Download Quixotic Order and Broken Symmetry in the Quantum Hall Effect and

Quixotic Order and Broken Symmetry
in the Quantum Hall Effect
and its analogs
Siddharth Ashok Parameswaran
A Dissertation
Presented to the Faculty
of Princeton University
in Candidacy for the Degree
of Doctor of Philosophy
Recommended for Acceptance
by the Department of
Physics
Adviser: Shivaji L. Sondhi
September 2011
c Copyright by Siddharth Ashok Parameswaran, 2011.
All rights reserved.
Abstract
We study several quantum phases that are related to, or inspired by, the quantum
Hall effect. Our initial focus is on paired quantum Hall states in the weak-coupling
limit, where we construct a Landau-Ginzburg description akin to that of a standard
BCS superconductor; using this, we point out that the behavior in this limit is characteristic of a new kind of ‘Type I’ quantum Hall liquid, named in analogy to the
corresponding regime of a superconductor. We generalize this to other quantum Hall
states, provide a prescription of how to generate phases with Type I behavior both
within long-wavelength effective field theories and from microscopic Hamiltonians,
and discuss the experimental relevance of the same.
Next, we examine a pair of quantum Hall ferromagnets where the quantum Hall
ordering occurs simultaneously with a spontaneous breaking of an internal symmetry
associated with a semiconductor valley index, in each case deriving effective descriptions from microscopic Hamiltonians and symmetry arguments. Our first example,
relevant to aluminium arsenide quantum wells, exhibits nematic ordering that has
observable consequences for transport in both dirty and clean systems; the second,
applicable to bilayer graphene, hosts doubly-charged topological excitations that are
an intriguing example of charge binding in a purely repulsive system, and may be
observable via scanning probes.
We then broaden our discussion to the study of antiferromagnetic lattice spin systems with quantum-disordered valence-bond-solid ground states. After mapping to a
classical companion model on the same lattice – similar to Laughlin’s plasma analogy
in the quantum Hall problem – we use a combination of analytical results and numerical simulations to demonstrate a variety of ordered and disordered states on the threedimensional diamond and pyrochlore lattices and (formally infinite-dimensional) random graphs.
iii
We close with a pair of miscellaneous results: the first argues that symmetrybreaking transitions on random graphs occur in the absence of Goldstone modes, while
the second is a computation of the superconducting susceptibility of an interacting
Fermi liquid within the framework of the renormalization group.
iv
Acknowledgements
First of all, I’d like to thank my adviser, Shivaji Sondhi. From our first conversation
as I contemplated taking the plunge into condensed matter, he has been a steady
source of encouragement, and his dry humor and thoughtful criticism have made the
growing pains of graduate school easier to bear. Shivaji has been the ideal adviser:
never impatient, taking the time to teach me various subtleties that a less punctilious
scientist would have swept under the rug, and educating by example – always framing
just the right question to ask or suggesting the most concise calculation to do. The
sheer breadth of his intellectual ambit, both within physics and without, has always
illuminated much more than just the problem at hand, and I am grateful to him as
a teacher, counselor, colleague, and friend.
Apart from Shivaji, I have been lucky enough to collaborate with an extraordinary
group of physicists, all of whom have immeasurably improved my education. Daniel
Arovas worked with me on my first project and set an example of attention to detail
that I have tried hard to emulate. Nearly everything I know about quantum Hall ferromagnets and how to think about experiments I learned from Dmitry Abanin, and the
two papers we wrote together will always be favorites of mine. Ramamurti Shankar
has taught me many things about renormalization and field theories, and apart from
his unfailingly clear explanations, he has been generous with advice, friendship, and
(occasionally terrible) puns. I have spent many hours working with Boris Spivak,
at a blackboard in Seattle and on Skype virtually everywhere else; I am grateful for
his willingness to patiently explain what to him is often immediately obvious. Steve
Kivelson is at once a collaborator and my adviser once removed; the month I spent
at Stanford visiting him was among the happiest times of my graduate career. More
recently, Steve Simon and Ed Rezayi have helped fill in gaps in my understanding of
the Hall effect. Finally, there’s Chris Laumann: scientific sibling, occasional hiking
buddy, and friend. While our robust debates often drove us to distraction, the rev
sulting work was far better for it, and I will miss having him next door to keep my
mathematics honest and my hand-waving restrained.
The Princeton physics department has been a wonderful and stimulating environment. David Huse always seems to know the answer to any question put to him, and
a conversation with him is invariably enlightening; Duncan Haldane is inspiring in his
unwillingness to settle for a less than complete understanding of any problem, and I
am honored that he agreed to be a reader of this thesis; Ravin Bhatt’s encyclopedic
lectures on disordered systems placed many things, for the first time, in perspective;
Andrei Bernevig, first as a postdoc and then as a professor, has at once been a useful
source of advice on physics, and a convenient upper bound on productivity, energy,
and enthusiasm; Michael Aizenman’s knowledge of Cayley trees and Cheeger bounds
has been invaluable; Mansour Shayegan’s experiments and his willingness to explain
their minutiae led to a chapter of this; Igor Klebanov agreed to serve on my committee alongside David and Mansour; Will Happer and his group, particularly Brian
Patton, supervised my experimental project; and Steve Gubser, Dan Marlow, Lyman
Page and David Huse were supportive when I assisted in their courses.
Among the condensed matter postdocs, Ronny Thomale has on countless occasions served as a sounding board and Bryan Clark and Zlatko Papic have explained
numerics to me with impressive patience. Discussions with fellow students Fiona
Burnell, Anushya Chandran, Amir Erez, Hyungwon Kim, Mike Kolodrubetz, Akshay
Kumar, Charles Mathy, Arijeet Pal, Colin Parker, and Bo Yang have played a major
role, both in working out specific ideas as well as in my broader education.
During summers, I indulged in the ‘Leisure of the Theory Classes’, and am grateful
to the Insitute for Mathematical Sciences in Chennai, the Max Planck Institute in
Dresden and the Ecole de Physique in Les Houches for their hospitality. At these
places and elsewhere, I have benefited tremendously from conversations with Maissam
Barkeshli, Yafis Barlas, Ganpathy Baskaran, Rudro Biswas, Benoit Estienne, Eduardo
vi
Fradkin, Maria Hermanns, Tom Jackson, Roderich Moessner, Joel Moore, Vadim
Oganesyan, Srinivas Raghu, Gil Refael, Nicolas Regnault, Rahul Roy, Todadri Senthil,
Kirill Shtengel, Oleg Tchernyshyov, Ashvin Vishwanath, and many others.
Sarada Rajeev, Dan Watson, Carl Hagen and Nick Bigelow taught me and gave me
research opportunities as an undergraduate and without their early encouragement
this thesis would have been impossible. I’d also like to acknowledge administrators
that have made bureaucratic nightmares vanish: Toni Sarchi, Pat Barwick, Regina
Savadge, Charlene Borsack, Kim Fawkes, Jessica Heslin and most of all Laurel Lerner,
all at Princeton, and the incomparable Janet Fogg, at Rochester.
I’ve had a great group of physicist-roommates: Aakash Pushp, Tibi Tesileanu,
and Fiona Burnell at 7V Magie, and Thomas Dumitrescu, Sasha Rahlin, Will East,
and Eduardo da Silva Neto at 85 Red Hill Road. Conversations with them – on topics
from beer to bosons and Pacino to politics – have been among the highlights of the
past five years. Also, Alex Dahlen, XinXin Du, Josh Ruderman, Natalie Kostinski,
Cynthia Chiang, Bart McGuyer, Ben Olsen, Richard Saldanha, Silviu Pufu, Fabio
Rocha, Abhi Nellore, Justin Brown, Audrey Sederberg, Shannon Hughes, Sri IyerBiswas, Jess Bernevig, Justin Curry and the dBar gang have made Princeton fun.
My survival owes much to welcome distractions provided by friends and family:
Mike Weiss, Dan Nice, Danielle Friedman, Bob & Christina Forties, Melissa McClure
and Shweta Krishnan; Bhavik Sanghvi, Meghna Sukumar, Indira Ramesh, Madhukar
Sivakumar, Aditya Rathi and Saloni Shah; Rajiika Ramalingam & Raghu Boggaram;
Anita Vishwanath; Sarah, Harish, Naveen, Courtney & Sarada Nataraj; Ram Ramachandran; the Subramanians, Sivaramakrishnans, Ramans and Ramnarayans; my
late uncle Nataraj; both grandmothers; and various others too numerous to list here.
