when t/U ! 1, it holds hP̂n P̂m i = 1.
Now we turn our attention to symmetric phases which respect the odd-bond inversion
symmetries. The two resulting ground states are shown in FIG.2.8 (b). Because {Iˆ2n+1 , P̂n } =
0 it follows that hP̂n i = 0 for any symmetric ground state | 0 i with Iˆ2n+1 | 0 i = ±| 0 i,
hP̂n i = hIˆ2n+1 P̂n Iˆ2n+1 i =
hP̂n i = 0.
(2.39)
Thus the parity P̂n can no longer distinguish the ground states. As we will argue now, they
can however be distinguished by a topological invariant ⌫, protected by the odd-bond inversion
symmetry. In particular this implies that they can not be transformed one into each other
without breaking the inversion symmetry or closing the particle-hole energy gap to excitations.
Before we proceed, let us emphasize that – clearly – the two states in FIG.2.8 (b) can
0 defined for even-bond instead of odd-bond
be distinguished by a local parity operator P̂m
inversions. We note that, firstly, this is not in contradiction to the possibility of a (closely
related) topological classification. Second, we argue now that such a topological classification
gives new insights to the problem. Finally we remark that our investigation is more general
in the case when the even-bond inversion symmetry is explicitly broken, see discussion above.
In such a case the topological classification still applies, while states can not be distinguished
by (spontaneous) even-bond symmetry breaking7 .
To proof rigorously that the two states in FIG.2.8 (b) belong to di↵erent inversion-SPT
phases, we construct a quantized topological invariant ⌫ to distinguish them. It can only
change its value when the bulk many-body gap closes, i.e. at a quantum phase transition.
The definition of this topological invariant is the same as in the case of the half filling superlattice Bose-Hubbard model discussed in the last Section 2.3, and it may be interpreted as
a local Berry phase [84, 153]. We modify the hopping t2 on the odd bond around which our
system is inversion symmetric, by multiplying to it a phase factor ei# . When this phase is
adiabatically changed from # = 0 to # = 2⇡, the many-body state picks up a many-body
Berry phase ⌫ defining the topological invariant,
⌫=
7
Z
2⇡
0
d# A(#).
(2.40)
Note that a parity operator constructed to witness even-bond inversion-symmetry breaking can still take
a non-vanishing value. It is determined solely by the terms in the Hamiltonian which explicitly break the
even-bond inversion symmetry, however.
68
CHAPTER 2. TOPOLOGY IN THE SUPERLATTICE BOSE HUBBARD MODEL
Here A(#) = h (#)|[email protected]# | (#)i denotes the Berry connection and | (#)i is the ground state
many-body wavefunction at a given value of #. Because of the inversion symmetry around
the odd-bond, ⌫ is quantized to take only values ⌫ = 0, ⇡.
To calculate the value of the invariant for the phases shown in FIG.2.8 (b), we consider the
integrable limit defined in Eq.(2.32). In this case the Hamiltonian (2.34) is exactly solvable
in terms of a Gutzwiller product state of bosons occupying every second energetically lower
dimer. Then a simple calculation shows that ⌫ = 0 if no boson occupies the central dimer
(upper case in FIG.2.8 (b)) and ⌫ = ⇡ if there is a boson (lower case in FIG.2.8 (b)). Because
⌫ is quantized and changes only at a quantum phase transition, the derived values of the
invariant classifying the two phases are correct even beyond the integrable limit of the model.
In Table 2.1 we summarize the resulting classification scheme based on the prescription
by Chen et al.[6]. There are two symmetry breaking phases, which we label by their parity
hP̂n i = ±1. Within the symmetric sector it holds hP̂n i = 0 and di↵erent topologically distinct
short range entangled phases can exist. We label them by the value of the topological invariant
⌫ = 0, ⇡, which is quantized due to the inversion symmetry. To show that they can not be
transformed into each other by a local unitary transformation, we note that if they could,
we would also find an inversion-symmetric local Hamiltonian that transforms them into each
other adiabatically. However, because in this process the many-body gap would not close, the
invariant ⌫ can not change. Hence we can never go from a ⌫ = 0 to a ⌫ = ⇡ state adiabatically.
2.5.3
Topological excitations
Now we turn our attention to the topological excitations of the extended super-lattice Bose
Hubbard model at filling ⇢ = 1/4. They are elementary excitations of the system, whose
characteristic properties are dictated by the topology of the corresponding bulk configuration.
As such, topological excitations provide information about the topological order in the bulk.
In the particular case of the super-lattice Bose Hubbard model they provide direct information
about the di↵erence of the topological invariants left and right of the topological excitation.
On the one hand this makes topological excitations an important tool to detect topological
order. On the other hand their properties, like e.g. their fractional charge, may be of interest
in their own right.
In the system under consideration there are two sorts of topological excitations, those
localized at the edges and those which can be delocalized over the bulk. As we have shown in
the previous section 2.4, edge excitations do not necessarily correspond to protected quantum
degrees of freedom8 , but can be characterized by the quantized fractional part of their charge.
This feature is deeply related to the topology of the SSH model, and we thus expect it to
carry over to the ⇢ = 1/4 phase. On the other hand, because there exist degenerate ground
state configurations in the bulk, there is the possibility of domain-wall excitations which also
carry a fractionally quantized charge. They connect two distinct ground state configurations,
and this e↵ect is thus related to spontaneous symmetry breaking.
Integrable limit
We start by constructing all possible excitations in the integrable model (2.34) where t ! 0,
i.e. we derive all low-energy eigenstates. Then we discuss their topological properties. To
this end we fill up the dimers described by the e↵ective Hamiltonian (2.34). This allows us to
determine the charges of the excitations, which are topological invariants determined by the
bulk. In fact, by the King-Smith - Vanderbilt relation (see Sec.1.2.2), the charges are directly
8
Edge-states are no longer zero-energy mid-gap states but may disappear in the incompressible region.
2.5. EXTENDED SUPERLATTICE BOSE HUBBARD MODEL
(a) ...
...
...
...
(d)
1/2 TP
(b)
...
...
...
...
(c)
...
...
...
...
69
...
...
...
...
1/2 TP
1/2 TP
1/2 TP
+1
...
...
Figure 2.9: In the integrable limit of the extended super-lattice Bose Hubbard model all
eigenstates can be constructed explicitly. The degenerate CDW ground states in the bulk are
shown in the topologically trivial (a) and non-trivial dimerization (b) of the SSH model. They
are connected by half a Thouless pump (TP) cycle. The topological domain-wall excitations
in the bulk of the CDW ground states are shown in (c). These quasiholes/-particles carry
half-quantized fractional charges qb = ±1/2. In (d) some possible edge excitations are shown.
They are characterized by the fractional part qe of the charge localized at the edge, taken
modulo 1 here. They can be constructed from the trivial configuration (top) by applying half
TP cycles and adding (or subtracting) particles.
related to the topological invariant ⌫ defined in Eq.(2.40). Because ⌫ is strictly quantized
by the inversion symmetry of the model, the charges are robust even beyond the integrable
limit of the model. This is true arbitrarily far away from the integrable regime, as long as no
quantum phase transition takes place where the bulk gap closes. Hence we can understand
all topological properties of the super-lattice Bose Hubbard model from the integrable limit.
To construct all low-energy eigenstates of the integrable model (2.34), we employ a generalized Pauli principle as introduced in the context of the fractional quantum Hall e↵ect
[184]. To this end we note that there is an energy cost w whenever two neighboring dimers
are occupied, and U/2 when a dimer is doubly occupied, see Eq.(2.34). To remain in the
low-energy sector with vanishing interaction energy, the generalized Pauli principle reads:
There is no more than one particle per two consecutive dimers (or orbits).
To construct quasiparticle states we assume a separation of energy scales, U/2
w. Then
every violation of the generalized Pauli principle with a single pair of two neighboring dimers
being occupied leads to an excited state of energy w.
Using the generalized Pauli principle, we can immediately construct the two bulk CDW
configurations shown in FIG.2.9 (a) and (b). To construct the excited states, two strategies
may be pursued. Firstly bosons can be distributed over dimers, ensuring that the generalized Pauli principle is obeyed. More elegantly, Laughlin’s idea of constructing topological
excitations by flux insertion may be employed [47]. To this end a full Thouless pumping
cycle can be introduced in the integrable model, but only to the right, say, of a given dimer.
Because thereby the Hamiltonian is mapped back onto itself, a new eigenstate is obtained
by adiabatically following the corresponding time evolution. This is in direct analogy to the
gauge invariance ingeniously employed by Laughlin to explain the strict quantization of the
quantum Hall e↵ect [173] (when introducing magnetic flux through a thin solenoid).
A comment is in order about the action of the Thouless pump in the integrable model.
Because of the absence of tunneling, t ! 0, neither bulk nor edge excitations become dispersive. As soon as finite tunneling is considered, the actual dynamics of the Thouless pump
becomes more involved, and the question of adiabaticity is a more subtle one. In the case of
edge states, finite tunneling may lead to melting of excitations into the bulk, as discussed in
70
CHAPTER 2. TOPOLOGY IN THE SUPERLATTICE BOSE HUBBARD MODEL
the previous section 2.4.1 at half-filling. This, however, only modifies the energies of excited
states while their topological properties – which we are ultimately interested in – (such as
their fractional charge introduced below) are unmodified. We furthermore note that the same
subtleties occur if Laughlin’s argument is employed for quantum Hall systems on a lattice,
where the lowest Landau level acquires a finite, albeit small, band width.
Before we proceed by actually constructing the topological excitations of the model, we
emphasize that there is a connection to the topological excitations in the quantum Hall e↵ect.
This was also noticed in Ref.[185]. In the integer quantum Hall e↵ect, too, topological excitations can be constructed by filling up orbitals. In that case orbits corresponds to the states
(labeled e.g. by their angular momentum) of the lowest Landau level. In the fractional quantum Hall e↵ect the many-body wavefunctions can be constructed from the classical orbital
occupations by using Jack polynomials [186, 184, 187, 188].
Bulk excitations.– In FIG.2.9 (c) the two bulk excitations of the ⇢ = 1/4 MI are shown,
correspond to domain-wall excitations of a half-filling CDW in the lowest SSH band. They
can be constructed starting from one of the incompressible ground states e.g. by applying
one Thouless pump cycle, but only to the right of the location of the excitation. When
the Thouless pump is directed to the right (left) a quasihole (quasiparticle) excitation is
created. Hence the charge of these quasiparticle states | qp i, which is defined relative to the
incompressible ground state | 0 i,
X
qqp =
h qp |n̂j | qp i h 0 |n̂j | 0 i = ±1/2,
(2.41)
j
is fractional. It is a genuine fractional charge because its fluctuations vanish. Due to the
particle-number super-selection rule, fractionalized excitations can not be created locally.
The fractional part of the quasiparticle charge
qqp
mod 1 = 1/2
(2.42)
is topologically protected (in an SPT sense). This can be seen by comparison of the bulk
polarizations left and right of the domain wall, which di↵er by d ⌫/2⇡ (d is the size of the
unit-cell). The di↵erence of the bulk topological invariants is quantized, ⌫ = ⇡, see Sec.2.5.2,
by inversion symmetry. This demonstrates why it is useful to invoke topology to distinguish
CDW phases, rather than conventional order parameters like the parity in Eq.(2.38).
Bulk excitations can be investigated in a system with periodic boundary conditions. If
the number of dimers is even there are two degenerate ground states, corresponding to the
CDWs shown in FIG.2.9 (a). If, on the other hand, the number of dimers is odd, there is a
half-quantized excess charge in the system. In the simplest case either one quasihole (or one
quasiparticle) excitation exists. By adding or subtracting particles, the fractional part of the
charge can not be modified, and it constitutes a topological invariant of the states.
When a finite dispersion t 6= 0 is considered in the e↵ective model (2.34), bulk topological
excitations have kinetic energy. Therefore the eigenstates are eigenstates of the momentum
operator, which are delocalized over the entire bulk.
Edge excitations.– Next we construct the topological edge excitations, as shown in FIG.
2.9 (d). In this case we apply Thouless pump half -cycles to the entire system. As described
in the previous section 2.4.2, even half-cycles change the polarization of the state only by a
strictly half-integer quantized amount due to the inversion symmetry.
2.5. EXTENDED SUPERLATTICE BOSE HUBBARD MODEL
71
In FIG.2.9 we start from the trivial CDW configuration and introduce half a unit of the
Thouless pump flux. Because the polarization of the bulk changes by 1/4 (using a four-site
unit cell), the fractional part of the charge which is localized at the edge is quantized to
qe = 1/4. After introducing a second half Thouless pump flux quantum, a qe = 1/2 edge
excitation is created. In the next paragraph we will investigate the stability of such a state
beyond the integrable model, i.e. when t 6= 0, where it has the possibility to decay into a
half-charged bulk excitation.
In FIG.2.9 (d) all possible edge excitations are shown. By adding or subtracting additional
particles also other states can be constructed. Note however that, already by the polarizations
of the four possible bulk configurations shown in FIG.2.9 (a) and (b), the fractional part of
the charges localized at any edge is strictly quantized to integer multiples of 1/4,
qe
mod 1 = 1/4.
(2.43)
Numerical simulations
Now we investigate the topological excitations in the super-lattice Bose Hubbard model numerically. Unlike in the integrable limit of the model, we are now interested also in the
energies of di↵erent eigenstates, which determine their stability in the actual physical system.
As pointed out in the proceeding paragraph, the topological quantum numbers – i.e. the
charges qe,qp of the excitations – are determined by the invariant ⌫, see Eq.(2.40), which is
protected by the inversion symmetry. Hence they can not change when entering the nonintegrable regime of our model unless the bulk gap closes and a quantum phase transition
takes place.
To investigate topological excitations quantitatively, Richard Jen performed numerical
DMRG calculations of the grand-canonical Hamiltonian (2.31). We used open boundary
conditions, cutting the system at di↵erent inequivalent sites. We will now present phase
diagrams which demonstrate that various of the topological excitations predicted by the integrable model can actually be stabilized in a grand-canonical phase diagram. The observed
phase transitions are classified by two topological quantum numbers, the fractional charge qb
delocalized over the entire bulk, and the fractional charge qe localized at the edge.
In FIG.2.7 we showed the grand-canonical phase diagram for a system in the topologically
trivial dimerization and with an even number of lattice sites which is not dividable by four
(i.e. an odd number of dimers). We find a single grand-canonical ground state inside the
loophole region, as can be understood from the integrable limit of the model. Because of the
repulsion between the particles, the open boundaries eliminate the bulk-degeneracy of the
ground state. The fact that point-like edges are capable of pinning the CDW demonstrates
that the system spontaneously breaks the inversion symmetry in the bulk, relating two CDW
states.
In FIG.2.10 we consider a more involved situation. The system is in the topologically
non-trivial dimerization, and in the e↵ective model Eq.(2.31) we have to include edge terms
(i.e. additional orbitals with energy t1 ) for both sides. In between there is an even number
of dimers. In the grand-canonical phase diagram we find three distinct phases, which can be
understood on a qualitative level from the e↵ective model Eq.(2.34). The first phase – 1
in FIG.2.10 – corresponds to a localized particle at the edge, with charge qe = +1/4. (This
state is degenerate with a state where a particle is localized on the other edge and the bulk
polarization is changed. Without loss of generality we ignore the second edge now.)
As the tunneling t1 increases we find that the particle becomes increasingly weakly localized, until it enters the bulk and forms a delocalized quasiparticle excitation. This state –
72
CHAPTER 2. TOPOLOGY IN THE SUPERLATTICE BOSE HUBBARD MODEL
(a)
(b)
1
3
(c)
2
1
2
3
Figure 2.10: (a) The grand-canonical phase diagram of the super-lattice Bose Hubbard model
in a non-trivial dimerization is shown. The configuration is sketched in (c), up to an addition
of an even number of dimers in the middle. From the dependence of the density ⇢ = N/L
on the chemical potential µ the phase boundaries were determined. This is illustrated in the
inset of (a), where a mid-gap kink can be recognized inside the incompressible Mott plateau.
The latter gives rise to the red (midgap) line separating the ⇢ = 1/4 Mott lobe in two. In the
upper part another transition from an occupied edge (1) to an un-occupied edge (2) can be
observed. The divergence of the corresponding localization length ⇠p is demonstrated in (b).
⇠p was determined from the participation ratio [172]. In (c) the incompressible phases are
explained using the integrable e↵ective model described in the text. The DMRG simulations
were performed by Richard Jen. Parameters are t1 = 5t2 , V1 = U/5 and V2 = U/10.
2 in FIG.2.10 – has qe = 1/4 localized at its edge and carries qb = 1/2 fractional charge
in its bulk. To leading order in t we expect the transition to occur at t1 /U = 1/12 for the
parameters used in the figure, which is in rather good agreement with our findings, given how
poorly the conditions for a perturbative treatment are satisfied. In FIG.2.10 (b) we also show
the localization length ⇠p of the particle, which diverges around t1 /U ⇡ 0.06.
The melting of the particle-edge state into the bulk is analogous to the melting observed
for the simpler filling ⇢ = 1/2 MI phase in the previous section 2.4.1. There are some crucial
di↵erences, however. In particular, we observe in FIG.2.10 that the critical chemical potential
µe , which splits the Mott lobe into two, does not approach the critical chemical potential of
the bulk, unlike in the ⇢ = 1/2 SSH model.
To understand this, let us first discuss the third phase below µe – 3 in FIG.2.10. There,
one particle is removed from the system and a quasihole forms in the bulk. Its charge is
qb = 1/2, whereas the edge still carries fractional charge qe = 1/4. In its ground state the
quasihole forms a zero-momentum standing wave in order to minimize its kinetic energy, and
it becomes completely delocalized as did the quasiparticle state9 . Now, in view of the phase
9
Note that something quite interesting happens at this point. From a canonical point of view – i.e. at a
fixed total particle number – it is clear that both the quasihole and quasiparticle states are gapless because an
2.6. THOULESS PUMP CLASSIFICATION OF INVERSION-SYMMETRIC MODELS IN
ONE DIMENSION
73
diagram in FIG.2.10 (a), the question arises how both the quasihole and quasiparticle states
can simultaneously support a particle-hole gap.
The particle hole gaps of quasihole and -particle states can easily be derived from our
e↵ective model (2.34) in the integrable limit t ⌧ w. Adding one hole to a system (i.e. adding
two quasiholes) costs zero energy. Adding one particle to a system with one quasihole leads to
the formation of one quasiparticle, with an energy w. Therefore the particle-hole gap of the
quasihole state is µ = w. Adding one particle to the quasiparticle state requires formation of
two additional quasiparticles, with energy 2w. Hence the particle-hole gap of the quasiparticle
state is also µ = w. This result is in excellent agreement with the observation in FIG.2.10
(a) that the critical chemical potential µe is indeed midway between the upper and lower
1/4
critical chemical potentials µ± of the bulk.
The statement that both the quasihole and the quasiparticle states have a finite particlehole gap can be reformulated as follows: The function N (µ) shows a step of N (µ) = 1, for
a chemical potential inside the incompressible region. This can be observed e.g. in the inset in FIG.2.10 (a). We will now generalize this observation and proof the following statement:
Consider a translationally invariant system without edges. Assume that the particle number
N (µ) changes by a finite number (O(L0 ) in the system size L) for a chemical potential µ from
an incompressible region µ < µ < µ+ . Then the system has fractionally charged excitations.
To simplify the proof we introduce some concrete notations. µ± are upper and lower
critical chemical potentials between which there always exists a finite particle-hole gap in the
thermodynamic limit. Assume N is the total number of particles in the grand-canonical
ground state | i for µ < µ < µ0 . At µ0 the particle number changes by an integer number
n, to N+ = N + n. This is the particle number of the grand-canonical ground state | + i
for µ0 < µ < µ+ . In the most general case the state | + i has an excitation which carries
a charge q+ , and | i has an excitation with charge q = q+ n. However, if q+ was an
integer, we could add a second excitation of the same kind – and with the same energy in the
thermodynamic limit. (It is here that we require translational invariance of the system.) This
is in contradiction to the assumption that | + i is incompressible. Similarly it follows that q
can not be an integer. Hence both states | ± i carry fractionally charged excitations.
2.6
Thouless pump classification of inversion-symmetric models in one dimension
So far the topological classifications that we carried out were focused on the specific superlattice Bose Hubbard model. Now we want to consider a larger class of models, making
ˆ To adiabatically connect
assumptions only about their behavior under spatial inversion I.
di↵erent inversion-symmetric states, enabling us to compare their properties, we have to relax
the inversion-symmetry constraints. This will allow us to derive all possible Chen-type [6]
classifications in one-dimensional (1D) inversion-symmetric models.
For concreteness we restrict our analysis to systems with n = 2 low-energy states (below
the bulk-gap of the system) on both sides of one critical point or region, see FIG.2.11 (b). Our
goal is to derive all possible classifications of the resulting four di↵erent phases (two on each
arbitrarily small amount of kinetic energy can be added. In the grand-canonical setting, meanwhile, we clearly
observe that both the quasihole and quasiparticle states 2 and 3 are at the same time incompressible (they
have a particle-hole gap), i.e. it costs a finite amount of energy to add another particle or hole to the system.
In this context, this peculiar behavior can only occur because the excitations are fractionally charged.
74
CHAPTER 2. TOPOLOGY IN THE SUPERLATTICE BOSE HUBBARD MODEL
(a)
inv.-sym.
model
(b)
gapless
region
inv.-sym.
model
gapless
region
Figure 2.11: We consider generic inversion symmetric one-dimensional models which we characterize by a parameter SY , plotted on the x-axis of our parameter space. To derive the
most general classification of their di↵erent ground states, additional symmetry breaking
terms parametrized by a second variable SB can be added. They are chosen such that the
gapless critical region (shaded) in the center of parameter space is bounded. Specifically we
discuss cases with n = 1 (a) and n = 2 degenerate (b) gapped ground states on both sides of
the critical region.
side of the critical point). Specifically, we consider a model Hamiltonian Ĥ( SY ) described by
a parameter SY , which is inversion symmetric for all values of SY . Further we assume that
around SY = 0 there exists a critical point or region in parameter space where the ground
state is gapless. In the gapped region at sufficiently large | SY | there are two low-lying states
| A,B i with an energy di↵erence well below the gap . In the case of the (extended) superlattice Bose Hubbard model, SY = t1 t2 and the two low-lying states correspond to the
two CDW patters shown in FIG.2.8 (b), depending on the values of t1 , t2 .
We will now show that one out of three possible classification schemes applies to distinguish the four phases to the left and right of the gapless region in FIG.2.11 (b). The three
possibilities are given by combinations of topological quantum numbers and symmetry-related
order parameters, as summarized in Table 2.2. In the first scheme – top ⇥ top – we introduce
two topological quantum numbers ⌫1,2 = 0, ⇡, the possible combinations of which label all
four states. In the second scheme – top ⇥ SB – there is one topological quantum number (⌫)
and one symmetry-related order parameter (P ). The extended super-lattice Bose Hubbard
model discussed in Sec.2.5 is of this type. In the last scheme – SB ⇥ SB – there are two order
parameters P1,2 related to symmetry-breaking (SB).
To carry out the general classification, we make use of the concept of a quantized Thouless
pump. To this end we add additional (local) terms in the Hamiltonian which break the
inversion symmetry and which are parametrized by SB . We choose them such that the
critical region is bounded in the extended phase diagram, see FIG.2.11. Note that symmetry
breaking terms with this property can always be found in 1D systems, because there exist no
long-range entangled states in 1D [6]. The critical point or region in the so-obtained higherdimensional parameter space can be classified by a Chern number C [174], which is related
classification
top ⇥ top
top ⇥ SB
SB ⇥ SB
quantum numbers
(⌫1 , ⌫2 )
(⌫, P )
(P1 , P2 )
Table 2.2: All possible classifications of inversion symmetric one-dimensional models with
two-fold degenerate ground states on both sides of one critical point (or region) in parameter
space. ⌫1,2 = 0, ⇡ denote topological quantum numbers and P1,2 symmetry-related order
parameters.
2.6. THOULESS PUMP CLASSIFICATION OF INVERSION-SYMMETRIC MODELS IN
ONE DIMENSION
75
to the quantized current pumped across the 1D system when the critical region is encircled
adiabatically [87]. Importantly C depends only on the topology of the chosen path, and it is
thus independent of the concrete symmetry-breaking terms SB that we choose. We will now
demonstrate how the Chern number characterizing the critical region can be used to classify
the inversion-symmetric 1D systems at SB = 0.
Purely topological case n = 1.– To understand how inversion symmetry can be used to
derive an SPT invariant ⌫ from the Chern number C of the critical region, we first go back one
step and illustrate how this works when there is only n = 1 state on each side of the critical
region. Because this state is unique, it is inversion-symmetric; Hence a topological invariant
instead of a symmetry-related order parameter has to be used to distinguish di↵erent sides
of the critical region. Although this case was discussed previously in [81] [P2], we find it
instructive to present a simplified picture now. As depicted in FIG.2.11 (a), we start with
the state | A i on the left side of the critical region and apply half a Thouless pump. By
breaking inversion symmetry we find a path L through parameter space along which we can
0 i on the right side of the phase diagram.
adiabatically transform | A i into the ground state | A
In this process, the polarization [83, 80] of the 1D system changes by ⌫d/2⇡, where ⌫ = 'Zak
is a di↵erence of (many-body) Zak phases and d denotes the lattice constant.
Next, we construct a second path IL by applying the inversion operator Iˆ to each state
of the first path. Because there is only n = 1 state at every inversion-symmetric point of
ˆ A i = ±| A i and I|
ˆ 0 i = ±| 0 i. This is not true, however,
parameter space now, it holds I|
A
A
for any non-zero value of SB 6= 0 where inversion symmetry is broken explicitly. Hence IL
connects the same states as L but along an alternative route. It follows from its definition
that on IL the polarization of the 1D system changes by '0Zak =
'Zak (in units of d/2⇡).
This is where the Chern number comes into play: By applying first L to go from | A i to
0 i and then
0 i to |
| A
IL (the reverse of IL) to return form | A
A i, we encircle the critical
ˆ
region. (We assumed that SB !
SB under inversion I.) The total change of polarization
'Zak
'0Zak = 2 'Zak is thus related to the Chern number by
2 'Zak = 2⇡C.
(2.44)
To define the topological invariant ⌫ which distinguishes the two phases left and right of
the critical region, we start by choosing an inversion-symmetric gapped reference state | 0 i
which is (by definition) topologically trivial with ⌫ = 0. The choice of this state is somewhat
arbitrary, in practice it is often motivated by the properties of the edges in the system. For
0 i is
concreteness let us choose | A i as a reference state. Then the topological invariant of | A
0 := C⇡. Note that this definition of the
defined as the Chern number of the critical region, ⌫A
0 i is unique if we require to choose the simplest possible
topological invariant for the state | A
Thouless pump cycle with exactly one winding around the critical region.
Topological and symmetry-breaking case for n = 2.– Now we will generalize the
above construction to cases with n = 2 ground states on each side of the critical region. For
0
notational convenience we will label them by | A,B i and | A,B
i respectively (without referring
to their physical properties here). These states no longer have to be inversion symmetric, but
instead they can break this symmetry spontaneously. By explicitly including this possibility
in our construction we now derive the Chen-type classification scheme shown in Table 2.2.
In FIG.2.12 (a-c) we show all three possibilities how inversion symmetry can spontaneously
be broken by the four states. In addition, we sketched how half Thouless-pumping cycles interconnect these states. Note that these connections are uniquely defined (up to an interchange
76
CHAPTER 2. TOPOLOGY IN THE SUPERLATTICE BOSE HUBBARD MODEL
(a)
(b)
SY
SY
(c)
SY
SB
SB
SB
Figure 2.12: The phases of 1D inversion symmetric models can be classified by half-Thouless
pumping cycles in combination with spontaneous symmetry breaking. Symmetric phases (SY)
are marked green and symmetry-broken phases (SB) red. From half a Thouless pump cycle
connecting state | A i to its partner on the other side (solid blue), its inverted cycle is obtained
by application of the inversion operator Iˆ (solid red). The changes of polarization are 'A
Zak
and
'A
Zak , respectively. Similarly, two half-cycles are constructed by starting from state
0 i (dashed blue) and inverting it (dashed red), with polarization changes ± 'B .
| A
Zak
0 i and | 0 i), using only that I|
ˆ A i = ±| A i in the symmetric case and
of the labels for | A
B
ˆ
I| A i / | B i in the symmetry-broken case. Furthermore, each half-cycle of a Thouless pump
corresponds to a change in polarization 'Zak of the 1D system, as described in FIG.2.12.
Now we can derive the classification in Table 2.2 by including the constraint on the polarization changes 'A,B
Zak due to the Chern number of the critical region [174]. In the first
case, when the inversion symmetry is not broken spontaneously, we find two closed cycles of
states turning into each other adiabatically in the pumping protocol. Thus for each cycle the
argument from the n = 1 case discussed above can be repeated, yielding
2 'A
Zak = 2⇡CA ,
2 'B
Zak = 2⇡CB .
(2.45)
Here the Chern numbers CA,B of the two cycles define two topological invariants ⌫A,B =
⇡CA,B , and this case corresponds to the sector top ⇥ top.
In the second case only one closed cycle exists, for which the total change of polarization
is given by
'Zak = 2⇡Ctot .
(2.46)
The total Chern number Ctot is defined via the U (2) Berry curvature [91] in this case. From
the construction in FIG.2.12 we obtain the condition
2
'A
Zak +
'B
Zak = 2⇡Ctot .
(2.47)
Hence ⌫ := ⇡Ctot defines a topological invariant, which is related to the sum of polarizations
B
'AB
'A
Zak =
Zak + 'Zak through the last equation. It allows to distinguish the states | A i and
| B i from each other, whose polarization di↵ers by precisely the amount 'AB
Zak = ⌫. (This
can be seen by starting from | A i and applying two half Thouless-pump cycles to reach | B i.)
0
The states | A,B
i on the symmetry-breaking side of the critical region can be distinguished
by conventional order parameters describing how the the symmetry is broken. This case thus
corresponds to the second class top ⇥ symmetry-breaking (SB) in Table 2.2.
In the last case inversion symmetry is broken everywhere in the phase diagram, either
explicitly or spontaneously. Therefore we do not expect any SPT order to appear. Indeed,
from FIG.2.12 we conclude that there are no additional constraints on the polarization changes
'A,B
Zak which allow to distinguish any two states. In FIG.2.12 (c) we find two closed cycles
again, for which the Chern numbers are constraint to
'A
Zak +
'B
Zak = 2⇡C1 = 2⇡C2 ,
(2.48)
2.7. SUMMARY AND OUTLOOK
77
such that this case can only appear when Ctot = C1 + C2 = 2C1 is an even number. Moreover,
B
the sum 'A
Zak + 'Zak does not distinguish between any two states, since it only appears
0 i back to |
in a full cycle e.g. from | A i via | A
A i. Hence this last case is characterized by
two conventional symmetry-breaking order parameters P1,2 and belongs to the last class SB
⇥ SB in Table 2.2.
2.7
Summary and Outlook
In summary, we have discussed topological properties of the paradigmatic one dimensional
super-lattice Bose Hubbard model with alternating hopping rates t1 and t2 . In the limit
of infinite interaction U ! 1 and at half-filling ⇢ = 1/2 this model corresponds to the
SSH model of free fermions, which is known to possess topologically non-trivial insulating
phases for t1 6= t2 . First we studied the Bose Hubbard model, still at half-filling, but with
soft-core bosons U < 1. We introduced a many-body generalization of the Zak phase as a
symmetry-protected topological order parameter, which remains quantized as a consequence of
the inversion symmetry of the model. Next we considered the extended Bose Hubbard model
at quarter filling ⇢ = 1/4 and with nearest neighbor interactions, where an inversion symmetry
can spontaneously be broken. This complicates the topological classification scheme, which
however can still be carried out for the symmetric phases. We showed that the corresponding
symmetry-protected topological order parameter leads to a quantization of the charges of all
topological excitations.
A particular focus of this Chapter was on the topological edge states, which we simulated
at an open boundary. Unlike in the fermionic SSH model, where zero-energy edge states are
protected by the chiral symmetry, we showed that there is no strict bulk-boundary correspondence for soft-core bosons U < 1, in the sense of a quantum degree of freedom (i.e. a qubit)
localized at the edges. The chiral symmetry is broken by the interactions between bosons.
Instead, however, we showed that the quantized topological invariants of the bulk are directly
related to a quantization of the fractional part of the charge qe mod 1 = 0, 1/2 localized at
the edges. This is generalization of the bulk-boundary correspondence for inversion-symmetric
one-dimensional systems.
In the extended Bose Hubbard model the topological excitations have a richer structure. In addition to edge states with a quantized fractional part of their charge qe mod 1 =
0, 1/4, 1/2, 3/4, there are fractionally charged bulk excitations (domain walls), with quantized
charges qb = ±1/2. Our grand-canonical DMRG calculations showed that transitions between
states with di↵erent topological excitations can be induced in a system with open boundaries
by tuning the model parameters. In particular we identified a characteristic mid-gap kink in
the grand-canonical phase diagram and showed that it is a unique fingerprint for the existence
of fractional excitations in the system.
The system under consideration can be realized in ultra cold quantum gases. While sharp
open boundaries may be difficult to realize in such experiments, we showed that an interface
between a ⇢ = 1 and a ⇢ = 1/2 MI can be created, where two topologically distinct phases are
in contact. The required potential step can be realized by an admixture of a second atomic
species, which is heavy and thus immobile. We found that similar edge states emerge as in
the case of open boundary conditions. These edge states are characterized by a density dip
at the edge below 1/2 and a density peak at the edge with local particle number exceeding
1. These features allow a simple detection of the edge states and thus a verification of the
di↵erent topological nature of the MI phases in cold-atom experiments.
The discussion in this section was mostly limited to static defects. An interesting future
78
CHAPTER 2. TOPOLOGY IN THE SUPERLATTICE BOSE HUBBARD MODEL
direction would be the study of mobile defects, which are fractionally charged solitons as
discussed in the original paper by Su, Schrie↵er and Heeger [12]. First steps in this direction
have been taken, see e.g. Ref.[189]. From this point of view, the scope of this chapter was to
establish a topological classification scheme for paradigmatic Bose-Hubbard systems, which
can furthermore be realized in ultra cold quantum gases.
Chapter 3
Realization of Fractional Chern
Insulators in the Thin-torus Limit
3.1
Outline and Introduction
In this chapter we continue addressing the question how Hamiltonians can be engineered in
cold atom laboratories, which support topologically ordered ground states. As discussed in
the introduction, see Sec.1.2.4, the two dimensional (2D) Hofstadter-Bose-Hubbard model
provides such a Hamiltonian. Furthermore, it can be implemented in current experiments
[108, 109]. However, the preparation of its ground states requires extremely low temperatures,
which are not yet achievable [110, 124]. Here, instead, we consider the possibility of realizing
the Hofstadter-Bose-Hubbard model in a conceptually simpler quasi one-dimensional (1D)
ladder system. Its short range entangled ground states are closely related to Mott insulating
states in 1D which we discussed extensively in the preceeding chapter. This, we believe, will
simplify their preparation, making their experimental realization feasible in the near future.
An alternative perspective would be to start from 1D models with SPT order, as discussed
in chapter 2, and extend their size to approach the two-dimensional case. We achieve this by
considering a two-leg ladder. Because strictly speaking this system is still one-dimensional,
fundamentally no new physical e↵ects can be expected. Nevertheless, because with our model
we approach the prominent 2D Hofstadter-Bose-Hubbard Hamiltonian, this scenario is of
considerable experimental interest. A similar approach has recently been suggested [190] and
experimentally implemented [191, 192] with ultra cold atoms, where a few internal atomic
states were employed to realize a small additional synthetic dimension. This allowed to
observe, for the first time, skipping orbits of a quantum Hall system at a sharp boundary.
Here we propose a setup for the realization of topologically ordered states of strongly
interacting bosons in quasi 1D ladder systems, see FIG.3.1. We consider the thin-torus-limit
[193, 194] of the 2D Hofstadter-Bose-Hubbard model on a cylinder, which is obtained when the
circumference of the cylinder becomes small. We show that the Laughlin-type fractional Chern
insulators existing in the 2D square lattice [14, 102], which support intrinsic topological order,
turn into symmetry protected topological phases in the quasi 1D system under consideration.
Furthermore we demonstrate how the model can be implemented in current experiments with
ultra cold atoms [15].
Specifically we consider the Hofstadter-Bose-Hubbard model with ↵ = 1/4 units of magnetic flux per plaquette, placed on a cylinder with a circumference of two lattice sites. The
resulting ground state is a charge density wave (CDW) along the thin cylinder, at average
filling ⇢ = 1/8 per lattice site, related to the ⌫ = 1/2 Laughlin state [47] in the 2D limit. It can
79
CHAPTER 3. REALIZATION OF FRACTIONAL CHERN INSULATORS IN THE
THIN-TORUS LIMIT
80
be interpreted as well as a symmetry protected topological phase [195] (protected by inversion symmetry). In addition our model includes the possibility of twisted periodic boundary
conditions around the short perimeter of the torus, with a fully tunable twist angle ✓x , see
FIG.3.1 (b). Adiabatically changing this twist angle by 2⇡ realizes a many-body version of a
Thouless pump [87, 174], which is fractionally quantized, see FIG.3.1 (c).
Our results should be seen in light of the general examination of fractional Chern insulators
in the thin-torus limit. Although the intrinsic topological order of quantum states is lost
when going towards a 1D system, the anatomy [188] of the topological states can remain
unchanged. Indeed, by a detailed analysis of fractional Chern insulators in the thin-torus
limit, Bernevig and Regnault [195] demonstrated that many of the characteristic properties
of their wavefunctions survive in the 1D limit. Specifically the quasihole counting, which may
serve as a fingerprint for a particular topological order, and the generalized Pauli principle
discovered for fractional quantum Hall states [184] were shown to remain valid in 1D. Therefore
a detailed investigation, theoretically and experimentally, of the thin-torus limit of a model
can provide new insights about intrinsic topologically ordered states.
The chapter is based on the publication [P7], and it is organized as follows. In Sec.3.2 we
introduce the model and show that it is identical to the thin-torus-limit of the 2D HofstadterHubbard model. Possible experimental realizations are discussed. In Sec.3.3 we elucidate on
the topological properties in the non-interacting case, and show that they enable a realization
of a Thouless pump. In Sec. 3.4 we return to the discussion of interacting bosons, where
density matrix renormalization group (DMRG) results for the melting of the CDW at weak
interactions and the system in a harmonic trap are presented. The topological classification
of the ground state is carried out in Sec. 3.5, before we close our discussion with a summary
and an outlook in Sec. 3.6.
3.2
Model
We consider the following Bose-Hubbard type model of bosons hopping between the links of
a 1D ladder, see FIG.3.1 (a),
Ĥ =
J
L/2 ⇣
X
n=1
L ⇣
X
j=1
L X
⌘ UX
⇣
â†j+1,L âj,L + ( 1)j â†j+1,R âj,R + h.c. +
â†j,µ âj,µ â†j,µ âj,µ
2
t1 â†2n 1,L â2n 1,R
j=1 µ=L,R
+
t2 â†2n,L â2n,R
⌘
+ h.c. + V
L X ✓
X
j=1 µ=L,R
j
L+1
2
◆2
⌘
1
â†j,µ âj,µ . (3.1)
Here âj,µ annihilates a boson on the left (µ = L) or the right (µ = R) leg of the ladder, at
the horizontal link j. The first line describes hopping along the ladder (vertical links) with
amplitude J. On the right leg an additional phase ⇡ is picked up on every second bond,
for
= +1 from even j to odd j + 1 and for
= 1 from odd j to even j + 1. Along
the horizontal bonds the tunneling rates are alternating between t1 and t2 , both assumed to
be real-valued and positive. In the third line we added on-site Hubbard-type interactions of
strength U everywhere. The model is completed by an external harmonic trapping potential
in the fourth line.
Below we will show that the model (3.1) is equivalent to the thin-torus-limit of the 2D Hofstadter Hubbard model. Afterwards we discuss possible experimental realizations with ultra
cold atoms using currently available technology, based on the experimental setups described
in Refs.[196, 108, 109, 110].
3.2. MODEL
81
Figure 3.1: Using commensurate optical lattice potentials, interacting bosons in 1D ladder
systems (a) can be realized. When the hopping elements across the ladder alternate between
values t1 and t2 , see Eq.(3.3), and every second hopping on one leg along the ladder has a
phase ⇡, the thin torus limit of the 2D Hofstadter Hubbard model (at flux per plaquette
↵ = 1/4) can be realized (b). Which of the hoppings on the right leg have a non-trivial phase
⇡ depends on the value of = ±1. The boundary conditions are periodic across the ladder,
with a tunable twist angle ✓x , corresponding to magnetic flux ✓x /2⇡ threading the smaller
perimeter of the torus. The ground state is an incompressible CDW with average occupation
⇢ = 1/8 per lattice site, related to the 1/2 Laughlin state of the 2D model. The density
distribution hn̂j,L i = hn̂j,R i is shown in two unit-cells, chosen from a larger system with open
boundary conditions, for di↵erent values of ✓x in (c). This corresponds to a quarter-cycle of
the fractionally quantized Thouless pump.
3.2.1
Relation to the thin-torus-limit of the Hofstadter-Hubbard model
Now we show how the model (3.1) can be related to the thin-torus-limit of the 2D HofstadterHubbard model with flux per plaquette ↵ = 1/4 (in units of the flux quantum). The latter is
described by a Hamiltonian (see Sec.1.2.4)
ĤHH =
J
X⇣
hi,ji
â†i âj e
i
i,j
⌘ UX
+ h.c. +
n̂j (n̂j
2
1) ,
(3.2)
j
where n̂j = â†j âj and i,j are Peierls phases picked up when hopping from site j to a neighboring site i. When a particle is hopping around a plaquette these phases sum up to ⇡/2,
which can be achieved e.g. with the gauge choice i,j shown in FIG.3.2 (a). Next we consider this model on a torus of size Lx ⇥ Ly with twisted boundary conditions along x, i.e.
(xm + Lx ) = ei✓x (xm ) where xm is the coordinate of the m-th particle, m = 1, ..., N . Such
boundary conditions can be implemented by adding additional phases ✓x to the hoppings from
(Lx , jy ) to (1, jy ) for all jy = 1, ..., Ly .
We perform the thin-torus-limit by setting the length Lx = 2 equal to two lattice sites,
yielding an e↵ective ladder system as shown in FIG.3.2 (b). Because of the periodic boundary
conditions along x there are two possibilities how a boson can tunnel from the left to the
right leg of the ladder, originating from paths across the top and bottom of the torus. When
summing up these contributions, we obtain complex hoppings across the ladder ⌧1 = J(1 +
e i✓x ) from (2n 1, L) to (2n 1, R) and ⌧2 = iJ(1 e i✓x ) from (2n, L) to (2n, R) (site-labels
as in Eq.(3.1)), see FIG.3.2 (c). The hopping elements along the legs of the ladder are real
and given by J at links (2n 1, R) to (2n, R) and J at all other links.
To show the equivalence of the thin-torus Hofstadter Hubbard model to the Hamiltonian
82
CHAPTER 3. REALIZATION OF FRACTIONAL CHERN INSULATORS IN THE
THIN-TORUS LIMIT
iθ
(b) Je J
x
(a)
J
J
J
J
y
J
-J
J
J
J
J iπ/2 J iπ/2 J iπ/2
e J e J e J
J
J
iJ
-J
J
Φ1
τ2
J
Φ2
τ1
J
J
Φ1
τ2
J
J iJe
-J
J Jeiθ
x
iθ
J iJe
x
J
iJ
x
τ1
J
iθx
J iπ/2 J iπ/2 J iπ/2
e J e J e J
(c)
Figure 3.2: (a) For the 2D Hofstadter Hubbard model at flux per plaquette ↵ = 1/4 we
make a gauge choice leading to a two-by-two magnetic unit-cell. Supplemented by twisted
boundary conditions in x-direction (twist-angle ✓x ), an e↵ective 1D ladder model is obtained
when the thin-torus-limit is considered (b). For notational simplicity, the imaginary unit
i = ei⇡/2 is used to express the complex hopping elements in (b). When an additional gauge
transformation is applied, leaving invariant the magnetic flux 1,2 in every plaquette, the
ladder model described by Eq.(3.1) is obtained (c).
(3.1) we now define t1,2 := |⌧1,2 |, yielding
p
t1 = J 2 (1 + cos ✓x ),
t2 = J
p
2 (1
cos ✓x ),
(3.3)
such that the absolute values of all hopping amplitudes in both models coincide. Thus, we
only have to calculate the magnetic fluxes through each of the plaquettes of the ladder and
show that they coincide in both models. In the thin-torus-limit of the 2D model we obtain
fluxes 1 = (1 + sign (sin ✓x )) /4 and 2 = (1 sign (sin ✓x )) /4 in units of the magnetic flux
quantum, see FIG.3.2 (b), (c). Choosing
=
sign (sin ✓x )
(3.4)
in Eq.(3.1) we obtain the same fluxes in the 1D ladder model. Therefore the thin-torus-limit
of the ↵ = 1/4 Hofstadter Hubbard model is equivalent to the model (3.1), up to a unitary
gauge transformation Û (✓x ) which depends explicitly on the twist-angle ✓x .
Quasi 1D ladder systems were investigated before, and it was shown that e↵ective gauge
fields give rise to interesting physical e↵ects such as Meissner currents [197, 15]. A di↵erent
version of the thin-torus-limit of the Hofstadter model at arbitrary flux per plaquette ↵ was
also studied already in [198]. In contrast to our model, this work did not take into account
periodic boundary conditions across the ladder, resulting in a homogeneous flux 1 = 2 = ↵
per plaquette and equal hopping amplitudes corresponding to |t1 | = |t2 | = |J| using our
notations. As a consequence the authors could study interesting edge states, carrying chiral
currents along opposite edges of the ladder. Within our model on the other hand, we can study
the e↵ect of tunable twisted boundary conditions across the ladder. This allows, in particular,
to study a Thouless pump driving a quantized current along the ladder, by dynamically
inserting flux through the small perimeter of the torus. In addition, we study interactions
between bosons.
3.2.2
Possible experimental implementation
Next we discuss a possible experimental realization of our scheme. The first required ingredient
is a superlattice for creating ladders (with a four-site unit-cell), which has been implemented
3.2. MODEL
83
Figure 3.3: (a) Possible realization of the ladder system with half a magnetic flux-quantum
piercing every second plaquette, cf. [110]. By interference of standing waves from a red (r)
and a blue (b) detuned sideband of the long-lattice laser in x direction, with corresponding
slightly detuned running waves along y direction, two independent lattice modulations (upper
blue and lower red plot in (b)) are created. Each acts on a single leg of the ladder and
they move in opposite directions along y, as shown by the amplitude of the modulations for
di↵erent times in (b).
experimentally, see e.g. [199, 15]. The second ingredient is a (staggered) artificial gauge field,
which has been experimentally demonstrated as well [196, 108, 109]. The implementation of
the Hamiltonian (3.1) is motivated and very closely related to the recent experiment [110],
but we think that alternative realizations should also be possible.
To begin with, our scheme requires a cubic lattice created by standing waves with short
wavelength S both in x and y direction. We choose the origin such that lattice sites are
centered at xj,L = 0, xj,R = S /2, yj,µ = (j 1) S /2. Additional standing waves with
long wavelength L = 2 S are required in both directions. The strong long-lattice along x
separates the individual ladders, whereas the weaker long-lattice along y induces a staggered
potential of strength along the legs, as indicated by white and grey filled sites in Fig. 3.3(a).
This staggered potential is required for realizing the artificial magnetic field. We will denote
the bare hopping elements in the so-obtained ladder by Jy (along the ladder) and Jx (across
the ladder).
The alternating flux 1,2 = 0, 1/2 (in units of the magnetic flux quantum) can be realized
with a similar configuration as in the experiment [110], where a homogeneous flux of ↵ = 1/4
was realized via laser assisted tunneling [113, 118]. Bare hopping along the legs is strongly
suppressed by the staggered potential
Jy , and has to be restored by resonant modulation
of the potential landscape. This can be achieved by a time-dependent potential of the form
V (x, y, t) = V0 cos( t + gx,y ), and as pointed out by Kolovsky [118] the freedom in choosing
the phase-shifts gx,y allows to implement Peierls phases – and thus to create artificial gauge
fields. To implement a suitable time-dependent potential experimentally, two side-bands of
the long-wavelength laser can be employed. They make up four additional beams, two reddetuned ones with frequencies !r1 and !r2 , and two blue-detuned ones at frequencies !b1
and !b2 . When the red sidebands are sufficiently far detuned from the blue sidebands, i.e.
!rj !bj
for both j = 1, 2, interference terms between them can be neglected and they
can be treated separately form each other.
We now move on by constructing suitable interference patterns between the red-detuned
and blue-detuned pairs of beams respectively, with relative frequencies !i = !i2 !i1 where
i = r, b. These beat-notes give rise to the required modulation of the potential at frequency
84
CHAPTER 3. REALIZATION OF FRACTIONAL CHERN INSULATORS IN THE
THIN-TORUS LIMIT
, and we chose them to be !r =
, !b = . As in [110] both beams r1 and b1 are
retroreflected in x direction to form standing waves, see FIG. 3.3 (a), and they interfere with
running waves r2 and b2 in y direction. This configuration gives rise to the time-dependent
interference patterns shown in FIG.3.3 (b)
⇥
Vr (x, y, t) = V0 /4 1 + 4 cos2 (kL x)
+ 4 cos(kL x) cos(kL y + t
⇥
Vb (x, y, t) = V0 /4 1 + 4 sin2 (kL x)
+ 4 sin(kL x) cos(kL y
t
⇤
⇡/4) ,
⇤
⇡/4) .
(3.5)
(3.6)
From FIG.3.3 (b) we recognize that the phase of the retroreflected red sideband is chosen such
that the resulting modulation is restricted to the left leg of the ladder and moves in negative
y direction in time. The blue sideband, vice-versa, leads to a modulation restricted to the
right leg of the ladder which is moving in positive y-direction in time. This counter-directed
movement of the potential modulation introduces angular momentum into the system, which
mimics the e↵ect of a magnetic field. Note that the additional standing waves in Eq.(3.6)
sum up to a constant overall energy shift only. The described setup is completely analogous
to the one implemented in [110], except for the phases chosen for the di↵erent laser beams.
Now, as desired, every lattice site is subject to a time-dependent modulation of the local
potential Vj,µ = V0 cos( t + gj,µ ) (with µ = L, R). From Eq.(3.6) we read o↵ the phase shifts,
which are given by gj,L = 3⇡/4+j⇡/2 and gj,R = 3⇡/4 j⇡/2. To proceed and calculate the
resulting Peierls phases, let us consider the simplified case when two lattice sites 1 and 2 are
coupled by a hopping element Jy , where the second site is detuned by an energy
Jy from
the first one. Resonant periodic modulations of the local potentials V1 , V2 with frequency
and phases g1 and g2 restore strong hopping. p
Indeed, the e↵ective tunneling matrix element
ig
ig
1
2
from 1 to 2 is given by Je↵ = J(e
e
)/ 2 [110, 118], where we defined the amplitude
J as
p
J = Jy V0 /( 2 ).
(3.7)
Returning to the ladder model, in this way we find for the induced hoppings
J(j,L),(j+1,L) =
eij⇡ Jy V0 /(2 ) eigj,L
eigj+1,L
= Jeij⇡/2 ,
J(j,R),(j+1,R) =
e
= Je
ij⇡
(3.8)
Jy V0 /(2 ) e
ij⇡/2
igj,L
e
igj+1,L
.
(3.9)
The above configuration can be mapped to Eq.(3.1) via a gauge transformation.
Finally we turn to the implementation of the hoppings t1,2 connecting the legs of the
ladder. Without further modifications of the described setup, p
they are given by t1 = t2 =
Jx . Choosing the modulation strength V0 such that J = Jx / 2 readily realizes the cases
✓x = ±⇡/2. They are of special relevance, because the resulting model is inversion symmetric
around the center of links on the legs. As will be shown below, the model supports inversionsymmetry protected topological phases at these points. In order to realize arbitrary values
of ✓x , the hoppings t1,2 can be manipulated with a second independent square lattice rotated
by 45 , such that the potential barrier along every second horizontal bond of the ladder is
increased when the lattice is properly adjusted. To this end an additional sideband of the
short wavelength laser is added in x and y direction and both beams are retroreflected. The
resulting interference pattern realizes the required rotated square lattice with lattice constant
3.3. TOPOLOGY IN THE NON-INTERACTING SYSTEM – THOULESS PUMP
85
p
S/
2. Lastly we note that the periodic modulation used to restore hoppings along the ladder
reduces the tunneling
t1,2 between the legs. This gives rise to e↵ective hoppings
p amplitudes
2 /(4 2 ), but the e↵ect can be neglected for large
te↵
/t
=
1
(2
2)V
.
1,2 1,2
0
3.3
Topology in the non-interacting system – Thouless pump
We start the analysis of the model (3.1) by investigating non-interacting bosons. In this case
all properties of the bandstructure immediately follow from the 2D Hofstadter model [96]. The
lowest band of the 2D Hofstadter Hamiltonian at ↵ = 1/4 is characterized by a Chern number
C = 1, which gives rise to a quantized Hall current perpendicular to an applied external force.
We show below that such quantized particle transport survives along the 1D ladder in the
thin-torus-limit, when the external force is induced by inserting magnetic flux through the
smaller perimeter of the torus. Experimentally this corresponds to an adiabatic change of the
twisted boundary conditions, @t ✓x 6= 0. A change of ✓x by 2⇡ can also be interpreted as one
cycle of a Thouless pump [87].
Now we discuss the relation between Bloch wavefunctions of the one- and two-dimensional
models. At ↵ = 1/4 the 2D Hofstadter model has a four-site unit-cell. Using the gauge
choice introduced earlier in FIG.3.2 (a), we can calculate the Bloch Hamiltonian Ĥ(kx , ky ) of
the 2D model, resulting in Bloch wavefunctions |u(kx , ky )i (see e.g. Supplementary Material
in [110] for a concrete calculation). When performing the thin-torus-limit as described in
3.2.1, the Bloch wavefunctions do not change, except that the quasimomentum kx across the
resulting ladder is replaced by the angle ✓x defining the twisted periodic boundary conditions.
The Bloch wavefunction in the thin-torus-limit thus reads |u(✓x , ky )i. By applying the gauge
transformation Û (✓x ) (see 3.2.1) we also obtain the Bloch function of the 1D ladder model
(3.1), |v(✓x ; ky )i = Û (✓x )|u(✓x , ky )i, at a given twist angle ✓x . Consequently the Bloch bands
✏n (kx , ky ) labeled by n = 1, ..., 4 (i.e. the eigenenergies of the Bloch Hamiltonian) of the 2D
Hofstadter Hubbard model coincide with those of the 1D ladder model (3.1), ✏n (✓x ; ky ).
From the Bloch wavefunctions we will now derive the topological properties of the 1D
ladder model (3.1). To this end we calculate the Zak phase [81] 'Zak (✓x ) for a path through
the Brillouin zone (BZ) along ky and for a given value of ✓x . Because the Zak phase is invariant
under the gauge-transformation Û (✓x ), the 1D ladder model reproduces the result 'Zak (✓x )|H
of the 2D Hofstadter model, 'Zak (✓x ) = 'Zak (✓x )|H . Zak phases can be measured in ultra
cold atom systems using Ramsey interferometry in combination with Bloch oscillations [13]. A
similar measurement in the simplified 1D model would thus allow to study the Berry curvature
of the 2D Hofstadter model, see also [200, 201].
A characteristic feature of the Hofstadter model at ↵ = 1/4 is that its lowest band is
topologically non-trivial, with a Chern number CH = 1. The Chern number can be directly
related to the winding of the Zak phase, see Sec.1.2.2 or [82],
Z
1
CH =
dkx @kx 'Zak (kx )|H .
(3.10)
2⇡ BZ
For the 1D ladder model the winding of the Zak phase equivalently defines the Chern number,
1
C=
2⇡
Z
2⇡
0
d✓x @✓x 'Zak (✓x ).
(3.11)
Now we discuss the physical consequences of the non-trivial Chern numbers C = CH = 1. In
the case of the 2D Hofstadter model, it is related to the Hall current induced by a constant
CHAPTER 3. REALIZATION OF FRACTIONAL CHERN INSULATORS IN THE
THIN-TORUS LIMIT
86
external force [27, 25]. This current was recently measured with essentially non-interacting
atoms, in ultra cold Fermi gases [124] and also with ultra cold bosons homogeneously populating the lowest Bloch band [110]. In the thin-torus limit of the Hofstadter model, a constant
force around the short perimeter of the torus can be applied by adiabatically changing the
twist angle in the boundary conditions, F / @t ✓x . Like in the 2D model, this leads to a Hall
current perpendicular to the induced force – i.e. along the ladder – which is quantized and
proportional to the Chern number C.
An alternative interpretation of the quantized current in the 1D model (3.1) is given by the
concept of a Thouless pump [87]. To understand this, we recall that the Zak phase is related
to the macroscopic polarization through the King-Smith - Vanderbilt relation P = a'Zak /2⇡,
where a is the extent of the magnetic unit-cell in y-direction [83]. Now, by definition (3.11),
it follows that the Zak-phase changes continuously from 0 to C ⇥ 2⇡ when the parameter ✓x
is adiabatically changed by 2⇡ in a time T . This corresponds to a quantized change of the
polarization by P = Ca, or a quantized current Ca/T . Below (in 3.5) we give an intuitive
explanation of the microscopic mechanism of this e↵ect in the 1D ladder system.
Experimentally the Thouless pump can be detected by loading non-interacting ultra cold
atoms (bosons or fermions) into the lowest Bloch band. Then, comparing in-situ images of the
atomic cloud before and after adiabatically changing ✓x by 2⇡ reveals the quantized current.
This is similar to the measurements performed recently on the 2D Hofstadter model [124, 110].
Now we turn to the discussion of interacting atoms, where we will give an example for a fractionally quantized Thouless pump corresponding to a Chern number C = 1/2. There we will
make use of Wannier orbitals with a finite support, which allow us to study fractionalization
even in the presence of, fundamentally, purely local interactions.
3.4
Interacting topological states
As a next step we include local Hubbard-type interactions between the bosons in the investigation of our model. In the 2D limit of the Hofstadter-Hubbard model the existence of an
incompressible Laughlin-type ground state has been established numerically for fluxes within
a range ↵ = 0...0.4 [14, 102]. For ↵ = 1/4 studied in this paper we thus expect a fractional
Chern insulator at a magnetic filling ⌫ = N/N = 1/2, where N is the number of particles
and N the number of flux quanta in the system. The average occupation number of each
lattice site is thus ⇢ = 1/8 in this phase. Now we will show using DMRG calculations1 that
in the thin-torus limit an incompressible CDW survives at the same filling. We study the
robustness of this phase when the interaction strength U is lowered. In a harmonic trap the
incompressible phase is shown to be robust enough to form plateaus of constant density.
3.4.1
Grand-canonical phase diagram
We use a Matrix Product State (MPS) based algorithm to find the ground state of finite
size ladder systems with open boundary conditions (obc) along the ladder [202]. MPS are
very well suited to approximate the CDW like states we expect in the incompressible phase
and can – with increased resources – also describe the melting of the CDW at fillings near
⇢ = 1/8. By varying the chemical potential we have determined the ground state energy for
particle numbers around N ⇡ Ly /4 (corresponding to ⇢ = 1/8) and three di↵erent interaction
strengths U/J = 2, 5, 1. Due to symmetries it is sufficient to consider twist angles from the
parameter space ✓x 2 [0, ⇡/2].
1
The DMRG calculations were performed by Michael Höning.
3.4. INTERACTING TOPOLOGICAL STATES
87
Figure 3.4: (a) The particle-hole gap CDW of the incompressible phase at filling ⇢ = 1/8 for
varying interaction strength U/J = 1, 5, 2 extrapolated from finite system size calculations
at Ly = 18, 24, 32. This gap corresponds to the plateaus in the ⇢(µ) diagrams shown in (b)
for U/J = 1 at system size Ly = 48. Note that for ✓x = ⇡/2 the plateau has a kink in its
middle where ⇢(µ) changes, corresponding to the addition of a single particle. This is not a
bulk e↵ect, however, because the additional particle is localized at the edge of the system.
We now define the critical chemical potentials µ1/8± as the upper and lower boundaries of
the incompressible CDW phase. In Fig. 3.4(a) we show the corresponding particle hole gap
µ1/8 extrapolated to thermodynamic limit from finite system results at
CDW = µ1/8+
Ly = 16, 24, 32. Strong interactions stabilize the non-trivial CDW that is protected by a gap
on the order of CDW ⇡ J/6 at U = 1. At moderate interaction U = 5J the incompressible
phase is still protected by CDW ⇡ J/12, whereas for U = 2J the gap almost closes and the
CDW phase becomes unstable.
The topological nature of our system and the presence of edges (recall that we use obcs)
have to be taken into account when analyzing the dependence of the particle number N (µ)
on the chemical potential µ, as shown in Fig. 3.4(b). At ✓x = 0 we observe a single plateau
at filling ⇢ = 1/8, however at ✓x = ⇡/2 this plateau is split in two by the addition of a single
particle at intermediate chemical potential. This is an edge e↵ect and strongly dependent on
the choice of boundary conditions, allowing us to interpret the full plateau as an incompressible
bulk phase. For a detailed discussion of this e↵ect, see Sec.2.5.3.
3.4.2
Harmonic trapping potential
The incompressible Mott-insulating phases at integer filling of the conventional Bose-Hubbard
model can be observed in harmonically trapped systems. The emerging Mott-insulating
plateaus of constant density are surrounded by superfluid regions, leading to the formation of
the so-called ”wedding-cake” structure. Here we show that in a similar fashion the non trivial
CDW phase on the ladder can be visualized in harmonically trapped gases.
Using the MPS code we have calculated the density distribution in traps of sizes up to
Ly = 128 for fixed global chemical potential µ. The trap depth V is chosen such that,
from local density approximation, we expect a wedding cake structure with quarter filling
in the center, an compressible transition region and a large incompressible region of filling
⇢ = 1/8 in an outer ring before entering the vacuum. This picture is validated by our
numerical simulation, where the local density hn̂j,L i reveals the CDW structure. To proof the
P
incompressibility of the phases, we calculated an averaged density nj = 18 j+2
i=j 1 hn̂i,L + n̂i,R i,
which we show in Fig. 3.5. It illustrates the two expected density plateaus, which lie within
88
CHAPTER 3. REALIZATION OF FRACTIONAL CHERN INSULATORS IN THE
THIN-TORUS LIMIT
Figure 3.5: Density of hard-core bosons (i.e. U/J = 1) at ✓x = 0 in a harmonic trap centered
around j = 64.5. Other parameters are V = 8.4 ⇥ 10 5 J, µ = 2.4J and Ly = 128. While the
density hn̂j,L i along the left leg (blue squares) demonstrates the CDW character, the averaged
density nj (black solid line) reveals incompressible phases at fillings ⇢ = 1/4 and ⇢ = 1/8.
The vertical red lines indicate the phase boundaries between compressible and incompressible
(blue shading) phases calculated from local density approximation.
the regions predicted by the local density approximation for the position-dependent chemical
potential µ(x) = µ+Vtrap (x). The critical chemical potentials µ1/8,± calculated in the previous
section agree well with the observed phase transitions.
The outer incompressible phase is the CDW state at filling ⇢ = 1/8, corresponding to
a half-filled lowest Bloch band, which we are most interested in. The inner incompressible
phase at quarter-filling ⇢ = 1/4 corresponds to a completely filled lowest Bloch band. It is
similar to the Mott phase of bosons in the lowest band of the 1D Su-Schrie↵er-Heeger model
[P2] discussed in the previous Chapter 2.
3.5
Topological classification and fractional Thouless pump
Now we discuss the topological properties of the ⇢ = 1/8 CDW phase. We distinguish two
cases for the classification of the phase, the 1 + 1D model where the second dimension is
defined by the twist-angle ✓x = 0...2⇡, and the 1D model at points of highest symmetry
✓x = ±⇡/2. In the first case, robust topological properties carry over from the ⌫ = 1/2
LN state in the 2D Hofstadter-Hubbard model. In the second case, the CDW constitutes
a (inversion-) symmetry protected topological phase, which is not robust against symmetrybreaking perturbations however.
3.5.1
1 + 1D model and fractional Thouless pump
The ⌫ = 1/2 LN state in the 2D Hofstadter-Hubbard model is characterized by a fractionally
quantized Chern number C = 1/2 [88, 102], and as will be shown shortly this carries over to
the 1 + 1D gapped CDW state. Before going through the details of the calculation, however,
let us give an intuitive physical picture.
As mentioned above, the Chern number is directly related to the quantized Hall current
in the 2D model (on a torus, say). If one unit of magnetic flux is introduced through the
perimeter of the torus, i.e. ✓x = 2⇡, a quantized Hall current is induced around the induced
flux. While in the case of an integer-quantized Chern number C = p the state returns to
3.5. TOPOLOGICAL CLASSIFICATION AND FRACTIONAL THOULESS PUMP
θx=-π/2
π
θx=0
θx=π/2
θx=π
π
π
π
89
π
π
Figure 3.6: Approximate Wannier orbitals (blue shaded) at the points ✓x of highest symmetry.
itself immediately, when C = p/q takes a fractional value, the state returns to itself only after
introduction of q flux quanta, ✓x = q ⇥ 2⇡.
As discussed in the non-interacting case, an integer-quantized Thouless pump still exists
in the thin-torus limit when the twist-angle ✓x is adiabatically increased. This mechanism
carries over to the ⇢ = 1/8 CDW, as can be understood from a simple Gutzwiller-ansatz. To
this end we approximate the CDW by a product state
Y †
|CDWi =
b̂2n (✓x )|0i,
(3.12)
n
where b̂†j (✓x ) creates a boson in the Wannier orbital corresponding to unit-cell j. To understand how the Wannier orbitals depend on ✓x , we approximate them at the points of highest
symmetry, ✓x = 0, ⇡/2, ⇡, .... To this end we search for the state of lowest energy within each
unit-cell, and note that in principle the residual coupling between unit-cells could be treated
perturbatively. The result is illustrated in FIG. 3.6. At ✓x = 0 the hoppings are J, t1 = 2J
and t2 = 0, such that Wannier orbitals are localized on every other rung, with an energy
of t2 = 2J to zeroth order in the described perturbation
theory. At ✓x = ±⇡/2 on the
p
other hand, the hoppings read J and t1 = t2 = 2J such that considering only rungs is
not sufficient. Instead we compare the energy of a particle hopping around a single four-site
plaquette with zero and
p ⇡ flux respectively. While in the latter case there are two degenerate
states with
energy
3J, for vanishing flux we find a non-degenerate state with lower energy
p
1 + 2 J.
Although we consider only local Hubbard-type interactions, the CDW state (3.12) is stabilized by a finite gap CDW to any excitations (it can also be interpreted as a Mott insulator).
This is due to a hopping-induced finite range interaction. If we calculate the Wannier orbitals
beyond the zeroth order approximation introduced above, nearest and next-nearest neighbor
orbitals acquire a finite overlap. Thus, if the twist-angle ✓x is adiabatically varied, the state
(3.12) follows the modified Wannier orbitals. Because they re-connect to their neighbors after
a full pumping cycle, see FIG. 3.6, a quantized atomic current flows along the ladder. Because – assuming periodic boundary conditions along y – the state only returns to itself after
two full pumping cycles, the Thouless pump is fractionally quantized, with a coefficient (the
Chern number) C = 1/2. This quantization is robust against any perturbations which are
small compared to the gap CDW . The Thouless pump is also shown in FIG.3.1 (c).
Now we turn to a more formal topological classification of the 1+1D model, following [174].
To this end we calculate the many-body Chern number of the ⇢ = 1/8 CDW state. Because
periodic boundary conditions are required along the ladder (in y-direction), we restrict our
analysis in this section to exact diagonalization of small systems (instead of performing DMRG
calculations as above). Before getting started, we note that on a torus the CDW ground state
is two-fold degenerate (in thermodynamic limit Ly ! 1), as expected from the topologically
90
CHAPTER 3. REALIZATION OF FRACTIONAL CHERN INSULATORS IN THE
THIN-TORUS LIMIT
1
0.75
0.5
0.25
0
0
0.2
0.4
0.6
0.8
1
Figure 3.7: The U (2) Wilson-loop phase 'W (✓x ) = Im log detŴ (✓x ) is shown for the ⇢ = 1/8
CDW, the winding of which gives the total Chern number. We used exact diagonalization for
a system of size Lx = 2, Ly = 12 with periodic boundary conditions and N = 3 particles for
N = 6 flux quanta.
protected two-fold ground state degeneracy of the 1/2 LN state in the 2D Hofstadter model.
Naively this degeneracy can be understood in 1D from the obvious ambiguity in the choice
Q
of occupied orbitals in Eq.(3.12): Choosing odd orbitals instead, |CDW0 i = n b̂†2n+1 (✓x )|0i,
yields an equivalent but orthogonal CDW state. We saw already in the discussion above,
that |CDWi can be adiabatically transformed into |CDW0 i without closing the bulk gap by
applying one full Thouless pumping cycle.
The many-body Chern number is defined, in analogy to the single-particle case, by employing twisted boundary conditions. In the 1D thin-torus limit model we thus have a 2D
parameter space spanned by the external parameter ✓x = 0...2⇡ and the twist angle ✓y = 0...2⇡
of the 1D ladder model. However, because the ground state is two-fold degenerate (for some
values ✓x,y this is true even in a finite system), only the total Chern number of both states
can be defined. It can most conveniently be calculated as the winding of the U (2) Wilson
loop Ŵ , which is a non-Abelian generalization of the Zak phase, see Sec.1.2.3. Here we use
the following definition,
✓ Z 2⇡
◆
Ŵ (✓x ) = P̂ exp
i
d✓y Â(✓) ,
(3.13)
0
where Â(✓) is the non-Abelian Berry connection [91] and P̂ denotes path-ordering. Then
its winding yields the total Chern number Ctot , which divided by the number of degenerate
states Ndeg – in our case Ndeg = 2 – yields the fractional Chern number,
1
1
C=
⇥
Ndeg 2⇡
Z
2⇡
0
d✓x @✓x Im log detŴ (✓x ) .
|
{z
}
(3.14)
='W (✓x )
The Wilson loop phase can easily be evaluated numerically in a gauge-independent way, see
e.g. [94], and in FIG.3.7 we show 'W (✓x ) for the thin-torus model. We observe a winding by
2⇡, which as expected results in a many-body Chern number C = 2⇡/(2 ⇥ 2⇡) = 1/2.
3.5.2
1D model and SPT CDW
At special values of the twist angle ✓x = ±⇡/2 the model (3.1) is inversion-symmetric around
the center of links on the legs of the ladder. In this case, the CDW phase can be understood as
3.6. SUMMARY AND OUTLOOK
91
a SPT phase [195]. To come up with an elegant formal classification, the spontaneous breaking
of inversion symmetry by the CDW has to be carefully accounted for. This is discussed in
detail in Sec.2.5.2. Here we restrict ourselves to the definition and calculation of a topological
invariant ⌫, which is quantized to ⌫ = 0, ⇡ and protected by inversion symmetry.
The topological invariant we employ is the many-body Zak or Berry phase defined by
twisted boundary conditions along the ladder [84] [P2]. Like in the case of the Chern number
in the 1 + 1D case, we introduce the twist angle ✓y , however now the second parameter
✓x = ±⇡/2 is fixed. In practice the most convenient way to implement twisted boundary
conditions numerically is to multiply the hopping elements from the last to the first sites of
the ladder (which realize periodic boundary conditions) by the complex phase ei✓y . Then the
eigenstate | (✓y )i depends on ✓y and the Berry phase can be calculated as usual,
⌫=
Z
2⇡
0
d✓y h (✓y )|[email protected]✓y | (✓y )i.
(3.15)
From inversion symmetry it follows that ⌫ = 0, ⇡ is strictly quantized [81, 153].
To calculate the topological invariant ⌫, we restrict ourselves to the simple representation
(3.12) of the CDW state | i. Then we distinguish four di↵erent cases, characterized by the
value of ✓x = ±⇡/2 and by which of the two states |CDWi and |CDW0 i we use. To begin
with we note that only for, say, ✓x = ⇡/2 the link with the complex phase ei✓y is part of an
atomic orbital, as defined in the discussion of FIG.3.6. Then in the trivial case ✓x = ⇡/2,
| i is independent of ✓y and thus ⌫ = 0 vanishes for both CDW states. For ✓x = +⇡/2 on
the other hand, we have to distinguish between CDW and CDW’. Only for one of the two
states – say for |CDWi – the link with the complex phase ei✓y is part of an occupied atomic
orbital. Thus for the state described by CDW’ the wavefunction | i is independent of ✓y
and ⌫ = 0 again. Finally we will show that the state CDW is topologically non-trivial with
⌫ = ⇡. To this end, note that there is an occupied atomic orbital on the link connecting
the last and the first rung of the ladder. The energy of this orbital can not be changed by
the complex phase ei✓y , which is merely a gauge transformation, but the eigenfunction of the
orbital m (✓y ) (with m = 1, ..., 4 labeling the four sites),
on ✓y . In fact, a simple
R 2⇡depends
P
⇤ [email protected]
calculation shows that the corresponding Berry phase is 0 d✓y m m
✓y m = ⇡. Because
|CDWi is a simple product state it follows that ⌫ = ⇡ in this case.
3.6
Summary and Outlook
In summary, we have proposed and analyzed a realistic setup for the realization of a topologically non-trivial CDW state (at filling ⇢ = 1/8) of strongly interacting bosons in a 1D
ladder geometry. Although fundamentally the interactions are purely local, in the lowest
Chern band they become e↵ectively non-local because of the finite extend of the Wannier
orbitals. Our model was derived by taking the thin-torus limit of the 2D Hofstadter-Hubbard
model at flux ↵ = 1/4 per plaquette. The ⌫ = 1/2 Laughlin-type fractional Chern insulator
in this 2D model is directly related to the 1D CDW at filling ⇢ = 1/8. As a consequence,
the CDW has interesting topological properties: When adiabatically introducing magnetic
flux ✓x /2⇡ through the small perimeter of the thin torus, which can be realized by changing
the hoppings in our model, a fractionally quantized Hall current is induced along the ladder.
Alternatively, the CDW phase can be interpreted as inversion symmetry-protected topological phase, characterized by a quantized topological invariant taking values ⌫ = 0, ⇡. We
used DMRG calculations to determine the particle-hole gap of the CDW and found values of
CDW ⇠ 0.1J, a sizable fraction of the bare hopping J. When placed in a harmonic trap,
92
CHAPTER 3. REALIZATION OF FRACTIONAL CHERN INSULATORS IN THE
THIN-TORUS LIMIT
the wedding cake structure of the density provides a clear signature of the appearance of the
topological CDW state.
Investigating the thin-torus limit of fractional Chern insulators is a promising route to
gain understanding of more complicated, but closely related, topologically ordered states in
2D systems [193, 194, 195, 203]. In this chapter we showed how the thin-torus limit can be
realized experimentally with ultracold atoms, including the possibility of fully tunable twisted
boundary conditions. Similar ideas can be carried over to photonic systems, where synthetic
gauge fields can also be implemented [204, 121, 122, 111] and strong non-linearities on a
single-photon level are realized e.g. using Rydberg atoms [126, 129]. Therefore an interesting
future direction for such experiments would be the observation of more complicated thin-torus
models, going beyond the analogue of the simple 1/2 LN state and including for instance states
related to the non-Abelian Read-Rezayi series [50, 194, 203].
Once a system like the one described in this paper is realized, an important question is how
to witness its topological properties. The quantized transport connected to the Chern number
could be measured by taking in-situ images of the atomic cloud. A more direct measurement
of the topological invariant would be desirable, which should also be able to measure the
invariant ⌫ characterizing the symmetry-protected topological order. Such measurements
have been performed in non-interacting systems [13, 201] using a combination of Ramseyinterferometry and Bloch oscillations, and they could be extended to interacting systems in
the future. This question how to detect topological invariants will be addressed in detail in
Chapters 6 - 8.
Chapter 4
Fractional Quantum Hall E↵ect
with Rydberg Interactions
4.1
Summary and Introduction
This is the last Chapter in which we deal with the question of how to engineer Hamiltonians
with topologically ordered ground states. Unlike in the previous chapters, our attention
will be entirely focused on the interactions now. To this end we study ultra cold Rydbergdressed Bose gases subject to homogeneous artificial gauge fields in the fractional quantum
Hall (FQH) regime. We investigate how the characteristics of the Rydberg interaction gives
rise to interesting many-body ground states, di↵erent from standard FQH physics in the lowest
Landau level (LLL).
In the context of the fractional quantum Hall e↵ect (FQHE) interesting many body ground
states may exist, as we discussed already in the introduction, see 1.2.4. For an experimental
study of the FQHE a high degree of control, strong magnetic fields and large interactions are
required. Two dimensional ultra cold quantum gases in artificial magnetic fields have been
suggested as potential candidates [100]. E↵ective magnetic fields can be generated either by
rotation [115, 205], employing light-induced gauge potentials [79, 206, 119, 116, 117], or in
lattices with complex hopping amplitudes [113, 196, 108, 109].
When a weakly interacting superfluid Bose gas is rotated sufficiently fast, an ordered
vortex-lattice forms [207, 134] and the lowest Landau level (LLL) regime can be reached
[135]. Although for bosonic atoms there is no direct analogue of the integer quantum Hall
e↵ect based on the Pauli principle, repulsive interactions can give rise to highly correlated FQH
states for fillings ⌫ . 6 [100] when only the LLL is occupied [137, 140, 101]. However, despite
the impressive experimental progress, strongly correlated phases have not been realized yet.
In part this can be attributed to the rather small interaction energies in atomic gases.
In this Chapter we propose an alternative approach using 1/r6 van-der-Waals (vdW)
interactions between Rydberg states. The associated energies can be orders of magnitude
larger than those achievable with contact or magnetic dipole-dipole interactions [127], making
Rydberg interactions ideal candidates for the realization of FQH physics. Rydberg interactions
can not only be implemented in atomic systems, but it is moreover possible to hybridize
Rydberg atoms with light in form of so-called Rydberg polaritons [208, 209, 129]. This
opens the possibility to realize FQH physics with photons, where artificial gauge fields can be
generated either by rotation of a suitable medium [204], using wave-guide lattice structures
[122, 121] or employing dipolar interactions in e↵ective spin systems [210].
In the present chapter we theoretically analyze the phase diagram of particles with a
93
94
CHAPTER 4. FRACTIONAL QUANTUM HALL EFFECT WITH RYDBERG
INTERACTIONS
1
0
0
1
2
Figure 4.1: (a) An ultra-cold Bose gas is dressed with Rydberg-excitations |ri by illuminating
with a far o↵-resonant (detuning ) laser beam ⌦. Rotating the gas with frequency !0
and applying a weak harmonic trap !? in radial and a tight one in the transverse direction
(!z
!? ) the quasi-2D FQH regime can be reached. (b) Interaction potential for Rydbergdressed atoms.
Rydberg interaction in the LLL using exact numerical diagonalization (ED) techniques as
well as variational methods. First we consider the crystallization transition at small densities.
Similar to dipolar gases with 1/r3 interactions, which have been studied extensively in the
past [211, 212, 213], for very dilute systems we find a transition from Laughlin (LN) liquids
[47] to crystalline ground states. However, due to the strength of 1/r6 Rydberg interactions
at small inter-particle spacing, we identify an extended region of filling fractions ⌫ where
only composite particles can crystallize at zero temperature. In contrast to dipolar gases the
di↵erence in variational energy between composite crystals and a simple Wigner crystal of
bare bosons is rather large.
Second, we study the e↵ect of an additional length scale, the so-called Rydberg blockade
radius aB , which competes with the magnetic length `B . Below the blockade radius the
typical Rydberg interaction potential saturates. As we will show, this gives rise to a clustering
mechanism for bosons at large aB . As a consequence bubble crystal ground states exist in this
regime for fillings ⌫  1/4. For larger fillings, on the other hand, we find strong indications
for exotic cluster liquids with non-Abelian excitations in the regime of large blockade radii.
This chapter is based on the publication [P1] which is an extension of the diploma thesis
of the author [55]. The chapter is organized as follows. In Sec.4.2 we introduce the model and
calculate the Haldane pseudopotentials corresponding to the Rydberg interactions. Ground
states of the model at small blockade radii aB ⌧ `B are discussed in Sec.4.3 for di↵erent
filling fractions. In particular we investigate the Wigner crystallization transition. In Sec.4.4
we investigate the e↵ect of large blockade radii and speculate about non-Abelian cluster liquids
at large fillings. We close with a summary and an outlook in Sec.4.5.
4.2
The Model
We consider ultra cold bosons of mass M confined to a two-dimensional plane which are
subject to a homogeneous magnetic field. As discussed in Sec.1.2.4 they can be described by
a Hamiltonian of the form
Z
Z
( irr A(r))2 ˆ
2
†
ˆ
Ĥ = d r (r)
(r) + d2 rd2 r 0 ˆ† (r) ˆ† (r 0 )V (r r 0 ) ˆ(r 0 ) ˆ(r), (4.1)
2M
4.2. THE MODEL
95
see Eq.(4.1). Possible experimental implementations of the single-particle Hamiltonian were
discussed in Sec.1.2.4. Here, to be specific, we discuss a system of rotating ultra cold atoms
placed in parabolic confining potentials which confine the atoms to a two-dimensional plane
and cancel the centrifugal force, see FIG.4.1(a). The Coriolis force, on the other hand, mimics
the e↵ect of a magnetic field B = r ⇥ A(r).
The interaction between the bosons is described by a potential V (r), which can be realized
by utilizing Rydberg states. In general it also contains a local contribution gBB (r) due
to microscopic vdW interactions between ground state atoms. In the following, however,
we assume that gBB is negligible compared to the large Rydberg interactions. Below we
discuss the specific form of the Rydberg interactions and calculate the corresponding Haldane
pseudopotentials.
When the Rydberg-dressed Bose gas is set into rotation with angular frequency !0 in
a radial trap of frequency !? , the characteristic oscillator length scale is given by `c =
(2M !? /~) 1/2 , see e.g. [26]. The competition of the two length scales `c and aB can lead
to interesting physics. Already in the mean-field regime, where the rotation frequency is well
below the deconfinement limit (!? > !0 ), a non-trivial phase transition from a vortex lattice
to a supersolid was predicted [214]. Here, in contrast, we are interested in the regime of
strong correlations. To this end we consider a Rydberg-dressed Bose gas in the LLL regime
for fillings ⌫ < 1, assume a quasi-2D gas, neglect finite-thickness e↵ects and use !? = !0 . In
the latter case `c is replaced by the magnetic length `c ! `B . For simplicity we consider only
even values of 1/⌫ where bosonic LN states can be defined.
4.2.1
Rydberg dressing
Recently there has been considerable interest in atoms excited to high-lying Rydberg states,
and there has been impressive experimental progress to make these systems accessible in the
laboratory [215, 216, 217, 218, 219, 220, 221]. Of particular interest in the present context are
atoms excited by far-o↵ resonant laser radiation (detuning from the resonance), see FIG.4.1
(a). In this case, called Rydberg dressing, the atoms essentially remain in their ground state
and acquire only a small admixture of the excited Rydberg state. As a consequence they show
an e↵ective interaction [222, 223]
C̃6
V (r) = 6
,
aB + r 6
aB =
✓
C6
2~
◆1/6
,
(4.2)
see FIG.4.1 (b). The interaction strength is determined by C̃6 = (⌦/2 )4 C6 , where C6
denotes the bare vdW coefficient and ⌦ ⌧ | | is the Rabi-frequency of the excitation laser.
For large particle separations r the interaction potential is of vdW type, showing the
characteristic power-law behavior V (r) ⇠ r 6 . At small distances below the blockade radius
r . aB , on the other hand, the interaction potential (4.2) flattens o↵, thereby defining a new
characteristic length scale. We will demonstrate in the following how these two characteristic
features gives rise to interesting many-body behavior of the fractional quantum Hall system.
Let us note before, however, that the same behavior of the interaction potential is found for
Rydberg polaritons [224], which allows to apply many of the results obtained here to FQH
physics of photons hybridized with Rydberg atoms.
96