Most of all, I owe my parents, Sarojini and Ashok, more than I can ever repay
for their love, encouragement, and support over the years; words are insufficient to
acknowledge such a debt. This thesis is, of course, dedicated to them.
vii
Relation to Previously Published Work
Chapter 2 is based on [185]; Chapter 4 on [4] and work in progress [186]; Chapter
5 is a slightly modified version of [5]; Chapter 6 is based on [188]; Chapter 7 on [134];
Chapter 8 on [133]; and Chapter 9 on [187]. Most of these bear slight modifications
from the published versions; nevertheless, we note that the copyright to the original
articles rests with the American Physical Society.
Chapter 3 is derived from work in progress with S.L. Sondhi, B. Z. Spivak, S. H.
Simon, S. A. Kivelson and E. H. Rezayi.
viii
To my parents, who knew I could, even when I didn’t.
Coming back to where you started is not the same as never leaving.
– Terry Pratchett
quixotic:
adj. 1. Of an action, attribute, idea, etc.: characteristic of or
appropriate to Don Quixote; demonstrating or motivated by
exaggerated notions of chivalry and romanticism; naively
idealistic; unrealistic, impracticable; (also) unpredictable,
capricious, whimsical.
– The Oxford English Dictionary
ix
Contents
I
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
Relation to Previously Published Work . . . . . . . . . . . . . . . . . . . .
viii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xvi
Introductory Material
1
1 The Quantum Hall Effect
2
1.1
A Brief History, 1879-1984 . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2
Samples and Probes
. . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.3
The Integer Effect
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
1.4
1.5
1.3.1
Single-particle physics: Landau levels
. . . . . . . . . . . . .
10
1.3.2
Semiclassical Percolation . . . . . . . . . . . . . . . . . . . . .
12
Why is the Quantization Robust? . . . . . . . . . . . . . . . . . . . .
15
1.4.1
Laughlin’s Argument
15
1.4.2
Hall Conductance as a Topological Invariant
. . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . .
17
. . . . . . . . . . . . . . . . . . . . . . . . . .
21
1.5.1
Trial Wavefunctions . . . . . . . . . . . . . . . . . . . . . . . .
21
1.5.2
Plasma Analogy . . . . . . . . . . . . . . . . . . . . . . . . . .
26
1.5.3
Neutral Excitations and the Single-Mode Approximation . . .
27
1.5.4
Fractionally Charged Excitations . . . . . . . . . . . . . . . .
28
The Fractional Effect
x
1.5.5
Life on the Edge . . . . . . . . . . . . . . . . . . . . . . . . .
30
1.5.6
Hierarchies
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
Half-Filled Landau Levels . . . . . . . . . . . . . . . . . . . . . . . .
37
1.6.1
Composite Fermions and the Halperin-Lee-Read Theory . . .
38
1.6.2
Paired States of Composite Fermions . . . . . . . . . . . . . .
45
Landau-Ginzburg Theories of the Quantum Hall Effect . . . . . . . .
48
1.7.1
Composite Boson Chern-Simons theory . . . . . . . . . . . . .
49
1.7.2
A Landau-Ginzburg Theory for Paired Quantum Hall States .
51
1.7.3
Off-Diagonal Long Range Order in the lowest Landau level . .
54
1.8
Type I and Type II Quantum Hall Liquids . . . . . . . . . . . . . . .
57
1.9
ν = 1 is a Fraction Too: Quantum Hall Ferromagnets . . . . . . . . .
58
1.9.1
Spin Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
1.9.2
Skyrmions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
1.9.3
Low-energy Dynamics . . . . . . . . . . . . . . . . . . . . . .
63
1.9.4
Other Examples . . . . . . . . . . . . . . . . . . . . . . . . . .
65
1.6
1.7
1.10 Antiferromagnetic Analogs and AKLT States
II
. . . . . . . . . . . . .
66
1.11 This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
Type I Quantum Hall Liquids
2 The Weakly Coupled Pfaffian as a Type I Quantum Hall Liquid
72
73
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
2.2
Landau-Ginzburg theory . . . . . . . . . . . . . . . . . . . . . . . . .
76
2.3
Vortex structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
2.4
Vortex Binding and Optimal Droplets . . . . . . . . . . . . . . . . . .
80
2.5
Phase diagram near ν = 5/2 . . . . . . . . . . . . . . . . . . . . . . .
80
2.6
Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
2.7
Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
xi
3 More Type I Quantum Hall Liquids
84
3.1
Chern-Simons Landau-Ginzburg Theory . . . . . . . . . . . . . . . .
84
3.2
Paired quantum Hall states . . . . . . . . . . . . . . . . . . . . . . .
88
3.3
Microscopic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
3.4
Type I plateau formation . . . . . . . . . . . . . . . . . . . . . . . . .
91
3.5
Experimental Realizations . . . . . . . . . . . . . . . . . . . . . . . .
91
3.6
Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
III
Valley Ordered Quantum Hall States
4 Nematic Valley Ordering in Quantum Hall Systems
94
95
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
4.2
Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
4.3
Ising anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
4.4
Thermal properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.5
Domain walls and quasiparticles . . . . . . . . . . . . . . . . . . . . . 102
4.6
Properties of the clean system . . . . . . . . . . . . . . . . . . . . . . 104
4.7
Length scales from weak disorder . . . . . . . . . . . . . . . . . . . . 104
4.8
Intrinsic resistive anisotropy . . . . . . . . . . . . . . . . . . . . . . . 106
4.9
Domain walls and the QHRFPM . . . . . . . . . . . . . . . . . . . . 108
4.10 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.11 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.12 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5 Charge 2e skyrmions in bilayer graphene
112
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.2
Landau levels of Bilayer Graphene . . . . . . . . . . . . . . . . . . . . 115
5.3
Splitting of the zero-energy Landau level . . . . . . . . . . . . . . . . 116
xii
5.4
Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.4.1
Skyrmion energy . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.5
Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.6
Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
IV
Antiferromagnetic Analogs of the Hall Effect
6 Order and Disorder in AKLT Antiferromagnets
126
127
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.2
AKLT States: A Brief Review . . . . . . . . . . . . . . . . . . . . . . 130
6.3
Mean Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.4
Unfrustrated Lattices: Simple Cubic and Diamond
6.5
6.6
. . . . . . . . . . 133
6.4.1
Mean Field Transition . . . . . . . . . . . . . . . . . . . . . . 133
6.4.2
Monte-Carlo Simulations . . . . . . . . . . . . . . . . . . . . . 134
Frustrated Lattice: The Pyrochlore . . . . . . . . . . . . . . . . . . . 137
6.5.1
Single-Tetrahedron Ground States . . . . . . . . . . . . . . . . 138
6.5.2
Ground States on the Full Lattice . . . . . . . . . . . . . . . . 139
6.5.3
Bounds on Tc . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7 AKLT Models with Quantum Spin Glass Ground States
144
7.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.2
AKLT and arbitrary graphs . . . . . . . . . . . . . . . . . . . . . . . 147
7.3
Transfer Matrix Solution of the Classical Problem on Trees . . . . . . 148
7.4
Variational Bounds on the Gap . . . . . . . . . . . . . . . . . . . . . 153
7.5
AKLT model on regular random graphs . . . . . . . . . . . . . . . . . 156
7.6
Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
xiii
V
Miscellany and Marginalia
166
8 There are no Goldstone bosons on the Bethe lattice
167
8.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
8.2
The Classical Spherical Model . . . . . . . . . . . . . . . . . . . . . . 170
8.3
8.4
8.2.1
Phase transition . . . . . . . . . . . . . . . . . . . . . . . . . . 171
8.2.2
Ordered phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
8.2.3
Spatial correlations . . . . . . . . . . . . . . . . . . . . . . . . 175
The Quantum Spherical Model . . . . . . . . . . . . . . . . . . . . . 176
8.3.1
Quantum phase transition . . . . . . . . . . . . . . . . . . . . 177
8.3.2
Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
9 The Superconducting Susceptibility of a Fermi Liquid
184
9.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
9.2
Review of Results from the RG . . . . . . . . . . . . . . . . . . . . . 187
9.3
Calculation of the Beta Function and Anomalous Dimension . . . . . 192
9.4
Derivation of the Callan-Symanzik Equations
. . . . . . . . . . . . . 194
9.4.1
Fundamental Fermi field correlators . . . . . . . . . . . . . . . 194
9.4.2
Cooper pair correlator . . . . . . . . . . . . . . . . . . . . . . 197
9.5
Solution of the Cooper Pair Callan-Symanzik Equation . . . . . . . . 202
9.6
Another route . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
9.7
Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
A Impurity-induced random Zeeman field
206
A.1 Characterizing the disorder potential from the mobility . . . . . . . . 208
B Transfer Matrix for the AKLT Model
210
C Stability against spin glass ordering on a regular random graph
212
xiv
Bibliography
214
xv
List of Figures
1.1
Transport data in the quantum Hall regime
. . . . . . . . . . . . . .
6
1.2
Sample geometry for the quantum Hall effect. . . . . . . . . . . . . .
8
1.3
Density of states of 2DEGs in high fields . . . . . . . . . . . . . . . .
13
1.4
Laughlin’s argument in Corbino geometry . . . . . . . . . . . . . . .
15
1.5
Flux insertion argument for fractional charge. . . . . . . . . . . . . .
30
1.6
Edge excitations of a quantum Hall droplet. . . . . . . . . . . . . . .
31
1.7
Skyrmion spin configuration. . . . . . . . . . . . . . . . . . . . . . . .
63
2.1
Schematic phase diagram of the Pfaffian . . . . . . . . . . . . . . . .
75
4.1
Model band structure for AlAs quantum wells . . . . . . . . . . . . .
99
4.2
Phases of nematic ordering in dirty and clean systems . . . . . . . . . 105
5.1
Bilayer graphene lattice. . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.2
Schematic of the splitting of the zeroth Landau level
6.1
Transition temperature for classical companion models . . . . . . . . 136
6.2
The quadripartite pyrochlore lattice. . . . . . . . . . . . . . . . . . . 138
7.1
Different realizations of the Bethe lattice. . . . . . . . . . . . . . . . . 145
7.2
Phases of AKLT models on trees . . . . . . . . . . . . . . . . . . . . 158
8.1
Support of Gij (ω) at zero temperature for z = 3. . . . . . . . . . . . . 180
8.2
Analytic structure of Gij (ω) in the complex plane. . . . . . . . . . . . 181
xvi
. . . . . . . . . 118
9.1
Kinematics of the couplings F and V . . . . . . . . . . . . . . . . . . . 190
9.2
One-loop diagrams for the beta function and anomalous dimension . . 193
xvii
Part I
Introductory Material
1
Chapter 1
The Quantum Hall Effect
1.1
A Brief History, 1879-1984
In 1879, Edwin Hall, a twenty-four-year-old graduate student at Johns Hopkins University, was confounded by two dramatically different points of view on the behavior
of a fixed, current-carrying wire placed in a magnetic field. The first, espoused by
no less authority than Maxwell [149], was that the electromagnetic forces acted not
on currents but on the conductor itself, so that if the latter were immobile there
would be no effect whatsoever once transients died down. The second [55] held that
the forces acted on moving charges, and so there should be measurable consequences
on transport through the wire even if it were held fixed. Understandably confused,
Hall consulted his doctoral advisor, Henry Rowland, and with his help designed an
experiment in favor of the latter view [89], to wit: “If the current of electricity in
a fixed conductor is itself attracted by a magnet, the current should be drawn to one
side of the wire, and therefore the resistance experienced should be increased.” With
this succinct observation – and the experimental tour de force that followed – Hall
became the first to study his eponymous effect. As the modern theory of metals was
developed in the mid-twentieth century, Hall effect measurements were applied to a
2
variety of problems: they served not only as a means to measure the sign of charge
carriers in different materials, but also in constructing magnetometers and sensors for
various uses.
Beginning in the 1930s, a series of experiments began to probe quantum mechanical phenomena in the transport of electrons. The Shubnikov – de Haas and de Haas
– van Alphen effects were the first in a series of ‘quantum oscillations’ in various
quantities – resistivity and magnetization respectively in the initial examples, but
eventually many others – observed as an applied external magnetic field was varied.
Seminal work by Landau [128] on the quantization of cyclotron orbits of quadratically dispersing electrons in magnetic fields, and semiclassical extensions to more
complicated situations allowed a unified explanation of the different measurements.
This work also led to an appreciation of the fact that quantum oscillations provide an
extremely precise technique for measuring the shapes of Fermi surfaces [177]. Experiments progressed rapidly1 , and with each successive refinement increasingly baroque
Fermi surfaces were mapped out, enhancing greatly the understanding of various
metallic phenomena.
Roughly in parallel with these developments, the technological applications of
solid state physics developed, at a pace that multiplied tremendously following the
invention of the transistor. Increasingly elaborate semiconductor devices were engineered; originally these were intended solely for industrial applications, but gradually
it was recognized that there was interesting and fundamental physics to be mined,
for quantum-mechanical phenomena become visible in such devices, particularly if
they confine electrons in extremely clean structures of reduced dimensionality. As a
harbinger of things to come, in 1966 Shubnikov–de Haas oscillations were observed in
a two-dimensional electron gas (2DEG) in a silicon metal-oxide-semiconductor fieldeffect transistor (MOSFET) [70].
1
For an entertaining account of the historical development of the field, see [214].
3
Just over a century after Hall’s experiments, von Klitzing, Pepper and Dorda made
careful measurements of the Hall effect, in a silicon MOSFET [238]. At magnetic
fields sufficiently high that that the characteristic energies of Landau quantization
were larger than the ambient temperature scale – the ‘extreme quantum limit’ – they
observed that the Hall resistance was quantized in integer multiples of the fundamental resistance quantum2 h/e2 : rather than show a smooth linear rise with changing
field, the Hall resistance trace described a series of plateaus. Within each plateau,
the longitudinal resistance was nearly zero, but had sharp peaks at each step between
plateaus. That step-like features were seen in experimental observations was not particularly surprising, given Landau’s work: the centers of the plateaus occured when
the number of electrons was an integer multiple of the number of available eigenstates
at a given energy, with this integer – known as the ‘filling factor’ ν – setting the Hall
resistance. However, it rapidly became apparent that the existing theory of transport
in metals was unable to account for the fantastic accuracy with which the quantization occurred, especially as samples were tuned away from the ‘magic’ commensurate
points. Such universality, independent of microscopic details, hinted strongly that
some deeper principle was at work, ‘protecting’ the Hall conductance from correction
by such experimental complications as sample imperfections, field inhomogeneities
and electron density differences.
In 1981, Laughlin gave a beautiful explanation of the universality of the experimental observations in terms of adiabatic cycles in the space of Hamiltonians [130].
Subsequently refined by Halperin [90], his argument rests on a simple fact: if we
thread a quantized flux through the hole of a non-simply connected sample – for concreteness, say in the shape of an annulus – the Hamiltonian (and hence the spectrum)
returns to itself. In physical terms, the only net result of this adiabatic cycle could be
that an integer charge was transferred from one edge to the other, thereby making it
2
Subsequently renamed the von Klitzing constant; perhaps more so than in any other branch of
condensed matter physics, eponyms flourish in quantum Hall physics.
4
possible to do work against a potential gradient in a direction transverse to the electric
current induced by the changing flux. This gives rise to a Hall conductance quantized
at integer values. These arguments hold quite generally – immune to details of the
disorder, inhomogeneities and so on – as long as the chemical potential is in a mobility
gap, i.e. if the electronic states at the Fermi level are all localized3 . Eventually, it
was realized [229, 20, 175, 15] that the Laughlin argument could be reformulated in
a manner which made it clear that the Hall conductance was a topological invariant,
further explaining its universal nature.
Almost simultaneously with this understanding of the importance of gauge invariance in explaining the quantization, Tsui, Störmer and Gossard performed similar
experiments as von Klitzing’s group, in extremely clean gallium arsenide (GaAs) heterostructures [235]. They found, in addition to the integer plateaus, additional steps
in the Hall resistance at fractional values of the filling factor, at ν = 13 , 51 and so on.
The theoretical obstacle to explaining these features was stark and immediate: when
ν < 1, there are more available degenerate electronic states than electrons in the
system, so that perturbation theory is useless to treat this problem, which quickly
became known as the fractional quantum Hall effect to distinguish it from its integer
predecessor.
It was Laughlin who once again came to the rescue, by proposing a truly remarkable trial wavefunction to describe the correlated electronic state at the heart
of the fractional effect4 [131]. He was able to show by numerically solving few-body
examples that his ansatz, besides the obvious feature of being commensurate, had
extremely high overlap with the true ground state5 at ν = 1/3. He was even able to
construct exact wavefunctions for excited states – ‘quasiholes’ and ‘quasielectrons’ –
3
Note however that for a nonzero Hall conductance it is essential that at least one electronic
eigenstate below the Fermi level is extended [90].