CHAPTER 4. FRACTIONAL QUANTUM HALL EFFECT WITH RYDBERG
INTERACTIONS
n
aB [µm]
V0 /2⇡ [kHz]
40
0.97
2.8
50
1.5
5.3
60
2.0
7.9
70
2.7
9.9
80
3.4
11.2
90
4.3
11.8
100
5.2
12.1
Table 4.1: Realistic blockade radii aB and energies V0 for Rb with !c ⇡ 2⇡ · 130Hz (i.e.
`B ⇡ 1µm), = 10⌦ = 2⇡ · 1GHz as a function of the principle quantum number n.
4.2.2
Rydberg-interaction pseudopotentials in the LLL
The interaction projected to the LLL is described by Haldane’s two-particle pseudopotentials
Vm [133], where the integer m denotes the relative angular momentum. In disc geometry the
pseudopotenials are given by
Z 1
1
2
Vm = 2m+1
dr r1+2m V (r)e r /4 ,
(4.3)
2
m! 0
where for bosons (fermions) only m = 0, 2, 4, ... (m = 1, 3, 5, ...) are relevant.
The relevance of the pseudopotentials becomes apparent if one considers the (bosonic) LN
wavefunction for the ground state of N particles at filling ⌫ = 1/n, with n being an (even)
integer. Denoting the coordinate of the j-th particle in the two-dimensional plane by the
normalized complex variable zj = (xj + iyj )/`B , the LN wave function reads [47]
⌫=1/n
z1 , . . . , z N
⇠
Y
zi
zj
i<j
n
e
1
4
P
k
|zk |2
.
(4.4)
In the absence of interactions and of any confinement potential all states are degenerate
and have zero energy. When considering the contribution of the interaction Hamiltonian to
Q
the energy of the LN states one recognizes that the Jastow factors i<j (zi zj )n eliminate
furthermore all contributions from pseudopotentials Vm with m = 0, 2, . . . , n 2. As a consequence the energy scale of the ground state at filling ⌫ = 1/m is set by the largest unscreened
pseudopotential, i.e. by Vm .
For typical interactions (point-like, Coulomb, dipolar) the largest energy scale within
the LLL is determined by the lowest pseudopotential V0 , while the higher-order terms fall
o↵ quickly. This is also true for the vdW interactions (4.2) under consideration. To give an
impression of the corresponding energy scales in a Rydberg system we list some explicit values
of V0 for Rubidium (Rb) atoms in tables 4.1 and 4.2. There we assume a fixed magnetic length
`B ⇡ 1µm (corresponding to !c ⇡ 2⇡ ⇥ 130Hz which is a realistic value [225]) as well as a fixed
ratio ⌦/ = 0.1. For the 60S1/2 Rydberg state in Rb, C6 (n = 60)/2⇡ = 0.14THzµm6 and
we used the scaling law C6 (n) / n11 [226]. In table 4.1
= 2⇡ ⇥ 1GHz is constant and the
principle quantum number of the Rydberg state n is varied. In table 4.2 on the other hand,
n = 46 is fixed and the detuning
is varied. Note that by varying
in time, it is possible
to tune the interaction potential adiabatically in an experiment.
[MHz]
aB [µm]
V0 /2⇡ [kHz]
1.0
9.0
3.94
2.73
0.0744 0.574
80
1.90
3.75
700
1.32
20.8
20 · 103
0.76
240.0
Table 4.2: Realistic blockade radii aB and energies V0 for Rb with !c ⇡ 2⇡ · 130Hz (i.e.
`B ⇡ 1µm), n = 46 as a function of the detuning of the dressing laser . Note that the Rabi
frequency ⌦ was chosen as ⌦ = 0.1 .
4.3. GROUND STATE FOR SMALL BLOCKADE RADII
97
10
2
-6
-5
8
0
6
-5
4
⌥2
-4
-3
2
⌥4
-1
0
4
8
12
16
20
Figure 4.2: Contour plot of pseudopotentials Vm for the potential (4.2), where m was treated
as continuous parameter for better illustration. Black solid lines denote contours of constant
(continuous) pseudopotential.
From the data in tables 4.2, 4.1 one recognizes rather large values of V0 up to hundreds
of kHz, which can easily become comparable to – or even exceed – the typical LL splitting
!c . Thus, in contrast to all previously discussed interactions in cold gases, the characteristic
energy scales for Rydberg-Rydberg interactions should be limited from above in order for the
LLL-approximation to be valid (i.e. V0 < ~!c , see [132]).
The pseudopotentials Vm for the vdW interaction (4.2) are plotted in FIG.4.2, together
with contour lines of the continuous function V (m) = Vm for m 2 R defined as in (4.3) above.
One recognizes that, at given value p
of aB , the first pseudopotentials Vm are approximately
equal until the radial extend Rm = 2m`B of the wavefunction in the relative p
coordinate is
⇠ aB . The subsequent pseudopotentials decrease quickly. Therefore the curve 2m `B = aB
separating the two regions in the contour plot of pseudopotentials FIG.4.2 also separates two
parameter regions with qualitatively di↵erent physics
p
p
2/⌫ `B < aB , and
2/⌫ `B > aB ,
(4.5)
for a given filling fraction ⌫. The first region corresponds to a situation with at most one
particle per blockade area on average, while in the second region more than one particle is
found in that area.
4.3
Ground state for small blockade radii
Now we discusspthe ground state of the Hamiltonian (4.1) in the case of a small blockade
radius aB < `B 2/⌫. In this case the interaction takes the form of a pure vdW interaction
which is merely regularized by aB at small distances. We discuss separately the regimes of
fillings ⌫ 1/4, where Laughlin type states are stabilized, and dilute systems where ⌫  1/6.
In particular we study the Wigner crystallization transition at small fillings.
4.3.1
Ground states at ⌫ = 1/2 and ⌫ = 1/4
We start by considering the limit aB /`B ! 0, where the first relevant pseudopotentials diverge
3 C̃6
V0 ⇡
8 `6B
✓
`B
aB
◆4
,
1
V2 ⇡ 6
2
✓
0.571
aB
ln
`B
◆
C̃6
.
`6B
(4.6)
98
CHAPTER 4. FRACTIONAL QUANTUM HALL EFFECT WITH RYDBERG
INTERACTIONS
aB /`B
6.1
5.1
4.1
3.1
2.1
1.1
0.1
⌫ = 1/2
1.690 · 10 09
4.935 · 10 10
7.303 · 10 10
2.594 · 10 03
0.9940
0.9998
0.999999999978
⌫ = 1/4
0
0
0
0.9026
0.9696
0.9960
0.99988
Table 4.3: Overlaps squared of the ground state to the Laughlin states at fillings ⌫ = 1/2, 1/4.
They were obtained using ED at N = 6 in spherical geometry. We write 0 when our value is
below the numerical precision, i.e. < 10 15 .
Meanwhile, all higher-order pseudopotentials Vm>2 are convergent. Therefore the ⌫ = 1/2
and 1/4 bosonic LN states are exact ground states in this limit.
To understand the e↵ect of finite (but small) blockade radii we performed exact diagonalization (ED) calculations in spherical geometry for small particle numbers N (see Appendix
B for details). Table 4.3 lists overlaps to the LN states for N = 6 particles and several values
of aB . The results remain valid even for larger particle numbers: We obtain overlaps squared
of e.g. 0.992 for ⌫ = 1/2 and N = 10 and 0.974 for ⌫ = 1/4 and N = 7.
4.3.2
Ground states at small fillings
Now we address the regime of small fillings, i.e. ⌫ < 1/4, again for aB . `B . Due to the
non-local interactions, a natural ground state candidate in this regime is the non-correlated
Wigner crystal (NWC) [227] where bosons are maximally localized within the LLL, forming
a regular lattice structure. The NWC is described by the wavefunction
NWC (z)
=S
Y
e (|zj
Rj |2 +zj Rj⇤ zj⇤ Rj )/4
,
(4.7)
j
where Rj 2 C defines the lattice in the plane of complex coordinates zj = (xj + iyj )/`B and
S stands for complete symmetrization.
We will now investigate under which conditions (4.7) is a valid variational ansatz for
the many-body ground state. To this end we start by calculating its variational energy and
compare it to the variational estimate obtained from a LN state at the corresponding filling.
Then we estimate the stability of the Wigner crystal, which yields a contradictory result for
the critical filling ⌫c where crystallization takes place. To clarify what the actual physics is
at small fillings, we present numerical results from ED studies. In the following subsection
we will then resolve the contradictory result from the variational analysis by considering a
refined wavefunction including additional correlations.
Variational ground state energy
To see whether the NWC can be a ground state, we compare the corresponding variational
energy per particle ✏NWC to that of the LN state. To this end we generalize the expression
for ✏NWC derived for fermions by Maki and Zotos [227] to bosons,
✏NWC =
2
1 C̃6 X e |Rj | /4
· K(⇠ = |Rj |),
2
2 `6B
1 + 12 e |Rj | /2
j6=0
(4.8)
4.3. GROUND STATE FOR SMALL BLOCKADE RADII
99
⇧5
x 10
1
10
1.6
⇧1
1.4
10
1.2
⇧3
10
1
0.8
⇧5
10
5
10
15
0.6
20
12
14
16
Figure 4.3: Comparison of the variational energies per particle ✏0 (at fillings ⌫ =
1/2, 1/4, 1/6, ... corresponding to the bosonic Laughlin states) for the di↵erent trial wavefunctions at aB = 0.2`B (LN - Laughlin states, NWC - non-correlated Wigner crystal of bare
particles.)
where we defined
K(⇠) =
Z
1
0
2
dr re r /4
r6 + (aB /`B )6
✓
I0
✓
r⇠
2
◆
+ J0
✓
r⇠
2
◆◆
.
(4.9)
Here I0 , J0 denote Bessel functions. The integral in the last equation can be handled numerically and the lattice sum in Eq.(4.8) can easily be performed. We find that, like for electrons
[228, 227] and dipolar fermions [145], the lattice geometry minimizing the NWC energy is a
hexagonal one, see FIG.4.3.
In FIG.4.3 the energies of di↵erent NWCs are compared to those of the LN liquid, for small
aB = 0.2`B and for even values of 1/⌫. The energy of the LN liquid was calculated using the
Metropolis Monte Carlo (MC) algorithm [229] based on the standard plasma analogy [47] for
N = 100 particles for all examined ⌫. We find a transition from LN liquid to a NWC for
⌫  ⌫NWC =
1
,
14
(4.10)
although for ⌫ = 1/12 the energy di↵erence is so small that we can not exclude a transition
there. Interestingly this value of the critical filling is far below the corresponding value of
⌫ ⇡ 1/7 found for pure Coulomb and dipolar interacting fermions [228, 145, 230]. Due to
the more local nature of the vdW interaction potential, this is not surprising: For point-like
interactions there is no crystallization at all.
Stability of the Wigner-crystal
In an alternative approach we can determine the stability of the Wigner crystal using the
phenomenological
Lindemann criterion1 . To this end we calculate
the Lindemann parameter
p
p
2
2
⌘ h u i/a, which compares the average displacement h u i of the atoms from their
lattice sites {Rj } to the lattice spacing a. The atomic displacement u can be obtained from
the phonon spectrum in harmonic and nearest-neighbor approximation following [231].
The phenomenological Lindemann criterion states that the crystal melts when exceeds
1
We would like to thank M. Baranov for drawing our attention to the Lindemann approach for estimating
the transition point from liquid to crystal.
100
CHAPTER 4. FRACTIONAL QUANTUM HALL EFFECT WITH RYDBERG
INTERACTIONS
x10 4
3
2
1
0
1
0
0.1
0.2
0.3
0.4
0.5
Figure 4.4: Gap to collective excitations E at ⌫ = 1/4, 1/6, 1/8 and for aB = 0.2`B . Note
that for ⌫ = 1/4 energies were scaled down by a factor of 100. ED in spherical geometry
was used and finite-size corrections were performed as described in Appendix B. Solid lines:
quadratic fits.
p
a critical value c . In our case we obtain = 0.57 ⌫ for aB < `B and at zero temperature.
Comparing to the universal critical value c ⇡ 0.28, see [232, 145], we expect that the crystal
is thus stable for
1
⌫ < ⌫Ld = .
(4.11)
4
This is similar to the result found in dipolar systems [212, 145], but the value di↵ers significantly from the transition point to a NWC, ⌫NWC = 1/14, found above. To clarify where the
crystallization transition takes place, we now present the results of exact numerical calculations for small systems.
Gap to collective excitations and spatial correlations
For small system sizes and small aB . `B the exact ground state has relatively large overlap
to the LN liquid and the spectrum shows a low-lying exciton branch. On the other hand, the
analysis of the previous section suggests that, in the thermodynamic limit, the true ground
state may not be a LN liquid. Thus to investigate the nature of the ground state in the region
of intermediate fillings we calculate the energy gap E of LN states to collective excitations
(the exciton) using ED in spherical geometry. If E remains finite for large N the LN state
is stable. However, if E vanishes at some finite N this signals an instability of the LN state.
More specifically E is the di↵erence from the ground to first excited state, where the
latter can well be described as a density wave and exhibits a roton minimum [233, 234].
In Fig.4.4 E is plotted for ⌫ = 1/6, 1/8 for finite size systems and extrapolations to the
thermodynamic limit are shown. Also shown are the results for ⌫ = 1/4 for comparison,
which clearly yield a positive gap in the thermodynamic limit. Although the system sizes
which we were able to reach with our ED are not very large, they do allow for an extrapolation
to the infinite size limit and indicate negative gaps – i.e. instability of the LN liquid – for
⌫ = 1/6 and ⌫ = 1/8. From the full energy spectra on the sphere we find that the energy
gap becomes minimal at a finite angular momentum L2 , indicating an instability towards a
crystalline phase.
The transition to a crystalline phase is characterized by the onset of long-range spatial
4.3. GROUND STATE FOR SMALL BLOCKADE RADII
101
3.5
3
2.5
2
1.5
1
0.5
0
0
3
6
9
12
Figure 4.5: The second order correlation functions g (2) (r) of the ground state for van-derWaals (vdW) interactions are shown at aB = 0.2`B and ⌫ = 1/8 (solid). For comparison the
corresponding correlations are shown for LN liquids. ED in spherical geometry was used and
the system size was N = 6 (thin) and N = 5 (thick).
correlations. To check whether our results for finite systems show the expected behavior, we
calculate the second order correlations g (2) (r) = h ˆ† (0) ˆ† (r) ˆ(r) ˆ(0)i using ED. Our results
are shown in FIG. 4.5 for ⌫ = 1/8 and aB = 0.2`B . We observe strongly enhanced oscillations
for N = 6, signalizing a trend towards long-range order. For N = 5 this e↵ect is absent,
which can be attributed to a more general finite-size e↵ect: Already for the gaps (FIG. 4.4)
we observe a slightly di↵erent behavior for N even/odd and ⌫ = 1/6, 1/8. For N = 2, 4, 6,
E is smaller than for N = 3, 5 and we find the same behavior for the ground state energy.
These observations are in agreement with the formation of a crystal, because we find the same
oscillations of the ground state energy per particle for a classical vdW crystal on a sphere
(due to incommensurability).
Our numerical findings for small systems thus support the result of the Lindemann stability
analysis, that crystallization occurs below ⌫ = 1/4. This raises the question about the nature
of the ground state for fractional fillings in the range 1/4 > ⌫ 1/14.
4.3.3
Correlated Wigner crystal of composite particles
To understand the physics of the ground state at small fillings, we now make use of a refined
variational wavefunction. Instead of a NWC, the ground state in the region ⌫Ld > ⌫ ⌫NWC
could be a crystal of particle-flux composites, so-called composite particles (CPs) [143]. This
state was introduced by Yi and Fertig [235] and it is termed a correlated Wigner crystal
(CWC) [236]. As for the LN states, the first pseudopotentials are screened for CP states, and
only the e↵ects of the long-range tails of the repulsive interaction potential remain. As the
latter should be the same for CPs as for bare particles the stability arguments given above
remain valid and we expect formation of a CWC already for ⌫ < ⌫Ld = 1/4.
CWC variational wavefunctions of the form
Y
(µ)
(z)
=
P
(zj zk )µ NWC (z)
(4.12)
LLL
CWC
j<k
were investigated in [235] using MC simulations based on the plasma analogy, and the critical
filling ⌫ ⇡ 1/7 in electronic, i.e. fermionic systems was correctly predicted. Here PLLL is
102
CHAPTER 4. FRACTIONAL QUANTUM HALL EFFECT WITH RYDBERG
INTERACTIONS
1.1
1.05
1
0.95
0.9
0.85
2
4
6
8
10
Figure 4.6: Variational energy per particle of the hexagonal CWC of µ CP, ✏CWC (µ), in units of
the LN state energy ✏LN (⌫) at filling ⌫. Results were obtained in Metropolis MC simulations
with N = 91 particles.
a projector to the LLL and (zj zk )µ are Jastrow factors describing the formation of µcomposite bosons µ CB (µ- composite fermions µ CF) for µ even (odd). We here performed
similar MC simulations and calculated the variational energy of hexagonal CWCs for di↵erent
µ. From the plasma analogy it can be concluded [235] that the lattice constant a used for the
NWC has to be rescaled in the definition of the CWC wavefunction in order to produce the
desired density at filling ⌫:
a ! a (1 µ · ⌫) .
(4.13)
In the MC simulations we take into account only direct but no exchange terms, and we have
checked that the first exchange corrections are negligible. Our results are shown in FIG.4.6.
We observe that for ⌫  1/6 CWCs are energetically favorable compared to LN states.
We thus expect crystalline order already for ⌫ < ⌫Ld = 1/4 in agreement with the stability
analysis and long before the simple NWC becomes energetically favorable at ⌫ = ⌫NWC . More
specifically we expect a 4 CB CWC ground state at ⌫ = 1/6 and ⌫ = 1/8. For ⌫ = 1/10 the
4 CB and 5 CF CWCs are equal in energy within our MC errors and we can not make a final
conclusion about the underlying CPs. Most importantly we find that the variational energy of
the composite crystals is substantially below the value of the non-correlated Wigner-crystal,
for a large range of fractional fillings.
The formation of crystals from composite particles for ⌫ < 1/4 is further supported by our
ED results for the second-order correlation function plotted in FIG.4.5. For larger distances r
one recognizes enhanced oscillations of g (2) (r) as compared to those expected for LN liquids.
On the other hand, for small r, the correlations still decay according to a power law. We find,
for instance, that g (2) (r) ⇠ r11 for N = 6 at ⌫ = 1/8. This is direct evidence that the ground
state physics of these systems is dominated by µ CPs with µ between 4 and 6, as predicted by
our variational analysis2 .
Q
From a µ CP wavefunction of the form (zj ) = i<j (zi zj )µ CP (zj ), where the CP wavefunction CP (zj )
lives in the LLL and contains neither additional Jastrow factors nor poles ⇠ (zi zj ) 1 , the g (2) function scales
as g (2) (r) ⇠ r2µ for small r.
2
4.4. EFFECTS OF FINITE BLOCKADE RADIUS
103
2.5
2
1.5
1
0.5
0
0
2
4
6
8
10
Figure 4.7: The second-order correlation function g (2) (r) of the ground state is shown for
increasing values of aB for N = 8 particles at ⌫ = 1/2 (solid) and for N = 6 at filling ⌫ = 1/6
(dashed). The translational symmetry is broken for large aB (i.e. L 6= 0, thick lines), whereas
for small aB the system is symmetric (L = 0, thin lines).
4.4
E↵ects of finite blockade radius
Now we discuss how the saturation of the e↵ective vdW interaction potential Eq. (4.2) (for
distances less than the blockade radius aB ) a↵ects the FQH physics. As we will show the
competition of the magnetic length `B with the new length scale aB gives rise to new physics
which has similarities with the physics of higher LLs in the solid state context. Specifically
we consider the case p
where there are more than two particles per blockade area AB = ⇡a2B ,
i.e. the regime aB & 2/⌫`B .
We start with a variational analysis and compare crystalline states of clusters of bosons
to LN liquids. For small fillings we predict a transition to such a bubble crystal phase. For
larger fillings ⌫
1/2 on the other hand, such a transition is not possible. Instead we find
indications for an instability towards a cluster liquid, possibly with non-Abelian topological
excitations.
4.4.1
Bubble crystal at small fillings
We begin our analysis by presenting exact
p numerical results on a sphere. As discussed in
the previous section 4.3, for small aB . 2/⌫`B and at ⌫ = 1/4 and 1/2 we find ground
states with large overlaps to LN states. In particular their total angular momentum L = 0
vanishes, indicating that they describe a liquid phase. For ⌫  1/4,pwhen increasing the
blockade radius aB we find a transition to an L 6= 0 state around aB ⇡ 4/⌫`B . By mapping
the sphere (radius R) to the plane (linear momentum k), with the relation kR = L [237],
this indicates the spontaneous breaking of the translational invariance. Therefore we expect
a crystalline phase at large blockade radii and for ⌫  1/4.
To gain further insight into the ground state properties at large aB we calculate the twoparticle correlation function g (2) (r) for di↵erent ratios of aB /`B . Our results for ⌫ = 1/4
are shown in FIG.4.7. For small aB (L = 0 phase) we find LN-like correlations, whereas for
large aB (L 6= 0 phase) we find particle bunching at r = 0, which indicates cluster formation.
Clustering of k particles can be seen more clearly by considering, in addition, the k +1st-order
104
CHAPTER 4. FRACTIONAL QUANTUM HALL EFFECT WITH RYDBERG
INTERACTIONS
8
5
6
4
4
3
2
2
0.2
1
2
6
10
14
18
Figure 4.8: Phase diagram (for 1/⌫ 2 2N) obtained from the comparison of variational energies
of LN state (N = 100 using Metropolis MC) and bubble crystal. The optimal number k opt of
particles per cluster (color code) was determined, as indicated in the BC region. The squares
with error bars show the phase transition from a LN to a symmetry-breaking state obtained
numerically (ED, spherical geometry).
density-density correlation function,
g (k+1) (z) ⌘
D⇣
ˆ † (0)
⌘k
⇣
⌘k E
ˆ † (z) ˆ (z) ˆ (0)
.
(4.14)
For example for ⌫ = 1/2 and N = 8 particles at aB = 2.9`B we find that k = 2 particles
cluster, resulting in a very small g (3) (0) ⇡ 2 ⇥ 10 5 (⌫/2⇡)3 while g (2) (0) = 0.9 (⌫/2⇡)2 is still
sizable. See also FIG.4.9 (a). Similar results hold for smaller values of ⌫  1/4.
p
Our numerical findings can be understood as follows: For aB > 4/⌫`B where there are
more than two particles per area AB , the long-range contribution to the interaction energy
can be reduced by bringing the particles inside AB closer together. At the same time there is
no energy penalty from interactions inside AB because the potential (4.2) is almost constant
for r . aB . This provides a pairing mechanism in the purely repulsive potential. Assuming
that all particles within AB undergo clustering, one expects a transition from k 1 to k
particles per cluster at
p
(k)
aB = 2k/⌫ · `B .
(4.15)
By analogy we expect the formation of a NWC consisting of clusters when the reduced
filling of the system with clusters ⌫cl = ⌫/k is small. This state is referred to as bubble crystal
(BC) and it was considered previously for electrons in a weak magnetic field where ⌫ > 1,
i.e. beyond the LLL, see [238]. The situation for electrons in a higher LL is comparable to
our case, because after projecting the interaction into a higher LL it is “smeared out” and
saturates around r = 0. This makes it qualitatively similar to our interaction in Eq. (4.2)
projected to the LLL. A BC phase was moreover predicted for dipolar bosons at filling ⌫ = 1/2
with large finite-thickness e↵ects [239]. (Finite-thickness e↵ects similarly leads to a smeared
e↵ective interaction potential.)
To verify that BCs are indeed good ground state candidates for large blockade radii we
calculate the variational energy per particle by generalizing (4.8). This yields
✏BC (k; ⌫) = k · ✏NWC (⌫ = ⌫cl ) + (k
1) V0 ,
(4.16)
where the second term describes the binding energy per particle required for the cluster
4.4. EFFECTS OF FINITE BLOCKADE RADIUS
105
−3
(a)
(b)
3
4
x 10
3
2
2
1
0
0
1
2
4
6
8
0
0
1
2
3
4
5
Figure 4.9: (a) Higher-order correlation functions g (k+1) (r) are shown for di↵erent k = 1, 2, 3
and at di↵erent blockade radii aB . Exact numerical calculations were performed in spherical
geometry at filling ⌫ = 1/2 for the indicated particle numbers. (b) The exciton spectrum
E(k), corresponding to the energy of a collective density wave (or exciton) excitation on
a ⌫ = 1/2 LN state is shown as a function of its linear momentum. It was calculated using
bosonic Girvin-MacDonald-Platzman theory [233], see [55].
formation. In FIG.4.8 the ground state phase diagram is shown determined by comparing the
variational energies of the LN liquid, the NWC and the BC. Also shown are the transition
points between symmetry conserving (L = 0) and symmetry breaking (L 6= 0) states obtained
from ED. One recognizes that BC ground states with k
2 particles per cluster exist for
(2)
⌫  1/4 and aB & aB .
4.4.2
Large filling – indications for cluster liquids
Our variational analysis in the previous subsection showed that for ⌫ = 1/2 the BC is higher in
energy than the LN state, for all values of aB . On the other hand we know that for sufficiently
(2)
large aB & aB the LN state is no longer a good ground state candidate, see e.g. table 4.3.
Now we will examine in more detail what the nature of the ground state is at large aB and for
filling ⌫ = 1/2. We mostly rely on our exact numerical calculations up to N = 10 particles,
making it difficult to draw any definite conclusions. Yet our preliminary studies show that
this parameter regime could be a very interesting one.
As a first step we examine more closely the instability of the gap to collective excitations
above the LN state. In the previous subsection we mentioned that for large aB the energy gap
E closes, signaling an instability of the LN state. For finite-size systems we moreover observed this instability at a finite angular momentum L2 = L(L+1) > 0. For ⌫ = 1/2, however,
the lowest energy states at L = 0 and L = 2 are very close in energy, so it is not clear whether
spontaneous breaking of the translational symmetry takes place in a thermodynamically large
system in this case.
To shed more light on the behavior of the collective excitation gap E of the LN state
at large aB , we calculate it in the thermodynamic limit using Girvin-Macdonald-Platzman
theory [233]. This approach was generalized to bosonic systems in the diploma thesis of the
author [55]. In FIG.4.9 the resulting excitation gap E(k) is shown as a function of the
linear momentum k of the collective excitation. Indeed, for large aB ⇡ 3.2`B we observe
an instability and E ! 0 vanishes. Interestingly, for the large aB under consideration the
familiar roton minimum at finite k > 0 is absent in the exciton spectrum. Hence the instability
takes place at k = 0, indicating the absence of symmetry breaking. This strongly suggests
CHAPTER 4. FRACTIONAL QUANTUM HALL EFFECT WITH RYDBERG
INTERACTIONS
106
⌦4
1
1.2
x 10
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0
2
0.2
2.5
3
3.5
0
0
0.1
0.2
0.3
0.4
Figure 4.10: ED results in spherical geometry. (a) Overlap of the numerically determined
ground states to the di↵erent trial wavefunctions. (b) Finite size approximation of the gap to
collective excitations for the ground state at the system size of the Hf (at aB = 3`B ). Finite
size corrections were carried out following [241]. The di↵erent behavior of for N even/odd
is another indication of cluster formation with k = 2 particles per cluster.
that a quantum phase transition to a distinct liquid phase takes place at large blockade radii.
We will examine this possibility more closely now.
Besides clustered crystalline states of the type discussed in 4.4.1, many di↵erent correlated
cluster liquids have been proposed and discussed in the context of the FQHE in higher LLs.
Most prominent are the Read-Rezayi (RR) Zk parafermion states [50] which have recently
been generalized using totally symmetric Jack polynomials [184]. The Ha↵nian (Hf) [240]
is another example not contained in the Jack series. These states are characterized by the
number of particles per cluster k (e.g. k = 2 for Hf) and the exponent r which determines
the short-distance behavior of the correlations g (k+1) (z) ⇠ |z|2r . For example the Hf is
characterized by r = 4, while for all RR states r = 2. For filling ⌫ = 1 the Moore-Read
Pfaffian (Pf) [7] (which is the k = 2 RR state) was shown to be the ground state for dipolar
[213] and contact [101] interactions corresponding to aB ⌧ `B . We also find a large overlap
to the Pf of e.g. |h (N = 12)|Pfi|2 = 0.90 for aB = `B . Since the potential (4.2) provides a
clustering mechanism also for ⌫ < 1, the cluster liquids are reasonable trial wavefunctions for
our situation as well.
In the following we will focus on filling ⌫ = 1/2, where the BC is not a ground state. We
investigate the three simplest ⌫ = 1/2 cluster liquids, namely the LN state (k = 1, r = 2),
the Ha↵nian (k = 2, r = 4) and the k = 3, r = 6 Jack polynomial. We will refer to the
latter as 3-6 BH (Bernevig, Haldane) state. As in the BC case, we expect a transition from
(k)
k 1 to k particles per cluster at aB . This estimates the transition from LN to Hf to be at
(2)
(3)
aB ' 2.8`B and from Hf to 3-6 BH at aB ' 3.5`B . In FIG.4.10 (a) we show the numerically
obtained overlaps of the ground state wavefunctions with the trials. (The 3-6 BH state is
obtained from the “Jack generator” [188].) Note that the calculations were done for spheres
of the di↵erent sizes supporting the respective trial ground states (see Appendix B). We find
that for every aB at least one of the overlaps takes a substantial value. This finding must be
taken with care however: It is known that when a trial wavefunction has a large overlap to
the exact ground state for small systems, this does not necessarily imply a good description
in the thermodynamic limit. The transitions between the di↵erent trial states in FIG.4.10 are
reasonably well described by the above estimates. Coexistence of phases is a finite size e↵ect,
Graduate School o
V. SUMMARY
4.5.
Summarizing, we have shown that Rydberg-dressed Bose
gases AND
can give
rise to extremely strong interactions and a
SUMMARY
OUTLOOK
APPEND
107
In
addition to v
of-the-art ED stud
we did all our syst
We work on stand
to the following p
ν
Nmax
dim.
FIG. 10. (Color online) Qualitative form of the LLL phase
Figure 4.11: Qualitative form of the LLL phase diagram.
diagram.
The dimension
angular momentu
explicitly exploit
043628-7
at least partly caused by the di↵erent system sizes.
We close our discussion of cluster liquids by investigating numerically the ground state
excitation gap. In FIG.4.10 (b) finite size approximations are shown for the Hf state which
suggest that it is an incompressible ground state in the thermodynamic limit. This is a
surprising result since the Hf is generally believed to describe compressible states. We note
that the arguments given by Green [240] that the Hf lies on a phase transition and should be
gapless rely on the fact that the e↵ective 2 CB interaction is purely repulsive. It was shown
for composite fermions ( CFs) that a flattening of the bare 1/r potential leads to attractive
inter- CF interactions [242]. By analogy we speculate that for our extremely flat potential
(4.2) the inter- 2 CB interactions becomes attractive as well, which will indeed lead to 2 CBpairing [243]. We also note that even our bare potential has a small attractive component in
momentum-space. The question whether or why the pairing symmetry should have a d-wave
character, as required for the Hf to be a valid description, will be devoted to future work.
We conclude this paragraph by noting that this parameter regime, which may be easier
accessible experimentally than the low-filling regimes discussed above, is potentially a very
interesting one. In order to make more definite statements however, further studies are needed.
4.5
1
14
194 668
Summary and Outlook
Summarizing, we have shown that Rydberg-dressed Bose gases can give rise to extremely
strong interactions and a variety of interesting correlated phases not found in the standard
fractional quantum Hall physics of the lowest Landau level. This has two reasons: the rapid
fall-o↵ of the interaction potential with distance and the competition of two involved length
scales, the blockade radius aB and magnetic length `B . A qualitative picture of the resulting
phase diagram is shown in FIG.4.11. In the limit of pure van-der-Waals interactions (aB /`B !
0) the ⌫ = 1/2 and 1/4 Laughlin states are exact ground states. Although we mainly discussed
filling fractions with even integer values of 1/⌫, the entire region is denoted by “FQHE“ since
we expect the standard bosonic fractional quantum Hall physics [101] for all fractional fillings
larger than ⌫ = 1/6. For ⌫  1/6 we find correlated Wigner crystal ground states of particleflux composites, while a non-correlated
Wigner crystal wavefunction predicts crystallization
p
only at ⌫ = 1/12. For aB & 4/⌫`B and ⌫  1/4 a transition to a bubble crystal is expected.
Finally, for larger fillings and blockade radii the nature of the ground state is an open question,
but for filling ⌫ = 1/2 we found indications for interesting cluster liquid states, in particular
108
CHAPTER 4. FRACTIONAL QUANTUM HALL EFFECT WITH RYDBERG
INTERACTIONS
for the Ha↵nian [240].
In an actual experiment, addressing the low-filling fractional quantum Hall regime with
ultra cold atoms is challenging. However, the recent implementation of strong artificial gauge
fields in optical lattices [108, 109] constitutes an important step in this direction. At this
point, the experimental progress is limited by the achievable temperatures, making it hard
to prepare topologically ordered ground states at low fillings [124, 110]. Although Rydberg
atoms are routinely used in modern experiments and strong interactions have been observed
(see e.g. [215, 218, 244, 128, 129]), Rydberg dressing has not been demonstrated yet. This is
partly due to technical challenges and to the complicated level structure of a pair of Rydberg
atoms in close vicinity to each other [245]. Nevertheless several groups are pursuing this goal.
On the theoretical side, our model has been studied both the mean-field regime at large
filling [214] and (in this thesis) in the fractional quantum Hall regime at ⌫  1 [P1]. The
transition region at intermediate fillings ⌫ ⇡ 6 [100, 239] has not been investigated, however.
It would be particularly interesting to understand how the supersolid phase predicted in Ref.
[214] connects to the fractional quantum Hall regime. Such an approach, starting from large
fillings, would also be promising to understand in more detail the possible cluster liquid phases
discussed in this chapter.
Chapter 5
Topological Growing Scheme for
Laughlin States
5.1
Outline and Introduction
In this chapter we address the question how topologically ordered ground states of ultra cold
atoms or photons can be prepared in a realistic experimental setting. Topological states of
matter have attracted a great deal of interest recently, mostly due to the astonishing physical
properties of their excitations (like fractional charge and statistics) which are of potential
practical relevance for quantum computation [10, 11]. To first probe these properties experimentally and second utilize them for engineering new quantum technologies, a reliable scheme
needs to be developed for the preparation of topologically ordered ground states with high
fidelity. While in electronic systems powerful cooling mechanisms are available allowing to
prepare the ground state in thermal equilibrium, this is currently a major limitation for experiments with photons and, in particular, ultra cold atoms [110, 124]. This seems surprising,
given the ultra low temperatures achievable with cold atoms [53]. However compared to the
relevant excitation energies these temperatures are still rather large.
Although sufficiently quick thermalization processes are lacking for cold atoms and photons, lasers with narrow line-widths allow for a completely di↵erent avenue towards the preparation of extremely pure quantum states with topological order. For instance, it was suggested
to use the coherence properties of lasers to directly excite two- (and more) photon Laughlin
(LN) states in non-linear cavity arrays [246], where the laser plays the role of a coherent
pump. However, this approach has the inherent problem of an extremely small multi-photon
transition amplitude. While this might be acceptable for small systems of N = 2, 3 photons, it makes the preparation of true many-body states with N
2 practically impossible.
Moreover, the prepared states in this case contain superpositions of di↵erent photon-numbers
rather than being Fock states.
In this chapter we demonstrate how these limitations of coherent pumps can be overcome.
We suggest an alternative scheme for the preparation of topologically ordered states of strongly
interacting bosons or fermions. The basic idea is to grow the states by adding particles to the
system one-by-one. This is in complete analogy with the growth of crystals, except that instead
of keeping the long-range spatial ordering intact in every step, we add particles such that they
respect the non-local topological order. This is achieved by combining three key elements: a
topological pump creating topological hole excitations; a coherent pump replenishing these
holes; and an interaction blockade ensuring that only one particle is added at a time.
Specifically we discuss LN-type fractional Chern insulators in this chapter. Their topo109
110
CHAPTER 5. TOPOLOGICAL GROWING SCHEME FOR LAUGHLIN STATES
logical order can be understood, for example, from the composite fermion (CF) picture
[141, 142, 143, 26], which we employ to explain our growing scheme. We show how our protocol can be implemented with state-of-the-art technology in the Hofstadter-Hubbard model,
see Sec.1.2.4. In this case the topological pump is realized by local flux insertion in the spirit
of Laughlin’s argument for the quantization of the Hall conductivity xy [173]: Introducing magnetic flux in the center of the system leads to a quantized outwards Hall current
⇠ xy @t , leaving behind a hole, see FIG. 5.2 (a). The hole can be refilled by a local coherent
pump supplemented with an interaction-blockade mechanism.
Previously most of the schemes suggested to prepare topologically ordered ground states
relied on the adiabatic crossing of a topological phase transition in a finite system [247, 14, 125].
If this phase transition is of first order, the excitation gap
scales exponentially with the
system size, ⇠ e N , and the adiabatic schemes are impractical for large N . It was suggested
recently, however, that continuous phase transitions can exist which separate topologically
trivial from non-trivial regions [248] of the phase diagram. In such a case polynomial scaling
of the adiabatic protocols can be achieved, making them useful in practice. A complementary
scheme to prepare LN states in a driven non-equilibrium situation has also been suggested
recently [249].
A key advantage of our scheme is the ability to grow LN states with a size increasing
linearly in time. To reach N particles with given fidelity 1 ", the protocol has to be carried
out sufficiently slow to avoid errors in the re-pumping protocol. For " ⌧ 1 the total required
time scales like
N 3/2
T ⇠
,
(5.1)
1/2
LN "
where LN is the bulk many-body gap. Importantly, T only grows algebraically with N . By
using more than one pair of topological and coherent pumps the performance of our scheme
can be further enhanced.
This chapter is based on the publications [P6], [P11]. It is organized as follows. In Sec.5.2
the growing scheme is introduced using a simplified picture of non-interacting CFs to describe
the fractional quantum Hall e↵ect. In Sec.5.3 we introduce the concrete model which we use
to investigate our scheme in detail. We carry out the analysis in a continuum model in Sec.5.4,
which we generalize to lattice systems in Sec.5.6. We present estimates for the performance of
our scheme in Sec.5.5 and discuss them in connection with possible experimental realizations
in Sec.5.6.3. In Sec.5.7 we close with an outlook.
5.2
Growing quantum states with topological order
We start by presenting the general concept of our growing scheme. To this end we specify
on topologically ordered systems in two dimensions that are characterized by a Chern number1 . Furthermore we assume that interacting states in this setting can be described by the
introduction of non-interacting CFs. In the most important class of systems, the fractional
quantum Hall e↵ects, the CF description was developed in Refs. [141, 142, 143, 26]. In
a nutshell, it states that the fractional quantum Hall e↵ect of interacting particles can be
understood as an integer quantum Hall e↵ect of CFs.
Topological systems characterized by a non-vanishing Chern number come along with
an intrinsic topological – so-called Thouless – pump [87]. To see this, let us express the
1
This includes (1 + 1)-dimensional systems, where one dimension corresponds to an external parameter
determining the Hamiltonian of a system in one spatial dimension. In this case, Chern numbers characterize
quantized topological Thouless pumps [87, 174].
5.2. GROWING QUANTUM STATES WITH TOPOLOGICAL ORDER
111
Chern number as a winding of the polarization P ( ) over the 2⇡ (say) periodic variable ,
see Eqs.(1.21), (1.19). Hence, by changing adiabatically by 2⇡ a quantized current P
is pumped across the system and a topological excitation is created (particle- or hole-type
depending on the sign of the Chern number). In the quantum Hall e↵ect is the magnetic
flux through an infinitesimal solenoid and this argument goes back to Laughlin [173].
Loosely speaking, composite fermions (or bosons) are defined as bound states of bare
particles (either bosons or fermions) to quanta of the flux . That is, particles can only come
along with topological defects (holes) created by the Thouless pump described above. This can
Q
be exemplified using the ⌫ = 1/2 LN wavefunction, LN = i<j (zi zj )2 , where any particle
Q
zi can only appear in the center w = zi of a vortex j6=i (zj w)2 (of topological charge two)
seen by the remaining particles. The binding to such vortices screens the interactions between
the particles, and it can lead to statistical transmutations.
The introduction of composite particles in this way serves two purposes. On the one
hand, it allows a description of the fractional quantum Hall e↵ect as an uncorrelated integer
quantum Hall e↵ect of CFs, see [143, 26] for details. In particular, a 1/m Laughlin (LN)
state is equivalent to an integer quantum Hall state of m 1 CFs, i.e. CFs binding m 1 flux
quanta. On the other hand, the composite-particle description specifies how the many-body
wavefunction is topologically ordered [4]. Now we will make use of both these insights to
construct a growing scheme for topologically ordered LN states.
To keep the topological order intact in every step of the growing scheme, we simply have
to make sure to add CFs (rather than bare particles) to the system. As described above, this
can be achieved by the combination of two elements:
(i) Topological pump: Starting from an N -particle 1/m LN state in a system with N flux
quanta, one additional flux quantum is adiabatically introduced in the center of the
system, N ! N + 1, creating a m 1 CF hole excitation.
(ii) Coherent pump (& blockade): The hole is refilled by a single m 1 CF, i.e. by a bare
particle bound to m 1 flux quanta. First, the m 1 fluxes are introduced by applying
the topological pump (i) another m 1-times, i.e. N + 1 ! N + m. Then a single
fermion or boson is added in the center of the system, N ! N + 1. In the case of
fermions, Pauli blockade ensures that only one particle is added. In the case of bosons
a coherent pump can be combined with an interaction-blockade such that states with
more than one additional particle are energetically forbidden.
These two steps are continuously repeated. The growing scheme is summarized in FIG.5.1 for
the ⌫ = 1/2 bosonic LN state.
The case of the quantum Hall e↵ect discussed so far is special because the Landau level is
completely dispersion-less. More generically, in particular in lattice systems, the lowest singleparticle band has a finite bandwidth E. Although, typically, the flatness ratio f = / E
(where is the band gap) is large, the finite dispersion of the lowest band leads to additional
dynamics in the system which have to be considered in the protocol.
On the one hand, the hole excitation created in the center of the system acquires its
own di↵usive dynamics and starts to propagate outwards. This (a) leads to a dephasing
mechanism for the coherent pump, thereby reducing its performance. On the other hand,
di↵usive dynamics set in at the edge of the many-body system. They lead to di↵usion of
particles into the vacuum surrounding the incompressible liquid which has already been grown.
This may be interpreted as (b) an evaporation at a rate evap ' E 1 , destabilizing the
liquid. Vice versa, (quasi-) hole excitations can di↵use into the liquid from the vacuum. This
(c) makes the liquid compressible.
112
CHAPTER 5. TOPOLOGICAL GROWING SCHEME FOR LAUGHLIN STATES
(b)
(a)
magn.
flux
CF
(c)
Figure 5.1: The key idea of our scheme is to grow LN states by introducing weakly interacting
CFs (a) into the system. This is achieved by adding magnetic flux (arrows) in the center
and replenishing the arising hole by a new boson (red bullet), see (b). The quantum states
grown in this way are shown in the CF basis, see (c). Here, concretely, the scheme for growing
⌫ = 1/2 LN states is shown.
Note that, in a many-body context, it is important to distinguish between the edge of a
system and its boundary. The edge is determined by the quantum state, and it is defined as
the position in space where the particle density vanishes. The boundary, on the other hand, is
determined by the Hamiltonian, and it is defined by the position in space where the confining
potential rises. If the entire system is filled by the liquid, edge and boundary coincide. During
the growing scheme, however, the edge of the system is contained within its boundaries, which
it slowly approaches.
Once the boundary is reached, yet another e↵ect sets in. Because the boundary of the
system supports gapless states connecting Wannier states from the lowest to those of the
higher bands, the topological pump (i) creates excitations in the higher bands. Furthermore,
since the total Chern number of a finite-dimensional Hilbertspace is zero, the pump will (d)
subsequently pump back particles into the center of the system through the higher bands.
This may not only cause modifications of the liquid state grown previously, but once particles
occupy the entire higher bands the topological pump (i) is no longer able to create hole
excitations in the center of the system.
All the e↵ects (a) - (d) destroy our growing scheme in the case of finite flatness ratio f < 1,
and hence have to be suppressed. We suggest the following additions to the ingredients (i)
and (ii) of the growing scheme, which provide generic solutions.
(iii*) Hole trapping: To prevent (a) – di↵usive dynamics of the hole excitations created in
the center – a trapping potential of strength gh can be employed. To overcome di↵usion
gh & E is required, and to keep the topology of the spectral flow intact it has to be
ensured that gh ⌧ (where
is the bulk gap of the incompressible liquid).
(iv*) Smooth confinement: To avoid (b), (c) – evaporation of the liquid – the system can be
placed in a smooth overall confining potential, where the local density approximation
can be applied. A harmonic trapping potential is ideally suited for this purpose, see e.g.
[250] and [P3].
(v*) Dissipative boundaries: To avoid (d) – excitations of higher bands at a sharp edge of
the system – sufficiently strong dissipation channels can be introduced at the edge.
We will now investigate an implementation of our scheme in a specific model of the fractional quantum Hall e↵ect. Realizations of the ingredients (i), (ii) and (iii*) will be discussed
in detail, and we show how (iv*) and (v*) can be dispensed in finite systems without edges.
5.3. MODEL
113
Figure 5.2: We consider the Hofstadter-Hubbard model (flux ↵ per plaquette). Additional flux
can be introduced in the center by adiabatically changing the complex phase of the hoppings
marked with a box. Furthermore, the central site is assumed to be externally accessible for
coupling it to a coherent drive (Rabi frequency ⌦).
5.3
Model
We consider a 2D lattice with complex hopping elements (amplitude J) realizing an e↵ective magnetic field, supplemented by Hubbard-type on-site interactions (strength U ). This
Hubbard-Hofstadter model can be described by the following Hamiltonian (see FIG.5.2),
Ĥint + Ĥ0 =
UX
n̂m,n (n̂m,n
2 m,n
1)
J
Xh
e
i2⇡↵n †
âm+1,n âm,n
m,n
i
+ â†m,n+1 âm,n + h.c. , (5.2)
where n̂m,n = â†m,n âm,n and we used Landau gauge. Implementations of the model are
discussed in Sec.(1.2.4).
Local flux insertion (i) can most easily be realized by changing the hopping elements from
site (m 0, n = 0) to (m, 1) by a factor ei , see FIG.5.2. These links are thus described by
i
Xh
Ĥ = J
e i â†m,1 âm,0 + h.c. ,
(5.3)
m 0
modifying the total magnetic flux through the central plaquette to ↵
/2⇡. Ĥ is motivated
by recent experiments with photons [121, 111], where the hopping-phases can locally and
temporally be manipulated [251].
To replenish the system with boson (ii), we place a weak coherent pump (⌦ ⌧ 4⇡↵J) in
the center,
Ĥ⌦ = ⌦e i!t â†0,0 + h.c..
(5.4)
In the following we present the details of our scheme, neglecting linear losses (rate ) for the
moment. We include losses afterwards in the discussion of the performance of our scheme.
5.4
Protocol – continuum
We begin by discussing the continuum case when the magnetic flux per plaquette ↵ ⌧ 1 is
small, allowing us to make use of angular momentum Lz as a conserved quantum number.
The continuum can be described by Landau levels, which are eigenstates of Ĥ0 in the limit
↵ ! 0 with energies
En = (n + 1/2)!c ,
n = 0, 1, 2, ...
!c = 4⇡↵J,
(5.5)
114
CHAPTER 5. TOPOLOGICAL GROWING SCHEME FOR LAUGHLIN STATES
Figure 5.3: The first three steps of the growing scheme for ⌫ = 1/2 bosonic Laughlin states
are illustrated in the continuum. The many-body eigenstates of the fractional quantum Hall
Hamiltonian (1.46), between which transitions are driven, are shown as a function of the
total conserved angular momentum Lz . Starting from the vacuum |0i, the LN state with
N = 3 bosons is grown. After adding one boson in the central orbital, two flux quanta are
adiabatically inserted and the resulting hole excitation is refilled by a second boson. This
can be interpreted as replenishment of the system by a composite fermion, hence keeping the
topological order of the system intact.
where !p
c denotes the cyclotron frequency, see e.g.[26]. The magnetic length is defined as
`B = a/ 2⇡↵, where a denotes the lattice constant. In symmetric gauge the single particle
states of the lowest Landau level (LLL) are labeled by their angular momentum quantum
number l = 0, 1, 2, ... [26] and we define boson creation operators of these orbitals as
Z
2
2
†
b̂l = Nl d2 z ˆ† (z)z l e |z| /4`B .
(5.6)
Here Nl is a normalization constant and the complex variable z = x + iy parametrizes coordinates x, y in the 2D plane.
Now we discuss the preparation of filling ⌫ = N/N = 1/2 LN states starting from
vacuum |0i, but the generalization to other fillings is straightforward. We start by an explicit
derivation of the first few steps of the protocol, as summarized in FIG.5.3. Then we show
what generalizations are required for large systems.
To create the first excitation from vacuum |0i, we switch on the coherent pump (5.4) and
choose the frequency ! = !c /2 to be resonant with the LLL. When the Rabi frequency ⌦ is
small compared to the cyclotron frequency |⌦| ⌧ !c , we can project the corresponding term
in the Hamiltonian into the LLL, i.e.
Ĥ⌦ ⇡ P̂LLL Ĥ⌦ P̂LLL = b̂†0 e
i!t
(1)
⌦e↵ + h.c..
(5.7)
p
(1)
Here the reduced Rabi frequency is given by ⌦e↵ = ⌦ ↵. Note that only bosons with zero
angular momentum l = 0 can be created by Ĥ⌦ because of the rotational symmetry of the
system when ↵ ! 0.
The pump (5.7) tries to create a coherent superposition of di↵erent boson number states
(b̂†0 )N |0i. This can be prevented by non-linearities induced by boson-boson interactions (5.2),
see e.g. Ref. [252]. Indeed, the two-boson state b̂†2
0 |0i has an eigenenergy (!c + V0 ), where the
contribution from the interactions is given by Haldane’s zeroth-order pseudopotential [133],
(1)
V0 = U ↵/2. Hence if the e↵ective Rabi frequency is sufficiently weak, ⌦e↵ ⌧ V0 , the blockade
prevents occupation of states with more than one boson.
We end up with an e↵ective two-level system consisting of the states |0i and b̂†0 |0i. To
prepare b̂†0 |0i = |LN, 1i – which we can think of as the one-particle LN state – from |0i, a
5.4. PROTOCOL – CONTINUUM
115
(1)
(1)
Rabi-⇡ pulse can be used. To this end the pump is switched on for a time T⇡ = ⇡/2⌦e↵ .
Note that such a scheme is not very robust, but it has the advantage of being quick. We
believe that this is most important to overcome linear losses which are present in realistic
systems. We come to a discussion of this further below.
Next, we adiabatically introduce two units of magnetic flux into the center of the system.
Thereby the initial state |LN, 1i attains two units of angular momentum and we end up in
|2qh, 1i = b̂†2 |0i. This state can be interpreted as a one-particle LN state with two quasiholes
localized in the center. It has a ring-structure with a hole in its center, which – repeating the
first step of our protocol – can now be replenished by an additional boson. Note that above
we assumed that the Hamiltonian (5.3) for flux insertion is for simplicity replaced by one with
a symmetric gauge choice preserving rotational symmetry, see e.g. Ref. [P3]. Because the
vector potential A(t) is modified in time, a di↵erent gauge choice leads to a di↵erent electric
field E = @t A. We treat such a case in an e↵ective model of non-interacting CFs [P11].
The coherent pump (5.7) directly couples to the state b̂†0 |2qh, 1i and, again due to the
interaction blockade, states with more particles are forbidden. Moreover, the pump only
couples to the empty center of the system and can not reduce the total particle number. The
state b̂†0 |2qh, 1i, however, is not an eigenstate of Ĥint and has a finite interaction energy. To
understand why this is the case, we note that there is only precisely one state with in total
N = 2 particles at total angular momentum Lz = 2 which has vanishing interaction energy:
the two-particle LN state |LN, 2i. It di↵ers from b̂†0 |2qh, 1i by an interference term,
|LN, 2i = (z1
2
2
2
z2 )2 e (|z1 | +|z2 | )/4`B = z12 + z22
b̂†0 |2qh, 1i = z12 + z22 e (|z1
|2 +|z
2
|2
)
/4`2B
2
2
2
2z1 z2 e (|z1 | +|z2 | )/4`B
.
(5.8)
(5.9)
Because the two-particle LN state has vanishing interaction energy and lies in the correct
total angular momentum sector accessible by the pump (5.7), it can be prepared. Since
all other states in this sector are gapped by the bulk LN gap LN (which is of the order of
LN ⇠ V0 ), we can again define an e↵ective two-level system within which the pump is acting.
It consists of |2qh, 1i at energy !c /2 and |LN, 2i at !c . In contrast to the first step, because
(2)
of the additional correlations present in the LN state, the e↵ective Rabi frequency ⌦e↵ for
the transition |2qh, 1i ! |LN, 2i is further reduced by a many-body Franck-Condon factor
(FCF),
p
(2)
⌦e↵ = hLN, 2|b̂†0 |2qh, 1i ↵⌦.
(5.10)
The following steps are shown in FIG.5.3.
Having established our protocol for two bosons, the extension to N -particle LN states
|LN, N i is straightforward. In this case, starting from |LN, N 1i, local flux insertion is
used to create the state with two quasiholes |2qh, N 1i. The resulting hole is subsequently
refilled by the coherent pump. It is true for any number of particles N that the total angular
momenta Lz of |2qh, N 1i and |LN, N i coincide. |LN, N i is the unique N -particle ground
state at zero energy in this Lz sector, while all states with N + 1 particles in the same Lz
sector have an energy above the bulk LN gap LN . Hence the coherent pump (5.7) couples
|2qh, N 1i resonantly to |LN, N i in an e↵ective two-level system. The corresponding Rabi
(N )
frequency ⌦e↵ is reduced by a many-body FCF,
(N )
⌦e↵ = hLN, N |b̂†0 |2qh, N
p
1i ↵⌦.
(5.11)
(N )
For the growing scheme to work efficiently, it is crucial that the coupling ⌦e↵ remains
116
CHAPTER 5. TOPOLOGICAL GROWING SCHEME FOR LAUGHLIN STATES
(a)
(b)
1
1.5
0.8
0.6
1
0.4
0
0.5
ED
lin. fit
0.2
0
0.2
0.4
0.6
0.8
1
0
0
ED
quadr. fit
0.2
0.4
0.6
0.8
1
Figure 5.4: (a) The many-body Franck-Condon factor reducing the re-pumping efficiency
|hLN, N |b̂†0 |2qh, N 1i| is shown as a function of 1/N . Calculations were performed in disc
geometry using exact numerical diagonalization. (b) The non-universal factor ⇤N determines
the performance of the coherent pump. We present results from analyzing all states in the
many-body Hilbertspace in the LLL at finite N and using disc geometry.
(N )
finite as N ! 1. At first glance, it may not be obvious that this should be the case: ⌦e↵
is reduced by the overlap of two highly correlated many-body wavefunctions. In addition, as
it stands the coherent pump (5.7) adds bosons in a non-correlated way. Therefore we have
to check that the blockade mechanism, which can enforce local correlations in the prepared
(N )
wavefunction, is actually sufficient to make ⌦e↵ finite in the thermodynamic limit.
To check this, we used exact numerical diagonalization of small systems (N = 1, ..., 9) in
(N )
disc geometry. We calculate |⌦e↵ | for all accessible N , see FIG.5.4 (a), and find that it is
nearly constant as a function of N . In fact it even increases slightly with N . Extrapolation
p
(1)
to N ! 1 yields |⌦e↵ | ⇡ 0.70 |⌦| ↵, and we conclude that our pump works equally well
for large and small boson numbers.
A natural explanation why highly correlated many-body states can be grown in such
a relatively simple fashion is provided by the CF picture: LN states are separable (Slater
determinant) states of non-correlated CFs filling the CF-LLL [143]. Thus, introducing CFs
one-by-one into the orbitals of this LLL, LN states can easily be grown. The key to building
up long-range entanglement in the wavefunction is provided by the flux insertion which creates
global topological excitations.
In our considerations so far we neglected bulk losses, which result in a finite particle life1 . When included in the growing scheme, we expect the mean density ⇢(r) to decay
time
with the distance r from the center. We estimate an equilibrium distribution
✓
◆
1
r2
⇢(r) ⇡
exp
T0 2 ,
(5.12)
4⇡`2B
4`B
with T0 being the duration of a single step of the protocol. In Refs. [P11] and [253] we study
larger systems using a simplified model of non-interacting CFs on a lattice and investigate
the e↵ect of losses (and edge e↵ects) in greater detail.
5.5
Performance
High fidelity is a prerequisite for measuring e.g. braiding phases of elementary excitations,
which play a central role for topological quantum computation [11]. The fidelity of a quantum
5.5. PERFORMANCE
117
state | i grown by our scheme after N steps is defined by its overlap to the targeted LN state,
FN = |h |LN, N i|.
(5.13)
Now we estimate the fidelity achieved by our protocol, including errors in the coherent pump
as well as homogeneous linear losses (rate ). Then we optimize our scheme for obtaining the
highest possible fidelity at given set of system parameters.
In our estimation of the fidelity we take into account couplings between low-energy states
of the N and N + 1 particle sectors, induced by the coherent pump (5.4). We calculate the
population pn in a wrong state | n i after applying the ⇡-pulse perturbatively2 to the leadingorder in ⌦2 / 2n . Here n is the gap of the state | n i from the desired LN state. Moreover, pn
is determined by a many-body Franck-Condon factor, i.e. pn = |FCn |2 ⌦2 / 2n . After summing
over all possible undesired states we obtain an error rate cp of the coherent pump
cp
=
X
pn /T⇡ =
n
⇤2N
2 T3.
LN ⇡
(5.14)
(N )
This expression is valid only for ⇤2N ⌧ 2LN T⇡2 . Here T⇡ = ⇡/2⌦e↵ is the duration of a
⇡-pulse in the growing scheme. ⇤N denotes a non-universal factor which contains the FranckCondon factors and the gaps n / LN weighted by the LN gap, summed over all states n. We
determined ⇤N in the continuum for point interactions in the LLL by using exact numerical
diagonalization. Our resulting finite-size scaling is shown in FIG.5.4 (b), from which we
extrapolate
⇤ = lim ⇤N ⇡ 1.4.
(5.15)
N !1
The errors made during the flux insertion have the same algebraic from and scale with
see Refs. [P11] and [253] for details. Moreover we observe in the exact numerics that
the absolute error made during the flux insertion is smaller than the one from the coherent
pump. This is easy to understand by noting that the cyclotron gap !c ⇠ J is about one
order of magnitude larger than the LN gap LN ⇡ 0.1J in a realistic situation. We take these
observations as motivation to neglect errors due to flux insertion in the following.
2
LN ,
Next we include linear particle losses. Assuming there are N particles in the quantum
state after N steps of the growing scheme, the probability that one of them is lost after N
P
steps is pN = N
1)/2. This leads to an error rate
= (N 1)/2.
n=1 n T⇡ = T⇡ N (N
Note that the decay of a single particle is sufficient to make the resulting state orthogonal to
the desired one, such that the fidelity decays to zero in this case. We assume that the protocol
can not recover from any such error.
By combining both linear losses and pumping errors, we obtain for the fidelity FN =
exp[ ( cp + ) N T⇡ /2]. The additional factor of 1/2 in the exponent takes into account
that the fidelity describes overlaps of the wavefunctions, whereas the error rates correspond
to probabilities (i.e. overlaps squared). Thus we arrive at the following expression for the
2
Let us emphasize here that the algebraic dependence of the error rates on ⌦/ LN is due to the fact that
we have chosen a ⇡-pulse to add bosons to the system by the coherent pump. Had we used an adiabatic
sweep (in the absence of dissipation), a non-analytic expression with exponential error suppression similar to
the Landau-Zener result could be obtained. Our primary objective here is to use a quick protocol in order to
overcome the dominant linear losses. I am indebted to Leonid Glazman for pointing this out. Moreover, it was
shown that in the presence of dissipation, the quantum interference responsible for the exponential suppression
in the Landau Zener formula is destroyed [254]. In that case, errors scale algebraically again. I am indebted
to Michael Fleischhauer for pointing this out.
118
CHAPTER 5. TOPOLOGICAL GROWING SCHEME FOR LAUGHLIN STATES
fidelity of the growing scheme,
FN ' exp
✓ ✓
⇤2
N
2 T 2 + T⇡ 2
LN ⇡
◆
N
2
◆
.
(5.16)
In the second term we neglected sub-leading contributions of order N . Note that a similar
result holds true after inclusion of flux-insertion errors, see Refs. [P11] and [253].
Eq.(5.16) describes a competition between losses / T⇡ and errors of the pump / 1/T⇡2 .
To optimize our protocol we will now fix the desired infidelity " = 1 FN at the end of the
growing scheme. This defines a fixed relation between the duration of one step in the growing
scheme T⇡ and the particle number N (T⇡ ), which for " ⌧ 1 is defined by
✓
◆
⇤2
N N
= ".
(5.17)
2 T 2 + T⇡ 2
2
LN ⇡
!
By optimizing @T⇡ N (T⇡ ) = 0 we find a maximum possible particle number
Nmax = 1.365 "
3/5
✓
LN
⇤
◆2/5
(5.18)
which can be grown at a given infidelity ". The required time to do so calculates to
3/2
T = Nmax T⇡ = 1.22 Nmax
"
1/2
⇤/
LN .
(5.19)
This yields Eq.(5.1) from the introduction.
5.6
Protocol – lattice
To ensure a sizable cyclotron gap !c in an actual experiment [111, 108, 109], not too small
flux per plaquette ↵ ⇡ 0.2 is desirable [14]. In this regime lattice e↵ects become important,
caused by the non-vanishing dispersion of the lowest Chern bands. We will now discuss this
regime in detail and present exact numerical simulations of our protocol for preparing ⌫ = 1/2
LN-type fractional Chern insulators of bosons [14, 102].
The basic ideas for the growing scheme directly carry over from the continuum to the lattice
case. Because the many-body Chern number is strictly quantized, Laughlin’s argument shows
that a hole excitation can still be created by local flux insertion. However, due to the formation
of magnetic sub-bands in the Hofstadter Hamiltonian Ĥ0 (↵) [96], such a quasihole becomes
dispersive and will propagate away from the center. This leaves us only a restricted time to
refill the hole-defect, and leads to a reduced efficiency of re-pumping. As suggested in Sec.5.2
we introduce a trap for quasiholes to circumvent this problem. In addition a smooth harmonic
trapping potential should be superimposed on the lattice in order to prevent evaporation of
the LN liquid. An alternative approach avoiding the additional trapping potential would be
to include carefully chosen long-range hoppings leading to a completely flat band [255].
To trap quasiholes in the center of the system, we suggest to use a repulsive Gaussian
potential of the form
X
g
2
2 2
2
p h
Ĥpot =
e (m +n )a /2`B â†m,n âm,n .
(5.20)
2⇡`B /a
m,n
We note that a simpler, local trapping potential acting only on one lattice site would be
5.6. PROTOCOL – LATTICE
(a)
(b)
N
119
(c)
7.8
7.9
8
8.1
8.2
S
8.3
0
1
2
0
1
2
Figure 5.5: (a) The Hofstadter-Hubbard model on a buckyball lattice: Bosons can tunnel on
the links (amplitude J) and pick up an Aharonov-Bohm phase 2⇡↵ whenever they encircle a
plaquette (consisting of either 5 or 6 sites). Additional magnetic flux can be adiabatically
introduced at the north pole. (b) At = 0 the low-energy spectrum of the N = 3 BuckyballHofstadter-Bose-Hubbard model is shown, where a gapped ground state can be observed.
There is no quasihole trapping potential gh = 0. The number of flux quanta is N = 4 + /2⇡.
When flux is introduced, a quasihole manifold is observed with the correct counting expected
for a Laughlin state. (c) When a weak trapping potential Eq.(5.20) is switched on (here
gh = J), the ground state is gapped for all . U = 10J was used in both cases and for = 0
( = 4⇡) we have ↵ = 0.125 (↵ = 0.1875).
insufficient to trap more than one quasihole. This is easy to see in the continuum limit, and
we verified it numerically in lattices with larger values of ↵ & 0.2. The potential (5.20) can
easily be implemented in photonic systems, where the resonances of individual cavities can
be engineered, and with ultra cold atoms by using the light-shift of a focused laser beam.
To check our model we will now perform a complete numerical simulation of the growing
scheme in a lattice. The limited calculations resources of classical computers restrict us to
small systems of up to N = 3 bosons on about 60 lattice sites. This forces us to make simplifications in order to get reliable results. As a first step we truncate the bosonic Hilbertspace
at a maximum particle number of Nmax  3. As a second step, we eliminate both edgeand boundary e↵ects from the problem. Thus there is no need to add a smooth confinement
potential as described in Sec.5.2.
5.6.1
Buckyball-Hofstadter-Bose-Hubbard model
To eliminate edge e↵ects, we suggest to use a spherical geometry. Because, at the same time,
we want to include lattice e↵ects we simulate the Hofstadter-Hubbard model on a buckyball
lattice, see FIG.5.5 (a). We prefer this to the usual torus geometry – which also eliminates
edges – because it avoids spurious topological degeneracies, which are absent in an infinite
system (disc). Due to the curvature of the sphere, choosing a square lattice is impossible.
We define the model on the buckyball by allowing bosons to tunnel on the links of the
lattice, with amplitude J. The phases of the complex hopping elements are chosen such that
each plaquette encircles ↵ units of magnetic flux. This also leads to Dirac’s famous half-integer
quantization constraint for the total number of flux quanta N 2 Z/2 piercing the surface
[256]. Finally, we add local Bose-Hubbard type interactions to the model and introduce a
Gaussian quasihole trapping potential of the type (5.20) at the north pole.
Furthermore, to eliminate boundary e↵ects also, we choose ↵(N ) as a function of N , such
120
CHAPTER 5. TOPOLOGICAL GROWING SCHEME FOR LAUGHLIN STATES
that an incompressible LN state with N particles exists. The corresponding value of ↵(N )
can easily be determined from the condition that N (N ) = (N 1)/⌫ for filling ⌫ LN states
on the sphere. Together with the observation that the buckyball has Np = 32 plaquettes, it
follows that ↵(N ) = (N 1)/32⌫. In FIG.5.5 (b) we illustrate that, indeed, using N = 3
bosons for this value of ↵ = 0.125 a gapped ground state can be observed. The excitation gap
LN ⇠ 0.1J is of the same order as expected in a square lattice [14].
To create a quasihole excitation by adiabatic flux insertion, the strength of the imaginary
monopole inside the buckyball can be increased. To do so continuously, the magnetic flux ↵n
through the plaquette at the north pole can be decreased, ↵n (t) = ↵ (1 1/Np ) (t)/2⇡,
where (t) changes from 0 to 2⇡ over time. To ensure Dirac’s quantization condition, all
remaining fluxes increase in time, ↵(t) = ↵ + (t)/2⇡Np . If Np ! 1 becomes large, this is
merely a finite-size e↵ect and in the thermodynamic limit only the flux at the north pole ↵n
changes by one unit.
As a second check that the ground state has LN-type topological order (besides the observation of incompressibility at the correct value of N ), we investigate the quasihole counting.
To this end flux is adiabatically introduced into the system, creating a quasihole excitation. In FIG.5.5 (b) the low-energy part of the many-body spectrum is shown as a function
of . After introducing one flux quantum, = 2⇡, we find the correct number of quasihole
states (Nqh = 4 in this case). This is still the case when a second flux quantum is inserted,
= 4⇡, where N2qh = 10. At this point ↵ ⇡ 0.19 is already substantial and observe a sizable
broadening of the two-quasihole manifold, which is absent in the continuum.
Last, we investigate the e↵ect of the quasi-hole trapping potential. To this end we repeat
the flux-insertion calculation for finite gh > 0, starting from the incompressible LN-type state.
By adiabatically increasing the strength of the trap from 0 to gh at = 0 we checked that the
excitation gap remains finite, such that the topological order of the corresponding LN-type
ground state is preserved. Meanwhile FIG.5.5 (c) shows that choosing gh = J the true ground
state of the system is gapped for all values of = 0...4⇡, i.e. two quasiholes can be trapped.
The size of the gap is of the order min
( ) ⇡ LN /2 ⇡ 0.05J. For smaller values of gh . J
we find that ( = 0) is larger whereas ( = 4⇡) is smaller, and vice-versa for gh & J.
5.6.2
Numerical Simulation
As a proof of principle demonstration that our protocol works, we will now present a full
numerical simulation of our growing scheme on the buckyball lattice. To this end we implemented the full Hamiltonian matrix numerically, and solved its dynamics using standard
solvers for ordinary di↵erential equations.
In FIG.5.6 we present our results. We start from particle vacuum |0i and N = 0 flux
quanta. In the first step the coherent pump Eq.(5.4) is switched on for a time T⌦ = 6⇡/⌦
(with ⌦ = 0.05J fixed) and one boson is inserted, with an overlap close to unity to the targeted
N = 1 ground state. The driving frequency ! is chosen to be resonant on the transition from
the N = 0 to the N = 1 ground state. After introducing two more flux quanta in a time
2 ⇥ 20⇡/J, of the order 2⇡/ LN ⇡ 60/J, the whole protocol is repeated twice and we arrive
close to a three particle LN-type ground state. Note that in our numerics we determined the
resonance condition for the coherent pump in each step, whereas the duration of each cycle
T⌦ was kept fixed.
We find that the overlaps of the prepared states to the targeted N particle ground states
|gsN i are close to one after all steps, and the overlaps conditioned on having the correct particle
number N (occurring with probability PN ) are even larger. To check the performance of the
coherent pump, we also calculate the particle number fluctuations [hN̂ 2 i hN̂ i2 ]/hN̂ i. We
5.6. PROTOCOL – LATTICE
121
Figure 5.6: Simulation of the full protocol on a C60 buckyball as described in the text, for U =
10J and including the static potential (5.20) with gh = J. The overlaps (solid, conditioned on
the targeted particle number N - dotted) together with particle-number fluctuations (dashdotted) indicate the accuracy of our protocol.
observe that they are strongly suppressed after completing the first cycle, indicating that the
blockade mechanism is working.
The fidelity of the prepared N = 3 particle LN-type ground state is about F ⇡ 87%, see
FIG.5.6. By comparing to our performance estimate Eq.(5.16) we find that the non-universal
factor ⇤ ⇡ 10, quite a bit above the result in the continuum. We believe that this is partly
due to lattice e↵ects (for example the angular momentum is no longer conserved and there
are thus no selection rules left preventing undesired transitions). Furthermore imperfections
in carrying out the individual steps of the protocol can cause errors (for example we do not
perform perfect ⇡-pulses with the coherent pump). Nevertheless we think that the obtained
fidelity provides a good starting point for experiments aiming to detect the topological order
in the LN-type fractional Chern insulator.
5.6.3
Possible experimental realizations
Our protocol can be implemented in photonic cavity arrays [120, 121, 122, 246, 111], where
the main experimental challenges are the required large interactions U & J and small losses
⌧ LN /N 5/2 . Strong non-linearities can be realized e.g. by placing single atoms into the
cavities [120] or coupling them to quantum dots [111] or Rydberg gases [129, 111] [P1]. Most
promising are circuit-QED systems, where loss-rates = (0.1ms) 1 have been achieved [165]
(and
= 1ms 1 seems feasible). The strong coupling regime can be reached and singlephoton non-linearities U = 100MHz are realistic [257]. For the case when U ⇡ J and for
↵ ⇡ 0.1 the LN gap can be estimated to LN ⇡ 0.05U = 5MHz [102] which corresponds to
3
LN / ⇡ 3 ⇥ 10 . For an infidelity of ✏ = 0.1 this yields Nmax = 7.4 in a continuum system
(Nmax = 3.4 for ⇤ ⇡ 10 as in our simulation). To observe interesting many-body physics
on a qualitative level, ✏ = 0.5 should be sufficient which results in Nmax ⇡ 20 in continuum.
To reach even larger photon numbers, an array of multiple flux and photon pumps could be
envisioned.
Alternatively, our scheme could be realized in ultra cold atomic systems [108, 109], where
large interactions U and negligible decay are readily available [5]. In this case an idea for
realizing local flux insertion would be to use optical Raman beams with non-zero angular
122
CHAPTER 5. TOPOLOGICAL GROWING SCHEME FOR LAUGHLIN STATES
(a)
(b) 1
0.8
0.6
0.4
0.2
0
0
CF
0.05
0.1
0.15
0.2
0.25
magn. flux
Figure 5.7: (a) We expect that our growing scheme can be generalized to grow the Pfaffian
fractional quantum Hall state. It can be interpreted as a p-wave superfluid of composite
fermions. (b) Preliminary calculations of the many-body Franck-Condon factors relevant
for growing Pfaffian states with non-Abelian topological order. Contact interactions were
considered in lowest Landau level approximation and using the spherical geometry.
momentum [258], or as an alternative quasiholes could be introduced by placing a focused
laser-beam close to the edge of the system and increasing its intensity adiabatically [259].
5.7
Outlook – Beyond Laughlin states
Our scheme is not restricted to the preparation of Laughlin states. For example, we expect that
the ⌫ = 1 bosonic Moore-Read Pfaffian [7, 8, 101] supporting non-Abelian topological order,
can also be grown using our technique. In this case, too, our starting point is the CF picture.
The Moore-Read Pfaffian state can be interpreted as a topological p-wave superconductor
composed of Cooper pairs built from CFs. By introducing magnetic flux quanta into the
system locally, (quasi-) hole excitations can be created which may be refilled by additional
bosons.
We summarize the results of our preliminary calculations in FIG.5.7. We calculated
Franck-Condon factors determining the efficiency of possible coherent pumps for re-filling
hole excitations. We considered two routes, a direct two-boson transition and one with an
intermediate one-boson state. As in the case of Laughlin states, they do not drop for large N .
A closer examination of the growing scheme for non-Abelian states would be very interesting,
and it might even shed more light on their topological order.
Another interesting generalization of the growing scheme presented in this section is the
possibility of preparing bosons in higher Landau levels. This opens the possibility to simulate
exotic Haldane pseudo-potentials appearing in higher Landau levels. They mimicking the
e↵ect of long-range interactions without the need to implement these in first place.
Part II
Interferometry-based Detection of
Topological Invariants
123
Chapter 6
Introduction
6.1
Outline
In this second part of the thesis the question will be addressed how topological invariants
can be detected. By its very definition, topological order classifies quantum states by their
global properties. Therefore local measurements, which are often easiest to implement in
actual experiments, are insufficient to distinguish between topologically distinct phases. For
experiments aiming to realize quantum states with topological order, as described e.g. in
the first part of this thesis, this poses a serious problem. Without an efficient detection
mechanism for topological order at hand, it is hard to implement such a state. Even if this
is achievable, however, it is impossible to proof that the implementation has been correct. In
fact, the search for an isolated Majorana fermion in one-dimensional semiconducting nanowires
[40, 41, 42, 43, 44] is currently challenging precisely this fundamental problem. Although it
is generally believed that the correct model Hamiltonians have been implemented in the
experiments, it is impossible at this point to unambiguously detect an isolated Majorana
fermion.
Several di↵erent approaches exist how topological order can be detected. The first class
consists of measurements which essentially search for indicators of non-trivial topology rather
than attempting a direct detection. One example is the detection of edge states [260], the
characteristics of which are linked to the underlying topological order of the bulk through
the bulk-boundary correspondence. An other example is the detection of the topological
entanglement entropy [261]. A second class of approaches is based on bulk transport properties. For example the Chern number is directly related to the linear response xy of a
quantum Hall system to an applied electric field. This approach is not always applicable,
however, as the example of Z2 topological insulators with spin-orbit coupling demonstrates
[28, 262]. Furthermore, for systems with intrinsic topological order classified by category theory, several measurement methods attempt to directly observe non-Abelian braiding statistics
[263, 259, 264].
In this part of the thesis we continue developing a more direct method for the detection
of topological order. It is based on the use of interferometry in (quasi-) momentum space to
realize a measurement of topological invariants, which is in one-to-one correspondence with
their definition. In this part of the thesis we restrict ourselves to topological orders which can
be characterized through (Abelian or non-Abelian) geometric phases1 . The original idea was
to combine Bloch oscillations with Ramsey interferometry in order to measure geometric Zak
1
The author is not aware of any topological order in non-interacting systems that can not be characterized
by geometric phases.
125
126
CHAPTER 6. INTRODUCTION
phases [13]. Making use of their relation to the Chern number, see Sec.1.2.2, this furthermore
enables a direct measurement of the Chern number [200]. The interferometric approach is
particularly well suited to measure topological invariants in systems of ultra cold atoms. The
cleanliness of such systems along with the full coherent control gained in modern experiments
allows for long coherence times and makes interferometry easily accessible. Furthermore, by
the use of the time-of-flight method, observables in (quasi-) momentum space can be detected.
This provides an elegant way to implement non-local measurements.
The goal of this part of the thesis is two-fold. In a first step, we generalize the interferometric measurement scheme for topological invariants [13] to non-Abelian topological invariants.
In particular we discuss how Z2 invariants characterizing the quantum spin Hall e↵ect can
be directly measured in this way, in two and three spatial dimensions [P4]. As described
so far, the interferometric scheme is restricted to non-interacting systems with translational
symmetry. In the second step we relax these assumptions and generalize the protocol to
the measurement of many-body topological invariants in interacting systems. Concretely we
present a method for the detection of geometric phases characterizing topological excitations
in interacting many-body systems.
For the measurement of Z2 topological invariants we introduce two protocols. In both
cases the dynamics of atoms within multiple bands has to be kept track of. The first scheme
is based on a gauge-invariant expression for the Z2 invariant using U (2) Wilson loops [94]. For
the second scheme we derive a new formula for the Z2 invariant, expressing it as the winding
of the so-called continuous time-reversal polarization over half the Brillouin zone (BZ). We
show how continuous time-reversal polarization can be measured with ultra cold atoms.
To generalize the interferometric measurement scheme to interacting many-body systems
we introduce single impurity atoms as coherent probes. They are coupled to elementary
topological excitations (e.g. quasiholes) leading to the formation of so-called topological polarons. By carrying over the interferometric detection scheme to these topological polarons,
the topology of their e↵ective band structure can be determined. We discuss in detail how
this topology is related to the underlying topological order of the incompressible groundstate
in the case of Laughlin states.
This part of the thesis is organized as follows. In the remainder of this first chapter we
introduce the fundamental concepts underlying the interferometric detection scheme for topological invariants. This includes a brief review of the Z2 invariant and time-reversal invariant
topological insulators in general. Next, in Chap. 7.2 we develop an interferometric scheme for
the measurement of Z2 topological invariants. Implementations in ultra cold quantum gases
are discussed. We generalize our approach to interacting many-body topological invariants
in Chap.8. There we discuss how symmetry-protected invariants can be measured in onedimensional systems, and how intrinsic topological order can be detected in two-dimensional
settings.
6.2
Fundamental Concepts
Now we introduce the fundamental concepts underlying the interferometric detection scheme
for topological invariants. We start by a brief review of the interferometric measurement
of Zak phases [13]. The generalization to Chern numbers, and two-dimensional systems in
general, is also discussed. In Sec.6.2.2 we continue our discussion from Sec.1.2.3 and introduce
the Z2 topological invariant. We focus, in particular, on its gauge-invariant representations
which underly our measurement protocols.
6.2. FUNDAMENTAL CONCEPTS
6.2.1
127
Interferometric Measurement of Topological Invariants
Measurement of the Zak phase
Consider weakly interacting ultra cold bosons loaded into a one-dimensional (1D) band structure. At sufficiently small temperatures they will condense in the band minimum, forming a
Bose Einstein condensate (BEC). To measure the Zak phase of the corresponding 1D Bloch
wavefunction, see FIG.6.1 (a), a superposition of the BEC in two internal spin states is created.
Note that a macroscopic condensate is used here for the investigation of the single-particle
band structure merely to enhance the experimental signal. The condensate behaves as a
macroscopic quantum mechanical particle.
The fundamental idea of the interferometric measurement technique for Zak phases is
to apply a magnetic field gradient, leading to opposite forces acting on the spin states. As
a consequence they undergo Bloch oscillations in opposite directions through the BZ. By
recombining them at the edge of the BZ, a Ramsey interferometer is created which measures
the geometric Zak phase picked up by the two wave packets.
In Ref.[13] this scheme was implemented in an experiment realizing the band structure of
the Su-Schrie↵er-Heeger model [12], see also Chap. 2. Because of its inversion symmetry, the
Zak phase measured in this model is quantized to 'Zak = 0, ⇡, which was confirmed by the
experiment. Now we will give a brief summary of some technical details of the interferometric
scheme that had to be considered for the experiment.
The first point concerns the dependence of the Zak phase on the choice of the unit cell,
see Sec.1.2.2. As pointed out below Eq.(1.20) the Zak phase can also be interpreted as the
expectation value of the potential energy, 'Zak = T hF x̂i (integrated over time T ) when the
force F is applied to drive Bloch oscillations. Hence the value of the Zak phase expected in the
experiment is determined by the position in space where this potential vanishes. Concretely,
in the experiment [13] it depends on the zero of the applied magnetic field.
In a realistic experiment magnetic field fluctuations are unavoidable, leading to a fluctuating value of the Zak phase by the argument above. To eliminate such fluctuations from the
measured signal, Atala et al. [13] have implemented a more involved interferometric sequence.
Instead of recombining spins at the edge of the BZ, they apply a quench which switches the
dimerization of the model. Afterwards the atoms are moved back into the center of the
BZ, where the spin-states are recombined. The obtained phase thus measures the di↵erence
'Zak = 'D1
'D2
Zak
Zak of the Zak phase corresponding to the two di↵erent dimerizations D1,
D2.
'Zak = ⇡ is still quantized by the inversion symmetry, and it is independent of the
choice of the unit-cell2
The second point concerns dynamical phases which are picked up besides the desired
geometric phase. In the setup of Ref. [13] two dynamical phases contribute. Firstly, when
moving through the BZ atoms pick up a dynamical phase which is determined by the shape
of the dispersion relation ✏(k). This phase can be eliminated by using an inversion-symmetric
model where ✏(k) = ✏( k) and implementing a symmetric protocol (i.e. starting from the
band minimum at k = 0 and using exactly opposite forces ±F acting on opposite spins).
In this case dynamical phases picked up by opposite spin components are exactly equal and
cancel each other at the end. Secondly, the spin components pick up opposite dynamical
2
Note that the dimerization-switching process takes place non-adiabatically. Thus atoms are excited into
the higher band of the Su-Schrie↵er-Heeger model in the middle of the protocol. Therefore, precisely speaking,
the Zak phase 'D2
Zak measured during the second half of the protocol corresponds to the excited Bloch band
of the Su-Schrie↵er-Heeger model. However, in a any two-band model the sum of the Zak phases of the two
bands is trivia, '1Zak + '2Zak = 0 mod 2⇡. Hence, up to multiples of 2⇡ which can not be detected, 'D2
Zak is the
same for both Bloch bands.
128
CHAPTER 6. INTRODUCTION
(a)
1D
(b)
2D
(c)
2D - TR
Figure 6.1: A combination of Ramsey interferometry with Bloch oscillations allows interferometric measurements of topological invariants in bulk topological insulators: (a) 1D systems
(whose first BZ is depicted here) are classified by the geometric Zak phase. (b) The Chern
number classifies 2D systems (again the first BZ is shown) and its relation to the Zak phase
can be used for its measurement. (c) Time-reversal (TR) invariant 2D systems are classified
by the winding of time-reversal polarization P̃✓ (precise definition is given in Eq.(7.2) below)
which can be measured as a Zak phase along twisted paths in the BZ [P4]. These twists
correspond to Rabi ⇡-pulses applied between the two bands. The upper half of the 2D BZ is
depicted here.
Zeeman phases ±T gµB B, where T is the duration of the protocol, µB is the Bohr magneton
and g is the g-factor. To cancel this dynamical contribution, Atala et al.[13] have implemented
an additional ⇡-pulse to flip the spins at the edge of the BZ. Therefore Zeeman phases in the
first and second part of the protocol are equal but opposite and cancel each other in the final
signal.
Measurement of the Chern number
The Chern number characterizes the topology of Bloch bands in two dimensional lattices. As
introduced in Eq.(1.21) it is given by the winding of the Zak phase over the BZ. Hence, by
measuring the Zak phase 'Zak (ky ) for paths crossing the BZ at quasimomentum ky the Chern
number can be extracted, see FIG.6.1 (b). This idea was proposed by Abanin et al.[200],
where a detailed discussion can be found. More generally this scheme allows to map out the
Berry curvature, which has lead to the recent observation of the quantized ⇡ Berry phase
picked up by atoms encircling a Dirac cone [201].
6.2.2
Z2 topological invariant
In this section we review di↵erent formulations of the Z2 topological invariant, which characterizes time-reversal (TR) invariant band structures of half-integer spin systems. For concreteness we consider spin one-half systems, where the TR operator takes the form
✓ˆ = Kiˆ y
(6.1)
with K denoting complex conjugation. ✓ˆ is an anti-linear operator, i.e. ✓ˆ (↵| 1 i + | 2 i) =
ˆ 2 i, because of the complex conjugation K. Moreover it is anti-unitary, i.e.
↵⇤ | 1 i + ⇤ ✓|
†
ˆ
ˆ
✓ ✓ = 1. Most importantly, because ✓ˆ2 = 1, Kramer’s theorem applies:
6.2. FUNDAMENTAL CONCEPTS
129
Figure 6.2: The evolution of Wannier centers (solid and dashed lines respectively) in a twodimensional Brillouin zone is shown, as a function of ky , in di↵erent physical situations. (a)
Two time reversed copies (labeled I, II) of a Chern insulator: The reversed copy (dashed lines)
carries a Chern number of opposite sign, CII = CI = 1. (b) Time-reversal (TR) invariant