4
A wonderfully candid discussion of the toy computations, intuitive leaps and occasional missteps
that led to his insight is in Laughlin’s Nobel autobiography [132].
5
Laughlin proposed states for fillings ν = 1/m, m odd; we explicitly discuss the example of the
ν = 1/3 state here for convenience.
5
Figure 1.1: Transport data in the quantum Hall regime
Longitudinal (ρxx ) and Hall (ρxy ) resistance traces in the quantum Hall effect; region
(a) of the left panel is shown in magnified form in the right panel. Notable features include integer and odd-denominator
incompressible fractions, even-denominator com
1 3
pressible states ν = 2 , 2 as well as an incompressible state at ν = 52 . Reprinted
with permission from R. L. Willett, et. al., Phys. Rev. Lett. 59, 1776 (1987) [248].
c 1987 by the American Physical Society.
Copyright and compute their energies. Finally, and most strikingly, he pointed out – by mapping the problem to a classical Coulomb plasma – that his excited state described
particles with fractional electric charge, e/3, and argued that such ‘fractionalized’
quasiparticles were the natural excitations of the two-dimensional electron gas near
ν = 1/3. These ideas were soon extended by Haldane [86] and Halperin [91] to explain
the ‘hierarchy’ of other fractional quantum Hall phases descending from the Laughlin
states, and by Arovas, Schrieffer and Wilczek [18] to show that that the excitations
had not only fractional charge, but also fractional statistics. The fractionalization
of quantum numbers was later recognized by Wen [246] as a characteristic of what
he termed topological order [239, 241], which has since been the subject of much
investigation.
6
Since the early 1980s, when much of the groundwork for the present was laid, there
has been a steady improvement in our understanding of the fractional quantum Hall
effect. Powerful techniques from conformal field theory and the ever-growing power
of modern computers have been brought to bear on the problem of constructing and
studying increasingly elaborate trial wavefunctions for the current zoo of observed
quantum Hall fractions. Besides the states with quantized Hall conductance, the
global phase diagram of the quantum Hall effect includes Fermi liquid-like states
and modulated stripe and bubble phases. The dramatic improvement in sample
mobilities in the past thirty years has meant that much of the progress has been
experimentally driven: several of these phases were first seen in transport studies
before they were understood theoretically. The quantum Hall effect has been shown
to have a remarkable analogy with the theory of superconductivity [78, 198, 258]; it
underlies perhaps the best understood itinerant ferromagnet [218]; it provided the first
example6 of the fractionalization of quantum numbers in dimensions higher than one.
It has also played a central role in the modern understanding of topological phases,
with analogs in magnetic systems [239], band insulators [165, 74, 206], superfluids
[200] and superconductors [95]. This thesis is devoted to exploring just a few of the
many exciting and fascinating possibilities that owe their ultimate inspiration to the
strange behavior of electrons in high magnetic fields.
1.2
Samples and Probes
The quantum Hall effect is is a transport phenomenon, observed when twodimensional electron gases (2DEGs) are placed in quantizing magnetic fields transverse to the plane in which electrons are free to move (for a sketch of the geometry,
see Fig. 1.2.) The 2DEGs are typically realized in semiconductor heterostructures,
6
More precisely the first example understood to be such. As pointed out to a greater or lesser
degree by various authors [241, 121, 95] the oldest known fractionalized phase is the venerable
superconductor!
7
B
E
j
Figure 1.2: Sample geometry for the quantum Hall effect.
The electron gas is confined in a two-dimensional plane, and a perpendicular magnetic
field B is applied. By means of contacts, current is passed through the sample and
Hall and longitudinal voltage measurements can be made. In the figure we show the
direction of the current j and the Hall electric field E.
where by suitable epitaxial growth techniques it is possible to build quantum wells
that confine the transverse motion of electrons and thereby render them effectively
two-dimensional. The thickness of the confining well controls both the spread of the
electronic wavefunctions in the transverse direction as well as the spacing between
the different ‘subbands’ for motion in this direction. While in some experiments the
well thickness is increased in order to modify electron-electron interactions or make
it favorable to fill multiple subbands and enhance the tunability of the system, for
the purposes of this thesis we focus on the single subband case and in the remainder
shall assume that the electrons are purely two-dimensional.
The density of electrons in the 2DEG is controlled primarily by doping with donor
impurities, situated in a layer set back a fixed distance from the plane of the 2DEG;
some samples may also permit some degree of tunability of density using electrostatic
gates. The impurities serve as the major source of disorder: when screened by the
electrons they give rise to a smooth random potential. There may also be some
amount of short-range impurity scattering from imperfections at or near the plane of
the 2DEG.
8
More recently, quantum Hall plateaus – both integer and fractional – have been observed in high-mobility two-dimensional semimetals, graphene and bilayer graphene,
either in suspended structures, or over a variety of substrates. Owing to the robustness of the quantum Hall effect in these materials, and their accessibility to surface
probes, they are likely to play a significant role in future experimental studies.
The electrons in the 2DEG may have additional internal degrees of freedom, such
as their spin, valley pseudospin in the case of degenerate conduction band minima
or layer index in double quantum wells. There are additional symmetries associated
with these degrees of freedom, and the complex of phenomena that accompany the
breaking of these is the subject of Part III of this thesis. In the remainder of this
introductory chapter, we shall generally assume that the internal degrees of freedom
are ‘polarized’ –effectively, absent – and discuss spinless electrons unless explicitly
stated otherwise.
The primary experimental probe of the quantum Hall effect is transport; measurements of currents and voltages in the Hall bar can be made by means of contacts
(usually gold) on the edges of the sample. The most striking observation is the quantization of the Hall resistance into a series of plateaus, and the near-vanishing of
longitudinal resistance except at a series of sharp peaks when the Hall resistance is
between two different quantized values. In addition, transport through quantum point
contacts [51, 54] and double point contacts [38, 250] ) serve as probes of quasiparticle
charge (via shot noise) and statistics (via interference measurements) respectively.
In certain specific cases, the Hall effect lends itself to study by other means besides transport. Nuclear Magnetic Resonance (NMR) measurements are particularly
useful in studying the spin textures associated with quantum Hall ferromagnets [24];
surface-acoustic-wave (SAW) absorption is an important experimental signature of
the Fermi-liquid like state at ν =
1
2
[249]; measurements of the charging spectra of
disorder-induced compressible puddles may permit the determination of fractional
9
quasiparticle charge in various Hall plateaus [236]; optical absorption experiments
may be useful probes of spin-polarization of the ν = 5/2 state [224]; and tunneling
spectroscopy, both in the bulk [220] and at the edge [45] can reveal various correlation
effects; to name a few. We shall comment on the relation of work in the thesis to
these and other experiments where appropriate.
1.3
The Integer Effect
A natural place to begin our discussion is with the integer quantum Hall effect. We
first introduce the problem of noninteracting electrons in high magnetic fields, and
explain the reorganization of the spectrum into Landau levels. We then show how to
explain the experimental features of the integer effect via a semiclassical percolation
picture.
1.3.1
Single-particle physics: Landau levels
An electron with effective mass m∗ and charge −e, moving in the xy-plane under the
influence of a magnetic field B = B ẑ is described by the Hamiltonian
1
H=
2m∗
2
eA
p+
c
(1.1)
where B = ∇ × A. Choosing Landau gauge, we have Ax = 0, Ay = Bx, so that we
preserve translational invariance in the y-direction. The eigenstates of H can then
be classified by their y momentum and a Landau level index n, since the eigenvalue
problem reduces to that of a one-dimensional simple harmonic oscillator. We have,
explicitly
1
ψn,ky (x, y) = p eiky y ϕn x + ky ℓ2B
Ly
10
(1.2)
with En = n +
1
2
~ωc , where ℓB =
~c 1/2
eB
is the magnetic length, and ωc =
eB
m∗ c
is the cyclotron frequency. The wavefunctions are written in terms of ϕn (x) =
2
2
x
√ 1
H
e−x /2ℓB with Hn a Hermite polynomial, which follows from solvn
ℓB
1/2
n
2 n!π
ℓB
ing the one-dimensional oscillator. The wavefunction is localized around a guiding
center coordinate Xky = −ky ℓ2B .
It is immediately apparent that there is a large degeneracy, since the energy depends only on the index n and not on ky . To determine the degeneracy of a given
energy level – henceforth referred to as a Landau level – we consider a finite rectangular strip of size Lx × Ly . Keeping periodic boundary conditions in y retains ky as a
good quantum number, but it takes on discrete values, ky =
2πm
,m
Ly
= 0, ±1, ±2, . . ..
These correspond to eigenfunctions centered around x = 0, ±2πℓ2B /Ly , ±4πℓ2B /Ly , . . .;
the finite extent in the x direction restricts the allowed ky values, and it then is a
simple matter to count states to find a degeneracy equal NΦ , the number of flux
quantum threading the sample7 .
The knowledge of the degeneracy of a Landau level allows us to determine the
filling factor, ν, which is simply the number of filled Landau levels, ν =
N
NΦ
where
N is the total number of electrons. As a result of this relation, the density of the
two-dimensional electron gas can be expressed entirely in terms of the filling factor
and the magnetic length, n =
ν
.
2πℓ2B
It is a simple exercise to show that when exactly ν Landau levels are filled, the
2
Hall conductance σxy = ν eh . However, it is not clear why it should remain tied to
this value for a a finite range of filling factors about commensuration; indeed, in the
absence of interactions and with translational invariance, we can show8 that the Hall
conductance cannot deviate from its classical value of B/nec. If we wish to explain
7
L L
BL L
x y
x y
The degeneracy is 2πℓ
2 =
hc/e = NΦ , since the numerator is the total flux and the denomiB
nator is Φ0 , the flux quantum.
8
The argument rests on the ability, in a translationally invariant system, to boost to a frame
comoving with the current, in which the electrons are stationary but see an electric field E =
B
nec j × B where j is the current in the lab frame. The classical value of the Hall conductance follows
immediately [76].
11
the integer effect without interactions, it is necessary to consider the quantum Hall
problem in a disordered system, in which translational invariance is lost.
Heuristically, it is easy to see why this is so: in the absence of disorder, the
density of states of noninteracting electrons consists of a series of δ-function peaks at
the energies of the Landau levels. Were we able to fix the chemical potential precisely
in a gap between Landau levels, and maintain it there for a finite range of fillings
about commensuration, then a finite-width plateau in the Hall conductance would
automatically follow. However, it is impossible to keep the chemical potential in a
gap while changing the electron density.
When disorder is present, the Landau level spectrum broadens, as electronic states
become localized in the random potential. A band of extended states remains near the
center of each level, but is now flanked on either side by localized states separated from
it by mobility edges (see Fig. 1.3). As long as the chemical potential remains in the
resulting mobility gap while the density is varied, the Hall conductance is unchanged
since the electronic states being filled are all localized, and do not contribute to
transport. In the following section, we sketch an argument for how this structure
arises in the Landau level spectrum, using a semiclassical model of electron dynamics
in a smooth random potential.
1.3.2
Semiclassical Percolation
A simple picture of the integer quantum Hall effect in the presence of a smooth
disorder potential can be obtained in terms of a semiclassical percolation problem
[232]. Let us recall the basic features of semiclassical dynamics of electrons in an
external potential, in the presence of a strong magnetic field [19]. First, the electronic
motion can be separated into two parts: a slow drift along equipotential lines of the
external potential, and fast cyclotron motion about these orbits. Each mode of the
fast degree of freedom corresponds to a different Landau level, while the slow drift
12
ρ(E)
ρ(E)
extended
localized
!ωc
2
3!ωc
2
5!ωc
2
E
!ωc
2
(a.)
3!ωc
2
5!ωc
2
E
(b.)
Figure 1.3: Density of states of 2DEGs in high fields
(a.) Without disorder we have a series of δ-function peaks at the Landau level energies.
(b.) When (bounded) disorder is included, the δ-functions broaden on a scale set by
the disorder potential and all states except for those in a narrow band centered on
the Landau level energy are localized.
is the motion of the guiding center coordinate of the previous section. Second, these
statements become increasingly precise in the limit of B → ∞, since the potential
becomes increasingly smooth on the length scale ℓB of the cyclotron orbits, which
tends to zero in this limit, and it is the ratio of these scales that determines the
validity of the semiclassical approximation.
For concreteness consider a sample finite in the x direction, but infinite (or at
least very long) in the y direction. As the chemical potential is swept through the
Landau level, different equal-energy surfaces are traced out. When it is at the bottom
of the disorder potential, most of the semiclassical orbits surround low-energy regions
and are therefore localized; if an electric field in the x-direction is turned on, the
orbits are perturbed only weakly, and processes that involve charge transfer across the
sample in the x-direction would require a conspiracy of hopping processes between the
semiclassical orbits, and are hence strongly suppressed. At high chemical potential,
most of the orbits surround high-energy regions, and are hence also localized, and
a similar argument goes through: there is no current in the x-direction. However,
13
the set of semiclassical orbits now includes the orbits that lie along the edge of the
sample that are extended in the y-direction. Applying a field in the x direction
leads to a chemical potential difference between current carrying states in the ±y
directions, and therefore there is a finite current transverse to the potential gradient,
i.e. a nonzero σxy . For intermediate chemical potential, there is a point at which the
semiclassical orbits percolate through the sample; orbits that carry edge currents in
the ±y directions approach arbitrarily close to each other. An infinitesimal electric
field can now couple the two edge states, leading to finite longitudinal resistance,
accounting for the jump in Rxx at the plateau transition; once the percolation point
is passed, a new set of edge currents has been added and σxy will therefore show
a jump. This leads to the sequence of integer quantum Hall plateaus: once the
chemical potential reaches the top of the disorder potential within one Landau level,
we commence filling low-energy states in the next Landau level (corresponding to the
next eigenenergy of the ‘fast’ cyclotron motion). It can be shown in the high-field limit
and for particle-hole symmetric disorder that the nth plateau transition occurs when
the chemical potential crosses the energy of the nth Landau level µ∗n = (n + 1/2)~ωc .
As the statistical mechanics of percolation are well known, they can be used to
make estimates for different critical exponents for the integer quantum Hall plateau
transition. However, in general these are modified by quantum tunneling between
different semiclassical orbits near saddle points of the disorder potential; augmenting
the semiclassical treatment with suitable corrections to account for this leads to the
network model proposed by Chalker and Coddington [43].
14
Φ
VH
I
Figure 1.4: Laughlin’s argument in Corbino geometry
The quantum Hall liquid is confined to an annulus with a voltmeter connected between
the edges. A flux Φ is adiabatically inserted through the hole; when this is equal to a
multiple of the flux quantum Φ0 , the result is that an integer number of charges are
transferred from the inner to the outer edge.
1.4
1.4.1
Why is the Quantization Robust?
Laughlin’s Argument
We have so far provided a reasonable description of the features of the integer plateaus,
at least in the limit of smooth disorder and high field. However, this still does
not explain why the Hall conductance is so precisely and robustly quantized. We
now explain this through a slightly modified version of Laughlin’s argument more or
less identical to that presented in [113], which is valid both at T = 0 and at finite
temperature, and which is readily extended (with a few caveats) to the fractional case.
Consider a quantum Hall liquid confined in an annulus, with a voltmeter connected
between the inner and outer edges, that measures a voltage VH . This is commonly
referred to as the Corbino geometry. Now, imagine adiabatically inserting a flux
Φ through the annulus, without letting it penetrate the sample itself. Recall that
for a system interacting with an external vector potential A and described by a
15
δHA
. In an instantaneous
Hamiltonian HA , the current operator is given by j(r) = c δA(r)
eigenstate, HA |ψA i = EA |ψA i, using the fact that |ψA i is normalized it follows that
A
A
|ψA i = c δE
. Taking a thermal average over eigenstates,
hjiA = chψA | δH
δA
δA
X δEα,A
hji = c
δA
α
e−Eα /kT ≡ c
δhHA i
δA(r)
(1.3)
which defines the adiabatic derivative in which we keep the Boltzmann weight of the
state α fixed while changing HA . This is equivalent to the thermodynamic requirement
of constant entropy, hence the name. Specializing to our example, and working in
Φ
polar coordinates centered on the annulus, we may write A = θ̂ 2πr
, so that
∂hHA i
=c
c
∂Φ
δhHA i ∂A
dr
·
=
δA(r) ∂Φ
Ω
Z
2
Z
Ω
d2 r
hjθ i
=I
2πr
(1.4)
where I is the azimuthal current flowing in the annulus. We ultimately wish to relate
this to the voltage drop across the annulus; so far all we have shown is that the
current is the adiabatic derivative of the energy with respect to the flux. Within
the annulus the flux is pure gauge, and so localized states are unaffected. However,
states extending around the hole see an Aharonov-Bohm phase of 2πΦ/Φ0 due to the
inserted flux, but as long as the latter is an integer multiple of Φ0 , the Hamiltonian
(and hence the spectrum) is the same, upto a gauge transformation, as at Φ = 0.