topological insulator: At TR invariant momenta (TRIM) kyTRIM = 0, ⇡ each Wannier center
(solid lines) has a degenerate Kramers partner (dashed lines). In the upper half of the BZ
di↵erent Kramers partners evolve independently in general. (The lower half of the BZ is
obtained by reflecting on the x-axis and exchanging solid and dashed codes, see (a).) In
this topologically non-trivial case, Wannier centers change partners when going from ky = 0
to ky = ⇡. (c) Symmetry protected topology: When additional symmetries are present,
Wannier centers can change partners at intermediate 0 < ky < ⇡ (left). When all symmetries
except TR are broken, Wannier centers can not exchange partners except at TRIM (right).
This situation is topologically trivial and it illustrates why the quantum spin Hall phase is
characterized by a Z2 invariant only.
For half-integer spin systems with TR symmetry all eigenstates are (at least) two-fold degenerate. If | i is an eigenstate with eigenenergy ✏ of a TR invariant Hamiltonian ✓ˆ† Ĥ✓ˆ = Ĥ,
ˆ i is a non-equivalent eigenstate of Ĥ with the same energy ✏.
then ✓|
ˆ i is non-equivalent (i.e. nonThe only non-trivial part of this theorem is to show that ✓|
ˆ i = | i for
parallel) to | i. To this end, let us assume the two states were equivalent, ✓|
2
ˆ
ˆ
some 2 C. By applying ✓ on both sides of this equation and using ✓ = 1 it is easy to
show that this implies | |2 | i = | i, which is a contradiction because | |2 > 1.
The quantum spin Hall phase was constructed by Kane and Mele [28] starting from two
uncoupled time-reversed copies (spin " and #) of a Chern insulator, each realizing the quantum
Hall e↵ect. The corresponding Wannier centers, encoding the underlying topological order,
are sketched in FIG.6.2 (a). Since time-reversal inverts the quasimomentum ky but not x,
the Wannier centers of the second spin are obtained from those in FIG.1.2 (a) by reflection
around the x-axis. Consequently the Chern numbers have opposite signs and cancel to give
a vanishing total Chern number. Meanwhile, the underlying topology of the system can be
classified by the di↵erence of the two Chern numbers,
⌫2D =
1
(C"
2
C# ) .
(6.2)
The Z2 invariant is a generalization of this expression to situations where spin-orbit coupling
(SOC) mixes di↵erent spin states.
130
CHAPTER 6. INTRODUCTION
Z2 invariant and time-reversal polarization
In the generic case with SOC mixing the spins ", #, spin is no longer a good quantum number,
[Ĥ, ˆ z ] 6= 0. In this case we will label the two energetically lowest Bloch bands by I and II in
the following. As a consequence of TR (Kramer’s theorem) they are related by
|uII ( k)i = ei
(k) ˆ
✓|uI (k)i.
(6.3)
Here the phase (k) accounts for the independent gauge degree of freedom at opposite quasiTR
momenta ±k in the BZ, and we used that by TR momenta are inverted, k ! k.
The two bands I and II are characterized by a Z2 topological invariant ⌫2D , which was
originally formulated by Kane and Mele using a classification scheme for real vector bundles
[28]. Fu and Kane pointed out in [265] that, like the Chern number, ⌫2D can be understood
from the topology of the Wannier centers. To see how this works, let us first discuss a generic
TR invariant band structure (see also FIG.6.1 (c) below). TR invariance requires the Bloch
Hamiltonian Ĥ(k) to fulfill
✓ˆ† Ĥ(k)✓ˆ = Ĥ( k).
(6.4)
As a consequence there are two 1D subsystems at fixed kyTRIM = 0, ⇡ (referred to as timereversal invariant momenta, TRIM) which are TR invariant as 1D systems, i.e. ✓ˆ† Ĥ(kx )✓ˆ =
Ĥ( kx ). Within these two 1D systems there are in total four momenta kTRIM = (kxTRIM , kyTRIM )
(also referred to as TRIM) where the Bloch Hamiltonian is TR invariant itself, ✓ˆ† Ĥ(kTRIM )✓ˆ =
Ĥ(kTRIM ). At these four points kTRIM Kramers theorem requires eigenvalues to come in degenerate pairs.
FIG. 6.2 (b) illustrates the corresponding Wannier centers for a generic – but topologically
non-trivial – case. The underlying TR symmetry requires Wannier centers to come in Kramers
pairs at TRIM kyTRIM = 0, ⇡, again as a consequence of Kramers theorem. When these
Kramers pairs switch partners upon going from ky = 0 to ky = ⇡ the system is topologically
non-trivial, while it is trivial otherwise [265].
Using the change of polarizations of the two states P I,II as indicated in FIG.6.2 (b), we
see that the topology is described by the integer invariant P✓ = P I
P II . Fu and Kane
[265] coined the name time-reversal polarization (TRP) for the quantity
P✓ (ky ) = P I (ky )
P II (ky ).
(6.5)
Using their language, the Z2 invariant is given by the change of TRP over half the BZ, i.e.
⌫2D = P✓ (⇡)
P✓ (0)
mod 2.
(6.6)
It can easily be checked that the TRP P✓ (ky ) is integer quantized for ky = kyTRIM = 0, ±⇡.
In TR constrained gauge, where (k) = 0 is chosen in Eq. (6.3), P I = P II at kyTRIM = 0, ±⇡ as
a direct consequence of TR symmetry Eq.(6.3). Since gauge transformations can only change
polarizations by an integer amount it follows that in a general gauge
P✓ (kyTRIM ) 2 Z,
kyTRIM = 0, ⇡.
(6.7)
Importantly, in the definition (6.6) a continuous gauge choice has to be used in the entire
BZ, since otherwise P✓ (⇡) and P✓ (0) could independently be changed by discontinuous gauge
transformations. Note that such a gauge choice is possible when the total Chern number
vanishes [62]. We conclude from TR symmetry that this is indeed the case, CI + CII = 0.
6.2. FUNDAMENTAL CONCEPTS
131
Finally we discuss why only a Z2 classification survives. To this end we note that for a
general Hamiltonian without accidental degeneracies, TRP can only change by P✓ = 0, ±1
between ky = 0, ⇡. This is because otherwise there exists some intermediate ky 6= 0, ±⇡ with
P I = P II , and as pointed out by Yu et.al.[94] small TR invariant perturbations can split this
degeneracy (of polarizations) away from Kramers degeneracies, see FIG.6.2(c). Moreover since
P = 1 and P = +1 only di↵er by exchanging up and down spins, they are topologically
equivalent. Therefore the topological invariant can only take two topologically distinct values
P = 0, 1 and we end up with Eq.(6.6).
Z2 invariant and Wilson loops
In Chap.1.2.3 we introduced U (M ) Wilson loops as natural generalizations of Abelian Zak
phases to multiple bands. The authors of [94] derived various formulas for the Z2 invariant,
expressing it in terms of U (2N ) Wilson loops. Here we consider the case N = 2 and we will
focus on one particular relation which reads
✓
◆
Z
1
1 ⇡
⌫2D =
'W
dky @ky (ky )
mod 2.
(6.8)
⇡
2 0
The terms on the right hand side are related to eigenvalues of Wilson loop operators as
explained in the following. In Appendix C a proof is given for Eq.(6.8). We also note that
Wilson loops have proven useful as a tool to classify other symmetry protected topology [266].
For the discussion of the Z2 invariant, TR invariant Wilson loops play a special role. (With
TR invariant Wilson loops we mean Wilson loops at TRIM.) Such TR invariant U (2) Wilson
loops reduce to U (1) phase factors [94],
ŴTR = e
i'W
Î2⇥2
(6.9)
as a consequence of Kramers theorem. 'W will be referred to as the Wilson loop phase.
Since Eq.(6.9) will be important later on, we quickly prove it here. To this end we choose
a special gauge where (k) = 0 in Eq.(6.3) (known as the TR constraint [265]). In this gauge
one has ✓ˆ† Â(k)✓ˆ = Â( k) which leads to ✓ˆ† Ŵ ✓ˆ = Ŵ † . Since Wilson loops are gauge invariant
this holds for an arbitrary gauge. Moreover it implies doubly degenerate eigenvalues: Assume
ˆ = ✓ˆŴ † |ui = e i'W ✓|ui
ˆ
Ŵ |ui = e i'W |ui and thus also Ŵ † |ui = ei'W |ui. Therefore Ŵ ✓|ui
ˆ
and besides |ui also ✓|ui
is eigenvector of Ŵ . These two eigenvectors can not be parallel
ˆ
however, i.e. we can not write ✓|ui
= ⌧ |ui with a complex number ⌧ 2 C, since this would
2
⇤
2
ˆ
ˆ
imply |ui = ✓ |ui = ⌧ ✓|ui = |⌧ | |ui =
6
|ui.
For the first term in Eq.(6.8) we employ Eq.(6.9) and write
Ŵ (kyTRIM ) = e
In Eq.(6.8) the Wilson loop phase di↵erence
i'W (kyTRIM )
Î2⇥2 .
'W appears, which is defined as
'W := 'W (⇡)
'W (0).
The second term in (6.8) is the winding of the total Zak phase,
Z ⇡
(ky ) := tr
dkx Âx (k) ⌘ 'IZak (ky ) + 'II
Zak (ky ),
⇡
across half the BZ in ky direction.
(6.10)
(6.11)
(6.12)
132
CHAPTER 6. INTRODUCTION
Alternative formulations of the Z2 invariant
Now we comment on alternative formulations of the Z2 invariant. In [265] the Z2 invariant
was expressed as an obstruction to continuously defining a gauge in the BZ. This leads to a
formulation of ⌫2D entirely in terms of Berry’s connection and Berry’s curvature (see [265])
which, however, is valid only when the TR invariant gauge (i.e.
(k) = 0 in Eq.(6.3))
is used. We emphasize that the formula Eq. (6.8) we discussed here also only involves
Berry’s connections, but without any restriction of the gauge. The relation between the
two expressions is shown in Appendix F of [P4]. Finally the Z2 invariant is also related
to the systems response to spin dependent twisted boundary conditions, which lead to the
classification in terms of a Chern number matrix [262].
Z2 invariant of a 3D system
In 3D two kinds of topological invariants exist [32]. There is one strong topological invariant,
which is protected against TR invariant (non-magnetic) disorder. It can be written as a
product of 2D invariants for subsystems at di↵erent kz = 0, ⇡:
( 1)⌫3D = ( 1)⌫2D (kz =0) · ( 1)⌫2D (kz =⇡) .
(6.13)
On the other hand, there are also 3 additional weak topological invariants which are not
protected against any kind of disorder. They may as well may be formulated in terms of 2D
invariants of di↵erent subsystems:
( 1)⌫i = ( 1)⌫2D (ki =⇡) ,
i = x, y, z.
(6.14)
Consequently, measuring 3D Z2 invariants only requires the measurement of the Z2 invariants
of di↵erent 2D subsystems within the 3D BZ.
Chapter 7
Interferometric Measurement of Z2
Topological Invariants
7.1
Outline and Introduction
Cold atom experiments o↵er a large degree of control, making them an ideal playground for
the quantum simulation of paradigmatic models previously encountered only in solid state
systems. The precise tunability of model parameters over a wide range [5] fuels the hope to
explore new states of interacting bosons or fermions in the future. Besides the fractional Chern
insulators discussed in Sec.1.2.4, TR invariant topological insulators are promising systems to
search for new physics with ultra cold atoms.
For the simulation of the quantum spin Hall e↵ect (QSHE) – or, more generally, a
Z2 topological insulator – with ultra cold atoms, artificial spin-orbit coupling (SOC) is required. This has recently been implemented experimentally [267, 268] using similar approaches
as developed for the generation of artificial gauge fields. Subsequently, di↵erent SOC schemes
have lead to several proposals for the implementation of two [269, 270, 271, 272] and three
dimensional [271] TR invariant topological insulators. In a recent experiment by Aidelsburger
et al. [108] Abelian SOC has successfully been implemented, which is sufficient for a realization of the QSHE. Also the recent experiment at MIT by Miyake et al. [109] allows an
implementation of Abelian SOC, see [273].
Motivated by these exciting experimental developments, we propose measurement schemes
for Z2 topological invariants in TR invariant topological insulators of ultra cold atoms in two
and three dimensions. Our method uses one of the most important technical strengths of
cold atom experiments, the ability to perform interferometric measurements. This goes to
the heart of topological states, whose topological nature is encoded in the overlaps of Bloch
wavefunctions. We discuss formulas relating the Z2 invariant to simple non-Abelian Berry
phases and show how they can be measured.
In this chapter, we generalize the ideas of Refs.[13, 200] to interferometric measurements
of Z2 invariants in TR-symmetric optical lattices. The key challenge in this case is to keep
track of two Kramers degenerate bands, related by TR symmetry. Defining the topological
properties of such bands requires understanding how Bloch eigenstates in the two bands
relate to each other. We argue that the Bloch/Ramsey sequence used in Ref.[13] should
be supplemented by band switchings, as shown schematically in FIG.6.1 (c). The obtained
interferometric signal not only depends on the phase accumulated when adiabatically moving
within a single band, but also on the phase picked up during the transition from one band to
the other. Hence this phase, too, has to be taken into account. Experimentally the described
133
CHAPTER 7. INTERFEROMETRIC MEASUREMENT OF Z2 TOPOLOGICAL
INVARIANTS
134
band switchings can be implemented by applying an oscillating force at the frequency matching
the band energy di↵erence. The relevant band gap can be measured experimentally using e.g.
Stückelberg interferometry1 . We show that, when applying this particular band switching
protocol, a geometric phase for the Bloch cycle is obtained, the winding of which (over half
the BZ) yields the Z2 invariant.
We also present an alternative approach based on measurements of Wilson loops. Their
eigenvalues are directly related to the Z2 invariant, as was shown by Yu et al. [94]. The
measurement of Wilson loops requires moving atoms non adiabatically in the BZ in two
directions and relies on keeping track of two-band dynamics of atoms. We show how this can
be achieved using currently available experimental techniques.
Other methods suggested to investigate topological properties of cold atom systems mostly
focused on detecting characteristic gapless edge states [274, 275, 276, 260, 250]. Even for
typical smooth confinement potentials present in cold atom systems a theoretical analysis
showed [250] that these edge states should still be observable, although steeper than harmonic
potentials are favorable. To detect Z2 topological phases of cold atoms, a spin-resolved version
of optical Bragg spectroscopy was suggested in Ref. [270]. A di↵erent approach to measure
Chern numbers makes use of the Streda formula, relating them to the change in atomic
density when a finite magnetic field is switched on [277, 278]. Extensions of this method for
the detection of Z2 topological phases were suggested [269, 270], however they only work in
cases when the Chern numbers for individual spins are well-defined (which is generally not
the case [262]). Recently also an interferometric method has been suggested to measure the
Z2 invariant of inversion-symmetric TR invariant topological insulators [279]. Our method in
contrast does not rely on any assumptions about the system’s symmetry other than TR.
This chapter is based on the publication [P4] and will be organized as follows. In section
7.2 we explain the basic idea of our measurement schemes. In section 7.3 the first of our two
measurement schemes (twist scheme) is presented. The experimental realization of this scheme
is discussed and we show that it can easily be implemented in the experimental setup proposed
in [270]. In section 7.4 we present the Wilson loop scheme and discuss its experimental
feasibility. Finally in section 7.5 we conclude and give an outlook how our scheme can easily
be generalized to measure topological invariants of three dimensional topological insulators.
7.2
Interferometric measurement of the Z2 invariant
In the following we introduce the basic ideas of our measurement protocols for the Z2 topological
invariant. In a nutshell, we try to express the Z2 invariant in terms of simpler Zak phases
which can be measured experimentally [13]. Our starting point will be the formulation of the
Z2 invariant as the change of TRP over half the BZ. Although this expression contains only
Zak phases, we will show now that it is insufficient for a measurement of the Z2 invariant
because TRP is discontinuous at the TR invariant momenta kyTRIM = 0, ±⇡. To circumvent
this problem we develop two alternative approaches.
In this entire chapter we consider a generic TR invariant band structure, as shown in
FIG.7.1 (a), with Kramer’s degeneracies at TR invariant momenta. In principle there can
be additional accidental degeneracies of the two bands I, II. However in this chapter we
will restrict ourselves to the simpler case without any further degeneracies besides the four
Kramers degeneracies.
1
I am grateful to Immanuel Bloch and his group for explaining this scheme to me.
7.2. INTERFEROMETRIC MEASUREMENT OF THE Z2 INVARIANT
135
Figure 7.1: (a) Typical band structure at TRIM kyTRIM = 0, ⇡, consisting of two Kramers
partners I and II (red and blue lines respectively). During Bloch oscillations the Zak phases
'I,II
Zak are picked up. (b) When small TR breaking terms are present away from the TR invariant
momenta kyTRIM = 0, ⇡, Kramers degeneracies become avoided crossings. (In principle there
can be additional accidental degeneracies of the two bands I, II. However in this chapter we
will restrict ourselves to the simpler case without any further degeneracies besides the four
Kramers degeneracies.) The band labels were chosen such that I (II) denotes the energetically
upper u (lower l) band. The color code indicates the similarity to the corresponding bands
I, II at ky = 0: while band I at kx = ⇡/2 is similar to band I at ky = 0, band I at kx = ⇡/2
is similar to band II at ky = 0. This illustrates why TRP is discontinuous as a function of
ky around ky = 0, ⇡. To obtain a continuous version of TRP the twist scheme introduces ⇡
pulses (green) in the middle and at the end of the Bloch oscillation cycles. Then atoms follow
the twisted paths i (gray dashed) and ii (gray dotted). For ky = 0 (a) twisted paths coincide
with the bands i = I and ii = II, while for ky 6= 0 (b) twisted paths i, ii are a mixture of I, II.
7.2.1
Discontinuity of time-reversal polarization
Naively one might think that, with the formulation of ⌫2D Eq.(6.6) entirely in terms of polarizations (i.e. by (1.19) in terms of Zak phases), we have an interferometric scheme at hand.
According to Eqs.(6.5) and (6.6) it would be sufficient to measure the di↵erence of Zak phases
'IZak (0) at ky = 0 and 'IZak (⇡) at ky = ⇡ and repeat the protocol for the second band II. Zak
phases, however, can only be measured up to 2⇡. Typically the problem of 2⇡ ambiguities
of Zak phases can be circumvented by rewriting their di↵erence as a winding over some continuous parameter. As pointed out previously, this strategy works out for the case of Chern
numbers, see Sec.6.2.1.
R
However we can not simply replace the change P✓ of TRP by its winding dky @ky P✓ (ky ),
because TRP is not continuous over the BZ. This discontinuity is a direct consequence of
Kramers degeneracies: Let us consider the Zak phase 'IZak (0) at kyTRIM = 0, see FIG.7.1 (a).
According to Eqs.(1.16), (1.9) 'IZak (0) is determined by the Berry connection AI (kx , 0) within
band I (note that band I crosses band II at the two Kramers degeneracies.) Now let us imagine
going to some slightly larger 0 < ky ⌧ 2⇡ and measure the Zak phase of band I here, see
FIG.7.1 (b). Because there is no longer any true band crossing, we now always have to follow
the energetically upper band. This means however, that the Zak phase 'IZak (ky ) is determined
by the Berry connection AI (kx , ky ) ⇡ AI (kx , 0) from kx < 0 and by AI (kx , ky ) ⇡ AII (kx , 0)
(note the exchanged index!) from kx > 0 2 . Then, because in general AI (k) 6= AII (k), we
obtain a very di↵erent result, 'IZak (ky ! 0) 9 'IZak (0) in general.
2
We can assume A(k) to be continuous on the small patch [ ⇡, ⇡) ⇥ [0, ky ] in the BZ, with 0 < ky ⌧ 2⇡.
136
CHAPTER 7. INTERFEROMETRIC MEASUREMENT OF Z2 TOPOLOGICAL
INVARIANTS
Let us add that as a consequence of the discontinuity of TRP, the meaning of Wannier
centers in FIG.6.2 (a)-(c) has to be taken with care. What is shown are the eigenvalues of the
polarization operator appearing in the non-Abelian King-Smith - Vanderbilt relation (1.36).
7.2.2
The twist scheme
The basic idea of our first interferometric scheme for the measurement of the Z2 invariant is
to circumvent the discontinuity of TRP discussed above, while keeping all Bloch oscillations
completely adiabatic. To do so, we want to add band switchings at the end and in the middle
of the sequence. Then close to the Kramers degeneracy at kx = 0, instead of staying in the
energetically upper band I, atoms will be transferred to the energetically lower band II. These
band switchings correspond to applying Ramsey ⇡ pulses, as indicated in FIG.7.1(b).
After finishing the entire Bloch cycle and applying a second Ramsey ⇡-pulse, the atoms
will finally return to the band they initially started from. The two possible twisted paths in
energy-momentum space will be labeled i and ii and they are illustrated in FIG.7.1. Path i
corresponds to atoms starting in band I, while ii corresponds to atoms starting in II.
In this process atoms pick up geometrical Zak phases '˜i,ii
Zak . We will refer to these as
twisted Zak phases, because they consist of Zak phases from the movement within bands I, II
as well as additional geometric phases from the Ramsey ⇡-pulses. The key idea of the twist
scheme is to measure these twisted Zak phases.
We note that for TR invariant kyTRIM = 0, ⇡ no band switchings are required and twisted
Zak phases coincide with their conventional counterparts,
I(II)
i(ii)
'Zak (kyTRIM ) = '˜Zak (kyTRIM ).
(7.1)
Moreover we will see that twisted Zak phases '˜Zak (ky ) are continuous as a function of ky ;
This is because we added band switchings by hand, right where conventional Zak phases fail
to follow the desired path. Like all geometric phases, twisted Zak phases are by definition
gauge invariant up to integer multiples of 2⇡.
Twisted Zak phases thus allow us to define a continuous version to TRP (which we will
refer to as cTRP) by
⇤
1 ⇥ i
P̃✓ (ky ) =
'˜Zak (ky ) '˜iiZak (ky ) .
(7.2)
2⇡
For TR invariant momenta, cTRP reduces to TRP see (7.1). Thus, starting from the definition
of the Z2 invariant as di↵erence of TRP Eq. (6.6) and using continuity of cTRP, we can express
⌫2D as the winding of cTRP:
Z ⇡
⌫2D =
dky @ky P̃✓ (ky ) mod 2.
(7.3)
0
This formulation is fully gauge invariant.
7.2.3
The Wilson loop scheme
A natural question to ask, from our interferometric point of view, is what happens in the
limit of very strong driving when the Bloch oscillation frequency exceeds all energy spacings
between bands I, II. Let us still assume a large energy gap separating I, II from other bands,
such that non-adiabatic transitions into the latter can be neglected.
The multi-band Bloch dynamics in the strong driving limit (period T ! 0) is characterized
by a geometric quantity depending solely on the path within the BZ. Since there is generally
7.2. INTERFEROMETRIC MEASUREMENT OF THE Z2 INVARIANT
137
strong mixing between bands I and II, the U (1) Zak phase we encountered in the singleband case generalizes to a U (2) unitary matrix acting in I II space, the U (2) Wilson loop.
In Appendix D we derive the general propagator Û describing Bloch oscillations within a
restricted set of N bands. From that derivation one can easily see that Wilson loops indeed
emerge as the propagators describing Bloch oscillations in the limit of infinite driving force,
ÛF =1 = Ŵ .
Our second interferometric scheme (Wilson loop scheme) is based on Eq.(6.8) from the
previous chapter. The basic idea is to measure both terms, the Wilson loop phase 'W and
the total Zak phases (as defined below Eq.(6.8)) separately. Both these quantities can be
obtained from measurements of simpler Zak phases.
To obtain the winding of total Zak phase (ky ) we suggest to use the tools developed
for the measurement of the Chern number, see 6.2.1. The only complication is that now two
bands have to be treated. This can be done by adiabatically moving within only a single band
(say I) and repeating the same measurement for the second band II. An alternative protocol
allowing non-adiabatic transitions between bands I and II will also be presented in 7.4.3.
To obtain the di↵erence of Wilson loop phases 'W = 'W (⇡) 'W (0) mod 2⇡ we
suggest to use a direct spin-echo type measurement. Like any interferometric phase, the
obtained result is only known up to integer multiples of 2⇡. The key to the Wilson loop
scheme is that knowledge of 'W mod 2⇡ is sufficient in Eq.(6.8). I.e. if 'W is replaced
by 'W + 2⇡ in that equation, the resulting Z2 invariant ⌫2D ! ⌫2D + 2 = ⌫2D mod 2 does
not change.
7.2.4
Relation between Wilson loops and TRP
Before proceeding to the detailed discussion of our two interferometric protocols, we want to
point out the relation between the corresponding formulations of the Z2 invariant. This will
also shed more light on the relation between Z2 invariant and Wilson loops given in Eq.(6.8).
Let us start by rewriting the winding of the total Zak phase in terms of polarizations.
Using Eq.(1.19) and setting the lattice constant a = 1 we obtain
Z ⇡
1
dky @ky (ky ) = P I (⇡) + P II (⇡) P I (0) P II (0).
(7.4)
2⇡ 0
Meanwhile the formulation of the Z2 invariant in terms of TRP reads
⌫2D = P I (⇡)
P II (⇡)
P I (0) + P II (⇡)
mod 2,
(7.5)
see Eq.(6.6). After clever adding and subtracting terms in the last equation we can write
X
⌫2D = 2 P I (⇡) P I (0)
(P s (⇡) P s (0)) mod 2.
(7.6)
s=I,II
In the second term of this equation we recognize the winding of the total Zak phase
discussed before. The first term on the other hand denotes the di↵erence of Zak phases at
ky = 0 and ⇡,
1
P I (⇡) P I (0) =
'I (⇡) 'IZak (0) .
(7.7)
2⇡ Zak
Here, as a consequence of TR invariance, the Zak phases of the two bands I, II are equal,
explaining why only the polarization P I appears. What’s more, these Zak phases are given
138
CHAPTER 7. INTERFEROMETRIC MEASUREMENT OF Z2 TOPOLOGICAL
INVARIANTS
by the Wilson loop phase 'W , i.e. we obtain
P I (⇡)
P I (0) =
1
'I (⇡)
2⇡ W
'IW (0) =
'W
.
2⇡
(7.8)
Combining Eqs.(7.4), (7.8) in Eq.(7.6) we have thus derived Eq.(6.8).
Now the two terms in Eq.(6.8) have a clear physical meaning: The winding of total
Zak phase is related to the translation of the center of mass of the two Wannier centers,
i.e.
P I + P II . (Here
denotes the di↵erence of the quantity across half the BZ.)
The di↵erence of Wilson loop phases stands for the change of polarization of a single band,
'W /2⇡ = P I = P II mod 1.
In FIG.6.2 (a)-(c) these changes of polarization can easily be read o↵ from the plotted
Wannier centers. A word of caution is in order, however. As a consequence of the discontinuity
of TRP, FIG.6.2(b) has to be taken with a grain of salt: Although appealing, the idea that
each line (solid/dashed) shows the polarization of a single band is wrong. As explained by
Yu et.al.[94], what is shown are the eigenvalues of the position operator X̂ projected on the
two bands I, II and its non-commutative quantum mechanical nature plays a crucial role in
resolving the discontinuity of TRP. Yu et.al. showed that the eigenvalues of X̂ are given by
the angle (in the complex plane) of the U (1) Wilson loop eigenvalues. Because Wilson loops
include non-adiabatic band-mixings they are in general continuous as a function of ky - and
so is their spectrum.
7.3
Twist scheme
In this section we discuss the twist scheme in detail. We start by introducing the concrete
protocol and show how to get rid of dynamical phases. We proceed by giving the theoretical
derivation of the phases to be measured; Then we show their relation to the Z2 invariant
and present a mathematical formulation of continuous time-reversal polarization (cTRP). We
close the section by discussing cTRP using the example of the Kane-Mele model [28].
7.3.1
Interferometric sequence
As discussed in Sec.7.2.2, the basic idea of the twist scheme is to measure twisted Zak phases
using a combination of Bloch oscillations and Ramsey interferometry. Twisted Zak phases
were defined by introducing band-switchings in the middle (kx = 0) and at the end (kx = ⇡) of
the interferometric sequence, see FIG.7.1 (b). These band switchings correspond to Ramsey
⇡ pulses between the bands, and along with them come additional geometric phases which
will be discussed at the end of this section.
Note that since only a continuous function interpolating between TRP P✓ (⇡) and P✓ (0)
is required, the two band switchings (labeled 1, 2) can be performed at any intermediate
kx = f1,2 (ky ). The only requirements are that f1 (0) = f1 (⇡) = 0 and f2 (0) = f2 (⇡) = ⇡ as
well as continuity of f1,2 (ky ). This most general case only leads to a redefinition of twisted
Zak phases, while keeping their relation to the Z2 invariant Eq.(7.3) unchanged. We will
therefore not discuss it in the following.
To realize the Ramsey ⇡ pulses between the bands we suggest to drive Bloch oscillations
with a time-dependent force, see FIG.7.2 (a), described by a Hamiltonian
Z
Ĥrf (t) = d2 r ˆ † (r) cos(!rf t)F0 · r ˆ (r).
(7.9)
7.3. TWIST SCHEME
139
(a)
(b)
Figure 7.2: Ramsey pulses by lattice shaking: (a) The lattice is tilted and the slope reverses
its sign in each cycle. Therefore (b) atoms localized in momentum space around kx = 0 can
only perform Bloch oscillations in the direct vicinity of kx = 0 if !Frf0 ⌧ 2⇡. When the driving
!rf equals the transition frequency
Ramsey pulses can be realized.
Here ˆ (r) is a pseudo-spinor (components ", #) annihilating a particle at position r and !rf
is the (typically radio-frequency, rf) driving frequency. Note that in this way only motional
degrees of freedom are coupled, independent of the (pseudo) spin state of the atoms. This
turns out to be crucial for the scheme to work. For simpler realizations with a direct coupling
between the pseudospins, additional information about the Bloch wave functions is required.
We discuss this issue in detail in Appendix E.
The equations of motion for the Hamiltonian Eq.(7.9) are derived in Appendix D. According to Eq.(D.7) in that Appendix we obtain a modulation of momentum
k(t) = k(0)
sin(!rf t)F0 /!rf .
(7.10)
Dynamics of this kind have been studied before, see e.g. [280]. FIG.7.2(b) illustrates the e↵ect
of this driving in momentum space: particles undergo Bloch oscillations within a restricted
0|
area ± |F
!rf around their mean value of kx .
Therefore, when |F0 | ⌧ !rf (with lattice spacing a = 1), we may approximate the Berry
connection (and equivalently the Bloch Hamiltonian) by A (k(t)) ⇡ A (k(0)). Taking into
account only the two Kramers partners I, II and applying the rotating wave approximation
we obtain the Hamiltonian in the frame rotating at frequency !rf
✓
◆
0
F0 · Au,l (k)
Ĥrf (k) =
.
(7.11)
F0 · Al,u (k)
(k) !rf .
l
l
The basis of the rotating frame is defined as |l, kie iE t and |u, kie i(E +!rf )t , and = E u E l
denotes the band-gap between the upper (u) and lower (l) of the two bands. For the rotating
wave approximation to be valid, we require
|F0 · Au,l (k)| ⌧ !rf ⇠
.
(7.12)
Band-switchings
We note that the phase of the e↵ective driving field in Eq.(7.11), defined through
'A (k) := arg Al,u (k) =
arg Au,l (k),
(7.13)
is determined by the non-Abelian Berry connection (where in the second step we employed
† = Â). This is important because the latter encodes information about the underlying
topology of the two bands I, II. We will come back to this point below.
140
CHAPTER 7. INTERFEROMETRIC MEASUREMENT OF Z2 TOPOLOGICAL
INVARIANTS
0.4
0.35
12
0.3
10
8
0.25
6
0.2
4
0.15
2
0.1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
Figure 7.3: Absolute value of the o↵-diagonal Berry connection |Alu
x | in units of the lattice
constant a (a = 1 in the main text). Calculations were performed on the Kane-Mele model [28]
discussed below in the main text. Parameters (corresponding to a topologically non-trivial
phase) were chosen as v = 0.1t, R = 0.05t, SO = 0.06t with notations from [28].
One might be afraid that the resulting Rabi frequency is too small for the method to be
practically applicable. However we find e.g. for the Kane-Mele model [28] (which will be
discussed in more detail below in 7.3.5) that |Au,l | takes substantial values in the entire BZ,
see FIG. 7.3.
Note that the edges of the BZ are not shown in FIG.7.3 since |Au,l | diverges around the
Kramers degeneracies. (The reason is that the lower-band Bloch function continuously evolves
into the upper one at the Kramers degeneracy, such that hl, kx |l, kx i ! 0 for kx ! 0 and
thus |hu, kx |@kx |l, kx i| ! 1 at kx = 0.) In this case of too large |Au,l |, according to Eq.(7.12)
the rotating wave approximation is not applicable, but the band switching protocol can be
replaced by a quick Landau-Zener sweep across the avoided crossing.
Sequence
Now we introduce the interferometric sequence which allows one to measure twisted Zak
phases '˜i,ii
Zak , and therefore cTRP Eq. (7.2) directly. To this end we assume that atoms are
located initially in the upper band at kx = ⇡ and some fixed ky , i.e. | 0 i = |u, ⇡i and start
by applying a ⇡/2-pulse3 , see FIG.7.4. In the following we will ignore all dynamical phases
which will be discussed below in 7.3.2.
The ⇡/2 pulse creates a superposition state of atoms in the upper and lower band,
|
1i
1
= p |u, ⇡i
2
iei'A (⇡) |l, ⇡i .
(7.14)
In this step atoms in lower and upper band pick up the relative phase 'A (⇡) of the e↵ective
driving field, see Eqs. (7.11) and (7.13). Next, a Bloch oscillation half-cycle transports the
atoms from kx = ⇡ to kx = 0 and each component picks up geometric phases 'u,l
Zak, . These
incomplete Zak phases are defined for the lower (s = l) and upper (s = u) band as
'sZak,± (ky )
=±
Z
±⇡
0
dkx Ass (k),
s = u, l.
(7.15)
Note that incomplete Zak phases are not gauge invariant, and thus not physical observables.
However the interferometric signal we obtain at the end of our sequence will be fully gauge
3
Of course, in order to apply a ⇡/2 pulse, the band gap has to be measured before hand. As pointed out
above, this can for instance be achieved using Stückelberg interferometry.
7.3. TWIST SCHEME
141
Figure 7.4: General interferometric scheme: A ⇡/2-pulse creates a superposition of atoms in
the upper and lower band. When performing Bloch oscillations though the BZ they pick up
twisted Zak phases as a consequence of the ⇡-pulse in the middle of the sequence. Finally a
⇡/2-pulse serves to read out the accumulated phase.
invariant and observable.
The resulting state now reads
|
1 ⇣ i'uZak,
p
i
=
e
|u, 0i
2
2
⌘
l
iei('A (⇡)+'Zak, ) |l, 0i .
(7.16)
A ⇡-pulse at kx = 0 then exchanges populations of the upper and lower band such that the
corresponding wave function reads
|
3i
l
1 ⇣
= p ei('A (⇡)+'Zak,
2
'A (0))
|u, 0i
⌘
u
iei('A (0)+'Zak, ) |l, 0i .
After a second Bloch oscillation half-cycle the atoms reach kx = ⇡ =
up incomplete Zak phases 'u,l
Zak,+ .
(7.17)
⇡ mod 2⇡ and pick
Finally another ⇡/2-pulse is applied to read out the relative phase of the two components
|u, ⇡i, |l, ⇡i. This is achieved by a phase shift of the driving frequency, !rf t ! !rf t '⇡E in
Eq.(7.9). As a function of this shift the population in the upper band yields Ramsey fringes
 ⇣
⌘
⇡ 2
2 1
| u ('E )| = cos
2⇡ P̃✓ (ky ) '⇡E
.
(7.18)
dyn
2
Here dyn contains all dynamical phases from the Bloch oscillations as well as Ramsey pulses.
Most importantly, the incomplete Zak phases in combination with the phases 'A yield a full
expression for cTRP,
2⇡ P̃✓ (ky ) = 'uZak, (ky ) + 'lZak,+ (ky )
'lZak, (ky )
'uZak,+ (ky )
2 'A (⇡, ky )
'A (0, ky ) . (7.19)
At the end of this section we will give an explicit proof that the above equation (7.19) has
all desired properties of cTRP. In particular, it reduces to TRP at ky = 0, ⇡ and is continuous
throughout the BZ; therefore its winding yields the Z2 invariant, see Eq.(7.3).
7.3.2
Dynamical-phase-free sequence
Now we turn to the discussion of dynamical phases and present a scheme that completely
eliminates them. When performing Bloch oscillations, to move the atoms from e.g. kx (0) =
142
CHAPTER 7. INTERFEROMETRIC MEASUREMENT OF Z2 TOPOLOGICAL
INVARIANTS
Figure 7.5: Final interferometric sequence at fixed ky : A ⇡/2 pulse at kx = ⇡ creates a
superposition in the upper and lower band. Bloch oscillations move the atoms to kx = 0 where
a ⇡ pulse exchanges populations in the upper and lower band. After a second Bloch oscillation
half-cycle followed by a second ⇡ pulse the sequence is reversed to get rid of dynamical phases.
Finally at kx = ⇡ a ⇡/2-pulse can be used to read o↵ twice the cTRP from the Ramsey
signal.
⇡ to kx (T ) = +⇡ in time T , additional dynamical phases
BO
dyn,s (ky )
=
Z
T
dt E s (kx (t), ky )
(7.20)
0
contribute to dyn in Eq. (7.18). Here s = u, l denotes the band index and E s the corresponding energy.
To cancel them we use the opposite transformation properties of geometrical and dynamical phases when inverting the path taken in the BZ. From dk
dt = F we see that dynamical
phases do not depend on the orientation of the path,
Z
T
dtE(k(t)) =
0
Z
⇡
E(k)
dk
=
F
⇡
Z
⇡
dk
⇡
E(k)
.
F
Geometric phases on the other hand acquire a negative sign upon path inversion,
Z ⇡
Z ⇡
dk A(k) =
dk A(k).
⇡
(7.21)
(7.22)
⇡
Therefore, when reversing the interferometric sequence (F ! F ) after reaching kx = ⇡
(as indicated in FIG. 7.5), the Ramsey signal yields twice the continuous TR polarization
(7.19) while dynamical phases are canceled.
Experimentally, phases can only be measured up to 2⇡. As we argued above, the Z2 invariant
can be written as winding of cTRP, see Eq.(7.3). This winding is measured by summing up
small changes P̃✓ = P̃✓ (ky + ky ) P̃✓ (ky ). By choosing ky sufficiently small we may always
assume 2 P̃✓ ⌧ 1 and doubling the interferometric sequence still allows to infer the winding
of cTRP.
The complete sequence is summarized in FIG.7.5. The Ramsey signal in this case reads
h
i
(0)
| u ('⇡E )|2 = cos2 2⇡ P̃✓ '⇡E
(7.23)
dyn ,
where the remaining dynamical phase is picked up when
applying Ramsey
pulses. It only
⇣
⌘
(0)
3!rf (⇡)
!rf (0)
depends on the known driving parameters, dyn = ⇡ 4⌦rf (⇡) ⌦rf (0) .
7.3. TWIST SCHEME
7.3.3
143
Experimental realization and limitations
Our scheme can be applied in the experimental setup proposed in [270], where nano-wires on
an atom-chip are used to generate state-dependent potentials for di↵erent magnetic hyperfine
states. These could also be used to realize the band-switching Hamiltonian (7.9) and for
driving Bloch oscillations. In more conventional setups without atom chips, like e.g. the
experiment [108] and the proposals [269, 271, 273], Bloch oscillations can e.g. be driven using
magnetic field gradients [13] or optical potentials. This would also allow the realization of
Hamiltonian (7.9) for band-switchings.
The main advantage of the twist scheme is that - although it makes use of interferometry
- no additional degrees of freedom are required besides the pseudospins ", # needed for the
realization of the QSHE. This is of practical relevance, since already the realization of two
pseudospins for the QSHE is a non-trivial task.
The applicability of our scheme is somewhat limited in that we did not consider accidental degeneracies besides the four Kramers degeneracies. If such additional degeneracies
are present, the definition of cTRP has to be modified. The scheme for the Ramsey pulses
presented in subsection 7.3.1 is also not applicable when the o↵-diagonal Berry connections
become too small. Let us also add however, that cTRP contains more information about the
band structure than only the Z2 invariant, since it resolves the two TR partners individually.
We would like to stress that currently the key limitation for implementing our interferometric scheme with ultra cold atoms is the lack of an efficient spin-orbit coupling scheme
in optical lattices. We will now argue that once a generic model of the QSHE, akin to the
Kane-Mele Hamiltonian, will be realized, the twist scheme can be applied. From the author’s
perspective, which is based on several discussions with experimentalists working in the field,
this makes the scheme not only of academic interest but also of significant practical relevance.
For concreteness let us consider now the experimental setup described in Ref.[201], and
let us further assume that spin-orbit coupling can be implemented to realize the Kane-Mele
Hamiltonian. To apply the twist-scheme, the band gaps at all band-switching points have to
be known. Measurements of band gaps have readily been achieved in the setup described in
[201] using Stückelberg interferometry4 . Next, an oscillating force has to be applied in order
to realize the band switchings. This can be achieved by a time-dependent optical potential,
which due to fast piezo-elements allows quick modulations. Next, a constant force has to be
applied to move the atoms adiabatically through the BZ, which is equal for both spins. This
has readily been demonstrated in the setup [201] using optical potentials. The relevant adiabaticity conditions can be fulfilled. Note that the setup described above avoids completely the
use of magnetic field gradients, which have slow reaction times only. This is a major advantage
of the twist scheme. Furthermore, we note that the twist scheme measures a relative (twisted)
Zak phase, which is thus gauge-invariant. This eliminates temporal fluctuations of the geometric phase due to a fluctuating magnetic field, as described in Ref.[13]. (Such fluctuations
change the origin relative to which Zak phases are measured.) It has been demonstrated experimentally that relative Zak phases can be measured by inter-band interferometry [13]. To
read out the resulting phase of an interferometric experiment, band-mapping schemes have to
be employed. They are standard tools which have been implemented in the setup described
in Ref. [201] and can be generalized to be spin-resolved, see e.g. [268]. Finally we comment
on the diabatic Landau-Zener sweeps required close to TRIM. In the setup described in Ref.
[201] the strong-gradient regime where the evolution becomes fully diabatic within the lowest
bands has readily been achieved, without substantial population of higher bands 4 .
4
Tracy Li and Immanuel Bloch, private communication.
144
CHAPTER 7. INTERFEROMETRIC MEASUREMENT OF Z2 TOPOLOGICAL
INVARIANTS
7.3.4
Formal definition and calculation of cTRP
In this section we will give a formal proof that our scheme presented above does indeed
measure the Z2 invariant; I.e. we will derive Eq.(7.19). Instead of starting from this explicit
expression for cTRP however, we will introduce the concept of cTRP in a formal way and
derive it independently.
Definition of cTRP
We will now formally define a generalization of TRP P✓ (ky ) that we will refer to as P̃✓ (ky ); We
require this quantity to fulfill the following properties, making it suitable for an interferometric
measurement of the Z2 invariant. It has to
(i) reduce to TRP at the end points kyTRIM = 0, ⇡, i.e. P˜✓ (kyTRIM ) = P✓ (kyTRIM ), and
(ii) be continuous as a function of ky .
Any such function P̃✓ (ky ) will be called continuous time-reversal polarization (cTRP). To
assure that cTRP constitutes a physical observable it should furthermore
(iii) be gauge-invariant, at least up to an integer at each ky .
Finally, from a practical point of view, we want cTRP to
(iv) be measurable in an interferometric setup consisting of a combination of Bloch oscillations and Ramsey interferometry.
In the following subsection we will explicitly construct cTRP and subsequently prove all its
desired properties (i)-(iv). We will always consider a generic 2D TR invariant band structure
consisting of two time reversed Kramers partners, see FIG.7.1.
Our construction of cTRP is motivated by the experimental sequence described earlier in
this section. It will reproduce the expression (7.19) obtained from our interferometric protocol
and thus (iv) follows naturally. Let us add that as a direct consequence of the properties (i)
and (ii) the winding of cTRP yields the Z2 invariant, see Eq.(7.3).
Discretized version of continuous time-reversal polarization
We start by discretizing momentum space for fixed ky into N equally spaced (spacing k)
points kx0 , ..., kxN 1 . The discrete version of the Zak phase in a single gapped band |u, kx i is
then given by
'Zak =
⇢✓NY2
◆
⌦
↵ ⌦
j
j+1
lim arg
u, kx |u, kx
u, kxN
N !1
j=0
1
|u, kx0
↵
.
(7.24)
Here argz denotes the polar angle of the complex number z. One obtains the continuum
expression Eq.(1.16) for the Zak phase by using that
⌦
↵
s, kxj |s0 , kxj+1 ⇡
s,s0
0
i kx As,s (kxj ).
(7.25)
Here s and s0 denote band indices (the single band above was labeled s = s0 = u) and the
Berry connection A was defined in Eq.(1.31).
For kyTRIM = 0, ⇡ TRP is given by the di↵erence of the Zak phases of bands I and II which
– unlike u and l – are defined continuously at the Kramers-degenerate points, see Eqs.(6.6)
7.3. TWIST SCHEME
145
Figure 7.6: Definition of the (discretized) cTRP at fixed ky . The dashes, numbered by
j = 0, ..., M, ..., N 1, stand for Bloch functions of the upper (|u, kx i) and lower (|l, kx i) band
at di↵erent kxj . The solid lines connecting them correspond to the scalar products appearing
in the product of equation (7.27).
and (1.19). Due to the presence of Kramers degeneracies the discretized versions of these Zak
phases contain cross terms between the energetically upper (u) and lower (l) band,
'IZak
=
⇢✓N/2
◆
Y2⌦
↵
j
j+1
lim arg
u, kx |u, kx
hu, kxN/2
N !1
j=1
⇥
✓
N
Y2
j=N/2+1
⌦
1
|l, kxN/2+1 i
l, kxj |l, kxj+1
◆
↵ ⌦
l, kxN
1
|u, kx1
↵
, (7.26)
and equivalently for 'II
Zak . This discrete product is shown in a graphical form in FIG. 7.6
with the mid point choose to be in the center, M = N/2, which is assumed to be integer.
Note that in order to avoid ambiguities in the definition of the wavefunctions at the Kramers
degeneracies we did not include kxTRIM = 0, ⇡ in the product. This is justified when taking
the limit N ! 1.
The above discrete expression can readily be generalized to non-TRIM 0 < ky < ⇡. To
this end we introduce a discrete version of twisted Zak phases '˜Zak (twisted polarization P̃ )
for given ky in the BZ as
'˜iZak
i
= 2⇡ P̃ (ky ) =
lim
N !1
M/N const.
◆
⇢✓MY2
⌦
↵ ⌦
j
j+1
arg
u, kx |u, kx
u, kxM
j=1
⇥
✓ NY2
⌦
j=M +1
l, kxj |l, kxj+1
1
|l, kxM +1
◆
↵ ⌦
l, kxN
1
↵
|u, kx1
↵
. (7.27)
Here i is a the band index labeling the twisted contour introduced in Sec.7.2.2, see also FIGs.
7.6 and 7.1; M denotes the index of some intermediate band switching point, see FIG.7.6.
Analogously we can define twisted polarization P̃ ii (ky ) (twisted Zak phase '˜iiZak (ky ) of the
second band ii, which is obtained from i by exchanging energetically upper (u) and lower (l)
band indices.
Like in Sec.7.2.2 we can now define the discretized version of cTRP using twisted polarizations, see Eq.(7.2),
P̃✓ (ky ) = P̃ i (ky ) P̃ ii (ky ).
(7.28)
In the following we will check all its desired properties (i)-(iv) listed above.
By construction it is clear that (i) P̃✓ (kyTRIM ) reduces to standard TRP provided that
146
CHAPTER 7. INTERFEROMETRIC MEASUREMENT OF Z2 TOPOLOGICAL
INVARIANTS
M = N/2 is chosen, cf. (7.26). To check (ii), i.e. continuity of P̃✓ (ky ), we notice that all
scalar products are continuous as a function of ky for fixed discretization into N points along
kx . Therefore the discrete version of cTRP is continuous as a function of ky , assuming that
also the band switching point labeled by M changes continuously with ky . Finally P̃ i,ii (ky ) –
and thus P̃✓ (ky ) – are gauge invariant up to an integer. This can be seen by considering U (1)
gauge transformations in momentum space, |s, kx i ! |s, kx iei#s (kx ) . Since all wavefunctions
appear twice in (7.27), once as a bra hs, kx | and once as a ket |s, kx i, all U (1) phases drop
out. A 2⇡Z ambiguity of '˜Zak remains since arg is only well-defined up to 2⇡ (unless Riemann
surfaces are considered).
We point out that cTRP can also be used for numerical evaluation of the Z2 invariant. In
subsection 7.3.5 we demonstrate this for the specific example of the Kane-Mele model [28].
Incomplete Zak phases and continuum version of cTRP
To derive a continuum version of cTRP Eq.(7.28) constructed above, we use Eq.(7.25) to replace scalar products by Berry connections. Between the band switching points, for simplicity
assumed to be located at kx = 0, ⇡, we obtain e.g.
M
Y2
j=1
⌦
↵
⇥
⇤
u, kxj |u, kxj+1 ! exp i'uZak, (ky )
(7.29)
with the incomplete Zak phase 'uZak, defined in Eq.(7.15).
We are now in a position to formulate the discontinuity problem discussed in the introduction in a more precise way. For TRIM kyTRIM there are two band-crossings right where we
switch from one ('Zak, ) to the other ('Zak,+ ) incomplete Zak phase, see FIG. 7.1 (a). Here
TRP can be written in terms of incomplete Zak phases,
P✓ (kyTRIM ) = 'uZak, + 'lZak,+
'lZak,
'uZak,+ .
(7.30)
Away from TR invariant lines, ky =
6 0, ⇡, gaps open in the vicinity of the Kramers degeneracies,
see FIG. 7.1 (b). Consequently the incomplete Zak phases belong to bands that no longer
cross, and their relation to TRP is strikingly di↵erent,
P✓ (kyTRIM ) = 'uZak, + 'uZak,+
'lZak,
'lZak,+ .
(7.31)
To obtain↵ a complete continuum description of cTRP, we note that cross terms like
l, kxN 1 |u, kx1 between energetically upper and lower band are related to o↵-diagonal elements of the non-Abelian Berry connections according
to Eq.(7.25).
(Note that care has to
⌦
↵
0
be taken in the case ky = kyTRIM = 0, ⇡ where s, kxN 1 |s0 , kx1 / (1
s,s0 ) for s, s = u, l as a
consequence of the Kramers degeneracies.) For non-TRIM ky 6= kyTRIM we thus have
⌦
⌦
arg l, kxM
1
⇣
⌘
↵
|u, kxM +1 ! arg i kx Al,u (0, ky ) .
(7.32)
In terms of the phase 'A of Al,u introduced in Eq.(7.13) we obtain the continuum expression of twisted polarization,
P̃ i =
1 h u
'
(ky ) + 'lZak,+ (ky )
2⇡ Zak,
i
'A (⇡, ky ) + 'A (0, ky ) ,
(7.33)
7.3. TWIST SCHEME
147
and analogously for P̃ ii . This finally leads to the continuum description of cTRP,
P̃✓ (ky ) =
1 h u
'
(ky ) + 'lZak,+ (ky )
2⇡ Zak,
'lZak, (ky )
'uZak,+ (ky )
2 'A (⇡, ky )
i
'A (0, ky ) , (7.34)
which coincides with the Ramsey signal of our interferometric protocol, see Eq.(7.19).
All desired properties of P̃✓ (ky ) listed in 7.3.4 carry over from its discretized version. To
get a better understanding of the physical meaning of the di↵erent terms, we now show that
twisted polarization Eq.(7.33) is gauge invariant up to an integer. To this end we consider a
gauge-transformation,
|s, kx i ! e i s (kx ) |s, kx i
s = l, u.
(7.35)
Under this transformation the diagonal of the Berry connection obtains additional summands,
As,s (kx ) ! As,s (kx ) + @kx s (kx ), whereas o↵-diagonal terms in the Berry connection obtain
additional factors, Au,l ! Au,l ei( u l ) , as can be seen from
⇣
Au,l (kx ) = hu, kx |[email protected] |l, kx i ! Au,l (kx ) + hu, kx |l, kx i (@kx
|
{z
}
=0
⇥e
i(
u (kx )
l (kx ))
⌘
(k
))
⇥
l x
= Au,l (kx )ei(
u (kx )
l (kx ))
. (7.36)
Incomplete Zak phases from Eq.(7.15) alone or 'A from Eq.(7.13) alone are not gaugeinvariant because e.g.
'uZak, ! 'uZak, +
'A (0) ! 'A (0) + (
u (0)
u(
⇡)
u (0)
l (0))
mod 2⇡,
mod 2⇡.
(7.37)
(7.38)
However using s ( ⇡) = s (⇡) mod 2⇡ (s = u, l) we find that twisted polarization Eq.(7.33)
is a gauge invariant quantity, transformations of incomplete Zak phases and phases 'A cancel.
7.3.5
Example: Kane-Mele model
We will now illustrate that the winding of cTRP indeed gives the Z2 invariant by explicitly
calculating it for the Kane-Mele model [28]. The physical system described by this model is
(a)
(b)
Figure 7.7: (a) Kane-Mele model on the honeycomb lattice. All coupling elements between
the lattice sites are shown. (b) In k-space there are four time-reversal invariant momenta
marked by black dots. Continuous time-reversal polarization can be defined for paths (solid
blue line) in the upper half of the unit cell (blue shaded). Blue crosses on dashed blue lines
denote the band switching points. Bands within the lower half of the unit cell are related to
those in the upper part by TR symmetry.
148
CHAPTER 7. INTERFEROMETRIC MEASUREMENT OF Z2 TOPOLOGICAL
INVARIANTS
1.2
1
0.8
0.6
0.4
0.2
0
0.2
0
0.2
0.4
0.6
0.8
1
Figure 7.8: Continuous time-reversal polarization P̃✓ (cTRP) in the Kane-Mele model [28]
as a function of lattice momentum y in the upper half of the BZ. Parameters: R = 0.05t,
SO = 0.06t with notations from [28]. In the topologically trivial phase ( v = 0.4t, dashed)
the winding of cTRP is zero, while it is one in the non-trivial phase ( v = 0.1t, solid). For
the calculation the discretized form of cTRP was used, see Eqs. (7.27), (7.28), with the band
switching point M = N/2 at x = ⇡ for all y .
sketched in FIG.7.7 and its Hamiltonian reads
X †
X
X †
Ĥ = t
ĉi ĉj + i SO
⌫ij ĉ†i sz ĉj + i R
ĉi (s ⇥ dij ) · ez ĉj +
hi,ji
hhi,jii
hi,ji
v
X
⇠i ĉ†i ĉi ,
(7.39)
i
with the same notations as in [28]; The spin indices of ĉ†i , ĉj were
p suppressed and s denotes
the vector of Pauli matrices for the spins. Moreover ⌫ij = 2/ 3 (d1 ⇥ d2 ) · ez = ±1 with
ez the unit vector along z-direction and d1 , d2 being unit vectors along the two bonds which
have to be traversed when hopping between next nearest neighbor sites j and i.
Kane and Mele started from a Hamiltonian describing two copies ", # of the Haldane model
[103] on a honeycomb lattice (first line in Eq.(7.39)). Importantly, the magnetic flux seen by
" is opposite to that seen by # which is realized by a spin-dependent next nearest neighbor
hopping with amplitude ±i SO . They also included TR invariant Rashba SOC terms / R
as well as a staggered sublattice potential / ± v characterized by ⇠i = ±1.
In order to define cTRP we use a non-orthogonal basis in k-space labeled by x , y , see
FIG. 7.7 (b). In this basis the unit cell is given by x ⇥ y = [0, 2⇡] ⇥ [0, 2⇡] and TRIM are
found at x = 0, ⇡ and y = 0, ⇡. The fact that we use a non-orthogonal basis does not a↵ect
the definition of 1D Zak phases nor their relation (7.3) to the Z2 invariant.
Using Eq.(7.27) we calculate cTRP P̃✓ (y ) for band switchings at x = 0 as indicated
in FIG.7.7(b). The result is shown in FIG. 7.8 for v = 0.1t ( v = 0.4t) corresponding
to a topologically non-trivial (trivial) phase. As predicted by Eq.(7.3) P̃✓ does not wind in
the topologically trivial case whereas it does so in the topologically non-trivial case. The
example also demonstrates that the derivative @y P̃✓ (y ) generally takes finite values which
is important to make measurements of the winding experimentally feasible.
7.4
Wilson loop scheme
As we discussed in Sec.6.2.2, Wilson loops are related to the Z2 invariant [94] by Eq.(6.8), i.e.
✓
◆
1
1
⌫2D =
'W
mod 2.
(7.40)
⇡
2
7.4. WILSON LOOP SCHEME
149
We identified two terms,
R ⇡ the di↵erence of Wilson loop phases 'W and the winding of the
total Zak phase
= 0 dky @ky (ky ) constituting the Z2 invariant.
Our second interferometric scheme (Wilson loop scheme) for the measurement of the
Z2 invariant consists of treating these two terms ( 'W and
) separately. The basic idea
of our protocol is to express them in terms of simple Zak phases which can be measured using
Ramsey interferometry in combination with Bloch oscillations [13, 200].
In the entire section we will assume that, when driving Bloch oscillations, non-adiabatic
transitions from the valence bands I, II to conduction bands are suppressed. From the adiabaticity condition (given in Appendix D, Eq. (D.10)) we find that this is justified as long as
the band gap band 5 is smaller than the Bloch oscillation frequency aF (with a the lattice
constant),
aF ⌧ band .
(7.41)
We start this section by discussing the relation of TR Wilson loops (7.4.1) and total Zak
phase (7.4.2) to simpler geometric Zak phases. Then we show in 7.4.3 how this leads to a
realistic experimental scheme and discuss necessary requirements.
7.4.1
TR Wilson loops and their phases
As we pointed out in Sec.7.2.3, U (2) Wilson loops correspond to propagators describing completely non-adiabatic or diabatic (i.e. infinitly fast) Bloch oscillations within the two bands
I, II,
ÛF =1 = Ŵ .
(7.42)
This can be seen directly by comparing the general propagator Û derived in Appendix D Eq.
(D.10) with the definition of the Wilson loop Ŵ Eq.(1.33).
An infinite driving force corresponds to the condition I II ⌧ aF that the energy spacing
I II of the two bands I, II is always much smaller than the Bloch oscillation frequency. If
this condition can be met, the Wilson loop phase can directly be measured experimentally, see
Eq.(6.9). We will show below however that even when this condition is violated the Wilson
loop phase 'W can still be measured, provided that TR symmetry is present.
To this end we consider TR invariant Bloch oscillations of finite speed within the two
valence bands. With TR invariant Bloch oscillations we mean that the driving forces at
momenta ±k(T /2 ± t) related by TR coincide, F (T /2 t) = F (T /2 + t). For simplicity
we will further restrict ourselves to a constant movement through the BZ in the following
calculations,
k(t) = (F t, 0)T + k(0),
(7.43)
which is TR invariant in the above sense.
The e↵ect of TR invariant Hamiltonian dynamics within the two bands I, II is just a U (1)
phase 'U , without any residual band mixing between I, II. I.e. the propagator describing one
Bloch oscillation cycle reads
TRIM )
Û (kyTRIM ) = ei'U (ky
Î2⇥2 .
(7.44)
For an exact proof, which is a generalization of the calculation performed by Yu et al.
[94], we refer the reader to the Appendix F while here we only outline the basic idea. The
5
The band gap band is defined as the minimum energy spacing from the two bands I or II to any further
(conduction) bands. Here we assume that the band gap is larger or comparable to the width of the valence
band I II , i.e. I II . band .
150
CHAPTER 7. INTERFEROMETRIC MEASUREMENT OF Z2 TOPOLOGICAL
INVARIANTS
Figure 7.9: TR Wilson loops within two TR bands yield only phase factors: The SU (2)part of the propagator at +kx (i.e. the amount of band mixing) reverses the action of the
corresponding SU (2) part at kx . The U (1) parts (i.e. phases) on the other hand add up.
propagator for propagation from kx to kx + kx is given by Û (kx ) = exp
Eq.(D.10) in Appendix D, with
B̂x (kx ) = Â(kx ) +
Ĥ(kx )
.
F
⇣
⌘
i kx B̂x (kx ) , see
(7.45)
From TR symmetry it follows
from kx
kx to kx
⇣ that the corresponding propagator
⌘
U (1)
ˆ
is given by Û ( kx ) = exp +i kx Bx (kx ) 2i kx Bx (kx ) up to a gauge-dependent phase
factor. (Following Yu et.al. [94] we used that ✓ˆ† ˆ j ✓ˆ = ˆ j for j = x, y, z while ✓ˆ† Î2⇥2 ✓ˆ =
+Î2⇥2 . Here ✓ˆ = Kiˆ y denotes the TR operator.) This shows that band mixings at kx are
reversed at +kx , while phases at ±kx add up. This is depicted in FIG. 7.9.
For the U (1) phase 'U characterizing the propagator in Eq.(7.44) we obtain (see Eq.(F.26)
in Appendix F)
Z ⇡
1
TRIM
TRIM
'U (ky
) = 'W (ky
)+
dkx trĤ(k),
(7.46)
2F
⇡
which can be measured in an interferometric setup. The last term on the right hand side
/ 1/F is a dynamical phase6 and can in principle be inferred by comparing 'U taken at
di↵erent driving forces F .
Before turning to a more detailed discussion of a possible experimental protocol in subsection 7.4.3, let us comment on the relation between the Wilson loop phase 'W and the Zak
phases 'Zak of the time reversed bands I, II. Since the geometric phase 'W in the propagator
Eq.(7.44) is independent of the speed F of Bloch oscillations, we can consider the case of
infinitesimal driving force F ! 0. In this limit, as a consequence of the adiabatic theorem,
an atom starting in say band I remains in this band. The geometric phase it picks up in this
process is therefore given by the Zak phase 'IZak of the corresponding band. At the same time
we can calculate this phase using the general result Eq.(7.46) from which we conclude that
the geometric phase picked up by the atoms is given by the Wilson loop phase 'W . Because
these two phases must coincide we have
'W = 'IZak = 'II
Zak
mod 2⇡.
(7.47)
We note that since there is a priori no fixed relation between the Zak phases at ky = 0
and ⇡, Wilson loop phases 'W may take any value between 0 and 2⇡ in general. A particular
example is sketched in FIG.6.2 (b). In [94] it was claimed that TR Wilson loops “ are
proportional to unity matrix, up to a sign”; This statement is not correct (already the Kane6
When a single band is considered the dynamical phase reduces to the well-known result; RTaking Ĥ(k) =
⇡
diag (E(k), E(k)) we obtain for the dynamical phase in Eq.(7.46) 'U (kyTRIM )+'W (kyTRIM ) = F1
dkx E(k) =
⇡
R 2⇡/(aF )
dt E(k(t))
0
7.4. WILSON LOOP SCHEME
151
Mele model [28] provides counter examples), and in general 'W can take arbitrary values.
Let us furthermore mention that the results Eqs. (7.44) - (7.47) are relevant for the twist
scheme presented in section 7.3: To measure the Zak phase 'IZak = 'II
Zak at TR invariant
momenta ky of the two time reversed partners I, II, adiabaticity is only required with respect
to the conduction bands. The gap I II = |E I E II | may be arbitrarily small compared to the
Bloch oscillation frequency aF .
7.4.2
Zak phases
In the following we will discuss how to measure the change of total Zak phase
= (⇡)
(0)
which is required (besides the Wilson loop phases 'W ) to obtain the Z2 invariant from
Eq.(6.8). The basic idea is, as in the Chern number protocol [200], to express it as a winding
(which is well-defined not only up to 2⇡):
Z ⇡
X
=
dky @ky (ky ) ⇡
(ky + ky )
(ky ).
(7.48)
0
ky
Since (ky ) is the sum of two Zak phases 'I,II
Zak , see Eq. (6.12), the latter can simply be
measured independently, provided that the bands of interest are separated by a sufficiently
large energy gap from each other. However, when accidental degeneracies are present or the
gap is simply too small to follow adiabatically (which is always the case close to the Kramers
degeneracies at the four TRIM), we can still infer the total Zak phase from non-Abelian loops.
For this purpose let us consider the general propagator Û (T ) within the (restricted) set
of bands to which the dynamics is constrained. In practice these will be the two Kramers
partners I, II and non-adiabatic transitions to the conduction bands can be neglected. Like in
the case of a single band, a geometric and a dynamical U (1) Berry phase can be identified,
i log det Û (T ) =
I
dk · trÂ(k) +
Z
T
0
dt tr Ĥ(k(t)),
(7.49)
when the time-dependent parameter k(t) returns to its initial value after time T . The proof
of this statement is a simple non-Abelian generalization of Berrys calculation [78] for the
(Abelian) Berry phase.
When k denotes quasi-momentum we will call the corresponding geometric phase the total
Zak phase,
I
=
dk · trÂ(k).
(7.50)
This, of course, is exactly the definition we gave in Eq.(6.12) already. Therefore we see that
it is sufficient to measure the determinant of the propagator,
(ky ) =
i log det Û (ky ) +
Z
T
0
dt tr Ĥ(kx (t), ky ).
(7.51)
Here, as in Eq.(7.46), the dynamical U (1) phase still has to be determined by varying the
driving force.
For a generic two-band model the propagator is given by a generic unitary matrix
✓
◆
⇤
↵
i⌘
Û = e ·
, |↵|2 + | |2 = 1,
(7.52)
↵⇤
152
CHAPTER 7. INTERFEROMETRIC MEASUREMENT OF Z2 TOPOLOGICAL
INVARIANTS
Figure 7.10: Spin-echo type measurement of the Wilson loop phase 'W = 'W (⇡) 'W (0).
Half the BZ is shown, with black dots denoting TRIM. All relevant propagators are shown.
such that i log det Û = 2⌘. We will discuss below how ⌘ can be measured using a combination
of interferometry and Bloch oscillations.
7.4.3
Experimental realization
We begin this subsection by commenting on the necessary degrees of freedom to realize the
Wilson loop scheme. In general, to perform interferometry one needs (at least) two auxiliary
”interferometric” pseudospin degrees of freedom. The first one (referred to as | *i) picks up
a phase '* that is to be measured while the second one (| +i) picks up '+ and serves for
comparison afterwards. The interferometric signal is '* '+ . Therefore '+ has to be known
(it may also be a suitable known function of '* ).
Note that the interferometric pseudospin degrees of freedom | *i, | +i have to be distinguished from the “spin” pseudospin degrees of freedom | "i, | #i which mimic the electron spin
of the QSHE. Therefore the Hilbert space in general consist of
| *i ⌦ | "i,
| *i ⌦ | #i,
| +i ⌦ | "i,
| +i ⌦ | #i.
(7.53)
Each of these sectors also contains motional degrees of freedom and we assume that the QSHE
is at least realized in the sector | *i ⌦ {| "i, | #i}.
We note that the twist scheme presented in section 7.3 relies only on interferometry between the bands. Therefore in this case linear combinations of | "i, | #i yield the interferometric pseudospins | *i and | +i, which are exactly the eigenstates of the Bloch Hamiltonian.
In the following we will discuss the case of two equivalent copies of the QSHE realized in
the two sectors defined by | *i and | +i.
Wilson loop phase
We start by discussing the measurement of the Wilson loop phase 'W = 'W (⇡) 'W (0).
The essential idea of this part is based on the schemes [200, 13] for measuring Zak phases within
a single band. To make the measurement more robust, we suggest a spin-echo type measurement as depicted in FIG. 7.10. In the movements along ky , * (+) atoms pick up geometric
Wilson loop phases 'W (⇡) ('W (0)), while geometric phases corresponding to movements
along kx cancel.
We assume an initial wavepacket of atoms in some superposition state | 0 , ki of bands
I, II at quasi-momentum k = ( ⇡, ⇡/2), and in the internal state | *i. A ⇡/2-pulse between
the internal states | *i, | +i then creates a superposition
|
1i
1
= p (| *i + | +i) ⌦ |
2
0 , ki.
(7.54)
7.4. WILSON LOOP SCHEME
153
A Zeeman field gradient for interferometric spins | *i, | +i,
Z
⇣
⌘
ĤZ = d2 r f0 · r ˆ †* (r) ˆ * (r) ˆ †+ (r) ˆ + (r)
(7.55)
with f0 / ey moves * (+) atoms to ky = ⇡ (ky = 0) at fixed kx = ⇡ and the state is given
by
⌘
1 ⇣
(+)
( )
(7.56)
| 2 i = p | *iÛ* | 0 , ( ⇡, ⇡)i + | +iÛ+ | 0 , ( ⇡, 0)i .
2
(±)
Here Û*,+ denote the propagators of the corresponding paths, see FIG.7.10.
Next, an equal potential gradient along ex is applied such that atoms move from kx = ⇡
at time t1 to kx = ⇡ at time t2 . We assume this to be done in a TR invariant fashion, i.e.
✓
◆
✓
◆
t2 t1
t2 t1
kx
t = kx
+ t ,
(7.57)
2
2
where kx (t) is a function of time t. Thereby atoms only pick up the U (1) phases 'U (kyTRIM )
from Eq.(7.46) as discussed in subsection 7.4.1 and their quantum state is described by
|
3i
1 ⇣
(+)
= p ei'U (⇡) | *iÛ* |
2
0 , (⇡, ⇡)i
( )
+ ei'U (0) | +iÛ+ |
0 , (⇡, 0)i
⌘
.
(7.58)
As pointed out in Sec.7.4.1 adiabaticity is only required with respect to the conduction band
in this step.
Finally, reversing the first part of the protocol and moving the atoms back to k =
(⇡, ⇡/2) = ( ⇡, ⇡/2) mod 2⇡ yields the final state
|
1 ⇣ i'U (⇡)
( ) (+)
p
i
=
e
| *iÛ* Û* |
4
2
(+)
( )
i'U (0)
| +iÛ+ Û+ |
0 , (⇡, ⇡)i + e
⌘
,
(⇡,
0)i
.
0
(7.59)
Note that dynamical Zeeman-phases due to the di↵erent Zeeman fields felt by *, + Eq.(7.55)
cancel when the protocol applied at kx = ⇡ reverses that at kx = ⇡.
( ) (+)
To realize a Ramsey interferometer, we have to make sure that Û* Û* = ei'y,* and
(+)
( )
Û+ Û+ = ei'y,+ only constitute dynamical phases but not geometric phases or band-mixing
between I and II. This can be realized either by a completely non-adiabatic protocol (with
aF
I II ) or a completely adiabatic protocol (with aF ⌧
I II ). In the former case
dynamical phases are negligible while non-Abelian geometric U (2) propagators cancel, i.e.
'y,*/+ ⇡ 0. In the latter case in contrast, there is no band-mixing between I, II and geometric
Zak phases cancel while non-vanishing dynamical U (1) phases 'y,*/+ / 1/F are picked up.
The Ramsey signal R , given by the phase di↵erence between the + and * components in
Eq.(7.59), thus yields R = 'U (0) 'U (⇡) + 'y,+ 'y,* . Using Eq. (7.46) we find that the
geometric part of the Ramsey signal is given by the Wilson loop phases,
R
=
'W + 'dyn .
|{z}
(7.60)
/1/F
Here 'dyn summarizes all dynamical phases, being inversely proportional to the driving
force F . Therefore repeating the whole cycle after rescaling the time-scale by some factor
allows to measure the dynamical phases, as long as adiabaticity with respect to the conduction
band is still fulfilled. Moreover we can see that symmetries of the band structure might be
helpful to minimize these dynamical phases and should be considered in a concrete setup.
154
CHAPTER 7. INTERFEROMETRIC MEASUREMENT OF Z2 TOPOLOGICAL
INVARIANTS
Figure 7.11: Spin-echo type measurement of the total Zak phase
= (ky + ky )
(ky ).
The two bands and the relevant propagators are shown. Note that two periods are shown in
kx -direction.
Total Zak phase
Now we turn to the measurement of total Zak phase winding Eq.(7.48). We discuss spin-echo
type measurements which directly yield the di↵erence (ky + ky )
(ky ) while canceling all
dynamical phases. The sequence described in the following is depicted in FIG. 7.11.
We assume starting with atoms in the upper band |ui at k = (0, ky ) in the state
p
| 1 i = |u, (0, ky )i ⌦ (| *i + | +i) / 2.
(7.61)
Then a Zeeman field gradient Eq.(7.55) along f0 / ex for *, + can be used to move the * atoms
in positive kx direction to k = (2⇡, ky ) and the + atoms in opposite direction to k = ( 2⇡, ky ).
After a displacement by ky using a potential gradient (equal for both interferometric spins
*, +) the sequence is reversed at ky + ky . The final state is given by
|
2i
⌘
1 ⇣
= p | *i ⌦ Û* |ui + | +i ⌦ Û+ |ui .
2
(7.62)
From Eq. (7.49) we find that dynamical phases vanish (including Zeeman phases from the
di↵erent potential gradients) and the total accumulated phase yields twice the change of the
total Zak phase,
I
⇣
⌘
i log det Û+† Û* = tr dk · Â ⌘ 2
.
(7.63)
C
Here C denotes the (counterclockwise) contour through the BZ shown in FIG. 7.11. Consequently it is sufficient to measure only det Û+† Û* , and according to Eq. (7.52) we have
⇣
⌘
i log det Û+† Û* = 2⌘+
2⌘* .
(7.64)
Next we assume that the two bands |u, li are individually addressable experimentally; This
is feasible with current experimental technology, see e.g. [268]. The population in the upper
band of the final state Eq. (7.62) is described by the wave function
|
ui
1
= p ei⌘* ↵* | *i + ei⌘+ ↵+ | +i .
2
(7.65)
After measuring the populations |↵*,+ |2 standard Ramsey pulses between the spin states | *i,
| +i can be used to obtain the phase-di↵erence,
u
= ⌘* + arg(↵* )
⌘+
arg(↵+ ).
(7.66)
7.5. SUMMARY AND OUTLOOK
155
Analogously one finds for the populations in the lower band when also starting in the lower
band
1
| l i = p ei⌘* ↵*⇤ | *i + ei⌘+ ↵+⇤ | +i
(7.67)
2
and the corresponding phase di↵erence is given by
l
= ⌘*
arg(↵* )
⌘+ + arg(↵+ ).
(7.68)
Finally combining these equations, we find that the change of the total Zak phase is
2
=
u
+
l.
(7.69)
Note that if ↵ is too small one may use a protocol which starts from atoms in the lower
band again but detects the resulting wave function in the upper band. A similar calculation
as above can be done and one can again infer the total Zak phase 2
.
7.5
Summary and outlook
Summarizing, we have shown that the Z2 invariant classifying time-reversal invariant topological insulators can be measured using a combination of Bloch oscillations and Ramsey
interferometry. No further assumptions about the system’s symmetry are required. The interferometric signal yields direct information about the topology of the bulk wavefunctions.
We presented two schemes which are both applicable to realizations of topological insulators
in ultra-cold atoms in optical lattices without the need of introducing sharp boundaries and
resolving any edge states. Similar schemes have already been realized experimentally in 1D
[13] and 2D [201] systems and discussed theoretically for 2D Chern numbers [200]. Unlike
these situations the measurement of the Z2 invariant requires non-Abelian Bloch oscillations
(i.e. some form of band switchings) and makes the interferometric protocol more involved.
Our first scheme (”twist scheme”) uses the fact that the Z2 invariant is the di↵erence
of time-reversal polarization at kyTRIM = 0 and kyTRIM = ⇡, which itself is a di↵erence of
Zak phases. Since standard time-reversal polarization is discontinuous however, its di↵erence
can not be formulated as a winding. To circumvent this issue we developed a continuous
generalization of time-reversal polarization P̃✓ , the winding of which gives the Z2 invariant,
Z ⇡
⌫2D =
dky @ky P̃✓ (ky ) mod 2.
(7.70)
0
We further laid out a measurement protocol for continuous time-reversal polarization,
employing a combination of Abelian (i.e. adiabatic) Bloch oscillations with Ramsey pulses
between the two valance bands required by TR symmetry. Such Ramsey pulses can be realized
by shaking the optical lattice and using the coupling of the bands through non-Abelian Berry
connections. We also pointed out that a general coupling scheme realizing the required Ramsey
pulses does not work since the phases of the corresponding coupling constants at di↵erent
points in the BZ are generally unknown. Our scheme is readily applicable in the suggested
experimental setup [270]. Most importantly, it does not require any additional degrees of
freedom to perform Ramsey interferometry.
The second scheme (”Wilson loop scheme“) uses a formulation of the Z2 invariant in terms
of non-Abelian Wilson loops. In particular our protocol relies on an expression which involves
156
CHAPTER 7. INTERFEROMETRIC MEASUREMENT OF Z2 TOPOLOGICAL
INVARIANTS
eigenvalues of Wilson loops along with total Zak phases,
✓
◆
1
1
⌫2D =
'W
mod 2.
⇡
2
(7.71)
The Wilson loop phase 'W is the di↵erence of polarizations at ky = ⇡ and ky = 0. We
showed that to measure the polarization of a band at time-reversal invariant momentum ky ,
the existence of the second (partly degenerate) Kramers partner can be ignored. This is a
direct consequence of TR symmetry.
Secondly the winding
of the total Zak phase is required. The total Zak phase is the
sum of the Zak phases of the two Kramers partners and therefore continuous throughout
the BZ. When the bands (the two Kramer’s partners) are separated by a sufficiently large
energy gap they can be measured independently, but we also showed how one can still reliably
measure their sum when Abelian Bloch oscillations are not applicable e.g. due to accidental
degeneracies. The experimental realization of the Wilson loop scheme requires a second copy
of the quantum spin Hall e↵ect that can independently be controlled, making it harder to
implement in some of the existing proposals.
Although for the formulation of the two protocols we restricted ourselves to two spatial
dimensions, our scheme is applicable to 3D TR invariant topological insulators as well. The
reason is that the 3D Z2 invariants (one strong and three weak ones) can be expressed as
products of 2D Z2 invariants corresponding to specific 2D planes within the 3D Brillouin zone
[32] (see Sec.6.2.2). These constituting 2D invariants can straightforwardly be measured with
our scheme.
Chapter 8
Interferometric Measurement of
Many-Body Topological Invariants
8.1
Outline and Introduction
Locally, interacting topological phases can not be distinguished from each other, making
their classification and detection challenging. Currently researchers are trying to classify
all topological orders [75, 76, 6], and to find physical realizations in setups ranging from
fractional quantum Hall systems to quantum spin liquids [10, 69, 281]. A major challenge
for experiments with such systems is to probe their non-local topological order. While it is
already challenging to identify topological order in numerical simulations, it is even harder to
do so in actual experiments. Many of the theoretically developed tools, like the entanglement
entropy [75] and spectrum [76], the topological ground state degeneracy on a torus or manybody topological invariants based on geometric phases [78, 27, 88] are hard to address directly
in an experiment.
In this chapter we focus on systems of interacting quantum gases, where individual atoms
can directly and fully coherently be controlled. We consider gapped topological phases of ultra
cold atoms and develop a new method for the detection of their topological order, see FIG.8.1.
To this end we suggest to investigate the topological excitations of the systems. In some cases
these excitations carry fractional charge or statistics [47, 72, 73, 7], which are characteristic for
the topological order in the incompressible ground state, and previously it was argued that
by direct manipulation of the excitations these properties can be measured experimentally
[263, 259, 264]. In addition, however, the e↵ective quasiparticle (qp) band structure can have
non-trivial topology, encoded in its geometric phases. (Topological superconductors [38] are
an example, where the topology is in fact defined through their Bogoliubov excitations.) Here
we show by a direct calculation that for integer and fractional Chern insulators (or quantum
Hall systems) the qp band structure is characterized by the same topological invariant as the
many-body system. It is unclear at present whether this is generically the case. We propose to
measure the topology of the qp band structure by coupling qp excitations to mobile impurities.
To investigate the topology of the qp band structure experimentally, we generalize the
interferometric scheme [13] described in Sec.6.2.1 for weakly interacting Bose-Einstein condensates. The single topological excitation takes the role of the condensate moving through
the BZ. In order to perform Ramsey interferometry, internal (pseudo) spin degrees of freedom
of the qp are required. They can be realized by coupling an impurity which carries spin
to the topological excitation, such that an impurity-qp bound state is formed. Because the
impurity is considered to be mobile and interacting with the many-body system, we refer
157
158
CHAPTER 8. INTERFEROMETRIC MEASUREMENT OF MANY-BODY
TOPOLOGICAL INVARIANTS
(a)
(b)
Figure 8.1: We suggest an experimental scheme for the measurement of many-body topological
invariants of interacting states with topological order. It can be applied e.g. to measure the
Chern number characterizing Laughlin states, as illustrated in (a). We couple a topological
excitation, e.g. a quasihole, to a mobile spin-1/2 impurity. When the impurity is tightly bound
to the excitation, it forms a topological polaron shown in (a). It has two internal spin states
(red and green), and can be labeled by its (quasi-) momentum q. The resulting band structure
is depicted in (b) for a generic 1D case. We suggest to measure the topological properties
of this band structure using tools developed for non-interacting systems by a combination of
Bloch oscillations and Ramsey interferometry. The resulting geometric Zak or Berry phase
picked up by opposite spin-components (b) yields the many-body topological invariant.
to the resulting bound state as a topological polaron (TP), in analogy to mobile impurities
interacting with a bath of phonons [17]. The two internal (pseudo) spin degrees of freedom
of the impurity translate into two spin states of the TP. Now the rest of the protocol can be
straightforwardly generalized from a non-interacting two-component BEC to a spin-1/2 TP
in a topologically non-trivial band structure, see FIG.8.1.
In this chapter we consider phases where topological invariants are explicitly known that
can distinguish states from di↵erent topological classes. For example the one-dimensional
Su-Schrie↵er-Heeger type models are characterized by a quantized geometric phase [81, 13]
which can be generalized to interacting systems [P2]; similarly the quantum Hall e↵ect is
characterized by the Chern number [27, 88], see Chap.1.2.
More generally we consider gapped phases which are characterized by a topological invariant ⌫0 that can be expressed in terms of geometric Berry phases. We show explicitly for
di↵erent models of interacting topological phases in one and two dimensions that the topological invariant ⌫TP characterizing a TP is directly related to the topological invariant ⌫0 of
the underlying many-body phase,
⌫TP ' ⌫0 .
(8.1)
Precise definitions of ⌫0 and ⌫TP will be given separately for every model, and the meaning
of the relation ”'” will be clarified. We explain in detail how the measurement described in
FIG.8.1 yields ⌫TP . This allows to unravel the topological order of the many-body state.
The chapter is based on results which are currently prepared for publication [P13], and it
is organized as follows. In Sec. 8.2 we introduce a generic model and solve it using the strong
coupling approximation. We define topological invariants on a general ground and explain
how they relate to one another. In Sec. 8.3 we apply our theory to long-range entangled
Chern insulators and the quantum Hall e↵ect at integer magnetic filling. In Sec. 8.4 we
generalize to fractional quantum Hall states and Chern insulators where qps are fractionally
charged. In Sec. 8.5 we discuss short-range entangled Mott insulating phases which can be
realized in current experiments with ultra cold atoms. We give an example for a many-body
Thouless pump, characterized by a many-body Chern number that can be measured using
our scheme. Additionally we show how (inversion-) symmetry-protected topological invariants
can be measured. In Sec. 8.6 we close with concluding remarks and an outlook.
8.2. THEORETICAL FRAMEWORK
8.2
159
Theoretical Framework
In this section we set up a general framework to describe our measurement scheme for manybody topological invariants. We present an approximate solution in the strong coupling limit.
This leads us to the definition of the topological invariants characterizing the system, based on
geometric phases. Making use of the approximate wavefunction we explain how TPs enable
a measurement of the bulk topological invariants of the underlying many-body system as
described in FIG.8.1.
8.2.1
The Model
We consider a gapped phase of a many-body system described by a local Hamiltonian Ĥ0 in
its bulk. We assume that a unique ground state | 0 i of Ĥ0 exists1 . Furthermore we assume
that we can create an excited state | a i containing a single qp a in the system2 . Assuming
translational symmetry of the system, we can label all qp states | a (q)i by their (quasi)
momentum q. We suggest to unravel the topology of the many-body system by measuring
the properties of these qp states. Some phases, like for instance the Moore-Read Pfaffian
fractional quantum Hall state [7], have di↵erent types of qp excitations a1 , a2 , ..., but here for
simplicity we shall not be concerned with such additional complications.
To investigate the topological properties of the qp excitation, we introduce a mobile impurity carrying spin one-half. It is described by a Hamiltonian
ĤI = ĤI0
ˆ z F · r̂,
(8.2)
where ĤI0 describes its kinetic energy and r̂ is the position operator of the impurity. It has
two internal states " and # which experience opposite forces ±F , as described by the Pauli
matrix ˆ z in Eq.(8.2). To bind the impurity to the qp a local attractive interaction Ĥint is
introduced between them. Its concrete form di↵ers from model to model, but for simplicity
we will assume throughout that it is independent of the impurity spin. Thus our system is
described by
Ĥ = Ĥ0 + ĤI + Ĥint .
(8.3)
In equilibrium, i.e. for F = 0, its ground state | TP (q, )i describes a TP, and it can be
labeled by (quasi) momentum q and the impurity spin =", #. The external force couples to
the (quasi) momentum q of the TP. For sufficiently small force this drives Bloch oscillations
d
where the (quasi) momentum changes according to dt
q = z F . By generalizing the noninteracting scheme [13] this allows to map out the topology of the TP band structure as
described in FIG.8.1. In particular this can be used to measure the TP topological invariants
Z
1
1D
⌫TP
=
dq h TP (q, )|irq | TP (q, )i,
(8.4)
2⇡ C
where C is a closed loop in quasimomentum space.
Below we will discuss how the TP invariant (8.4) can be measured and how it is related to
the topology of the underlying many-body state. To this end we find it convenient to formulate
1
What we have in mind here is a many-body system on a closed surface such that edge e↵ects due to gapless
boundary modes can be ignored. If the genus g 6= 0 of the surface di↵ers from zero, the ground state can have
a finite topological degeneracy Ntop . In this case | 0 i should be thought of as a short-hand notation of the
(n)
entire ground state manifold | 0 i, n = 1, ..., Ntop .
2
This may require a non-local operation on the ground state | 0 i if the quantum numbers of elementary
excitations fractionalize.
160
CHAPTER 8. INTERFEROMETRIC MEASUREMENT OF MANY-BODY
TOPOLOGICAL INVARIANTS
Eq. (8.4) in terms of twisted boundary conditions, allowing us to change the momentum q of
the TP by coupling to the gauge field. Later we discuss exact models of TPs where Eq.(8.4)
can be directly applied.
8.2.2
Strong coupling approximation
The strong coupling approximation is inspired by Landau and Pekar’s treatment of the conventional polaron problem in solids [17, 18]. Essentially we assume the impurity to be light
enough and its interaction with the qp to be strong enough to follow the qp adiabatically.
To make this more precise, let us assume that the energy scale associated with qp dynamics
is given by Ja . Typically, in systems with a large flatness ratio3 which we want to consider
here, Ja ⌧ 0 is well below the bulk gap 0 of the many-body phase. The characteristic
energy scale of free impurity dynamics in ĤI0 will be denoted by JI , and it is proportional
to the inverse of the impurity mass. We will denote the scale of impurity-qp interactions by
V . Then the strong coupling approximation amounts to the assumption that dynamics of the
impurity happens on much shorter timescales than for the qp, i.e.
Ja ⌧ JI , V
(strong coupling).
(8.5)
In addition, to prevent the impurity from coupling to high-energy excitations in the bulk of
the many-body system, we assume
JI , V ⌧ 0 .
(8.6)
In the strong coupling regime (8.5) we can treat the qp dynamics perturbatively. Thus we
can start from a local basis set {| a (Rj )i}Rj of qp states, which we will not specify further at
this point4 ; Rj label the sites around which the basis states are localized. When constructed
as a Wannier basis from qp states in (quasi) momentum space | a (q)i,
|
a (Rj )i
1 X
:= p
e
N q qi
iqi ·Rj
|
a (qi )i,
(8.7)
we can furthermore assume that the localized states are orthogonal h a (Ri )| a (Rj )i = i,j .
Here Nq is the number of discrete momentum states qj , i.e. j takes values 1...Nq .
Now we make an ansatz for the TP wavefunction, where the impurity adiabatically follows
the qp,
| TP (Rj , )i = | a (Rj )i ⌦ | I (Rj , )i.
(8.8)
To determine the impurity wavefunction | I (Rj , )i for a given hole wavefunction | a (Rj )i
we use the variational principle. Minimizing the energy h TP |Ĥ| TP i in equilibrium, i.e. for
F = 0, yields
⇣
⌘
ĤI0 + h a (Rj )|Ĥint | a (Rj )i EI | I (Rj , )i = 0.
(8.9)
The second term on the left-hand side describes an e↵ective potential seen by the impurity.
For the TP to exist we require this equation to have a bound state solution with the TP
binding energy EI (assuming that the free impurity has zero energy). There can also be
excited bound states, but we will not discuss them any further here. Note that because of the
translational symmetry of the problem, the TP energy EI is independent of Rj .
3
The flatness ratio f = /! of a Bloch band is defined as the ratio of the band gap
and the band width
!. A large flatness ratio is a crucial ingredient e.g. to stabilize fractional Chern insulators.
4
Note, however, that the meaning of local is rather weak here, and may include Wannier functions which
decay slower than exponential.
8.2. THEORETICAL FRAMEWORK
161
Using the strong coupling bases {| TP (Rj , )i}j in position- and {| TP (qi , )i}i in (quasi)
momentum-space, we conclude that the e↵ective TP Hamiltonian at low energies can be
expressed as
ĤTP = Ea + EI +
X
✏TP (qi )|
TP (qi ,
)ih
i,
F
TP (qi ,
X
j,
)|
Rj ( 1) |
TP (Rj ,
)ih
TP (Rj ,
)|. (8.10)
Here ✏TP (q) denotes the e↵ective TP dispersion relation and Ea is the energy of the free qp.
( 1) = 1 for =" and ( 1) = 1 for =#. In the second line we assumed that the TP
bound state is centered around Rj , i.e. h I (Rj , )|r̂| I (Rj , )i = Rj . In the rest of this
section, for simplicity, we only treat the case =" with z = 1 and drop the spin index.
8.2.3
TP invariant
As outlined in the introduction, the essence of our scheme is to investigate qp excitations in
a many-body system via an impurity. Using the strong coupling approximation, we will now
explain how the topology of the many-body system can be mapped out by TPs. We show
that the measurement scheme described in FIG.8.1 yields a geometric phase 2⇡⌫TP consisting
of two parts,
⌫TP = ⌫TPext + ⌫TPint .
(8.11)
The internal TP invariant ⌫TPint reflects the internal structure of the TP bound state.
More concretely, in the strong coupling approximation, it is given by the geometric phase
picked up by the impurity wavefunction | I (t)i when the force F is applied. It may even
contain contributions from ĤI0 , but for simplicity we will consider only cases where the band
structure of ĤI0 is trivial.
The external TP invariant ⌫TPext maps out the topology of the qp excitation in the manybody system. Concretely, in the strong coupling approximation, it is given by the geometric
phase picked up by the qp wavefunction | a (t)i when the boundary conditions of the manybody system are twisted. We will explain here that the force F acting on the impurity has
exactly the same e↵ect as twisting boundary conditions of the many-body system when it is
placed on a torus.
We thus find that TPs probe di↵erent topological aspects of the qp excitation at the same
time. In Secs.8.3, 8.4 and 8.5 we will discuss examples where the full TP invariant ⌫TP directly
probes the many-body topological invariant ⌫0 . In the case of dimerized Mott insulators in
a Su-Schrie↵er-Heeger type super-lattice model (see Sec.8.5) the external invariant vanishes
and the internal TP invariant directly probes the system’s topology. For fractional quantum
Hall systems and Chern insulators discussed in Secs. 8.3 and 8.4 in contrast, the internal TP
invariant vanishes and the external (or qp) invariant carries information about the system’s