The only possible change induced by the adiabatic insertion of a quantized flux is
to carry the system from one eigenstate to another. This process of changing a
variable adiabatically so that the spectrum returns to itself is termed an adiabatic
cycle. A simple example is furnished by the noninteracting problem, without disorder.
In symmetric gauge, the eigenstates are labeled by angular momentum m, and are
localized at successively higher radii with increasing m. In this basis, it is easily
16
verified that the flux insertion procedure shifts each9 single particle eigenfunction
from m to m + 1. The net result of a cycle is that in each filled Landau level a single
electron is adiabatically transported from one edge of the annulus to the other. For p
filled Landau levels, the total energy required is ∆E = peVH where the Hall voltage
is by definition the energy to move a unit test charge between edges through a weakly
coupled external circuit. If we now approximate
∂hHA i
∂Φ
by
∆E
∆Φ
and set ∆Φ = Φ0 , which
2
is the minimal flux for an adiabatic cycle, the current follows as I = p eh VH . The Hall
conductance is therefore σxy =
pe2
.
h
This can be extended to disordered systems by
postulating that if the Fermi energy lies in a mobility gap, the only states that can be
excited by the adiabatic insertion involve charge transfer from one edge to the other.
Since weak perturbations cannot easily move the Fermi energy out of the mobility
gap, the Hall conductance remains tied to its quantized value.
A further extension to the fractional effect is possible, if we postulate that in the
fractional case adiabatic cycles require the insertion of an integer number q > 1 flux
quanta10 to transfer p electrons between edges. The net result is a Hall conductance
σxy =
1.4.2
p e2
.
q h
Hall Conductance as a Topological Invariant
Shortly after Laughlin’s elucidation of the role played by gauge invariance in the universality of the Hall conductance, several workers [229, 20, 175, 15] extended his ideas
to rewrite the Hall conductance as a topological invariant. The original treatment by
Thouless, Kohmoto, den Nijs and Nightingale [229] considered noninteracting electrons in a periodic potential; the approach we shall follow is more or less identical
9
In this case, all the single-particle states extend around the hole and therefore see the AharonovBohm phase
10
This requirement is intimately connected to the fact that fractional quantum Hall states are
topologically ordered and hence have a ground state degeneracy on a topologically nontrivial manifold, such as an annulus.
17
to that of Niu and Thouless [175] which extends the result to interacting systems
without periodicity.
Let us return to our 2DEG of the previous section, now with interactions as well
as an electric field E x̂ in the plane. The linear-response Hall conductivity follows
from a Kubo formula,
σxy
e2 ~ X hΨ0 |v̂x |Ψn ihΨn |v̂y |Ψ0 i − hΨ0 |v̂y |Ψn ihΨn |v̂x |Ψ0 i
=i
Lx Ly n6=0
(E0 − En )2
(1.5)
where 0 and n label the ground and excited many-body eigenstates. The velocity
operators appearing in the Kubo formula are, in the same gauge as used earlier, given
by
N
N
X
X
1
∂
∂
1
−i~
, v̂y =
−i~
+ eBxi
v̂x =
m
∂x
m
∂y
i
i
i
i
i=1
i=1
(1.6)
Before we can use the Kubo formula, we require appropriate boundary conditions
under which to solve the eigenvalue problem. Real samples have edges, and thus
periodic boundary conditions are appropriate only in the y direction. However, since
we are interested in the bulk contribution11 we may make the system periodic in
x direction as well with the appropiate y-dependent phase factor necessitated by
translation in the magnetic field. The boundary conditions are then relaxed to
Ψ ({xi + Lx }) = eiαLx e−i(eB/~)yi Lx Ψ ({xi })
Ψ ({yi + Ly }) = eiβLy Ψ ({yi })
(1.7)
Note that we work explicitly in Landau gauge. It is possible to reformulate the entire problem explicitly in gauge-covariant form, but we shall continue to work in a fixed
gauge for clarity of presentation. In any event, our final result for σxy will be in manifestly gauge-invariant form, as appropriate to a physical observable. If we now make a
11
For a discussion of possible subtleties, see [175].
18
unitary transformation on the many-body eigenstates Ψ̃n = e−iα
PN
i=1
xi −iβ
e
PN
i=1
yi
Ψn ,
(1.5) becomes
2
σxy = i
e
~Lx Ly
ED E D ED E
D H̃ H̃ Ψ̃n Ψ̃n ∂∂βH̃ Ψ̃0 − Ψ̃0 ∂∂βH̃ Ψ̃n Ψ̃n ∂∂α
Ψ̃0
X Ψ̃0 ∂∂α
(E0 − En )2
n6=0
(1.8)
where H̃ is the transformed Hamiltonian. It is then straightforward to reexpress
the Hall conductance purely in terms of the transformed many-body ground state
wavefunction12
σxy
e2
=i
h
"*
+ *
+#
∂ Ψ̃ ∂ Ψ̃
∂ Ψ̃ ∂ Ψ̃
−
∂θ ∂ϕ
∂ϕ ∂θ
(1.9)
where θ = αLx , ϕ = βLy , and each takes values on [0, 2π).
At this point, we have simply rewritten the Hall conductance as a response of
the ground state wavefunction to changes in boundary conditions; we have as yet
given no reason for its quantization. We now make a crucial assumption: that there
is always a finite energy gap between the ground state and the excitations under any
given boundary conditions of the form (1.7). Note that it is reasonable to assume that
the Kubo conductance is insensitive to boundary conditions, as long as there no longrange correlations in the ground state, which is true for the case of an incompressible
liquid. As a result, we may equate the Hall conductance to its average over boundary
conditions,
σxy = σxy
e2
=
h
Z
0
2π
Z
2π
0
1
dθdϕ
2πi
"*
+ *
+#
e2
∂ Ψ̃ ∂ Ψ̃
∂ Ψ̃ ∂ Ψ̃
−
=
C
∂ϕ ∂θ
∂θ ∂ϕ
h
(1.10)
The above expression shows that the dimensionless Hall conductance, σxy /(e2 /h) is a
topological invariant, known as the Chern number (C) [169], of the family of ground
state wavefunctions. This explains why the quantization is robust: as it is a discrete
12
Assuming it is nondegenerate; this is not true for the fractional effect.
19
topological index, the Hall conductance cannot be changed by small perturbations.
Adding disorder leads to a Hall plateau at the quantized values of σxy as before.
The Chern number is an integer, and therefore the foregoing discussion satisfactorily explains the integer quantum Hall effect. The assumption that forced integer
quantization was that the ground state was nondegenerate: this enabled us to rewrite
σxy as a property solely of the ground state wavefunction. For fractional quantization,
we must require that the ground state be degenerate. The generalization of (1.10) to
the fractional case is
σxy = σxy
Z
d Z
e2 X 2π 2π
1
=
dθdϕ
hd K=1 0
2πi
0
"*
+ *
+#
∂ Ψ̃K ∂ Ψ̃K
∂ Ψ̃K ∂ Ψ̃K
−
(1.11)
∂ϕ ∂θ
∂θ ∂ϕ
where d is the degree of degeneracy, and the ΨK are orthogonal and span the ground
state subspace.
Unlike in the integer case, the integrals in (1.11) are not topological invariants,
since a cycle in θ or ϕ need not return each of the degenerate states to itself. However,
it is possible to show that the summation over the integrals, and hence σxy , is a
topological invariant. The fractional Hall conductance is then simply a fractional
multiple of a Chern number, with the fraction related to the ground-state degeneracy
on the torus. For instance, for the Laughlin states with ν = 1/m which are m-fold
degenerate on the torus, the argument gives σxy = e2 /mh [175].
We close by relating the formulation above to the gauge argument. This follows
immediately if we compute the current induced by the adiabatic flux insertion, and use
the Kubo formula for the Hall conductance to show that the total charge transported
in an adiabatic cycle is simply related to the averaged σxy [175].
20
1.5
The Fractional Effect
In the previous section, we argued that we can relate the Hall conductance to a
topological invariant as long as there is a gap to bulk particle-hole excitations; adding
disorder then provides a mobility gap so that the Fermi energy can vary through a
band of localized states while keeping the Hall conductance unchanged, leading to a
plateau in σxy . For the integer Hall effect, the first step is logically straightforward,
since a filled Landau level is automatically gapped to particle-hole pairs.