topology.
Internal TP invariant
The internal TP invariant describes how the wavefunction | I (Rj )i is modified when the force
F is applied to the impurity. The impurity wavefunction is defined as the ground state of the
strong coupling impurity Hamiltonian in Eq. (8.9). From the static case F = 0 we obtain the
TP bound state, but when a non-vanishing force F is applied – along ex say – for some time
162
CHAPTER 8. INTERFEROMETRIC MEASUREMENT OF MANY-BODY
TOPOLOGICAL INVARIANTS
(a)
(b)
Figure 8.2: (a) A force F is acting on a qp excitation of a topologically ordered system placed
on a cylinder via an impurity. The impurity is tightly bound to the qp to form a TP. (b) The
force F from (a) is equivalent to twisted boundary conditions in the many-body system with
a time-dependent twist angle ✓˙x / F .
t, the system picks up an additional phase
2⇡⌫TPint =
tF h I (Rj )|x| I (Rj )i,
(8.12)
see Eq.(8.2). At first, this expression appears to be a dynamical phase. However, it should
rather be interpreted as a geometrical phase which does not vanish when extrapolated to
t ! 0 or F ! 1, because the product F t always takes a universal value. It describes the
change of TP momentum, F t = qx , which takes a value qx = 2⇡/ax when the TP crosses
one BZ (with ax denoting the lattice constant in x-direction), see FIG.8.1 (b).
External TP invariant
The external TP invariant describes the geometric phase ' = 2⇡⌫TPext picked up by the manybody qp wavefunction | a (t)i when the force F is applied to the TP via the impurity. To derive
its value we will now relate it to the response of the many-body system to twisted boundary
conditions. These are commonly used by theorists to detect topological order in numerical
simulations, and we thus demonstrate that this established theoretical measure can be carried
over to experiments by the use of TPs. In the following Subsection 8.2.4 we will summarize
concrete formulas defining topological invariants (for TP, qp and the incompressible ground
state) by twisted boundary conditions.
The starting point for our discussion is the strong coupling TP Hamiltonian in Eq.(8.10).
It implies that, through the impurity, the constant force F is acting on the qp. Its dynamics
is described by an e↵ective Hamiltonian
X
Ĥa = Ĥ0 F ·
Rj | a (Rj )ih a (Rj )|.
(8.13)
j
Note that in this expression we neglected the renormalization of the qp dispersion due to the
polaronic dressing by the impurity, which is still present in Eq.(8.10). This will be sufficient,
however, to derive the geometric phase of the qp wavefunction which has a purely topological
origin.
To describe the e↵ect of the force F acting on the qp, let us perform a Gedanken experiment and place the many-body system on a cylinder, see FIG.8.2 (a). Then we move the qp
around its perimeter adiabatically, which requires an amount of energy E = Fx Lx where
Lx is the circumference of the cylinder. Alternatively we can replace the force F acting on
the qp by an electric field F̃ which couples directly to the particles of the many-body system.
We set the elementary charge q0 = 1 equal to one such that electric field and forces can be
8.2. THEORETICAL FRAMEWORK
163
R
interchanged. Hence the coupling reads F̃ · dd r r ˆr† ˆr , where d is the dimensionality of
the system and ˆr is the quantum field describing the many-body system. Now the amount
of energy which is required to move the qp around the cylinder adiabatically is given by
E = qa F̃x Lx . Here qa is the charge of the qp excitation, which is defined as
Z
qa = q0 dd r (⇢a (r) ⇢0 ) .
(8.14)
The density of the incompressible ground state |
qp state | a i by ⇢a (r). Therefore if we choose
0i
was denoted by ⇢0 and the density of a
F̃ = F /qa
(8.15)
we can describe the force acting on the qp by an electric field acting on the incompressible
many-body system.
H The electric field F̃ acting on the many-body system has a non-vanishing0 circulation
L dr · F̃ 6= 0 for a loop L around the cylinder. On the other hand, for a loop L which does
not wind around the cylinder the circulation vanishes. This field configuration is equivalent to
d
magnetic flux ✓x threading the cylinder, at a constant rate of change dt
✓x / F̃x , see FIG.8.2
(b). This magnetic flux generates twisted boundary conditions [88], with a twist angle
✓x (t) = Fx Lx t/qa .
(8.16)
By using the same argument for the other directions all twist angles ✓x , ✓y , ... can be related
to the respective components of the force F .
8.2.4
Topological invariants: general considerations
Now we formally define the topological invariants of our system in terms of geometric phases
picked up when twisting boundary conditions of the many-body system. We distinguish
between three di↵erent invariants, ⌫0 defined for the incompressible ground state | 0 i, ⌫a
defined for the qp state | a i and the external contribution to the TP invariant ⌫TPext . In
the following sections we will discuss the relations between them for di↵erent specific models.
Our results suggest that they are deeply related to one another.
One-dimensional systems
Ground state invariant.– We start with 1D systems, where we assume that the incompressible ground state | 0 (✓x )i depends on some parameter ✓x = 0...2⇡ from a compact periodic
parameter space. Then the 1D topological invariant is defined as the geometric Berry phase
which is picked up when ✓x is varied adiabatically from 0 to 2⇡,
⌫01D
1
=
2⇡
Z
2⇡
0
d✓x h
|
0 (✓x )|[email protected]✓x | 0 (✓x )i .
{z
=:A0 (✓x )
}
(8.17)
Without assuming further symmetries, ⌫01D is not quantized but takes arbitrary values from
the interval [0, 1). Imposing additional symmetries leads to constraints on possible values of
⌫01D , see e.g. [153].
More specifically, we want to consider 1D systems on a closed ring and impose twisted
CHAPTER 8. INTERFEROMETRIC MEASUREMENT OF MANY-BODY
TOPOLOGICAL INVARIANTS
164
boundary conditions [88, 282]. That is the many-body wavefunction fulfills
(x1 , ..., xj + Lx , ..., xN ) = ei✓x (x1 , ..., xj , ..., xN )
(8.18)
for all j = 1, ..., N with N denoting the total particle number and Lx the system size. Then
the 1D topological invariant is obtained by using the twist angle ✓x in the definition (8.17).
In this case it has been shown that e.g. inversion symmetry can lead to a quantization of the
invariant to values ⌫01D = 0, 12 [81, 13] [P2].
Quasiparticle invariant.– The topological invariant of a qp state | a (q; ✓x )i can be defined
analogously as in Eq.(8.17). Following this definition however, the qp invariant would be
q-dependent, which makes sense as long as q is a good quantum number. In the particular
setting with twisted boundary conditions, however, we notice an interesting relation between
states in di↵erent q-sectors. The e↵ect of twisting the boundary conditions can equivalently
be interpreted as insertion of magnetic flux through the center of the ring (when we promote
the 1D system into 3D for now). This induces a circular electric field around the 1D ring,
corresponding to a homogeneous force acting on the particles. Therefore we expect the (quasi)
momentum q of the qp to change upon twisting the boundary conditions. More precisely
this statement means: starting from a state | a (q; 0)i at ✓x = 0 and inserting magnetic flux
adiabatically creates the state | a (q; 2⇡)i ' | a (q+ q; 0)i up to a phase factor. Note, however,
that the value q depends on the particular state we consider, and figuring out its exact value
may be difficult.
As an example, let us apply the argument to an incompressible Mott insulating state | 0 i
in 1D. Without any qp excitation there is only exactly one low-energy state | 0 i at q = 0 and
the above statement is trivial with q = 0. For the simplest example of a free particle in a
1D lattice, on the other hand, the relation can easily be proven and one finds q = 2⇡/Lx .
This scenario also captures hole type excitations | a (q)i in a 1D Mott insulator of hard-core
bosons on a lattice (the hole can be described by a free particle-like excitation).
In the last paragraphs we argued that low-energy sectors of di↵erent (quasi) momentum
| a (q; ✓x )i can transform into each other upon twisting boundary conditions. Thus we can
consider the entire set of qp states {| a (q; ✓x )i}q and define a single topological invariant as
⌫a1D =
1
2⇡
Z
2⇡
0
d✓x trq Âa (✓x ),
Ai,j
a (✓x ) = h
a (qi ; ✓x )|[email protected]✓x | a (qj ; ✓x )i.
(8.19)
Here trq denotes a trace over all discrete momentum states {| a (qj ; ✓x )i}j=1...Nq .
In Eq.(8.19), by taking the trace, we essentially added up the invariants of all states
defined at values of qj . This makes sense if we discard the possibility of sub-band formation:
We can imagine cases where the band {| a (qj )i}j=1...Nq breaks up into two (or more) classes
{| a (qj )i}j2J and {| a (qi )i}i2I such that states from the first class J can never transform into
states of the second class I upon twisting boundary conditions. This can have two reasons.
Firstly, it can be an intrinsic topological e↵ect, expected for instance when the topological
phase has non-Abelian excitations. Twisting the boundary conditions can not transform one
topological superselection sector (for a definition see e.g. [11]) into another. Here we will not
consider this possibility however. A second reason for splitting of the qp band can be pure
energetics. In a general system qp dynamics have an associated energy scale Ja which can
lead to the formation of multiple bands separated by sub-gaps with characteristic scale sub .
As an example consider a (2D) free fermion Chern insulator in a Hofstadter model at flux
↵ = 1/s per plaquette, with s > 2 an integer. When the lowest band, corresponding to the
lowest Landau level in the limit ↵ ! 0, is filled we can create a hole excitation by removing
8.2. THEORETICAL FRAMEWORK
165
one fermion. Because of the particle-hole symmetry, the single-hole spectrum is described
by the Hofstadter butterfly at the same value of ↵ = 1/s. When ↵ is modified slightly to
↵0 = ↵ ✏ = r0 /s0 for incommensurate r0 and s0 , the many-body bulk gap does not close, but
the hole band splits up into sub-bands with di↵erent Chern numbers at r0 6= 1 [96]. The sum
of these Chern numbers, however, is the same as the Chern number of the lowest Landau level
(namely one), independent of the precise value of ↵0 .
When there is a term in the Hamiltonian which mixes these sub-bands, only their total
topological invariant (8.19) is relevant. In our case the sub-gaps are assumed to be small
compared to the coupling of the impurity to the qp, i.e.
sub
⇠ Ja ⌧ JI , V,
(8.20)
see Eq.(8.5), such that sub-bands will hybridize in general. Thus, in the measurement described in FIG.8.1, sub-gap e↵ects can not be resolved in general.
TP invariant.– Now we can give a precise definition of the external TP invariant by considering the e↵ect of twisting boundary conditions in the many-body system. In straightforward
generalization of Eq.(8.19) it can be written as
1D
⌫TPext
1
=
2⇡
Z
2⇡
0
d✓x trq Âext
TP (✓x ),
(8.21)
where
Aext
TP (✓x )
i,j
=h
TP (qi ; ✓x )|[email protected]✓x | TP (qj ; ✓x )i.
(8.22)
Here the TP wavefunction | TP (qj ; ✓x )i depends on the twist angle ✓x in the many-body
system.
Eq. (8.21) is still sufficiently di↵erent from Eq.(8.4), which was the main motivation for
the measurement scheme. The main di↵erence lies in the way how we couple to the total TP
momentum q. In Eq.(8.4) we assumed that q can be adiabatically changed by anticipating its
coupling to the force F acting on the impurity, q̇ = F . In Eq.(8.21), in contrast, we couple to
the total system momentum q (which equals the TP momentum) only indirectly via the gauge
field ✓x . Most importantly, the gauge field ✓x couples to the underlying incompressible manybody system. Ultimately both approaches only di↵er in a gauge transformation, yielding the
same e↵ective force acting on the TP. We will discuss the e↵ects of such a gauge choice in
detail for integer Chern insulators in Sec.8.3.1.
Two-dimensional systems
We want to consider 2D systems where topological invariants can be defined in terms of
geometric Berry phases. We discuss two scenarios, both of which correspond to an extension of
the parameter space to ✓x ! (✓x , ✓y ) by inclusion of a second periodic dimension ✓y = 0...2⇡.
In the first, truly 2D case the physical system is realized in two spatial dimensions. In
this case we consider many-body systems on a torus, where ✓y is the angle defining twisted
periodic boundary conditions along the y-direction. Second we consider the (1+1)D case
where the many-body system is realized in one spatial dimension, while the Hamiltonian
is characterized by a second externally tunable parameter ✓y . Such (1+1)D systems are
characterized by topologically protected quantized pumps, first considered by Thouless [87]
and later generalized to 1D models of interacting bosons [174]. Both times the Hamiltonian
is assumed to be 2⇡-periodic, Ĥ0 (✓y ) = Ĥ0 (✓y + 2⇡), and in both cases the corresponding
topological invariant is a many-body Chern number ⌫02D = C [88], see also Sec.1.2.2.
166
CHAPTER 8. INTERFEROMETRIC MEASUREMENT OF MANY-BODY
TOPOLOGICAL INVARIANTS
The 2D or (1+1)D topological invariants (Chern numbers) can be defined as windings of
the corresponding 1D invariants [82],
⌫⌧2D
=
Z
2⇡
0
d✓y @✓y ⌫⌧1D (✓y ),
⌧ = 0, a, TP.
(8.23)
In this expression, together with the wavefunctions | (q, ; ✓x , ✓y )i, the 1D invariants ⌫ 1D (✓y )
depend explicitly on the second parameter ✓y .
Note that in the last equation the integral can formally be solved,
⌫⌧2D = ⌫⌧1D (2⇡)
⌫⌧1D (0).
(8.24)
This shows that the 2D Chern numbers are quantized, because 1D invariants are defined via
Berry phases which are only well-defined up to 2⇡. Hence ⌫01D (2⇡) = ⌫01D (0)mod1 and in order
to calculate (or measure) the 2D invariants, the integrand in Eq.(8.23) has to be evaluated
explicitly everywhere and not only at the integration boundaries.
8.2.5
Strong coupling external TP invariant
Now we will discuss in more detail the external TP invariant, using the strong coupling
approximation. In particular we show that it is identical to the qp invariant,
⌫TPext = ⌫a .
(8.25)
Hence the external part of the TP invariant maps out the qp topology and – provided the
internal TP invariant is known – this enables a measurement of the qp invariant using TPs.
To derive the last equation, we use the strong coupling wavefunction from Eq.(8.8),
| TP (Rj , ; ✓x )i = | a (Rj ; ✓x )i ⌦ | I (Rj , ; ✓x )i where ✓x is the twist angle in the manybody system. The U (Nq )-invariance of the 1D invariant (8.21) allows us to evaluate it in
an arbitrary basis. Choosing the localized states and using their orthogonality as well as
1D
trq 1 = Nq , we find that ⌫TPext
= ⌫a1D + ⌫I1D , where an additional geometric phase of the
impurity wavefunction appears,
⌫I1D
Z
1 X 2⇡
=
d✓x h I (Rj ; ✓x )|[email protected]✓x | I (Rj ; ✓x )i.
2⇡
0
(8.26)
j
Our claim is that ⌫I1D = 0. To show this, we note that the impurity wavefunction | I (Rj ; ✓x )i
depends on ✓x only via the e↵ective potential h a (Rj ; ✓x )|Ĥint | a (Rj ; ✓x )i, see Eq.(8.9). This
potential is determined by the density distribution ⇢(r; ✓x ) of the many-body system, which is
a physical observable and thus gauge invariant. From this gauge invariance it follows that, in
the thermodynamic limit (i.e. for large system size), the density distribution is independent
of the twist angle, @✓x ⇢(r; ✓x ) = 0, and hence also @✓x | I (Rj ; ✓x )i = 0.
8.3
Integer Chern insulators and Integer Quantum Hall e↵ect
We will now consider experimentally relevant models of long-range entangled topological
phases of fermions in 2D, with integer-quantized topological invariants. We focus on the
integer quantum Hall e↵ect in a homogeneous magnetic field, and its cousin, the integer
Chern insulators. We start by presenting a concrete tractable model of an integer Hofstadter
Chern insulator of non-interacting fermions, with a TP consisting of a hole coupled to an
8.3. INTEGER CHERN INSULATORS AND INTEGER QUANTUM HALL EFFECT 167
impurity. This serves as a check of the strong coupling approximation developed in the last
section. Then we briefly discuss how our treatment can be generalized to systems of interacting fermions. We compare exact numerical results from the di↵erent approaches. Finally we
use the strong coupling approximation to investigate TPs for the integer quantum Hall e↵ect
in a homogeneous magnetic field.
To illustrate our method, we first discuss models of free fermions ĉq,n where the Fermi
energy lies within an energy gap of the single-particle band structure. When q is the quasimomentum and n = 1, 2, ... labels the single-particle bands, the incompressible many-body
state is given by a product state over all occupied bands
Y Y
| 0i =
ĉ†q,n |0i.
(8.27)
q n occ
For Chern insulators, q labels quasimomenta, whereas in the quantum Hall e↵ect these are
Landau sites (the concrete meaning depends on the gauge choice). The topology of such a
system is characterized by the Chern numbers Cn of the occupied bands, and they can be
defined by twisting boundary conditions for a system on a torus. The topological invariant
P
is given by the sum over Chern numbers of occupied bands, ⌫02D = n occ. Cn . Note that
even in the presence of interactions, the state (8.27) provides a reasonable description of the
true ground state, as long as the interaction energy is small compared to the bulk gap. In the
following, for simplicity, we consider only the case when a single band is occupied by fermions.
To detect the many-body Chern number of the state | 0 i we suggest to measure the Chern
number of a hole-type excitation in the system. A hole with momentum q can be described
by the operator â†q = ĉq for which the incompressible state | 0 i acts as the vacuum (we
dropped the band-index n = 1 here). Now we will discuss how the Chern number ⌫a2D of a
hole state | a (q)i is related to the Chern number ⌫02D of the incompressible ground state | 0 i.
Specifically we will show that the qp Berry phase has the form
⌫a2D = ⌫02D N
⌫02D ,
(8.28)
where N denotes the quantized number of orbitals in the occupied band.
The first term in Eq.(8.28) is a bulk contribution depending only on two universal numbers,
the bulk Chern number ⌫02D and the number of available orbitals. This contribution would
be present even in the absence of a hole, and it is an artifact originating from the coupling
to the TP via a gauge field acting on the many-body system. We may thus discard this bulk
contribution. Much more interesting is the second term, which is independent of the system
size and up to a minus sign equal to the bulk topological invariant. The minus sign in this
expression corresponds to the negative charge of the quasihole excitation, qa = 1. Hence,
ignoring the irrelevant bulk contribution, the TP invariant is given by
2D
⌫TP
=
⌫02D .
(8.29)
By exact numerical calculations in small systems we proof that this relation holds true even
when interactions between the fermions are taken into account.
8.3.1
Topological invariants
To understand the e↵ect of twisted boundary conditions, we consider single-particle Bloch
states on a torus
2⇡
ik·x
uk (x),
k=
n,
(8.30)
k (x) = e
L
168
CHAPTER 8. INTERFEROMETRIC MEASUREMENT OF MANY-BODY
TOPOLOGICAL INVARIANTS
(a)
(b)
ky
ky
kx
kx
Figure 8.3: (a) Integer Chern insulator with 9 non-interacting fermions filling up all Nx ⇥Ny =
9 sites. When applying a full twist of boundary conditions (✓x = 0 ! 2⇡, ✓y = 0 ! 2⇡, ✓x ! 0,
✓y ! 0, in this order) each fermion encircles a plaquette of size 2⇡/L ⇥ 2⇡/L in reciprocal
space (blue shaded for upper right fermion). (b) When a single fermion is removed from the
insulator, 3 ⇥ 3 = 9 low- energy hole states can be constructed, one of which is shown here.
When applying a full twist of boundary conditions, each fermion again covers a small area of
the BZ (blue shaded). The hole, too, encircles only the same small part of the BZ (lower left
corner).
with n = (nx , ny ) being a vector of two integers and L denoting the linear system size.
These Bloch wavefunctions fulfill periodic boundary conditions, i.e. k (x + L) = k (x). To
construct the correct eigenfunctions for twisted boundary conditions (twist angles ✓ = (✓x , ✓y ))
we displace the quasimomentum by ✓/L,
k (✓, x)
= ei(k+✓/L)·x uk+✓/L (x).
(8.31)
These wave functionsfulfill k (✓, x + L) = k (✓, x)ei✓·L/L , and one easily checks that they
are proper eigenfunctions of the lattice Hamiltonian. Most importantly, we observe that a 2⇡
twist – along x say – results in an adiabatic change of the momentum by kx = 2⇡/L.
From FIG.8.3 (a) we can now read o↵ the topological invariant of the many body ground
state | 0 i. When boundary conditions are adiabatically twisted, first by 2⇡ in x-direction
then by 2⇡ in y-direction, followed by 2⇡ in x -direction and then 2⇡ in y-direction, each
fermion covers an area (2⇡/L)2 and picks up a geometric phase which is given by the integral
of the Berry curvature F(k) over this area. Hence the geometric phase picked up by the
many-body wave function is
Z
2⇡⌫02D =
BZ
d2 k F(k),
(8.32)
which is another way of expressing the Chern number [82], see Eq.(1.24).
In the case of a single hole excitation of the Chern insulator, see FIG.8.3 (b), there are
Nx ⇥Ny low-energy hole states (with Nx(y) the number of momentum-states in x(y) direction).
In order to transform the hole state into itself adiabatically by twisting boundary conditions,
Nx (Ny ) twists by 2⇡ are required in x (y) direction. The geometric phase ' picked up when
twisting first by 2⇡Nx along x, then by 2⇡Ny along y, followed by 2⇡Nx along x and then
2⇡Ny along y, is given by the total Chern number of the set of hole states
'=
XZ
q
2⇡
0
d✓x d✓y Fq (✓) = 2⇡⌫a2D .
(8.33)
Here Fq (✓) = r✓ ⇥ h a (q; ✓)|ir✓ | a (q; ✓)i is the Berry curvature of the hole state.
On the other hand, from FIG.8.3 (b) we can read o↵ ' by noting that every fermion covers
the area of the BZ and thus picks up a phase 2⇡⌫02D . Hence
' = (Nx Ny
1)2⇡⌫02D ,
(8.34)
8.3. INTEGER CHERN INSULATORS AND INTEGER QUANTUM HALL EFFECT 169
and by combining this result with the last equation we obtain
✓
◆
2D
2D
⌫ a = ⌫ 0 1 Nx Ny .
| {z }
(8.35)
=N
To explain the second term in the last equation, which depends on the system size N ,
let us consider the relation between the Chern number ⌫a2D and the Hall current Ix which is
pumped across the system when ✓y changes by Ny ⇥ 2⇡. In the incompressible ground state
| 0 i, a macroscopic current Ix / Nx Ny ⌫02D flows through the system. Next, consider the
quasihole state and apply the electric field along y again (i.e. twist ✓y by Ny ⇥ 2⇡). Still,
the corresponding current is macroscopic, but it will be reduced by the microscopic amount
⌫a2D because one fermionic state is not occupied. This corresponds to the hole, and we are
interested only in this microscopic correction to the ground state result.
An alternative interpretation of this result, valid in thermodynamic limit Nx , Ny ! 1, is
to assign a Chern number ⌫a2D = ⌫02D to the hole band defined by the states | a (q)i = â†q | 0 i.
We will now adopt this picture and show how this Chern number can be measured.
8.3.2
TP in the Hofstadter Chern insulator - single hole approximation
We will now discuss a concrete model of a TP in a Chern insulator. To this end we restrict ourselves to a single hole excitation and neglect particle-hole fluctuations caused by the impurity.
Then we make use of the particle-hole symmetry in the free-fermion problem and formulate
an e↵ective single-particle Hamiltonian for the hole excitation. Specifically we consider the
Hofstadter model [96] for fermions at magnetic filling ⌫ = 1. The only free parameter in this
model is the magnetic flux per plaquette ↵ (in units of the magnetic flux quantum).
The following discussion serves two main purposes. First, in a simple two-particle problem
we illustrate the mathematical techniques used to describe TPs. Second, we show that the
single-hole approximation is a powerful tool for predicting TP properties like e.g. their binding
energy.
The Model
The Hamiltonian describing a single hole in a Hofstadter Chern insulator reads (in Landau
gauge)
i
Xh
Ĥ0 = t
e i2⇡↵n â†m+1,n âm,n + â†m,n+1 âm,n + h.c. .
(8.36)
m,n
Here t is the hopping element between the sites of the square lattice, which are located at
rm,n = maex + naey where m, n are integers and a is the lattice constant. Note that we use
second quantization for notational convenience, although we restrict ourself to only a single
P
†
hole, i.e.
m,n âm,n âm,n = 1 for all relevant states.
Next we couple the hole to the impurity atom, which is described by operators b̂m,n . We
assume the impurity to be confined to the same lattice as the fermions (and the hole), but to
be insensitive to the magnetic flux. Thus the free impurity Hamiltonian reads
i
Xh †
X
ĤI = J
b̂m+1,n b̂m,n + b̂†m,n+1 b̂m,n + h.c.
F·
rm,n b̂†m,n b̂m,n ,
(8.37)
m,n
m,n
where J denotes the impurity hopping and F is the externally applied force. We omitted
the spin-index at this point because it is only required for carrying out the interferometric
170
CHAPTER 8. INTERFEROMETRIC MEASUREMENT OF MANY-BODY
TOPOLOGICAL INVARIANTS
protocol later but does not a↵ect the physics of the TP directly. We will model the interaction
between impurity and hole by a local attractive delta-potential,
X
Ĥint = V
â†m,n âm,n b̂†m,n b̂m,n .
(8.38)
m,n
The model described above can be realized with ultra cold atoms. The Hofstadter Hamiltonian (8.36) has been implemented [108, 109, 110] and fermions have successfully been loaded
into the Chern band of the related Haldane model [124]. In such setups spin-1/2 impurities
could be realized by di↵erent internal (e.g. hyperfine) states |±i of the atoms [283]. Then
by choosing the optical beams required for implementing an artificial gauge field such that
the impurity states |±i are not a↵ected, the Hamiltonians (8.36), (8.37) can be realized with
t = J. When local s-wave interactions between fermions of di↵erent spins are included, also
the term in Eq.(8.38) can be implemented, and the interaction strength V could be tuned by
using Feshbach resonances [284].
Solution in the excitation-centered polaron frame
To solve the two-particle problem Eqs. (8.36) - (8.38) we apply the Lee-Low-Pines (LLP or
polaron) unitary transformation [19], taking us into the so-called polaron frame. This allows
us to explicitly identify the quasimomentum of the TP as a conserved quantity and to calculate
the TP dispersion relation as well as its topological invariant.
The excitation-centered polaron transformation allows us to switch into a frame where
the hole is localized in the central magnetic unit cell. This is achieved by translating the
impurity in real space, by an amount defined by the hole position. In addition we perform a
time-dependent gauge transformation to include the e↵ect of the external force F ,
h
i
ÛLLP (t) = exp iR̂a · (p̂I + F t) .
(8.39)
Here p̂I denotes the impurity momentum, which is the generator of infinitesimal translations
of the impurity. The hole position operator R̂a on the other hand generates translations of
the hole, but in momentum-space.
The detailed calculation in Appendix G shows that the e↵ective Hamiltonian in the polaron
frame is of the form,
X †
TP
ĤTP =
âk,µ âk,µ0 ⌦ Ĥµµ
(8.40)
0 (k),
k,µ,µ0
where the TP band Hamiltonian is given by
TP
Ĥµµ
0 (k) = hµµ0 (k
Ft
p̂I ) +
µµ0
⇣
ĤI
⌘
V b̂†0,µ b̂0,µ .
(8.41)
This form will be particularly usefull to understand the strong coupling approximation, as
well as for numerical calculations. It contains the hole Bloch Hamiltonian hµµ0 (k), where µ, µ0
label sites within the magnetic unit-cell. The impurity is subject to a localized potential of
strength V in the central unit cell, where b̂j,µ annihilates the impurity on site µ in the j-th
magnetic unit cell. The strength of the potential furthermore depends on the hole population
P †
k âk,µ âk,µ on sites of type µ. We emphasize that Eq.(8.40), although written in second
quantization, is only valid for a single hole.
The most important feature of the e↵ective Hamiltonian (8.40) is its factorization into
individual momentum sectors which are decoupled from one another. The band Hamiltonian
8.3. INTEGER CHERN INSULATORS AND INTEGER QUANTUM HALL EFFECT 171
TP (k) depends on the TP momentum k. From its concrete from in Eq.(8.41) we conclude
Ĥµµ
0
TP (k)
that the force F acting on the impurity couples to the TP momentum, because Ĥµµ
0
depends only on the di↵erence k(t) = k F t. We also note that the operator-valued impurity
momentum p̂I appears in the argument of the free hole Bloch Hamiltonian hµµ0 . This can be
understood as additional kinetic energy of the impurity in the polaron frame and corresponds
to hole-hopping in the laboratory frame.
We have implemented the TP Hamiltonian (8.40) numerically and present results below,
after discussing the solution in the strong coupling approximation. In this way we obtain
the static (F = 0) TP ground state | TP (k)i in the polaron frame as a function of the TP
momentum k. By calculating overlaps between the TP state at di↵erent discrete momenta
kj we obtain the TP topological invariant ⌫TP , see e.g. [80].
A word of caution is in order. In Eq.(8.41), although we did not make it explicit, the
impurity Hamiltonian still depends on the driving force F , see Eq. (8.37). Hence the wave1D , cf.
function picks up an additional phase corresponding to the internal TP invariant ⌫TPint
1D
Eq. (8.12). In Appendix H we show that while ⌫TPint
in general modifies geometric phases of
the TP measured in the BZ, it does not a↵ect the resulting TP Chern number.
8.3.3
Solution in strong coupling approximation
To derive the strong coupling TP solution, we start by revisiting the TP Hamiltonian in the
excitation centered polaron frame Eq.(8.41). Under strong coupling conditions, see Eq.(8.5),
there is a separation of time scales. Assuming a flat hole band and a large cyclotron gap to the
second band, we see that the first term on the right-hand side of Eq.(8.41) restricts the hole
to its lowest Chern band. When this band is flat we can furthermore ignore the additional
kinetic energy due to coupling of the hole momentum k to the impurity momentum p̂I .
This leaves us with a simple Hamiltonian for the impurity alone, with a local potential
of strength V in the origin caused by the quasi static hole. The so-obtained strong coupling
Hamiltonian is sufficient to describe the binding of impurity to hole in the TP. To include
dynamics of the TP in our description we would have to treat the coupling term of p̂I to k e.g.
perturbatively. Having an exact (numerical) description at hand, however, we will content
ourselves with the lowest-order treatment here.
Motivated by the discussion above, we now put the strong coupling ansatz on a variational
basis as described previously in Sec. 8.2.2. To this end we start from a maximally localized hole
state (around some lattice site m0 , n0 ) in the lowest Chern band | a (m0 , n0 )i = P̂LCB â†m0 ,n0 |0i
and write the strong coupling wavefunction as
|
TP i
=|
a (m0 , n0 )i
⌦ | I i.
(8.42)
The lowest Chern band projection (P̂LCB ) is carried out for a general gauge in Appendix I.
The impurity wavefunction | I i can then be obtained numerically as the ground state of the
strong coupling Hamiltonian,
X
ĤSC = ĤI0 V
b̂†m,n b̂m,n hâ†m,n âm,n im0 ,n0 .
(8.43)
m,n
Here h·im0 ,n0 denotes an average with respect to the hole state | a (m0 , n0 )i and ĤI0 is the
kinetic part of the impurity Hamiltonian. The binding energy of the impurity in the e↵ective
potential above, which determines the TP gap TP , can be obtained numerically.
The calculation of the TP invariant in the strong coupling approximation is straightfor-
CHAPTER 8. INTERFEROMETRIC MEASUREMENT OF MANY-BODY
TOPOLOGICAL INVARIANTS
172
ward. The external TP invariant is given by the hole invariant, see Subsection 8.2.5, which coincides with the Chern number of the filled Hofstadter band, see Eq.(8.35). I.e. ⌫TPext = ⌫0
in thermodynamic limit. The internal TP invariant can easily be calculated from Eq.(8.43)
by complementing the purely kinetic impurity Hamiltonian ĤI0 by additional driving terms
included in ĤI , see Eq.(8.2). Assuming that the internal ground state TP wavefunction is
close to a Gaussian, we expect that the internal TP invariant vanishes, ⌫TPint = 0. We will
check these predictions by comparison to exact numerical results in Subsection 8.3.5.
8.3.4
Interacting fermions
Next we generalize our discussion from the previous subsections and include fermion-fermion
interactions in our model. We show how the TP problem can be solved numerically with the
help of the impurity centered polaron transformation. In the following Subsection 8.3.5 we
present our numerical results.
The model we consider here is similar to the one presented in 8.3.2. The fermion Hamiltonian includes Peierls phases in its kinetic part, and we add nearest neighbor interactions of
strength U ,
i
Xh
X
Ĥ0 = t
e i2⇡↵n ĉ†m+1,n ĉm,n + ĉ†m,n+1 ĉm,n + h.c. + U
ĉ†m,n ĉm,n ĉ†m0 ,n0 ĉm0 ,n0 .
m,n
h(m,n),(m0 ,n0 )i
(8.44)
The impurity model is identical to Eq.(8.37) and the repulsive fermion-impurity interaction
reads
X
Ĥint = V
ĉ†m,n ĉm,n b̂†m,n b̂m,n .
(8.45)
m,n
To make use the conserved total TP (quasi) momentum, we apply the impurity centered
polaron transformation,
h
i
ÛLLP (t) = exp iR̂I · (p̂c + F t) .
(8.46)
Here, in contrast to Eq.(8.39), the fermion-system is translated in real space (using its total
momentum operator p̂c ), such that the impurity is located in the central unit-cell in the
resulting polaron frame. With R̂I we denote the impurity position operator, see Appendix G
for the precise definitions. A similar calculation as for the two-body problem discussed above
shows that the TP Hamiltonian now reads
X †
TP
ĤTP =
b̂k,µ b̂k,µ0 ⌦ Ĥµµ
(8.47)
0 (k).
k,µ,µ0
The impurity centered TP band Hamiltonian is given by
⇣
TP
I
Ĥµµ
F t p̂I ) + µµ0 Ĥ0
0 (k) = hµµ0 (k
⌘
V ĉ†0,µ ĉ0,µ ,
(8.48)
where hIµ,µ0 (k) is the impurity single-particle band Hamiltonian in the magnetic unit-cell
defined by the fermions. From the corresponding groundstate | TP (k)i in the polaron frame,
we calculate the TP invariant in the usual way cf. Eq.(8.4),
Z
1
1D
⌫TP (ky ) =
dkx h TP (k)|[email protected] | TP (k)i.
(8.49)
2⇡ BZ
Eq.(8.48) demonstrates that the force F directly couples to the TP momentum. Using
8.3. INTEGER CHERN INSULATORS AND INTEGER QUANTUM HALL EFFECT 173
4
-6.5
3
-7
2
-7.5
1
-8
0
-1
-8.5
-2
-9
-3
-9.5
-4
-10
-5
-6
-1
-0.5
0
0.5
1
-10.5
-1
-0.5
0
0.5
1
Figure 8.4: The spectrum of a TP in an integer Chern insulator for ky = 0, calculated using the
single hole approximation (a) and including particle hole fluctuations (b). Note the di↵erent
energy scales. The thick red lines in (a) correspond to the spectrum of the free hole. We used
the following parameters J/t = 0.5 and V /t = 2 to simulate N = 3 fermions on a torus with
4 ⇥ 4 sites and N = 4 flux quanta.
exact diagonalization we will solve the TP band Hamiltonian for di↵erent values of k now.
This allows us to extract the TP invariant for systems with a small number of fermions.
8.3.5
Numerical results
Now we present numerical results for TPs in integer Chern insulators. In FIG.8.4 the spectra
of the TP band Hamiltonians (8.41) and (8.48) are compared. We find that the original bands
of the free hole are broadened substantially due to the coupling to the mobile impurity. In
the many-body spectrum FIG.8.4 (b) we recognize a number of TP bands at low energies,
before the scattering states corresponding to an unbound impurity and a free hole appear at
higher energies. The pattern of bound TP states predicted by the single hole approximation
(a) coincides with the exact result including particle hole fluctuations (b). At the discrete
momenta kx ax /⇡ = 1, 1/2, 0, 1/2 and 1 corresponding to the quantization due to finite
system size (Lx = 4ax ), we find good agreement for the TP gap predicted by the two methods.
In between we recognize sizable finite-size e↵ects leading to discrepancies.
In FIG.8.5 we compare the TP invariants predicted by the di↵erent approaches. The
2D = 1, in all
winding of the 1D invariants – i.e. the Chern number – is equal to one, ⌫TP
cases (for the conventions used in the numerics, the free particles have a Chern number
C = 1). In its least sophisticated form, the strong coupling approximation predicts that
the TP invariant is identical to the free hole result, and the result is shown in FIG.8.5. The
single hole approximation predicts an accumulation of Berry curvature at the edges of the
1D (✓ ). This
BZ (i.e. around ✓y = ±⇡), manifested in a steeper slope of the 1D invariant ⌫TP
y
calculation includes the internal TP invariant, and in Appendix H we show that the external
TP invariant is almost identical to the free hole result. The full many-body calculation yields
1D (✓ ) close to the single hole result, but with a slightly larger redistribution
TP invariants ⌫TP
y
of Berry curvature towards the edges of the BZ.
CHAPTER 8. INTERFEROMETRIC MEASUREMENT OF MANY-BODY
TOPOLOGICAL INVARIANTS
174
0.5
0.25
0
-0.25
-0.5
-0.5
-0.25
0
0.25
0.5
1D (✓ ) is shown as a function of the TP momentum k .
Figure 8.5: The 1D TP invariant ⌫TP
y
y
2D = 1. We simulated
Di↵erent methods are compared which all predict a Chern number ⌫TP
the same system as in FIG.8.4.
The full many-body calculation can easily be extended to included nearest neighbor
fermion-fermion interactions. In FIG.8.6 we show that the TP invariant is robust against
such interactions. The TP can still be used to measure the many-body topological invariant.
8.3.6
TP in the integer quantum Hall e↵ect
We will close this section by discussing the TP problem in the integer quantum Hall e↵ect.
As in the case of Chern insulators, we couple a single hole excitation to a light impurity atom.
For simplicity we restrict our analysis to the strong coupling approximation. We will derive
the strong coupling TP gap TP and show that the TP invariant ⌫TP = ⌫a is given by the hole
invariant. Using our results from Subsection 8.3.1 we thus conclude that the TP measures
the bulk many-body topological invariant, ⌫TP = ⌫0 , in the thermodynamic limit.
We start by briefly describing our model. We consider an impurity of mass MI to which
0.5
0.25
0
-0.25
-0.5
-0.5
-0.25
0
0.25
0.5
Figure 8.6: The TP invariant is calculated exactly for a few-fermion system as in FIG.8.5.
We included nearest neighbor fermion-fermion interactions of strength U , and show that they
barely a↵ect the TP invariant.
8.3. INTEGER CHERN INSULATORS AND INTEGER QUANTUM HALL EFFECT 175
a constant force F can be applied,
ĤI =
r2
2MI
F · rI .
(8.50)
The fermions of mass MF will be described in two dimensions by a field operator ˆ(r), and
they are placed in a homogeneous magnetic field,
Z
1
ĤF = d2 r ˆ† (r)
[ ir + A(r)]2 ˆ(r).
(8.51)
2MF
Here A(r) denotes the vector potential with r ⇥ A(r) = Bez . We describe the impurityfermion interaction by a local contact potential,
Z
ĤIF = d2 r ˆ† (r) ˆ(r)gIF (r rI ).
(8.52)
The wavefunction | a (r0 )i of a hole localized around r0 in the integer quantum Hall e↵ect
can be constructed explicitly [285]. To calculate the strong coupling Hamiltonian Eq.(8.9)
only the fermion density ⇢(r) = h a (r0 )| ˆ† (r) ˆ(r)| a (r0 )i is required, yielding an e↵ective
potential V (rI ) = gIF ⇢(|rI r0 |) for the impurity. For the quantum Hall liquids with magnetic
filling fraction ⌫ discussed in this chapter the fermion density of a state with a hole at r0 = 0
is given by [285]
⌘
⌫ ⇣
r 2 /2`2B
⇢(r) =
1
e
.
(8.53)
2⇡`2B
Here `B denotes the magnetic length of the fermions in the magnetic field. For the integer
quantum Hall e↵ect we have ⌫ = 1.
To calculate the TP gap TP separating the ground from excited polaron states, we solve
the two-dimensional Schrödinger equation for the impurity placed in the Gaussian potential
Eq.(8.53). Using the harmonic potential approximation we obtain
s
gIF MF
⌫
!c ,
(8.54)
TP =
2⇡`2B MI
where !c denotes the cyclotron frequency of the fermions in the magnetic field. For the strong
coupling approximation to hold, consistency requires that
TP
⌧ !c .
(8.55)
Finally we show that the internal TP invariant vanishes, ⌫TPint = 0. The external TP
invariant, on the other hand, is given by the Chern number and we obtain ⌫TP = ⌫TPext =
C, see Subsection 8.3.1. Using the harmonic approximation as above, the impurity wave2
2
function is given by a Gaussian I (r) = e r /2` (up to normalization). Here ` denotes
the oscillator length and without loss of generality we have placed the hole in the origin,
r0 = 0. The e↵ect of a driving force F along x can be described as a gauge transforr2 /2`2 +iF xt . The corresponding geometric phase vanishes,
mation,
R suchR that ⇤ I (r, t) = e
= d(F t) dx I (r, t)[email protected] t I (r, t) = 0. Choosing a hole located at a di↵erent position r0 ,
the resulting phase = F · r0 t corresponds simply to a Zeeman phase which has to be taken
into account in the experimental sequence [13], see Sec.6.2.1.
176
8.4
CHAPTER 8. INTERFEROMETRIC MEASUREMENT OF MANY-BODY
TOPOLOGICAL INVARIANTS
Fractional Quantum Hall e↵ect and Fractional Chern Insulators
In this section we apply our theory to the fractional quantum Hall e↵ect and its cousins the
fractional Chern insulators. In particular we focus on filling ⌫ = 1/m Laughlin states with
fractionally charged excitations [47] (where m
2 is an integer). We show that also the
topological invariants characterizing the quasihole band structure fractionalize, taking values
⌫a = 1/m. Because the force F acting on the TP only couples to the fractional charge
qa = 1/m of the quasihole excitations, we find that the TP invariant takes a value ⌫TP = m.
This enables a direct experimental measurement of the fractionalization of Laughlin quasiholes
using TPs. Most of the analysis in this section relies on the strong coupling approximation and
we will mainly be concerned with the calculation of the quasihole topological invariant ⌫a . At
the end of the section we briefly discuss how our protocol can be implemented experimentally.
8.4.1
Topological invariants
In what follows we argue that the topological invariant ⌫a of a quasihole excitation in a filling
1/m Laughlin state is of the form
⌫a = ⌫0 N a
1 .
(8.56)
Here ⌫0 is the bulk Chern number of the Laughlin state, which is given by ⌫0 = 1/m [88].
The integer m = 2, 3, ... is even for bosonic and odd for fermionic Laughlin states. We denote
the number of flux quanta required for the single quasihole state | a (q)i by N a , which on a
torus takes the value N a = mN + 1 (where N is the number of particles). Afterwards we
show what implications Eq.(8.56) has for the TP invariant.
Quasihole invariant
To derive Eq.(8.56), we start by discussing the leading-order term in the system size N a .
To this end let us consider an incompressible 1/m Laughlin state | 0 i on a sphere [133], see
FIG.8.7 (a). There are in total N = N a 1 flux quanta piercing the surface of the sphere.
To create a quasihole excitation we make use of Laughlin’s argument and introduce magnetic
flux ↵ (in units of the flux quantum) through one of the poles. After introducing a full flux
quantum ↵ = 1 we end up with a quasihole excitation | a i located at the pole [47], see FIG.8.7
(b). In this process the bulk gap of the system (which is well-defined in the thermodynamic
limit) does not close. Hence the Chern number of the new ground state | a i at ↵ = 1 can
not change and it is therefore given by ⌫0 . Because there are in total N a quasihole states, see
FIG.8.7 (c), the total Chern number reads ⌫a = N a ⌫0 + O(L 2 ) .
From the last paragraph we know that the quasihole invariant is of the form ⌫a = N a ⌫0 +u,
where u is a universal contribution independent of the system size. We can derive its value by
using the precise integer quantization of the invariant ⌫a . From N a = N m + 1 and ⌫0 = 1/m
it follows that ⌫a = N + 1/m + u 2 Z. Assuming that the integer part of u vanishes, we
conclude that u = 1/m. In the following Subsection 8.4.2 we will proof this result by an
exact numerical simulation of many-body systems in the torus geometry5 .
5
An alternative way to derive Eq. (8.56) is using the superposition principle. The total Chern number Ctot
of a discrete manifold of N single particle states in a Chern band is given by the Chern number C of this
band. Hence on average every state carries a Chern number C/N . For a quasihole manifold with N a flux
quanta and N particles we thus obtain a Chern number per state N ⇥ C/N a . Hence the total Chern number
of the quasihole manifold reads ⌫a = N C = (mN + 1)/m 1/m.
8.4. FRACTIONAL QUANTUM HALL EFFECT AND FRACTIONAL CHERN
INSULATORS
(a)
(b)
177
(c)
Figure 8.7: In a thought experiment an incompressible Laughlin state can be prepared on a
sphere with a magnetic monopole in its center (a). By adiabatically introducing magnetic
flux ↵ through its pole, a quasihole excitation can be created (b). In this process the manybody Chern number remains unchanged because the bulk gap does not close (c). See also
supplementary material of Ref. [P6] for a full many-body calculation on a buckyball lattice.
TP invariant
Now we discuss the implications of the fractional qp invariant in Eq.(8.56) for the measurable
TP invariant. As for the integer quantum Hall e↵ect, we assume that the internal TP invariant
vanishes, ⌫TPint = 0 see Subsection 8.3.6. In Subsection 8.2.5 we have shown that in this case
the TP invariant is identical to the qp invariant, where we assumed that the twist angles ✓x ,
✓y are adiabatically changed by 2⇡ respectively. This is achieved by the force F acting on the
impurity, and we will now discuss how fractional charges a↵ect this coupling.
To measure the topology of the e↵ective TP band structure, its Berry curvature has to
be integrated over the entire BZ. Thus when the TP encircles its BZ it picks up a geometric
phase 2⇡⌫TP . This requires an introduction of TP momentum Fx tx = 2⇡/ax along x and
Fy ty = 2⇡/ay along y. Because the TP ground state is non-degenerate in general, and the
TP Hamiltonian ĤTP (k) is 2⇡/ax,y periodic in its quasi momentum components kx,y , the TP
invariant is strictly integer quantized, ⌫TP 2 Z [27]. This is true even when the qp invariant
is fractionalized.
This paradox can be resolved by noting that the force F coupling directly to the TP
has an enhanced coupling to the many-body system by the inverse qp charge 1/qa . From
Eq.(8.16) we can derive the change of the twist angles ✓x,y . When the force Fx,y is applied to
the impurity for a time tx,y = 2⇡/Fx,y ax,y , the twist angles change by
✓x,y =
Lx,y
1
2⇡ .
ax,y qa
respectively. Therefore when the TP covers its BZ once, the qp state |
1/qa2 times.
(8.57)
a (q)i
covers its BZ
Let us discuss the concrete example of 1/m Laughlin states. When the TP covers its BZ
once, the qp covers its BZ m2 times. Because it picks up the Chern number ⌫a = 1/m each
time, see Eq.(8.56), the total phase picked up by the TP wavefunction is 2⇡⌫TP where
⌫TP = m2 ⌫a =
m.
(8.58)
Therefore an integer TP invariant larger than one can be an indicator of fractionalization of
the topological qp excitation.
178
CHAPTER 8. INTERFEROMETRIC MEASUREMENT OF MANY-BODY
TOPOLOGICAL INVARIANTS
0.5
0
-0.5
0
0.2
0.4
0.6
0.8
1
Figure 8.8: The topological invariant ⌫a1D (✓y ) of single quasihole manifolds in Laughlin states
on a torus are shown. We simulated two fermions (solid) on a torus of size 4 ⇥ 7 with
nearest neighbor interactions (U = 10t in Eq.(8.44)) and three bosons (dashed) with contact
interactions (Hubbard U equal to U = 5t) on 6⇥6 sites. In both cases the number of magnetic
flux quanta is N = 7, corresponding to a single quasihole in a 1/3 fermionic and 1/2 bosonic
Laughlin state, respectively.
8.4.2
Fractional Chern insulators
We will now check our theoretical predictions Eqs.(8.56), (8.58) from the last subsection by
exact numerical simulations of small systems of bosons and fermions. Specifically we will
calculate the qp topological invariant ⌫a2D for such models by twisting boundary conditions.
We also present exact numerical results for a TP in a fractional Chern insulator.
We calculated the topological invariants ⌫a1D of quasihole manifolds, as defined in Eq.(8.19)
using the twist of boundary conditions along x, for Laughlin states of fermions at magnetic
filling 1/3 and for bosons at magnetic filling 1/2. To this end we used the interacting Hofstadter model described in Subsection 8.3.4 and generalized it to bosons. The winding of
⌫a1D (✓y ) when the twist angle ✓y is changed from 0 to 2⇡ defines the 2D qp invariant ⌫a2D . For
all accessible system-sizes we found that the qp invariant
⌫a2D = N
(8.59)
is given by the particle number. In FIG.8.8 we show an example for quasiholes in Chern
insulators with N = 2 fermions and nearest neighbor interactions and for N = 3 bosons with
contact interactions. From our finding Eq.(8.59) it follows that ⌫a2D = ⌫0 (N a 1), as claimed
in Eq.(8.56) above.
To check our predictions for the TP invariant, we performed an exact numerical calculation
of a TP in a 1/3 Laughlin state with N = 2 fermions. The calculations were performed in
the impurity centered polaron frame described in Subsection 8.3.4. In FIG.8.9 we present
2D = 3. This confirms our prediction that
our result, showing a TP with a Chern number ⌫TP
the fractionalization of quasihole excitations manifests itself in an enlarged Chern number of
the TP. The appearance of a Chern number larger than one in this system is clear signature
of fractionalization, which can not otherwise be explained. Indeed the Chern number of the
lowest Bloch band is C = 1 in this case with flux ↵ = 1/4 per plaquette (in units of the flux
quantum).
8.4. FRACTIONAL QUANTUM HALL EFFECT AND FRACTIONAL CHERN
INSULATORS
179
0.5
0.25
0
-0.25
-0.5
-1
-0.5
0
0.5
1
1D (k ) in a 1/3 fractional Chern insulator of
Figure 8.9: The TP topological invariant ⌫TP
y
fermions is shown. Exact many-body calculations in the impurity-centered polaron frame
were performed to obtain this result. The winding of the TP invariant yields the TP Chern
2D = 3 – an indicator of fractionalization. We used the following
number which takes a value ⌫TP
parameters J/t = 0.5, U/t = 10 and V /t = 2 to simulate N = 2 fermions on a torus with
4 ⇥ 7 sites and N = 7 flux quanta.
8.4.3
Fractional quantum Hall e↵ect
A fractional quantum Hall system can be described as in Subsection 8.3.6 but with the inclusion of interactions,
Z
int
Ĥ0 = d2 rd2 r 0 ˆ† (r) ˆ† (r 0 )V (r r 0 ) ˆ(r 0 ) ˆ(r).
(8.60)
Here the field operator ˆ(r) can be either fermionic or bosonic. When the interaction potential
V (r) decays fast enough the ground state of the many-body Hamiltonian Ĥ = Ĥint + ĤF (see
Eq.(8.51)) is well described by a Laughlin state.
We construct a TP in this system by coupling a single fractionally charged quasihole
to the mobile impurity by a local contact interaction as in Eq.(8.52). To apply the strong
coupling theory the quasihole density is required, which determines the e↵ective TP binding
potential seen by the impurity. For 1/m Laughlin states the same result is obtained as in
Eq.(8.53), but at a magnetic filling ⌫ = 1/m [285]. Accordingly the TP gap TP (in harmonic
approximation) is the same as in Eq.(8.54), again at ⌫ = 1/m. As discussed above, the TP
topological invariant is given by ⌫TP = m in this case.
8.4.4
Experimental considerations
In systems of ultra cold atoms, impurities can be realized by a second atomic species, or by
di↵erent internal states. To obtain independent TPs, the impurity density has to be much
smaller than one particle per cyclotron orbit.
To realize TPs and use them for the measurement of many-body topological invariants,
a major challenge is their preparation. This is even more complicated in fractional Chern
insulators, where TPs can not be created by local operations because of the fractional charge
carried by qp excitations. Therefore TPs have to be built into the state as defects right from
the beginning. One approach is to start from a two-component system with large population
imbalance and cool it into its ground state. If the densities of the two components are
CHAPTER 8. INTERFEROMETRIC MEASUREMENT OF MANY-BODY
TOPOLOGICAL INVARIANTS
180
correctly chosen this leads to the formation of TPs. Finding a reliable cooling scheme is an
open problem even for preparing the bulk fractional Chern insulator.
An alternative approach is introduce one unit of magnetic flux adiabatically to create a
single fractionalized quasihole excitation in the spirit of Laughlin’s argument for the quantization of the Hall e↵ect [173]. Then an impurity atom can be added at the same place
to create a TP in a systematic way. A similar scheme was suggested for the preparation of
the entire Laughlin type state in a fractional Chern insulator [P6]. To avoid the problem
of fractionalization another option would be to consider TPs where the impurity binds more
than one qp. However we expect that this would lower the robustness of the TP state with
respect to perturbations in the system, and the fractionalization can no longer be detected.
8.5
Mott insulators and symmetry protected topological order
In this section we discuss examples of interacting topologically ordered systems in one dimension, where all states are short-range entangled [6] but may have SPT invariants. We
demonstrate that the concept of topological polarons can be employed to unravel the topology and detect the many-body SPT invariants of these systems. The methods developed in
this section could e.g. be applied to describe TPs in Hofstadter-Hubbard models in the quasi
1D thin-torus-limit [P7]. Here we discuss the simpler class of 1D super-lattice Bose Hubbard
models which can be realized with ultra cold atoms in current experiments [P2]. We propose a
concrete experimental setup, see FIG.8.10, for which exact numerical results will be presented.
In parameter space, in the simplest case, the super-lattice Bose Hubbard model has a
superfluid (SF) and a Mott insulating (MI) phase. By adiabatically encircling the SF region,
a Thouless pump can be realized (see FIG.8.10 (b)) pumping a quantized current through the
system, given by a many-body Chern number [87, 174]. We will show how this Chern number
can be detected interferometrically using TPs, without the need of single-site resolution to
measure the quantized current. On a 1D line in parameter space the super-lattice Bose
Hubbard model is inversion symmetric. In this symmetric subspace two gapped MI phases
exist, which are topologically distinct by an inversion-symmetry protected many-body Berry
phase [P2]. We will demonstrate that this SPT invariant, too, can be measured using TPs.
This Section is organized as follows. In Subsection 8.5.1 we introduce the many-body
Hamiltonian and describe the proposed protocol for an interferometric measurement of the
TP invariant. In Subsection 8.5.2 we develop an exact mathematical theory which allows
us to solve the many-body TP problem using exact diagonalization techniques. We apply
this method to obtain numerical results for the TP dispersion relation and its topological
invariant, which we discuss in Subsection 8.5.3. We supplement our results by approximate
theoretical calculations including a strong coupling treatment.
8.5.1
The model
Many-body system
We consider the 1D superlattice Bose Hubbard model described by the following Hamiltonian,
ĤB =
X⇣
j
t2 b̂†2j+1 b̂2j + t1 b̂†2j b̂2j
⌘
+
h.c.
+
1
2
X
i
( 1)i b̂†i b̂i +
UX † ⇣ †
b̂i b̂i b̂i b̂i
2
i
⌘
1 . (8.61)
In this section b̂i denotes a boson annihilation operator on lattice site i, and we label unit
cells by indices j. The parameters t1 , t2 describe alternating hopping rates of the bosons, as
8.5. MOTT INSULATORS AND SYMMETRY PROTECTED TOPOLOGICAL ORDER
181
(a)
(b)
2a
[min( , )]
−2
j
j
−1
2j 2j+1
0
SF
1
MI
2
TP
−2
−1
0
1
2
[min( , )]
Figure 8.10: We consider a 1D model of interacting bosons in a super-lattice potential at half
filling, solid red in (a). For sufficiently large | | and |t2 t1 | a gapped MI phase is realized,
while for small | | or |t2 t1 | the system is SF, see (b). By adiabatically changing
as
well as t1 and t2 along a loop within the MI phase (parametrized by ', solid blue in (b))
a topological Thouless pump can be realized which is characterized by a many-body Chern
number. To measure this Chern number we couple a hole excitation of the MI to a spin1/2 impurity in a conventional lattice, leading to the formation of a TP (a). Then, using
a combination of Ramsey interferometry and Bloch oscillations, the many-body Zak phase
(indicated by arrows in (b)) can be measured. Its winding over parameter space yields the
many-body Chern number. Results in (b) are perturbative in min (t1 , t2 ) (up to first order).
in the Su-Schrie↵er-Heeger model [12]. We included a staggered potential of amplitude
well as local Hubbard-type interactions of strength U .
as
In its half-filling MI phase the superlattice Bose Hubbard model supports inversionsymmetry protected topological phases for
= 0 [P2]. They can be characterized by quantized values of the polarization P (for a review of polarization see e.g.[80]). Polarization is
closely related to Zak phases, P = a Zak /⇡ [83, 282], where 2a is the size of the unit cell, see
FIG.8.10 (a). Many-body Zak (or Berry) phases Zak = 2⇡⌫ 1D are defined as the geometric
phases picked up when boundary conditions are twisted [88, 282], see Subsection 8.2.4. For
= 0, because of inversion symmetry, only the two values Zak = 0, ⇡ are possible [P2]. We
will show below how this topological invariant Zak can be measured using TPs.
When arbitrary values of the staggered potential
are allowed, the model (8.61) breaks
inversion symmetry and realizes a many-body topological Thouless pump [87, 174] as depicted
in FIG.8.10 (b). It is uniquely characterized by the winding of polarization P (') over one
pumping cycle, which we parametrize by the angle ' = 0...2⇡ around the SF region in the
phase diagram, see FIG.8.10 (b). This winding can also be interpreted as a many-body Chern
number C characterizing the Thouless pump [87, 174],
1
C=
2⇡
Z
2⇡
d' @'
Zak (').
(8.62)
0
To obtain this many-body Chern number we suggest to measure individual many-body Zak
phases Zak (') at a set of system parameters {'m }. When these are chosen sufficiently dense
the winding can be calculated as a sum,
C=
1 X
2⇡ m
Zak ('m+1 )
Zak ('m ).
(8.63)
Thus in the rest of the Chapter we will mostly be concerned with the measurement of Zak
phases in interacting 1D systems.
182
CHAPTER 8. INTERFEROMETRIC MEASUREMENT OF MANY-BODY
TOPOLOGICAL INVARIANTS
TP Hamiltonian
To measure many-body Zak phases, we introduce a TP by coupling an impurity to an elementary hole excitation in the MI. We use a spin one-half impurity which is confined to the
lowest band of a long-wavelength lattice of period 2a (see FIG.8.10 (a)) and can be described
by a Hamiltonian
⌘
X
X⇣ †
ĤI = J
ĉj, ĉj+1, + h.c.
F
2ajĉ†j,⌧ ( z )⌧, ĉj, .
(8.64)
j,
j,⌧,
The last term describes a spin-dependent driving force ±F acting on the impurity, where
, ⌧ =", # are spin indices. In practice it can be realized e.g. by a magnetic field gradient.
To realize strong coupling between the impurity and the hole excitation we consider repulsive interactions between the impurity and the bosons, described by
⇣
⌘
X †
ĤIB = V
ĉj, ĉj, b̂†2j b̂2j + b̂†2j+1 b̂2j+1 .
(8.65)
j,
We only take into account the local interaction between impurity site j and its two nearest
neighbors i = 2j and i = 2j + 1 within the boson lattice, see FIG.8.10(a).
Experimental considerations
The TP Hamiltonian introduced above
Ĥ = ĤI + ĤB + ĤIB
(8.66)
can be realized with ultra cold quantum gases using current technology. Cold atoms in a superlattice potential have readily been realized and the Mott insulating regime can be accessed
by tuning the potential depth to increase Hubbard-type interactions among the bosons, see
e.g. [13, 286, 5].
Impurities can be realized as di↵erent hyperfine states of the atoms. If the bosons b̂i are
chosen to be mF = 0 states in a F = 1 manifold of hyperfine states, the impurity states
can take two pseudo spin values ", # corresponding to mF = ±1. By applying a magnetic
field gradient @x B(x) 6= 0, the spin-dependent force on the impurity follows from the Zeeman
potential gB(x)µB mF where gF is the g-factor of the hyperfine states and µB the Bohrmagneton. This has also been used in previous experiments to measure band topology [13].
When the short-wavelength optical lattice (period a = s /2) is realized as a state-dependent
lattice which is only seen by the mF = 0 bosons, our model (8.66) can readily be implemented.
Assuming that bosons interact via local contact interactions independent of their spin-state
mF , the interaction strenghts U and V are determined only by the overlap of Wannier orbitals
in the lowest Bloch bands of the normal and super-lattice potential. State dependent optical
lattices of multi-component quantum gases have also been realized experimentally [287].
The experimental protocol for the measurement of the TP is identical to the scheme developed for weakly interacting BECs, see Sec.6.2.1. Similarly it can be generalized to measure the
many-body Chern number (8.62) characterizing the Thouless pump. If the impurity atoms
are realized as di↵erent electronic states of the bosonic atoms, radio frequency pulses can
be used to create a small initial concentration of TP wave packets with a given spin. We
will assume that these TPs are initially at rest, i.e. their mean momentum is q = 0. The
experimental feasibility of a similar preparation protocol has been demonstrated [288].
To measure TP Zak phases, unlike in the case of the Hofstadter problem discussed in
8.5. MOTT INSULATORS AND SYMMETRY PROTECTED TOPOLOGICAL ORDER
183
Sec.8.3, we make use of the internal TP invariant which essentially probes the polarization of
the MI locally within the bulk. For this scheme to work we find two conditions. Firstly, the
impurity has to be sufficiently mobile (see eq.(8.79) below) in order to introduce the required
momentum into the system. Secondly, the impurity-boson interaction V has to be sufficiently
small not to destroy the topological phase completely in the vicinity of the impurity. We will
make these statements more precise now.
8.5.2
Polaron transformation
A powerful tool developed in polaron theory is the Lee-Low-Pines (LLP) unitary transformation [19], which we employed already in our treatment of the Hoftstadter TP problem in
Subsection 8.3.2. Here we use a slightly di↵erent version of this exact transformation, which
allows us to make use of the conservation of TP momentum in a true many-body setting. The
basic idea is to translate the entire Bose system by the impurity position,
Z
X
X †
†
iŜ
Û = e ,
Ŝ =
2jaĉj ĉj
dk k
b̂k,↵ b̂k,↵ .
(8.67)
j
|
BZ
↵=1,2
{z
}
=:P̂B
Hence in the so-obtained polaron frame the impurity is located in the center (note the di↵erence to the Hofstadter problem where the hole excitation was fixed to the center). In the last
equation ↵ = 1, 2 is a band index and b̂k,↵ denotes the k-th Fourier component of the Boson
operators b̂2j (↵ = 1) and b̂2j+1 (↵ = 0) respectively. Here and in the following we moreover
suppress the spin label of the impurity.
Before applying the LLP transformation let us simplify the impurity Hamiltonian,
Z
ĤI = 2J
dq cos(2aq !B t)ĉ†q ĉq .
(8.68)
BZ
Here we introduced the Bloch oscillation frequency !B = 2aF and eliminated the linear
potential by performing a time-dependent gauge transformation, a standard trick to treat
Bloch oscillations in finite systems with periodic boundary conditions. Note that this is
equivalent to imposing twisted boundary conditions (8.18) for the impurity, with the twist
angle given by ✓x = !B t.
Applying the
R LLP transformation (8.67) to the full Hamiltonian (8.66) with exactly one
impurity (i.e. BZ dq ĉ†q ĉq = 1) yields
†
H̃ = Û ĤÛ =
Z
BZ
h
⇣
⌘
dq ĉ†q ĉq ĤB + V b̂†0 b̂0 + b̂†1 b̂1
⇣
2J cos 2aq
!B t
P̂B
⌘i
=:
Z
BZ
dq ĉ†q ĉq Ĥq (t). (8.69)
In the static case F = 0 the TP quasi momentum q is a conserved quantity and Ĥq describes
the TP band Hamiltonian with many-body degrees of freedom. The ground state of Ĥq is
the TP state and its energy !TP (q) yields the TP dispersion relation. The gap to the first
excited state, TP (q), can be interpreted as the TP binding energy. When the force F is
switched on the TP changes its quasi momentum in time, Ĥq (t) = Ĥq F t (0), but because of
the translational symmetry in the system there are no transitions between di↵erent momentum
sectors.
184
CHAPTER 8. INTERFEROMETRIC MEASUREMENT OF MANY-BODY
TOPOLOGICAL INVARIANTS
(a)
2
1.5
(b)
TP, ED
10
bulk, ED
8
TP, pert.
6
1
4
0.5
0
0
2
0.2
0.4
0.6
0.8
1
0
0
0.2
0.4
0.6
0.8
1
Figure 8.11: The numerical solution of the TP Hamiltonian (8.69) for model parameters
chosen along the Thouless pumping cycle parametrized by ' are presented. (a) Many-body
Zak phases of the bulk (solid) for 5 bosons and the TP (dashed) for 4 bosons. For comparison
the perturbative result for the TP is shown (dotted). It relies on the single-hole approximation
and the insets depict the corresponding e↵ective single hole models. (b) Gaps of the manybody ground states, with the samep
notations as in (a). We simulated 10 sites and used
U = 20, V = 10, J = 2 and a radius |t2 t1 |2 + 2 = 5 for the Thouless pump, all in units
of min (t1 , t2 ).
8.5.3
Results
The transformation leading to Eq. (8.69) is of great importance for an exact numerical
treatment of the TP problem. The key advantage over simulating the full Hamiltonian Ĥ is
that for a given momentum q the Hamiltonian Ĥq has a single ground state which is separated
by an energy gap from further states. This allows in particular, by considering a discrete set
of momenta q0 ,q1 ,....,qM , to calculate the many-body Zak phase TP
Zak which is measured by
our protocol. To do so, we use the standard discretization of Berry phases, see e.g. [80].
Thouless pump Chern number
In FIG.8.11 we show the results obtained from exact diagonalization (ED) of Ĥq . We find
that the TP Zak phase qualitatively follows the many-body Zak phase of the bulk, see (a).
Importantly the windings of both Zak phases over the Thouless pump cycle (i.e. the manybody Chern numbers of bulk and TP) coincide. In (b) we compare the TP gap and find that
it is of the same order as the bulk gap of the MI and takes sizable values everywhere. This
is crucial because it demonstrates that the TP can be adiabatically moved through the BZ
without creating excitations. A simple perturbative analysis, which will be explained in detail
below in 8.5.4, also yields reasonable results. We find substantial deviations only for the TP
gap when the bulk gap is smaller than the perturbative result. This can be understood from
the fact that in our perturbative analysis we do not take into account particle-hole fluctuations
in the MI.
In FIG.8.12 (a) we show a heavy impurity with much smaller hopping J = 0.2 min(t1 , t2 )
than in FIG.8.11. In this case the TP Zak phase does not wind in the pumping cycle ', and
the TP is thus unable to reproduce the bulk result. A simple explanation for this e↵ect will
be given below, but essentially the TP is not mobile enough and it is far from the strong
coupling regime. In FIG.8.12 (b) we investigate the interaction dependence at inversioninvariant points,
= 0. We observe that in the MI regime the TP Zak phase correctly
predicts the bulk topological invariant, Zak = ⇡ in this case. On the SF side on the other
hand (we estimated the transition perturbatively), particle-hole fluctuations destroy the TP.
Its gap closes and the TP Zak phase indicates the occurrence of topological phase transitions.
8.5. MOTT INSULATORS AND SYMMETRY PROTECTED TOPOLOGICAL ORDER
185
(a)
(b)
2
TP, ED
6
1.5bulk, ED
TP, pert.
1
4
0.5
2
0
0.45
0.5
0.55
bulk, ED
TP, ED
SF
0
MI
gap TP, ED
gap bulk, ED
gap bulk, pert.
−1
10
0
10
1
10
2
1
0
Figure 8.12: (a) The TP Zak phase of a heavy impurity (bullets, N = 4 bosons) deviates from
the bulk many-body Zak phase (solid, N = 5 bosons). We have chosen a small hopping J =
0.2 min (t1 , t2 ) here and otherwise used the same parameters as in FIG.8.11. For comparison
we show the perturbatively obtained TP Zak phase (dotted). (b) Quantized many-body Zak
phases (thick) and excitation gaps (thin) are shown for t1 = 5t2 and = 0, where the system
is inversion symmetric. We used ED and varied the interaction U . Parameters were V = 10,
J = 2 in units of min (t1 , t2 ) and we used 10 sites with 4 bosons for the TP (5 bosons for
bulk).
We find that these appear at smaller values of U when V is lowered. Note, however, that in
this regime the concept of a TP is no longer expected to be valid because the impurity can
not couple to a well-defined hole excitation. We further expect large finite-size e↵ects in our
numerics on the SF side of the phase diagram in FIG.8.12.
Symmetry protected topology
In [P2] it was pointed out that the half-filled super lattice Bose Hubbard model supports an
(inversion-) symmetry protected topological phase. For = 0 this is the case and the manybody Zak phase of the bulk MI can only have two quantized values, Zak = 0, ⇡. Because for
= 0 the full TP Hamiltonian (8.66) is also inversion symmetric, the same arguments as in
[81, 153] [P2] can be used to show that the TP Zak phase is quantized, TP
= 0) = 0, ⇡.
Zak (
This explains why TP and bulk Zak phases at
= 0 (i.e. ' = 0, ⇡) in FIGs. 8.11(a) and
8.12(a) can only take quantized values. For light impurities we find that the TP protocol can
be used to measure this symmetry protected quantized Zak phase. In the following we will
give an explanation of our results by introducing approximate theoretical descriptions of the
TP, and discuss precise conditions under which the TP protocol can be used to measure the
many-body topological invariant.
8.5.4
Approximate descriptions
Now we use approximate methods to gain more understanding for the numerical results presented above and to clarify how the TP measures the many-body topological invariant. We
start by applying the strong coupling approximation from Subsection 8.2.2 and show that in
this case the internal TP invariant maps out the topology of the MI. For simplicity we restrict
ourselves to the inversion symmetric case here. Then we present a refined perturbative treatment based on the LLP transformation Eq.(8.67), which we carry out also for non-symmetric
cases.
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CHAPTER 8. INTERFEROMETRIC MEASUREMENT OF MANY-BODY
TOPOLOGICAL INVARIANTS
(a)
(b)
2j
2j+1
TP
2j
2j+1
TP
Figure 8.13: We distinguish two TP configurations. In the topologically trivial case (a) the
hole excitation in unit cell j interacts only with the impurity at site j. In the non-trivial case
(b) the hole interacts also with an impurity in site j + 1. Whether the system is in a trivial
or non-trivial phase is determined by the relative orientation of the impurity lattice to the
super-lattice seen by the bosons.
Strong coupling approximation
To apply the strong coupling approximation we restrict ourselves to inversion symmetric
Hamiltonians with
= 0. In addition, to obtain simple analytic results and focus on the
essential physics, we consider only the case of a deep super-lattice when one of the hoppings
vanishes, say t1 ⇡ 0. Hence, at half filling, the many-body state | a (j)i with a single hole
localized in unit cell j is a simple product state,
|
a (j)i =
⌘
Y 1 ⇣ †
p b̂2n + b̂†2n+1 |0i.
2
n6=j
(8.70)
To understand the dynamics of such hole excitations we can treat the hopping t2 as a perturbation. This gives rise to a nearest neighbor hopping model between adjacent unit-cells, with
tunneling amplitude t2 /2. Because the topological invariant of a tight-binding band vanishes,
⌫a = ⌫TPext = 0 and the external TP invariant is trivial, see Subsection 8.2.5.
To proceed and calculate the impurity wavefunction in the strong coupling approximation,
we have to distinguish two cases. In the first, topologically trivial case the hole excitation in
unit cell j interacts only with the impurity atom at site j, see FIG.8.13 (a). In the second,
topologically non-trivial case the impurity lattice is shifted by half a period (i.e. by a in our
notations), see FIG.8.13 (b). Hence the hole in unit-cell j interacts both with an impurity at
site j and j + 1. We will now show that in the former case the internal TP invariant is trivial
⌫TPint = 0 whereas it is non-trivial in the latter case, ⌫TPint = 1.
To this end we make yet another simplification and assume that the impurity-boson interaction V
J exceeds the impurity hopping. Then in the topologically trivial case the
impurity wavefunction
| I (j)i = ĉ†j |0i + O(|Je i!B t |2 /V )
(8.71)
is given by an impurity localized at site j. Hence to zeroth order in 1/V the internal TP
invariant is trivial, ⌫TPint = 0 because the impurity wavefunction is independent of the twist
angle ✓x = !B t induced by the driving force. Because the internal TP invariant is quantized to
⌫TPint = 0, ⇡ due to inversion symmetry, this result is valid for a larger range of model parameters than can be captured accurately by our perturbation theory. This topologically trivial
case corresponds to ' = 0 in FIG.8.11, where indeed the TP invariant vanishes identically.
In the topologically non-trivial case, to leading order in 1/V the impurity wavefunction
reads
⌘
1 ⇣ i!B t †
†
p
| I (j)i =
e
ĉj + ĉj+1 |0i + O(|Je i!B t |2 /V )
(8.72)
2
8.5. MOTT INSULATORS AND SYMMETRY PROTECTED TOPOLOGICAL ORDER
187
because the impurity can tunnel resonantly between the two sites j and j + 1 which both
experience an e↵ective attractive potential V /2 due to the presence of the hole excitation.
By calculating the geometric phase picked up by the impurity wavefunction in one Bloch
cycle t = TB we obtain the internal TP invariant ⌫TPint = /2⇡ = 1 in this non-trivial case.
This explains our observation in FIG.8.11 that the TP invariant ⌫TP = 1 (or equivalently
Zak = ⇡) when ' = ⇡. This result, again, holds beyond the regime of validity of the
perturbation theories in t2 and 1/V .
Perturbative single hole approximation
Now we will refine our strong coupling analysis from the last section and treat the e↵ect of
hole-hopping on the TP wavefunction by a well-controlled perturbation theory. In particular
we will derive how light the impurity has to be for a measurement of the TP invariant. The
main approximation in our calculations will be to neglect particle-hole fluctuations in the MI.
Hence we can use the localized hole states | a (j)i from Eq.(8.70) as a truncated basis for the
many-body system. After projecting the exact LLP-transformed Hamiltonian Eq. (8.69) into
this basis we will solve it perturbatively.
For notational convenience, we introduce hole creation operators by â†j |MIi = | a (j)i. (As
for the impurity we prefer to use second quantization although we consider only a single hole,
P
i.e. j â†j âj = 1.) Using a lowest order cell strong coupling perturbative expansion technique
[169, 170] we calculated the phase diagram and polarization vectors in FIG.8.10(b) and obtain
the e↵ective Hamiltonian
⌘
X⇣ †
ĤB ⇡ th
âj+1 âj + h.c. ,
(8.73)
j
where the hole hopping amplitude is given by
t 1 t2
th = p 2
4t1 +
2
.
(8.74)
Here we assumed that t2 < t1 is the smaller of the two hoppings.
Next we transform into the polaron frame and use the fact that the incompressible MI
ground state has vanishing
total momentum, P̂B |MIi = 0. In the single-hole subspace we
R
may thus write P̂B = BZ dk k â†k âk , where âk is the Fourier transform of âj . We find for the
approximated kinetic part of the TP band Hamiltonian
X
Ĥqkin =
tTP (q)â†j+1 âj + h.c.,
(8.75)
j
where the e↵ective TP hopping is given by
tTP (q) = th + Jei(q
!B t)
.
(8.76)
This TP hopping depends on the value of the TP momentum q, which can be changed in
time by applying the force F to the impurity. This will allow to introduce the analogue of
twisted boundary conditions for the TP. Before discussing this in more detail, let us include
the potential part Ĥqpot to the TP band Hamiltonian,
Ĥq = Ĥqkin + Ĥqpot .
(8.77)
To this end we have to distinguish again between the two impurity lattice configurations in
FIG.8.13. In the trivial case, the e↵ective hole potential reads Ĥqpot = V n̂0 , where n̂j = â†j âj
188
CHAPTER 8. INTERFEROMETRIC MEASUREMENT OF MANY-BODY
TOPOLOGICAL INVARIANTS
denotes the hole density in unit cell j. Note that these expressions are already in the polaron
frame, see Eq.(8.69). In the non-trivial case in contrast, the hole at both j = 0 and j = 1
is a↵ected by the potential, Ĥqpot = V0 n̂0 V1 n̂1 . To zeroth order in t2 , the corresponding
potentials on unit cells j = 0, 1 are given by
✓
◆
q
V
V0,1 =
1 ± / 4t21 + 2
(8.78)
2
p
and this perturbative result is valid for V ⌧ 4t21 + 2 . The e↵ective hole potentials are
also depicted in the inset of FIG.8.11 (a).
Now we recognize a very similar structure of the e↵ective Hamiltonian Ĥq to the strong
coupling case discussed above. When the impurity is confined to a single lattice site (trivial
case in FIG.8.13) we expect a vanishing TP invariant. On the other hand, when the impurity
can tunnel more or less freely between two lattice sites (non-trivial case in FIG.8.13) it picks
up a non-trivial Berry phase when the complex phase of the hopping element is adiabatically
changed by 2⇡. From Eq.(8.76) we note, however, that it is only possible to twist the phase
of the TP hopping tTP by application of the force F on the impurity if
th < J
(light impurity).
(8.79)
For th > J the impurity is too heavy, and the TP invariant can not reproduce the bulk topological invariant. This condition exactly matches our expectation when the strong coupling
theory, predicting that ⌫TP = ⌫0 , can be applied. In FIG.8.12 (a) an example of a heavy impurity is shown where the TP fails to measure the bulk topological invariant. In this concrete
example J/th = 0.4 and the impurity is indeed too heavy.
8.6
Conclusions and Outlook
In conclusion, we proposed an interferometric scheme for the measurement of many-body
topological invariants and discussed it in detail. The key idea was to investigate the topological
invariants characterizing quasiparticle excitations, which we related to the invariants of the
underlying groundstates in specific cases. By coupling an impurity with two (pseudo) spin
states to this elementary excitation, interferometric schemes developed for non-interacting
particles can be applied. We call the resulting composite object, consisting of a quasiparticle
bound to the impurity, a topological polaron and showed for specific examples that its band
structure is characterized by the same topological invariant as the underlying many-body
system.
We developed a general theoretical framework to describe topological polarons and their
properties. We applied our theory to integer and fractional Chern insulators and quantum
Hall systems. We also showed that the topological properties of short-range entangled 1D
systems can be measured using our approach, including the possibility to obtain many-body
symmetry protected topological invariants and proof their quantization experimentally. In the
case of fractional Chern insulators we demonstrated that fractionalization of the underlying
Chern number manifests itself in a Chern number larger than one of the topological polaron.
The method presented in this Chapter may become a powerful experimental technique
for detecting topological order in interacting many-body systems. It is ideally suited for cold
atom experiments, which o↵er a rich toolbox with precise and fully coherent control over
individual atoms. The main idea to witness topological order by investigating topological
excitations, however, is of broader relevance. It may also be useful in numerical simulations
8.6. CONCLUSIONS AND OUTLOOK
189
and for defining topological order in the first place, as is done in topological quantum field
theories.
The concept of topological polarons as experimental probes for topology can be generalized
to other systems, including for example quasiparticle excitations on topological superconductors. Another interesting direction would be to probe the (non-Abelian) braiding statistics of
anyons by coupling them to impurities to form topological polarons.
190
CHAPTER 8. INTERFEROMETRIC MEASUREMENT OF MANY-BODY
TOPOLOGICAL INVARIANTS
Part III
Polaron Physics with Ultra Cold
Atoms
191
Chapter 9
Introduction
9.1
Summary and Overview
In this Chapter we turn our attention to one of the oldest problems in many-body quantum mechanics, the polaron problem. It concerns an impurity particle interacting with a
surrounding many-body system of bosons or fermions. The polaron problem is distinct from
other famous impurity models, like the Kondo e↵ect [289], the Anderson impurity model [290]
or the spin-boson problem [291], in that the impurity is mobile and thus has its own dynamics.
For a detailed understanding of many physical phenomena in solids, polaronic e↵ects play a
crucial role. In this case the impurity is typically represented by a conduction electron, which
interacts with the phonon modes of the host crystal. Thus the resulting electron-phonon
interaction, first derived by Fröhlich [16], is the starting point for many polaron problems.
It is now established that polaronic e↵ects play a role in situations of great practical relevance [292, 293], including for instance the high-Tc superconducting cuprates [294, 295, 296]
or electron transport in organic materials [297].
Despite the fact that its roots reach back to the beginning of modern quantum field theory,
the polaron problem is still a field of active research. In particular non-equilibrium problems
of mobile impurities are still rather poorly understood, and the questions about the fate of
the orthogonality catastrophe in such a situation [298] provides an interesting example. In
addition, the experimental progress made with ultra cold atomic systems now allows clean
realizations of the polaron problem, with tunable model parameters like e.g. the potential
landscape or the statistics of the atoms and their mutual interaction strengths. This has
enabled a detailed experimental [299, 300, 301, 302, 303, 304] and theoretical [305, 306, 307,
308, 304] study of the Fermi polaron problem, where impurity atoms are immersed in a Fermi
sea of ultra cold atoms. Also the problem of Bose polarons, where the Fermi sea is replaced
by a BEC, is under active theoretical research [24, 309, 310, 311, 312, 313, 314, 315, 316, 317,
318, 319, 320, 321] [P5], and first experiments have also started to explore physics connected
to the Bose polaron [322, 323, 324, 283, 325]. Most importantly, the long achievable coherence
times combined with the versatile toolbox allowing coherent detection and manipulation of
individual atoms, make ultra cold quantum gases prime candidates for the investigation of
non-equilibrium polaron physics [326, 313, 327, 328, 329, 330, 283, 331, 325, 283], out of reach
in solid-state systems.
In this part of the thesis we investigate di↵erent aspects of the Bose polaron problem
describing a mobile impurity atom immersed in a BEC. We address two fundamental questions,
first, how to describe non-equilibrium dynamics of the Bose polaron and, second, how to treat
the intermediate coupling regime of the Fröhlich Hamiltonian. Finally, we also address the
193
194
CHAPTER 9. INTRODUCTION
combined question about the non-equilibrium dynamics of an intermediate-coupling polaron.
To answer the first question about non-equilibrium dynamics, we start by considering a
weak-coupling Fröhlich polaron in a BEC and calculate its spectral function [P5]. To this
end we develop a time-dependent variational ansatz, allowing us to treat non-equilibrium
polaron problems on the mean-field (MF) level. Then the radio-frequency (RF) spectrum of
the impurity can be calculated by solving a particular non-equilibrium problem. As a main
result we identify universal high-energy properties of the RF-spectrum, which are connected
to two-particle scattering with universal contact interactions [332, 333]. In the second part we
consider a di↵erent Bose polaron problem, when the impurity is confined in an optical lattice
[P8]. We show that, as a consequence of the polaronic dressing, the polaron dispersion relation
can become strongly renormalized. After treating the equilibrium problem in a weak-coupling
approximation, we apply a similar time-dependent variational ansatz as in the first part [P5]
to investigate the non-equilibrium problem of polaron Bloch oscillations. As a main result we
derive a closed semi-analytical expression for the incoherent polaronic current resulting from a
constant applied force, which predicts a sub-Ohmic current-force relation in the weak-driving
limit (for dimensionality d > 1 of the BEC).
To address the second question concerning the intermediate-coupling Fröhlich polaron, we
develop a renormalization group (RG) formalism on a Hamiltonian level [P9]. Our starting
point is MF polaron theory, and from the perturbative RG we obtain RG flow equations for
the renormalized impurity mass, the total phonon momentum as well as the polaron groundstate energy. By a comparison of the polaron energy with numerical diagrammatic Monte
Carlo calculations [22], we show that our method works excellent in the intermediate coupling
regime. We show how properties of the polaron ground state, like the renormalized polaron
mass, can easily be calculated from our RG formalism. As a main result we predict sizable
deviations of the polaron mass from the MF value, and find a smooth cross-over from weakto strong-coupling regime. This is expected on general grounds [20, 334] and di↵ers from the
prediction by Feynman’s path integral method [24, 23]. While most polaron observables are
well-behaved when the ultra-violet (UV) cut o↵ is sent to infinity, we show analytically that
the polaron energy is logarithmically UV divergent. This divergence is related to quantum
fluctuations of the impurity, introduced by the entanglement between phonons, and we develop a regularization scheme based on the Lippmann-Schwinger equation. In addition we
supplement our RG results with a closely related Gaussian variational wavefunction and find
excellent agreement of the variational with the RG approach. In the last part of this Chapter we generalize the RG formalism to non-equilibrium polaron problems captured by the
Fröhlich Hamiltonian. We apply this method to calculate corrections to the RF spectrum of
an impurity immersed in a BEC, which shows strong deviations from the MF result in the
intermediate- and strong-coupling regimes.
This part is organized as follows. Below in Sec.9.2 we start by deriving the polaron Hamiltonian describing an impurity in a BEC, and discuss its experimental realization. Afterwards
we introduce the fundamental concepts for the treatment of the polaron problem. In Sec.10 we
consider a Fröhlich polaron in the weak-coupling regime and derive its full spectral function
by solving a non-equilibrium problem. In Chap.11 we add a tight optical lattice confining
the impurity atom, and investigate the non-equilibrium problem of polaron Bloch oscillations.
In Chap.12 we return to the Fröhlich polaron Hamiltonian and develop an all-coupling theory based on a RG formalism. This treatment is generalized to non-equilibrium situations in
Chap.13, where we calculate the full spectral function of polarons in the intermediate coupling
regime.
9.2. FUNDAMENTAL CONCEPTS
9.2
195
Fundamental Concepts
In this section we lay out the fundamental theoretical concepts required for a description
of the Bose polaron in a BEC. We start by deriving the Fröhlich polaron Hamiltonian in
9.2.1, based on the discussion in [315, 24]. Next, in 9.2.2, we discuss the applicability of the
model to describe realistic experiments, and address the question under which conditions the
intermediate-coupling regime can be reached. In the remainder we introduce the standard
theoretical approaches to solve for the groundstate of the polaron Hamiltonian. To this end
we first discuss the Lee-Low-Pines (or polaron-) unitary transformation in 9.2.3, allowing to
switch to a basis where the polaron momentum is explicitly conserved [19]. Then we describe
in detail the MF theory of the polaron in 9.2.4, valid in the weak-coupling regime, which is the
starting point for all discussions in the following sections. In 9.2.5 we finish this introductory
section by a brief summary of the strong-coupling treatment of the polaron due to Landau
and Pekar [18, 17], which was historically the first approach to the polaron problem.
This section is intended as a review of the existing literature about Bose polarons, coined
by the personal perspective of the author. All relevant references will be given in the text.
9.2.1
Polaron Hamiltonian for Impurities in a BEC
The starting point for our discussion is a d-dimensional BEC of Bosons (B) with impurity
atoms (I) immersed in it, see FIG. 9.1 (a). The density of the BEC will be denoted by n0 .
For concreteness, in the rest of this Chapter we will restrict ourselves to three dimensional
systems, d = 3, but we return to the general case in Chap.11. We denote the mass of bosons
by mB and describe them in second quantization by the field operator ˆ(r). The impurity
atoms on the other hand are assumed to carry a mass M and we describe them also in second
quantization by the field operator ˆ(r). Note that in the following we always restrict our
analysis to the case of small impurity density (such that the average distance between two
1/3
impurities is much larger than the polaron radius and the inter-boson distance n0 ), allowing
us to treat a single-impurity problem. Then we will sometimes switch to first quantization
to describe the impurity, but here we find it convenient to keep the formalism general. To
describe interactions between an impurity and the bosons, as well as between two equivalent
bosons, we employ contact pseudopotentials with strength gIB and gBB respectively. We thus
arrive at a microscopic many-body Hamiltonian (setting ~ = 1)
Ĥ =
Z
d3 r ˆ† (r)

r2
gBB ˆ†
+ VB (r) +
(r) ˆ(r) ˆ(r)+
2mB
2

Z
r2
3
†
ˆ
+ d r (r)
+ VI (r) + gIB ˆ† (r) ˆ(r) ˆ(r), (9.1)
2M
where in addition we considered potentials VI,B (r) for both the impurities and the bosons.
In typical cold-atom experiments both VI,B (r) include harmonic confinement potentials, but
since they are typically weak we will ignore them in the following. In Chapter 11 we will
consider the case when VI (r) describes a deep species-selective optical lattice seen only by the
impurity. For the discussion in this Chapter, however, we assume VI,B (r) ⌘ 0.
Fröhlich Hamiltonian in a BEC
It was shown by Tempere et al. [24] that Eq.(9.1) gives rise to an e↵ective Fröhlich polaron
Hamiltonian, describing the interactions of the impurity atoms with the collective phonon
196
CHAPTER 9. INTRODUCTION
(b)
(a)
n0
z
(c)
8
4
q
y
0.5
2
0
0
x
1
6
1
2
0
3
1
0
2
3
Figure 9.1: We consider a single impurity immersed in a three dimensional homogeneous BEC
(a). The total momentum q of the system is conserved, and the interaction of the impurity
with the Bogoliubov phonons of the BEC leads to the formation of a polaron. The dispersion
relation of the Bogoliubov phonons in the BEC is shown in (b) and their scattering amplitude
Vk with the impurity in (c).
excitations of the BEC. As pointed out by Bruderer et al. [315], this is only true however
when the local depletion of the condensate in the vicinity of the impurity is negligible, resulting
in a condition on the interaction strength,
|gIB | ⌧ 4c⇠ 2 .
(9.2)
Here the healing length and the speed of sound in the BEC are denoted by
p
p
⇠ = 1/ 2mB gBB n0 ,
c = gBB n0 /mB
(9.3)
respectively. Note that Eq.(9.2) is always fulfilled when gBB ! 0, but in this limit the
Fröhlich model approaches the subsonic- to supersonic transition (even at zero momentum)
and becomes critical. In this regime a more careful analysis is required.
Before presenting a self-contained derivation of the Fröhlich Hamiltonian (and condition
(9.2)) at the end of this Subsection, let us give an overview first. By treating the BEC and its
excitations in Bogoliubov approximation, and keeping only scattering terms involving both
the impurity and the condensate, we arrive at
Ĥ = gIB n0 +
Z
d3 k
"Z
⇣
d3 r ˆ† (r) ˆ(r)Vk eik·r âk + â
†
k
⌘
+
!k â†k âk
Z
#
2
r ˆ
d3 r ˆ† (r)
(r), (9.4)
2M
which we will refer to as the Bogoliubov-Fröhlich Hamiltonian. Here we made a choice of
the overall energy scale such that the BEC in the absence of impurities has zero energy,
E(gIB = 0) = 0. The term in the second line includes Bogoliubov phonons, which are described
(†)
by operators âk obeying canonical commutation relations (CCRs), [âk , â†k0 ] = (k k0 ). The
corresponding Bogoliubov dispersion is given by
r
1
!k = ck 1 + ⇠ 2 k 2 ,
(9.5)
2
which is shown in FIG.9.1 (b). In addition there are impurity-phonon interactions, which are
9.2. FUNDAMENTAL CONCEPTS
197
characterized by the scattering amplitude [24]
Vk =
p
n0 (2⇡)
3/2
gIB
✓
(⇠k)2
2 + (⇠k)2
◆1/4
,
(9.6)
which is shown in FIG.9.1 (c).
Pseudopotentials and the Lippmann-Schwinger equation
Experimentally, the interaction strengths gIB and gBB can not directly be measured because
they are only convenient parameters in the simplified model (9.1). The actual inter-atomic
potentials in contrast are far more complicated, and in most cases not even known. Nevertheless there is a fundamental reason why these interactions can be modeled by simple contact
interactions, specified by only a single parameter g. When performing two-body scattering
experiments, the measured scattering amplitude fk takes a universal form in the low-energy
limit, see e.g. [5] and references therein,
fk =
1
,
1/a + ik
(9.7)
which is determined solely by the scattering length a (here we restrict ourselves to s-wave
scattering for simplicity). Since scattering amplitudes can be directly accessed in cold atom
experiments, the scattering lengths aIB and aBB fully characterize the interactions between
low-energetic atoms.
To connect the value of a scattering length a to the contact interaction strength g in the
simplified model, the following logic is applied. The algebraic form of the scattering amplitude
(9.7) in the low-energy limit is universal, independent of the particular interaction potential.
The complicated microscopic potential Vmic (x) may thus be replaced by any pseudopotential
Vpseudo (x), as long as it reproduces the same universal scattering length a. In particular, we
can chose the simple 3D contact interaction potential Vpseudo (x) = g (x), where the interaction
strength g is chosen such that the correct scattering length
a=
lim fk (g)
k!0
(9.8)
is reproduced. This equation implicitly defines g(a) as a function of a.
To calculate the interaction strength gIB from the scattering length aIB , which is assumed
to be known from a direct measurement, we now have to calculate the scattering amplitude
fk (gIB ) by solving the two-body scattering problem on the pseudopotential gIB (r). The
easiest way to accomplish this is to use the T -matrix formalism, see e.g. [335], where the
scattering amplitude is given by
✓
◆
mred
k2
fk =
hk, s|T̂ ! =
|k, si.
(9.9)
2⇡
2mred
Here |k, si denotes a spherical s-wave (zero angular momentum ` = 0 of the relative wavefunction) with wave vector k, and we introduced the reduced mass mred of the two scattering
partners. In the case of impurity-boson scattering,
mred1 = M
1
+ mB 1 ,
(9.10)
whereas for boson-boson scattering we would have mred = mB /2. To calculate the T -matrix,
198
CHAPTER 9. INTRODUCTION
the Lippmann-Schwinger equation (LSE) has to be solved,
⇣
⌘
T̂ = 1 + T̂ Ĝ0 V̂ ,
(9.11)
where the free propagator is given by
Ĝ0 (!) =
Z
d3 k
!
|kihk|
k2
2mred
+ i✏
.
(9.12)
To zeroth
order the T -matrix is simply given by the scattering potential, T̂0 = V̂ where
R
V̂ = d3 r |rigIB (r)hr|, and all higher orders can be solved by a Dyson series.
From the first order LSE we can easily derive the familiar relations [5]
aIB =
mred
gIB ,
2⇡
aBB =
mB
gBB ,
4⇡
(9.13)
which we will use to calculate the scattering amplitude Vk in Eq.(9.6) as well as the BEC
properties in Eq.(9.3). Later we will show that proper regularization of the polaron energies
requires the solution of the second-order impurity-boson LSE [24, 321] [P5]. Although we
will not elaborate on this further at this stage, let us already derive the second order result.
Formally the second order result reads T̂ = T̂0 + T̂0 Ĝ0 T̂0 , and using the expressions for T̂0
and Ĝ0 one easily derives
aIB
gIB
=
mred
2⇡
gIB
=
mred
2⇡
Z ⇤0
2
2gIB
1
mred
dk k 2 2
+ O(a3IB )
3
(2⇡)
k /2mred
0
2
4gIB
m2 ⇤0 + O(a3IB ).
(2⇡)3 red
(9.14)
(9.15)
Note that the integral on the right-hand-side of this equation is UV-divergent, and in order
to regularize it we introduced a sharp momentum cut-o↵ at ⇤0 .
Characteristic Scales and the Polaronic Coupling Constant
To get a rough understanding of the physics captured by the Bogoliubov-Fröhlich Hamiltonian,
it is instructive to work out the relevant length and energy scales of di↵erent processes. The
comparison of di↵erent energy scales will then lead to the definition of the dimensionless
coupling constant ↵ characterizing the Fröhlich polaron.
To begin with, we note that the free impurity Hamiltonian is scale-invariant and no particular length scale is preferred. This is di↵erent for the Bose-system, where the boson-boson
interactions give rise to the healing length as a characteristic length scale. This can be seen for
instance from the Bogoliubov dispersion, which has universal low-energy behavior !k ⇠ ck and
high-energy behavior !k ⇠ k 2 /2mB , with a cross-over around k ⇡ 1/⇠. Note that the same
length scale shows up in the impurity-boson interactions, where Vk saturates for momenta
k & 1/⇠. Thus we find it convenient to measure lengths in units of ⇠.
Regarding energy scales, the free impurity problem, again, is scale-invariant. We saw that
the Bogoliubov dispersion is scale invariant asymptotically around k ! 0 and 1, but at a
characteristic energy
Eph ' c/⇠
(9.16)
its behavior is non-universal. The impurity-phonon interactions in the Fröhlich model on the
other hand have an own associated energy scale, which can be derived as follows. Imagine
9.2. FUNDAMENTAL CONCEPTS
199
a localized impurity (M ! 1), such that ˆ† (r) ˆ(r) ! (r) can formally be replaced by a
delta-function in the microscopic Hamiltonian (9.1). To estimate the magnitude of ˆ† ˆ, we
recall that the interaction terms in the Fröhlich Hamiltonian (9.4) originate from scatterings
involving both phonon excitations and bosons from the condensate. We can thus estimate
ˆ† by the BEC density pn0 and ˆ by the phonon density pnph (or vice versa), such that
p
the typical impurity-phonon interactions scale like gIB n0 nph . Since the characteristic length
scale associated with phonons is ⇠, we find the characteristic impurity-boson interaction scale
p
EIB = gIB n0 ⇠ 3 .
(9.17)
Now we can proceed and characterize the Bogoliubov-Fröhlich Hamiltonian by a single
dimensionless coupling constant ↵. Essentially the coupling strength in the Fröhlich model is
determined by the ratio of the impurity-boson interactions EIB to the characteristic phonon
energy scale Eph . We note, however, that the Fröhlich Hamiltonian is invariant under sign
changes of gIB ! gIB (i.e. EIB ! EIB ), up to an overall energy shift due to the BEC-MF
term gIB n0 . Thus we would like to define the dimensionless coupling constant ↵ such that it
depends on ↵ ' (EIB /Eph )2 only. In the literature, the definition of ↵ has been established
in terms of the impurity-boson scattering length aIB [24], which we will now motivate.
To this end we start by calculating the ratio EIB /Eph in the limit M ! 1. In this case
the reduced mass reads mred = (1/M + 1/mB ) 1 = mB and we make use of the relation
between the interaction
strength gIB and the universal scattering length Eq.(9.13). Now we
p
p
obtain EIB /Eph = 8⇡ 2 aIB n0 ⇠, which leads us to the definition
↵ := 8⇡a2IB n0 ⇠
(9.18)
introduced previously by Tempere et al.[24]. In the infinite mass case M ! 1 the last equation reduces to ↵ = (EIB /Eph )2 /⇡, where an additional numerical factor ⇡ 1 was introduced
to recover Tempere et al.’s result (9.18).
We emphasize again that in the following we use the convention that ↵ is always defined
through the scattering length aIB as in Eq.(9.18), independent of the impurity mass1 . We
would also like to mention that Tempere et al. formulated the coupling constant in a slightly
di↵erent – but equivalent – way as
a2
↵ = IB .
(9.19)
aBB ⇠
While the simplicity of this expression is appealing, we think that Eq.(9.18) is more closely
related to the physics of the Fröhlich Hamiltonian.
Finally we discuss the characteristic mass scale in the problem. From the typical energy
and length scales c/⇠ and ⇠ we derive a characteristic mass scale m0 by c/⇠ = (2m0 ⇠ 2 ) 1 .
From
1
p
c⇠ =
(9.20)
mB 2
p
we obtain m0 = mB / 2. Therefore the impurity mass M should be compared to the boson
mass mB in order to distinguish between light, M < mB , and heavy impurities, M > mB .
1
When M < 1 the relation of ↵ to EIB /Eph is slightly modified because EIB depends explicitly on gIB ,
which relates to aIB via mred , thus depending on M .
200
CHAPTER 9. INTRODUCTION
Derivation of the Fröhlich Hamiltonian
Now we turn to the formal derivation of the Fröhlich Hamiltonian (9.4) from the microscopic
model (9.1). To this end we start by considering periodic boundary conditions for the bosons
(period L in each direction) and take the limit L ! 1 in the end. Thus the boson field
(†)
operator can be expressed in terms of a discrete set of bosonic modes dˆk ,
ˆ(r) = L
3/2
X
eik·r dˆk .
(9.21)
k
Because the bosonic field obeys CCRs, [ ˆ(r), ˆ† (r 0 )] = (r
operators dˆk obey CCRs,
[dˆk , dˆ†k0 ] = k,k0 .
r 0 ), also the discrete set of
(9.22)
The bosonic part of the Hamiltonian (9.1) is now discrete in momentum space and can easily
be computed from the above relations,
ĤB =
X
k
k2
1 X gBB ˆ†
dˆ†k dˆk
+ 3
dk+ k dˆ†k0
2mB L
2
0
ˆ 0 dˆk .
k dk
(9.23)
k,k , k
To describe the BEC we apply standard Bogoliubov theory (see e.g. [336]), valid for
weakly interacting condensates, and introduce macroscopic population of the zero-mode,
p
dˆ0 = N0 .
(9.24)
Here N0 is the extensive number of atoms inside the condensate. The Hamiltonian of the
bosons thus reads
ĤB = E0 +
X
k
⌘
k2
gBB N0 X ⇣ ˆ† ˆ
† ˆ†
ˆ
ˆ
ˆ
dˆ†k dˆk
+
2
d
d
+
d
d
+
d
d
k
k ,
k k
k
k
2mB
2L3
(9.25)
k6=0
with E0 the MF energy of the BEC. This Hamiltonian can be diagonalized by means of a
Bogoliubov transformation,
sinh ✓k ↠k ,
dˆk = cosh ✓k âk
(9.26)
(†)
which defines the Bogoliubov phonons âk . From Eq.(9.22) one easily checks that they, too,
obey CCRs. For the diagonalized Hamiltonian ĤB we obtain
X
ĤB = E00 +
!k â†k âk ,
(9.27)
k
where E00 is obtained from E0 by a constant energy shift due to the Bogoliubov transformation.
The mixing angles in the Bogoliubov transformation (9.26) are given by
s
1
k 2 /2mB + gBB n0
cosh ✓k = p
+ 1,
(9.28)
!k
2
s
1
k 2 /2mB + gBB n0
sinh ✓k = p
1.
(9.29)
!k
2
Next we derive the e↵ective boson-impurity interaction. To this end we rewrite the mi-
9.2. FUNDAMENTAL CONCEPTS
(a)
201
(b)
gIB
(c)
(d)
Vq
Figure 9.2: Di↵erent vertices describing the BEC-MF shift of the impurity (a) and impurityphonon interactions (b-d). Solid (black) lines denote the impurity field, dotted (blue) lines
the macroscopic condensate (density n0 ) and dashed (red) lines the Bogoliubov phonons. The
last two kinds of vertices (c,d) are neglected to obtain the Fröhlich Hamiltonian describing an
impurity in a BEC.
croscopic density-density contact interaction in the Hamiltonian as
Z
Z
X
gIB
3
†
†
ˆ
ˆ
ˆ
ˆ
ĤIB = gIB d r (r) (r) (r) (r) = 3
d3 r ˆ† (r) ˆ(r)
ei(k
L
0
k0 )·r ˆ† ˆ
dk dk 0 .
(9.30)
k,k
Taking into account the macroscopic occupation of the condensate yields
2
Z
⌘
X⇣
X
p
gIB
ĤIB = 3
d3 r ˆ† (r) ˆ(r) 4N0 +
eik·r N0 dˆ†k + h.c. +
ei(k
L
0
k,k 6=0
k6=0
k0 )·r
3
dˆ†k dˆk0 5 .
(9.31)
The di↵erent vertices described by this Hamiltonian are summarized in FIG.9.2, where the
discrete boson modes dˆk were rewritten in terms of Bogoliubov phonons.
To proceed, we would like to neglect boson-boson scattering terms O(N0 )0 which do not
involve the condensate 2 . This is justified when the last term, scaling with the (real-space)
p
phonon density nph , is small compared to the second term scaling with n0 nph . We thus
obtain the condition
r
nph
⌧ 1.
(9.32)
n0
We will return to this condition and derive a simpler expression below. Neglecting twoboson terms and replacing the discrete boson modes dˆk by Bogoliubov phonons according to
Eq.(9.26) yields the discretized version of the Fröhlich Hamiltonian,
0
1
Z
⇣
⌘
2
X
X
N
r
0
Ĥ =
!k â†k âk + gIB 3 + d3 r ˆ† (r) @
+
Vkdisc e ik·r âk + ↠k A ˆ(r). (9.33)
L
2M
k
k6=0
In this step we introduced the convenient choice of the overall energy scale such that E = 0
in the absence of an impurity. Moreover, the scattering amplitude Vkdisc can be obtained from
the mixing angle ✓k describing the Bogoliubov rotation. After a straightforward simplification
it reads
✓
◆1/4
gIB p
k 2 /2mB
disc
Vk = 3 N 0
.
(9.34)
L
2gBB n0 + k 2 /2mB
Note, however, that this quantity is not well-behaved when taking the limit L ! 1.
Now, as a last step, we want to consider the thermodynamic limit by taking L ! 1,
while the BEC density N0 /L3 = n0 remains fixed. To do so, continuous operators â(k) have
p
Note that although N0 is a macroscopic number and thus N0
1 in the thermodynamic limit,
P this is
not sufficient to drop two-boson vertices ⇠ dˆ†k dˆk0 . The discrete operators scale like dˆk ⇠ L 3/2 and k ⇠ L3 ,
P
such that L 3 k,k0 dˆ†k dˆk0 ⇠ 1.
2
202
CHAPTER 9. INTRODUCTION
to be defined, which are no longer dimensionless but become densities in momentum space
instead. We will show that, as a consequence, the scattering amplitude Vkdisc will be modified
and the expression (9.6) for Vk is obtained, which is well-behaved in thermodynamic limit.
To be precise, let us introduce continuous phonon operators,
â(k) :=
✓
L
2⇡
◆3/2
âk .
(9.35)
d3 k,
(9.36)
Taking the continuum limit by replacing
X
k
L3
!
(2⇡)3
Z
we find that continuous phonon operators obey CCRs, i.e.
[â(k), ↠(k0 )] = (k
k0 ).
(9.37)
Plugging the continuous operators into the discrete Fröhlich Hamiltonian (9.33), we obtain the
continuum version of the Fröhlich Hamiltonian Eq.(9.4). (Note that in Eq.(9.4) we denoted
the continuous operators â(k) by âk again for a more convenient notation.)
Finally, let us use the Fröhlich Hamiltonian (9.4) to simplify the condition in Eq. (9.32).
In order to estimate the phonon density in the vicinity of the impurity, we consider the e↵ect of
quantum depletion due to one-phonon vertices in Eq.(9.31), see also FIG.9.2 (b). To this end
we calculate the phonon number Nph due to these processes perturbatively (in the interaction
strength gIB ) from the Fröhlich Hamiltonian,
Nph =
Z
3
d k
✓
Vk
!k
◆2
n0 g 2
= p IB .
2⇡ 2 c2 ⇠
(9.38)
To understand on which length scales quantum depletion takes place, let us consider the
asymptotic expressions for Vk and !k ,
(
(
k 0 = 1 k & 1/⇠
k k & 1/⇠
Vk ⇠ p
,
!k ⇠
.
(9.39)
k
k . 1/⇠
k 2 k . 1/⇠
Thus for small k . 1/⇠ the phonon density k 2 Vk2 /!k2 ⇠ k, whereas for large k & 1/⇠ it
scales like k 2 Vk2 /!k2 ⇠ k 2 . Thus most of the depleted phonons have momentum k ⇡ 1/⇠,
and the phonon cloud has a spatial extend on the order of ⇠. Consequently we can estimate
nph ⇡ Nph ⇠ 3 and the condition (9.32) reads
|gIB | ⌧ ⇡21/4 c ⇠ 2 = 3.736... c ⇠ 2 ,
(9.40)
which is essentially the same condition as Eq.(9.2) suggested in [315].
9.2.2
Experimental Considerations
The microscopic model in Eq.(9.1) applies to a generic mixture of cold atoms. Such a situation
is realized experimentally either by mixing atoms of di↵erent species [337, 338] (including the
possibility of chosing di↵erent isotopes [339, 340]), or by using one species but with di↵erent
meta-stable (e.g. hyperfine) groundstates [288, 283]. The model (9.1) can be used as well to
describe a charged impurity immersed in a BEC, as realized e.g. with + Ba and + Rb ions
9.2. FUNDAMENTAL CONCEPTS
203
[323]. The experiments cited above provide only a few examples, and in fact a whole zoo of
Bose-Bose [341, 324, 342, 343, 344, 345] and Bose-Fermi [346, 347, 340, 339, 348, 349, 338,
350, 351, 352, 325, 337, 353, 354] mixtures have been realized. Below we will discuss whether
these experiments can be understood from the Fröhlich polaron model by interpreting atoms
from the minority species as impurities. To this end we derive conditions on the experimental
parameters under which the Fröhlich Hamiltonian (9.4) can be derived from the microscopic
model (9.1). In addition we present typical experimental parameters for such a case.
To get an impression of typical length and energy scales in current experiments, let us
plug in realistic numbers for the example of a Rb-Cs mixture [324, 342, 343, 344]. For a
typical 87 Rb BEC with a density of n0 ⇡ 1013 cm 3 and aBB ⇡ 100a0 [341, 284] (with a0 the
Bohr radius) we obtain values for the healing length ⇠ ⇡ 0.9µm and for the speed of sound
c ⇡ 0.6mm/s. The characteristic time scale associated with phonons is thus ⇠/c = 1.5ms. For
the 133 Cs impurity the inter-species scattering length is aRbCs = 650a0 [344], leading to a
dimensionless coupling constant ↵ = 0.25. The mass-ratio of M/mB ⇡ 1.5 is of the order of
unity.
Now we turn to the discussion of condition (9.2), which requires sufficiently weak impurityboson interactions gIB for the Fröhlich Hamiltonian to be valid. On the other hand, to reach
the interesting intermediate coupling regime of the Fröhlich model, coupling constants ↵ larger
than one ↵ & 1 – i.e. large interactions gIB – are required 3 . We will now discuss under which
conditions both ↵ & 1 and Eq. (9.2) can simultaneously be fulfilled. To this end we express
both equations in terms of the experimentally relevant parameters aBB , mB and M which
are assumed to be fixed, and we treat the BEC density n0 as well as the impurity-boson
scattering length aIB as variable parameters. Using the first-order Born approximation result
gIB = 2⇡aIB /mred Eq.(9.13), condition (9.2) reads
⇣
!
p
mB ⌘
✏ := 2⇡ 3/2 1 +
aIB aBB n0 ⌧ 1,
M
(9.41)
p a2IB pn0
↵ = 2 2⇡ p
.
aBB
(9.42)
and similarly the polaronic coupling constant can be expressed as
Both ↵ and ✏ are proportional to the BEC density n0 , but while ↵ scales with a2IB , ✏ is
only proportional to aIB . Thus to approach the strong-coupling regime aIB has to be chosen
sufficiently large, while the BEC density has to be small enough in order to satisfy Eq.(9.41).
When setting ✏ = ✏max ⌧ 1 and assuming a fixed impurity-boson scattering length aIB , we
find an upper bound for the BEC density,
n0 
nmax
0
= (1 + mB /M )
2
✓
aIB /a0
100
◆
2✓
aBB /a0
100
◆
1
✏2max ⇥ 5.45 ⇥ 1016 cm
3
,
(9.43)
where a0 denotes the Bohr radius. For the same fixed value of aIB the coupling constant ↵
takes a maximal value
p
2
aIB
max
↵
= ✏max
(1 + mB /M ) 1
(9.44)
⇡
aBB
compatible with condition (9.2).
Before discussing how Feshbach resonances allow to reach the intermediate coupling regime,
3
We assume mass ratios M/mB ' 1 of the order of one throughout this work. For very light impurities,
however, the strong-coupling regime is expected to begin already for smaller values of ↵.
204
CHAPTER 9. INTRODUCTION
aRb-K /a0
284.
994.
1704.
2414.
3124.
3834.
max
↵Rb-K
0.26
0.91
1.6
2.2
2.9
3.5
nmax
[1014 cm 3 ]
0
2.8
0.23
0.078
0.039
0.023
0.015
aRb-Cs /a0
650.
1950.
3250.
4550.
5850.
7150.
max
↵Rb-Cs
0.35
1.0
1.7
2.4
3.1
3.8
0.18
0.02
0.0073
0.0037
0.0022
0.0015
nmax
[1014 cm
0
3]
Table 9.1: Experimentally the impurity-boson scattering length aIB can be tuned by more than
one order of magnitude using a Feshbach-resonance. We consider two mixtures( 87 Rb 41 K,
133 Cs, bottom) and show the maximally allowed BEC density nmax along
top and 87 Rb
0
with the largest achievable coupling constant ↵max compatible with the Fröhlich model, using
di↵erent values of aIB and choosing ✏max = 0.3 ⌧ 1.
we estimate values for ↵max and nmax
for typical background scattering lengths aIB . Despite
0
the fact that these aIB are still rather small, we find that keeping track of condition (9.41) is
important. To this end we consider two experimentally relevant mixtures, (i) 87 Rb (majority)
- 41 K [345, 322] and (ii) 87 Rb (majority) - 133 Cs [344, 324]. For both cases the boson-boson
scattering length is aBB = 100a0 [284, 341] and typical BEC peak densities realized experimentally are n0 = 1.4 ⇥ 1014 cm 3 [345]. In the first case (i) the background impurity-boson
scattering length is aRb-K = 284a0 [284], yielding ↵Rb-K = 0.18 and ✏ = 0.21 < 1. By
setting ✏max = 0.3 for the same aRb-K , Eq.(9.43) yields an upper bound for the BEC density
max = 0.26.
nmax
= 2.8⇥1014 cm 3 below the value of n0 , and a maximum coupling constant ↵Rb-K
0
For the second mixture (ii) the background impurity-boson scattering length aRb-Cs = 650a0
[344] leads to ↵Rb-Cs = 0.96 and ✏ = 0.83 < 1. Setting ✏max = 0.3 for the same value of aRb-Cs
max = 0.35. We thus note that already for small values
yields nmax
= 0.18 ⇥ 1014 cm 3 and ↵Rb-Cs
0
of ↵ . 1, Eq.(9.41) is often not fulfilled but has to be kept in mind.
The impurity-boson interactions, i.e. aIB , can be tuned by the use of an inter-species
Feshbach resonance [284], available in a number of experimentally relevant mixtures [347, 342,
350, 351, 352, 353, 354]. In this way, an increase of the impurity-boson scattering length by
more than one order of magnitude is realistic. In Table 9.1 we show the maximally achievable
coupling constants ↵max for various impurity-boson scattering lengths. We consider the two
41 K and 87 Rb
133 Cs), where broad Feshbach resonances
mixtures from above ( 87 Rb
are available [322, 350, 351, 342]. We find that coupling constants ↵ ⇠ 1 in the intermediate
coupling regime can be realized, which are compatible with the Fröhlich model and respect
condition (9.2). The required BEC densities are of the order n0 ⇠ 1013 cm 3 , which should
be achievable with current technology. Note that if Eq.(9.2) would not be taken into account,
couplings as large as ↵ ⇠ 100 would be possible, but then ✏ ⇠ 8
1 indicates the importance
of the phonon-phonon scatterings shown in FIG.9.2 (c,d). In summary we think that a faithful
realization of the intermediate-coupling (↵ ⇠ 1) Fröhlich polaron in ultra cold quantum gases
is possible with current technology.
9.2.3
The Lee-Low-Pines Transformation
In the remaining part of this introductory Chapter, we present the standard approaches which
can be applied to solve polaron Hamiltonians (like the one due to Fröhlich) in the limits of
weak and strong coupling. We start by introducing a very powerful exact method which
was developed by Lee, Low and Pines (LLP) [19] to make the conservation of the polaron
9.2. FUNDAMENTAL CONCEPTS
205
momentum q explicit. It provides a useful starting point for a further approximate treatment.
In the following we will assume a translationally invariant polaron model (it can be a
discrete translational invariance), and show how the conserved (quasi) momentum can be
derived. For concreteness we will discuss the Fröhlich model (9.4). To this end a unitary
transformation is applied which translates bosons (phonons) by an amount which is chosen
such that the impurity is localized in the center of the new frame. Translations of the bosons
are generated by the total phonon-momentum operator,
Z
P̂ph = d3 k kâ†k âk .
(9.45)
Next we would like to
ourselves to a single impurity, which in second-quantized
R restrict
3 r ˆ† (r) ˆ(r) = 1. The position operator of the impurity reads
notation
means
that
d
R
R = d3 r r ˆ† (r) ˆ(r) and the LLP transformation thus reads
ÛLLP = eiŜ ,
Ŝ = R · P̂ph .
(9.46)
To calculate the transformed Fröhlich Hamiltonian, let us first treat the phonon operators.
Because â†k changes the phonon momentum by k, and R can be treated as a C-number from
the point of view of phonons, it follows that
†
ÛLLP
â†k ÛLLP = â†k e
ik·R
.
(9.47)
For the impurity we also want to calculate the transformation of momentum operators, defined
in the usual way as
Z
ˆ† = (2⇡) 3/2 d3 r e iq·r ˆ† (r).
(9.48)
q
Now it is easy to show that the LLP transformation corresponds to a momentum kick for the
impurity, i.e.
†
ˆ† ÛLLP = ˆ†
ÛLLP
.
(9.49)
q
q+P̂ph
With the equations above the Fröhlich Hamiltonian in the polaron frame – defined by
application of ÛLLP – is quickly derived. It takes a product form,
Z
†
H̃ := ÛLLP ĤÛLLP = d3 q ˆq† ˆq ⌦ Ĥq ,
(9.50)
where the momentum q is a conserved quantum
We stress again, however, that
R 3number.
† ˆ
ˆ
this result is true only for a single impurity, d q q q = 1. The q-dependent phonon
Hamiltonian Ĥq reads
Ĥq = gIB n0 +
Z
3
d k
h
!k â†k âk
⇣
+ Vk â
†
k
+ âk
⌘i
1 ⇣
+
q
2M
Z
3
d k
kâ†k âk
⌘2
. (9.51)
This Hamiltonian involves only phonon degrees of freedom. In the original frame, the momentum q corresponds to the total momentum of the combined system of phonons and impurity.
The elimination of the impurity from the problem comes at the cost of a non-linear term in
the phonon operators. It describes interactions between the phonons, mediated by the mobile
impurity. The corresponding interaction parameter is given by the (inverse) impurity mass
M 1.
206
9.2.4
CHAPTER 9. INTRODUCTION
Weak-coupling or Mean-Field Polaron Theory
We will start by solving the polaron problem in the weak-coupling limit. To understand when
the system is weakly coupled, let us study the Fröhlich Hamiltonian in the polaron frame
Eq.(9.51) more closely. We observe that in two cases (9.51) is exactly solvable. The first
case corresponds to small interaction strength ↵ ⌧ 1, and in this case we may approximate
Vk ⇡ 0. Hence [Ĥq , â†k âk ] = 0, and the Hamiltonian is diagonalized in the phonon-number
basis (in k-space). The second case corresponds to large mass M ! 1, when the phonon
non-linearity can be discarded. Let us estimate under which conditions this is justified. To
this end we consider the case q = 0 for simplicity and note that the characteristic phonon
momentum is ⇠ 1 , see 9.2.1. We may thus neglect the last term in Eq.(9.51) provided that
1/(2M ⇠ 2 ) ⌧ c/⇠, where c/⇠ = Eph is the characteristic phonon energy, see Eq.(9.16). Thus
we derived that
p
↵⌧1
or
M
mB / 2
(weak-coupling),
(9.52)
where for the second estimate we used
p
2c⇠ = mB 1 .
To solve for the groundstate of Eq.(9.51) in the weak-coupling regime, we start from the
case when M ! 1 is large. In this case we arrive at a quadratic Hamiltonian, which can be
solved exactly by a coherent displacement operator. That is, we write the groundstate as
✓Z
◆
Y
†
3
| MF i = exp
d k ↵k âk h.c. |0i =
|↵k i,
(9.53)
k
where in the infinite-mass case the complex amplitudes of the coherent states |↵k i are given
by ↵k = Vk /!k . In the case of a finite mass M < 1 we can use the mean-field (MF) wavefunction | MF i from above as a variational ansatz, with variable amplitudes ↵k . Minimizing
Q
the variational MF energy functional HMF (q) = k h↵k |Ĥq |↵k i yields the self-consistency
equations for the ground state polaron
↵k =
Vk
.
MF
⌦k [↵ ]
(9.54)
Here we introduced the renormalized phonon dispersion in the polaron frame, which reads
⌦MF
k = !k +
k2
2M
1
k· q
M
MF
Pph
[↵k ] ,
(9.55)
MF denotes the MF phonon momentum,
where Pph
MF
Pph
[↵k ]
=
Z
d3 k k|↵k |2 .
(9.56)
That ⌦MF
can indeed be interpreted as phonon dispersion in the polaron frame becomes
k
apparent when fluctuations around the MF polaron state are taken into account in Chap.12.
In the following paragraphs, we will discuss di↵erent aspects of the weak-coupling MF
solution. After simplifying the MF self-consistency equations, we calculate the polaron mass
and its energy and finally introduce the MF theory of the subsonic- to supersonic phase
transition. From now on, for simplicity, we consider only real scattering amplitudes Vk 2 R,
which is the case for our model – see Eq. (9.6), and without loss of generality we assume that
q = (q, 0, 0)T is directed along ex with q > 0.
9.2. FUNDAMENTAL CONCEPTS
207
Self-consistency equation
The infinite set of self-consistency equations (9.54) can be simplified into a single equation
MF , where P MF = P MF e . Using cylindrical symmetry of the problem and performing
for Pph
ph
ph x
angular integrals analytically in spherical coordinates yields
MF
Pph
=
2⇡M 2
(q
MF )2
Pph
Z
⇤0
0

dk Vk2 k ⇣
⇣
2 !k +
!k +
k2
2M
k2
2M
⌘2
⌘
k
M
⇣
⇣
MF
Pph
q
k
M (q
0
+ log @
⌘
MF )
Pph
!k +
k2
2M
!k +
k2
2M
⌘2 +
k
M
+
k
M
⇣
⇣
q
MF
Pph
q
MF
Pph
⌘1
⌘ A . (9.57)
MF .
This equation can easily be solved numerically for Pph
As suggested by our physical intuition, the phonon momentum never exceeds the total
MF  q. To show this also mathematically, we note that Eq.(9.57)
system momentum, i.e. Pph
MF = f (q
MF ) where f is some complicated function. Furthermore we
is of the form Pph
Pph
MF
MF ). We will now
read o↵ from Eq.(9.57) that f is anti-symmetric, f (Pph
q) = f (q Pph
MF
assume that a solution Pph > q exists and show that this leads to a contradiction. To this
MF = 0 < q is the solution. Then,
end let us first note that for vanishing interactions, Pph
MF depends continuously on the interaction strength g
assuming that the solution Pph
IB ⇠ Vk ,
MF
we conclude that at some intermediate value of gIB the solution has to be Pph0 = q. This
MF = f (q
MF ) = f (0) = f (0) = 0 6= q.
leads to a contradiction since Pph0
Pph0
Polaron energy
The first physical observable which we easily obtain from the MF solution is the variational
polaron energy. At the saddle-point, i.e. for ↵k from Eq.(9.54), the energy functional HMF (q)
evaluates to
⇣
⌘2
MF
Z ⇤0
P
2
ph
Vk2
q
HMF (q) =
+ gIB n0
d3 k MF
.
(9.58)
2M
2M
⌦k
12
10
M/mB=1
M/mB=2
8
M/mB=5
6
M/mB=20
4
2
0
0
1
2
3
4
Figure 9.3: Polaron ground state energy E0 as a function of the coupling constant ↵ for
n0 = 1 ⇥ ⇠ 3 , q = 0 and using a sharp momentum cut-o↵ at ⇤0 = 2 ⇥ 103 /⇠. We show
MF results Eq.(9.59) for various impurity-boson mass ratios, M/mB = 20 (bottom curve),
M/mB = 5, M/mB = 2 and M/mB = 1 (top curve).
208
CHAPTER 9. INTRODUCTION
Note that we introduced a sharp momentum cut-o↵ at ⇤0 in the integral on the right hand
side. This is necessary because the integral in Eq.(9.58) is UV-divergent, scaling linearly
with
the energy [24, 321] [P5], we note that the polaronic contribution
R 3 ⇤0 .2 ToMFregularize
2
d k Vk /⌦k / gIB yields terms of order a2IB , where we used the first-order LSE (9.13) to
express gIB in terms of aIB . Consistency requires evaluation of the BEC-MF shift gIB n0 to
second order in aIB , which is achieved by using the second-order LSE (9.15). Thus
HMF (q) =
q2
2M
⇣
MF
Pph
2M
⌘2
+ 2⇡aIB n0
✓
◆
1
1
+
mB M
✓
◆
Z ⇤0
Vk2
1
1
3
2
d k MF + 4aIB
+
⇤0 n0 , (9.59)
mB M
⌦k
and one easily checks that this expression leads to a UV convergent polaron energy.
As a sanity-check, we also calculate the polaron energy E0 perturbatively in the coupling
constant ↵. To leading order we obtain the following expression,
q2
E0 (q) =
+ 2⇡aIB n0
2M
✓
1
1
+
mB M
◆
+
4a2IB
✓
◆
1
1
+
⇤0
mB M
Z ⇤0
Vk2
d3 k
k2
!k + 2M
qkx
M
+ O(↵2 ), (9.60)
where, as in the MF case, we employed the second order LSE to evaluate the BEC-MF shift.
MF = O(↵), see Eq.(9.59).
This result is correctly reproduced by MF polaron theory because Pph
Moreover we see that in the spherically symmetric case of vanishing polaron momentum q = 0
MF = 0 and thus MF theory is equivalent to lowest-order perturbation theory.
it follows that Pph
In FIG.9.3 we show the resulting polaron energies as a function of the interaction strength
and using di↵erent mass ratios M/mB . After the regularization of the power-law divergence
discussed above and for positive scattering length4 aIB > 0 the polaron energy is always
positive. This has to be the case for any consistent solution of the microscopic Hamiltonian
(9.1), which is positive definite for repulsive interactions aIB > 0. For attractive interactions
aIB the BEC-MF shift gIB n0 can become negative, but for sufficiently large values of ↵ the
MF energy HMF (q) is always positive because terms linear in ↵ / a2IB dominate.
Polaron mass
In the following we show how the phonon momentum Pph can be used to calculate the polaron
mass Mp . To this end, we note that the polaron velocity is given by vp = q/Mp , where we
assumed that in the ground state all momentum in the system q is carried by the polaron
(instead of being emitted). On the other hand, polaron and impurity position coincide at
all times, so vI = vp . The momentum qI carried by the bare impurity can be calculated
from the total momentum q by subtracting the phonon momentum Pph , qI = q Pph . This
is a direct consequence of the conservation of total momentum. Furthermore, the average
4
We note that for positive impurity-boson scattering lengths aIB > 0, an energetically lower molecular
impurity-boson bound state exists. Thus, the polaron becomes unstable in this regime due to three-body
scattering into the bound-state [321]. We can avoid this problem by considering negative scattering lengths
aIB < 0, where the polaron is stable.
9.2. FUNDAMENTAL CONCEPTS
209
impurity velocity is given by vI = qI /M and we thus arrive at the MF expression
MF
Pph
M
=1
MpMF
q
.
(9.61)
An alternative way of calculating the polaron velocity is to assume that the polaron forms a
wavepacket with average momentum q. Thus vp = @q HMF (q), and using the self consistency
equations (9.54) and (9.56) a simple calculation yields
vp =
dHMF (q)
1
=
q
dq
M
MF
Pph
.
(9.62)
Assuming a quadratic polaron dispersion HMF (q) = q 2 /(2Mp ) we obtain (9.61) again and this
result completely agrees with the semiclassical argument given above. Note however that the
semiclassical argument is more general because it does not rely on the particular variational
state we have chosen.
To check our results, we can also calculate the polaron mass perturbatively in ↵. To this
end we expand the perturbative ground state energy Eq.(9.60) to second order in q (around
q = 0) and obtain
Z
k 4 Vk2
8⇡ ⇤0
2
Mp = M +
dk ⇣
(9.63)
⌘3 + O(↵ ).
3 0
k2
!k + 2M
The same expansion can straightforwardly be derived from Eq.(9.61) employing the MF exMF .
pression (9.56) for the total phonon momentum Pph
In FIG.9.4 we show the momentum dependence of the MF polaron mass Eq.(9.61) for
various cut-o↵s ⇤0 . We notice that for small q . M c the momentum dependence is rather
weak. The e↵ective mass increases with increasing cut-o↵ ⇤0 , due to dressing with more and
more high energy phonons. In FIG. 9.5 we show the dependence of the MF polaron mass on
the coupling strength ↵. We find a linear dependence
MpMF
M / ↵.
(9.64)
=2000/⇥
0
1.3
=200/⇥
0
1.25
=20/⇥
0
1.2
=2/⇥
0
1.15
1.1
1.05
1
0
0.5
1
1.5
Figure 9.4: The polaron mass Mp /M calculated from MF theory is shown as a function
of polaron momentum q (in units of M c) for a coupling constant ↵ = 2 and a mass ratio
M/mB = 1.5. Results are shown for di↵erent sharp momentum cut-o↵s ⇤0 and convergence
is found for around ⇤0 & 200/⇠. The MF curves end at the critical momentum qcMF (see
Eq.(9.70)) where the transition to the supersonic regime takes place.
210
CHAPTER 9. INTRODUCTION
When decreasing the bare impurity mass, the ratio M/MpMF decreases as well, i.e. the smaller
the bare impurity mass the larger the renormalization of the polaron mass. This is in accordance with our expectation that the polaron becomes strongly coupled when the mass ratio
mB /M becomes large. Comparison to the perturbative polaron mass in FIG.9.5 shows strong
qualitative di↵erences to MF, unlike for the overall polaron energy. Around some critical
coupling strength, in this case ↵ ⇡ 6, the perturbative result shows an unphysical divergence
of the polaron mass.
Subsonic to supersonic transition
The last aspect that we discuss about the MF Fröhlich polaron is the transition from the
subsonic- to the supersonic regime. It takes place when the polaron momentum q, entering the
polaron Hamiltonian (9.51) as an external parameter, exceeds some critical value qc . We will
show explicitly that this transition is a reformulation of Landau’s criterion for superfluidity.
While this is obvious in the case of a quasi-free impurity, it is instructive to derive Landau’s
condition also for a strongly interacting impurity (at least on the MF level).
In the non-interacting case the subsonic- to supersonic transition can easily be understood
from the original Fröhlich Hamiltonian (9.4). Obviously its global groundstate has zero total
momentum and corresponds to the phonon vacuum. To construct the groundstate at finite
momentum q we compare the energy costs of adding impurity and phonon momentum in
FIG.9.6. For small momenta adding impurity momentum is advantageous because @q !q >
@q q 2 /2M . This changes when the speed of sound c = @k !k |k!0 is smaller than the impurity
velocity vI = q/M , where q is the impurity momentum. Thus we find a phase-transition
at a critical total momentum qc = M c in the non-interacting case. Reformulated in terms
velocities we recognize Landau’s criterion: there will be no phonon-emission as long as the
impurity velocity vI < c is below the speed of sound. (Note that this is not true in systems
where the impurity is driven by an external force, as we will discuss in Chap.11.) Because the
groundstate energy E0 (q) changes smoothly across the transition and its derivative @q E0 (q)
is discontinuous, this is a second-order quantum phase transition.
Now we return to the discussion of an interacting impurity, which we treat on the MF
level. We will show that above the critical momentum qc the MF saddle point equations do
5
4
3
2
1
0
0
2
4
6
8
10
Figure 9.5: Polaron mass Mp /M as a function of the coupling strength ↵ for di↵erent mass
ratios M/mB = 1 (red) and M/mB = 5 (green). Other parameters are q/M c = 0.01 and
⇤0 = 2 ⇥ 103 /⇠. We show MF calculations (solid) and compare them to the perturbative
result (dashed) from Eq.(9.63), which shows an unphysical divergence, and strong coupling
(SC) results according to Eq.(9.78).
9.2. FUNDAMENTAL CONCEPTS
211
(a)
(b)
2.5
supersonic
2
1.5
1
subsonic
0.5
0
0
1
2
3
4
Figure 9.6: (a) The subsonic- to supersonic phase transition takes place when the total system
momentum q exceeds a critical value qc . In the non-interacting case ↵ = 0 the transition point
can be derived by comparing the energy cost of adding impurity versus phonon momentum
to the system. (b) MF polaron phase diagram: While for small momenta q < M c the polaron
is subsonic for all couplings ↵, it becomes supersonic for large enough momenta q > qc (↵).
We used a mass ratio of M/mB = 1 and a momentum cut-o↵ at ⇤0 = 2 ⇥ 103 /⇠.
not have a solution. By relating this point to Landau’s criterion of superfluidity, formulated
for the polaron however, we identify it with the subsonic- to supersonic transition. We start
by showing that for any solution ↵k of the saddle-point equation (9.54) the phonon dispersion
⌦MF
0 is positive everywhere. To this end let us assume the opposite, i.e. we consider a
k
solution ↵
˜ k such that ⌦MF
↵ ] < 0 somewhere, and show that this leads to a contradiction.
k [˜
If ↵
˜ k exists, this would also require the existence of an entire 2D surface S in k-space where
⌦MF
↵ ] = 0 for k 2 S, because asymptotically for k ! 1 we have ⌦MF
k [˜
k > 0 and we assume
MF
that ⌦k [˜
↵ ] is a continuous function of k. Such zeros of the dispersion correspond to poles
of ↵
˜ k , see Eq.(9.54), and in their vicinity one generally has ⌦MF
↵ ] ⇠ k? @? ⌦MF
where
k [˜
k
k? denotes momentum in normal direction to S. Consequently we obtain diverging physical
quantities in general,
Z
Z
Z
Z
1
2
2
2
d k dk? |↵
˜k | ⇠
d k dk? 2 = 1.
(9.65)
k?
S
S
MF [˜
In particular the phonon momentum Pph
↵ ] ! 1 diverges, which means that the amplitude
MF
↵
˜ k = Vk /⌦k [˜
↵ ] ! 0 vanishes. This leads to a contradiction because for ↵
˜ k = 0 the total
MF [˜
phonon momentum vanishes, Pph
↵ = 0] = 0 6= 1.
We have just shown that ⌦MF
k > 0 for any self-consistent MF solution of the saddle point
equations (only for k=0 we have ⌦MF
= 0). Closer inspection of the k-dependence of ⌦MF
k
k ,
see Eq.(9.55), shows that this implies
vP =
1
q
M
MF
Pph
< c.
(9.66)
In the first equality we used Eq.(9.62) for the MF polaron velocity vP . This is exactly Landau’s
criterion for superfluidity, but formulated for the polaron which has a larger mass than the
bare impurity. Thus we identify the breakdown of MF theory (in the sense that saddle-point
equations have no solutions anymore) with the subsonic- to supersonic phase transition of
the polaron. This is further supported by the fact that on the supersonic side the e↵ective
phonon dispersion is required to take negative values, ⌦MF
< 0, signaling a breakdown of
k
dissipationless impurity (or polaron) current because emission of phonons is energetically
212
CHAPTER 9. INTRODUCTION
favorable. The MF polaron phase diagram is presented in FIG. 9.6(b). We observe that for
q/(M c) < 1 the polaron is subsonic for all interactions, whereas it becomes supersonic for
q/(M c) > 1 and for small enough couplings.
In the remainder of this section we discuss how the MF phase diagram in FIG.9.6 was calMF )/M = c.
culated. To determine where the polaron becomes supersonic, we solve vP = (q Pph
To this end we find it convenient to introduce the normalized coupling
Vk =
Vk
p ,
gIB n0
(9.67)
p
which is independent of the interaction strength gIB n0 . From Eq.(9.57) we then obtain the
critical interaction strength (at fixed momentum q)
2
gIB
n0
c
=
2⇡
R ⇤0
0
(q
M c) c2
2
dk k V k [A(k) + L(k)]
,
where in the denominator we have an algebraic and a logarithmic term
⇣
⌘
!
k2
k2
2ck !k + 2M
!k + 2M
+ ck
A(k) = ⇣
> 0,
L(k) = log
< 0.
⌘2
k2
k2
!
+
ck
2
2
k
2M
!k + 2M
c k
(9.68)
(9.69)
Alternatively we can introduce the critical momentum qcMF (at fixed interaction strength)
MF (q ). From Eq.(9.68) we
above which the polaron becomes supersonic, i.e. qcMF = M c Pph
c
read o↵
Z
2⇡ ⇤0
qcMF = M c + 2
dk k Vk2 [A(k) + L(k)] .
(9.70)
c 0
We find that A(k) + L(k) > 0 for all k
0 and thus there is no MF solution only when
q
qcMF > M c. This can be understood from the simple physical picture that dressing the
impurity with phonons leads to an increase of its mass and hence to an increase in the critical
momentum qcMF above the non-interacting value M c.
9.2.5
Strong-coupling polaron theory
In the last part of this introductory chapter we discuss briefly the strong-coupling polaron
wavefunction, initially introduced by Landau and Pekar [17, 18]. We will summarize its
application to the problem of impurities in a BEC, discussed by Casteels et al. [319], and
derive some expressions which we return to in the chapters below. We also discuss why
the strong-coupling approximation yields an inconsistent result for the polaron energy in the
specific case of an impurity immersed in a BEC.
The idea in the strong-coupling treatment is to assume that phonons adiabatically follow
the slow dynamics of the heavy impurity. An other way of putting it, is to say that phonons
can instantly adjust to a change of the impurity position r. This is similar to the physics
of molecular binding, where electrons instantly adjust to the positions of the atomic nuclei.
For such a description to apply, we require either slow impurity dynamics – i.e. a large mass
M
mB – or fast impurity-phonon interactions – i.e. large ↵
1,
↵
1
or
M
mB
(strong-coupling),
(9.71)
Note that this implies that strong- and weak-coupling regimes, as defined in this thesis, are
9.2. FUNDAMENTAL CONCEPTS
213
identical in the case of large impurity mass M
mB .
Mathematically the starting point for the strong-coupling treatment is the Fröhlich Hamiltonian Eq.(9.4). The strong-coupling wavefunction has a product form,
Z
| SC i = | ph ia ⌦ d3 r SC (r) ˆ† (r)|0iI ,
(9.72)
where | ph ia is the phonon wavefunction and | SC iI the impurity state. The first step consists
of solving the phonon problem for an arbitrary impurity state | SC iI . The e↵ective phonon
Hamiltonian has a form
Z
⇣
⌘
Ĥph = d3 k Ṽk âk + ↠k + !k â†k âk ,
(9.73)
where the e↵ective interaction is defined by Ṽk = Vk I h SC |eik·r | SC iI , with Ṽ k = Ṽk⇤ . This
Hamiltonian is quadratic and can easily be solved by coherent state wavefunctions,
|
ph ia
=
Y
k
|
k i,
k
Ṽk
.
!k
=
(9.74)
R 3
The phonon energy is given by Eph [| SC iI ] =
d k |Ṽk |2 /!k , and it depends on the particular impurity wavefunction. This has the e↵ect of an e↵ective potential, depending on the
state of the impurity.
To make further progress, a variational ansatz is made for the impurity wavefunction.
Motivated by the idea that the e↵ective potential due to the phonon cloud can localize the
impurity, one introduces a Gaussian wavefunction to approximate the polaron groundstate,
SC (r)
=⇡
3/4
3/2
e
r2
2 2
.
(9.75)
Here is a variational parameter, describing the spatial extend of the impurity wavefunction.
The total energy
Z
r2
E0SC = gIB n0 + h SC |
| SC i
d3 k |Ṽk |2 /!k
(9.76)
2M
can be expressed in terms of , see [319],
p

3
2
1
SC
p ↵µ
E0 ( ) = gIB n0 +
1
4M 2
M ⇠
⇡
p
⇡ e
⇠
2 /⇠ 2
(1
erf ( /⇠)) .
(9.77)
which can then easily be minimized numerically. Here erf(x) denotes the error function and
the dimensionless quantity µ is defined as µ = M mB /(4m2red ). One comment is in order about
this energy. The polaronic contribution to the energy in Eq.(9.77) is UV convergent and the
cut-o↵ ⇤0 = 1 was used in the calculations. At first sight this might seem desirable, however
consistency requires to use an expression for gIB which is accurate at least to order a2IB . As
discussed in Sec.9.2.1 the LSE yields a BEC-MF shift gIB n0 which is UV divergent to order
a2IB . Unlike in the MF case, this divergence is not canceled by a similar term in the polaronic
contribution to the energy. Thus the resulting energy (9.77) should be taken with care, and it
can not be compared one-to-one to properly regularized polaron energies. Nevertheless it can
be hoped that Eq.(9.77) gets the dependence on model parameters like the coupling constant
↵ about right at low energies.
By modifying the variational wavefunction (9.75) to a wavepacket describing finite polaron
214
CHAPTER 9. INTRODUCTION
momentum, Casteels et al. [319] also derived an expression for the polaron energy. Using their
equations we arrive at the following expression for the strong-coupling polaron mass,

✓
◆
2
p
↵ ⇣
mB ⌘2
2 2
SC
p
Mp = M + mB
1+
2 + ⇡ 1 + 2 2 e /⇠ (1 erf ( /⇠)) . (9.78)
M
⇠
⇠
3 2⇡
Most notably at this point is that the strong-coupling polaron mass follows the same powerlaw MpSC M / ↵ as MF theory for large ↵ (this can be checked by expanding in powers of
↵ 1 ). Thus in the strong-coupling regime both predictions di↵er only in the prefactor of this
power-law. We show some numerical results for the strong-coupling polaron mass in FIG.9.5.
Chapter 10
RF Spectra of Fröhlich Polarons in
a BEC
10.1
Summary
In this chapter we start our investigation of out-of-equilibrium properties of Fröhlich polarons
in a BEC on the mean-field (MF) level. Specifically we will describe non-equilibrium polaron
formation at finite momenta and calculate the full spectral function of the impurity, which
can experimentally be measured using radio-frequency (RF) spectroscopy [355, 356]. In the
case of Fermi-polarons such experiments have successfully been carried out [299], and they
can be carried over to Bose polarons directly. Momentum dependent RF spectra allow to
extract a number of equilibrium properties of the polaron, see FIG.10.1 (a). For example the
position of the coherent polaron peak is described by the polaron energy E0 and its amplitude
encodes the quasiparticle weight Z. By comparing RF spectra at di↵erent polaron momenta
q the dispersion relation E0 (q) can be measured, which gives access to the polaron mass Mp .
(a)
(b)
0.1
0.08
0.06
rf
0.04
0.02
0
0
0
0
1
2
3
4
5
6
Figure 10.1: In this chapter we calculate RF spectra of an impurity immersed in a BEC, as
shown in the inset of (b). The interacting impurity is described by the Fröhlich Hamiltonian.
(a) The characteristic properties of an impurity RF spectrum, taken for a given impurity
momentum q, allow to extract equilibrium properties of the polaron. (b) Inverse RF spectra
calculated from MF theory are shown for di↵erent coupling strengths ↵ in and using the q = 0
groundstate. We plotted only the incoherent part Iincoh ( !) as a function of the frequency
! from the coherent polaron peak. Other parameters were M/mB = 0.26 and ⇤0 ⇡ 200/⇠
(we used a soft cut-o↵ to avoid unphysical high-frequency oscillations of the time-dependent
overlaps Aq (t) resulting from a sharp cut-o↵).
215
216
CHAPTER 10. RF SPECTRA OF FRÖHLICH POLARONS IN A BEC
The full polaron spectrum moreover contains information about additional excitations
of the system. To calculate the RF spectrum, we solve the non-equilibrium problem of an
interaction quench and keep track of the subsequent phonon dynamics [P5]. To this end we
develop a time-dependent MF theory in the polaron frame, which is mathematically exact for
localized impurities. We apply this approach to investigate the physics of polaron formation
both in the time- and frequency domain. In the frequency domain we obtain the RF spectrum,
in which we identify a power-law tail at high energies linked to universal two-body physics
[332, 333]. In the time domain we investigate how momentum is transferred from an impurity
to the phonons in the process of polaron formation. We identify pronounced oscillations of
the impurity velocity which we interpret as a signature of the polarons internal structure.
This chapter is intended as an overview of the publication [P5], where a more detailed
discussion of some points can be found that will be of less importance for the rest of this
thesis. The chapter is organized as follows. In Sec.10.2 we give a brief overview of RF spectra
and their generic properties. Afterwards we introduce our concrete model and show how RF
spectra can be interpreted as non-equilibrium probes of the system. In Sec.10.3 we describe
the time-dependent MF ansatz used to solve non-equilibrium polaron problems. In Sec.10.4
we discuss the resulting RF spectra, both for the ”direct” and ”inverse” method. In Sec.10.5
we present results for non-equilibrium polaron dynamics and discuss possible connections to
the internal polaron structure.
10.2
RF spectra
The definition of the (momentum-resolved) spectral function I(!, q) is motivated by its relation to the RF-spectrum of an impurity atom. Experimentally, to obtain RF-spectra, impurities with internal states – labeled | "i and | #i in the following – are required. In practice
they would correspond to hyperfine states of the atoms. Furthermore, let us consider here
the case when only one of the two internal states – say | "i – interacts with the phonon
environment. Thus, the ground state where the system has total momentum q and the atom
is in its interacting | "i-state can be described in the polaron frame by | "0 i = | "i ⌦ |qi ⌦ | q i,
where | q i is the phonon wavefunction, see Sec.9.2.3. Analogously, when the impurity is in
its non-interacting state | #i, the systems ground state reads | #0 i = | #i ⌦ |qi ⌦ |0i with the
phonons in their vacuum state |0i. The energy of the non-interacting state will be labeled E#0 .
The RF-spectrum is then obtained by driving a spin-flipping transition from the noninteracting state | #0 i to an interacting state | "n i (both at total momentum q), with frequency
!. Here n labels all possible interacting eigenstates, the energy of which we denote by E"n .
More specifically, I(!, q) is defined as the absorption cross-section describing the probability
for such a spin-flip to occur, and it can easily be calculated using Fermi’s golden rule,
X
I(!, q) =
|h "n | "ih# | #0 i|2 ! (E"n E#0 ) .
(10.1)
n
In a realistic experimental setup, the
R absorbed intensity Pabs (!) of an RF beam at frequency !
will be proportional to Pabs (!) / d3 q I(!, q)nq , where nq is the initial impurity momentumdistribution.
The RF-spectroscopy described above, where a non-interacting state is flipped into an
interacting one, is also referred to as inverse RF. In the direct RF protocol, in contrast, the
interacting ground state | "0 i is coupled to non-interacting states | #n i. In this case, Eq.(10.1)
still holds after exchanging spin-labels in the expression (" to # and # to "). In the rest of
this section, for concreteness, we will only be concerned with the inverse RF spectrum. We
10.2. RF SPECTRA
217
checked that its qualitative properties are the same as those of the direct RF protocol. In
particular, both spectra allow to measure characteristic polaron properties. We note that,
from an experimental point of view, the inverse RF protocol has the conceptual advantage
of being less sensitive to a finite polaron life-time: Before applying the spin-flip, the noninteracting impurity is stable and has time to equilibrate into its ground state without being
lost. Strongly interacting impurities close to a Feshbach-resonance, on the other hand, can
have a finite lifetime due to energetically lower molecular impurity-boson bound states [321].
Before discussing how to calculate the spectral function, let us summarize its key properties
in the case of the polaron problem. As an example we show RF spectra in FIG.10.1. The
Bogoliubov-Fröhlich polaron is a stable quasiparticle with an infinite lifetime (at least in the
idealized model (9.1)) which corresponds to a coherent delta-peak in the spectrum. The
finite intrinsic line width of the RF transition, and other experimental limitations [299], are
discarded here.
The coherent peak is located at the polaron ground state energy, ! = !0 + E0 (q), where
!0 is the bare transition frequency between | #i and | "i in the absence of the BEC. Thus
a measurement of the position of the polaron peak is sufficient to obtain the polaron energy
E0 (q). Moreover, because the impurity is coupled to phonons by the impurity-boson interaction, we obtain a broad spectrum of phonon excitations at larger frequencies ! !0 > E0 (q).
These features can be identified in FIG. 10.1. Thus the spectral function takes the form
I(!, q) = Z (!
(!0 + E0 )) + Iincoh (!, q),
(10.2)
as can be shown using a Lehmann expansion [P5]. The spectral weight of the coherent part
Icoh (!, q) = Z (! (!0 + E0 (q))) is given by the quasiparticle weight Z. The quasiparticle
weight describes the amount of free-impurity character of the polaron and can be calculated
from the groundstate phonon wavefunction in the polaron frame,
Z = |h0|
q i|
2
.
The incoherent part Iincoh (!, q) is non-vanishing only for !
following sum-rule (which follows from Eqs.(10.1), (10.2))
Z
d! Iincoh (!, q) = 1 Z.
(10.3)
!0 > E0 (q), and it fulfills the
(10.4)
Using this sum-rule the quasiparticle weight Z can be determined from the absorption spectrum Pabs (!), even when the non-universal prefactor relating Pabs (!) and I(!) is unknown.
The RF spectrum of an impurity can serve as a fingerprint of polaron formation, and it
was used to proof polaron formation of impurities in a Fermi sea [299]. Let us summarize
the required signatures to claim that a polaron has formed. First of all the presence of a
coherent (delta)-peak in the RF spectrum proofs the existence of a long-lived quasiparticle in
the system. Secondly, to distinguish a non-interacting impurity from a polaron – i.e. a dressed
impurity – the spectral weight Z of the coherent peak has to be measured. Only when Z < 1
the quasiparticle has a phonon cloud characteristic for the polaron. Alternatively, because of
the sum-rule Eq.(10.4), it is sufficient to show the existence of an incoherent tail in the RF
spectrum to conclude that Z < 1.
218
CHAPTER 10. RF SPECTRA OF FRÖHLICH POLARONS IN A BEC
10.2.1
Model
In the following we calculate the RF spectrum of a Fröhlich polaron in a BEC, see Sec.9.2.1.
To this end we consider an atomic mixture with a minority (= impurities) and a majority
species (= BEC). We assume the impurities to be sufficiently diluted, allowing to treat them
1/3
independently of each other. This may safely be assumed when their average distance nI
1/3
(where nI denotes their average density) is well below their radius, i.e. for nI
⌧ ⇠.
Moreover we assume that impurity atoms have two spin states =", # which interact with
the bosons with di↵erent interaction strengths gIB . We assume that, in the absence of the
BEC, they are separated by an energy di↵erence !0 from each other. The Hamiltonian for
this system thus reads
ĤRF =
!0
2
z
"
#
+ | "ih" | ⌦ Ĥ(gIB
) + | #ih# | ⌦ Ĥ(gIB
) + ⌦ei!t | #ih" | + h.c. ,
(10.5)
where Ĥ(gIB ) denotes the Fröhlich Hamiltonian (9.4) for the interaction strength gIB . For
#
simplicity we assume gIB
= 0 in the rest of the chapter. The last term describes a RF
transition with strength ⌦, which conserves the total momentum q. (The momentum transfer
kRF ⌧ 1/⇠ can safely be neglected.) When starting in the state | #i at momentum q the
transition rate into the second spin-state | "i is given by ",# = 2⇡|⌦|2 I(!, q) according to
Fermi’s golden rule.
10.2.2
Formulation as a non-equilibrium problem
Having discussed how the spectral function can be measured using rf-spectroscopy, we now
turn to the question how it can most conveniently be calculated. To this end we reformulate
the problem as a dynamical one, rather than a static one as it stands in Eq.(10.1). Using
standard manipulations [357], one can write
Z
1 1
I(!, q) = Re
dt ei!t Aq (t),
(10.6)
⇡ 0
where the time-dependent overlap (related to the Loschmidt-echo, see e.g. [358]) is defined as
0
Aq (t) = eiE# t h0|e
iĤq t
|0i.
(10.7)
We formulated the last expression (10.7) already in the polaron frame, i.e. Ĥq denotes the
polaron Hamiltonian (9.51) at total momentum q and after application of the LLP transformation. The state |0i, meanwhile, denotes the phonon vacuum in this frame and we emphasize
that |0i is not an eigenstate of Ĥq but a superposition of highly excited states of the interacting Hamiltonian. Equations (10.6) and (10.7) make apparent that the spectral function
I(!, q) is related to a non-equilibrium quantity. The function Aq (t) is defined as the overlap
of the time-evolved vacuum state e Ĥq t |0i to the initial state itself. Because |0i is not an
eigenstate of Ĥq , the time-evolution above is highly non-trivial in general.
In the remainder of this chapter we will make use of the representation Eqs.(10.6), (10.7)
to calculate the spectral function I(!, q). To this end we calculate the time-evolved state
e Ĥq t |0i which yields the time-dependent overlap Eq.(10.7). Let us also mention that the
time-dependent overlap itself can also be accessed experimentally in real-time dynamics using
a Ramsey-type protocol [298].
10.3. TIME-DEPENDENT MF THEORY
10.3
219
Time-dependent MF theory
In this section we introduce the time-dependent MF theory. It is a straightforward generalization of the static theory presented in Sec.9.2.4 to time-dependent variational wavefunctions.
To this end the time evolution in Eq.(10.7) is approximated by time-dependent coherent
phonon states,
Y
e Ĥq t |0i ⇡ e i q (t)
|↵k (t)i.
(10.8)
k
Now the (complex) amplitudes ↵k (t) are time-dependent and in addition we have to take into
account a global (real) phase degree of freedom q (t). In the following subsection 10.3.1 we
use Dirac’s time-dependent variational principle to derive the following equations of motion
for the variational parameters,
[email protected] ↵k (t) = ⌦k [↵ (t)]↵k (t) + Vk ,
⇣
⌘2
MF [↵ (t)]
Z
P
2

ph
q
@t q (t) =
+ gIB n0 + Re d3 k Vk ↵k (t).
2M
2M
(10.9)
(10.10)
MF [↵ (t)] denotes the time-dependent MF phonon momentum, see Eq.(9.56).
Here Pph

The time-dependent overlaps Aq (t), required for the calculation of the spectral function
(10.6), can easily be calculated from the variational wavefunction from Eq.(10.8). We obtain