For the fractional effect, this is no longer the case. In the absence of interactions,
we have a highly degenerate set of states, and there is no obvious reason to privilege a commensurate filling over any other13 . Indeed, the low-energy Hilbert space
of the problem, consisting of states exclusively belonging to the n = 0 energy level
– commonly referred to as the lowest Landau level approximation – is completely
degenerate, and in the absence of interactions there cannot be a fractional Hall conductance. Worse, as a result of this degeneracy, there is no good parameter in which
one can construct a perturbation expansion to systematically include interactions.
Given this all-or-nothing feature, it is a formidable task to construct a gapped many
body ground state at each fractional filling. The solution – as is generally the case –
is inspired guesswork, to which we now turn.
1.5.1
Trial Wavefunctions
Our focus in this section is the trial wavefunction approach to the the quantum Hall
problem. In its essence, the method rests on making a more or less physical guess
for the form of the many-body wavefunction at a given filling. In some, but by
no means all, cases this is an exact groundstate of a special, typically short-range,
13
In fairness, a commensurate charge-density-wave (CDW) was originally proposed as a ground
state, but this is incompatible with the strictly linear I − V curves in experiments and the cusp
singularity in the ground state energy at commensuration observed in numerical studies. See [234]
and references therein for a discussion.
21
model Hamiltonian which may involve 3- and higher-body interactions. The form
of the ground-state wavefunctions often suggests natural choices for excited-state
wavefunctions corresponding to quasielectrons and quasiholes.
There are many different ways in which trial wavefunctions can be motivated.
Laughlin’s original guess blended a study of few-body examples with an intuitive leap
to the N-body problem. More systematic approaches include the Jain construction,
which builds fractional quantum Hall trial wavefunctions from filled pseudo-Landau
levels of composite fermions; guessing trial states from conformal blocks of conformal
field theories; and the Haldane-Halperin ‘hierarchy’ construction at various fillings,
which rests on forming quantum Hall states from the quasiparticles of a parent quantum Hall liquid. Many of the model wavefunctions can also be understood in a unified
fashion within a recent formulation based on the properties of Jack symmetric polynomials [26]. There are also various ‘parton’ approaches [104, 242, 244]. which build
in fractionalization at the outset. We shall discuss composite fermions in Section 1.6,
and in this section we briefly summarize the hierarchy construction; the conformal
block technique, the Jack polynomial approach and the parton constructions, while
extremely important to our understanding of the quantum Hall effect, are somewhat
peripheral to our concerns in this thesis and will be omitted for brevity.
Once constructed, model wavefunctions can be used to determine a variational
upper bound on the ground state energy; alternatively, one can obtain the exact
ground state for small systems by numerically diagonalizing the many-body problem14 , and compute the overlap with the trial state. Often, the overlap is extremely
high, a strong indication that the ansatz captures most of the essential many-body
correlations of the fractional quantum Hall phase under investigation. When there
are competing states at a given filling – for instance, ν = 2/5 has variously been described as a hierarchy state [86, 91], a composite fermion state [103], or the so-called
14
Perhaps surprisingly, many quantum Hall systems appear to converge to the thermodynamic
limit with only ∼ 10 electrons.
22
‘Gaffnian’ state [216]– such numerical studies of overlaps and energetics may be able
to settle the question of which alternative is more likely to be stable in real systems.
On occasion, trial states with very different physics have extremely high overlap – for
instance, the Gaffnian and the composite fermion states in this example. Comparing
the ground state entanglement spectrum has been proposed as a resolution to such
issues [202]. The entanglement spectrum can also be used to systematically show
adiabatic continuity between model Hamiltonians and the more realistic Coulomb
interaction case [228].
As an illustrative example, we shall consider Laughlin’s wavefunction for the ν =
1
m
states. We shall be fairly concise, as the trial wavefunction approach has been the
subject of several reviews, e.g. [192]; our approach shall hew closely to that of [76].
Also, we pick the pedagogically simpler example of the disk geometry, although most
numerical studies seek to avoid the complication of edges by studying the problem on
the sphere or the torus. Finally, we shall work in symmetric gauge, as this is ideally
suited to wavefunction studies of the lowest Landau level.
With these preliminaries, we are ready to begin our discussion. In symmetric
gauge, working in two-dimensional complex coordinates z = x+iy the wavefunction of
an N-electron system that is confined to the lowest Landau level can always be written
as the product of a function f analytic in all the electron coordinates z1 , z2 , . . . , zN
with a Gaussian factor
15
,
1
Ψ(z1 , z2 , . . . , zN ) = fN [z]e− 4
PN
i=1
|zi |2
,
(1.12)
with the obvious requirement from the Pauli principle that that f is totally antisymmetric. Here and below we have chosen to set the magnetic length, ℓB = 1. As the
Gaussian factor is set by the cyclotron degree of freedom, the Hilbert space of the
lowest Landau level reduces to the space of analytic functions in N complex variables.
15
For brevity, we shall denote functions of all the electron coordinates f (z1 , z2 , . . . , zN ) as fN [z].
23
An orthonormal basis of single-particle states for the lowest Landau level is thus provided by functions of the form ϕk (z) =
k
2
√ z
e−|z| /4
2π2k k!
[77]. These states each have
angular momentum k and it is easy to show that their probability density is peaked
√
at radius 2k.
Laughlin made the following guess for the wavefunction at ν =
fNm [z] =
N
Y
i<j
1
,
m
m odd:
(zi − zj )m .
(1.13)
Since m is an odd integer, analyticity and antisymmetry are immediate. In addition,
the m = 1 state is simply a Slater determinant built out of the single-particle states
ϕk , i.e.
ϕ0 (z1 )
ϕ0 (z2 ) . . . ϕ0 (zN )
ϕ1 (z1 )
ϕ1 (z2 ) . . . ϕ1 (zN )
Ψ1N [z] = ..
.
ϕN −1 (z1 ) ϕN −1 (z2 ) . . . ϕN −1 (zN )
(1.14)
which clearly corresponds to a filled lowest Landau level. For m > 1, we verify that
Ψm
N has the correct filling as follows. Since the highest degree of any of the zi s in
f is m(N − 1), it follows that this is the highest possible angular momentum in the
decomposition of Ψm
N in the single-particle basis. The area of the droplet described
by Ψm
N is thus A = 2πm(N − 1); from this, the filling factor is ν =
have already verified ν = 1 for m = 1, it follows that ν =
1
m
2πN
.
A
Since we
in the general case of
arbitrary odd m [113]. Thus, we have produced a trial wavefunction that has the
correct filling and lives entirely in the lowest Landau level.
The Laughlin wavefunction has an additional and extremely useful property: we
can show that it is the exact ground state for a short-range model Hamiltonian. To
see this we observe, following Haldane [86], that we can expand any translationally
and rotationally invariant two-body interaction projected onto a single Landau level
24
as16 [233, 86]
V =
∞ X
X
vm′ Pm′ (ij)
(1.15)
m′ =0 i<j
where Pm (ij) are operators that project onto states such that particle i and j have
relative angular momentum m. The coefficients of the expansion, vm , are known as
Haldane pseudopotentials and depend on the Landau level under consideration; for
repulsive interactions they are all positive. Since the projectors for different angular
momenta do not commute, this rewriting does not immediately simplify the problem.
However, if we consider a model Hamiltonian defined by vm′ > 0 for m′ < m and
zero otherwise, then it is clear that the Laughlin state Ψm
N [z] is an exact, zero-energy
eigenstate for any N, since any two electrons have a relative angular momentum of
at least m. It is also possible to show that the model Hamiltonian has a gap, since
any excitation involves reducing the relative angular momentum of at least one pair
of electrons and therefore costs positive energy.
It would be truly remarkable if every trial wavefunction had a corresponding model
Hamiltonian for which it is the exact ground state. Unfortunately, this is not the
case; there exist several different trial states for which no simple model Hamiltonian
is known. However, an infinite family of states belonging to the so-called Read-Rezayi
sequence [201] can be shown to be exact ground states of model Hamiltonians, albeit
ones involving n-body interactions with n > 2. These include the Laughlin states as
well as the Moore-Read state for even-denominator fillings. We shall say more about
the Read-Rezayi sequence when we discuss Type I quantum Hall liquids in Chapter
3.