q2
1 MF
Aq (t) = exp it
i q (t)
N (t) ,
(10.11)
2M
2 ph
Z
MF
Nph (t) =
d3 k |↵k (t)|2 ,
(10.12)
MF (t) is the phonon number of the time-dependent MF state (10.8), and we have
where Nph
chosen our overall energy scale such that E#0 = q 2 /2M .
10.3.1
Equations of motion – Dirac’s variational principle
To derive equations of motion for the coherent amplitudes ↵k (t) and the phase q (t) in
Eq.(10.8), we employ Dirac’s time-dependent variational principle, see e.g. [359]. It states
that, given a possibly time-dependent Hamiltonian Ĥ(t), the dynamics of a quantum state
| (t)i (which can alternatively be described by the Schrödinger equation) can be obtained
from the variational principle
Z
dt h (t)|[email protected] Ĥ(t)| (t)i = 0.
(10.13)
We reformulate this in terms of a Lagrangian action L = h (t)|[email protected]
Z
dt L = 0.
Ĥ(t)| (t)i and obtain
(10.14)
When using a general variational ansatz | (t)i = | [xj (t)]i defined by a set of some timedependent variational parameters xj (t), we obtain their dynamics from the Euler-Lagrange
equations of the classical Lagrangian L[xj , ẋj , t]. We note that there is a global phase degree
of freedom: when | (t)i is a solution of (10.13), then so is e i (t) | (t)i because the Lagrangian
changes as L ! L + @t (t). To determine the dynamics of q (t) in Eq.(10.8) we note that for
220
CHAPTER 10. RF SPECTRA OF FRÖHLICH POLARONS IN A BEC
the exact solution |
ex (t
0 )i
of the Schrödinger equation it holds
Z
t
0
dt0 L(t0 ) = 0,
(10.15)
for all times t, i.e. L = 0. This equation can then be used to determine the dynamics of the
overall phase for variational states.
The equations of motion for ↵k (t) can now easily be derived from the Lagrangian
Z
i
⇤
⇤
⇤
L[↵k , ↵k , ↵˙ k , ↵˙ k , t] = @t q HMF [↵k , ↵k ]
d3 k (↵˙ k⇤ ↵k ↵˙ k ↵k⇤ ) ,
(10.16)
2
where we used the following identity valid for coherent states |↵i,
h↵|@t |↵i =
1
(↵↵
˙ ⇤
2
↵˙ ⇤ ↵) .
(10.17)
The MF variational energy functional in Eq.(10.16) evaluates to
HMF [↵k , ↵k⇤ ]
=
q2
2M
⇣
MF
Pph
2M
⌘2
+ gIB n0 +
Z
⇥
⇤
2
⇤
d3 k ⌦MF
k |↵k | + Vk (↵k + ↵k )
(10.18)
MF
where the renormalized phonon dispersion ⌦MF
k [↵k ] and the phonon momentum Pph [↵k ] are
defined in Eqs.(9.55) and (9.56), and they depend on the amplitudes ↵k themselves (unless
MF = 0). Using the Eulerq = 0 and the initial state is spherical symmetric, i.e. when Pph
Lagrange equations associated with the Lagrangian (10.16) the equations of motion (10.9)
follow, and from L = 0 we obtain (10.10).
10.4
Discussion of RF spectra
In this section we discuss the MF polaron spectra in detail and present numerical results. We
also discuss generic features of the RF spectrum in the high-frequency limit.
In FIG.10.1 (b) we show inverse RF spectra, where the initial impurity state | #i is noninteracting while the final state | "i is. We compare results for di↵erent interaction strengths ↵.
For increasing couplings we observe that spectral weight moves from small to high frequencies.
Meanwhile, the total weight Z in the coherent peak (1 Z in the incoherent part) decreases
(increases). The position of the maximum of I(!) is essentially unchanged. In FIG.10.2
sample spectra are shown for the direct RF protocol, where the initial impurity state | #i is
interacting while the final state | "i is not. It is assumed that initially the impurity is in its
polaronic groundstate, which can adiabatically be prepared in an experiment before the RF
pulse is switched on. On a qualitative level the behavior is very similar to the inverse RF
spectra. For very large couplings ↵ ⇡ 40 we observe a shift of the intensity maximum towards
higher frequencies. Although this is plausible, we note that in this regime MF theory is no
longer trustworthy. The strong-coupling regime will be investigated further in Chap.13.
10.4.1
Leading-order expansions
From our result (10.11) above, we can analytically derive the spectral functions I(!, q) in the
weak coupling limit for the inverse RF protocol. To this end we consider the limit ↵ ! 0,
when the MF prediction becomes exact, allowing us to expand around Nph = 0. Then using
10.4. DISCUSSION OF RF SPECTRA
221
Eq.(10.6) and assuming spherical symmetry q = 0 for simplicity, we arrive at
Iincoh (!
HMF ) = 4⇡
k 2 Vk2
2
⌦MF
k
@k ⌦MF
k
2
+ O(Nph
),
(10.19)
where k is related to ! by the condition that
HMF = ⌦MF
k ,
!
(10.20)
with HMF the MF polaron energy, see Eq.(9.59). Note that Eq.(10.19) only applies when !
HMF , whereas for ! < HMF the resonance condition (10.20) can not be fulfilled and Iincoh = 0
in this regime. The coherent part of the spectrum is simply given by Icoh = ZMF (! HMF ),
where ZMF = |h0| MF i|2 is obtained from the equilibrium MF wavefunction (9.53).
Plugging the asymptotic expressions for ⌦MF
and Vk into Eq.(10.19), we arrive at the
k
low-frequency expansion
Iincoh (!
Here mred1 = mB 1 + M
limit on the other hand
1
Iincoh (!
10.4.2
(! HMF )
HMF ) = ↵ p
+ O(↵2 ),
4 2⇡c4 m2red
! ⌧ c/⇠.
(10.21)
denotes the reduced impurity-boson mass. In the large-frequency
HMF ) = ↵
HMF ) 3/2
+ O(↵2 ),
p
1/2
2 2⇡⇠mred
(!
!
c/⇠.
(10.22)
Universal high-energy RF tail
From the asymptotic behavior of the RF spectrum in 3D we saw that I(!) ⇠ ! 3/2 in the
high-frequency limit, see Eq.(10.22). The numerical results in FIGs.10.1 and 10.2 are found
Figure 10.2: Direct RF spectra I(!) are shown for di↵erent values of the coupling constant ↵
of the initial state | #i. The final state | "i is non-interacting in this protocol. The mass ratio
was chosen to be M/mB = 2.5; These spectra were calculated by Aditya Shashi.
222
CHAPTER 10. RF SPECTRA OF FRÖHLICH POLARONS IN A BEC
to be in good agreement with this prediction for all values of ↵. Similarily, in 2D we obtain a
power-law I(!) ⇠ ! 2 . These characteristic power-laws have been observed by various authors
in di↵erent systems of interacting fermions and bosons [360, 361, 362, 363, 364]. Naively one
expects high-energy physics to be independent of collective phenomena related to the BEC,
which happen on longer time-scales ⇠/c. Indeed it was pointed out that the universal highfrequency power-law is directly linked to universal two-body physics [362, 363, 364] [P5], as
can be shown by relating the RF spectrum to the universal momentum distribution at large
momenta [332, 333].
10.5
Non-equilibrium polaron dynamics
In this section we shortly address non-equilibrium dynamics of polaron formation. To this
end we use the same variational wavefunction as required for calculating RF spectra, which
approximates the full time-evolution of the many-body system. Then we calculate the expectation values of experimentally relevant observables with respect to this state, going beyond
the RF spectrum.
MF (t) in an interaction
Specifically we consider the impurity momentum pI (t) = q Pph
quench as discussed for the inverse RF protocol. I.e. we start from a non-interacting impurity
MF (t = 0) = 0. Although the time-dependent variational
with momentum q and no phonons, Pph
approach works even for supersonic impurities, its reliability is unclear. Therefore we restrict
our analysis to subsonic impurities here, with q < qc .
Then at time t = 0 interactions are switched on and momentum is transferred from the
impurity to the surrounding phonons. In FIG.10.3 (a) we show how the impurity momentum
decreases over time, which corresponds to a slow-down of the impurity because its average
velocity is given by vI = pI /M . For not too large coupling we observe a slight drop in the
impurity velocity, until an asymptotic value is reached. In FIG.10.3 (b) the dependence of
this asymptotic impurity momentum on the coupling strength is shown. Compared to the
(a)
(b)
GS
NESS
Figure 10.3: (a) The impurity momentum pI is shown as a function of time, for an interaction
quench from a non-interacting to an interacting system at t = 0. For large coupling ↵
pronounced oscillations can be observed. The impurity momentum reaches an asymptotic
value, which is shown in (b) as a function of the coupling (solid, NESS – non-equilibrium
steady state). We compare this value of the impurity momentum to the groundstate value in
a system of total polaron momentum q = pI (t = 0) below the critical momentum (dashed, GS
– groundstate). These curves were calculated by Aditya Shashi.
10.5. NON-EQUILIBRIUM POLARON DYNAMICS
223
polaron ground state at the same total momentum q, the impurity retains more momentum.
This can be understood by noting that when out-of-equilibrium, the system is in an excited
state such that the impurity keeps more kinetic energy.
Interestingly, for very large couplings ↵ ⇡ 600 we observe pronounced oscillations of
the impurity momentum in FIG. 10.3 (a). Similar e↵ects were observed in a semi-classical
treatment of the polaron problem [331]. We interpret these oscillations as a indicator for a
long-lived excited state of the polaron, which is populated in the interaction quench. To judge
whether this picture is correct, further investigation is required. Moreover, at such large values
of the coupling constant, we can no longer trust MF theory. We will return to this question
in the last Chapter 13 where an all-coupling theory for polaron dynamics is introduced.
224
CHAPTER 10. RF SPECTRA OF FRÖHLICH POLARONS IN A BEC
Chapter 11
Weak-coupling theory of polaron
Bloch oscillations in optical lattices
11.1
Summary and Introduction
Now we turn our attention to an experimental setting where the impurity atom is confined to
the lowest Bloch band of an optical lattice, as shown in FIG. 11.1 (a). We assume that the
entire impurity lattice is immersed in a homogeneous BEC. Due to the dressing of the impurity
with Bogoliubov excitations of the BEC the ground state of this system can be understood
as a polaron. In this chapter we develop a weak-coupling theory describing Bose polarons
in a lattice. Then we extend our formalism to describe non-equilibrium polaron dynamics,
focusing in particular on the problem of Bloch oscillations (BO) of polarons when applying a
constant driving force to the impurity. We investigate in detail how polaron BO decohere (or
dephase) as a function of the driving strength.
We will also apply our theory beyond the weak-coupling regime. While it is not completely
clear how well justified this is, we will show in chapter 13 for continuum polaron models
that a time-dependent weak-coupling MF theory can capture the physics at large couplings
surprisingly well on a qualitative level.This serves as a motivation here to investigate also the
regime of moderate to large couplings.
It is well known that an isolated quantum mechanical particle in a lattice will undergo
(a)
(b)
3D BEC
n0
z
5
a
10
y
x
0.15
0.1
15
1D impurity
lattice
20
0
0.05
5
10
0
Figure 11.1: In this chapter we consider an impurity (blue) constrained to the lowest Bloch
band of a 1D optical lattice, immersed in a homogeneous 3D BEC (red), see (a). Strong
interactions with the Bose gas lead to polaron formation and a modified dispersion. Applying
a constant force to the impurity results in polaron Bloch oscillations, see (b). Although the
speed of sound c is never exceeded, Bloch oscillations are superimposed by a constant drift
velocity vd as well as di↵usion of the polaron wavepacket.
225
CHAPTER 11. WEAK-COUPLING THEORY OF POLARON BLOCH OSCILLATIONS
226
IN OPTICAL LATTICES
coherent BO when subject to a constant force. However when a polaron is subject to a constant
force it is less clear that coherent Bloch oscillations can be sustained because coupling to the
phonon bath provides a dephasing mechanism. Thus, if the phonons are integrated out, the
impurity can decohere. Here we establish that the Bose polaron can indeed undergo BO, see
FIG.11.1 (b). By calculating the renormalized shape of the dispersion relation of the polaron,
we show that phonon dressing has a pronounced e↵ect on BO, which can be observed in
experiments by measuring the real time dynamics of impurity atoms. Such experiments can
be done using recently developed quantum gas microscopes with single atom resolution [160,
161, 162]. Single-site resolution in the optical lattice is not a necessary requirement to observe
polaron BO however. In principle any mixture of ultra cold atoms in which polarons can be
realized, see Sec.9.2.2, could be used to probe the physics described in this chapter.
Polarons in optical lattices were considered earlier by Bruderer et al. [315, 313] using a
strong-coupling ansatz in the limit of the so-called “small” polaron where impurity hopping
is sub-dominant to phonon coupling. This corresponds to a large e↵ective impurity mass, and
as discussed in Secs.9.2.4, 9.2.5 in this regime both weak- and strong-coupling polaron theory
can be applied, cf. Eqs.(9.52), (9.71). The small polaron regime was further studied in [316],
and it is also the traditional approach to lattice polarons [365, 366] in solid state systems.
We use the weak-coupling MF theory introduced in Sec.9.2.4, which we modify to describe
impurities in a lattice. This approach has the advantage that it does not only cover the limit
of small polarons at large e↵ective mass, but also the regime of small e↵ective mass (large
hopping) and weak phonon coupling (corresponding to small ↵ . 1) can be captured. As our
discussion in Sec.9.2.2 has shown, this regime is most relevant for current experiments with
ultra cold atoms. As described in the last chapter 10.3, MF theory can easily be extended
to non-equilibrium situations. Similarly we develop a time-dependent MF theory for lattice
polarons and apply it to calculate the full non-equilibrium dynamics of polaron BO.
From our simulations we find that polaron formation takes place on a timescale ⇠/c set
by the BEC, and subsequently we observe pronounced BO, see FIG.11.1 (b). Additionally,
to gain insight into the nature of the coherent polaron dynamics, we introduce an analytic
“adiabatic approximation” which correctly predicts the predominant characteristics of the
polaron trajectory in the subsonic regime, e.g. overall shape, and frequency of oscillations.
As an added advantage of our approach, and in contrast to the strong coupling polaron
approximation, we can approach the supersonic regime, near which we find strong decoherence
of the BO in connection with a large drift velocity vd , i.e. a net polaron current. Such
incoherent transport can only be sustained in the presence of decoherence mechanisms, and
consistently we observe phonon emission in this regime.
Historically, the study of the interplay between coherent BO and inelastic scattering (e.g.
on phonons) was pioneered in the solid state context by Esaki and Tsu [367], who derived
a phenomenological relation between the driving force F and the net (incoherent) current
vd . In particular they proposed a generic Ohmic regime for weak driving, i.e. vd ⇠ F . The
precision of ultra-cold atom experiments allowed a detailed verification of the Esaki-Tsu model
in thermal gases [368], and thus triggered theoretical interest in this topic [369, 370]. While
all these works focused on non-condensed gases, Bruderer et al. [313] considered a 1D BEC
where the phonons provide an Ohmic bath (see e.g. [371]) and established a finite current (i.e.
vd 6= 0) even for subsonic impurities, with a current-force relation vd (F ) of a shape similar to
that predicted by Esaki and Tsu.
At the end of this chapter we address the question how polaron BO decohere, and in
particular how the polaron drift velocity depends on the driving force, for condensates in
arbitrary dimensions d = 1, 2, 3, .... In the weak driving regime, we find that the drift current
11.2. THE MODEL
227
strongly depends on dimensionality d and deviates from the Ohmic behavior predicted by
the phenomenological Esaki-Tsu relation. We show that a quantitative description of polaron
drift can be obtained by applying Fermi’s golden rule to calculate the phonon emission of
oscillating impurity atoms. This analysis correctly reproduces the current-force relation vd ⇠
F d observed in our numerics for weak driving.
The chapter is based on the publication [P8] and it is organized as follows. In Sec.11.2
we introduce the concrete model and employ a lattice version of the Lee-Low-Pines unitary
transformation to make use of the discrete translational invariance (by a lattice period) of our
problem. Then in Sec.11.3 we discuss the ground state of the impurity-Bose system in the
presence of a lattice, and calculate the renormalized polaron dispersion. We also present the
MF phase diagram which shows where the subsonic to supersonic transition takes place. In
Sec.11.4 we discuss polaron BO within the adiabatic approximation. We also show that direct
imaging of real-space impurity trajectories reveals the renormalized polaron dispersion. How
non-adiabatic corrections modify BO is studied in Sec.11.5 using a time-dependent variational
wavefunction. In Sec.11.6 we discuss incoherent polaron transport and present numerical as
well as analytical results for its dependence on the driving force.
11.2
The Model
In this section we present our theoretical model, starting from a microscopic Hamiltonian
in 11.2.1. We subsequently simplify the latter by applying Bogoliubov theory for the BEC
as well as the nearest-neighbor tight-binding approximation for the free impurity. Then we
discuss the corresponding impurity-boson interaction, which requires careful treatment of the
two-particle scattering problem in order to make connection with experiments. Having established the connection to microscopic properties, we discuss realistic numbers and introduce a
dimensionless polaron coupling constant. In the second part 11.2.2 we apply the lattice LeeLow-Pines transformation to our model, which is at the heart of our formalism and makes
conservation of the polaron quasimomentum explicit.
Here, as well as in the subsequent three sections, we will focus on the case of a three
dimensional BEC (d = 3), but an analogous analysis can be done for dimensions d = 1, 2. We
will discuss the di↵erence in dynamics of systems of di↵erent dimensionality in Section 11.6.
11.2.1
Derivation from microscopic model
Our starting point is the microscopic Hamiltonian (9.1) at zero temperature, where we treat
the BEC as homogeneous and neglect the confinement potential for bosons VB (r) ⌘ 0. The
impurity is confined to a deep species-selective optical lattice which is completely immersed
in the surrounding Bose gas. For concreteness we assume the lattice to be one-dimensional
(pointing along ex ), but our analysis can easily be carried over to arbitrary lattice dimensions.
To study transport properties of the dressed impurity, we will furthermore consider a constant
force F acting on the impurity. In experiments this force can e.g. be applied using a magnetic
field gradient [372, 373, 13]. The impurity is assumed to be confined to a 1D tube y 2 + z 2 .
( /2)2 and in the potential VI (r) a linear term F x describs the constant force acting on
the impurity. With k0 = 2⇡/ we denote the optical wave vector used to create the lattice
potential.
We will now show that the polaron Hamiltonian describing this system can be written in
CHAPTER 11. WEAK-COUPLING THEORY OF POLARON BLOCH OSCILLATIONS
228
IN OPTICAL LATTICES
the form
Z
n
⇣
X †
Ĥ = d3 k !k â†k âk +
ĉj ĉj eikx aj â†k + â
k
j
+ gIB n0
⌘
J
o
Vk +
X⇣
ĉ†j+1 ĉj + h.c.
j
⌘
F
X
ja ĉ†j ĉj . (11.1)
j
Here ĉ†j creates an impurity at lattice site j, Vk is the scattering amplitude in the lattice, J is
the impurity hopping and a the lattice constant. The second term in the first line of Eq.(11.1)
⇠ ĉ†j ĉj describes scattering of phonons on an impurity localized at site j (with amplitude Vk ).
This term thus breaks the conservation of total phonon momentum (and number), and we
stress that phonon momenta k can take arbitrary values 2 R3 , not restricted to the Brillouin
zone (BZ) defined by the impurity lattice 1 . Below we derive separately the terms in (11.1)
and discuss conditions under which they are valid.
Free Hamiltonians
We start by discussing the free Hamiltonians of our model, in the absence of impurity-phonon
interactions. We will assume the optical lattice to be sufficiently deep to employ nearestneighbor tight-binding approximation. Then the operator ĉ†j creates an impurity at site j,
ĉ†j
=
Z
d3 r wj (r) ˆ† (r).
(11.2)
The corresponding Wannier functions can be approximated by local oscillator wavefunctions,
wj (r) = ⇡`2ho
3/4
e
(r jaex )2 /(2`2ho )
,
(11.3)
p
p
where `ho = 1/ M !0 is the oscillator length in a micro trap of the lattice and !0 = 2 V0 Er
the corresponding micro trap frequency, determined by the recoil energy Er = k02 /2M [5].
This gives rise to an e↵ective hopping J between lattice sites, such that – after inclusion of
the uniform force – the free impurity Hamiltonian reads
⌘
X⇣ †
X
ĤI = J
ĉj+1 ĉj + h.c.
F
ja ĉ†j ĉj .
(11.4)
j
j
In the absence of the impurity, bosons condense and form a BEC. In the spirit of Refs. [315,
313, 24] we will assume that the impurity-boson interaction does not significantly alter the
many-body spectrum of the bath, allowing us to treat the bosons as an unperturbed homogeneous condensate within the Bogoliubov approximation [336]. As shown in Sec.9.2.1 this is
justified when
|gIB | ⌧ 4c⇠ 2
(11.5)
and the free boson Hamiltonian is given by
Z
ĤB = d3 k !k â†k âk ,
1
(11.6)
Only if also the bosons were subject to an optical lattice potential, the phonon momenta k appearing in
Eq.(11.1) would be restricted to the corresponding BZ. This is not the case for the model of a continuous BEC
considered in this work.
11.2. THE MODEL
229
where k R2 R3 is Rthe 3D phonon momentum (with k denoting its absolute value). In this
1
chapter d3 k =
1 dkx dky dkz denotes the integral over all momenta from the entire kspace.
Impurity-Boson interaction
In the discussion of the interaction Hamiltonian describing impurity-boson scattering, we
restrict ourselves to the tight binding-limit. This allows us to expand the impurity field in
P
terms of Wannier orbitals, ˆ(r) = j ĉj wj (r). Using this decomposition, Eq.(9.1) yields the
following expression for the impurity-boson Hamiltonian,
X † Z
ĤIB = gIB
ĉj ĉj d3 r |wj (r)|2 ˆ† (r) ˆ(r),
(11.7)
j
where we neglected phonon-induced hoppings (the validity of this approximation will be discussed further below). Rewriting this equation in terms of Bogoliubov phonons yields to
lowest order in phonon operators âk ,
Z
⇣
⌘
X †
ĤIB = gIB n0 + d3 k
ĉj ĉj eikx aj â†k + â k Vk ,
(11.8)
j
where the scattering-amplitude is given by
Vk = (2⇡)
3/2 p
n0 gIB
✓
(⇠k)2
2 + (⇠k)2
◆1/4
e
k2 `2ho /4
.
(11.9)
In contrast to the free-space result Eq.(9.6) this expression includes a smooth UV cut-o↵
around k = ⇤0 ⇡ 1/`ho .
An important question is how the interaction strength gIB in the simplified model (11.7)
above relates to the measurable impurity-boson scattering length aIB . While for unconfined
impurities this relation is usually derived from the Lippmann-Schwinger equation describing
two-particle scattering, see Sec.9.2.1, it is more involved for an impurity confined to a lattice.
In a lattice the scattering length ae↵
IB has to be distinguished from its free-space counter part,
and can even be substantially modified due to lattice e↵ects [374, 375]. Furthermore, also
e↵ of the interaction between a free boson and an impurity confined to
the e↵ective range rIB
a lattice can be modified by the lattice. We can take this e↵ect into account in our model
e↵ ) of the Wannier functions entering Eq.(11.7). In the
by choosing a proper extent `ho (rIB
e↵
follwing we will not calculate the numerical relation between ae↵
IB (or rIB ) in the lattice and its
free-space counterpart aIB . Instead we assume that these numbers are known – either from
numerical calculations [374, 375] or from an experiment [376] – and work with the e↵ective
e↵
model Eq.(11.7). In the Appendix J we discuss in detail how ae↵
IB and rIB relate to our model
parameters gIB and `ho , in the tight-binding case.
Before proceeding with our model, we briefly discuss under which conditions phononinduced tunneling events can be neglected. They originate form the impurity-boson interactions and are described by
⇣
⌘
X † Z
(i j)
ĤJ ph =
ĉi ĉj d3 k eikx aj â†k + â k Vk
+ h.c.,
(11.10)
i>j
CHAPTER 11. WEAK-COUPLING THEORY OF POLARON BLOCH OSCILLATIONS
230
IN OPTICAL LATTICES
where the scattering (or hopping) amplitudes in this equation read
R 3
d r w⇤ (r naex )eik·r w(r)
hwn |eik·r |w0 i
(n)
R
Vk = Vk
=:
V
k
hw0 |eik·r |w0 i
d3 r eik·r |w(r)|2
(11.11)
for integer n = ..., 1, 0, 1, ... . Phonon-induced tunneling modifies both the impurity-phonon
interactions and the impurity hopping. Thus, in order to neglected this contribution in the
Hamiltonian, we have to ensure that Eq.(11.10) only yields subdominant contributions to
both e↵ects. By comparing impurity-phonon interactions we obtain the condition
(n)
|Vk |
|hwn |eik·r |w0 i| !
=
⌧ 1.
|Vk |
|hw0 |eik·r |w0 i|
(11.12)
ik·r
It
R is3 typically fulfilled in the tight-binding limit, as can be seen by estimating |hwn |e |w0 i| 
d r |w(r naex )w(r)|. It is also automatically fulfilled when a ⌧ ⇠ and assuming the extend
of a Wannier orbit w(r) is of the order of a. In this case an expansion of eik·r yields a leadingorder contribution |hw1 |k · r|w0 i| ⇡ a/⇠ ⌧ 1. By comparing the phonon-induced nearest
neighbor hopping to the bare impurity hopping J we obtain the condition
Z
(1)
d3 k Vk Vk /!k ⌧ J,
(11.13)
where we estimated âk ⇡ ↵k ⇡
Vk /!k .
Coupling constant and relation to experiments
In this chapter we slightly modify the definition of the dimensionless coupling constant. The
disadvantage of the definition given in Eq.(9.18) is that there ↵ depends on the (free-space)
impurity-boson scattering length aIB . Since we will express all results in this chapter in terms
of gIB – the only coupling constant appearing in the Hamiltonian (11.1) (which has to be
determined from the in-lattice scattering length ae↵
IB ) – we define
2
ge↵
2
n0 gIB
=
=
⇠c2
✓
EIB
Eph
◆2
.
(11.14)
If we re-express aIB in terms of gIB using the free-space Lippmann-Schwinger equation, we
2 . Thus our coupling constant (11.14) is related to ↵ by
find ↵ = ⇡2 mred2 n0 gIB
↵=
1h
mB i
1+
⇡
M
2
2
ge↵
.
(11.15)
Because in this expression the impurity mass M enters as an additional parameter, which is
not required to calculate ge↵ , we prefer to use ge↵ instead of ↵ in this work.
For experimentally realized Bose-Bose [345, 342, 324] or Bose-Fermi mixtures [348, 346]
we find that background interaction strengths are of the order ge↵ ⇠ 1, but using Feshbach
resonances values as large as ge↵ = 15 may be within reach for sufficiently small mass ratios
M/mB . For standard Rubidium BECs characteristic parameters are ⇠ ⇡ 1µm, c ⇡ 1mm/s
and for Rubidium in optical lattices one typically has hoppings J . 1kHz [5].
11.2. THE MODEL
11.2.2
231
Time-dependent Lee-Low-Pines transformation in the lattice
To make further progress, we will now simplify the Hamiltonian (11.1). To this end we make
use of the Lee-Low-Pines transformation (see Sec. 9.2.3), making conservation of polaron
quasi momentum explicit, and include the e↵ect of the constant force F acting on the impurity.
To do so, we start by applying a time-dependent unitary transformation,
0
1
X †
ÛB (t) = exp @i!B t
jĉj ĉj A ,
(11.16)
j
where !B = aF denotes the BO frequency of the bare impurity. Next we apply the LLP
transformation which takes the form
Z
X
iŜ
ÛLLP = e , Ŝ = d3 k kx â†k âk
ajĉ†j ĉj
(11.17)
j
in the presence of a lattice, cf. Eq. (9.46). As described in Sec.9.2.3, the action of the Lee-LowPines transformation on an impurity can be understood by noting that it can be interpreted as
a displacement in quasimomentum space. Such a displacement q ! q + q (modulo reciprocal
lattice vectors 2⇡/a) is generated by the unitary transformation ei qX̂ , where the impurity
P
position operator is defined by X̂ = j ajĉ†j ĉj . Comparing this to Eq.(11.17) yields q =
R 3
d k kx â†k âk , which is the total phonon momentum operator. Thus we obtain
†
ÛLLP
ĉq ÛLLP = ĉq+ q .
(11.18)
For phonon operators, on the other hand, transformation (11.17) corresponds to translations
in real space by the impurity position X̂ and one can easily see that
†
ÛLLP
âk ÛLLP = eiX̂kx âk .
(11.19)
Now we want to apply the time-dependent LLP transformation
✓Z
◆ X
†
iŜ(t)
3
ÛtLLP (t) = ÛLLP ÛB (t) = e
, Ŝ(t) =
d k kx âk âk + F t
ajĉ†j ĉj
(11.20)
j
to the Hamiltonian Eq.(11.1). In the new basis the (time-dependent) Hamiltonian reads
†
Ĥ(t) = ÛtLLP
(t)ĤÛtLLP (t)
†
iÛtLLP
(t)@t ÛtLLP (t).
(11.21)
To simplify the result we first write the free impurity Hamiltonian in quasimomentum space,
X
ĤI = 2J
ĉ†q ĉq cos(aq),
(11.22)
q2BZ
where we introduce the quasimomentum basis in the usual way,
X
ĉq := (L/a) 1/2
eiqaj ĉj .
(11.23)
j
Here L denotes the total length of the impurity lattice and q = ⇡/a, ..., ⇡/a is the impurity
quasimomentum in the BZ. The transformation (11.16) thus allows us to assume periodic
CHAPTER 11. WEAK-COUPLING THEORY OF POLARON BLOCH OSCILLATIONS
232
IN OPTICAL LATTICES
boundary conditions for the Hamiltonian (11.21), despite the presence of a linear potential
F x.
P
Next we make use of the fact that only a single impurity is considered, i.e. q2BZ ĉ†q ĉq = 1
in the relevant subspace, allowing us to simplify
ĉ†j ĉj eikx X̂ = ĉ†j ĉj eikx aj .
(11.24)
Note that although the operator X̂ in Eq.(11.24) consists of a summation over all sites of the
lattice, in the case of a single impurity the prefactor ĉ†j ĉj selects the contribution from site j
only. We proceed by employing Eqs.(11.18) - (11.24) and arrive at the Hamiltonian
Ĥ(t) =
†
ÛLLP
H̃ÛLLP
=
X
q2BZ
ĉ†q ĉq
⇢Z
h
⇣
⌘i
d3 k !k â†k âk + Vk â†k + âk
✓
2J cos aq
!B t
a
Z
3 0
d k
kx0 â†k0 âk0
◆
+ gIB n0 . (11.25)
P
Let us stress again that this result is true only for a single impurity, i.e. when q2BZ ĉ†q ĉq = 1.
P
We find it convenient to make use of this identity and pull out q2BZ ĉ†q ĉq everywhere to
emphasize that the Hamiltonian factorizes into a part involving only impurity operators and
a part involving only phonon operators. Notably the Hamiltonian (11.25) is time-dependent
and non-linear in the phonon operators. From the equation we can moreover see that, in
the absence of a driving force F = 0 (corresponding to !B = 0), the total quasi momentum
qR in the BZ is a conserved quantity. We stress, however, that the total phonon-momentum
d3 k kâ†k âk of the system is not conserved.
Even in the presence of a non-vanishing force F 6= 0 the Hamiltonian is still block-diagonal
for all times,
X
Ĥ(t) =
ĉ†q ĉq Ĥq (t),
(11.26)
q2BZ
and quasimomentum evolves in time according to
q(t) = q
F t,
(11.27)
i.e. Ĥq (t) = Ĥq(t) (0). This relation has the following physical meaning: if we start with
an initial state that has a well defined quasimomentum q0 , then the quasimomentum of the
system remains a well defined quantity. The rate of change of the quasimomentum is given
by F , i.e. q(t) = q0 F t. Thus states that correspond to di↵erent initial momenta do not
mix in the time-evolution of the system.
11.3
Weak-coupling Theory of Lattice Polarons
Now we develop the weak-coupling theory of lattice polarons. To this end we consider the
equilibrium problem at F = 0 and calculate the polaron groundstate as a function of total
momentum q. Because we employed the LLP canonical transformation above, quasimomentum q is explicitly conserved in the Hamiltonian (11.25). This enables us to treat every sector
of fixed q separately for the characterization of the equilibrium state. We begin the section
by introducing the MF polaron wavefunction in 11.3.1, where we also minimize its variational
energy. This readily gives us the renormalized polaron dispersion, the properties of which we
discuss in 11.3.2. There we moreover present the MF polaron phase diagram.
11.3. WEAK-COUPLING THEORY OF LATTICE POLARONS
11.3.1
233
Mean-field polaron wavefunction
To describe the polaron ground state we make the variational ansatz of uncorrelated coherent
phonon states, see Sec.9.2.4,
Y
| MF
|↵kMF i.
(11.28)
q i=
k
Here
|↵kMF i
denotes coherent states with amplitude ↵kMF 2 C,
h
i
⇤
|↵kMF i = exp ↵kMF â†k
↵kMF âk |0i.
(11.29)
We note that the wavefunction (11.28) is asymptotically exact in the limit of a localized
impurity, i.e. when J ! 0. As discussed in Sec.9.2.4 it can also be applied in the regime of
small ge↵ , but in the case of a lattice it is not known exactly where the strong-coupling regime
begins.
To obtain self-consistency equations for the polaron ground state we minimize the MF
variational energy HMF ,
MF !
HMF (q) = h MF
(11.30)
q |Ĥq | q i = min.
As shown in Appendix K, the MF energy functional can be written as
Z
⇥
⇤
H [↵ ] = 2Je C[↵ ] cos (aq S[↵ ]) + d3 k !k |↵k |2 + Vk (↵k + ↵k⇤ ) ,
(11.31)
where we introduced the functionals
C[↵k ] =
S[↵k ] =
Z
Z
d3 k|↵k |2 (1
cos(akx )),
(11.32)
d3 k|↵k |2 sin(akx ).
(11.33)
Eq.(11.30) together with (11.31) then yields the MF self-consistency equations for the polaron
ground state,
MF
↵kMF = Vk /⌦k [↵
],
(11.34)
where we defined yet another functional
h
⌦k [↵ ] = !k + 2Je C[↵ ] cos (aq
S[↵ ])
cos (aq
akx
i
S[↵ ]) .
(11.35)
MF ] can be interpreted as the renormalized phonon dispersion at total
This frequency ⌦k [↵
quasimomentum q.
Importantly for numerical evaluation, Eq.(11.34) reduces to a set of only two self-consistency
equations for C MF = C[↵kMF ] and S MF = S[↵kMF ]: Plugging ↵kMF from (11.34) into the definitions (11.32), (11.33) readily yields
C
MF
S
MF
=
=
Z
Z
Vk
MF
⌦k (C , S MF )
2
d3 k
Vk
MF
⌦k (C , S MF )
2
d3 k
1
cos(akx ) ,
sin(akx ).
(11.36)
(11.37)
Moreover, from the analytic form of ⌦k Eq.(11.35) we find the following exact symmetries of
CHAPTER 11. WEAK-COUPLING THEORY OF POLARON BLOCH OSCILLATIONS
234
IN OPTICAL LATTICES
0.6
0.6
0.4
0.4
0.2
0.2
0
0
J=0.7
J=0.7c/a
J=0.6c/a
J=0.4c/a
0.2
0.4
0
0.2
0.2
0.4
0.6
0.8
1
0.4
Figure 11.2: The MF polaron ground state at total quasimomentum q is characterized by
C MF (q) (upper thin lines) and S MF (q) (lower thick lines). These quantities are plotted for
various hoppings J, all in the subsonic regime. When approaching the transition towards
supersonic polarons (which takes place slightly above J = 0.8c/a in this case) the phase shift
S MF (q) develops a strong dispersion around q = ⇡/a. Atpthe same point a pronounced local
minimum of C MF (q) develops. We used ⇠ = 5a, `ho = a/ 2 and ge↵ = 10.
the solution under spatial inversion q !
q,
C MF ( q) = C MF (q),
11.3.2
S MF ( q) =
S MF (q).
(11.38)
Results: equilibrium properties
In FIG.11.2 we show the solutions C MF and S MF of the self-consistency equations (11.36),
(11.37) as a function of total quasimomentum q for di↵erent hoppings. For weak interactions
and not too close to the subsonic to supersonic transition we find S MF (q) ⇡ 0 while C MF (q) ⇡
const. In this limit the MF polaron dispersion becomes
!p (q) ⇡ Eb
2J ⇤ cos(qa),
(11.39)
MF
cf.(11.31). Here J ⇤ = Je C
describes the renormalized hopping of the polaron, and we
obtain a similar exponential suppression as reported in [313]. Eb describes the binding energy
of the polaron.
In FIG.11.3(a) we show the full polaron dispersion relation. For substantial interactions
ge↵ = 10 chosen in FIG.11.3 we find a transition from a subsonic to a supersonic polaron
around Jc ⇡ 0.8c/a. For hoppings close to this transition point the renormalized dispersion
deviates markedly from the cosine shape familiar from bare impurities, and we observe strong
renormalization at the edge of the BZ, q = ±⇡/a. At the same time the overall energy is
shifted substantially as a consequence of the dressing with high-energy phonons.
In FIG.11.3(b) we show the MF phase diagram. To this end we calculated the critical
hopping Jc where the maximal polaron group velocity in the BZ exceeds 90% of the speed
of sound c. (We only went to 90% because close to the transition to supersonic polarons,
solving the MF equations for C[↵kMF ] and S[↵kMF ] becomes increasingly hard numerically.)
We observe that for large interactions the polaron is subsonic, even for bare hoppings J one
(0)
order of magnitude larger than the non-interacting critical hopping Jc = c/2a. This is in
direct analogy to the strong mass renormalization predicted for free polarons, see e.g. [24, 377]
[P5]. Interestingly we observe di↵erent behavior for weakly and strongly interacting polarons;
11.3. WEAK-COUPLING THEORY OF LATTICE POLARONS
(a)
235
(b)
⌥8.5
5
⌥9
4
supersonic
⌥9.5
3
⌥10
2
⌥10.5
subsonic
1
⌥11
0
0.25
0.5
0.75
0
1
0
100
200
300
400
Figure 11.3: (a) MF polaron dispersion HMF (q) for di↵erent impurity hoppings J, where the
BEC MF shift gIB n0 was neglected (it depends not only on the coupling strength ge↵ but
also on the BEC density n0 which we did not specify here). For larger J & 0.8c/a ⇡ Jc
the group velocity vg = @q HMF (q) exceeds the speed of sound c for some quasimomentum
q. The interaction strength was ge↵ = 10. (b) Critical hopping J where the maximal group
2 . For large
velocity maxq vg (q) is 90% of c, as a function of interaction strength squared ge↵
interactions ge↵
1 the hopping where the polaron becomes supersonic is much larger than
in the non-interacting case (dashed line). Errorbars are due to the finite mesh-size
used to
p
raster parameter space. In both figures we have chosen ⇠ = 5a and `ho = a/ 2.
We fitted the critical hopping to the curve
Jc (vg = 0.9c) =
0.9Jc(0)
+
2
ge↵
C1
1+
✓
ge↵
c
ge↵
◆4 !
,
(11.40)
c . In this way we obtain a cross-over at g c = 14.2 for the parameters
varying parameters C1 , ge↵
e↵
from FIG.11.3.
We also consider the the quasiparticle weight Z, which is another quantity characterizing
the polaron ground state. Z is defined as the overlap between the bare and the dressed
impurity state,
Z = |h0|
q i|
2
,
(11.41)
and can e.g. be measured using radio-frequency absorption spectoscopy of the impurity [P5]
[321]. Within the MF approximation (11.28) | q i = | MF
q i, Z is directly related to the
number of phonons in the polaron cloud,
✓ Z
◆
ZMF = exp
d3 k |↵kMF |2 = e hNph i .
(11.42)
Note, however, that this relation between the quasiparticle weight and the number of excited
phonons is specific to the MF wavefunction and originates from its Poissonian phonon number
statistics.
In FIG.11.4 the dependence of the MF quasiparticle weight on quasimomentum is shown.
For the relatively strong coupling we have chosen, Z ⌧ 1 and the corresponding number of
phonons is Nph = log(ZMF ), taking values between Nph = 5 and Nph = 9 in the particular
case of FIG.11.4. Importantly, we observe that the polaron properties are strongly quasimomentum dependent. Especially close to the subsonic to supersonic transition (i.e. for larger
CHAPTER 11. WEAK-COUPLING THEORY OF POLARON BLOCH OSCILLATIONS
236
IN OPTICAL LATTICES
−3
x 10
5
J =0.7 c/a
J =0.5 c/a
J =0.4 c/a
4
3
2
1
0
0
0.1
0.2
0.3
0.4
0.5
Figure 11.4: Dependence of the quasiparticle weight Z of the polaron on quasimomentum
q. We used the static MF polaron ground state to calculate Z = ZMF , which according to
hNph i .
Eq.(11.42) is related to the average
p number of phonons in the polaron cloud, ZMF = e
We have chosen ⇠ = 5a, `ho = a/ 2 and ge↵ = 10 as in FIGs.11.2 and 11.3 (a).
hopping J), we find an abrupt change of the quasiparticle weight close to the edge of the BZ.
This is related to the peak observed in the renormalized polaron dispersion in FIG.11.3 (a).
We interpret both these features as an onset of the subsonic to supersonic transition, which
takes place at the edges of the BZ for strong impurity-boson interactions like in FIG.11.3.
11.4
Polaron Bloch Oscillations and Adiabatic Approximation
In this section we discuss how a uniform force acting on the impurity a↵ects coherent polaron
wavepacket dynamics. To this end we derive the equations of motion of a time-dependent
variational state, and give an approximate solution using the adiabatic principle. From the
latter we calculate real-space impurity trajectories. We close the section by pointing out
how these trajectories can be used to measure the renormalized polaron dispersion in an
experiment. In the following section we will check the validity of the adiabatic approximation
by solving full non-equilibrium dynamics.
11.4.1
Time-dependent variational wavefunctions
We now treat the fully time-dependent Hamiltonian from Eq.(11.25), allowing us to solve for
polaron dynamics. Our logic is as follows: we decompose the wavefunction of the impurityBEC system into di↵erent quasimomentum sectors, and use the conservation of quasimomentum of the polaron, which we established in Sec. 11.2.2, to treat each quasimomentum sector
independently.
To this end, at time t = 0, we consider a general initial wavefunction jin of the impurity2
when the force is switched o↵, and for simplicity we assume complete absence of phonons.
Thus, the initial quantum state reads
X
in †
| (0)i =
(11.43)
j ĉj |0ic ⌦ |0ia ,
j
where |0ic and |0ia denote the impurity and phonon vacuum respectively. Note that Eq.(11.43)
2
To be precise, jin denotes the projection
of the initial impurity wavefunction
P
basis function wj (r), i.e. Iin (r) = j jin wj (r)
in
I (r)
onto the j th Wannier
11.4. POLARON BLOCH OSCILLATIONS AND ADIABATIC APPROXIMATION
237
is true not only in the lab frame, but also in the polaron frame, i.e. after applying the Lee-LowPines transformation (11.16); Because in the absence of phonons we have â†k âk | (0)i = 0, for
the initial state from Eq.(11.43) it holds Ŝ| (0)i = | (0)i, with Ŝ defined in Eq.(11.17).
The initial state (11.43) considered in most of the remaining part of this chapter can be
realized experimentally by di↵erent means. For instance, if Feshbach resonances are used to
realize strong impurity-boson interactions one can quickly change the magnetic field strength
from a value far away from the resonance to a value very close to it at time t = 0. Therefore an
initially non-interacting impurity, immersed in a cold BEC, suddenly starts to interact strongly
with the surrounding phonons as the magnetic field approaches the Feshbach resonance.
Alternatively, if a di↵erent internal (e.g. hyperfine) state of the majority bosons is used
as an impurity like e.g. in [283], the initial state can be prepared by applying a microwave or
optical Raman pulse, which is possible also in combination with local addressing techniques
[288, 283]. In this case, however, the preparation of a phonon vacuum state like in Eq.(11.43)
is hard to achieve since a spin-flip always comes along with a local excitation of the BEC.
Nevertheless, the true initial state for this situation can be calculated exactly if after a local
spin-flip the impurity is tightly confined by an addressing beam [288, 283] until the dynamic
evolution is started at time t = 0. In fact, a sufficiently tight local confinement of the impurity
corresponds to vanishing hopping J = 0, and in this case the MF ansatz Eq.(11.28) yields the
(J=0)
exact phonon ground state with coherent state amplitudes ↵k
. Therefore, assuming the
system has enough time to relax to its ground state after preparation of the tightly confined
impurity on the central site j = 0, the initial state reads
Y (J=0)
| (0)i = ĉ†0 |0ic ⌦
|↵k
i.
(11.44)
k
Like the state from Eq.(11.43) this wavefunction is invariant under the Lee-Low-Pines transformation (11.16), but in this case because of a trivial action of the impurity position operator,
X̂| (0)i = 0.
Next, focusing on Eq.(11.43) again for concreteness, we decompose the initial state into
its di↵erent quasimomentum sectors, which is achieved by taking a Fourier-transform of the
impurity wavefunction,
1 X iqaj in
fq = p
e
(11.45)
j .
L/a j
When the force is switched on at time t = 0, all quasimomentum sectors evolve individually
without any couplings between them. As a consequence the amplitudes fq defined above are
conserved, and we may write the time-evolved quantum state in the polaron frame as
X
| (t)i =
fq ĉ†q |0ic ⌦ | q (t)i.
(11.46)
q2BZ
At given initial quasimomentum q(0) = q and for finite driving force F we can make a
variational ansatz for the phonon wavefunction similar to the MF case Eq.(11.28), but with
time-dependent parameters,
Y
| q (t)i = e i q (t)
|↵k (t)i.
(11.47)
k
This is completely analogous to the ansatz made in Sec.10.3 for Fröhlich polarons. To derive equations of motion for ↵k (t) we use Dirac’s time dependent variational principle (see
CHAPTER 11. WEAK-COUPLING THEORY OF POLARON BLOCH OSCILLATIONS
238
IN OPTICAL LATTICES
Sec.10.3.1) and arrive at
[email protected] ↵k (t) = ⌦k [↵ (t)] ↵k (t) + Vk .
(11.48)
Here ⌦k [↵ (t)] is the renormalized phonon dispersion, see Eq.(11.35), but evaluated for timedependent ↵ (t). Note that ⌦k explicitly depends on q(t) = q F t. Following the prescription
from Sec.10.3.1 we also derive an equation describing the dynamics of the global phases q (t),
Z
Y
i
@t q =
d3 k (↵˙ k⇤ ↵k ↵˙ k ↵k⇤ ) +
h↵k |Ĥq(t) |↵k i.
(11.49)
2
k
11.4.2
Adiabatic approximation
Before presenting the full numerical solutions of Eqs.(11.48), (11.49), we first discuss the
adiabatic approximation. It assumes that the polaron follows its ground state without creating
additional excitations, i.e. without emission of phonons. We may thus approximate the
dynamical phonon wavefunction by
|
q (t)i
⇡e
i
q (t)
|
MF
q(t) i.
(11.50)
The intuition here is that the time-scale for polaron formation is much faster than the dynamics of BO. In particular, | MF
q(t) i is simply the equilibrium polaron MF solution for quasimomentum q(t) obtained in Sec. 11.3.1, which changes in time according to
q(t) = q(0)
F t.
(11.51)
Additionaly, we allow for a time-dependence of the global phase, which we obtain from
Eq.(11.49),
Z t
(t)
=
dt0 HMF (q(t0 )).
(11.52)
q
0
11.4.3
Polaron trajectory
Next, we derive the real-space trajectory of the polaron. To this end we calculate the impurity
density, which can be expressed as
D
ĉ†j ĉj
E
=
1
L/a
X
eia(q2
q1 )j
Aq2 ,q1 (t)fq⇤2 fq1 .
(11.53)
q2 ,q1 2BZ
This formula is derived in Appendix L, and it requires knowledge of the time-dependent
overlaps
Aq2 ,q1 (t) = h q2 (t)| q1 (t)i.
(11.54)
They consist of two factors, Aq2 ,q1 = Aq2 ,q1 Dq2 ,q1 ; The phase factors obey |Aq2 ,q1 | = 1 and are
given by
Aq2 ,q1 (t) = exp [i ( q2 (t)
(11.55)
q1 (t))] ,
whereas the amplitudes Dq2 ,q1 , determined by phonon dressing, are
Dq2 ,q1 =
Y
k
h↵k (q2 , t)| ↵k (q1 , t)i .
(11.56)
Within the adiabatic approximation we set ↵k (q, t) = ↵kMF (q(t)). For non-interacting impurities the phases alone give rise to BO, while the amplitude is trivial, D = 1. When interactions
11.5. NON-ADIABATIC CORRECTIONS
239
of the impurity with the phonon bath are included, |D| < 1 and interference is suppressed.
To get an insight into the BO of polarons we begin by discussing a special case of a polaron
wavepacket prepared with narrow distribution in quasimomentum space. In particular we will
consider an initial ground state polaron wavepacket centered around q = 0, which is described
by
s
2LI X q2 L2 †
I ĉ |0i ⌦ | MF i,
| (0)i = p
e
(11.57)
c
q
q
2⇡ q2BZ
and where LI denotes its width in real space. We will assume LI
a in the analysis below,
such that all wavepackets carry a well-defined quasimomentum. Therefore, in Eq. (11.53)
only neighboring momenta |q2 q1 | ⌧ 2⇡/a contribute, allowing us to expand the exponent
of Aq2 ,q1 to second order in |q2 q1 |. In this way we obtain the adiabatic impurity density
(the detailed calculation can be found in Appendix M)
n(x, t) = e
(x X(t))2
2 L2 + 2 (t)
I
(
) ⇥2⇡ L2 +
I
2
(t)
⇤
1/2
.
(11.58)
Note that due to the large spatial extent assumed for the polaron wavepacket we treated
aj = x as a continuous variable here.
The center-of-mass coordinate of the polaron is determined by Aq2 ,q1 and it reads
X(t) = X(0) + [HMF (F t)
HMF (0)] /F.
(11.59)
The amplitude Dq2 ,q1 , meanwhile, leads to reversible broadening of the polaron wavepacket,
Z
⇣
⌘2
2
(t) = d3 k @q ↵kMF q= F t .
(11.60)
From Eq.(11.59) we thus conclude that a measurement of the polaron center X(t) directly
reveals the renormalized polaron dispersion relation. Although derived from a simplified
theory, we expect that this result holds more generally beyond MF approximation of the
polaron ground state.
11.5
Non-Adiabatic Corrections
In this section we study the full non-equilibrium dynamics of the driven polaron by numerically solving for the time-dependent MF wavefunction Eq.(11.47). We start from the phonon
vacuum and some initial impurity wavefunction jin , see Eq.(11.43), mostly chosen to be a
Gaussian wavepacket with a width LI of several lattice sites and vanishing mean quasimomentum q = 0. After switching on the impurity-boson interactions at time t = 0, we find
polaron formation and discuss the validity of the adiabatic approximation for a description
of the subsequent dynamics (in 11.5.1). We also briefly discuss the case of initially localized
impurities (in 11.5.2).
To solve the equations of motion (11.48) we employ spherical coordinates k, #, and make
use of azimuthal symmetry around the direction of the impurity lattice. We introduce a grid
in k # space (typically 170 ⇥ 40 grid points) and use a standard matlab solver for ordinary
di↵erential equations. From the so-obtained solutions ↵k,# (t) and q (t) we calculate Aq2 ,q1 (t)
using Eqs.(11.55), (11.56), giving access to impurity densities for arbitrary impurity initial
conditions, see Eq.(11.53).
CHAPTER 11. WEAK-COUPLING THEORY OF POLARON BLOCH OSCILLATIONS
240
IN OPTICAL LATTICES
0
5
0.15
10
15
0.1
20
25
0.05
30
35
0
10
20
30
40
50
60
0
Figure 11.5: Impurity density hĉ†j ĉj i (gray scale) with ja = x for a heavily dressed impurity.
The polaron dynamics, starting from phonon vacuum, is compared to the result from the
adiabatic approximation (red, dashed) as well as the trajectory of a non-interacting impurity
2
wavepacket
p (blue, dashed-dotted). The parameters are J = 1.7c/a, F = 0.1c/a , ge↵ = 17.32,
`ho = a/ 2 and ⇠ = 5a.
11.5.1
Impurity dynamics beyond the adiabatic approximation
To extend our analysis beyond the assumption that the system follows its ground state adiabatically, we now consider the full dynamical equations (11.48) and (11.49). We assume that
the system starts in the initial state (11.43) with the phonons in their vacuum state, and at
time t = 0 interactions between the impurity and the bosons are switched on abruptly. We
chose the initial impurity wavefunction jin to be a Gaussian wavepacket (standard deviation
LI ) like in the discussion of the adiabatic approximation, see Sec. 11.4.3. Thus the amplitudes
2
fq read fq = e (qLI ) (2LI )1/2 (2⇡) 1/4 , as in Eq.(11.45). The global phases vanish initially, i.e.
we set q (0) = 0 for all quasimomenta q.
In FIG.11.5 the evolution of the impurity density is shown for a strongly interacting case.
(0)
Although the impurity hopping J = 1.7c/a exceeds the critical hopping Jc = 0.5c/a where
a non-interacting particle becomes supersonic by more than a factor of three, we observe
well defined BO with group velocities of the wavepacket below the speed of sound c. By
investigating the mean phonon number we moreover find that polaron formation takes place
on a time-scale ⇠/c after which a quasi steady state is reached.
Along with the plot in FIG.11.5 we show the result of the adiabatic approximation. Although the latter can not capture the initial polaron formation, it is expected to be applicable
once a steady state is reached 3 . In the case shown in the figure, however, non-adiabatic
corrections play an important role and we observe a pronounced polaron drift in the direction
of the force F . Moreover irreversible broadening of the polaron wavepacket takes place. Nevertheless the shape of the BO trajectory, including its pronounced peaks and the amplitude
of oscillations, can be understood from the adiabatic result. For smaller hopping and smaller
interactions the adiabatic approximation compares even better with the full numerics, as is
shown in FIG.11.6 (left).
To perform a more quantitative analysis when adiabaticity may be assumed, we determine
P
the center-of-mass X(t) = j jhĉ†j ĉj i of the impurity wavefunction from the full variational
calculation and fit it to
vgfit
X(t) =
cos (⌦t + ') + vd t + X0 .
(11.61)
⌦
3
After the quench there is excess energy which will however be carried away by phonons. When tracing out
these emitted phonons, we expect the remaining state to be well described by a ground state polaron, provided
that equilibration mechanisms are available.
11.5. NON-ADIABATIC CORRECTIONS
241
1.2
0
0.15
10
1
0.8
0.1
20
0.05
30
0.6
0.4
0.2
adiab. approx.
bare impurity
40
0
10
20
30
0
0
0
0.5
1
1.5
Figure 11.6: Left: Impurity density hĉ†j ĉj i (gray scale) with ja = x for a weakly driven
polaron. For comparison the trajectory of a non-interacting impurity wavepacket is shown
(dashed-dotted). The polaron dynamics is well described by the adiabatic approximation
(dashed), which is given by the polaron dispersion relation,
see Eq.(11.59). The parameters
p
are J = 0.4c/a, F = 0.06c/a2 , ge↵ = 10, `ho = a/ 2 and ⇠ = 5a. Right: The impurity
center-of-mass X(t) obtained from our time-dependent variational calculation can be fitted
to the expression from Eq.(11.61). The dependencies of the fitting parameters vd , vgfit as well
as ⌦ on the hopping strength J are shown in this figure. In the subsonic regime (J . 1.2c/a)
the fitted maximum group velocity vgfit (red bullets) is compared to the result obtained from
the adiabaticpapproximation (solid line). The parameters were F = 0.2c/a2 , ge↵ = 10, ⇠ = 5a
and `ho = a/ 2.
Here vgfit denotes the maximum polaron velocity in the absence of a drift. In FIG.11.6 (right)
the resulting fit parameters are shown as a function of the bare hopping J. We compare the
value of vgfit to the polaron group velocity expected from adiabatic approximation vgfit |adiab. .
The latter is obtained by fitting Eq.(11.61) to the adiabatic trajectory. While the adiabatic
theory captures correctly the qualitative behavior, on a quantitative level it overestimates the
group velocity. This, however, is related to our initial conditions and not to a shortcoming of
the adiabatic approximation in general. When starting the dynamics from the MF polaron
state Eq.(11.57) instead of considering an interaction quench of the impurity, we find excellent
agreement, with deviations below 1%. This is demonstrated by a few data points in FIG.11.6
(right). The quench, on the other hand, leads to the creation of phonons, which are also
expected to contribute to the dressing of the impurity in general [315].
Close to the subsonic to supersonic transition around Jc ⇡ c/a, the polaron drift velocity
takes substantial values of ⇡ 0.2c. We also note that, in the entire subsonic regime, the
fitted BO frequency ⌦ is precisely given by the bare-impurity value !B (to within < 0.5%
in the numerics). However, once the polaron becomes supersonic we observe a decrease of
the frequency to ⌦ < !B . We attribute this e↵ect to the spontaneous emission of phonons
in regions of the BZ where the polaron becomes supersonic. Along with phonon emission
comes emission of net phonon momentum qph , which has to be replenished by the external
driving force, qph = F t. Thus an extra time t is required for each Bloch cycle and as a
consequence we expect the BO frequency of the polaron to decrease.
Within the adiabatic approximation we have shown that the wavepacket trajectory X(t)
allows a direct measurement of the renormalized polaron dispersion. We found that even
when non-adiabatic e↵ects are appreciable, the polaron dispersion can be reconstructed. BO
can therefore be used as a tool to measure polaronic properties, which are of special interest in
the strongly interacting regime. We emphasize that our scheme does not rely on the specific
variational method used above. As long as the ground state of the impurity interacting with
CHAPTER 11. WEAK-COUPLING THEORY OF POLARON BLOCH OSCILLATIONS
242
IN OPTICAL LATTICES
(a)
1
0
5
(b)
0.8
10
15
1
0.8
15
0.6
20
0.6
20
0.4
25
30
0.2
35
40
0
10
20
40
60
80
0
0.4
25
30
0
0.2
5
10
15
20
25
0
Figure 11.7: Impurity density hĉ†j ĉj i (gray scale) with ja = x for an initially localized state
on a single lattice site (x(0) = 20a in this concrete example). (a) Weak driving F = 0.06c/a2
2
and (b) stronger
p driving F = 0.2c/a . Other parameters are J = 0.3c/a, ge↵ = 10, ⇠ = 5a
and `ho = a/ 2 in both cases.
the phonons of the surrounding BEC, is described by a stable polaron band, the real-space
BO trajectory maps out the integrated group velocity, i.e. the band structure itself.
11.5.2
Beyond wavepacket dynamics
Motivated by their possible application for measurements of the renormalized dispersion, we
focused on polaron wavepackets so far. Our variational treatment, however, is applicable to
any initial wavefunction. In FIG.11.7 we show two examples starting from an impurity which is
localized on a single lattice site, still assuming phonon vacuum initially. Since all momenta are
occupied, we first observe interference patterns which are symmetric under spatial inversion
x ! x. For large enough interactions and sufficiently strong driving however, we observe
di↵usion of the polaron and the interference patterns disappear. The maximum impurity
density drops substantially and the symmetry under spatial inversion is lost. Moreover we
observe a finite drift velocity of the polaron.
11.6
Polaron Transport
In this section we discuss the polaron drift velocity vd , which is the most important nonadiabatic e↵ect and can also be interpreted as a manifestation of incoherent transport. After
some brief general remarks about the problem, we present our numerical results for the currentforce relation vd (F ). These are obtained, like in the last section, from the time-dependent
variational MF ansatz Eq.(11.47), requiring numerical solutions of Eqs.(11.48), (11.49). Next
we derive a closed, semi-analytical expression for the current-force relation vd (F ) from first
MF
principles in the limit of small polaron hopping J ⇤ = Je C
and show that our predictions
are in good quantitative agreement with the full time-dependent MF numerics. As a result, we
find that the polaron drift in the weak-driving limit strongly depends on the dimensionality
of the system. At the end of this section we discuss the connection between our results and
the Esaki-Tsu relation, which originates from a purely phenomenological model of incoherent
transport in a lattice potential. We find that, in the polaron case, this simplified model is
unable to capture many key features of our findings. In particular it completely fails in the
weak-driving regime and predicts a wrong dependence on the hopping strength J.
11.6. POLARON TRANSPORT
243
(a)
(b)
0.018
10
2
10
3
10
4
10
5
10
6
0.016
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
0.5
1
1.5
2
2.5
3
3.5
4
10
1
0
10
1
10
Figure 11.8: (a) Dependence of the polaron drift velocity vd (obtained from the fit Eq.(11.61))
on the driving force F , for interaction strength ge↵ = 3.16 and various hoppings (top: J =
0.5c/a, middle: J = 0.3c/a, bottom: J = 0.1c/a). For each J we also show the result from our
analytical model Eq.(11.65) of polaron transport (solid lines), free of any fitting parameters.
We find excellent agreement for J = 0.3c/a and J = 0.1c/a. We also plotted the prediction of
an extended model (dashed lines, solution of the truncated Hamiltonian Eq.(11.64)), which
for J = 0.5c/a yields somewhat better results. In (b) we show the same data (legend from
(a) applies), but in double-logarithmic scale. In the lower left corner we indicated an Ohmic
p
power-law dependence ⇠ F (thin solid line). For all curves we used ⇠ = 5a and `ho = a/ 2
and simulated at least three periods of BO assuming an initial Gaussian impurity wavepacket
with a width LI of three lattice sites, assuming a three-dimersional BEC.
11.6.1
General observations
The fundamental Hamiltonian (11.1) is manifestly time-independent, and thus total energy
is conserved. When the impurity slides down the optical lattice, the loss of potential energy
Ėpot = F vd requires a gain of radiative energy E in the form of phonons, Ė = F vd . (This
relation can also formally be derived from Eq.(11.21).) Therefore the non-zero drift velocity
of the polaron wavepacket observed e.g. in FIG.11.5 comes along with phonon emission, albeit
its velocity never exceeds the speed of sound c. Such phonon emission is not in contradiction
to Landau’s criterion for superfluidity, which is appropriate only for impurities (or obstacles
in general) in a superfluid moving with a constant velocity. However, the system considered
here is driven by an external force F which gives rise to periodic oscillations of the net
quasimomentum of the system q(t). We thus expect phonons to be emitted at multiples of
P
the BO frequency ! = n!B , with rates ph (n!B ). Using Ė =
n n!B ph (n!B ), we can
express the drift velocity as
X
vd = a
n ph (n!B ).
(11.62)
n
11.6.2
Numerical results
In FIG.11.8 we present numerical results for the current-force dependence at di↵erent hopping
strengths J, in linear (a) and double-logarithmic scale (b). These curves were obtained by
solving for the variational time-dependent MF wavefunction (11.47). Like in the last section,
we started from phonon vacuum and assumed a zero-quasimomentum impurity wavepacket
extending over a few lattice sites. The center-of-mass X(t) of the resulting polaron trajectory
was then fitted to Eq.(11.61) from which vd was obtained as a fitting parameter.
All curves have a similar qualitative form: For small force !B . c/⇠ the polaron current
CHAPTER 11. WEAK-COUPLING THEORY OF POLARON BLOCH OSCILLATIONS
244
IN OPTICAL LATTICES
increases monotonically with F . Somewhere around !B ⇡ c/⇠ the curvature changes and the
polaron drift velocity takes its maximum value vdmax for a force FNDC . For even larger driving
!B we find negative di↵erential conductance, defined by the condition dvd /dF < 0. The
maximum is also referred to as negative di↵erential conductance peak. Previously all these
features have been predicted by di↵erent polaron models for impurities in 1D condensates
[313, 327].
From the double-logarithmic plot in FIG.11.8 (b) we observe a sub-Ohmic behavior in the
weak driving regime. For the smallest achievable forces F , we can approximate our curves
by power-laws vd ⇠ F . The observed exponents in FIG.11.8 (b) are in a range = 3.0 (for
J = 0.3c/a, ge↵ = 3.16) to = 1.5 (for J = 0.5c/a, ge↵ = 3.16). While this behavior is clearly
sub-Ohmic, it is hard to estimate how well these power-laws extrapolate to the limit F ! 0.
Going to even smaller driving is costly numerically, because the required total simulation time
for a few Bloch cycles T ⇠ 1/F becomes large.
To our knowledge, the sub-Ohmic behavior in the weak-driving regime was not previously
observed. As we discuss at the end of this section, it goes beyond the phenomenological
Esaki-Tsu model for incoherent transport in lattice models. We show in the following that
it is moreover tightly linked to the dimensionality d of the condensate providing phonon
excitations. For 1D systems, which were studied in some depth in the literature [313, 327, 328],
we do in fact expect Ohmic behavior for F ! 0. This is in agreement with the results of
[313, 327, 328].
11.6.3
Semi-analytical current-force relation
Now we want to extend our formalism used to describe the static polaron ground state in
Sec.11.3.1 by including quantum fluctuations. To this end we apply the following unitary
transformation
⇣
⌘
Y
⇤
Û (q) =
exp ↵kMF (q)â†k
↵kMF (q) âk ,
(11.63)
k
where in the new frame âk describes quantum fluctuations around the MF solution in the
absence of driving, F = 0. In the case of a non-vanishing force F 6= 0, we can analogously
obtain corrections to the adiabatic MF polaron solution Eq.(11.47). To this end we have to
make the transformation (11.63) time-dependent, Û (t) := Û (q(t)), where q(t) = q(0) F t
(see Eq.(11.51)).
By applying Û (q(t)), defined by Eq.(11.63) above, to the polaron Hamiltonian Eq.(11.21)
we obtain the following time-dependent Hamiltonian describing quantum fluctuations around
the adiabatic MF polaron solution in the case of a d-dimensional condensate,
Z
Z
h
i
H̃(t) = dd k ⌦k (q(t))â†k âk + iF dd k @q ↵kMF (q(t)) â†k âk + O(J ⇤ â2 ).
(11.64)
Here we introduced J ⇤ (q(t)) := J exp C MF (q(t)) and O(J ⇤ â2 ) denotes terms describing
corrections to the adiabatic solution beyond the MF description of the polaron ground state.
The leading order terms have a form ⇠ J ⇤ âk âk0 and can be treated following ideas by Kagan
and Prokof’ev [378]. In the rest of this chapter, however, we will discard such terms and
assume that the MF polaron state provides a valid starting point to calculate corrections
to the adiabatic approximation. Note that the time-dependent ansatz (11.47) used for our
calculations of non-equilibrium dynamics includes corrections due to the additional terms of
order O(J ⇤ â2 ). We also mention that from Eq.(11.64) it becomes apparent why, in the absence
of driving, ⌦k describes the renormalized phonon dispersion in the polaron frame.
11.6. POLARON TRANSPORT
245
Results: analytical current-force relation
In the following we will employ Fermi’s golden rule to calculate non-adiabatic corrections,
corresponding to phonon excitations due to the terms in the second term of Eq.(11.64). To
leading order in J ⇤ we will derive (in 11.6.3) the following expression for the current-force
relation,
J ⇤2 k d 1 Vk2
vd (F ) = Sd 2 8⇡ 0 2
(11.65)
(1 sinc(ak)) + O(J0⇤ )3 ,
aF (@k !k )
where k is determined by the condition that !k = !B . Here J0⇤ := limJ!0 J ⇤ (q) is the
renormalized polaron hopping in the heavy impurity limit (which is independent of q), and
Sn = (n + 1)⇡ (n+1)/2 / (n/2 + 3/2) denotes the surface area of an n-dimensional unit sphere.
sinc(x) is a shorthand notation for the function sin(x)/x.
Importantly, our model yields the closed expression (11.65) for the current-force relation,
at least for heavy polarons. Although this limit has been considered before [313], we are not
aware of any such expression describing incoherent polaron transport and derived from first
principles. Our result is semi-analytic, in the sense that the prefactor J0⇤ has to be calculated
numerically from an integral, see Eq.(11.73) below.
In FIG.11.8 we compare our numerical results to the semi-analytical expression (11.65).
We obtain excellent agreement for both cases of small and intermediate hopping J = 0.1c/a
and J = 0.3c/a. For large J = 0.5c/a very close to the subsonic to supersonic transition,
larger deviations are found in the weak-driving limit aF . c/⇠, which in view of the fact
that our result Eq.(11.65) is perturbative in the hopping strength J ⇤ , does not surprise us.
Interestingly, for large force aF & c/⇠, our semi-analytical theory yields good agreement for
all hopping strengths. We will further elaborate on the conditions under which our model
works in Appendix N.
From Eq.(11.65) we can furthermore obtain a number of algebraic properties of the polaron’s current-force relation. To begin with, let us discuss the dependence of the drift velocity
2 ) we obtain
on system parameters. Because Vk ⇠ ge↵ and J0⇤ = J + O(ge↵
2
4
vd ⇠ ge↵
+ O(ge↵
).
(11.66)
Moreover, the leading order contribution in the hopping strength scales like
vd ⇠ J 2 + O(J 3 ).
(11.67)
In FIG.11.9 we investigate the position of the negative di↵erential conductance peak, obtained
from the full time-dependent variational simulations of the system. For small hopping J and
weak interactions ge↵ we identify power-laws whose exponents agree well with our expectations
(11.66), (11.67) derived above.
Next, we investigate the behavior in the weak-driving regime. A series expansion of
Eq.(11.65) around F = 0 yields
2
vd = F d ge↵
(J ⇤ )2 ⇠ 2
a3+d Sd 2
p
+ O(F d+1 , J0⇤3 ).
cd+1 6 2⇡ 2
(11.68)
This result describes correctly the strong sub-Ohmic behavior we found in 11.6.2, and furthermore shows that the latter strongly depends on the dimensionality d of the condensate. In
particular, for d = 1, we arrive at Ohmic behavior as found in [313, 327, 328]. The numerical
results for J  0.3c/a in FIG.11.8 are also consistent with the power-law vd ⇠ F 3 predicted
in Eq. (11.68). Note however that for larger J a comparison of the exponents is difficult
CHAPTER 11. WEAK-COUPLING THEORY OF POLARON BLOCH OSCILLATIONS
246
IN OPTICAL LATTICES
(a)
(b)
0.8
10
2
10
3
0.6
0.4
0.2
0
0
0.2
0.4
0.6
(c)
0.1
0.2
0.3
0.4
(d)
0.8
0.6
10
2
10
3
0.4
0.2
0
0
1
10
2
10
10
1
10
100
Figure 11.9: Dependence of the negative di↵erential conductance peak position, characterized
by FNDC and vdmax , on the system parameters; In (a) and (b) the hopping J is varied while the
coupling ge↵ = 3.16 is fixed. In (c) and (d), in contrast, the interaction strength ge↵ is varied
while keeping J = 0.4c/a fixed. In (b) and (d) a double-logarithmic scale is used, allowing
us to read o↵ the indicated power-law dependencies from best fits to the data (dashed lines),
2 (for small g ). The position F
vdmax ⇠ J 2 and vdmax ⇠ ge↵
NDC , in contrast, is only weakly
e↵
J-dependent (a) and we can not observe any clear interaction dependence in (c). The dashed
horizontal line in (c) shows the mean of our data. The indicated error bars in (a), (c) are due
to the finite mesh used for sampling the underlying current-force relations.
because, even for the smallest numerically achievable driving F , some residual curvature is
left and, more importantly, higher orders in J ⇤ can not simply be neglected.
For large driving, on the other hand, we arrive at the following asymptotic behavior in
the continuum limit `ho = 0 of the impurity lattice,
vd =
2d/4 1
Sd
⇡2
2
⇣ a ⌘d/2
c
2
⇠1
d/2 2
ge↵ (J0⇤ )2 F
3+d/2
+ O(F
4+d/2
, J0⇤3 ).
(11.69)
We can not compare our results in FIG.11.8 to this power-law, because non-vanishing `ho 6= 0
was considered in FIG.11.8. Interestingly from a theoretical perspective, as a consequence
of Eq.(11.69), in d
6 dimensions we expect the negative di↵erential conductance peak to
disappear. For non-vanishing `ho it reappears, but its position may be located at very large
F . This e↵ect, however, is simply connected to the absence of interacting phonons at the
Bloch frequency. Therefore in more than six spatial dimensions coherent Bloch oscillations
can never overcome incoherent scattering, in contrast to what we find in lower-dimensional
systems.
In the following (11.6.3) we will derive Eq.(11.65). We discuss its range of validity as well
as possible extensions in Appendix N.
11.6. POLARON TRANSPORT
247
Derivation of the current-force relation
To derive Eq.(11.65), we start by noting that the driving term in Eq.(11.64), i.e. F @q ↵kMF (q(t)) ,
is TB = 2⇡/!B periodic in time. We can thus expand it in a discrete Fourier-series,
@q ↵kMF (q(t))
1
X
=
(m)
Ak ei!B mt ,
(11.70)
m= 1
where the Fourier coefficients read
(m)
Ak
a
=
2⇡
Z
⇡/a
⇡/a
dq @q ↵kMF (q) eiamq .
(11.71)
Using partial integration and a series expansion of ↵kMF in J ⇤ , we find for m
(m)
Ak
=i
⇤ Vk
m,1 aJ0 2
!k
⇣
eikx a
0
⌘
1 + O(J ⇤ )2 .
(11.72)
Here we employed that C MF (q) = C0MF + O(J ⇤ ) and S MF (q) = O(J ⇤ ) and we used J0⇤ =
MF
Je C0 , where
Z
V2
MF
C0 = d3 k k2 (1 cos(akx )) .
(11.73)
!k
( m)
(m)⇤
The coefficients for m < 0 can be obtained from symmetry, Ak
= Ak .
Next, we want to apply Fermi’s golden rule to calculate phonon emission due to the driving term ⇠ F @q ↵kMF (q(t)) in Eq.(11.64). Before doing so, we notice that the renormalized
phonon frequency ⌦k (q(t)) has a time-dependent contribution. However, we can treat the
latter as a perturbation itself and find that to leading order in time-dependent perturbation theory (from which Fermi’s golden rule is obtained), it has a vanishing matrix element,
h0|â†k âk |0i = 0. Then, from Fermi’s golden rule we obtain
ph
=
1
X
m=1
2⇡F 2
Z
(m) 2
d d k Ak
(!k
m!B ) .
(11.74)
Plugging in Eq.(11.72) yields our result Eq. (11.65) if we make use of the fact that (to the
considered order) phonons are emitted only on the fundamental frequency !B , and using
Eq.(11.62), vd = a ph . In Appendix O a somewhat simpler derivation is presented, which,
however, only works in the weakly interacting regime where J ⇤ = J and provided that F is
sufficiently small.
11.6.4
Insufficiencies of the phenomenological Esaki-Tsu model
In this subsection we discuss the relation of our results to the phenomenological Esaki-Tsu
model [367]. While the latter explains some of the qualitative features of the observed currentforce relations, we find that it is insufficient for their detailed understanding. Nevertheless, a
comparison to this model clarifies how an impurity atom in an optical lattice immersed in a
thermal bath [368] di↵ers from a particle immersed in a superfluid, as discussed in this paper.
In the former case, the Esaki-Tsu relation is valid [368] and can even be rigorously derived
from microscopic models [369, 370].
We begin by a brief review of the Esaki-Tsu model and derive its basic predictions for the
polaron case. Afterwards we compare these expectations to our numerical results and discuss
CHAPTER 11. WEAK-COUPLING THEORY OF POLARON BLOCH OSCILLATIONS
248
IN OPTICAL LATTICES
their shortcomings.
Phenomenological Esaki-Tsu model
Esaki and Tsu considered an electron in a periodic lattice, subject to a constant electric
field. Using nearest-neighbor tight-binding approximation, the dispersion relation reads !q =
2J cos (qa). Because of the external field the particle undergoes Bloch oscillation, so long
as incoherent scattering is absent. To include decoherence mechanisms with a rate 1/⌧ , the
relaxation time approximation is employed and the following closed expression for the resulting
drift velocity was derived [367],
vd = 2Ja
!B ⌧
.
1 + (!B ⌧ )2
(11.75)
We will not re-derive this result here, however it is instructive to consider the limiting
cases F ! 0, 1. The essence of the relaxation time approximation is the assumption that
a wavepacket evolves coherently for a time ⌧ . Then, incoherent scattering takes place and
instantly the particle equilibrates in the state of minimal energy, i.e. at q = 0. In the
mean-time the distance traveled in real-space is
Z ⌧
2J
x=
dt @q !q =
(1 cos(!B ⌧ )) .
(11.76)
F
0
In the weak-driving limit !B ⌧ 1/⌧ we can expand the cosine and find vd = x/⌧ = J⌧ a2 F ,
which explains the Ohmic behavior in Eq.(11.75). In the strong driving limit !B
1/⌧ on
the other hand, we can average out the coherent part of the evolution and set cos(!B ⌧ ) ⇡ 0.
Then we obtain vd = x/⌧ = 2J/(F ⌧ ), which captures the large-force limit in Eq.(11.75).
Now we can naively adapt the Esaki-Tsu model to the polaron case, without specifying the
origin of the relaxation mechanism. It makes the following predictions for the current-force
relation.
(i) For weak driving F ! 0, Ohmic behavior vd ⇠ F is expected.
(ii) For strong driving, negative di↵erential conductance vd ⇠ 1/F is predicted.
(iii) For intermediate force, a negative di↵erential conductance peak appears, where dvd /dF =
0.
(iv) The polaron drift should depend linearly on the e↵ective hopping strength vd ⇠ J ⇤ ,
at least for small hopping J ⇤ ! 0 (for larger hopping, ⌧ might include J ⇤ -dependent
corrections).
In the following we will investigate our numerical results more carefully, and show that many
of them are not consistent with the simple Esaki-Tsu model, despite the fact that this model
has been applied in numerous polaron models before [313, 327, 328]. However, all these points
are correctly described by our analytical model of the polaron current.
Comparison to numerics
As discussed in 11.6.2 the Esaki-Tsu relation correctly predicts (ii) the existence of negative
di↵erential conductance and (iii) a corresponding peak at which vd takes its maximum value.
This is a direct manifestation of the interplay between coherent transport, which dominates for
large F , and its incoherent counterpart responsible for the weak-driving behavior. However,
11.6. POLARON TRANSPORT
7
x 10
249
3
6
5
10
3
10
4
10
5
4
3
10
1
2
1
0
0
2
4
6
8
10
12
Figure 11.10: Best fit of the Esaki-Tsu relation (dashed black line) to our numerically obtained
current-force relation vd (F ) (red squares), where both J and ⌧ were treated as free parameters
in Eq.(11.75). We also show our analytical result Eq.(11.65) (solid orange line), which was
obtained from first principles and without any free fitting parameters. While the Esaki-Tsu
model can reproduce the negative di↵erential conductance peak, it fits less well in the strongdriving regime. In the inset the same data is shown, but using a double-logarithmic scale.
Here the complete failure of the Esaki-Tsu model in the weak-driving
p limit becomes apparent.
The parameters are J = 0.3c/a, ge↵ = 3.16, ⇠ = 5a and `ho = a/ 2 and from the best-fit we
obtain ⌧ = 1.34a/c and J|fit = 0.0065c/a.
we also pointed out already that (i) is inconsistent with the sub-Ohmic behavior observed in
our numerics.
In FIG.11.10 we fitted Eq.(11.75) to the results of our full solution of the semiclassical
dynamical equations (11.48), (11.49). While for moderate driving F & c/a2 the shape of the
curve can be reproduced by the fit, the comparison for small force (in the inset of FIG.11.10)
clearly shows that the Esaki-Tsu relation can not capture the weak driving regime. Importantly, to get reasonable quantitative agreement, one should treat not only the relaxation
time ⌧ , but also the hopping strength J as a free parameter [313]. The resulting best fit J|fit
MF
always yields e↵ective hoppings exceeding the renormalized polaron hopping J ⇤ = Je C .
For instance, in the case shown in FIG.11.10 (ge↵ = 3.16, J = 0.3c/a) we find from fitting
J|fit /J = 0.022 whereas J ⇤ /J ⇡ 0.96 is almost two orders of magnitude larger. Therefore, on
a quantitative level, the Esaki-Tsu model completely fails here.
To get a better understanding why the quantitative result from the Esaki-Tsu relation
is so far o↵, we now investigate how the current-force relation vd (F ) depends on our system
parameters ge↵ and J. To this end we consider the position of the negative di↵erential conductance peak, which is characterized by FNDC and vdmax . From the Esaki-Tsu relation (11.75)
we would expect FNDC = 1/⌧ a and vdmax = J ⇤ a, see (iv).
In FIG. 11.9 (a), (b) we show how FNDC and vdmax depend on the hopping strength J. While
the e↵ect on FNDC is rather weak, a power-law very close to vdmax ⇠ J 2 is observed in (b). This
is in contradiction to the Esaki-Tsu model, which suggests vdmax ⇠ J ⇤ , since to leading order
J ⇤ ⇠ J. It shows that not only ⌧ , but also J should be considered as a fitting parameter in
order to describe the numerical curves by the Esaki-Tsu relation (11.75). Physically, however,
it is not clear why J should be a free parameter in this equation. Meanwhile, from our
analytical model we obtain the correct power-law vd ⇠ J 2 for small J, see Eq.(11.67).
CHAPTER 11. WEAK-COUPLING THEORY OF POLARON BLOCH OSCILLATIONS
250
IN OPTICAL LATTICES
In FIG. 11.9 (c), (d) we show the dependence of the negative di↵erential conductance peak
on the interaction strength. While no dependence of FNDC can be identified (c), we obtain
2 for sufficiently weak interactions. From Esaki-Tsu in contrast, we
a power-law vdmax ⇠ ge↵
would expect a decrease of the polaron drift with the interaction strength, because the latter
suppresses the polaron hopping J ⇤ . Again, our analytical model can explain the observed
power-law, see Eq.(11.66). It also predicts vd ⇠ J0⇤2 , such that we do indeed expect to find
indications of the polaronic dressing for sufficiently large interaction strength. This e↵ect can
be observed in (c), where for large ge↵ the incoherent polaron current reaches a maximum
value before it becomes strongly suppressed by interactions.
Thus, we have seen that on a quantitative level the Esaki-Tsu model is insufficient for
understanding the incoherent polaron current. We attribute the reasonable fit to our data
in the moderate driving regime simply to the fact that the Esaki-Tsu relation works on a
qualitative level, in the sense that it predicts a negative di↵erential conductance peak.
Chapter 12
All-coupling Theory of the Fröhlich
Polaron
12.1
Summary and Introduction
In this chapter we return to the Fröhlich polaron in the continuum. So far we have mainly
dealt with MF polaron theory, despite the fact that our discussion of experimentally relevant
parameters in Sec.9.2.2 demonstrated that the intermediate coupling regime ↵ & 1 can be
reached in state-of-the-art experiments. Extending the theoretical description of Fröhlich
polarons to intermediate coupling strengths requires taking into account correlations between
phonons due to quantum fluctuations, which are absent in the mean-field (MF) variational
wavefunction. The physical e↵ects of these fluctuations make up the essence of intermediateand strong-coupling physics. Here we develop an all-coupling theory for the Fröhlich polaron,
based on a renormalization group (RG) approach. For concreteness we apply our method to
describe mobile impurity atoms immersed in a BEC.
p
When the coupling strength ↵ & 1 becomes large and the mass ratio M/mB . 1/ 2 becomes small, MF theory can no longer be trusted. Indeed, recent quantum Monte Carlo (MC)
calculations by Vlietinck et al.[22] have shown extremely large deviations of the groundstate
energy from MF predictions already in the intermediate-coupling regime. A comparison for
the Bogoliubov-Fröhlich model is shown in FIG.12.1. While the MC method [21] can be considered one of the most powerful tools for the investigation of polaron problems, the analytical
insights gained are limited.
The RG approach developed in this chapter accomplishes two main tasks. Firstly it describes the polaron quantitatively all the way from weak to strong coupling. It does so much
more efficiently than previous approaches, thus making it a numerically cheap method. Secondly, it yields important new analytical insights. For example in the case of the BogoliubovFröhlich Hamiltonian (in three dimensions) it predicts an unphysical logarithmic ultra-violet
(UV) divergence. The analytic insights gained from the RG method allow us to construct a
regularization scheme, which is not available for numerical quantum MC methods.
Previously polaron theories have been developed for the limiting cases of weak- and strongcoupling polarons, see Secs. 9.2.4, 9.2.5. Feynman introduced a variational theory, based on
the path-integral formalism, which is able to describe accurately the Fröhlich polaron in the
solid-state context for all coupling strengths [23, 379]. In addition, diagrammatic MC methods were developed, relying on heavy numerics [21]. Furthermore it has been understood
that the impurity induces interactions between phonons, which lead to multi-mode squeezing
251
252
CHAPTER 12. ALL-COUPLING THEORY OF THE FRÖHLICH POLARON
0.6
0.5
0.4
0.3
0.2
0.1
0
−0.1
−0.2
0
0.2
0.4
0.6
0.8
1
Figure 12.1: The polaronic contribution Ep = E0 2⇡aIB n0 mred1 to the groundstate energy
of an impurity immersed in a BEC is shown, as a function of the coupling constant ↵. It
corresponds to the groundstate energy of the Fröhlich Hamiltonian, after the linear power-law
divergence predicted by MF theory is regularized. We compare results from Feynman pathintegral calculations and diagrammatic Monte Carlo (MC) calculations – both taken from
Vlietinck et al. [22] – to MF theory, our RG approach and to variational correlated Gaussian
wavefucntions (CGWs). The results are sensitive to the value of the UV momentum cut-o↵
⇤0 , choosen to be ⇤0 = 2000/⇠ here. Other parameters are M/mB = 0.26316 and q = 0.
between di↵erent phonon modes in the polaron cloud [380, 381]. The use of correlated Gaussian wavefunctions (CGWs) incorporating multi-mode squeezing was considered the most
advanced semi-analytical approach to the polaron problem [382, 383, 384]. Recently also a
fully consistent variational treatment including interactions between all phonon modes has
been developed in Ref.[P10], in collaboration with the author of this thesis.
Here we develop a unified approach to the polaron problem using an RG formalism [P9].
As demonstrated in FIG.12.1, our method yields polaron energies which are in excellent
agreement with MC results in the weak- and intermediate-coupling regimes. Our method
builds upon the analytical MF approach to the polaron problem discussed earlier. We include
quantum fluctuations on top of the MF polaron, and treat them using the RG formalism.
In this way correlations between phonon modes at di↵erent momenta are taken into account.
The RG formalism uses Schrie↵er-Wol↵ type unitary transformations to decouple high-energy
phonons from the problem step by step, allowing us to formulate the polaron groundstate
explicitly after each step. In a subsequent chapter we will show how this RG approach can
be generalized to non-equilibrium problems.
FIG.12.1 shows that also the variational CGW approach [P10] is in excellent agreement
both with RG and MC calculations. The CGWs emerge naturally from the RG as variational
wavefunctions. Remarkably the RG can even be used to construct the form of the variational
wavefunctions explicitly, and further it includes corrections to the squeezed coherent states.
All calculations for CGWs presented in this thesis were performed by Yulia Shchadilova.
In this chapter we apply the RG method to the Bogoliubov-Fröhlich model describing
an impurity immersed in a BEC. So far, in the intermediate coupling regime, Feynman’s
path integral approach [24, 322, 318] and the numerical MC method [22] have been applied
to this model. Beyond the Fröhlich model self-consistent T-matrix calculations [321] and a
variational analysis [385] have been performed.
Despite di↵erent attempts no clear picture of the physics in the intermediate-coupling
12.1. SUMMARY AND INTRODUCTION
253
regime of the Bogoliubov-Fröhlich model has been obtained so far. By comparing groundstate
energies, as in FIG.12.1, Vlietinck et al.[22] found substantial deviations of Feynman pathintegral results [24] from MC predictions. In Ref.[P9], relying on the analytical insights from
our RG method, we identified the key discrepancy between these predictions: The dependence
on the UV cut-o↵.
As will be explained later, the RG predicts analytically that the groundstate energy E0
of the Bogoliubov-Fröhlich model diverges to 1 logarithmically with the UV cut-o↵, even
after the linear power-law divergence of the MF contribution to the polaron energy has been
properly regularized. The RG prediction is verified by the variational CGW approach.
This insight provides the key for understanding the contradictory results from Feynman
path-integral calculations [24] and MC predictions [22]. Early on, Tempere et al.[24] noted that
the energy predicted by the path-integral method shows a peculiar dependence on the UV cuto↵. Later, Vlietinck et al.[22] showed conclusively that Feynman’s path-integral calculations
are UV convergent for a sufficiently large cut-o↵. In their MC data, however, Vlietinck et
al. overlooked the logarithmic divergence of the polaron groundstate energy predicted by the
RG.
The first main goal of our all-coupling theory is to calculate the polaron energy, which is
directly measurable in experiments with ultra cold quantum gases using e.g. RF spectroscopy,
see Chap.10. This requires careful regularization of the UV log-divergence described above.
Note that the data shown in FIG.12.1 does not provide a meaningful prediction for any experiment, because it depends explicitly on the value of the UV cut-o↵. Here we introduce a
regularization scheme for the log-divergence and make meaningful predictions for the polaron
energy expected in experiments with ultra cold atoms. The regularization is based on a refinement of the relation between the e↵ective interaction strength gIB and the impurity-boson
scattering length aIB , enabled by the analytical insights gained from the RG.
The second main goal of our all-coupling theory is to calculate the e↵ective polaron mass.
Using the RG we find a smooth cross-over from weak- to strong-coupling regime. For intermediate couplings we predict substantial deviations from Feynman’s variational path integral
method [24, 318]. We are not aware of any MC calculations of the BEC-polaron mass at this
point. Thus, to clarify which theory is correct, experiments will be needed. From momentumresolved RF spectra also the polaron mass can directly be obtained. Alternatively the e↵ective
polaron mass can be measured from the oscillation frequency in a harmonic trap, as has been
demonstrated for Fermi polarons [386].
Lastly we will employ our RG method to calculate other equilibrium polaron properties. These include the phonon number in the polaron cloud and the quasiparticle weight Z
characterizing the polaron peak. The latter can also be measured using RF spectroscopy [299].
This chapter is organized as follows. In Sec.12.2 we derive our model and summarize the
coupling constants flowing in the RG. In Sec.12.3 we develop our RG formalism and derive the
RG flow equations. The equilibrium properties of the polaron groundstate are derived from
the RG protocol in Sec.12.5. There we moreover discuss the logarithmic UV divergence of
the polaron energy and introduce a regularization scheme. In Sec.12.6 we discuss our results
in detail and compare to predictions by other polaron theories. In appendix R we present
an alternative check of the RG protocol by comparing to related calculations by Kagan and
Prokof’ev [378].
254
CHAPTER 12. ALL-COUPLING THEORY OF THE FRÖHLICH POLARON
12.2
Fröhlich Model and RG coupling constants
Our starting point is the Fröhlich Hamiltonian (9.51) after the Lee-Low-Pines transformation.
As discussed in Sec.9.2.4 the Hamiltonian can be solved approximately using MF theory. We
want to build upon the MF solution and include quantum fluctuations on top of it. To this
end we apply a unitary transformation
ÛMF = exp
✓Z
⇤0
◆
h.c. ,
d3 k ↵k â†k
(12.1)
which displaces phonon operators by the MF solution,
†
ÛMF
âk ÛMF = âk + ↵k .
(12.2)
As a result, from Eq.(9.51), we obtain the following polaron Hamiltonian,
H̃q =
†
ÛMF
Ĥq ÛMF
= HMF +
Z
⇤0
3
d k
†
⌦MF
k âk âk
+
Z
⇤0
d3 kd3 k0
k · k0 ˆ ˆ
: k k0 : .
2M
(12.3)
Here HMF is the MF groundstate energy, see Eq.(9.58), we defined ˆ k = ˆ †k as
ˆ k := ↵k (âk + ↠) + ↠âk ,
k
k
(12.4)
and : ... : stands for normal-ordering. The absence of terms linear in âk in Eq.(12.3) reflects
the fact that ↵k correspond to the mean-field (saddle point) solution of the problem. We
emphasize that (12.3) is an exact representation of the original Fröhlich Hamiltonian, where
the operators âk now describeRquantum fluctuations around the MF polaron. Above, as well
⇤ 3
as in the rest of this chapter,
d k stands for a three dimensional integral over a spherical
region containing momenta of length |k| < ⇤.
From Eq.(12.3) we notice that the only remaining coupling constant is the (inverse) impurity mass M 1 . This should be contrasted to the original Hamiltonian (9.51) before applying
the MF shift, where the interaction strength ↵ and the inverse mass M 1 define coupling
constants. Both are required to determine whether the polaron is in the weak-, strong- or
intermediate-coupling regime, see Eqs.(9.52), (9.71). In the following section 12.3.1 we will
carry out a simple dimensional analysis and show that the non-linear terms in (12.3) are
marginally irrelevant, allowing an accurate description by the RG (which is perturbative in
M 1 in every RG step).
To facilitate our subsequent RG analysis we generalize the form of the polaron Hamiltonian
(12.3) already at this point. The particular choice of terms will become apparent later on.To
this end we introduce all required coupling constants generated by the RG flow. The general
form of the Hamiltonian reads
Z ⇤
⇣
⌘ 1Z ⇤
†
†
3
H̃q (⇤) = HMF + E+
d k ⌦k âk âk + Wk (âk + âk ) +
d3 kd3 k0 kµ Mµ⌫1 k⌫0 : ˆ k ˆ k0 : .
2
(12.5)
In this expression the coupling constant Mµ⌫ (⇤) and the RG energy shift E(⇤) depend
on the cut-o↵ ⇤ which is running in the RG. Note that the interaction is now characterized
by a general tensor Mµ⌫1 , where the anisotropy originates from the total momentum of the
polaron q = qex , breaking the rotational symmetry of the system. The indices µ = x, y, z
label cartesian coordinates and they are summed over when occurring twice. Due to the
cylindrical symmetry of the problem the mass tensor has the form M = diag(Mk , M? , M? ),
12.2. FRÖHLICH MODEL AND RG COUPLING CONSTANTS
255
and we will find di↵erent flows for its longitudinal and the transverse components. While M
can be interpreted as the (tensor-valued) renormalized mass of the impurity, it should not
be confused with the mass of the polaron, see Appendix P. The first integral in Eq.(12.5)
describes the quadratic part of the renormalized phonon Hamiltonian. It is also renormalized
compared to the original expression ⌦MF
in Eq.(12.3),
k
1
⌦k = !k + kµ Mµ⌫1 k⌫
2
k
· (q
M
Pph ) ,
(12.6)
where the momentum carried by the phonon-cloud Pph acquires an RG flow describing corMF . In addition there is a term linear in the phonon operators,
rections to the MF result Pph
weighted by

✓
◆
kµ k⌫
k
µ⌫
MF
1
Wk = Pph Pph ·
+
Mµ⌫
↵k .
(12.7)
M
2
M
By comparing Eq.(12.5) to Eq.(12.3) we obtain the initial conditions for the RG, starting at
the original UV cut-o↵ ⇤0 where H̃q (⇤0 ) = H̃q ,
Mµ⌫ (⇤0 ) =
12.2.1
µ⌫ M,
MF
Pph (⇤0 ) = Pph
,
E(⇤0 ) = 0.
(12.8)
Towards the supersonic regime
So far we assumed that a MF polaron solution exists, providing a valuable starting point to
formulate a Hamiltonian for quantum fluctuations describing beyond-MF e↵ects. However,
as we have shown in Sec.9.2.4, for sufficiently large polaron momentum q > qcMF no such MF
solution can be found. Nevertheless, since we expect quantum fluctuations to give rise to
additional dressing of the impurity with phonons, the true critical polaron momentum should
be larger than the MF value, qc > qcMF .
To treat also the regime of polaron momenta exceeding the MF critical value, we expand
around the MF polaron defined at the MF critical momentum qcMF . That is, we still apply
the unitary transformation in Eq.(12.1), but with the coherent amplitudes
↵kc =
Vk /⌦ck ,
(12.9)
where the critical renormalized phonon dispersion reads
⌦ck = !k + k 2 /2M
ckx .
(12.10)
In this case we still find the universal polaron Hamiltonian (12.5), however with a modified
initial condition for the phonon momentum,
MF c
Pph (⇤0 ) = Pph
[↵k ] + qcMF
q = 2qcMF
q
M c,
(12.11)
MF , q point along the same direction. The impurity mass
assuming that all momenta Pph , Pph
is unmodified, Mµ⌫1 (⇤0 ) = µ⌫ M 1 . Meanwhile, the MF energy shift HMF entering Eq.(12.5)
is modified as compared to Eq.(9.59),
q 2 + qcMF
HMF (q) =
2M
2
+ 2⇡aIB n0
✓
1
1
+
mB M
Z ⇤0
◆
+
d3 k
q
Mc
M
1
M c2
2
◆
1
1
+
⇤0 n0 . (12.12)
mB M
qcMF
✓
Vk2
+ 4a2IB
⌦MF
k
256
12.3
CHAPTER 12. ALL-COUPLING THEORY OF THE FRÖHLICH POLARON
Renormalization Group Formalism for the Fröhlich model
Now we develop the RG formalism for the Fröhlich polaron. To keep track of all involved
basis transformations we summarize our treatment of the Bose polaron problem in FIG.12.2.
The essence of the RG is to separate a shell of fast phonon modes, with momenta in a thin
shell ⇤ ⇤ < |k| < ⇤, and integrate them out. This renormalizes the remaining Hamiltonian
for slow phonons with momtenta |p| < ⇤
⇤. The approach is justified by the separation
of time-scales associated with slow and fast phonons: In the spirit of the Born-Oppenheimer
approximation (familiar e.g. from the description of covalent molecular binding), slow phonons
appear as quasi-static classical variables from the point of view of fast phonons. In practice
this means that we may use 1/⌦k , with k a fast-phonon momentum, as a small parameter.
This allows to solve for the groundstate of fast phonons, now depending on the slow variables,
which introduces entanglement between di↵erent phonon modes in the polaron groundstate.
In practice the RG procedure can be implemented by the consecutive application of approximate unitary transformations Û⇤ . We will derive their form explicitly in 12.3.2 below.
Before, however, we perform a dimensional analysis of the Hamiltonian (12.5) in 12.3.1 to get
a simple understanding of the expected RG flow. At the end of this section, in 12.6.1, we
discuss solutions of the RG flow equations.
12.3.1
Dimensional analysis
We start our discussion of the RG by performing a dimensional analysis. Besides helping us
understand the behavior of our RG flow equations obtained in the next subsection, it shows
that there are di↵erent energy regimes where phonons contribute di↵erently to the polaron
properties.
In the following subsection we will perform a momentum-shell RG procedure, i.e. the
cut-o↵ ⇤ will be reduced and quantum fluctuations at larger momenta will be integrated out
successively. This gives rise to ⇤-dependent coupling constants, but in this subsection we will
ignore such dependence for simplicity. To understand the importance of the various terms in
Eq. (12.5), we instead assign a scaling dimension to the fluctuation field, âk ⇠ ⇤ . It is
Figure 12.2: Sketch of our treatment of the Bose polaron problem: After introducing Bogoliubov phonons to describe excitations of the weakly interacting BEC, a Fröhlich Hamiltonian is
obtained. It is translationally invariant and its groundstate with total momentum q is denoted
by | q i. The Fröhlich Hamiltonian can be further simplified by applying the Lee-Low-Pines
transformation. Conservation of total momentum becomes explicit and the groundstate reads
†
ÛLLP
| q i = |qi ⌦ | q i. Next we introduce the MF polaron solution, allowing us to transform
into a frame describing quantum fluctuations around the MF polaron. There the groundstate
†
can be written ÛMF
(q)| q i = |gsq i. To find the groundstate with quantum fluctuations, we
apply an RG procedure which consists of a series of unitary transformations Û⇤ . They lead
to a factorization of the groundstate in subsequent momentum shells, Û⇤† |gsq i = |0if ⌦ |gsq is .
12.3. RENORMALIZATION GROUP FORMALISM FOR THE FRÖHLICH MODEL 257
determined by defining the phonon dispersion ⌦k in Eq.(12.6) as the energy scale at ⇤,
Z
⇤
!
dd k â†k âk ⌦k ⇠ ⇤0 = 1.
(12.13)
In this step we moreover generalize our analysis to the case of d spatial dimensions.
A unique property of the Bogoliubov-Fröhlich Hamiltonian is the crossover of the phonon
dispersion from quadratic behavior (⌦k ⇠ k 2 ) for large momenta to linear behavior (⌦k ⇠ k)
for small momenta. The crossover takes place around k0 ⇡ min ⇠ 1 , Mk c . For heavy impurities k0 ⇡ 1/⇠ and the crossover is due to the Bogoliubov dispersion, whereas for light
impurities k0 ⇡ M c and the crossover is caused by impurity fluctuations (we approximated
Mk ⇡ M ). To understand the physics at mid-energy, i.e. before the crossover to the linear
regime takes place, we have to perform a dimensional analysis based on the quadratic dispersion. For the actual groundstate properties on the other hand, we will examine the regime of
linear dispersion.
In the following we distinguish three di↵erent regimes. In the linear low-energy regime we
have k . M c, ⇠ 1 such that
⌦k ⇠ k,
Vk ⇠ k 1/2 ,
↵k ⇠ k
In the quadratic light-impurity regime it holds M c . k . ⇠
⌦k ⇠ k 2 ,
Vk ⇠ k 1/2 ,
↵k ⇠ k
1/2
1
.
(12.14)
and we obtain
3/2
.
(12.15)
Finally in the quadratic heavy-impurity regime where ⇠ 1 . k (no matter whether k ? M c)
it holds
⌦k ⇠ k 2 , Vk ⇠ 1, ↵k ⇠ k 2 .
(12.16)
In Table 12.1 we summarize the resulting scaling dimensions of the di↵erent quantum fluctuation terms. We observe a markedly di↵erent behavior in the regimes of linear and quadratic
phonon dispersion. For medium energies (i.e. the low-energy sector of the quadratic dispersion regime) we find that all quantum fluctuations are relevant in three spatial dimensions,
d = 3, with the exception of the two-phonon term in the light-impurity case (12.15). Even for
d > 6 the quartic phonon term is always relevant. For low energies in contrast, in all spatial
dimensions quantum fluctuations are mostly irrelevant, only the quartic term is marginal. Its
pre-factor is given by the inverse mass M 1 which we expect to become small due to dressing with high-energy phonons, making the term marginally irrelevant. Therefore we expect
that the linear regime ⇤ ⌧ 1/⇠ is generically well described by MF theory on a qualitative
operator
â
Rk d d 0 0
0
0
R dd k d dk 0 kk 0↵k ↵k a2k ak
R dd k dd k 0 kk 0 ↵2k a2k ak0
d k d k kk ak0 ak
⇤ & 1/⇠
⇤ d/2 1
⇤d 4
⇤d/2 3
⇤ 2
1/⇠ & ⇤ & M c
⇤ d/2 1
⇤d 3
⇤d/2 5/2
⇤ 2
⇤ . M c, 1/⇠
⇤ (d+1)/2
⇤d
⇤d/2
⇤0
Table 12.1: Dimensional analysis of the Hamiltonian (12.5) in di↵erent energy regimes. In
the three columns to the right the scaling of various operators with momentum cut-o↵ ⇤
is shown for the three distinct cases described in the main text. The first line shows the
engineering
of the fluctuations, determined to make the bare phonon dispersion
R ddimension
2
scale as d k ⌦k ak ⇠ ⇤0 , see Eq.(12.13).
258
CHAPTER 12. ALL-COUPLING THEORY OF THE FRÖHLICH POLARON
level. On a quantitative level we also expect that corrections can well be captured by the RG
protocol introduced below, which is perturbative in M 1 in every step.
12.3.2
Formulation of the RG
Now we turn to the derivation of the RG flow equations for the coupling constants Mµ⌫1 (⇤)
and Pph (⇤). After introducing the basic idea of our scheme we will carry out the technical
part of the calculations. Then we summarize the resulting RG flow equations. Their solutions
will be discussed in the following subsection 12.6.1. In 12.3.2, as a side-product, we also obtain
the renormalization of the polaron groundstate energy E(⇤). The e↵ect on other important
observables (e.g. the polaron mass) will be discussed later in 12.5.
RG step – motivation
The essential idea of the RG in general is to make use of the separation of time scales,
which usually translates into di↵erent corresponding length scales. In our case, momentum
space provides us with a natural order of energy scales through the phonon dispersion ⌦k ,
see Eq.(12.6). The latter is mostly dominated by the bare Bogoliubov dispersion !k , which
ultimately allows us to perform the RG.
To make use of separation of time scales, we may formally split the Hamiltonian into
a slow (labeled s) and a fast part (labeled f), as well as coupling terms between these two
(labeled sf) [387],
Ĥ = Ĥf + Ĥsf + Ĥs .
(12.17)
In our case, Ĥf contains only phonons with momenta k from a high-energy shell ⇤
⇤
k  ⇤, where ⇤ is the sharp momentum-cut o↵ and ⇤ ! 0 the momentum-shell width. Ĥs
on the other hand contains the remaining phonons with momenta p, where p < ⇤
⇤.
In order to integrate out fast degrees of freedom, we decouple the latter from the slow
Hamiltonian. In practice this is achieved by applying a unitary transformation Û⇤ , which
has to be chosen such that the resulting Hamiltonian is diagonal in the fast phonon number
operators,
[Û⇤† ĤÛ⇤ , â†k âk ] = 0.
(12.18)
While solving this equation exactly for Û⇤ is a hopeless task in general, we may at least do
so perturbatively. Here we make use of the separation of time scales, namely we assume
||Ĥf ||
||Ĥsf ||, ||Ĥs ||.
(12.19)
As explained above, ||Ĥf ||/||Ĥs || ⇠ ⌦k /⌦p ⌧ 1 is automatically the case for most of the slow
phonons. The coupling ||Ĥsf ||, however, has to be sufficiently weak for our method to work.
For this reason, our RG protocol should be understood as a perturbative RG in the coupling
defined by the impurity mass M 1 .
We will present the perturbative solution of Eq.(12.18) for Û⇤ below. In the end, if we
evaluate the decoupled Hamiltonian in the fast-phonon groundstate |0if , what we obtain is
†
f h0|Û⇤ ĤÛ⇤ |0if
= Ĥs + Ĥs + O(⌦k )
2
.
(12.20)
This is the essence of the RG-step: (i) We decouple fast and slow phonon dynamics from each
other (perturbatively), and find the fast-phonon groundstate. This e↵ectively reduces the
cut-o↵ in the remaining slow-phonon Hamiltonian, ⇤ ! ⇤
⇤. (ii) We calculate the e↵ect
of fast phonons in their groundstate on the new slow degrees of freedom, which amounts to
12.3. RENORMALIZATION GROUP FORMALISM FOR THE FRÖHLICH MODEL 259
the replacement
Ĥs ! Ĥs + Ĥs = Ĥs0 .
(12.21)
We can now start over from the renormalized Hamiltonian Ĥs0 and integrate out its highest
energy shell in the next RG step. Like this, provided Ĥs0 has the same algebraic form as the
original Hamiltonian Ĥ, we obtain di↵erential RG flow equations for the coupling constants.
RG step – formal calculation
Now we turn to the actual calculation, the individual steps of which were discussed in the last
paragraph. Thereby we proof that (to the considered order in (⌦k ) 1 ) no more than the two
coupling constants Mµ⌫1 and Pph introduced in Eq.(12.5) are required. Readers who are not
interested in the technical details can skip this section and proceed directly to the RG flow
equations presented in the following paragraph.
To begin with, we bring the Hamiltonian (12.5) into a more transparent form by evaluating
the normal-ordering operator : ... : in the last term. To this end we make use of the identity
h
i
: ˆ k ˆ k0 := ˆ k ˆ k0
k k0 ˆ k + |↵k |2 ,
(12.22)
which yields the universal Hamiltonian
H̃q (⇤) = HMF +
E
Z
Z
k2
1 ⇤ 3
2
d k
|↵k | +
d k d3 k0 kµ Mµ⌫1 k⌫0 ˆ k ˆ k0
2M
2

✓
◆
Z ⇤
k2 ˆ
†
3
MF
MF kµ
+
d k âk âk ⌦k + (Pph Pph )µ
k . (12.23)
M
2M
⇤0
3
In the following we will perform the RG procedure described in the last paragraph on this
Hamiltonian (12.23).
We proceed by extracting the fast-phonon Hamiltonian from the general expression (12.23),
Ĥf =
Z
3
d k
f

â†k âk ⌦MF
k
✓
◆
1
+ (Pph
+ kµ Wµ⌫ k⌫ ˆ k
M
2
Z
Z
1
kµ Mµ⌫ k⌫
kµ Mµ⌫1 k⌫0
3
2
+ d k
|↵k | + d3 k d3 k0
: ˆ k ˆ k0 :, (12.24)
2
2
f
f
MF kµ
Pph
)µ
Here, and throughout this section, we use the definition
Wµ⌫ := Mµ⌫1
µ⌫ M
1
.
(12.25)
to make our expressions more handy. Note that in Eq.(12.24), unlike in Eq.(12.23), we wrote
the phonon-phonon interactions in a normal-ordered form again. As a consequence we observe
an energy shift (first term in the second line of (12.24)), which seems to reverse the e↵ect of
the corresponding term in Eq.(12.23) where we started from. Importantly, however, here the
renormalized mass Mµ⌫ appears instead of the bare mass M , thus yielding a non-vanishing
overall energy renormalization. Below we will furthermore show that the normal-ordered
double-integral in the second line of Eq.(12.24) yields only corrections of order O( ⇤2 ), and
may thus be neglected in the limit ⇤ ! 0 considered in the RG.
The slow phonon Hamiltonian Ĥs is simply given by Eq.(12.23) after replacing ⇤ ! ⇤
⇤
260
CHAPTER 12. ALL-COUPLING THEORY OF THE FRÖHLICH POLARON
in the integrals. Finally, for the coupling terms we find
Z
Z
3
Ĥsf = d k d3 p kµ Mµ⌫1 p⌫ ˆ k ˆ p ,
f
(12.26)
s
where use was made of the symmetry Mµ⌫1 = M⌫µ1 . Next we will decouple fast from slow
phonons. To this end we make an ansatz for the unitary transformation Û⇤ as a displacement
operator for fast phonons,
✓Z
h
i◆
†
†
3
Û⇤ = exp
d k F̂k âk F̂k âk .
(12.27)
f
Importantly, we assume the shift F̂k to depend solely on slow phonon operators, i.e.
[F̂k , âk ] = [F̂k , â†k ] = 0.
(12.28)
The e↵ect on fast phonons is simply
Û⇤† âk Û⇤ = âk
F̂k .
(12.29)
As a first consequence, normal-ordering of fast phonon operators is unmodified such that
Z
kµ Mµ⌫1 k⌫0
d kd k
: ˆ k ˆ k0 : |0if = O( ⇤2 ),
f h0|
2
f
3
3 0
(12.30)
where we used that F̂k is a smooth function of k. Since the decoupling unitary Û⇤ is chosen
such that |0if (with âk |0if = 0) is the fast phonon groundstate, we may neglect terms from
Eq.(12.30). This is the reason why fast phonon-phonon interaction terms in Eq.(12.24) can
be discarded.
The operator F̂k can be determined from the condition in Eq.(12.18), i.e. we demand
that terms linear in fast phonon operators âk vanish. To this end, let us perform a series
expansion in the fast phonon frequency ⌦k and note that F̂k = O(⌦k ) 1 (for ⌦k = 1, fast
and slow phonons are decoupled already before applying Û⇤ ). This allows us to make use of
the following identity, valid for any slow-phonon operator Ôs (with Ôs = O(⌦0k = 1)),
Z
n
o
†
Û⇤ Ôs Û⇤ = Ôs + d3 k âk [Ôs , F̂k† ] â†k [Ôs , F̂k ] + O(⌦k ) 2 .
(12.31)
f
In this way we derive the following equation for F̂k† , which is a sufficient condition for Û⇤† ĤÛ⇤
to decouple into fast and slow phonons,
Z
⇣
⌘
⌦k F̂k† = Wk + ↵k F̂k† kµ Mµ⌫1 d3 p p⌫ ˆ p + [Ĥs , F̂k† ] + O(⌦k ) 2 .
(12.32)
s
As expected, to zeroth order the solution F̂k of Eq.(12.32) vanishes, F̂k = O(⌦k )
orders can easily be solved iteratively and we obtain
1.
Higher


Z
Z
⇣
⌘
1
1
1
3
1
3
MF
†
F̂k =
Wk + ↵k kµ Mµ⌫ d p p⌫ ˆ p
↵
k
M
d
p
⌦
p
↵
â
â
⌫ p
p +
k µ
µ⌫
p
p
⌦k
⌦2k
s
s
✓
◆
Z
Z
1
1
3
ˆ
+ Wk + ↵k kµ Mµ⌫ d p p⌫ p k M
d3 p p ˆ p + O(⌦k ) 3 . (12.33)
s
s
12.4. POLARON GROUNDSTATE ENERGY
261
Using the last result Eq.(12.33) we can now calculate the full transformed Hamiltonian.
In a compact form it may be written as
Z
⇣
⌘
†
ˆ s (k) + Ĥs + Ĥs + O(⌦k ) 2 .
Û⇤ H̃q Û⇤ = d3 k â†k âk ⌦k + ⌦
(12.34)
f
Here the fast-phonon frequency is modified due to the coupling to slow degrees of freedom,
Z
1
ˆ
⌦s (k) = kµ Mµ⌫ d3 p p⌫ ˆ p ,
(12.35)
s
while the renormalization of the slow-phonon Hamiltonian reads
Z
kµ Mµ⌫1 k⌫
Ĥs = d k
|↵k |2
2
f
3
Z

Z
1
1
d k
Wk + ↵k kµ Mµ⌫ d3 p p⌫ ˆ p
⌦
k
f
s
2
3
+ O(⌦k )
2
.
(12.36)
This expression includes both, terms depending on the slow phonon operators leading to
renormalized coupling constants Mµ⌫1 and Pph , as well as real numbers describing a renormalization of the polaron groundstate energy. Finally, let us also mention that in order to
calculate Eq. (12.36), only first order terms ⇠ (⌦k ) 1 are required in F̂k .
RG flow equations
Now we are in a position to derive the RG flow equations for the groundstate RG protocol. To
this end we compare the Hamiltonian Ĥs0 obtained in the RG step Eq.(12.21) to the original
one in Eq.(12.23). Using the result Eq.(12.36) we find that in the new Hamiltonian the mass
0
term is renormalized, Mµ⌫1 ! Mµ⌫1 with
0
1
Mµ⌫ =
Mµ⌫1
2Mµ
1
Z
d3 k
f
|↵k |2
k k M
⌦k
1
⌫.
0
(12.37)
Rewriting formally Mµ⌫1 = Mµ⌫1 + ⇤ @⇤ Mµ⌫1 (⇤) and employing d3 k =
obtain the RG flow equation
@Mµ⌫1
= 2Mµ 1
@⇤
Z
d2 k
f
|↵k |2
k k M
⌦k
d2 k ⇤ 1 , we
1
⌫.
(12.38)
Analogously the RG flow of the µ-th component of the phonon momentum is obtained,
µ
@Pph
@⇤
=
2Mµ⌫1
Z
d2 k
f
h
MF
Pph
1
Pph · k + k
2
MM
1
k
i |↵ |2
k
k⌫ .
⌦k
(12.39)
In subsection 12.6.1 we will discuss solutions of these RG flow equations and present typical
numerical results.
12.4
Polaron Groundstate Energy
The first property of the polaron groundstate that will be discussed is its energy. During
the formulation of the RG protocol in the last section we already derived an expression for
the polaron energy. It consists of the MF term plus corrections from the RG, E0RG (⇤) =
1
Note the minus sign in d3 k =
d2 k ⇤, accounting for the fact that the RG flows from large to small ⇤.
262
(a)
CHAPTER 12. ALL-COUPLING THEORY OF THE FRÖHLICH POLARON
(b)
5
50
35
0
0
−50
30
−5
8
9
10
−100
−10
1
2
10
3
10
10
4
10
−150
0
2
4
6
8
10
Figure 12.3: (a) The polaronic contribution to the groundstate energy Ep = E0 2⇡aIB n0 mred1 )
in d = 3 dimensions is shown as a function of the UV momentum cut-o↵ ⇤0 in logarithmic
scale. We compare our results RG, MF theory and correlated Gaussian wavefunctions (CGWs)
[P10] to predictions by Vlietinck et al. [22] (diagrammatic MC - bullets, Feynman - dashed).
The data shows a logarithmic UV divergence of the polaron energy. Parameters are M/mB =
0.263158, q = 0 and ↵ = 3. (b) We show the polaron energy E0 as a function of the coupling
constant ↵ for two di↵erent UV cut-o↵s ⇤0 . We conclude that the polaron groundstate energy
E0 depends sensitively on the value of ⇤0 . For large cut-o↵s the groundstate energy becomes
negative, in contradiction to the fact that the microscopic polaron Hamiltonian (9.1) is positive
definite when aIB > 0. Parameters are M/mB = 0.26 and n0 = 1 ⇥ ⇠ 3 .
HMF +
E(⇤), which are given by
E(⇤) =
Z
⇤0
3
d k
⇤
⇢
|↵k |2 ⇥
kµ
2M
µ⌫
⇤
|Wk |2
M Mµ⌫1 (k) k⌫ +
⌦k
+ O(⌦k 2 ).
(12.40)
To obtain this equation we combined the energy shift from every RG step, see Eq.(12.36),
with the constant term arising from normal-ordering the original Hamiltonian, see Eq.(12.23).
Note that the fully converged energy is obtained from Eq.(12.40) by sending ⇤ ! 0.
The resulting polaron energy is plotted in FIG.12.1. There we found excellent agreement
with recent MC calculations [22], supporting our RG approach. However both the RG and MC
predict large deviations from MF theory, already for values of the coupling constant ↵ ⇡ 1.
The magnitude (of the order of 20%) of the deviations at such relatively small coupling
strength is rather surprising. In FIG.12.3 (b) we plot the polaron energy E0 again, but for
larger values of the coupling constant ↵. For a large value of the UV cut-o↵ ⇤0 = 3000/⇠
we make the bothersome observation that the polaron energy becomes negative, E0 < 0.
This is in contradiction to the fact that the microscopic Hamiltonian (9.1) is positive definite
for repulsive interactions aIB > 0 assumed here, signaling a break-down of the approximate
Fröhlich model for these values of the coupling constant. It will be shown below that these
unexpected results for the polaron energy are related to an improper treatment of the UV
cut-o↵.
In the following subsection we will analytically derive a log-divergence of the polaron
energy within the RG formalism and argue that it is related to zero-point fluctuations of
the impurity. At the end of this section we introduce a regularization scheme for the logdivergence, using the Lippmann-Schwinger equation again to renormalize the BEC-MF shift
gIB n0 .
12.4. POLARON GROUNDSTATE ENERGY
12.4.1
263
Logarithmic UV Divergence of the polaron energy
The cut-o↵ dependence of the energy of a Fröhlich polaron in a BEC in the strong-coupling
limit is a controversial matter. As pointed out in 9.2.5, the strong-coupling approximation is
even incapable of treating consistently the power-law divergence of the polaron energy. For
Feynman’s variational method it was reported [24] that the cut-o↵ dependence is weak when
1
the cut-o↵ is on the order of the inverse van-der-Waals length ⇤0 ⇡ `vdW
. Indeed the authors
1
2
of [24] suggested to use `vdW as a cut-o↵, which is of the order 10
103 ⇠ 1 . However our
results in FIG.12.3 demonstrate that this leads to inconsistent negative polaron energies.
In a recent work by Vlietinck et al. [22] the cut-o↵ dependence was investigated in more
detail, and it was shown that polaron energies predicted by Feynman’s variational method
are UV convergent. We summarize these results in FIG.12.3 (a). The numerical MC results
on the other hand are even more controversial. In [22] it was claimed that the polaron energy
is converged around a cut-o↵ ⇤0 & 3000/⇠. When their data is plotted in a logarithmic scale,
however, no clear convergence can be established, see FIG.12.3 (a). Instead the data suggests
a logarithmic UV divergence, within the errorbars of the MC calculations. In FIG.12.3 (b) we
also compare MC data to our RG prediction Eq.(12.40), which is clearly logarithmic divergent.
While the overall scale of the RG energy is somewhat o↵ from the numerical MC results, the
slope of the curves @⇤0 E0 (⇤0 ) are in excellent agreement with each other. Only for large
⇤0 & 3000/⇠ deviations are observed, which are of the order of the MC errorbars however.
In the remainder of this subsection we show by an explicit calculation that the groundstate
energy is logarithmically divergent. To check whether RG corrections E to the groundstate
energy, see Eq.(12.40), can become UV divergent we perform simple power-counting. To this
end the asymptotic behavior of the renormalized impurity mass M(⇤ ! 1) in the UV regime
will be required, which we now derive.
Asymptotic solution of impurity mass renormalization
First we consider the spherically symmetric case when q = 0, where the RG flow equation
(12.38) for the mass is exactly solvable. In this case because of the symmetry Mµ⌫ = µ⌫ M
and the flow equation reads
Z
@M 1
2
|↵k |2 2 8⇡
|↵⇤ |2 4
= M 2 d2 k
k =
M 2
⇤ .
(12.41)
@⇤
3
⌦k
3
⌦⇤
f
It is a separable di↵erential equation with the solution
8⇡
M(⇤) = M +
3
Z
⇤0
dk
⇤
|↵k |2 4
k .
⌦k
(12.42)
From Eq.(12.42) it is now easy to read o↵ the asymptotic behavior as ⇤0 = 1 and ⇤ ! 1.
Using the asymptotic expressions for Vk and ⌦k ,
⌦k =
k2
1 + O(k
2mred
we arrive at
Mµ⌫1 (⇤)
=
µ⌫
M

1
1
) ,
Vk2 = n0
a2IB
m 2 + O(k
2⇡ red
32
mred
n0 a2IB
⇤
3
M
1
+ O(⇤
2
)
2
),
(12.43)
(12.44)
More generally, a perturbative expansion of Mµ⌫1 in ⇤ 1 in the full RG flow equation (12.38)
shows that the last Eq. (12.44) is correct even for non-vanishing polaron momentum q 6= 0.
264
CHAPTER 12. ALL-COUPLING THEORY OF THE FRÖHLICH POLARON
Derivation of the log-divergence from the RG
By using the asymptotic expression for the renormalized impurity mass Eq.(12.44), valid in
the UV limit, we find that only the zero-point energy of the impurity,
EI0 =
4⇡
Z
⇤0
dk k
0
2 |↵k |
2
2M
k
2
✓
1
M
M(k)
◆
(12.45)
becomes logarithmically
UV divergent (recall that ↵k ⇠ 1/k 2 at high momenta k
⇠ 1 ). The
R 3
second term
d k |Wk |2 /⌦k in Eq.(12.40), on the other hand, is UV convergent because
1
Wk ⇠ k . From Eqs.(12.44) and (12.45) we derive the following exact expression for the UV
divergence,
128 mred 2 4
EUV =
n a log (⇤0 ⇠) .
(12.46)
3 M 2 0 IB
We find that the slope predicted by this curve is in excellent agreement with the MC data
shown in FIG.12.3 (a). We also point out the negative sign of the UV divergence, i.e. EUV !
1 as ⇤0 ! 1. Again, this is in contradiction to the fact that the microscopic Hamiltonian
(9.1) is positive definite for aIB > 0, indicating that additional terms besides the approximate
Fröhlich Hamiltonian have to be taken into account in the strong-coupling regime.
We note that similar log-divergencies are known to appear e.g. in the Casimir e↵ect in
quantum electrodynamics, or in relativistic polaron models [388]. In the high-energy context
divergencies can be resolved by renormalizing fundamental quantities like the electron mass
or charge entering the microscopic model, which can not directly be measured. In the context
of ultra cold quantum gases, however, divergencies may not be absorbed into fundamental
coupling constants, but correct energies should be derived from the available microscopic
models. In the following we present a proper regularization scheme from which we will derive
consistent UV convergent polaron energies for impurities immersed in a BEC.
12.4.2
Regularization of the Polaron Energy
To regularize the log-divergence we need to return to the BEC-MF energy shift gIB n0 again.
Here the interaction strength gIB is related to the scattering length aIB via the LippmannSchwinger equation (LSE), see Sec.9.2.1. Only the scattering length is a universal quantity
characterizing low-energy impurity-boson scattering, which is independent of the many-body
physics captured by the Fröhlich Hamiltonian. As discussed in Sec.9.2.4, to regularize the MF
polaron energy the LSE has to be calculated consistently up to second order in the BEC-MF
shift. By the RG we predict a logarithmic UV divergence proportional to ↵2 , see Eq.(12.46).
Thus consistency requires to go to even higher order in the LSE to calculate the BEC-MF
shift.
Instead of just going to higher and higher orders in the LSE, we make the following
observation. The BEC-MF shift gIB n0 describes the interaction of the impurity with the
surrounding atoms in the condensate. As demonstrated by our RG protocol, the impurity
becomes renormalized by its interactions with the Bogoliubov phonons in the BEC, specifically
its mass M increases. Thus to calculate the BEC-MF shift gIB n0 using the second-order LSE
we have to take into account this mass-enhancement. The impurity mass enters through
the impurity-propagator Ĝ0 (!), see Eq.(9.12), which we will now replace by the dressed
propagator,
Z
|kihk|
1
1
1
Ĝ⇤ (!) = d3 k
,
=
+
.
(12.47)
⇤
k2
mred (k)
M(k) mB
!
+ i✏
⇤
2mred (k)
12.4. POLARON GROUNDSTATE ENERGY
265
Since the renormalization of the mass M depends on aIB , see Eq.(12.42), this also increases
the accuracy used to evaluate the LSE. Here, for simplicity, we restrict our analysis to the
spherically symmetric case q = 0 when M(⇤) is just a scalar. The generalization to q 6= 0 is
straightforward, however, and one easily checks that E0 (q) E0 (0) is UV convergent.
Let us give another interpretation for our scheme of using the dressed propagator (12.47)
in the LSE for calculating the BEC-MF shift. To this end we recall our treatment of the Bose
polaron problem. Starting from the microscopic Hamiltonian Ĥ, Eq.(9.1), we derived the
Fröhlich Hamiltonian ĤF Eq.(9.4) by neglecting two-phonon terms Ĥ2ph , see FIG.9.2 (c,d).
However to calculate the relation between gIB and the scattering length aIB , we essentially
treated the two-phonon terms on a two-particle level in the first-order LSE (at high energies
the two-phonon terms are simple two-particle scattering terms underlying the LSE). Using
this first-order result we can then calculate corrections to the impurity properties (like its
renormalized mass M) from the Fröhlich terms in the Hamiltonian. Next we treat the twophonon terms at a higher order in aIB by using the second-order LSE with the renormalized
impurity mass, but still on a two-particle level. This yields a refined BEC-MF shift, and we
will show below that it cancels the log-divergence. Moreover this interpretation clarifies that,
in order to derive the corrections to the BEC-MF shift, we have to take into account the twophonon terms in the full Hamiltonian. This is a necessary requirement to resolve the problem
of negative energies showing up e.g. in FIG.12.3 (b). There we found that the groundstate
energy of the Fröhlich Hamiltonian hgs|ĤF |gsi < 0 becomes negative, which is not a problem
as long as the groundstate of the full Hamiltonian hgs|Ĥ|gsi = hgs|ĤF + Ĥ2ph |gsi 0 remains
positive.
Now we will employ the dressed propagator in the second-order LSE T̂ = T̂0 + T̂0 Ĝ⇤ T̂0 to
derive the corrected BEC-MF shift. The LSE reads
Z ⇤0
2
2gIB
gIB
aIB =
mred
mred
dk k 2 G⇤ (k),
(12.48)
2⇡
(2⇡)3
0
which can be inverted to solve for gIB . Using that m⇤red /mred
batively in aIB the following result 2 ,
4a2
4a2
2⇡
gIB (aIB ) =
aIB + IB ⇤0 + IB
mred
mred
mred
Z
⇤0
dk
✓
m⇤red (k)
mred
1 = O(a2IB ) we obtain pertur◆
1 + O(a5IB ).
(12.49)
The second term on the right hand side regularizes the power-law divergence of the MF
polaron energy, as described in Sec.9.2.1. The third term on the right hand side is new and
regularizes the log-divergence. To see this we employ the asymptotic solution of the RG flow
equation for the impurity mass M, Eq.(12.44). Plugging it into the last equation (12.49)
yields a logarithmically divergent contribution to the BEC MF shift,
n0 gIB
UV
=+
128 mred 2 4
n a log (⇤0 ⇠) =
3 M 2 0 IB
EUV ,
(12.50)
which exactly cancels the log-divergence from the Fröhlich Hamiltonian, cf. (12.46). Hence
the final expression for the impurity energy is now free of all UV divergencies.
The fully regularized ground state energy E0 is shown as a function of the coupling constant
↵ in FIG.12.4. Above ↵ & 1 we observe sizable deviations from MF theory, which are much
2
8a3
We find an additional contribution ⇡mIB
⇤20 = O(⇤20 a3IB ) to gIB , which also appears in the MF treatment of
red
the Fröhlich polaron. Its regularization requires a consistent treatment of the third-order LSE however, which
generates terms of order O(⇤20 a3IB ).
266
CHAPTER 12. ALL-COUPLING THEORY OF THE FRÖHLICH POLARON
60
50
40
30
20
10
0
0
2
4
6
8
10
Figure 12.4: The fully regularized RG ground state energy E0 is shown as a function of the
coupling strength ↵, and compared to the fully regularized MF prediction. Parameters were
n0 = 1 ⇥ ⇠ 3 , M/mB = 0.26316, q = 0 and ⇤0 = 2000/⇠.
less dramatic however than may be expected from FIG.12.1 without regularizing the logdivergence. Note that the polaron energy predicted by the fully regularized RG is above
the MF prediction. This result might be surprising at first glance, because MF polaron
theory relies on a variational principle and thus yields an upper bound for the groundstate
energy. However, this bound holds only for the binding energy, defined as the groundstate
energy of the Fröhlich Hamiltonian, and not for the entire impurity Hamiltonian including
the condensate-impurity interaction gIB n0 .
12.5
Other Groundstate Polaron Properties – Derivation
Now we turn to the question how polaron groundstate properties – other than its energy – can
be calculated from the RG protocol. In particular we discuss the e↵ective polaron mass Mp
(12.5.1), the phonon number Nph in the polaron cloud (12.5.2) and the quasiparticle weight
Z (12.5.3). In this section the RG-flow equations for these observables are derived and our
results will be discussed in the following section 12.6. All polaron observables in this section
are UV convergent, and not regularization is required.
To calculate polaron properties we find it convenient to introduce the following notations
for the groundstate in di↵erent bases used in this thesis. Our notations are summarized in
FIG.12.2. When | q i denotes the groundstate of the Fröhlich Hamiltonian Eq.(9.4) with total
momentum q, the corresponding groundstate of the Hamiltonian (9.50) in the polaron frame
†
reads |qi ⌦ | q i = ÛLLP
| q i. Analogously, the groundstate of the Hamiltonian (12.3) in the
†
frame of quantum fluctuations reads |qi ⌦ |gsq i = |qi ⌦ ÛMF
| q i. To keep our notation simple,
we always assume a fixed value of q and introduce the short-hand notation |gsi ⌘ |gsq i. In
the course of the RG, this groundstate factorizes in di↵erent momentum shells in every single
RG step. After the application of the RG unitary transformation Û⇤ , the groundstate in the
new frame reads |gs0 i := Û⇤† |gsi = |0if ⌦ |gsis and factorizes.
12.5.1
Polaron Mass
First we turn our attention to the polaron mass Mp . As we pointed out in section 9.2.4 it can
be determined from the total phonon momentum qph , see Eq.(9.61),
M
=1
Mp
qph
.
q
(12.51)
12.5. OTHER GROUNDSTATE POLARON PROPERTIES – DERIVATION
267
MF defined by Eq.(9.56). In the following we will
In MF approximation we used qph = Pph
include the e↵ect of quantum fluctuations to derive corrections to the polaron mass. To this
end we will calculate corrections to the phonon momentum first, which is defined as
qph =
Z
⇤0
†
MF
d3 k kx hgs|ÛMF
â†k âk ÛMF |gsi = Pph
+
Z
⇤0
d3 k kx hgs| ˆ k |gsi.
(12.52)
Here |gsi denotes the groundstate in the polaron frame and after the introduction of quantum
fluctuations around MF polaron, i.e. after application of ÛMF Eq.(12.1).
As shown in the Appendix Q, the phonon momentum qph , including corrections from the
RG, reads
qph = Pph (⇤ ! 0),
(12.53)
where Pph (⇤ ! 0) = lim⇤!0 Pph (⇤) denotes the fully converged RG coupling constant. The
last equation justifies the interpretation of Pph as the phonon momentum in the polaron cloud.
12.5.2
Phonon Number
Next we discuss the phonon number Nph in the polaron cloud. In the basis of Bogoliubov
phonons,
before applying the MF shift (12.1), the phonon number operator reads N̂ph =
R 3
†
d k âk âk . After application of the MF shift Eq.(12.1) we obtain
Nph =
MF
Nph
+
where the MF result reads
MF
Nph
=
Z
Z
⇤0
⇤0
d3 k hgs| ˆ k |gsi,
d3 k |↵k |2 .
(12.54)
(12.55)
The second term on the right hand side of Eq.(12.54) can be evaluated by applying an RG
rotation Û⇤ , and the calculation in Appendix Q leads to the following RG flow equation for
the phonon number,

Z
@Nph
↵k
k
2
=2 d k
Wk + ↵ k
· (Pph (0) Pph ) + O(⌦k 2 ).
(12.56)
@⇤
⌦k
M
f
MF . Note that in Eq.(12.56)
It should be supplemented with the initial condition Nph (⇤0 ) = Nph
the fully converged coupling constant Pph (0) = Pph (⇤ = 0) at cut-o↵ ⇤ = 0 appears and
Pph ⌘ Pph (⇤) should be evaluated at the current RG cut-o↵ ⇤.
12.5.3
Quasiparticle weight
The last observable we discuss here is the polaron quasiparticle weight Z, which is a key
property characterizing the polarons spectral function I(!, q), see Chap.10. It is defined by
the overlap of the polaron to the bare impurity, Z = |h q |0i|2 where | q i is the phonon
groundstate in the polaron frame (see FIG.12.2) and |0i denotes the phonon vacuum in this
frame. After applying also the MF shift Eq.(12.1) the quasiparticle weight reads
Y
Z = |hgs|
| ↵k i|2 ,
(12.57)
k
where we used that
†
ÛMF
|0i =
Y
k
|
↵k i.
(12.58)
268
CHAPTER 12. ALL-COUPLING THEORY OF THE FRÖHLICH POLARON
Q
Moreover, k includes all momenta 0 < |k| < ⇤0 in these expressions.
A characteristic feature of MF theory is that the polaron quasiparticle weight Z MF is
directly related to its phonon number,
Z MF = e
MF
Nph
,
(12.59)
see also Sec.11.3.2. This is a direct consequence of the Poissonian phonon statistics assumed
in the MF wavefunction, and indeed we find that – in general – it is no longer true for the
groundstate determined by the RG.
By introducing unities of the form 1̂ = Û⇤ Û⇤† into Eq.(12.57) for subsequent momentum
shells ⇤ > ⇤0 = ⇤
⇤ > ... we can formulate an RG for the quasiparticle weight. Our
calculation is presented in Appendix Q and it leads to the following RG flow equation,
@ log Z
=
@⇤
Z
2
d k ↵k
f

1
Wk
⌦k
↵k kµ Mµ⌫1
Z
2
s
d3 p p⌫ |↵p |2
+ O(⌦k 2 ).
(12.60)
Comparison of this expression with the RG flow of the phonon momentum Eq.(12.56) yields
✓Z
◆
Z
Mk
@
|↵k |2
1
2
3
2
(log Z + Nph ) = 2 d k
kx M k
d p px |↵p | + (Pph (0) Pph )
+ O(⌦k 2 ),
@⇤
⌦k
M
f
s
(12.61)
where we assumed q = qex points along x. Thus for q 6= 0 we find Z < e Nph , i.e. the phonon
correlations taken into account by the RG lead to a further reduction of the quasiparticle
weight, even beyond an increase of the phonon number.
12.6
Other Groundstate Polaron Properties – Results
Having obtained formulas in the last sections allowing us to calculate various polaron properties using the RG formalism, we will now discuss our results. To this end we compare
predictions from the RG analysis to results from MF polaron theory and to data from alternative polaron theories which we take from the literature. We start by presenting typical
solutions of the RG flow equations (in 12.6.1). Then we discuss the polaron mass (in 12.6.2),
where we find a smooth cross-over from weak- to strong-coupling regime. We also compare
our results to Feynman path integral predictions by Tempere et al. [24] at a small mass ratio
M/mB = 0.26, i.e. deep in the strong-coupling regime. Our results deviate markedly from
previous predictions, both qualitatively and quantitatively. We proceed by calculating the
phonon number in the polaron cloud (in 12.6.3) which again indicates a smooth cross-over
from weak- to strong coupling regime. Then we discuss the quasiparticle weight (in 12.6.4)
and its relation to the phonon number.
12.6.1
Solutions of RG flow equations
Now we will investigate solutions of the RG flow equations (12.38), (12.39). We find that
both the inverse mass Mµ⌫1 and momentum Pph are determined mostly by phonons from
the intermediate energy region k & 1/⇠. For smaller momenta the RG flow stops completely
and we obtain universal results for the coupling constants in this regime, in accordance with
expectations from our dimensional analysis, see Sec.12.3.1.
In FIG.12.5 (a) and (b) a typical RG flow of Mµµ1 and Pph (⇤) is calculated numerically
for di↵erent values of the coupling constant ↵. In both cases we observe that the coupling
constants only flow substantially in the intermediate regime, where ⇤ ⇡ 1/⇠. For smaller
12.6. OTHER GROUNDSTATE POLARON PROPERTIES – RESULTS
(a) 1
(b) 0.25
µ=y
µ=x
0.8
=1
=3
=6
=12
=18
0.2
0.6
0.15
=1
=3
=6
=12
=18
0.4
0.2
0
10
269
2
10
1
0
10
0.1
0.05
0
10
1
10
2
10
1
0
1
10
10
Figure 12.5: Typical RG flows of the (inverse) renormalized impurity mass M 1 (a) and the
MF along the direction of the system momentum q (b).
excess phonon momentum Pph Pph
Results are shown for di↵erent coupling strengths ↵ and we used parameters M/mB = 0.3,
q/M c = 0.5 and ⇤0 = 20/⇠ in d = 3 dimensions.
momenta ⇤ < 1/⇠, as we discussed in 12.3.1, all terms in the fluctuation Hamiltonian become
irrelevant (or marginal) which manifests itself in well-converged couplings as ⇤ ! 0. By comparing di↵erent ↵, as expected, we observe that the corrections of the renormalized impurity
mass M become larger for increasing ↵. Interestingly we observe a non-monotonic behavior
for the phonon momentum, which takes a maximum value between ↵ = 6 and ↵ = 12 in this
particular case.
In FIG.12.6 the renormalized phonon dispersion relation ⌦k is shown as a function of the
RG cut-o↵ k = ⇤. Around ⇤ ⇡ 1/⇠ we observe large deviations from the bare dispersion
!k + k 2 /2M . We find that the regime of linear dispersion is extended for large couplings ↵
as compared to the non-interacting case. The slope of the linear regime is given by the speed
of sound c and does not change.
12.6.2
Polaron Mass
In FIG.12.7 we show the polaron mass calculated using several di↵erent approaches. In the
weak-coupling limit ↵ ! 0 the polaron mass can be calculated perturbatively in ↵, and the
lowest-order result is shown in the figure. Around ↵ ⇡ 3 the perturbative result diverges and
(a)
(b)
8
2
10
6
4
0
10
2
0
0
0.5
1
1.5
2
10
2
10
2
10
1
0
10
1
10
Figure 12.6: The renormalized phonon dispersion relation ⌦⇤ is compared to the bare dispersion !⇤ + ⇤2 /2M for di↵erent coupling strengths ↵ in linear (a) and double-logarithmic
scale (b). Because the total momentum q = 0 vanishes there is no angle-dependence of the
dispersion. Parameters were M/mB = 0.26 and ⇤0 = 200/⇠.
270
CHAPTER 12. ALL-COUPLING THEORY OF THE FRÖHLICH POLARON
perturbation theory is no longer valid. We observe that in the limit ↵ ! 0 all approaches follow
the same line which asymptotically approaches the perturbative result. The only exception is
the strong-coupling Landau-Pekar approach, which only yields a self-trapped polaron solution
above a critical value of ↵, see Sec.9.2.5.
For larger values of ↵, MF theory sets a lower bound for the polaron mass. Naively
this would be expected, because MF theory does not account for quantum fluctuations due
to couplings between phonons of di↵erent momenta. These fluctuations require additional
correlations to be present in beyond-MF wavefunctions and should lead to an increased polaron
mass. Indeed, for intermediate couplings ↵ & 1 the RG approach predicts a polaron mass
MpRG > MpMF which is considerably di↵erent from the MF result.
In FIG.12.7 we present another interesting aspect of our analysis, related to the nature
of the cross-over [334, 20] from weak- to strong-coupling polaron regime. While Feynman’s
variational approach predicts a rather sharp transition, the RG results show no sign of any
discontinuity. Instead they suggest a smooth cross-over from one into the other regime, as
expected on general grounds [334, 20]. It is possible that the sharp crossover obtained using
Feynman’s variational approach is an artifact of the limited number of parameters used in the
variational action.
In FIG.12.7 we calculated the polaron mass in the strongly coupled regime for rather large
↵ while the mass ratio M/mB = 0.26 is very small. It is also instructive to see how the system
approaches the integrable limit M ! 1 when the problem becomes exactly solvable [P5], see
Sec.9.2.4. FIG.12.8 (a) shows the (inverse) polaron mass as a function of ↵ for di↵erent mass
ratios M/mB . For M
mB , as expected, the corrections from the RG are negligible and
MF theory is accurate. When the mass ratio M/mB approaches unity, we observe deviations
from the MF behavior for couplings above a critical value of ↵ which depends on the mass
ratio. Remarkably, for very large values of ↵ the mass predicted by the RG follows the same
power-law as the MF solution, albeit with a di↵erent prefactor. This can be seen more clearly
in FIG.12.8 (b), where the case M/mB = 1 is presented. This behavior can be explained
from strong-coupling theory. As shown in [319] the polaron mass in this regime is predicted
to be proportional to ↵, as is the case for the MF solution. However prefactors entering the
20
(a)
15
10
10
10
5
10
2
1
Casteels
0
(b)
0
0
0.5
1
2
3
4
5
1
2
4
7
6
Figure 12.7: The polaron mass Mp (in units of M ) is shown as a function of the coupling
strength ↵. We compare the RG method to MF, strong-coupling theory [319], correlated
Gaussian wavefunctions (CGWs) and Feynman’s variational path-integral approach. We are
grateful to Wim Casteels for providing his results of Feynman path-integral calculations [318].
We used parameters M/mB = 0.26, ⇤0 = 200/⇠ and set q/M c = 0.01. In (b) the same data
is shown as in (a) but in a double-logarithmic scale.
12.6. OTHER GROUNDSTATE POLARON PROPERTIES – RESULTS
(a) 100
271
(b)
2
10
−1
10
0
10
−2
10
−2
−1
10
0
10
1
10
2
10
10
−1
10
0
10
1
10
2
10
3
10
Figure 12.8: (a) The inverse polaron mass M/Mp is shown as a function of the coupling
strength ↵, for various mass ratios M/mB . We compare MF (dashed) to RG (solid) results.
The parameters are ⇤0 = 2000/⇠ and we set q/M c = 0.01 in the calculations. (b) The
polaron mass Mp /M 1 is shown as a function of the coupling strength for an impurity
of mass M = mB equal to the boson mass. We compare MF, perturbation and strongcoupling theories to the RG and to Feynman path-integral results by Wim Casteels [318],
which interpolates between the asymptotic results. We used parameters ⇤0 = 200/⇠ and
q/M c = 0.01.
weak-coupling MF and the strong coupling masses are di↵erent.
To make this more precise, we compare the MF, RG, strong-coupling and Feynman polaron
masses for M/mB = 1 in FIG.12.8 (b). We observe that the RG smoothly interpolates between
the strong- and the weak-coupling MF regime. While the MF solution is asymptotically
recovered for small ↵ ! 0 (by construction), this is not strictly true on the strong-coupling
side. Nevertheless, the observed value of the RG polaron mass in FIG.12.8 (b) at large ↵
is closer to the strong-coupling result than to the MF theory. In FIG.12.9 we investigate
the relation between RG and weak- and strong-coupling results more closely. As expected,
the MF result is accurate for large mass ratios or small ↵, and large deviations are observed
otherwise, see FIG.12.9 (a). The strong-coupling result, on the other hand, is not as well
reproduced by the RG for large ↵, but deviations are much smaller than for the MF theory
in this regime, see 12.9 (b).
Now we return to the discussion of the polaron mass for systems with a small mass ratio
M/mB < 1. In this case FIG.12.8 (a) suggests that there exists a large regime of intermediate
coupling, where neither strong-coupling approximation nor MF can describe the qualitative
behavior of the polaron mass. This is demonstrated in FIG.12.7, where the RG predicts values
for the polaron mass midway between MF and strong-coupling, for a wide range of couplings.
We find that in this intermediate regime to a good approximation the polaron mass increases
exponentially with ↵, over more than a decade. In this intermediate-coupling regime, the
impurity is constantly scattered between phonons, leading to strong correlations between
them. Here it acts as an exchange-particle mediating interactions between phonons. These
processes change the behavior of the polaron completely, until the impurity mass becomes so
strongly modified by phonons that a MF-like behavior of the renormalized impurity is restored
in the strong-coupling regime.
We conclude that measurements of the polaron mass rather than the binding energy should
be a good way to discriminate between di↵erent theories describing the Fröhlich polaron at
intermediate couplings. Quantum fluctuations manifest themselves in a large increase of the
e↵ective mass of polarons, in strong contrast to the predictions of the MF approach based on
the wavefunction with uncorrelated phonons. Experimentally both the quantitative value of
272
CHAPTER 12. ALL-COUPLING THEORY OF THE FRÖHLICH POLARON
(a)
(b)
8
1.5
6
4
2
2
1
2
0
4
10
1
1
3
4
2
10
0
10
5
3
0.5
4
4
10
2
10
5
Figure 12.9: Transition from weak- to strong-coupling regime: (a) The ratio of the RG e↵ective
polaron mass MpRG to the weak-coupling MF prediction MpMF is shown as a function of the
mass ratio M/mB and the coupling strength ↵. (b) The ratio of the RG e↵ective polaron
mass MpRG to the strong-coupling Landau-Pekar prediction MpSC [319] is shown as a function
of the mass ratio M/mB and the coupling strength ↵. We used parameters ⇤0 = 200/⇠ and
q/M c = 0.01. Note that the strong-coupling solution exists only for sufficiently large values
of ↵.
the polaron mass, as well as its qualitative dependence on the coupling strength can provide
tests of our theory. Experimentally the mass of the Fermi polaron has successfully been
measured using collective oscillations of an atomic cloud [386], and similar experiments should
be possible with Bose polarons in the near future.
12.6.3
Phonon Number
In FIG.12.10 (b) we plot the phonon number in the polaron cloud for one specific example.
We observe that for ↵ . 1 RG and MF are in good agreement with each other. For couplings
↵ > 1 the RG predicts more phonons than MF theory, as expected from the presence of
quantum fluctuations leading to additional dressing. The qualitative behavior of the phonon
number, however, does not change for larger couplings. For very large ↵ we find the same
power-law as predicted by MF theory, but with a di↵erent prefactor. This is another indicator
of the smooth transition from weak- to strong coupling regime.
To make this smooth cross-over even more apparent, we calculated the ratio of the RG
phonon number to the MF prediction in FIG.12.10. The ratio starts to grow around ↵ = 1
and eventually it saturates at very larger values of ↵ ⇡ 102 . The ratio of RG phonon number
to MF theory at the largest couplings ↵ increases with decreasing mass ratio. In the integrable
limit M ! 1 no deviations can be observed at all.
12.6.4
Quasiparticle weight
In FIG.12.11 (b) we show how the quasiparticle weight Z depends on the coupling strength
↵ for two di↵erent mass ratios. By plotting the data from FIG.12.11 (b) on a logarithmic
scale, we found that while MF yields an exponential decay of Z, the RG predicts faster than
exponential decay. In FIG.12.11 (b) we also compare Z to the MF-type expression e Nph .
Note that we calculate Nph using the RG in this case. For the smaller mass ratio M/mB = 0.26
we observe slight deviations of Z from this expression, indicating that the RG polaron includes
correlations between phonons going beyond the poissonian statistics of the MF state.
12.6. OTHER GROUNDSTATE POLARON PROPERTIES – RESULTS
(a)
273
(b)
2
M/mB=0.3
1.8
2
10
M/mB=1.5
MF
RG
M/m =5
B
1.6
M/m =20
B
1.4
0
10
1.2
1
0
1
10
2
10
10
10
1
0
1
10
10
2
3
10
10
RG in the polaron cloud and the
Figure 12.10: (a) The ratio between RG phonon number Nph
MF
MF result Nph is shown as a function of the coupling constant ↵ for various mass ratios.
Parameters are q = 0.01M c and ⇤0 = 2000/⇠. (b) The phonon number is plotted as a function
of the coupling ↵ on a double-logarithmic scale, using RG and MF theory. Parameters were
M/mB = 1, q = 0.01M c and ⇤0 = 2000/⇠.
In FIG.12.11 (a) it is also shown how the quasiparticle weight depends on the polaron
momentum q. For q < M c the polaron is subsonic for all couplings ↵ and the quasiparticle
weight Z decays as a function of ↵. For q M c and for sufficiently small couplings ↵ we find
a subsonic polaron where Z = 0. As shown in Sec.9.2.4 for ↵ = 0 the quasiparticle weight Z
jumps discontinuously from Z = 1 at q < M c to Z = 0 for q > M c. FIG.12.11 (a) indicates
that while Z(q) decreases on the subsonic side when approaching the supersonic polaron, it
still jumps discontinuously at the critical polaron momentum qc . For large ↵, however, the
polaron quasiparticle weight Z is exponentially suppressed and it is hard to distinguish a
smooth cross-over from a smooth transition. For large q > M c we find that the function
Z(↵) takes a maximum value at finite ↵ > 0. It is suppressed at smaller couplings due to the
proximity to the supersonic polaron, and at larger couplings due to the additional interactions.
(a)
(b)
1
0.8
1
0.6
0.5
0.4
0
0
1
2
3
4
20
15
10
5
0
0.2
0
0
5
10
15
20
Figure 12.11: (a) The quasiparticle weight Z of the polaron peak (calculated from RG) is
shown as a function of the polaron momentum q and the coupling strength ↵. For q
Mc
and sufficiently small ↵ the polaron becomes supersonic and the quasiparticle weight Z =
0 vanishes (red shaded area). Parameters are M/mB = 1.53 and ⇤0 = 200/⇠. (b) The
RG ⇣quasiparticle
weight Z is compared to MF predictions and to the MF-type expression
⌘
RG
exp
Nph for di↵erent mass ratios. Parameters are q = 0.75M c and ⇤0 = 2000/⇠.
274
CHAPTER 12. ALL-COUPLING THEORY OF THE FRÖHLICH POLARON
Chapter 13
Dynamical RG for
Intermediate-coupling Fröhlich
Polarons
In this chapter we return to the investigation of non-equilibrium polaron problems. We will
now extend our discussion from Chapters 10 and 11 and go beyond MF theory by adapting
the RG approach developed in the last chapter to dynamical problems. To this end we will
keep track of the dynamics of quantum fluctuations, which we integrate out shell-by-shell as in
the equilibrium RG approach. The mathematical formulation of the dynamical RG (dRG) in
terms of infinitesimal unitary transformations makes this generalization possible, essentially
without modifying the structure of the resulting flow equations. In contrast to the equilibrium
case, we can obtain dRG flow equations with an explicit time-dependence.
Dynamical problems in quantum many-body theory are still rather poorly understood.
This is partly due to the lack of efficient numerical methods for solving such problems in
more than one spatial dimension. Therefore the generalization of our RG approach to nonequilibrium situations constitutes an important step towards a deeper understanding of dynamical problems in general. Although our dRG approach will be formulated for a Fröhlich
Hamiltonian with arbitrary couplings, we apply it to impurities in ultra cold quantum gases
in this section. Because of the long coherence times in such systems, they are ideally suited
to investigate non-equilibrium dynamics experimentally.
We apply the dRG developed in this Chapter to revisit the non-equilibrium problems studied in Chapter 10. First, we calculate the spectral function of the polaron beyond MF, which
can be measured using RF spectroscopy. We find that MF theory yields surprisingly good results, on a qualitative level, even in the case of very strong couplings. In this regime the shape
of the RF spectra deviates substantially from weak-coupling results, with a large amount of
the spectral weight shifted to high frequencies. Second, we investigate the dynamics of polaron
formation when an impurity atom with a well-defined momentum is introduced in the BEC
and interactions are suddenly switched on. In the strong-coupling regime we predict interesting impurity trajectories, where the impurity bounces o↵ the suddenly created polarization
cloud. This prediction deviates substantially from time-dependent MF calculations.
This Chapter is organized as follows. First, in 13.1, we formulate the dRG and derive
the flow equations. A summary of our method, where technical calculations are skipped, is
provided in Appendix S. In 13.2, we present our results for the spectral function of the polaron
beyond MF theory. Finally in 13.3 we present results for the non-equilibrium dynamics of the
impurity momentum and discuss experimental signatures of strong-coupling physics.
275
276
CHAPTER 13. DYNAMICAL RG FOR INTERMEDIATE-COUPLING FRÖHLICH
POLARONS
13.1
Formulation of the dynamical RG
Now we develop an RG protocol for calculating time-dependent observables, including the
phonon momentum and phonon number (see13.1.1) and the time-dependent overlap Aq (t)
defined in Eq.(10.7) (see 13.1.2). The time-dependent overlap is not a conventional observable
and requires a generalization of our (d)RG approach to non-hermitian Hamiltonians. In this
section we will focus on the particular case when the initial wavefunction corresponds to
the complete phonon vacuum |0i, which simplifies the action of the infinitesimal unitary
transformations Û⇤ on the initial state. While this is useful for obtaining results more easily,
our scheme is not limited to this particular case and can in principle be applied to arbitrary
initial states.
This section is of a more technical nature and can be skipped by readers who are interested
only in our results. The most important technical steps are summarized in Appendix S.
The resulting dRG flow equations can be found in Eqs.(13.30), (13.31) and (13.33) for timedependent observables, and in Eqs.(13.65) and (13.73) for the time-dependent overlap.
13.1.1
Phonon number and momentum
Our goal in this section is to calculate the time-dependence of general observables Ô in the
polaron problem. For concreteness we restrict ourselves to observables which do not involve
correlations between di↵erent phonon momenta and can hence be written as
Ô =
Z
⇤0
d3 k Ôk .
(13.1)
As a first step, we formulate the problem in the frame of quantum fluctuations around the
MF polaron, i.e. we apply the unitary transformation ÛMF (see FIG.12.2) and obtain
O(t) =
=
Z
Z
⇤0
⇤0
†
†
d3 k h0|ÛMF eiÛMF Ĥq ÛMF t ÛMF
Ôk ÛMF e
d3 k h ↵ |eiH̃q t ôk e
iH̃q t
|
†
iÛMF
Ĥq ÛMF t
†
ÛMF
|0i
↵ i.
(13.2)
Here H̃q denotes the Hamiltonian from Eq.(12.5) describing dynamics of quantum fluctuations
†
around the MF polaron, and the observable in the MF frame reads ôk = ÛMF
Ôk ÛMF . In the
†
second line we also used Eq.(12.58) to express ÛMF
|0i = | ↵ i, where | ↵ i is short-hand
Q
for  | ↵ i.
dRG step
Now we will formally derive the dRG flow equations for the time-dependent observable ok (t) =
h ˆ k (t)i, from which both the phonon momentum and the phonon number can be derived. Our
starting point is Eq.(13.2) after the application of the MF unitary transformation. We start
every dRG step by performing the unitary transformation Û⇤ familiar from the equilibrium
RG approach, see Eq.(12.27),
h ˆ k (t)i =
Z
⇤
†
d3 k h ↵ |Û⇤ eiÛ⇤ H̃q Û⇤ t Û⇤† ˆ k Û⇤ e
†
iÛ⇤
H̃q Û⇤ t
Û⇤† |
↵ i.
(13.3)
13.1. FORMULATION OF THE DYNAMICAL RG
277
First of all, this leads to a simplification of the Hamiltonian H̃q describing dynamics,
Z
⇣
⌘
ˆ s (k) ↠âk + Ĥs + Ĥs ,
Û⇤† H̃q Û⇤ = d3 k ⌦k + ⌦
(13.4)
k
f
see Eq.(12.34). Note that the fast-phonon frequency ⌦k is flowing in the RG protocol. From
the expression (13.4), it was easy to read o↵ the fast-phonon ground state which is simply the
vacuum state, âk |0if = 0. However, since we have the full Hamiltonian at hand, we can as
well make an attempt to describe phonon dynamics within this formalism.
Secondly, as for the calculation of the quasiparticle weight Z, we make use of the fact that
the MF coherent states | ↵p i are eigenstates of ˆ p ,
†
↵p i = ˆ p ÛMF
|0is
⇣
†
= ÛMF â†p âp
ˆ p|
=
|↵p |2 |
(13.5)
⌘
|↵p |2 |0is
↵p i,
(13.6)
(13.7)
as derived already in Eq.(Q.18). Hence Û⇤† | ↵k i| ↵p i = | k i| ↵p i yields a coherent state
with amplitude k = fk ↵k , where

Z
1
1
fk =
Wk ↵k kµ Mµ⌫ d3 p p⌫ |↵p |2 .
(13.8)
⌦k
s
Here k and p denote fast and slow phonon momenta respectively. Wk was defined in Eq.(12.7).
We note that Eq.(13.8) showed up previously in the calculation of the quasiparticle weight Z,
see Appendix Q.3.
Next we separate Eq.(13.3) into contributions from fast and slow phonons,
h ˆ k (t)i = h ˆ k (t)if + h ˆ k (t)is .
(13.9)
For the slow-phonon part we obtain
h ˆ k (t)is =
Z
3
s
d ph
k (t)|h
↵p |e
⇣
⌘
R
ˆ s (k) t
i Ĥs + Ĥs + f d3 k â†k âk ⌦
⇥e
⇣
⌘
R
ˆ s (k) t
i Ĥs + Ĥs + f d3 k â†k âk ⌦
ˆ p⇥
|
↵p i|
k (t)i
+ O(⌦k 2 ) (13.10)
using Eq.(12.31) together with [F̂k , ˆ p ] = O(⌦k 2 ). To evaluate this expression, we will
R
ˆ s (k) / ⇤ in the exponow treat the fast-phonon frequency renormalization f d3 k â†k âk ⌦
nent perturbatively. This is achieved e.g. by using a formal Trotter decomposition of the
time-evolution, and we obtain the leading-order contribution
e
i(Ĥs + Ĥs )t
i
Z
t
d⌧ e
i(Ĥs + Ĥs )(t ⌧ )
0
Z
f
ˆ s (k)e i(Ĥs +
d3 k â†k âk ⌦
Ĥs )⌧
.
(13.11)
This expression appears in the scalar product h k (t)| · | k (t)i, which allows us to eliminate
the fast-phonon operators â†k âk by making them C-numbers:
h
†
k (t)|âk âk | k (t)i
=|
k|
2
.
(13.12)
278
CHAPTER 13. DYNAMICAL RG FOR INTERMEDIATE-COUPLING FRÖHLICH
POLARONS
In this expression the full dynamics of the fast-phonon coherent state, given by
e
i
R
f
d3 k ⌦k â†k âk t
Y

|
i
=
Y

|
e
i⌦ t
i⌘
Y

|
 (t)i,
(13.13)
was taken into account. This time-dependence has no e↵ect, however, because the coherent
state amplitude is conserved, | k (t)|2 = | k |2 = const. Further note that quartic terms
â†k âk â†k0 âk0 appear only in second order O( ⇤2 ) for the fast phonons and we can discard
them.
Using equation (13.12) and re-exponentiating the result, we obtain the slow-phonon contribution
Z
R 3
R 3
2ˆ
2ˆ
ˆ
h k (t)is = d3 p h ↵p |ei(Ĥs + Ĥs + f d k | k | ⌦s (k))t ˆ p e i(Ĥs + Ĥs + f d k | k | ⌦s (k))t | ↵p i.
s
(13.14)
Now we turn to the fast-phonon contribution,
Z
h ˆ k (t)if = d3 k h k (t)|h ↵p |ei(Ĥs + Ĥs )t Û⇤† ˆ k Û⇤ e i(Ĥs +
Ĥs )t
s
|
↵p i|
k (t)i
+ O(⌦k 2 ),
(13.15)
which is of order O( ⇤). This allowed us to neglect the fast-phonon frequency renormalization
in the exponents, which only yields corrections of order O( ⇤2 ). In this case we have to
transform ˆ k , which yields
⇣
⌘
⇣
⌘
⇣
⌘
Û⇤† ˆ k Û⇤ = âk ↵k F̂k† + â†k ↵k F̂k + â†k âk ↵k F̂k† + F̂k + O(⌦k 2 ).
(13.16)
Evaluating this expression in the scalar product h
equations, as will be shown below.
k (t)|
·|
k (t)i
will lead to the dRG flow
From Eqs.(13.9), (13.14), (13.15) we can now proceed and derive the RG flow equation
for the dynamics of the observable h ˆ k (t)i. This will be sufficient to determine the phonon
number and the phonon momentum in the polaron cloud. We expect the result to be modified
compared to a MF calculation because the phonon dispersion ⌦k acquires an RG flow, which is
not captured by a MF ansatz. Interestingly, the RG flow of the coupling constants also di↵ers
from the equilibrium case discussed previously. In Eq.(13.14) we recognize renormalization of
the slow phonon Hamiltonian Ĥs not only by the known result Ĥs , but in addition there is
a non-equilibrium contribution,
Ĥs ! Ĥs + Ĥs0
Z
Ĥs0 = Ĥs + d3 k |
f
(13.17)
k|
2ˆ
⌦s (k).
(13.18)
dRG flow equations I
From the last expression (13.18) the dRG flow-equation for the phonon momentum coupling
µ
constant Pph
can easily be read o↵, similar to the case of the equilibrium RG protocol (see
12.3.2). We obtain
µ
@Pph
@⇤
=
Mµ⌫1
Z
f
n
d2 k k⌫ M |
2
k| +
|↵k |2 h
MF
2 Pph
⌦k
Pph · k + k
MM
1
k
io
,
(13.19)
13.1. FORMULATION OF THE DYNAMICAL RG
279
cf. Eq.(12.39). The RG flow-equation (12.38) for the mass is unmodified, but since it depends
µ
on Pph
the dRG flow of Mµ⌫1 does in general di↵er from its groundstate counterpart. There
is one notable exception, however, when the system is spherically symmetric and the polaron
µ
momentum vanishes, q = 0. In this case Pph
(⇤) ⌘ 0 does not flow at all.
Now we turn to the dRG flow equation of concrete observables. We start by discussing
the phonon number, Ô = N̂ph , where Ôk = â†k âk . After the MF rotation we obtain
MF
Nph (t) = Nph
+
Z
⇤0
d3 k h ↵k |eiĤq t ˆ k e
iĤq t
|
↵k i.
(13.20)
Similarly, for the phonon momentum it holds
µ
qph
(t)
=
MF,µ
qph
+
Z
⇤0
d3 k kµ h ↵k |eiĤq t ˆ k e
iĤq t
|
↵k i.
(13.21)
Note that
R here we distinguish between the time-dependent observable phonon momentum
qph (t) = d3 k khâ†k âk i and the phonon momentum coupling constant Pph which appears
in the Hamiltonian. While these quantities were shown to be equal for the equilibrium case
(see Appendix Q.2), they will in general di↵er somewhat in the dynamical procedure because
qph (t) includes temporal (quantum) fluctuations. After deriving the dRG flow equations for
qph (t) below, we will return to the question how precisely the two quantities di↵er from one
another.
By applying a single dRG step, we obtain the following relations from Eqs.(13.14), (13.15),
Nph (⇤, t) = Ñph (⇤
⌫
qph
(⇤, t)
=
⌫
q̃ph
(⇤
⇤, t) + Nph (⇤),
⇤, t) +
⌫
qph
(⇤, t),
MF
Nph (⇤0 , t) = Nph
(13.22)
MF,⌫
qph
(13.23)
⌫
qph
(⇤0 , t)
=
where we defined
Ñph (⇤, t) =
⌫
q̃ph
(⇤, t)
=
Z
Z
⇤
⇤
d3 k h ↵k |eiH̃q (⇤)t ˆ k e
iH̃q (⇤)t
d3 k k⌫ h ↵k |eiH̃q (⇤)t ˆ k e
|
iH̃q (⇤)t
↵k i,
(13.24)
|
(13.25)
↵k i.
The dRG flow is described by the following contributions,
Z
n
Wk
Nph (⇤, t) =
⇤ d2 k | k (t)|2 2Re ( k (t) + ↵k )
+ 2↵k Re k (t)
⌦k
f
o
↵k
⌫
q̃ph
(⇤, t)M⌫µ1 kµ
2Re ( k (t) + ↵k ) ,
⌦k
Z
n
Wk
⌫
qph
(⇤, t) =
⇤ d2 k k⌫ | k (t)|2 2Re ( k (t) + ↵k )
+ 2↵k Re k (t)
⌦k
f
o
↵k
⌫
(⇤, t)M⌫µ1 kµ
2Re ( k (t) + ↵k ) .
q̃ph
⌦k
(13.26)
(13.27)
dRG flow equations II
⌫ (⇤, t) using similar
Next we derive self-contained dRG flow equations for Nph (⇤, t) and qph
techniques as developed for the equilibrium RG in Appendix Q.2. We start by applying the
same trick as in Eq.(Q.9) for the equilibrium RG, and rewrite the time-dependent phonon
280
CHAPTER 13. DYNAMICAL RG FOR INTERMEDIATE-COUPLING FRÖHLICH
POLARONS
momentum in the form
µ
µ
µ
µ
qph
(t) ⌘ qph
(⇤ ! 0, t) = qph
(⇤, t) + (⇤, t) q̃ph
(⇤, t).
(13.28)
This expression is valid for all values of the cut-o↵ ⇤, up to the considered order O(⌦k 1 ) of
µ
the dRG. Because (⇤, t) q̃ph
(⇤, t) = O(⇤) ! 0 flows to zero as ⇤ ! 0, it will be sufficient
µ
µ
to derive a dRG equation for qph
(⇤, t), which converges to qph
(t) as ⇤ ! 0. In addition we
require the following initial conditions,
(⇤0 , t) = 1,
µ
MF,µ
qph
(⇤0 , t) = qph
,
(13.29)
for all times t.
From the results in Eqs. (13.22) - (13.27) it is easy to derive the following closed set of
dRG flow equations,
⌫ (⇤, t)
@qph
Z
n
d2 k k⌫ |
Wk
2
2Re ( k (t) + ↵k )
+ 2↵k Re
k (t)|
@⇤
⌦k
f
Z
@ (⇤, t)
↵k
= 2 (⇤, t)Mk 1 (⇤) d2 k kx2
Re ( k (t) + ↵k ) .
@⇤
⌦
k
f
=
k (t)
o
(⇤, t),
(13.30)
(13.31)
Note that in the second line we assumed without loss of generality that the total polaron
momentum points along x, i.e. q = qex .
To obtain dRG flow equations for the phonon number as well, we derive the following
⌫ (⇤, t) from Eq.(13.28),
expression for q̃ph
⌫
⌫
q̃ph
(⇤, t) = qph
(t)
⌫
qph
(⇤, t)
1
.
(⇤, t)
(13.32)
⌫ (t) can
Note that from the full solution of Eqs.(13.30), (13.31) the fully converged result qph
be obtained. Then the dRG flow of the phonon number can be calculated from
Z
n
@Nph (⇤, t)
Wk
=
d2 k | k (t)|2 2Re ( k (t) + ↵k )
+ 2↵k Re k (t)
@⇤
⌦k
f
⌫ (t)
⌫ (⇤, t)
o
qph
qph
↵k
M⌫µ1 kµ
2Re ( k (t) + ↵k ) .
(13.33)
(⇤, t)
⌦k
Long-time limit: phonon momentum and coupling constant
Finally we discuss the relation of the phonon momentum coupling constant Pph to the timedependent observable qph (t). Specifically we will argue that these quantities coincide in the
long-time limit.
To this end we note that for the equilibrium RG we obtained (⇤) = M/Mk (⇤), see
Eq.(Q.13). To examine the dRG we define a new quantity ✓(⇤, t) by writing
⇣
⌘
(⇤, t) = Mk 1 (⇤) + ✓(⇤, t) M.
(13.34)
Using the dRG flow equation for the renormalized impurity mass (12.38) we obtain
Z
o
@✓(⇤, t)
↵k n
= M d2 k kx2
2✓(⇤, t)Mk 1 Re ( k (t) + ↵k ) + 2Mk 2 Re ( k (t)) ,
(13.35)
@⇤
⌦k
f
13.1. FORMULATION OF THE DYNAMICAL RG
281
and the phonon momentum relates to the coupling constant via
⌫ (⇤, t)
@qph
@⇤
=
⌫ (⇤)
@Pph
@⇤
Z
2
⇢
1
✓
d k k⌫ 2Re ( k (t)) Mk
↵k

✓
2
+ ✓(⇤, t) | k | + 2Re ( k (t)) ↵k
M
f
◆
Wk
+
⌦k
◆
Wk
Wk
2↵k
⌦k
⌦k
. (13.36)
Now we discuss what happens in the long-time limit. To this end let us remember that
the philosophy of the dRG was to solve the dynamics of fast phonons by averaging over slow
phonons. Therefore, from the point of view of slow phonons,
lim Re
t!1
k (t)
= 0.
(13.37)
This result is strictly true when we integrate over a sufficiently wide fast-phonon shell (width
⇤) before taking the long-time limit. Employing this result in Eq.(13.35) we conclude that
@✓(⇤, t ! 1)
= 2✓(⇤, t ! 1)Mk 1 M
@⇤
Z
f
d2 k kx2
↵k2
.
⌦k
(13.38)
Because ✓(⇤0 , t) = 0 for all times t, the solution of the last equation reads
✓(⇤0 , t ! 1) = 0.
(13.39)
If we employ the last result in Eq.(13.36) we find that
⌫
⌫
qph
(⇤, t ! 1) = Pph
(⇤),
(13.40)
i.e. the dRG flow of the time-dependent phonon momentum reduces to that of the static
coupling constant Pph of the dRG.
13.1.2
Time-dependent overlap
Now we turn to the discussion of the time-dependent overlap Aq (t). After applying the unitary
transformation ÛMF (see FIG.12.2) we obtain
Aq (t) ⌘ h0|e
iĤq t
= h0|ÛMF e
= h ↵k |e
|0i
†
iÛMF
Ĥq ÛMF t
iH̃q t
|
↵k i.
†
ÛMF
|0i
(13.41)
As for the time-dependent observables, we calculate Aq (t) shell-by-shell by applying infinitesimal transformations Û⇤ which decouple the Hamiltonian H̃q for fast and slow phonons in
every step. However, in contrast to the previous cases, the transformations Û⇤ will no longer
be unitary, although their form is closely related to the unitaries Û⇤ used so far.
dRG step
Now we will formally derive the dRG flow equations for the time-dependent overlap Aq (t). Our
starting point is Eq.(13.41). We start every dRG step by performing the dRG infinitesimal
transformation Û⇤ ,
1
Aq (t) = h ↵k |Û⇤ e itÛ⇤ H̃q Û⇤ Û⇤ 1 | ↵k i.
(13.42)
282
CHAPTER 13. DYNAMICAL RG FOR INTERMEDIATE-COUPLING FRÖHLICH
POLARONS
In contrast to the previous schemes, we only demand Û⇤ to be invertible but it no longer
has to be unitary. We will show that during the dRG flow the Hamiltonian H̃q is no longer
hermitian in general.
To ensure that in the transformed Hamiltonian Û⇤ 1 H̃q Û⇤ fast and slow degrees of freedom
are decoupled from one another, we choose Û⇤ to be of the following form,
✓Z
h
i◆
Û⇤ = exp
d3 k F̂k âk â†k .
(13.43)
f
This expression is very similar to the unitary transformations used previously, the only di↵erence being that âk is multiplied by F̂k instead of F̂k† , making the transformation non-unitary
in general. As before we assume that F̂k contains slow-phonon operators only.
Before proceeding with the calculation, let us derive some basic properties of the transformation in Eq.(13.43). For simplicity we will consider a single-mode expression,
h ⇣
⌘i
D̂f = exp f ↠â ,
f 2 C,
(13.44)
which can be interpreted as a non-unitary generalization of the coherent state displacement
operator D̂↵ = exp ↵↠↵⇤ â , where ↵ 2 C. Using similar manipulations as in the unitary
case we can show that
D̂f 1 âD̂f = â + f,
(13.45)
D̂f â D̂f = â + f,
(13.46)
1 †
†
and as in the unitary case, D̂f 1 = D̂ f .
Now we are in a position to generalize our RG protocol to non-hermitian Hamiltonians.
To this end we apply Eqs.(13.45), (13.46) and derive an equation for F̂k in Eq.(13.43) such
that slow and fast phonons are decoupled in the new Hamiltonian Û⇤ 1 H̃q Û⇤ . Demanding
that terms linear in â†k vanish, we obtain
Z
⇣
⌘
1
⌦k F̂k = Wk + ↵k F̂k kµ Mµ⌫ d3 p p⌫ ˆ p + [F̂k , Ĥs ] + O(⌦k 2 ),
(13.47)
s
similar to the equilibrium case Eq.(12.32). Meanwhile, for terms linear in âk to vanish we
obtain a separate equation,
Z
⇣
⌘
⌦k F̂k = Wk + ↵k F̂k kµ Mµ⌫1 d3 p p⌫ ˆ p [F̂k , Ĥs ] + O(⌦k 2 ).
(13.48)
s
Unlike in the equilibrium case this equation poses a condition on F̂k , instead of F̂k† . However, the last two equations for F̂k di↵er only in a minus sign in front of the commutator
[F̂k , Ĥs ], which leads to a second order contribution O(⌦k 2 ) only1 . Therefore the leading
order solution for F̂k is the same as in the equilibrium case,

Z
1
1
F̂k =
Wk + ↵k kµ Mµ⌫ d3 p p⌫ ˆ p + O(⌦k ) 2 .
(13.49)
⌦k
s
see also Eq.(12.33).
1
In order to decouple fast and slow phonons consistently to higher orders, a more general form
of unitary
transformation
of a unitary part, and a non-unitary part: Û⇤ =
hR
⇣
⌘ ⇣is required, consisting
⌘i
exp f d3 k F̂k âk â†k + F̂k† âk F̂k â†k . We will not discuss this further in this work, however.
13.1. FORMULATION OF THE DYNAMICAL RG
283
Because the required leading order solution F̂k is not modified compared to the equilibrium
case, we can check that the renormalized slow phonon Hamiltonian retains its form
Z
⇣
⌘
†
1
3
ˆ
Û⇤ H̃q Û⇤ = Ĥs + Ĥs + d k âk âk ⌦k + ⌦s (k) + O(⌦k 2 ),
(13.50)
see Eqs.(12.34) - (12.36). Next, in order to evaluate Eq.(13.42), we need to calculate the action
of Û⇤ 1 on | ↵k i. As a first step, we note that F̂k | ↵p i = fk | ↵p i, i.e. the slow-phonon
coherent MF states | ↵p i are eigenstates of F̂k , see Eq. (13.7).
Now we generalize the notion of coherent states to the non-unitary transformations (13.44).
To this end we define = (f, ↵)T , = (f ⇤ , ↵)T and
⇣
⌘†
| i := D̂f 1 D̂↵ |0i = D̂f ⇤ D̂↵ |0i.
| i := D̂f D̂↵ |0i,
(13.51)
In addition, we define a positive semidefinite scalar product by
✓
◆
⇤
† 1 1
:=
= (f ⇤ + ↵)⇤ (f + ↵) .
1 1
The last equation shows that formally we may set
(13.52)
= f ⇤ + ↵.
= f + ↵ and
Before proceeding with our calculation, we derive two more properties of the generalized
coherent states. The first concerns the time-dependent overlap for a single mode, for which
we find
h
i
†
⇤
A (t) = h |e i⌦â ât | i = exp
1 e i⌦t
.
(13.53)
The second concerns the time-dependent overlap including the number operator, for which
we find the following generalized expression,
n (t) = h |e
i⌦↠ât †
â â| i = A (t)
⇤
e
i⌦t
.
(13.54)
Now we continue evaluating Eq.(13.42). We start by noting that
Û⇤ 1 |
↵k i|
↵p i = |
k i|
↵p i,
k
= (fk , ↵k )T ,
(13.55)
while for the left-hand side of the scalar product we obtain
h ↵p |h ↵k |Û⇤ = h ↵p |h
k |.
(13.56)
Combining Eqs.(13.50), (13.55), (13.56) we can bring the time-dependent overlap Eq.(13.42)
into the following simplified form,
Aq (t) = h ↵p |h
it
k |e
hR
f
ˆ s (k))+Ĥs + Ĥs
d3 k â†k âk (⌦k +⌦
i
|
k i|
↵p i + O(⌦k 2 ).
(13.57)
To separate this equation into contributions from fast and slow phonons respectively, we
ˆ s (k) perturbatively (which is justified
treat the fast phonon frequency renormalization / ⌦
because this term is of order ⇤ only). In this way we obtain
h
Aq (t) = h ↵p | A k (t)e
it(Ĥs + Ĥs )
i
Z
t
d⌧
0
Z
d3 k e
f
i(t ⌧ )(Ĥs + Ĥs )
ˆ s (k)e
n k (t)⌦
i⌧ (Ĥs + Ĥs )
i
|
↵p i =
CHAPTER 13. DYNAMICAL RG FOR INTERMEDIATE-COUPLING FRÖHLICH
POLARONS
284
=
Y
k2f
A k (t)h ↵p | exp

✓
i Ĥs + Ĥs +
Z
3
d k
f
⇤
i⌦k t ˆ
⌦s (k)
k ke
◆
t |
↵p i,
(13.58)
up to corrections of order O(⌦k 2 , ⇤2 ).
Now we derived the following factorization,
Aq (t) = Af (t)As (t),
 ✓
Z
As (t) = h ↵p | exp i Ĥs + Ĥs + d3 k
⇤
i⌦k t
k ke
f
where the fast phonon contribution is given by
 Z
Y
Af (t) =
A k (t) = exp
d3 k 1
e
ˆ s (k) t |
⌦
i⌦k t
f
k2f
(13.59)
◆
⇤
k k
↵p i,
.
(13.60)
(13.61)
dRG flow equations I
The remaining slow phonon contribution in Eq.(13.60) is described by a t-dependent Hamiltonian. Let us emphasize however, that t should be interpreted as a given parameter which
defines Aq (t) rather than a dynamical variable during the dRG. In fact the renormalized slowphonon Hamiltonian in Eq.(13.60) is not time-dependent: If we rewrite the exponent which
determines the time-evolution of the system as an integral, we obtain
 Z t ✓
◆
Z
⇤
3
i⌦k t ˆ
exp i
d⌧ Ĥs + Ĥs + d k k k e
⌦s (k) ,
(13.62)
0
f
where the expression under the integral is ⌧ -independent. For the same reason, a time-ordering
operator is not required to make this expression valid 2 .
The crucial di↵erence to the (d)RG flows discussed previously is that the slow phonon
Hamiltonian becomes non-Hermitian,
Ĥs ! Ĥs + Ĥs00
Z
Ĥs00 = Ĥs + d3 k
f
(13.63)
⇤
i⌦k t ˆ
⌦s (k).
k ke
(13.64)
The second equation describes a time-dependent renormalization of the phonon momentum
µ
coupling constant Pph
(⇤, t), which acquires an imaginary contribution during the dRG flow,
µ
@Pph
(t)
@⇤
=
Mµ⌫1
Z
f
n
d2 k k⌫ M
⇤
i⌦k t |↵k |
+
k ke
⌦k
2h
MF
2 Pph
Pph ·k+k
MM
1
k
io
(13.65)
µ
Again the RG flow-equation (12.38) for the mass is unmodified, but since it depends on Pph
(t)
the dRG flow of Mµ⌫1 (t) becomes time-dependent as well and does in general di↵er from its
groundstate counterpart. As in the first dRG protocol for time-dependent observables, when
µ
the system is spherically symmetric the coupling constant Pph
(⇤, t) ⌘ 0 does not flow.
Before discussing the dRG flow equations further, let us point out an important connection
to the ground state RG flow. In the long-time limit we can formally set limt!1 e i⌦k t = 0,
i.e. quantum fluctuations have no e↵ect on average. Then comparison to the equilibrium flow
2
In fact, if we would add a time-order operator it would sort the di↵erent terms according to the value of
⌧ , not of the final value t.
.
13.1. FORMULATION OF THE DYNAMICAL RG
285
equations (12.38), (12.39) shows that the dRG is equivalent to the ground state flow in the
long-time limit.
To explain the time-dependence of the dRG flow, we simply note that quantum fluctuations
in the time-domain are taken into account in the dRG flow. Although it may be unexpected
at first sight that the time-dependent overlap is determined by a non-hermitian Hamiltonian
evolution, there is an intuitive explanation why this is the case: Unlike the time-dependent
observables O(t) discussed above, the time-dependent overlap describes the amplitude for
the state |0i to return to itself after a unitary time-evolution |0i ! e iĤq t |0i. Therefore
the information about any contribution which does not return to |0i is completely lost. The
corresponding decrease of the magnitude in Aq (t) is described by the imaginary part of the
Hamiltonian.3
dRG flow equations II
In the remaining part of this subsection, we use Eqs.(13.59) - (13.61) to derive dRG flow equations for the time-dependent overlap. Because it factorizes in every RG step, see Eq.(13.59),
it is more convenient to introduce the logarithm of Aq (t) as a fundamental quantity, which
(suppressing the index q for simplicity) we denote by
B(t) = log Aq (t).
(13.66)
Thus, after running the RG from the initial cut-o↵ ⇤0 down to a value ⇤, it holds
B(t) = B⇤> (t) + B⇤< (t).
(13.67)
In this expression, the yet unsolved dynamics at smaller momenta is accounted for by (see
Eqs.(13.60), (13.64))
B⇤< (t) = logh ↵p |e i(Ĥs +
Ĥs00 )t
|
↵p i,
p  ⇤.
(13.68)
On the other hand, the dynamics at larger momenta are captured by B⇤> (t), which flows in
the RG and starts from
B⇤>0 (t) = 0.
(13.69)
Let us emphasize again that the time t enters these expressions only as a fixed parameter,
while the dRG flow corresponds to a variation of model parameters with ⇤, described by a
di↵erential equation of the form @⇤ B⇤> (t) = ..., see Eq.(13.73) below. At the end of the dRG,
we will arrive at a fully converged value for the time-dependent overlap, B(t) = lim⇤!0 B⇤> (t)+
B⇤< (t). While lim⇤!0 B⇤> (t) will be determined from a dRG flow equation, B⇤< (t) contains
a C-number contribution E0 (t) flowing in the course of the dRG, plus corrections of order
O(⇤3 ). Therefore as ⇤ ! 0
B(t) =
iE0 (t)t + lim B⇤> (t),
⇤!0
(13.70)
which is the final result of the dRG. Here E0 (t) is the equivalent of the ground state energy
in the equilibrium RG, and it is determined form the t-dependent dRG flow of the coupling
3
Note that this statement seems to be in contradiction with the general theorem that, under a unitary
time-evolution, every quantum state will return to itself after sufficiently long time. However, the perturbative
argument we used to derive Eq.(13.64) assumes that the fast-phonon momentum shell is infinitesimally thin.
This is only consistent in an infinite system, where it takes infinitely long for the quantum state to return to
itself.
286
CHAPTER 13. DYNAMICAL RG FOR INTERMEDIATE-COUPLING FRÖHLICH
POLARONS
µ
constants Mµ⌫1 (t) and Pph
(t), Eqs.(12.38), (13.65). As shown above, in the long-time limit
the ground state RG flow is recovered, and consequently E0 (t) ! E0 converges to the ground
state polaron energy E0 as t ! 1.
With the notations introduced above we can now proceed by discussing the dRG flow
equations for B(t). For a single dRG step we obtain from Eqs.(13.59) - (13.60)
B(t) = B⇤> (t) + B⇤ + B⇤<
⇤ (t),
(13.71)
where we read o↵ (in generalization of the usual coherent states)
B⇤ = logh k |e
Z
=
d3 k
f
it
R
f
d3 k ⌦k â†k âk
⇤
k k
1
e
|
ki
i⌦k t
Thus, we arrive at the following dRG flow equation,
Z
@B⇤> (t)
⇤
=
d2 k k k 1 e
@⇤
f
.
i⌦k t
(13.72)
,
(13.73)
which determines the time-dependent overlap. The amplitudes read
k
= fk
↵k ,
k
= fk⇤
↵k ,
(13.74)
where fk is the same as in the previous dRG protocol for time-dependent observables, given
by Eq.(13.8); fk⇤ denotes the complex conjugate of fk .
Some comments are in order about the dRG flow equation (13.73). To begin with, we note
that in the limit t ! 1 the complex phase factors e i⌦k t vanish because of dephasing4 , and
we can e↵ectively set limt!1 e i⌦k t = 0. Thus, by comparing to the RG flow equation (12.60)
of the logarithm of the quasiparticle weight log Z and employing Eq.(13.70), we obtain
lim B(t) ⌘ lim log Aq (t)
t!1
t!1
=
iE0 t
log Z.
(13.75)
As discussed above, in the long-time limit the RG flows of Z and of the ground state energy
E0 can be calculated using the equilibrium flow equations. Thus Eq.(13.75) represents an
important consistency check for our dRG procedure: It can be shown rigorously, using a
standard Lehman-type spectral decomposition, that in the long-time limit the time-dependent
overlap Aq (t) is determined solely by ground state properties (see e.g. [P5] for a discussion).
A second important remark concerns the relation of our dRG flow equation (13.73) to the
MF result for the time-dependent overlap Aq (t). In Appendix T we discuss the spherically
symmetric situation (i.e. q = 0) and show that both expressions for Aq (t) (from RG and MF)
have an almost identical form in that case. To obtain the MF expression one has to merely
discard the dRG flow, i.e. replace ⌦k ! ⌦MF
and k , k ! ↵k in the dRG expression, and
k
drop energy corrections E in the expression for E0 due to the RG.
4
A naive argument for this dephasing is that for large times t ! 1 we may replace e i⌦k t by its longtime average h·it , i.e. e i⌦k t R! he i⌦k t it = 0. A more rigorous argument makes use of the fact that e i⌦k t
>
only shows up in an integral d3 k in physically relevant expressions (e.g. in B⇤
(t) after integrating the RG
flow equation (13.73)). Because the integral extends
over
a
broad
spectrum
of
momenta, and ⌦k is strongly
R 3
k-dependent, we generically expect that limt!1 d k e i⌦k t ... = 0.
13.2. RESULTS: SPECTRAL FUNCTION OF THE FRÖHLICH POLARON
13.2
287
Results: Spectral Function of the Fröhlich Polaron
Now we use the dRG equations derived above to calculate the spectral function of the polaron.
We investigate first how rf spectra change when the coupling ↵ is increased or the mass ration
M/mB is decreased. In both cases the strong coupling regime is entered, where the rf spectrum
di↵ers markedly from the shape familiar from weak coupling. Finally we compare predictions
from MF and dRG and find that MF theory yields qualitatively correct results even in the
strong coupling regime.
In FIG.13.1 (a) we present inverse rf spectra in the weak coupling regime for ↵ . 5, calculated using dRG. We checked that these spectra coincide with predictions by MF theory from
Chap.10. In the weak coupling regime, the shape of the spectrum is essentially unmodified
when ↵ is increased and only the overall spectral weight of the incoherent part increases.
However, for ↵ ⇡ 10 we observe a redistribution of spectral weight towards high frequencies.
Meanwhile the pronounced peak at ! ⇡ c/⇠ broadens and its maximum slightly shifts towards
larger frequencies.
In FIG.13.1 (b) the coupling constant ↵ is further increased beyond the weak-coupling
regime. The spectral weight continues to shift towards larger and larger energies and the
maximal spectral response decreases rapidly (note the modified overall scale). Most prominently, the frequency !max where the maximal weight is observed shifts to large energies. For
example for ↵ = 40 we find !max ⇡ 100c/⇠, two orders of magnitude above the characteristic
Figure 13.1: Inverse rf spectra of the polaron are shown in the weak (a) and strong (b)
coupling regimes for di↵erent values of ↵. They were calculated using our dRG scheme for
the time-dependent overlap. Parameters are M/mB = 2.5, P = 0 and ⇤0 = 2 ⇥ 103 /⇠. All
spectra are plotted relative to the respective polaron ground state energies E0RG (↵).
288
CHAPTER 13. DYNAMICAL RG FOR INTERMEDIATE-COUPLING FRÖHLICH
POLARONS
Figure 13.2: Inverse rf spectra of the polaron are shown in the weak (a) and strong (b) coupling
regimes for di↵erent values of M/mB . They were calculated using our dRG scheme for the
time-dependent overlap. Parameters in a (a) and (b) are ↵ = 3, P = 0 and ⇤0 = 2 ⇥ 103 /⇠.
All spectra are plotted relative to the respective polaron ground state energies E0RG (M/mB ).
frequency scale c/⇠ of the problem. This emergence of a new characteristic energy scale in the
problem is a direct consequence of the renormalization of the low-energy polaron Hamiltonian.
For even larger values of ↵ the maximum !max continues to shift to higher frequencies, while
for small frequencies ! < !max the spectral response becomes strongly reduced.
In FIG.13.2 we present inverse rf spectra for di↵erent mass ratios M/mB , again calculated
Figure 13.3: We compare inverse rf spectra of the polaron computed by MF theory (dashed)
and dRG (solid). Parameters are M/mB = 2.5, P = 0 and ⇤0 = 2 ⇥ 103 /⇠. All spectra are
plotted relative to the respective polaron ground state energies E0RG (↵).
13.3. RESULTS: DYNAMICS OF POLARON FORMATION
289
Figure 13.4: We show results for the time-dependent overlap A(t), calculated from MF and
dRG. In (a) its absolute value |A(t)| and in (b) the complex argument argA(t)/⇡ are shown.
Note that the noise-level drops substantially around t = 5⇠/c because for larger times we
performed a temporal average to obtain a cleaner signal. Parameters are M/mB = 2.5,
P = 0, ⇤0 = 2 ⇥ 103 /⇠ and ↵ = 50.
using dRG. For M/mB & 0.2 we recover the familiar weak-coupling shape of the rf spectrum,
with a maximum around !max ⇡ c/⇠. For smaller mass ratios the strong coupling regime is
entered, which shows the same characteristics as when ↵ is increased, cf. FIG.13.1.
Next we compare inverse rf spectra of the polaron calculated by MF theory to more reliable
dRG predictions, see FIG.13.3. As expected, in the weak coupling regime we find excellent
quantitative agreement. In the strong coupling regime we obtain quantitative deviations,
whereas the qualitative shape of the rf spectra is correctly predicted by MF theory. We thus
conclude that the dynamical MF theory is able to capture signatures of the strong-coupling
physics. This is in contrast to the static MF theory, which did not show any signs of strongcoupling physics in the polaron ground state (see e.g. FIG.12.7 for the polaron mass).
Finally we discuss our raw data for the time-dependent overlap A(t), from which the
spectral functions are obtained by Fourier transformation, see Eq.(10.6). In FIG.13.4 we
show results for A(t) in the strong coupling regime. On a time-scale between 10 2 ⇥ ⇠/c
and ⇠/c the magnitude |A(t)| drops substantially. Then it increases slightly and approaches
the quasiparticle weight Z in the long-time limit, which serves as an important consistency
check. In part (b) of the figure we show the complex phase of the time-dependent overlap.
We observe that it winds by about 2⇡ before it reverses and decays to zero eventually (we
subtracted the long-time oscillation at polaron ground state frequency). This is characteristic
for the strong coupling regime. For smaller couplings the complex phase always remains close
to zero and does not wind. For even larger couplings, on the other hand, several windings are
observed. We find the same qualitative behavior for both dRG and MF, where MF merely
deviaties from dRG on a quantitative level.
13.3
Results: Dynamics of polaron formation
Now we return to the discussion of non-equilibrium polaron dynamics. In particular, we
investigate polaron formation in a situation where the interactions between an impurity (momentum P ) and the surrounding bosons is suddenly switched on at time t = 0. A similar
problem was studied briefly in Sec.10.5 using time-dependent MF theory. There we found
interesting damped oscillations of the impurity velocity, albeit in a regime of enormously
large interactions. We will now revisit this regime and show that dRG can not confirm MF
predictions. Nevertheless, highly non-trivial polaron dynamics are predicted in this case.
290
CHAPTER 13. DYNAMICAL RG FOR INTERMEDIATE-COUPLING FRÖHLICH
POLARONS
Figure 13.5: The impurity momentum Pimp (t) is shown as a function of time (a) in the weak
coupling regime at ↵ = 10 and in (b) for extremely strong coupling ↵ = 628. Other parameters
are M/mB = 2.5, P = 0.5M c and ⇤0 = 20/⇠.
In FIG.13.5 (a) we show the evolution of the impurity momentum Pimp (t) = P qph (t) in
the weak coupling regime. In this case dRG and MF theory make similar predictions, although
dRG yields somewhat smaller impurity momentum. This is expected because quantum fluctuations lead to an enhanced amount of momentum carried by the phonons. The momentum
approached in the long-time limit is well below the corresponding ground state polaron momentum. Thus the mass of the excited polaron state which is realized in the long-time limit
is substantially larger than its ground state counterpart.
In FIG.13.5 (b) the impurity is shown again, but in an extremely strongly interacting
regime. In this case MF theory predicts damped oscillations of the impurity momentum,
as reported already in Chap.10.5. After a very short time ⌧MF & 10 2 ⇠/c the impurity
comes almost to a complete halt. In contrast, dRG does not predict any oscillations. Only
one dip is observed where the impurity momentum becomes negative. This happens on an
almost identical time scale as in MF theory. Afterwards, however, the impurity momentum
takes a sizable value again, before it eventually comes to a stop after a much longer time
scale ⌧dRG & ⇠/c. At intermediate time scales the e↵ective polaron mass Mp⇤ , defined by
Pimp = M vp = M P/Mp⇤ , is still of the order of M .
A comment is in order about the dRG results in FIG.13.5 (b). For small times the impurity momentum predicted by dRG does not approach the exact result P . This is because at
each step in time we solve a dRG flow equation, which is based on an approximate evaluation
of the infinitesimal unitary transformations Û⇤ . The error is of order O(⌦k 2 ), but for strong
interactions such additional terms may become relevant. Thus we conclude that at the extremely large coupling ↵ & 600 considered in FIG.13.5 (b) the leading-order dRG may not be
sufficient to describe the correct physics.
For smaller values of the coupling constant, however, the qualitative behavior of the impurity momentum is similar to FIG.13.5 (b), while the errors described in the last paragraph
become negligible. In FIG.13.6 we show the average impurity position,
ximp (t) =
Z
t
d⌧ Pimp (⌧ )/M,
(13.76)
0
for di↵erent couplings. MF theory predicts a gradual slow-down of the impurity velocity
until a non-vanishing constant value is reached. dRG, in contrast, shows that the impurity is
quickly slowed down, before it reverses direction and bounces back. Eventually it returns to
the origin again, and continues moving in the forward direction with a finite velocity. This
13.3. RESULTS: DYNAMICS OF POLARON FORMATION
291
Figure 13.6: The average impurity position ximp (t) is shown as a function of time. We compare
dRG (solid) and t-dependent MF theory (dashed), for di↵erent values of the coupling constant
↵. Other parameters are M/mB = 0.5, P = 0.5M c and ⇤0 = 20/⇠.
e↵ect can be observed for reasonable values of ↵ & 10, much smaller than the extreme case
shown in FIG.13.5 (b).
Our interpretation of this e↵ect is as follows. Initially the impurity is completely free.
When interactions are switched on, it suddenly gets dressed by high energy phonons which
can react on short time-scales. Now, in a simplified picture, we can think of the impurity which
is coupled to a rigid cloud of (high-energy) phonons. This is in the spirit of Feynman’s polaron
model [23]. Because in the strong coupling regime the e↵ective mass of the phonon cloud is
expected to exceed the impurity mass, from a classical picture we expected the impurity to
be scattered o↵ the phonon cloud. However, because the impurity is strongly interacting with
the phonon cloud, it can not leave completely but will be recaptured by the phonon cloud.
Meanwhile more and more low-energy phonons are created and eventually a polaron is formed
which travels with a reduced speed through the condensate.
292
CHAPTER 13. DYNAMICAL RG FOR INTERMEDIATE-COUPLING FRÖHLICH
POLARONS
Part IV
Appendices
293
Appendix A
Quantization of the fractional part
of the charge on the edge
In this appendix we give a more detailed theoretical argument for the quantization of the
fractional part of the charge,
q mod 1 = ⌫/⇡,
(A.1)
which is localized at the edges of the SSH model. Here ⌫ is the topological invariant defined
in Sec.2.3. More concretely, we consider grand-canonical ground states | D (µ)i for di↵erent
values of the chemical potential µ and compare di↵erent dimerizations D = I, II of the SSH
model. Later we add the possibility of having degeneracies in the bulk state, which applies to
systems with spontaneously broken symmetries.
Let us consider a completely generic model, which is defined by the following grandcanonical Hamiltonian,
Ĥ(µ, ') = Ĥbulk (') + Ĥedge + µN̂ .
(A.2)
Here N̂ denotes the particle number which is assumed to commute with the remaining terms.
Ĥbulk (') defines the Hamiltonian in the bulk of the system, of size L ! 1. Here ' is some
tunable parameter. Ĥedge defines the details of the edge of the system and it is restricted to
a finite size Le . This Hamiltonian includes all coupling terms to the bulk, which are assumed
to have a finite range only.
In the following we assume the bulk Hamiltonian Ĥbulk (') to be inversion-symmetric for
' = 0, ⇡ and to support incompressible ground states | ↵ (')i. The polarization di↵erences of
the ground states at ' = 0, ⇡ are related to the many-body Zak phases ⌫' by the King-Smith
- Vanderbilt relation, see Secs.1.2.2 and 1.2.3,
P =
X
↵
P↵ (⇡)
P↵ (0) =
d
(⌫⇡
2⇡
⌫0 ) = Cd/2,
C 2 Z,
(A.3)
where d is the lattice constant. In the last step we used that the many-body Zak phase of an
inversion-symmetric system is quantized in units of ⇡, see Sec. 2.31 .
At this point we furthermore note that the definition of the bulk Hamiltonian is not unique,
because there are several possibilities how to cut a given full Hamiltonian Ĥ(µ, ') into pieces.
However, ultimately we only need to find a Hamiltonian which provides a complete description
1
To proof this quantization, a Thouless pump argument can be made for the bulk Hamiltonian. The integer
C in Eq.(A.3) corresponds to the Chern number of such a pump. Note however, that here the use of a Thouless
pump is merely a clever way to make a strict statement about the static polarizations of the incompressible
ground state in the bulk.
295
APPENDIX A. QUANTIZATION OF THE FRACTIONAL PART OF THE CHARGE ON
296
THE EDGE
(a)
μ
bulk only
(b)
λ
μ
(c)
bulk +
edge
λ
μ
bulk +
edge
λ
Δq mod 1 = C/2
φ =0,π
φ =0
φ =π
Figure A.1: (a) Sketch of the grand-canonical phase diagram of the bulk Hamiltonian (at
' = 0, ⇡) with an incompressible phase (blue). µ is the chemical potential and
some
tunable system parameter. (b) When adding the edge Hamiltonian the grand-canonical phase
diagram generically includes more states which di↵er by their properties at the edge. Note
that the critical chemical potentials for the bulk phase are the same as in (a) and can not be
modified by perturbations at the edge. When changing the tunable parameter from ' = 0
to ' = ⇡, generically the grand-canonical phase diagram changes (c). During this process
all grand-canonical ground states acquire a half-integer quantized fractional portion of charge
q = C/2 mod 1 localized at the edge.
of the properties of our system in the bulk. Because we restrict ourselves to incompressible –
i.e. gapped
> 0 – phases, it follows that the ground state is short-range correlated in its
bulk. All correlations decay on a characteristic length-scale ⇠ ⇠ 1/ , the correlation length,
see e.g. Ref. [56]. Therefore, as long as L
⇠, we can capture all bulk physics of the full
model with a simplified Hamiltonian Ĥbulk ('). In fact, to describe the bulk, we may even
restrict ourselves to periodic systems of a size well above ⇠.
We want to start by discussing the case when the incompressible ground state | 0 (')i of
Ĥbulk (') is unique. First we examine the grand-canonical phase diagram of (A.2). Let us
start by discarding the edge part. In this case, we obtain a characteristic grand-canonical bulk
phase diagram with an incompressible region, as sketched in FIG.A.1 (a). The incompressible
region is determined by two critical chemical potentials,
µ < µ < µ+ ,
(A.4)
which may depend on an additional model parameter .
When including the edge Hamiltonian, the critical chemical potentials µ± of the bulk can
not change (in the limit L ! 1). Nevertheless, overall several ground states | 0n i may exist,
which di↵er at the edge. For example, they can have di↵erent total particle numbers N . In
this case, which of them is the actual grand-canonical ground state depends on the chemical
potential µ. This generic scenario is sketched in FIG.A.1 (b).
Our goal now is to compare grand-canonical ground states | 0n (')i at two di↵erent values
of ' = 0, ⇡. For simplicity we assume that the grand-canonical phase diagram of the bulk
at ' = 0 and ' = ⇡ is identical. Now we fix the chemical potential µ somewhere in the
incompressible regions, µ < µ < µ+ . Therefore at both values of ' = 0, ⇡ the bulks of the
actual grand-canonical ground states | 0 (µ, ')i are incompressible, for any allowed µ.
Now we compare the density distributions ⇢0 (x; µ, ') of the two ground states | 0 (µ, 0)i
and | 0 (µ, ⇡)i. To this end we first calculate the accumulated charge on the semi-infinite side
of the system,
Z 1
q1 (µ) =
dx ⇢0 (x; µ, ⇡) ⇢0 (x; µ, 0) = P/d = C/2,
(A.5)
Le
297
which is determined by the polarization in the bulk. Therefore the change of the charge
localized at the edge reads,
qe (µ) =
Z
Le
dx ⇢0 (x; µ, ⇡)
⇢0 (x; µ, 0).
(A.6)
0
It can now easily be derived from the super-selection rule,
Z 1
q1 (µ) + qe (µ) =
dx ⇢0 (x; µ, ⇡) ⇢0 (x; µ, 0) = m,
0
m 2 Z,
(A.7)
stating that the total charge (i.e. the total particle number) can only change by an integer
amount. Hence it follows that
qe (µ)
mod 1 = m
C/2
mod 1 = C/2,
(A.8)
for all values of µ within the incompressible region.
Finally we note that this discussion can easily be generalized to situations where the bulk
is not unique. In this case all possible polarization configurations of the bulk have to be
determined. Then the fractional part of the charges in the di↵erent ground states is given by
q↵n (') = P↵n (')/d mod 1. In this case the quantization condition Eq.(A.3) can not directly
be used to derive all possible charge di↵erences q↵, between di↵erent values of '. A more
careful analysis, including symmetry considerations has to be carried out in this case, see also
Sec.2.6.
APPENDIX A. QUANTIZATION OF THE FRACTIONAL PART OF THE CHARGE ON
298
THE EDGE
Appendix B
Exact diagonalization in the lowest
Landau level
In this appendix we summarize a few details about our numerical calculations in the lowest
Landau level. In addition to variational calculations, we performed state-of-the-art exact
diagonalization (ED) studies. To minimize the role of finite-size e↵ects we performed all
our systematic investigations in the spherical geometry. We worked on standard desktop
computers and have access up to the following particle numbers:
⌫
Nmax
dim.
1
14
194668
1/2
10
246448
1/4
7
48417
1/6
6
32134
1/8
6
118765
Here the dimension of the full Hilbertspace containing all angular momentum multiplets is
stated, since we do not exploit in our numerics that [H, L2tot ] = 0. In the numerics the filling
fraction ⌫ is adjusted by changing the magnetic flux N through the surface of the sphere.
For ⌫ = 1/2, 1/4, ... Laughlin states
N =
N
1
⌫
.
We eliminate leading-order finite size e↵ects by rescaling the magnetic length [241],
r
N ⌫
1
`B ! `B =
`B .
N
(B.1)
(B.2)
In simulations aB must be chosen
sufficiently small because when the blockade area fills half
p
a sphere, i.e. when aB
⇡ N /8`B , we expect the formation of two clusters on opposite
poles. We indeed found large overlaps to two-cluster trial wavefunctions in this case, and thus
no conclusions about the thermodynamic limit can be drawn.
In disc geometry we only performed systematic studies for small aB . `B and found similar
results as in spherical geometry. The accessible system sizes are the same as above, although
requiring slightly more CPU time.
299
300 APPENDIX B. EXACT DIAGONALIZATION IN THE LOWEST LANDAU LEVEL
Appendix C
Proof of the Wilson loop formula
for the Z2 invariant
In this appendix we will proof Eq.(6.8), using the same notations as in the main text. To
this end we first summarize the formulation of the Z2 invariant derived by Fu and Kane [265].
They assumed the most general gauge Eq.(6.3) which can be characterized by the so-called
sewing matrix,
ws,s0 (k) = hus ( k)| ✓ˆ |us0 (k)i,
(C.1)
where s, s0 are band indices (I, II). Their expression for ⌫2D reads
( 1)
⌫2D
=
4
Y
j=1
p
4
det w( j ) Y
⌘
Pfw( j )
j
.
(C.2)
j=1
Here Pf denotes the Pfaffian of an antisymmetric matrix, k = j denote the four TRIM in
the 2D BZ
(C.3)
1 = (0, 0),
2 = (⇡, 0),
3 = (⇡, ⇡),
4 = (0, ⇡),
and the branch of the square root in Eq.(C.2) has to be chosen correctly, see [265] for details.
Yu et.al. [94] calculated TR invariant two-by-two Wilson loops (time reversed bands I and
II) at kyTRIM = 0, ⇡ and found at kyTRIM = 0
Ŵ (0) = e
i
2
(0)
1
Î2⇥2 = e
2
i'W (0)
Î2⇥2 .
(C.4)
Here (ky ) denotes the total Zak phase, see Eq.(6.12). A similar formula holds for kyTRIM = ⇡
with (0) ! (⇡) and 1 2 ! 3 4 . (We generalized the proof given by these authors
from Wilson loops to arbitrary TR invariant propagators, and our generalized result can be
found in Eq. (F.24) in Appendix F.)
To proceed we note that since the determinant of an anti-symmetric matrix is given be
the square of its Pfaffian, det w( j ) = Pf2 w( j ),
can only take the two values ±1,
j
=
q
Pf2 w(
Pfw(
j)
j)
2 {±1} .
(C.5)
Therefore we may rewrite Eq.(C.2) as
ei⇡⌫2D = ( 1)⌫2D =
301
1
2
3
4
.
(C.6)
302
APPENDIX C. PROOF OF THE WILSON LOOP FORMULA FOR THE Z2
INVARIANT
Taking the product of the Wilson loop at kyTRIM = 0 and the inverse Wilson loop at kyTRIM = ⇡
we get according to Eq.(C.4)
ei('W (⇡)
Therefore we have
⌫2D
1
=
⇡
✓
'W (0))
=e
i( (0)
'W
1
( (⇡)
2
(⇡))/2
(0))
◆
ei⇡⌫2D .
mod 2,
(C.7)
(C.8)
from which our previously claimed equation (6.8) immediately follows using continuity of the
total Zak phase (ky ).
Appendix D
Bloch oscillation’s equations of
motion
Atoms in optical lattices undergo Bloch oscillations when a constant force F (t) is applied.
They can be described by the following Schrödinger equation,
[email protected] | (r, t)i = (H ± F · r) | (r, t)i.
|
{z
}
(D.1)
=HB
We assume that non-adiabatic conduction-band mixing is negligible. Using the Landau-Zener
probability for band-mixing one finds the following adiabaticity condition:
!B = a|F | ⌧
2
band 2⇡
(D.2)
I II
with band the band gap, I II the energy spacing between valence bands I, II, a the size of
the unit cell and !B the Bloch oscillation frequency. We may now decompose the wavefunction
into Bloch states | s,k (r)i:
X Z
| (r, t)i =
d2 k s,k (t)| s,k (r)i.
(D.3)
s=I,II BZ
For simplicity we only consider the case of two bands s = I, II here. Using orthogonality
Z
↵
d2 rh s,k (r)| s0 ,k0 (r) = (k k0 ) s,s0 ,
(D.4)
one obtains equations of motion for the amplitudes s,k (t):
Z
X Z
[email protected] s,k (t) =
d2 k0 s0 ,k0 (t) d2 r h s,k (r)|HB |
BZ
s0 =I,II
With the Bloch theorem, |
X Z
s0 =I,II BZ
d2 k
s0 ,k0 (t)F
s,k (r)i
s0 ,k0 (r)i.
(D.5)
s0 ,k0 (t)|us0 ,k0 (r)i
eik ·r . (D.6)
= eik·r |us,k (r)i, we find
0
· reik ·r |us0 ,k0 (r)i =
=i
X Z
s0 =I,II BZ
d2 k 0 F · rk 0
303
0
304
APPENDIX D. BLOCH OSCILLATION’S EQUATIONS OF MOTION
After defining the time-dependent quasi-momentum
k(t) = k0 ⌥
Z
t
F d⌧
(D.7)
0
and introducing the amplitudes at these k components,
s,k (t)
:=
s,k(t) (t),
(D.8)
it is easy to derive their equations of motion:
i
Xh
0
0
[email protected] s,k (t) =
±F (t) · As,s (k(t)) +H s,s (k(t))
s,k (t).
(D.9)
s0
Now each k-component sees a di↵erent t-dependent Hamiltonian but there is no mixing between di↵erent k. This is a direct consequence of the translational symmetry of the problem.
Formally these equations can be solved by a time-ordered exponential, which translates into
a path-ordered one when using Eq.(D.7).
The full propagator is thus given by
2
3
Uk2 ,k1
✓
◆7
6 Z k2
1
6
7
= P exp 6 i
dk A(k) ± H(k) 7 .
F
4
5
k1
|
{z
}
=B(k)
(D.10)
Appendix E
Non-universal Franck-Condon
factor phases
In this appendix we discuss general interferometric sequences realizing the twist scheme presented in Sec. 7.3. To this end we consider the most general coupling scheme realizing Ramsey
pulses between the two bands I, II. We show that, in general, additional phases are picked up
in the cycle which depend on the intrinsic properties of the Bloch functions. This rules out
many simpler schemes realizing Ramsey pulses between the two bands for the measurement
of cTRP.
We start by formalizing the idea of a band-switching, which is realized by some timedependent microscopic Hamiltonian
Ĥrf (t) = ei('E +!rf t) p̂,
(E.1)
with !rf the frequency of the (typically radio-frequency, rf) transition, 'E the phase of the
driving field and p̂ some microscopic operator coupling the two bands (called p̂ in analogy to
an atomic dipole operator in quantum optics).
In a rotating frame and in the Bloch function basis this Hamiltonian may generally be
described by
Ĥrf (k) = ⌦rf (k)|u, kihl, k| + h.c.,
(E.2)
where ⌦(k, t) = ei'⌦ (k) cos (!rf (k) t) is the Rabi frequency for atoms at quasi-momentum k.
The phase '⌦ = 'E + '˜FC of the driving is then determined by the phase of the driving field
'E relative to the phase '˜FC of the corresponding Franck-Condon (FC) factors
'˜FC = arghu, k|Ĥrf (0)|l, ki,
(E.3)
with arg denoting the polar angle of a complex number.
When transitions take place between given atomic (e.g. hyperfine) states one can make
use of the freedom in the choice of the global U (1) phase in order to eliminate the appearance
of FC factor phases. In our case however two FC phases appear at the two band switching
points kx = 0, ⇡ and only one of them may be eliminated using the global U (1) gauge freedom.
The di↵erence between FC phases at di↵erent momenta however carries information about
the band structure and can not be eliminated. In fact, it contains exactly those terms we
need to connect incomplete Zak phases from di↵erent bands u and l in a meaningful way. To
see this we decompose '˜FC into the gauge-dependent term 'A from Eq. (7.13) and a gauge
305
306
APPENDIX E. NON-UNIVERSAL FRANCK-CONDON FACTOR PHASES
invariant remainder
'FC := '˜FC
'A .
(E.4)
To prove gauge invariance of 'FC we note that the Hamiltonian (E.2) is invariant under
u
local U (1) gauge-transformations in momentum space, |u, ki ! ei# (k) |u, ki and analogously
l
u
for the lower band l. Therefore ⌦rf transforms as ⌦rf ! ⌦rf ei(# # ) , and one easily checks
that this is also how Au,l
x transforms. Since 'E is gauge-invariant this shows that so is 'FC ,
and from now on we can forget about '˜FC . Summarizing we have
'⌦ (k) = 'FC (k) + 'A (k) + 'E (k).
(E.5)
It is crucial for our measurement scheme to consider FC factor phases 'FC , which in
general take non-universal values. Let us illustrate this for a simple example. In experimental
schemes [269, 270, 271] the spin states ", # are typically proposed to be realized as hyperfine
states. In general the spins will be coupled in some way by the Bloch Hamiltonians Ĥ(k)
(realizing SOC) and the FC phases depend on the spin-mixture in the Bloch eigenfunctions.
We will consider a toy model of a two dimensional Hilbert space with the two orthogonal
bands |ui = ↵ei ↵ | "i + ei | #i and |li = e i | "i ↵e i ↵ | #i. Here the amplitudes ↵,
as well as the phases ↵ ,
are chosen to be real numbers.
The simplest rf Hamiltonian flips the spins but leaves spatial coordinates unchanged,
Ĥrf = ⌦rf | "ih# | + ⌦⇤rf | #ih" |.
(E.6)
According to Eqs.(E.3) and (E.4) we thus have 'FC = arg ↵2 e 2i ↵ ⌦rf + 2 e 2i ⌦⇤rf 'A .
We note that
= ↵
is gauge invariant (up to 2⇡) and from the last equation we
conclude that the FC phase 'FC generally depends on
. Therefore a simple Ramsey pulse
using rf transition between internal spin states Eq.(E.6) can generally not be used to realize
the band switchings required for the measurement of cTRP, unless for some reason the intrinsic
FC phases 'FC at the band switching points are known.
The scheme presented in Sec.7.3.1 yields universal FC phases, i.e. 'FC = 0 for the Hamiltonian given in Eq.(7.9). This was achieved by coupling only to the motional degrees of
freedom but not to the (pseudo) spins ", #.
Appendix F
TR invariant non-adiabatic
two-band dynamics
In this appendix we derive formulas for the propagators describing Bloch oscillations in 1D TR
invariant band structures. Our calculations straightforwardly generalize the results obtained
by Yu et.al. [94].
The generic form of the propagator describing Bloch oscillations within two bands I, II
between quasi momenta k1,2 is derived in Appendix D, and it is given by (see eq(D.10))
Û (k2 ; k1 ) = P exp
✓
i
Z
◆
k2
k1
dk B̂(k) ,
k2 > k 1 .
(F.1)
Here B̂(k) describes geometrical as well as dynamical contributions,
B̂(k) = Â(k) ±
Ĥ(k)
,
F (k)
(F.2)
and the sign ± corresponds to the direction of the driving force F , cf. Eq.(D.1). We will
consider a single Kramers pair, i.e. Â, B̂, Ĥ, Û are all two-by-two matrices in the band indices
I, II and ✓ˆ = K(iˆ y ) denotes TR. Furthermore we assume TR invariant driving of the Bloch
oscillations, i.e. forces at ±k are related by F ( k) = F (k).
In the context of the quantum spin Hall e↵ect these propagators correspond to nonadiabatic generalizations of Zak phases along kx at ky = 0, ⇡. More specifically, for infinite
driving F ! 1 (or equivalently kĤk ! 0) they correspond to the non-Abelian U (2) Wilson
loops, Û = Ŵ . For this case results were obtained in [94], and in the following we will
generalize the latter to finite F . Generally one expects F 6= 0 to cause qualitative changes
of the propagators since the commutator [Ĥ, Â] 6= 0 in general. However, as will be shown
below, when TR invariant loops are considered, non-zero F only yields a dynamical U (1)
phase factor (instead of a U (2) rotation).
In the following we will consider a general gauge characterized by (k), see Eq.(6.3).
Starting from |uI (k)i defined in some continuous gauge on the entire BZ ⇡ < k  ⇡, (k)
fixes
ˆ I ( k)i
|uII (k)i = ei ( k) ✓|u
(F.3)
for all k. We will without loss of generality assume a continuous gauge choice on the patches
⇡ < k < 0 as well as 0 < k < ⇡, whereas discontinuities of (k) are allowed at the sewing
points k = 0, ±⇡. We note that in the construction of the Bloch eigenfunctions the gauge307
308
APPENDIX F. TR INVARIANT NON-ADIABATIC TWO-BAND DYNAMICS
choice
|u(k + G)i = e
iGx
|u(k)i
(F.4)
was made with G 2 2⇡Z a reciprocal lattice vector, see [81]. This imposes a constraint on the
possible discontinuities of (k) at k = 0, ⇡ since
i2⇡x
|uI (⇡)i = e
=
=
=
e
i2⇡x i ( ⇡) ˆ
✓|uII (⇡)i
ˆ i2⇡x |uII ( ⇡)i
e i2⇡x ei ( ⇡) ✓e
ˆ i (⇡) ✓|u
ˆ I (⇡)i
ei ( ⇡) ✓e
= ei[
Therefore ( ⇡)
|uI ( ⇡i
e
( ⇡)
(⇡)]
|uI (⇡)i.
(F.5)
(⇡) 2 2⇡Z, and similarly around k = 0. Defining the di↵erence
⌘(k) := (k)
( k)
(F.6)
we thus obtain
⌘(0), ⌘(⇡) 2 2⇡Z.
(F.7)
Using the relation (F.3) we find by an explicit calculation
ˆ † Â(k)⌅(k)
ˆ
✓ˆ† Â( k)✓ˆ = ⌅
+ @k diag ( (k), ( k)) ,
k
where the gauge choice enters in the definition of the following unitary matrix
⇣
⌘
ˆ k = diag e i⌘(k)/2 , ei⌘(k)/2 .
⌅
(F.8)
(F.9)
Using TR invariance, ✓ˆ† Ĥ( k)✓ˆ = Ĥ(k) and F ( k) = F (k), together with the fact that
ˆ k ] = 0, we find that also
Ĥ(k) = diag (EI , EII ) such that [Ĥ, ⌅
ˆ † B̂(k)⌅
ˆ k + @k diag ( (k), ( k))
✓ˆ† B̂( k)✓ˆ = ⌅
k
(F.10)
This can be rewritten as
ˆ † B̂(k)⌅
ˆ k + i⌅
ˆ † @k ⌅
ˆ k + 1 @k ( (k) + ( k)) ,
✓ˆ† B̂( k)✓ˆ = ⌅
k
k
2
(F.11)
where the first two terms on the right hand side describe a gauge transformation of the
ˆ k is a continuous unitary matrix. This condition is indeed
e↵ective connection B̂ when ⌅
fulfilled on the two patches (0, ±⇡) since ⌘(k) Eq.(F.6) was chosen continuously there. From
the transformation properties of Wilson loops under this gauge transformation [95] we obtain:
ˆ † Û (k; 0)† ⌅
ˆk e
✓ˆ† Û (0; k)✓ˆ = ⌅
0
where ⇤ =
1
2
(0 ) + (0+)
( k)
i⇤
(F.12)
(k) .
Now we will derive a second expression for the transformation properties of Û (0; k) under
TR. Since B̂ † = B̂ we may write it as
X
B̂ = B U (1) Î2⇥2 +
B SU (2),j ˆ j ,
(F.13)
j=1,2,3
309
with B SU (2),j and B U (1) real numbers. From Eq.(F.13) and using ✓ˆ† ˆ j ✓ˆ =
obtain
⇣
⌘
✓ˆ†
iB̂(k) ✓ˆ = iB̂(k) + 2iB U (1) (k)Î2⇥2 ,
ˆ j for j 6= 0 we
(F.14)
and therefore we also find
✓ Z
†
ˆ
ˆ
✓ Û (k; 0)✓ = Û (k; 0) exp 2i
k
0
dk B
U (1)
◆
(k) .
(F.15)
Combining the results from Eqs. (F.12) and (F.15), we obtain for TR symmetric propagators from k to k:
Û (k; k) = Û (k; 0)Û (0; k)
h
ih
i
= ✓ˆ ✓ˆ† Û (k; 0)✓ˆ ✓ˆ† Û (0; k)✓ˆ ✓ˆ†
⇣
Rk
⌘
U (1)
2i 0 dk B
(k) i⇤ ˆ †
ˆ k ✓ˆ†
= ✓ˆÛ (k; 0)e
⌅0 Û (k; 0)† ⌅
✓
◆
Z k
U (1)
ˆ†⌅
ˆ ˆ†
= exp
2i
dk B
(k) + i⇤ ✓ˆ⌅
0 k✓ .
(F.16)
0
In the last step we used the fact that Û (k; 0) is unitary, as well as the integer quantization of
⌘(0) Eq.(F.7)
⇣
⌘
ˆ 0 = diag e i⌘(0)/2 , ei⌘(0)/2 = ( 1)⌘(0)/2⇡ Î2⇥2 .
⌅
(F.17)
The result can be further simplified by noting that
1
B U (1) ( k) = trB̂( k)
2
⌘
1 ⇣
= tr K B̂( k)K
(B̂ † = B̂)
2
⌘
1 ⇣
= tr ✓ˆ† B̂( k)✓ˆ
2
⌘
1 ⇣ˆ†
ˆ k + 1 @k (k) + ( k)
= tr ⌅
B̂(k)
⌅
k
2
2
1
U (1)
=B
(k) + @k (k) + ( k) .
2
(F.18)
Using this we have
2i
Z
k
0
dk B U (1) (k) =
and we thus obtain
Û (k; k) = e
⇣
i
Rk
k
i
Z
k
k
dk B U (1) (k)
dk BU (1) (k)
⌘
i⇤,
ˆ†⌅
ˆ ˆ†
✓ˆ⌅
0 k✓ .
(F.19)
(F.20)
Note that until here even the phases of the matrices are well defined (i.e. the above calculations
can be thought of being performed on a Riemann surface in the complex plane). We will now
drop this additional constraint and using Eq.(F.17) we finally obtain the full propagator as
Û (⇡; ⇡) = ( 1)
The factor ( 1)
⌘(0)+⌘(⇡)
2⇡
⌘(0)+⌘(⇡)
2⇡
e(
i
R⇡
⇡
dk BU (1) (k))
Î2⇥2 .
(F.21)
can be related to the Pfaffian-expressions Eq.(C.5). To this end we
310
APPENDIX F. TR INVARIANT NON-ADIABATIC TWO-BAND DYNAMICS
note that
✓
w(k) =
0
e
e
i ( k)
i ( k)
0
◆
(F.22)
and thus det w(k) = ei( (k)+ ( k)) as well as Pfw(k) = e i (k) . To evaluate Eq.(C.5) it
is important to choose the branch cut of the square root correctly [265]. To avoid these
difficulties we use the simpler but lengthy formula 0 ⇡ = ( 1)P✓ with the expression for TRP
[265]
Z ⇡
✓
◆
1
Pfw(⇡)
P✓ =
dk @k log det w(k) 2 log
2⇡i 0
Pfw(0)
1 h
=
(⇡)
( ⇡) + (0+) + (0 )
2⇡
i
i (⇡)+i (0+)
2 log e
=
1
[⌘(⇡)
2⇡
⌘(0)] + 2Z.
(F.23)
Therefore we end up with
Û (⇡; ⇡) =
0 ⇡
exp
✓
i
Z
⇡
⇡
dkx B
U (1)
◆
(kx ) Î2⇥2 .
By taking the limit F ! 1 in Eq.(F.24) we recover the Wilson loop phase
✓ Z ⇡
◆
e i'W = 0 ⇡ exp
i
dkx AU (1) (kx )
(F.24)
(F.25)
⇡
derived in [94]. Thus our final result for the propagator of general TR invariant Bloch oscillations within a single Kramers pair reads
✓
◆
Z ⇡
1
i'W
tr
Û (⇡; ⇡) = e
exp ⌥i
dkx Ĥ(kx ) .
(F.26)
2F
⇡
Appendix G
Hofstadter topological polaron in
the polaron frame
In this appendix we derive the TP Hamiltonian in the excitation centered polaron frame.
We apply the Lee-Low-Pines (LLP) transformation to the two body model from Sec.8.3 in
the main text and identify the total polaron (quasi) momentum as a conserved quantum
number of the system. This treatment is motivated by similar calculations performed for
more conventional polaron problems in lattices [P8]. The e↵ect of the external force F acting
on the impurity is also discussed. This allows an exact numerical solution of the two-particle
TP problem from Sec.8.3.
Our starting point is the static LLP transformation,
⇣
⌘
ÛLLP = exp iR̂a · p̂I ,
(G.1)
cf. Eq.(8.39). Dynamical e↵ects due to the driving force F will be included in the transformation below. To define the hole position operator R̂a and the impurity momentum operator
p̂I , we start by making a gauge choice. We introduce the magnetic unit cell for the hole
Hamiltonian of size ax ⇥ ay , which contains an integer number of flux quanta. Using the
Landau gauge as in Eq.(8.36) and assuming ↵ = r/s with r, s integers, we have ax = a and
ay = sa. Next we label sites within the unit cell by an integer µ and define the hole position
operator
X
R̂a =
(jx ax , jy ay )T â†j,µ âj,µ .
(G.2)
jx ,jy ,µ
Here the integers jx,y label unit cells and (j, µ) is just an alternative way of parametrizing the
site indices (m, n) which were used previously in the definition of the model. Hence we see
that R̂a measures only the position of the unit cell, but not the positions of individual sites
within one cell. Similarly the impurity momentum operator is defined by
X †
p̂I =
kb̂k,µ b̂k,µ ,
(G.3)
k,µ
where we introduced operators in momentum space,
r
ax ay X ik·(jx ax ex +jy ay ey )
b̂k,µ :=
e
b̂j,µ .
Lx Ly
(G.4)
j
The wave vector k takes values k = 2⇡ (ix /Lx , iy /Ly )T for integers ix = 1, ..., Lx /ax and
311
312
APPENDIX G. HOFSTADTER TP IN THE POLARON FRAME
iy = 1, ...., Ly /ay and with Lx,y denoting system size in x and y direction. Note that although
the impurity lattice has a smaller period of a we choose the larger magnetic unit cell of the
hole here. This is necessary to distinguish between inequivalent sites µ within one magnetic
unit cell for both the hole and the impurity.
We proceed by applying the LLP transformation defined above to the Hamiltonian
Ĥ = Ĥ0 + ĤI + Ĥint ,
(G.5)
see Eqs.(8.36) - (8.38). We start with the terms in the impurity Hamiltonian ĤI . Trivially the
LLP transformation commutes with the kinetic part of the impurity energy. Next we calculate
the e↵ect on the impurity position operator R̂I , which we define analogously to Eq.(G.2),
X
R̂I =
(jx ax , jy ay )T b̂†j,µ b̂j,µ ,
(G.6)
jx ,jy ,µ
by specifying only the position of the unit cell. Thus we have to carefully treat the driving
term,
0
1
X
X X †
F·
rm,n b̂†m,n b̂m,n = F · @R̂I +
rµ
b̂j,µ b̂j,µ A ,
(G.7)
m,n
µ
j
where rµ denotes the position of site µ within one unit cell while R̂I specifies the position of
P
the unit cell itself. Because p̂I commutes with j b̂†j,µ b̂j,µ and generates shifts of R̂I ,
†
ÛLLP
R̂I ÛLLP = R̂I + R̂a ,
(G.8)
we find that
†
ÛLLP
F·
Hence we arrive at
X
rm,n b̂†m,n b̂m,n ÛLLP =
m,n
†
ÛLLP
ĤI ÛLLP = ĤI
F·
X
rm,n b̂†m,n b̂m,n
m,n
F · R̂a .
F · R̂a .
(G.9)
(G.10)
Now we apply the LLP transformation to the kinetic part of the hole Hamiltonian. To
this end we note that
†
ÛLLP
âk,µ ÛLLP = âk+p̂I ,µ
(G.11)
follows from the definition of hole momentum operators
r
ax ay X ik·(jx ax ex +jy ay ey )
âk,µ :=
e
âj,µ .
Lx Ly
(G.12)
Written in momentum space, the kinetic hole Hamiltonian reads
XX
Ĥ0 =
hµ,µ0 (k)â†k,µ âk,µ0 ,
(G.13)
j
k µ,µ0
with hµ,µ0 (k) denoting the matrix elements of the Bloch Hamiltonian h(k). Under LLP it
transforms into
XX
†
ÛLLP
Ĥ0 ÛLLP =
hµ,µ0 (k p̂I )â†k,µ âk,µ0 .
(G.14)
k µ,µ0
Finally we apply the LLP transformation to the impurity hole interaction. For generality
313
we consider a density-density interaction of the form
X
Ĥint =
Vµ,⌫ (R̂I R̂a )b̂†j,µ b̂j,µ â†i,⌫ âi,⌫ ,
(G.15)
j,i,µ,⌫
where Vµ,⌫ (r) denotes an arbitrary potential. Using Eq. (G.8) the LLP transformation yields
†
ÛLLP
Ĥint ÛLLP =
X
Vµ,⌫ (R̂I )b̂†j,µ b̂j,µ â†i,⌫ âi,⌫ .
(G.16)
j,i,µ,⌫
That is the hole is located within the central unit cell and the impurity sees its static potential
Vµ,⌫ (R̂I ), which still depends on the hole populations of the sites ⌫ within the unit cell however.
We can further simplify the last expression by using
X †
X †
âi,⌫ âi,⌫ =
âk,⌫ âk,⌫ ,
(G.17)
i
which yields
†
ÛLLP
Ĥint ÛLLP =
k
X
Vµ,⌫ (R̂I )b̂†j,µ b̂j,µ
j,µ,⌫
X
â†k,⌫ âk,⌫ .
(G.18)
k
For the local interaction assumed in Eq.(8.38) it holds
Vµ,⌫ (r) =
V
r,0 µ,⌫ .
(G.19)
Combining the results from above Eqs.(G.10), (G.14), (G.18) we arrive at the following
Hamiltonian in the polaron frame,
XX
X †
X †
†
ÛLLP
ĤÛLLP = ĤI F · R̂a +
hµ,µ0 (k p̂I )â†k,µ âk,µ0 V
b̂0,µ b̂0,µ
âk,µ âk,µ . (G.20)
k µ,µ0
We proceed by eliminating the driving term
dependent gauge transformation,
µ
k
F · R̂a . To this end we introduce a time-
⇣
⌘
Û (t) = exp itF · R̂a .
(G.21)
Û † (t)âk,⌫ Û (t) = âk+F t,⌫ ,
(G.22)
In the new basis, hole operators are given by
and the TP quasimomentum changes continuously over time. Application of the time-dependent
LLP transformation
ÛLLP (t) := ÛLLP Û (t)
(G.23)
from Eq.(8.39) thus yields the e↵ective Hamiltonian
⇣
⌘
h
X †
†
ÛLLP
(t) Ĥ [email protected] ÛLLP (t) =
âk,µ âk,µ0 ⌦ hµ,µ0 (k
|
k,µ,µ0
Ft
⇣
p̂I ) + µ,µ0 ĤI
{z
=:ĤTP (µ,µ0 )
V b̂†0,µ b̂0,µ
⌘i
}
.
(G.24)
P
Most importantly it factorizes and the total TP momentum k,µ kâ†k,µ â†k,µ in the polaron
frame is a conserved quantity. Here we stress again, however, that the last expression is only
true if there is exactly one hole.
314
APPENDIX G. HOFSTADTER TP IN THE POLARON FRAME
Appendix H
Measurement of TP invariant in the
Hofstadter problem
In this appendix we discuss how the TP topological invariant can be calculated in the excitation centered polaron frame. As discussed already in the main text, one approach is to solve
the static TP band Hamiltonian Eq.(8.41) at F = 0 for di↵erent values of the TP quasimomentum k. Then we obtain the static TP ground states | TP (k)i, and the Chern number of
this manifold is defined by
Z
1
CTP =
d2 k r ⇥ h TP (k)|ir| TP (k)i.
(H.1)
2⇡ BZ
This TP Chern number can be measured by applying a force F to the impurity. From
Eq.(8.41) we find that the force indeed couples to the TP momentum k and can be used to
change it adiabatically in time, thus moving the TP through the BZ. However, in addition,
the impurity Hamiltonian ĤI contains terms which explicitly depend on the driving force F .
We will now discuss their implications for the measurement of the TP invariant. Afterwards
we show by a calculation in the adiabatic limit how such terms can be treated exactly.
H.0.1
E↵ect of driving terms on impurity
Let us consider a concrete measurement scheme for the TP Chern number where the Zak or
1D (k ) is measured at fixed values of k . Their winding when k goes from 0
Berry phase 2⇡⌫TP
y
y
y
to 2⇡/ay yields the Chern number, see Eq.(8.23). To measure 1D invariants a constant weak
force Fx along x is applied, which drives Bloch oscillations in this direction. When the Bloch
oscillation frequency !B = ax Fx ⌧ TP is small compared to the TP gap TP , non-adiabatic
transitions to excited states can be neglected.
In this case we can treat the driving terms in the impurity Hamiltonian, see Eq. (8.37),
perturbatively. First order perturbation theory yields an additional phase
X
'int (ky ) = Fx t
xm,n h TP ( Fx t, ky )|b̂†m,n b̂m,n | TP ( Fx t, ky )i,
(H.2)
m,n
which is directly related to the internal TP invariant ⌫TPint (ky ) = 'int (ky )/2⇡. Because
Fx t = 2⇡/ax after completing a full Bloch oscillation cycle, this phase can in principle take
a non-vanishing value even in the limit Fx ! 0. In fact, since 'int (ky ) depends on the
product Fx t rather than just the time t, it should be interpreted as a geometric rather than a
dynamical phase, see also Sec.8.2.3. Note that in the above expression | TP (k)i is evaluated
315
316
APPENDIX H. MEASUREMENT OF TP INVARIANT IN THE HOFSTADTER
PROBLEM
0.5
0.25
0
-0.25
-0.5
-0.5
-0.25
0
0.25
0.5
1D calculated from static TP wavefunctions (solid
Figure H.1: We compare the TP invariant ⌫TP
green) to the exact expression (within the single hole approximation) including the internal
TP invariant (dashed). For comparison the strong coupling result is shown (dash-dotted).
As in FIG.8.5 we used the following parameters J/t = 0.5 and V /t = 2 to simulate N = 3
fermions on a torus with 4 ⇥ 4 sites and N = 4 flux quanta.
in the polaron frame where the hole is localized in the center. Hence the additional phase
'int (ky ) describes how the center of mass of the impurity wavefunction in the TP bound state
depends on ky .
1D
The internal phase 'int will add up to the external contribution 'ext (ky ) = 2⇡⌫TPext
(ky )
which can thus not be measured alone. However, ultimately we are only interested in the Chern
1D
number, defined as the winding of the external invariant ⌫TPext
(ky ). From a measurement of
the winding of the total phase '(ky ) = 'int (ky )+'ext (ky ) this Chern number can be obtained,
CTP
1
=
2⇡
Z
2⇡/ay
0
dky @ky '(ky ).
(H.3)
The winding of the internal phase vanishes, 'int (2⇡) = 'int (0), because the center of mass of
the impurity in the polaron frame is 2⇡/ay -periodic in ky . Unlike in an extended crystal, the
impurity is always localized around the uniquely defined position of the single hole which it
is bound to.
H.0.2
Exact treatment of driving terms
Now we present an alternative approach which treats the additional driving terms in the
impurity Hamiltonian Eq. (8.37) exactly. We will use the same notations as in Appendix
G. We start from the TP band Hamiltonian (8.41) and perform a time-dependent gauge
transformation,
!
X
ÛI (t) = exp itF ·
rm,n b̂†m,n b̂m,n .
(H.4)
m,n
In the e↵ective Hamiltonian in the so-obtained rotating frame, the driving terms in the impurity Hamiltonian ĤI are eliminated and only the kinetic part H̃I0 remains. Compared to the
original kinetic impurity Hamiltonian ĤI0 it has modified hopping elements J˜x = Jx e iaFx t in
positive and J˜x⇤ = Jx eiaFx t in negative x-direction, and similarly for y-direction. In addition
317
the impurity momentum transforms as
ÛI† (t)p̂I ÛI (t) = p̂I
F t.
We end up with the transformed TP band Hamiltonian
⇣
TP
H̃µµ
p̂I ) + µµ0 H̃I0
0 (k) = hµµ0 (k
(H.5)
⌘
V b̂†0,µ b̂0,µ .
(H.6)
TP (k) essentially only by a gauge transformation.
It di↵ers from the original Hamiltonian Ĥµ,µ
0
From Eq.(H.6) we can now easily calculate the geometric phase which can be measured in
an experiment by changing the force adiabatically in time. To this end, for a given value of
k, the TP ground state | TP (✓x , ✓y )i can be calculated as a function of the phases
✓x = Fx ax t,
✓y = Fy ay t
(H.7)
of the hopping elements J˜x,y in positive x, y direction. When Fx t ⇥ Fy t covers the entire BZ
2⇡/ax ⇥ 2⇡/ay , we obtain the TP Chern number,
CTP
1
=
2⇡
Z
2⇡a/ax
d✓x
0
Z
2⇡a/ay
0
d✓y r✓ ⇥ h
TP (✓)|ir✓ | TP (✓)i.
(H.8)
In FIG.H.1 we compare the geometric phases of the static TP | TP (k)i and using the
full treatment of the driving terms | TP (✓)i for 1D trajectories at fixed momentum ky . As
expected, their winding over ky (✓y respectively) coincides and agrees with the strong coupling
result.
318
APPENDIX H. MEASUREMENT OF TP INVARIANT IN THE HOFSTADTER
PROBLEM
Appendix I
Lowest Chern band projection
In this appendix we construct a maximally localized hole state | a (m0 , n0 )i around lattice site
(m0 , n0 ) in the lowest Chern band of the Hofstadter model. We use a general gauge for the
construction and point out that the observable properties of the obtained state are independent
of the gauge choice. In particular the probability amplitude h a (m0 , n0 )|â†m,n âm,n | a (m0 , n0 )i
defining the strong coupling Hamiltonian (Eq.(8.43) in the main text) is gauge-invariant.
The maximally localized state is defined by starting from a localized hole on site (m0 , n0 )
and projecting it on the lowest Chern band, | a (m0 , n0 )i = P̂LCB â†m0 ,n0 |0i. To calculate
the projection, we find it convenient to introduce magnetic unit cells, labeled by a vector of
integers j, and label sites within the unit cell by an index µ, cf. Appendix G. Using this
notation, (m0 , n0 ) ⌘ (j0 , µ0 ). We solve the free hole Hamiltonian Ĥ0 by defining momentum
operators âk,µ as in Eq.(G.12) from Appendix G. Momentum components from di↵erent sites
µ are coupled by Ĥ0 , see Eq.(G.13), and by solving the single-hole Bloch Hamiltonian hµ,µ0 (k)
we obtain
XX
Ĥ0 =
✏⌫ (k)ã†k,⌫ ãk,⌫ .
(I.1)
k
⌫
The original momentum operators can be expressed in the eigenbasis,
X
âk,µ =
uµ (k, ⌫)ãk,⌫ ,
(I.2)
where ⌫ labels Hofstadter bands and the Bloch wavefunctions |u(k, ⌫)i fulfill
X
hµ,µ0 uµ0 (k, ⌫) = ✏⌫ (k)uµ (k, ⌫).
(I.3)
⌫
µ0
By inverting Eqs.(G.12) and (I.2) we obtain
r
ax ay X ik·(j0x ax ex +j0y ay ey ) X ⇤
†
âj0 ,µ0 =
e
uµ0 (k, ⌫)ã†k,⌫ .
Lx Ly
⌫
(I.4)
k
Now the projection operator P̂LCB simply amounts to a restriction to the lowest band ⌫ = 1.
We thus arrive at
ax ay X
P̂LCB â†j0 ,µ0 |0i =
'j,µ (j0 , µ0 )â†j,µ |0i,
(I.5)
Lx Ly
j,µ
where
'j,µ (j0 , µ0 ) =
X
u⇤µ0 (k, 1)uµ (k, 1)eik·((j0
x
k
319
jx )ax ex +(j0y jy )ay ey )
.
(I.6)
320
APPENDIX I. LOWEST CHERN BAND PROJECTION
Appendix J
Impurity-boson interactions in a
lattice
e↵
In the main text we mentioned that both the scattering length ae↵
IB and the e↵ective range rIB
of the impurity-boson interaction in a lattice are modified due to lattice e↵ects, see [389, 374,
375]. In the following we discuss in detail how our model parameters gIB and `ho (entering ĤIB
implicitly via the Wannier function w(r) in Eq.(11.7)), which characterize the impurity-boson
interaction within our simplified model Eq.(11.7), relate to the two universal numbers ae↵
IB and
e↵ . Since both ae↵ and r e↵ can be accessed numerically (see e.g.[374, 375]) or experimentally
rIB
IB
IB
(see e.g. [376]), this allows us to make quantitative predictions for actual experiments using
our model. Our treatment is analogous to that of [390], where a similar discussion can be
found.
To understand the connection between e↵ective model parameters, like gIB , and universal
numbers characterizing inter-particle interactions at low energies, like ae↵
IB , let us first recall the
standard procedure when both the impurity and the boson are unconfined [5], see Sec.9.2.1.
Already when writing the microscopic model in Eq.(9.1), we replaced the complicated microscopic impurity-boson interaction potential by a much simpler point-like interaction of
strength gIB . The philosophy here is as follows: when two-particle scattering takes place at
sufficiently low energies k ! 0, the corresponding scattering amplitude fk takes a universal
form which is characterized by only a hand-full of parameters – irrespective of all the microscopic details of the underlying interaction. In particular, for the smallest energies only the
asymptotic value of fk matters, defining the (s-wave) scattering length as = limk!0 fk .
In an e↵ective model describing low-energy physics only, it is sufficient to capture only
the s-wave scattering correctly. To this end one may replace the microscopic impurity-boson
potential by a simplified pseudo potential, characterized by only a single parameter gIB . Next,
one can calculate the scattering amplitude fk (gIB ) expected from this pseudo potential, and to
be consistent one has to choose gIB such that as (gIB ) = limk!0 fk (gIB ). This is an implicit
equation defining the relation between as and gIB .
In the case when one of the partners (in our case the impurity) is confined to a local
oscillator state (a tight-binding Wannier orbital), two-body scattering can be substantially
modified. For example, the possibility of forming molecules bound to the local trapping
potential gives rise to confinement induced resonances with diverging scattering length, quite
similar to Feshbach resonances [389, 374, 375]. Importantly for us, this case can be treated in
complete analogy to the scenario of free particles described above. The scattering amplitude
321
322
APPENDIX J. IMPURITY-BOSON INTERACTIONS IN A LATTICE
in the low energy limit is universally given by [374]
fk =
h
1/ae↵
IB + ik
e↵ 2
rIB
k /2 + O(k 3 )
i
1
,
(J.1)
e↵
where ae↵
IB denotes the s-wave scattering length and rIB is the e↵ective range of the interaction.
These two parameters can be calculated from the scattering lengths of unconfined particles,
as shown by Massignan and Castin [374], which however requires a full numerical treatment
of the two-body scattering problem. Doing so, these authors showed in particular that by
varying the lattice depth V0 , both parameters can be externally tuned.
Now, instead of going through the complicated microscopic calculations, we introduce
a simplified pseudo potential. Motivated by our derivation in the main text, we chose the
impurity-boson interaction from Eq.(11.7). It can be characterized by two parameters, firstly
the interaction strength gIB , and secondly the extent `ho of the involved Wannier functions. In
the following both will be determined in such a way that the universal scattering amplitude
Eq.(J.1) is correctly reproduced. To this end we calculate the latter analytically in Bornapproximation and obtain
fk =
mB gIB
1
2⇡
2
k 2 `2ho /2 + O(k 3 , gIB
).
(J.2)
Comparing Eq.(J.2) to the universal form Eq.(J.1) yields the following relations (valid within
Born-approximation),
2⇡ e↵
e↵ e↵
gIB =
a ,
`2ho = rIB
aIB ,
(J.3)
mB IB
which define our model parameters (see also [390]).
To derive Eq.(J.2) we assumed the impurity to be localized on a single Wannier site,
giving rise to a potential VIB (r) = gIB |w(r)|2 seen by the bosons. This is justified in the
tight-binding limit, when the hopping J can be treated as a perturbation after handling the
scattering problem. Then solving the Lippmann-Schwinger Equation of the scattering problem
for a single boson on VIB (r) (perturbatively to leading order in gIB ) yields our result Eq.(J.2).
In order to ensure that in the scattering process no higher state in the micro trap is excited,
we require the involved boson momenta k to be sufficiently small [374], k 2 /(2mB ) ⌧ !0 , where
!0 is the micro-trap frequency. Since the involved boson momenta are limited by k . 1/`ho
from Eq.(J.3) we obtain a condition for the interaction strength,
1
e↵ |ae↵
|rIB
IB
⌧ 2mB !0 .
(J.4)
A comment is in order about the use of the tight-binding approximation in this context.
Firstly, to study also cases with stronger hopping along the lattice, the full scattering problem
for this case has to be solved. Extending the calculations of [374, 375] to this case, we expect
to obtain the same universal form (J.1) of the scattering amplitude fk in the low-energy limit,
e↵
with modified values for ae↵
IB and rIB . Nevertheless, the relation Eq.(J.3) can still be used
to link the the new parameters to the e↵ective model parameters. Secondly, we note that
when we discuss approaching the subsonic to supersonic transition in the main text of the
paper, this is not necessarily in contradiction to the tight-binding approximation; In fact,
the subsonic to supersonic transition takes place around the critical hopping Jc a = c, which
is determined solely by properties of the Bose-system. In concrete cases, whether or not
tight-binding results may be used, has to be checked for each system individually.
Appendix K
Static MF polarons in a lattice
In this appendix we derive the MF self-consistency equation (11.34) from the main text. To
this end we have to calculate the variational energy,
Y
H [↵k ](q) =
h↵k |Ĥq |↵k i.
(K.1)
k
The main obstacle is the treatment of the non-linear term ⇠ cos ↠â in the Hamiltonian Ĥq
(11.25), for which we find
✓
◆
Z
Y
3 0 0
h↵k | cos aq a d k kx n̂k0 |↵k i = e C[↵ ] cos (aq S[↵ ]) .
(K.2)
k
The functionals C[↵ ] and S[↵ ] appearing in this expression were defined in the main text,
see Eqs.(11.32), (11.33).
R
To proof the result (K.2), let us first focus on a single mode and replace d3 k0 kx0 â†k0 âk0
by k↠â for simplicity. Next we write the cosine in terms of exponentials, for which it is then
sufficient to show that
h
i
h
⇣
⌘i
h↵| exp iak↠â |↵i = exp |↵|2 1 eiak .
(K.3)
This is most easily achieved by expanding coherent states |↵i in the Fock basis |ni,
|↵i = e
|↵|2 /2
1
X
↵n
p |ni,
n!
n=0
(K.4)
from which we can read o↵ the relation,
†
h↵|eiakâ â |↵i = e
|↵|2
1 iakn
X
e
|↵|2n
n=0
n!
= exp
h
⇣
|↵|2 1
eiak
⌘i
.
(K.5)
R
This result can easily be generalized to the multimode case with d3 k kx n̂k appearing in the
argument of the cosine, when use is made of the commutativity of phonon modes at di↵erent
momenta.
Using the result Eq.(K.2) we end up with the variational Hamiltonian
Z
⇥
⇤
⇤
C[↵ ]
H [↵ , ↵ ] = 2Je
cos (aq S[↵ ]) + d3 k !k |↵k |2 + Vk (↵k + ↵k⇤ ) .
323
(K.6)
324
APPENDIX K. STATIC MF POLARONS IN A LATTICE
The MF self consistency equations can now easily be obtained by demanding vanishing functional derivatives,
H [↵k , ↵k⇤ ]
H [↵k , ↵k⇤ ] !
=
= 0,
(K.7)
↵k
↵k⇤
which readily yields Eq.(11.34) from the main text,
↵kMF =
Vk
.
MF ]
⌦k [↵
(K.8)
Plugging this result into the definitions of C[↵ ] and S[↵ ] yields the coupled set of selfconsistency equations (11.36),(11.37) for C MF and S MF .
Appendix L
Impurity density in the lab frame
In this appendix we derive Eq.(11.53) from the main text, which allows us to calculate the
impurity density from the time-dependent overlaps Aq2 ,q1 (t). For the definition of the latter,
let us recall that we work in the polaron frame throughout, where the quantum state is of the
form
X
| (t)i =
fq ĉ†q |0ic ⌦ | q (t)ia .
(L.1)
q2BZ
Here |0ic denotes the impurity vacuum and | q (t)ia is a pure phonon wavefunction. The
corresponding time-dependent overlaps are defined as
Aq2 ,q1 (t) =
ah
q2 (t)|
q1 (t)ia ,
(L.2)
see also Eq.(11.54).
In order to calculate the impurity density in the lab frame, nj = hĉ†j ĉj i, we have to
transform the operator ĉ†j ĉj to the polaron frame first. Keeping in mind that we moreover
applied the time-dependent unitary transformation ÛB (t) Eq.(11.16), we thus arrive at
†
nj = hĉ†j ĉj ilab = h (t)|ÛLLP
ÛB† (t)ĉ†j ĉj ÛB (t)ÛLLP | (t)i.
(L.3)
We proceed by writing the impurity operators ĉj in their Fourier components, see Eq.(11.23),
and plug Eq.(L.1) into the last expression,
nj (t) =
a
L
X
e
i(q1 q2 )aj
q1 ,q2 2BZ
X
q3 ,q4 2BZ
⇥
ah
fq⇤3 fq4 ⇥
q3 (t)|c h0|
†
ĉq3 ÛLLP
ÛB† (t) ĉ†q2 ĉq1 ÛB (t)ÛLLP |0ic |
q4 (t)ia .
(L.4)
To simplify this expression, we note that
ÛB† (t)ĉ†q2 ÛB (t) = ĉ†q2 +!B t
(L.5)
and analogously for ĉq1 . Thus, by relabeling indices q1,2 ! q1,2 + !B t in Eq.(L.4), we can
completely eliminate ÛB (t) from the equations above.
To deal with the Lee-Low-Pines transformation, let us introduce an eigen-basis consisting
of states |P i where the total phonon momentum is diagonal,
Z
d3 k kx â†k âk |P i = P |P i.
(L.6)
325
326
APPENDIX L. IMPURITY DENSITY IN THE LAB FRAME
Of course, for each value of P there is a large number of states with the property (L.6),
which for simplicity we will all denote by |P i. Importantly, the Lee-Low-Pines transformation
Eq.(11.17) can now easily be evaluated in this new basis, where
hP |ÛLLP |P 0 i = eiX̂P
(L.7)
P,P 0
and for simplicity we used a discrete set of phonon modes. We can make use of this result by
formally introducing a unity in this basis,
X
|P ihP | = 1̂,
(L.8)
P
allowing us to write
†
ÛLLP
ĉ†q2 ĉq1 ÛLLP =
X
P,P 0
Next, using e
iP X̂ ĉ
q1 e
†
|P ihP |ÛLLP
ĉ†q2 ĉq1 ÛLLP |P 0 ihP 0 | =
iP X̂
X
P
|P ihP |e
iP X̂ †
ĉq2 ĉq1 eiP X̂ .
(L.9)
= ĉq1 +P , we obtain
†
ÛLLP
ĉ†q2 ĉq1 ÛLLP =
X
P
|P ihP |ĉ†q2 +P ĉq1 +P .
(L.10)
Using this identity after introducing unities (L.8) in Eq.(L.4), we find after relabeling summation indices q1,2 ! q1,2 + P that
nj (t) =
a
L
X
q1 ,q2 2BZ
e
i(q1 q2 )aj
X
q3 ,q4 2BZ
fq⇤3 fq4 c h0|ĉq3 ĉ†q2 ĉq1 ĉ†q4 |0ic
ah
q3 (t)|
After simplification of the impurity operators we obtain the desired result,
a X
nj (t) =
e i(q1 q2 )aj fq⇤2 fq1 Aq2 ,q1 (t).
L
q1 ,q2 2BZ
q4 (t)ia .
(L.11)
(L.12)
Appendix M
Adiabatic wavepacket dynamics
In this appendix we present the detailed calculation leading to the expression for the adiabatic
impurity density (11.58) given in the main text. To this end we first calculate the timedependent overlaps from Eqs.(11.55), (11.56),
Aq2 ,q1 (t) = h
q2 (t)|
q1 (t)i
= Aq2 ,q1 Dq2 ,q1 ,
and use Eq.(11.53) together with a suitable initial impurity wavefunction
(M.1)
in
j .
We start from an impurity wavepacket in the band minimum of the bare impurity, and
assume its width (in real space) LI
a by far exceeds the lattice spacing a. In this case the
width in (quasi-) momentum space is q ⇡ 2⇡/LI ⌧ 2⇡/a. Moreover we can treat aj ! x as
a continuous variable and write
✓
◆
x2
1/2
in
(x) = (2⇡) 1/4 LI
exp
,
(M.2)
4L2I
where the following normalization was chosen,
Z 1
dx | in (x)|2 = 1.
(M.3)
1
By Fourier-transforming the initial impurity wavefunction we obtain the amplitudes
Z 1
1
fq = p
dx eiqx in (x),
2⇡ 1
such that the impurity density (11.53) becomes
Z 1
dq2 dq1 i(q2
n(x, t) =
e
2⇡
1
q1 )x ⇤
fq2 fq1 Aq2 ,q1 (t).
(M.4)
(M.5)
Since the width
R q ⌧ 2⇡/a
R 1 of the wavepacket is much smaller than the size of the BZ, we
approximated BZ dq ⇡ 1 dq in this step.
Using the adiabatic wavefunction (11.50), the phases of Aq2 ,q1 (t) read
 Z t
Aq2 ,q1 (t) = exp i
dt0 HMF (q2 (t0 ))
0
327
HMF (q1 (t0 )) ,
(M.6)
328
APPENDIX M. ADIABATIC WAVEPACKET DYNAMICS
and the amplitude is given by
Dq2 ,q1 (t) = exp

1
2
Z
d3 k ↵kMF (q2 (t))
2
↵kMF (q1 (t))
.
(M.7)
Due to the small width q of the polaron wavepacket in quasimomentum space we can expand
the expressions in the exponents in powers of the di↵erence q2 (t) q1 (t) = q2 q1 . Note that
log Aq2 ,q1 (log Dq2 ,q1 ) is antisymmetric (symmetric) under exchange of q2 and q1 . To second
order in |q2 q1 | we obtain

Z t
Aq2 ,q1 (t) = exp i (q2 q1 )
dt0 @q HMF (q1 (t0 )) ,
0

Z
1
2
Dq2 ,q1 (t) = exp
(q2 q1 )
d3 k @q ↵kMF (q1 (t))
2
(M.8)
2
.
(M.9)
Since only q1 ⇡ q2 ⇡ 0 contributes substantially in fq⇤2 fq1 we further approximate
@q HMF (q1 (t0 )) ⇡ @q HMF ( F t0 )
and analogously in @q ↵kMF . Thus we obtain

Aq2 ,q1 (t) = exp i (q2 q1 ) X(t)
with
2 (t)
1
(q2
2
(M.10)
q1 ) 2
2
(t) ,
(M.11)
defined in Eq.(11.60) in the main text and
X(t) = X(0)
Z
t
0
dt0 @q HMF |q=
F t0
.
(M.12)
Evaluating this integral exactly yields the expression for X(t) given in the main text, Eq.(11.59).
Using the last expression for Aq2 ,q1 (M.11) to perform momentum integrals dq1 dq2 in (M.5)
finally yields the adiabatic impurity density
n(x, t) = e
as we claimed in the main text.
(x X(t))2
2 L2 + 2 (t)
I
(
) ⇥2⇡ L2 +
I
2
(t)
⇤
1/2
(M.13)
Appendix N
Discussion and extension of the
analytical current-force relation
In this appendix we will further discuss under which conditions our analytical prediction
(11.65) for the current-force relation vd (F ) is valid. In particular, we try to understand
FIG.11.8 (b) in more detail. To this end we suggest an extension of our model, beyond the
expression (11.74) obtained from Fermi’s golden rule.
To begin with, we investigate the e↵ect of higher order contributions in the polaron hopping J ⇤ . While an analytical series expansion is cumbersome, we note that the truncated
Hamiltonian (11.64), from which we started, is integrable. Since it does not couple di↵erent
phonon momenta k 6= k0 , we only have to solve dynamics of a driven harmonic oscillator at
each k. This can be done numerically using coherent phonon states, and takes into account all
orders in the renormalized hopping J ⇤ . Compared to a solution of the full time-dependent MF
dynamics, which includs couplings between di↵erent momenta, it is still cheaper numerically.
In FIG.11.8 (b) we also compare our results to such a full solution of the truncated
Hamiltonian (11.64) (dashed lines). While for the smallest hopping J = 0.1c/a only small
corrections to the result (11.65) from Fermi’s golden rule are obtained, we find large corrections
for J = 0.3c/a and J = 0.5c/a in weak driving regime aF  c/⇠ (deviations by up to two
orders of magnitude are observed).
To understand why this is the case, we first recall that to leading order (i.e. vd ⇠ J0⇤2 ) only
phonon emission on the fundamental frequency !B contributes, see Eq.(11.72). A higher order
series expansion moreover shows that to third order in J0⇤ , only phonons with frequencies !k =
2!B on the second harmonic contribute to vd . Therefore we expect higher order contributions
in J0⇤ to lead to phonon emission on higher harmonics. In FIG.N.1 we plot the energy density
of emitted phonons, calculated from the truncated Hamiltonian (11.64). Indeed, for large
hopping J = 0.5c/a and weak driving F = 0.048c/a2 we observe multiple resonances in
FIG.N.1 (a). For the same force but smaller hopping J = 0.1c/a in contrast, only the
fundamental frequency is relevant, see FIG.N.1 (c).
From the comparison in FIG.11.8, we moreover observe that the result Eq.(11.65) from
Fermis golden rule, which is perturbative in J ⇤ , works surprisingly well in the strong driving
regime (aF & c/⇠), even for hoppings as large as J = 0.5c/a close to the transition to the
supersonic regime. To understand why this is the case, we analyze the energy density of
phonons for large force F = 5.4c/a2 in FIG. N.1 (b) and (d). We find that in both cases
of large and small hopping, J = 0.5c/a in (b) and J = 0.1c/a in (d), only emission on the
fundamental frequency contributes. This is generally expected in the strong driving regime
aF > c⇠, as can be seen from a simple scaling analysis. Using Eq.(11.74) we expect the rate of
329
APPENDIX N. DISCUSSION AND EXTENSION OF THE ANALYTICAL
CURRENT-FORCE RELATION
330
(a)
(b)
(c)
(d)
Figure N.1: Phonon energy density ✏(k, t) in units of c of the truncated Hamiltonian (11.64)
as a function of time and radial momentum k = |k|. We integrated over the entire momentum
shell of radius
k and included the measure in the density, i.e. the total phonon energy is
R
Eph (t) = dk ✏(k, t). The results were obtained by solving full dynamics of the truncated
Hamiltonian Eq.(11.64) and starting from vacuum. Parameters are F = 0.048c/a2 and J =
0.5c/a in (a), F = 5.4c/a2 and J = 0.5c/a in (b), F = 0.048c/a2 and J = 0.1c/a in (c)
and F = 5.4c/a2 and J = 0.1c/a in (d). Positions of the first four resonances !k = n!B
for n = 1, 2, 3, 4 are indicated by dashed horizontal lines. Other parameters are ge↵ = 3.16,
⇠ = 5a and `ho = a/2 in all cases.
change of the energy density ✏(k, t) for driving with fixed frequency !B (in d = 3 dimensions)
to scale like
@
1
(m)
✏(k, t) ⇠ k 2 |Ak |2
.
(N.1)
@t
@ k !k
(m)
Estimating Ak
⇠ @q ↵kMF (q) ⇠ Vk /!k we find the following scalings with momentum,
@
✏(k, t) ⇠
@t
(
k
1
k3
if k ⌧ 1/⇠,
if k
1/⇠.
(N.2)
Thus for !B > c/⇠, i.e. for k > 1/⇠, phonon emission on higher harmonics !k = n!B with
n 2 is highly suppressed.
Finally, emission on the fundamental frequency !k = !B is captured by Fermi’s golden
(m)
rule (11.65) up to corrections of order J0⇤6 , as can be shown using a series expansion of Ak
to second order in J0⇤ . Thus in the strong driving regime, where mostly the fundamental
frequency contributes, only weak J-dependence can be expected. This is fully consistent with
FIG.11.9 (d) showing how the negative di↵erential conductance peak varies with J. Hardly
any deviations from the power-law Eq.(11.67) derived from Fermi’s golden rule can be observed
there.
Appendix O
Alternative derivation of polaron
current
In this appendix we give an alternative derivation of the analytical current-force relation
Eq.(11.65) introduced in the main text. The following treatment is somewhat simpler conceptually, however it is only valid in the limit of small force F and weak interactions ge↵ ! 0.
For simplicity we restrict our discussion to d = 3 dimensions, but all arguments can easily
generalized to arbitrary d.
The idea is to start from the Hamiltonian in Eq.(11.1) in the lab frame, i.e. before
applying the Lee-Low-Pines transformation. Then we can consider the limit ge↵ ! 0, where
to first approximation the impurity can be treated as being independent of the phonons. If
we moreover assume that the particle is sufficiently heavy, i.e. J is small, we may neglect
fluctuations of the impurity position and approximate the latter by its mean,
x(t) ⇡ hx(t)i =
2J
cos (!B t) .
!B
(O.1)
To describe the interactions of the impurity with phonons, we now plug the last equation
into Eq.(11.1) and obtain
Z
n
⇣
⌘ o
Ĥ(t) ⇡ d3 k !k â†k âk + eikx hx(t)i â†k + â k Vk .
(O.2)
Then we can expand the exponential in orders of the hopping, eikx hx(t)i ⇡ 1+ikx !2JB cos (!B t)+
O(J 2 ), and treat the resulting oscillatory term using Fermi’s golden rule. As a result we obtain,
using vd = a ph as described in the main text,
vd =
8⇡ 2 a J 2 k 4 Vk2
.
3 F 2 (@k !k )
(O.3)
Now as in the main text, we can perform a series expansion of the resulting expression
(O.3) in the driving force F . In the weak driving limit we obtain
vd =
a6
2 2 2 3
p ge↵
J ⇠ F + O(F 4 ).
c4 3⇡ 2
(O.4)
Notably, this is exactly the same expression as Eq.(11.68) from our calculation in the main
text, except that J appears instead of J0⇤ . In the strong driving limit, in contrast, we obtain
331
332
APPENDIX O. ALTERNATIVE DERIVATION OF POLARON CURRENT
a di↵erent power-law than in the main text Eq.(11.69),
vd =
21/4
c
3⇡
1/2 5/2
a
⇠
3/2 2 2
ge↵ J F
1/2
+ O(F
3/2
),
(O.5)
where we used the impurity continuum limit again, i.e. `ho ! 0. The reason why we do not
reproduce the result from the (more involved) calculation in the main text is that expanding
the exponential below Eq.(O.2) contains a small k approximation as well.
Appendix P
Renormalized impurity mass
In this appendix we would like to show that the coupling constant M in the RG protocol can
be interpreted as renormalized impurity mass. To this end we start from a Lee-Low-Pines
type polaron model with UV cut-o↵ ⇤0 and an impurity of mass M . Then we apply the RG
to integrate out phonons at momenta larger than ⇤, and show that the resulting low-energy
model is equivalent to a Lee-Low-Pines type polaron model with a UV cut-o↵ ⇤ and for an
impurity of mass M.
For simplicity we restrict ourselves to the spherically symmetric case q = 0. We start from
the Fröhlich Hamiltonian after the Lee-Low-Pines transformation,
Ĥ =
Z
⇤0
3
d k
h
!k â†k âk
+ Vk
⇣
â†k
+ âk
⌘i
1
+
2M
✓Z
◆2
(P.1)
k · k0 ˆ ˆ
: k k0 : ,
2M
(P.2)
⇤0
3
d kk
â†k âk
Next, as in the main text, we apply the MF shift ÛMF and obtain
H̃⇤0 :=
†
ÛMF
ĤÛMF
=
Z
⇤0
3
d k
†
⌦MF
k âk âk
+
Z
⇤0
d3 kd3 k0
where ⌦MF
= !k + k 2 /2M in this case. After the application of the RG from the initial UV
k
cut-o↵ ⇤0 down to ⇤, we end up with the Hamiltonian
H̃⇤ =
Z
⇤
3
d k
h
⌦k â†k âk
+ Wk
⇣
â†k
+ âk
⌘i
+
Z
⇤
d3 kd3 k0
k · k0 ˆ ˆ
: k k0 : ,
2M
(P.3)
2
where the frequency is renormalized, ⌦k = !k + k 2 /2M, and Wk = k2 M 1 M 1 ↵k .
Now, reversing the action of the MF shift, we end up with a Lee-Low-Pines type Hamiltonian
again, but at the reduced cut-o↵ ⇤,
†
ÛMF H̃⇤ ÛMF
=
Z
⇤
3
d k
h
!k â†k âk
+ Vk
⇣
â†k
+ âk
⌘i
1
+
2M
✓Z
⇤
3
d kk
â†k âk
◆2
+
E0 (⇤),
(P.4)
with E0 (⇤) describing a modified ground-state energy after integrating out phonon modes
at larger momenta. This model is equivalent to our original model, but with an impurity of
increased mass M(⇤) instead of M . That is, the e↵ect of the RG is to introduce a renormalized
impurity mass. Note that this mass-enhancement goes beyond MF, and is di↵erent from the
simple MF type enhancement of the polaron mass. The latter originates from the fact that
part of the polaron momentum q is carried by the phonons, whereas the enhancement of M
is due to quantum fluctuations of the impurity itself.
333
334
APPENDIX P. RENORMALIZED IMPURITY MASS
Appendix Q
Polaron Properties from RG
In this appendix we show in detail how polaron properties can be calculated using the RG
procedure introduced in the main text. In particular, we derive the RG flow equations of the
polaron phonon number Nph , the phonon momentum qph and the quasiparticle weight Z.
Q.1
Polaron phonon number
We start by the derivation of the RG flow equation (12.56) of the phonon number Nph . To
this end we split up the expression for Nph ,
Nph =
MF
Nph
+
Z
⇤0
d3 k hgs| ˆ k |gsi,
(see Eq.(12.54) in the main text) into contributions from slow and fast phonons,
Z
Z
MF
3
ˆ
Nph = Nph + d p hgs| p |gsi + d3 k hgs| ˆ k |gsi.
s
(Q.1)
(Q.2)
f
Next we apply the RG step, i.e. the unitary transformation Û⇤ , to simplify both integrals.
For the slow phonons, using Û⇤† ˆ p Û⇤ = ˆ p + O(⌦k 2 ), we find
Z
Z
d3 p hgs| ˆ p |gsi = d3 p s hgs| ˆ p |gsis + O(⌦k 2 ),
(Q.3)
s
s
where the ground state after applying the RG step factorizes,
|gsi = |0if ⌦ |gsis .
(Q.4)
Thus, in the subsequent RG step, we can treat the new term on the right hand side of Eq.(Q.3)
in the same way as we treated our initial term in Eq.(12.54).
The second term in Eq.(Q.2), corresponding to fast phonons, reads
Z
Z
d3 k hgs|Û⇤† ˆ k Û⇤ |gsi = 2 d3 k ↵k s hgs|F̂k |gsis + O(⌦k 2 )
(Q.5)
f
f
after the RG step. Here F̂k is defined in Eq.(12.33), from which we directly obtain the first
two terms in the square brackets of Eq.(12.56), along with an additional renormalization term
Nph ! Nph
Z
|↵k |2
2 d k
kµ Mµ⌫1
⌦k
f
3
335
Z
s
d3 p p⌫ s hgs| ˆ p |gsis .
(Q.6)
336
APPENDIX Q. POLARON PROPERTIES FROM RG
We will see below (in Q.2) that the integral over slow degrees of freedom in the second line of
Eq.(Q.6) appears as well in the expression for the phonon momentum, see Eq.(Q.9). Further,
from symmetry considerations it follows that only ⌫ = x can give a non-vanishing contribution,
assuming that q = qex points along x-direction. We can make use of Eq.(Q.9) by evaluating
it not only at our current cut-o↵ ⇤, but also at ⇤ = 0 where we obtain qph = Pph (0). From
the expression for qph at ⇤ (see Eq.(Q.9)) we thus obtain together with the result Eq.(Q.13),
Z
⇤
s
d3 p px hgs| ˆ p |gsi =
Mk (⇤)
[Pph (0)
M
Pph (⇤)] + O(⌦k 2 ).
(Q.7)
Now the inclusion of the remaining terms from Eq.(Q.6) in the RG flow of Nph Eq.(Q.5)
straightforwardly leads to our result, Eq.(12.56).
Q.2
Polaron momentum
To derive the result Eq.(12.53) stated the main text, i.e. qph (⇤) = Pph (⇤), we start by writing
the phonon momentum as
qph =
MF
Pph
+
Z
⇤0
d3 k kx hgs| ˆ k |gsi,
(Q.8)
see Eq.(12.52). Like in the first section of this appendix Q.1, we will decompose this expression
into parts including slow and fast phonons respectively. Before however, anticipating the e↵ect
of the RG, let us introduce a more general expression,
qph = P ph (⇤) + (⇤)
Z
⇤
d3 k kx hgs| ˆ k |gsi,
(Q.9)
where initially (i.e. before running the RG, ⇤ = ⇤0 ) we have
(⇤0 ) = 1,
MF
P ph (⇤0 ) = Pph
.
(Q.10)
Here P ph (⇤) is some function of the running cut-o↵ ⇤ for which we will now derive an RG flow
equation. In Eq.(Q.9) we observe that slow phonons contribute to this phonon momentum
P ph , but only with a reduced weight described by the additional factor (⇤).
Next, by applying the same steps as in Eqs.(Q.3) and(Q.5), we can easily derive the
following RG flow equations for ⌅(⇤),
!
Z
2 + k2 i
h
2
2
k
@P ph
|↵
|
1
M
k
y
z
k
x
MF
= 2
d2 k
kx kx Pph Pph
+ k2
+
, (Q.11)
@⇤
M f
⌦k
2
2 Mk
M?
as well as for (⇤),
@
=2
@⇤
Mk
Z
d2 k
f
|↵k |2 2
k .
⌦k x
(Q.12)
By comparing these RG flow equations to those of Mk 1 (⇤), see Eq.(12.38), and of Pph (⇤), see
Eq.(12.39), a straightforward calculation shows that the solutions can be expressed in terms
of the RG coupling constants,
P ph (⇤) = Pph (⇤),
(⇤) =
M
,
Mk (⇤)
(Q.13)
Q.3. QUASIPARTICLE WEIGHT
337
which apparently fulfill the required initial conditions at the initial cut-o↵ ⇤ = ⇤0 . Finally,
when ⇤ ! 0, from Eq.(Q.9) we obtain a fully converged phonon momentum qph = P ph (0) =
Pph (0) as claimed in the main text.
Q.3
Quasiparticle weight
Now we will derive the RG flow equation (12.60) of the logarithm of the quasiparticle weight,
log Z. To this end we introduce a unity 1̂ = Û⇤ Û⇤† into the definition Eq.(12.57), Z =
|h ↵k |h ↵p |gsi|2 , and obtain
2
Z = h ↵k |h ↵p |Û⇤ |0if ⌦ |gsis .
(Q.14)
Here we used that Û⇤† |gsi = |0if ⌦ |gsis and introduced the short-hand notation
|
↵k i|
↵p i ⌘
To evaluate Eq.(Q.14), we notice that |
the case, let us first use that
Y
k2f
|
↵k i ⌦
|↵p |2 |
=
p2s
|
↵p i.
(Q.15)
↵p i is an eigenstate of Û⇤† . To show why this is
†
↵p i = ˆ p ÛMF
|0is
⇣
(Q.19) †
= ÛMF â†p âp
ˆ p|
Y
(Q.16)
⌘
|↵p |2 |0is
↵p i.
(Q.17)
(Q.18)
In the second line we used that
†
ÛMF ˆ k ÛMF
= â†k âk
|↵k |2 .
(Q.19)
Thus, from the solution for F̂k Eq.(12.33) we obtain
F̂k |
where
↵p i = fk |

1
fk =
Wk
⌦k
↵p i + O(⌦k 2 ),
↵k kµ Mµ⌫1
Z
s
(Q.20)
d3 p p⌫ |↵p |2 .
(Q.21)
From the definition of Û⇤ Eq.(12.27), and noting that fk 2 R is real, it now follows that
Û⇤† |
↵p i = e
R
f
h
i
d3 k fk âk â†k +O(⌦k 2 )
|
↵p i.
(Q.22)
Using the last result, Eq.(Q.14) now factorizes, Z = Zs Zf , where
Zs = |h ↵p |gsis |2
(Q.23)
has the same form as Z but includes only slow phonons. The contribution from fast phonons
reads
h
i
R
Zf = |h ↵k |e
f
d3 k fk âk â†k +O(⌦k 2 )
|0if |2 ,
(Q.24)
338
APPENDIX Q. POLARON PROPERTIES FROM RG
and can be simplified by recognizing the displacement operator by fk ,
✓ Z
◆
2
2
2
3
2
Zf = |h ↵k | fk i| + O(⌦k ) = exp
d k |↵k fk | + O(⌦k ) .
(Q.25)
f
In summary, by applying a single RG step we can write
Z = |h ↵p |gsis |2 ⇥ e
R
f
d3 k |↵k fk |2 +O(⌦k 2 )
.
(Q.26)
I.e. in terms of the logarithm
log Z = log Zs
Z
f
d3 k |↵k
such that Eq.(12.60) immediately follows from
Z
@
log Z =
d2 k |↵k
@⇤
f
fk |2 + O(⌦k 2 ),
fk |2 + O(⌦k 2 ).
(Q.27)
(Q.28)
Appendix R
Alternative Check of the RG –
Kagan-Prokof ’ev theory
In the main text we have applied the RG protocol to obtain corrections to previously derived
MF results, and we established its validity by comparing to recent MC calculations [22]. In
this appendix we subject our RG procedure to a second, independent check by applying it to
another case which has been discussed before. To this end we simplify our model (in R.1) by
considering only vanishing polaron momentum q = 0 and dropping cubic as well as quartic
terms in the fluctuation Hamiltonian (12.3). Next, we show that the resulting simplified
Hamiltonian has a particular form which was discussed and solved by Kagan and Prokof’ev
[378]. After clarifying the relation between Kagan’s and Prokof’ev’s results to our case (in
R.2), we show (in R.3) that our RG protocol for the simplified model agrees reasonably with
the established results by comparing polaron groundstate energies (in R.4).
R.1
Simplified Model
In the subsequent analysis we want to restrict ourselves to a simplified model which is analytically tractable. To this end we neglect terms in the Hamiltonian (12.3) involving powers
larger than two in the quantum fluctuations âk . This amounts to the approximation
⇣
⌘⇣
⌘
†
ˆ k ˆ k0 ⇡ ↵k ↵k0 âk + â†
0 + â 0
â
(R.1)
k
k
k
in the last term of Eq.(12.3). Moreover we consider only the spherically symmetric case of
MF = 0. In this case, because of symmetry, there is
vanishing polaron momentum, i.e. q = Pph
no RG flow of the coupling constant Pph .
The starting point for our analysis is the following simplified Hamiltonian,
ĤKP = HMF
Z
Z
k2
†
2
d k
|↵k | + d3 k ⌦MF
k âk âk +
2M
Z
⇣
⌘⇣
⌘
1
↵k ↵k 0
+
d3 k d3 k 0
(k · k0 ) âk + â†k âk0 + â†k0 . (R.2)
2
M
3
It is obtained from Eq.(12.3) after normal-ordering the remaining terms, which gives rise to
the additional energy shift in the first line.
339
APPENDIX R. ALTERNATIVE CHECK OF THE RG – KAGAN-PROKOF’EV
THEORY
340
R.2
Relation to Kagan and Prokof ’ev theory
The solution of the simplified Hamiltonian (R.2) was derived by Kagan and Prokof’ev. Now
we discuss their expressions and show how they map to our model.
R.2.1
Kagan-Prokof ’ev theory
In the appendix A of [378], Kagan and Prokof’ev discuss a general Hamiltonian of the form
org
ĤKP
=
X
⌫k â†k âk +
X Bk,k0 ⇣
k,k0
k
2
âk + ↠k
⌘⇣
⌘
âk0 + ↠k0 ,
(R.3)
where indices k0 and k label phonon momentum states and Bk,k0 is an arbitrary function of
the latter. In their case a 1D system is considered, but at least formally we can map our 3D
system onto a lower-dimensional one and apply their theory.
Kagan and Prokof’ev show that the groundstate of this Hamiltonian can be written as
(see also [391])
0
1
X
0
⌘k,k
1
| 0 i = R exp @
↠↠0 A |0i.
(R.4)
2 0 ⌫k + ⌫k 0 k k
k,k
Here |0i is the vacuum, âk |0i = 0 and R denotes a normalization factor. The amplitude ⌘k,k0
has to be determined from the following self-consistency equation
⌘k,k0 = Bk,k0
X Bk,
q
X Bk 0 ,
q ⌘q,k0
⌫q + ⌫
k0
q
q ⌘q,k
⌫q + ⌫k
+
X
q,q 0
B q0 , q ⌘q,k0 ⌘q0 ,k
.
(⌫q + ⌫k0 ) ⌫q0 + ⌫k
(R.5)
Importantly, Kagan and Prokof’ev also derived a closed expression for the groundstate energy
org
EKP
of the Hamiltonian Eq.(R.3) in terms of the amplitude ⌘k,k0 ,
E0 |org
KP =
R.2.2
1X
Bk,
2
k
k
1 X Bk,k0 ⌘ k,
2 0 ⌫k + ⌫k0
k0
.
(R.6)
k,k
Polaron Hamiltonian
To solve for the groundstate energy of the approximate Hamiltonian Eq.(R.2), we now derive
a formal analogy between the latter and the Hamiltonian considered by Kagan and Prokof’ev
(R.3). This allows us to calculate the polaron groundstate energy in the following paragraph
and compare it to results from our RG protocol.
To make the analogy apparent, we perform a gauge transformation
✓
◆
Z
⇡
†
3
ÛKP = exp
i
d k âk âk
(R.7)
2
on our simplified Hamiltonian (R.2). Then, using also spherical symmetry of the MF solution
↵ k = ↵k at q = 0, we obtain
†
ÛKP
ĤKP ÛKP
= HMF
Z
Z
k 2 |↵k |2
†
d k
+ d3 k ⌦MF
k âk âk
2M
Z
⇣
↵k ↵k 0
d3 k d3 k 0
k · k0 â
2M
3
k
+ â†k
⌘⇣
â
k0
⌘
+ â†k0 . (R.8)
R.3. COMPARISON WITH RG
341
Comparison to Eq.(R.3) shows that the approximate polaron Hamiltonian can be solved by
Kagan-Prokof’ev theory. To this end we identify
Bk,k0 =
where we defined
R.2.3
⌦MF
k = ⌫k ,
Bk · Bk 0 ,
↵k
Bk = k p .
M
(R.9)
(R.10)
Application to polaron case
We are now in a position to apply Kagan-Prokof’ev theory to our simplified polaron Hamiltonian (R.8). Following Kagan’s and Prokof’ev’s tricks [378], we can derive simplified expressions
for the self-consistency equation (R.5) as well as the groundstate energy Eq.(R.6). To this
end we make an ansatz for the amplitudes
⌘k,k0 = ⌘k ⌘k0 ,
(R.11)
where – because of the spherical symmetry – only k = |k| appears. An analytic calculation
then leads to the following non-linear self-consistency integral equation in our 3D case,
⌘k = 1
1
3
Z
d3 p
p2 ↵p2 ⌘p
⌘k ,
M (⌫k + ⌫p )
(R.12)
which is very similar 1 to Eq.(A11) in the paper by Kagan and Prokof’ev [378].
Plugging the ansatz (R.11) into the expression for the groundstate energy Eq.(R.6), we
obtain the following energy E0 |KP for the polaron groundstate of Hamiltonian (R.8),
E0 |KP = HMF +
Z
d3 k
k2
|↵k |2 (⌘k
2M
1) .
(R.13)
Note that we included the second term on the right hand side of Eq.(R.2), which comes from
normal-ordering the polaron fluctuation Hamiltonian. Keeping this (UV divergent) term is
essential to cancel the unphysical power-law UV divergences of the polaron energy. Let us
further mention that the contribution ⇠ ⌘k in Eq.(R.13) has a very similar form to Eq.(A10)
in the paper by Kagan and Prokof’ev [378].
R.3
Comparison with RG
Having discussed the Kagan-Prokof’ev solution of our simplified model, we now summarize
our RG formalism for this case. The derivation of the RG flow equations goes completely
analogous to that presented in Section 12.3.2, so we will not repeat it here.
In our spherically symmetric case (q = 0) there is only a single parameter flowing in the
1
When comparing to Kagan and Prokof’ev’s result Eq.(A11) in [378], we note that our expression Eq.(R.12)
di↵ers by a minus sign. To see why this is so, note that in our case Bk,k0 = Bk ·Bk0 with = 1, see Eq.(R.9);
To obtain our result Eq.(R.12) from Eq.(A11) in [378] we have to use = +1 however. This discrepancy, it
appears, is due to the fact that Kagan and Prokof’ev consider only the case B k = Bk⇤ , whereas in our case it
holds B k = Bk⇤ see Eq.(R.10). To check that this is the reason for the discrepancy, we carefully re-derived
Eq.(A11) of Kagan and Prokof’ev from their expression (A8), but using the more general relation B k = ⌧ Bk⇤ .
Assuming ⌧ = ±1 this is consistent with Hamiltonian (R.3) being self-adjoint. After doing the math, we obtain
back their equation (A11), but with replaced by ⌧ .
342
APPENDIX R. ALTERNATIVE CHECK OF THE RG – KAGAN-PROKOF’EV
THEORY
RG, the mass term
µ⌫
Mµ⌫1 (⇤) =
M(⇤)
.
(R.14)
Applying our standard procedure yields the following RG flow equation,
@M
@⇤
1
=
2
M
3
2
4⇡
⇤4 |↵⇤ |2
,
⌫⇤
(R.15)
which can be solved analytically (cf. Sec.12.6.1)
8⇡
M(⇤) = M +
3
Z
⇤0
dp
⇤
p4 |↵p |2
.
⌫p
(R.16)
Here ⇤0 denotes a sharp momentum cut-o↵ used to regularize our model, and in the following
we consider the limit ⇤0 ! 1.
As for the full model, we obtain the polaron groundstate energy E0 |RG
KP for our simplified
model Eq.(R.2) directly from the formulation of the RG protocol,
✓
◆
Z
k2
M
RG
3
2
E0 |KP = HMF + d k
|↵k |
1 .
(R.17)
2M
M(k)
This expression has a remarkably similar form to the result derived from Kagan and Prokof’ev,
see Eq.(R.13). To obtain the RG result from the latter, we merely have to replace ⌘(k) by the
ratio M/M(k). This identification, in turn, gives a concrete physical meaning to the quantity
⌘(k) defined by Kagan and Prokof’ev.
R.4
Results: Kagan-Prokof ’ev versus RG
Now we compare results from Kagan-Prokof’ev theory to the RG formalism. We will show
that the RG is able to capture all key properties of the simplified polaron model (R.2). In
particular we compare the function ⌘k appearing in Kagan-Prokof’ev theory to the prediction
for the renormalized impurity mass M/M(k) from the RG. In addition groundstate energies
are calculated and the UV divergence is discussed in both formalisms.
R.4.1
Polaron mass term
We start by comparing the function ⌘k determined from Kagan-Prokof’ev theory to the mass
term M/M(k) from the RG. Before making a quantitative numerical comparison between
both models, we briefly discuss the asymptotic behaviors.
In the infrared (IR) limit ⇤ ! 0 we find from Eq.(R.16) for the RG result
M
=
M(0)
✓
8⇡
1+
3M
Z
1
0
k 4 |↵k |2
dk
⌫k
◆
1
.
(R.18)
The asymptotic expression for ⌘k in the IR reads
⌘0 =
✓
8⇡
1+
3M
Z
1
0
k 4 |↵k |2
dk
⌫k
◆
1/2
,
(R.19)
and a derivation of this result can be found in Appendix R.5.2. Remarkably, both theories
yield expressions of almost identical form, except for the di↵erent exponent.
R.5. ASYMPTOTIC SOLUTIONS OF KAGAN-PROKOF’EV THEORY
343
In the UV limit the situation is di↵erent. The asymptotic solution for the renormalized
impurity mass reads
M
32
mred 1
=1
n0 a2IB
+ O(1/k 2 ),
(R.20)
M(k)
3
M k
see Eq.(12.44). For ⌘k on the other hand we show at the end of this appendix in R.5 that
⌘(k) = 1
8⇡
mred 1
n0 a2IB
+ O(1/k 2 ),
3
M k
(R.21)
which di↵ers only by a numerical factor ⇡/4 in the first-order coefficient from the RG result.
In FIG.R.1 (a) we compare Kagan-Prokof’ev theory to the RG by calculating the parameters ⌘k and M/M(k) respectively, which determine the polaron groundstate energies. To this
end we solved numerically Eqs.(R.12), (R.16) for parameters in the strong-coupling regime.
We observe good qualitative agreement between both results, although quantitatively rather
large deviations are found. From the asymptotic behaviors of both quantities (shown in the
plot for the UV regime) we obtain good analytic understanding of these deviations, however.
R.4.2
Polaron energy
As we have seen in the last paragraph, the mass term M/M(k) of the RG is smaller than
the corresponding term ⌘(k) in Kagan-Prokof’ev theory. Because it is only these two terms
which determine the groundstate energy, see Eqs.(R.13), (R.17), we find that E0 |KP > E0 |RG
KP .
Indeed, in the concrete example shown in FIG.R.1 (b) we find that the RG predicts a smaller
energy than Kagan-Prokof’ev theory. The qualitative behavior, however, is the same in both
cases. By the inclusion of quantum fluctuations in the simplified model (R.2), both theories
predict smaller energies than the MF value.
We can gain further analytical insight into the polaron energy by deriving the logarithmic
UV divergence. Both models are UV divergent, but with di↵erent pre-factors. Using the
asymptotic expressions (R.20), (R.21) valid in the UV in the equations for the polaron energy
(R.17), (R.13) we derive the following log-divergence from the RG,
E0UV |RG =
2 ↵2 mred
log (⇤0 ⇠) .
3⇡ 2 M 2 ⇠ 2
(R.22)
This expression is equivalent to the log divergence of the full polaron model, see Eq.(R.22).
From Kagan-Prokof’ev theory on the other hand we obtain
E0UV |KP =
⇡ UV
E |RG ,
4 0
(R.23)
which di↵ers slightly from the RG by the numerical prefactor ⇡/4 = 0.785... . Most importantly the algebraic dependence on model parameters, like the dimensionless coupling constant
↵, are exactly the same for both theories.
R.5
Asymptotic solutions of Kagan-Prokof ’ev theory
In this last section of appendix R we derive the asymptotic behaviors of the solution ⌘(k) of
Kagan-Prokof’ev theory for the simplified polaron Hamiltonian Eq.(R.2). In the main text
we only mentioned our results, see Eqs.(R.21), (R.19).
The parameter ⌘(k) is determined by the non-linear equation (R.12). To derive its asymptotic values, we make use of an alternative form of this self-consistency equation. After using
APPENDIX R. ALTERNATIVE CHECK OF THE RG – KAGAN-PROKOF’EV
THEORY
344
(a)
(b)
0
10
0.7
0.6
10
0.5
25
2
20
0.4
10
0.3
15
4
0
2
10
0.2
10
4
10
10
5
0.1
0
0
30
5
10
15
20
0
0
1
2
3
4
5
Figure R.1: (a) Comparison of 1 ⌘(k) from Kagan and Prokof’ev theory (solid blue) to
1 M/M(k) determined from the RG protocol (dashed red). We obtain good qualitative
agreement, while quantitatively both results di↵er somewhat. For large momenta k we also
plotted the asymptotic behaviors in black (dash-dotted: 1 ⌘k , short-dashed: 1 M/M(k)),
and form the double-logarithmic plot of the data in the inset we find that they are indeed
asymptotically approached. The inset moreover shows that the overall form of both curves is
qualitatively the same. Parameters are ↵ = 6, M/mB = 0.26, q/M c = 0.01 and ⇤0 = 2⇥104 /⇠.
(b) Comparison of the polaron energy E0 from the simplified model Eq.(R.2), calculated using
di↵erent techniques. Neglecting quantum fluctuations all together yields the MF result HMF
(dashed-dotted). From Kagan-Prokof’ev theory we obtain Eq.(R.13), and the resulting curve
(KP, solid) shows deviations from MF. The RG result for the simplified model Eq.(R.17) is
also shown (dashed). Parameters are M/mB = 0.26, n0 = ⇠ 3 , q = 0 and ⇤0 = 200/⇠.
in the definition of the MF amplitude ↵k = Vk /⌫k and plugging the self-consistency equation
into itself following the prescription by Kagan and Prokof’ev, we arrive at
"
#
Z 1
Z 1
k 4 Vk2
k 4 Vk2
8⇡
4⇡
⌘k 0 1 +
dk
=
1
+
dk
⌘k .
(R.24)
3M 0
3M 0
⌫k2 (⌫k ⌫k0 )
⌫k ⌫k2 ⌫k20
This is a Fredholm integral equation for ⌘k0 , which is of the same form as derived by Kagan
and Prokof’ev (cf. Eq.(A13) in their paper [378]). Importantly, it is linear in ⌘k0 .
R.5.1
UV asymptotics
To derive the UV behavior of ⌘k0 as k 0 ! 1, we split up the integrals in Eq.(R.24) into
two separate regions. The IR region is defined by 0  k  ⇤1 with ⇤1
1/⇠, whereas the
UV region is composed of momenta k > ⇤1 . Let us moreover choose ⇤1 ⌧ k 0 , which may
consistently be assumed provided that k 0 is already sufficiently large. Moreover, we expand
⌘k around its asymptotic value,
⌘k = ⌘1
/k + O(1/k 2 ),
(R.25)
where the coefficients ⌘1 and are to be determined below.
We start by the IR parts of the integrals, which we may simply expand in 1/k 0 because
they do not include any poles. For the first integral we find
Z
⇤1
dk
0
⌫k
k 4 Vk2
= O(k 0 )
⌫k2 ⌫k20
0
4 k !1
! 0,
(R.26)
R.5. ASYMPTOTIC SOLUTIONS OF KAGAN-PROKOF’EV THEORY
345
whereas the second integral yields
Z
⇤1
dk
0
k 4 Vk2
⌘k = O(k 0 )
⌫k2 (⌫k ⌫k0 )
0
2 k !1
! 0.
(R.27)
When evaluating the UV part of the integrals, care has to be taken because of the appearance
of poles. Since ⇤1
1/⇠, we may furthermore use the asymptotic behaviors for Vk ⇡ V1 and
⌫k ⇡ Ak 2 , see Eq.(12.43). Next we analytically continue ⌘k0 by setting k 0 ! k 0 i✏, where
✏ > 0 will be sent to zero afterwards. Then, for the UV part of the first integral in Eq.(R.24),
we obtain
Z 1
2 Z 1
2 ⇡
k 4 Vk2
V1
k4
V1
1
dk
⇡
dk
=
(1 i) 0 .
(R.28)
4
2
2
3
3
4
0
A 0
A 4
k
⌫k ⌫k ⌫k 0
k
(k )
⇤1
Analogously we obtain for the second integral,
Z
1
⇤1
dk
k 4 Vk2
⌘k ⇡
⌫k2 (⌫k ⌫k0 )
⌘1
2 ⇡ i
V1
.
A3 2 k 0
(R.29)
Finally, combining the above results in Eq.(R.24), we find
⌘k 0 = 1
2 1
2⇡ 2 V1
( i⌘1 + i
3M A3 k 0
1) + O(k 0 )
2
.
(R.30)
Taking the limit k 0 ! 1 in this expression after employing Eq.(R.25) for ⌘k0 on the left hand
side, we obtain ⌘1 = 1 and the asymptotic relation Eq.(R.21) claimed in the main text.
R.5.2
IR asymptotics
To obtain the IR asymptotics, i.e. the value of ⌘0 , we start from Eq.(R.24) and get
⌘0 =
where we defined
4⇡
X=
3M
Z
1+X
,
1 + 2Y
1
dk
0
(R.31)
k 4 Vk2
⌘k ,
⌫k3
(R.32)
k 4 Vk2
,
⌫k3
(R.33)
which depends on the solution ⌘k , and
4⇡
Y =
3M
Z
1
dk
0
which is independent of ⌘k and can thus be calculated. Moreover, using the non-linear equation
(R.12) for ⌘k , we also find
1
⌘0 =
.
(R.34)
1+X
Combining the last two results Eqs.(R.31), (R.34) we obtain
⌘0 = (1 + 2Y )
1/2
,
corresponding to our result from the main text Eq.(R.19).
(R.35)
346
APPENDIX R. ALTERNATIVE CHECK OF THE RG – KAGAN-PROKOF’EV
THEORY
Appendix S
Summary of the dRG
In this appendix, a summary of the dRG technique is given. Here the technical mathematical
steps will be skipped, but references are provided to the respective places in the thesis where
calculations can be found. This appendix is intended for readers familiar with the equilibrium
RG protocol introduced in Chap.12, who want to understand only the key steps required to
generalize the scheme to non-equilibrium situations. Readers who prefer a completely selfcontained presentation are referred to the treatment provided in the main text, see Sec.13.1.
This appendix is divided into two parts. First, in S.1, we motivate the basic idea of our
dRG protocol for calculating time-dependent observables. Then, in S.2 we explain how timedependent overlaps can be calculated from the dRG, which are required to derive the spectral
function of the polaron.
S.1
Time-dependent observables
Our goal in this section is to calculate the time-dependence of observables Ô in the polaron
problem. For concreteness we restrict ourselves to observables which do not involve correlations between di↵erent phonon momenta and can hence be written as
Ô =
Z
⇤0
d3 k Ôk .
(S.1)
As a first step, we formulate the problem in the frame of quantum fluctuations around the
MF polaron, i.e. we apply the unitary transformation ÛMF (see FIG.12.2) and obtain
O(t) =
=
Z
Z
⇤0
⇤0
†
†
d3 k h0|ÛMF eiÛMF Ĥq ÛMF t ÛMF
Ôk ÛMF e
d3 k h ↵ |eiH̃q t ôk e
iH̃q t
|
↵ i.
†
iÛMF
Ĥq ÛMF t
†
ÛMF
|0i
(S.2)
Here H̃q denotes the Hamiltonian from Eq.(12.5) describing dynamics of quantum fluctuations
†
around the MF polaron, and the observable in the MF frame reads ôk = ÛMF
Ôk ÛMF . In the
†
second line we also used Eq.(12.58) to express ÛMF
|0i = | ↵ i, where | ↵ i is short-hand
Q
for  | ↵ i.
Next, we will motivate the basic idea of the non-equilibrium RG protocol. To this end, let
us start by recalling the equilibrium RG procedure from Sec.12.3. In that case, by performing
consecutive basis transformations Û⇤ , we simplified the Hamiltonian (12.5) which in the new
347
348
APPENDIX S. SUMMARY OF THE DRG
frame takes the form,
Û⇤† H̃q Û⇤
=
Z
d3 k
f
⇣
⌘
ˆ s (k) ↠âk + Ĥs + Ĥs ,
⌦k + ⌦
k
(S.3)
see Eq.(12.34). Note that the fast-phonon frequency ⌦k is flowing in the RG protocol. From
the expression (S.3), it was easy to read o↵ the fast-phonon ground state which is simply the
vacuum state, âk |0if = 0. However, sin