16
Note that within the lowest Landau level, the kinetic energy is quenched and the Hamiltonian
reduces to just the interaction term, H = V .
25
1.5.2
Plasma Analogy
The Laughlin wavefunction exemplifies another extremely useful property in common with several different wavefunctions that share its Jastrow (pair product) form17 ,
namely that the ground state correlations reduce to the finite-temperature equilibrium correlations of an interacting classical system in the same dimension. This is
accomplished by computing the ground state probability density and noting that it
can be written as a classical Boltzmann weight
2
−βHcl
|Ψm
N [z]| = e
where β =
2
,
m
(1.16)
and
Hcl = −m2
X
i<j
log |zi − zj | +
mX
|zk |2
4 k
(1.17)
corresponds to the energy of a two-dimensional Coulomb plasma of particles of charge
−m moving in a uniform background charge density (reinstating ℓB for clarity) ρB =
− 2πℓ1 2 . The long-range Coulomb forces18 enforce charge neutrality in the plasma,
B
which requires mn + ρB = 0, from which we recover the fact that ν = 2πℓ2B n = m1 .
P
2
ρq ρ−q upto unimportant
Working in momentum space, we have Hcl = 2L1 2 q 2πm
q2
self-energy corrections, where we take Lx = Ly = L. From this, we get ( at q ≫ ℓ−1
B )
that the density-density correlations in the Laughlin state are suppressed at long
wavelengths,
hρq ρ−q i =
17
L2 2
q .
2πm
(1.18)
These include several other quantum Hall trial states, as well the AKLT ground states that are
studied in Part IV of this thesis.
18
Note that these are purely fictitious; they always exist in the classical companion plasma to the
Laughlin state, even when the physical Hamiltonian is short-ranged. They are simply a means of
enforcing the correlations built into the Laughlin ansatz.
26
By performing Monte Carlo simulations of (1.16), we can verify both this result, as
well as the fact that the long-range plasma forces lead to liquid-like correlations.
It is worth mentioning that while it appears that the Laughlin construction works
for arbitrary odd m, in fact for m ≥ 7 the Coulomb interaction energy is minimized,
not by a quantum Hall liquid, but by a triangular electron Wigner crystal. The
plasma analogy also eventually leads to a crystal state, but at much lower filling; the
Laughlin state stops being a good variational state well before this.
1.5.3
Neutral Excitations and the Single-Mode Approximation
We turn now to the neutral collective excitations of the Laughlin liquid. In the absence
of additional degrees of freedom, the only neutral collective modes are phonon-like
density wave excitations. In keeping with the approach we have taken thus far, we
would ideally like to compute the neutral excitation spectrum variationally. To do
this, we use the Single Mode Approximation (SMA), originally employed to study
the collective mode spectrum of superfluid 4 He by Feynman [65] and Bijl [28]. In
its essence, the SMA approach relies on obtaining a variational upper bound for
excitation energies by constructing a trial wavefunction for the excited state, that
is orthogonal to the ground state at each wavevector k. Typically, this is done by
multiplying the ground state wavefunction with a density operator ρk . When the SMA
is applied to the fractional quantum Hall effect, we require an additional projection
of the trial excited state wavefunction into the lowest Landau level, in order to ensure
that we correctly restrict the Hilbert space and thereby capture the intra-Landau
level excitation scale set by the Coulomb energy e2 /εℓB . Explicitly, we have
m
Ψm;q
N [z] = ρ̄q ΨN [z]
27
(1.19)
where the bar denotes an operator projected into the lowest Landau level. While we
shall not present details here, it can be shown [79] that the SMA calculation always
gives a gap at all q. Near q = 0, for Coulomb interactions the result is
m;q
hΨm;q
N |V |ΨN i
e2
+ O(q 2 )
=c
∆SM A (q) ≡
m;q
m;k
εℓB
hΨN |ΨN i
(1.20)
where c is a numerical constant depending on details of the interaction. More careful
analysis over a greater range of wavevectors, as well as numerical studies, reveals that
the collective mode spectrum has a minimum at q ∼ ℓ−1
B , commonly referred to as
the magnetoroton in analogy with the roton minimum in superfluid Helium [79].
1.5.4
Fractionally Charged Excitations
In addition to neutral collective modes, quantum Hall states also have charged quasiparticle excitations, conventionally referred to as quasielectrons and quasiholes19 , corresponding to negative and positive charge respectively. These are nucleated when
the filling factor is altered from a commensurate value, either by varying the charge
density or the magnetic field. The wavefunction for a quasihole located at Z is [131]
Ψm
qh,N [z; Z]
N
Y
=
(zi − Z)Ψm
N [z].
(1.21)
i=1
What about a quasielectron? Naively, we would guess that to obtain a quasielectron
wavefunction we should simply replace the product in the quasihole wavefunction
with its complex conjugate, but this leads to a wavefunction that is not restricted to
the lowest Landau level. Projecting back to the restricted Hilbert space, the zi∗ s are
19
We shall reserve the term quasiparticle for situations when we mean ‘either quasielectron or
quasihole’.
28
mapped to derivatives, leading to [131]
Ψm
qe,N [z; Z]
N Y
∂
∗
− Z Ψm
=
2
N [z]
∂z
i
i=1
(1.22)
where the derivatives act only on the polynomial part of Ψm
N . Owing to the rather
complicated form of the quasielectron wavefunction, we shall work primarily with
quasiholes in the following20
The quasihole wavefunction can also be studied using the plasma mapping, which
leads to a classical Boltzmann weight of the form e−β(Hcl +Hi ) . Here Hcl and β have
the same values as in Section 1.5.2, and
Hi = −m
N
X
i=1
log |zi − Z|
(1.23)
is the energy of unit charge impurity at Z interacting with the mobile charge-m
particles of the plasma. Since the plasma attempts to maintain charge neutrality it
will screen the impurity. The resulting screening cloud has a net deficit of 1/m plasma
particles. When translated into a physical electric charge, the quasihole represents
an excitation with fractional electric charge,
q∗ =
e
m
(1.24)
An alternative way to see that the quasihole has fractional charge is to perform
the following gedanken experiment to produce a quasihole: drill a hole in the Laughlin
liquid at Z, and adiabatically insert a flux Φ0 (see Fig. 1.5). Consider a loop of radius
R encircling the point of flux insertion, and sufficiently far away from it (R ≫ ℓB )
that we can assume the quantum Hall fluid responds with the bulk σxy . By Faraday’s
20
As it happens, operators creating a quasihole are relatively easy to construct within conformal
field theory, but the same cannot be said for the quasielectron; for a discussion of the subtleties
involved, see [94].
29
Φ(t)
R
j(t)
E(t)
Figure 1.5: Flux insertion argument for fractional charge.
law, the changing flux induces an azimuthal electric field E(t) = − 1c dΦ
θ̂; this leads to
dt
a radial current density j = σxy Er̂ from the Hall response. Once the flux insertion is
complete, we see that a total charge
∗
q =
Z
dt
Z
j(t) · ds =
σxy
Φ0 = νe
c
(1.25)
flows into the area around the quasihole. Thus, the quasihole has a charge excess
whose magnitude is a fractional multiple of the electron charge, localized in a region
of size ℓB around Z. This argument makes it clear that the fractional charge is
inextricably linked with the fractional Hall conductance. For a lucid discussion of
some of the subtleties involved in thinking about the fractional charge of Hall effect
quasiparticles, we refer the reader to [76]. It can also be shown that the quasiparticles,
in addition to fractional charge, also possess fractional statistics [18].
1.5.5
Life on the Edge
So far, our discussion has exclusively focused on the bulk of the sample, where the
incompressibility of the fractional quantum Hall droplet leads to a gap both to neutral
30
h(x)
x
v
Figure 1.6: Edge excitations of a quantum Hall droplet.
collective modes as well as to quasiparticle excitations. The edge of a quantum Hall
droplet in contrast supports gapless excitations, whose field theory is that of a chiral
Luttinger liquid [240]. The subject of quantum Hall edge theories is vast and extremely technical, and is well beyond the scope of this introduction. Here, we content
ourselves with showing within a hydrodynamic approach that the edge of a
1
m
Laugh-
lin state is described by a chiral Luttinger liquid, closely following the presentation
of [243]. The central point [138, 225] is to note that the only low-lying excitations of
an incompressible, irrotational droplet that is gapped in the bulk are surface waves
along the edge of the droplet. By identifying these with the edge excitations of the
quantum Hall liquid and with an appropriate quantization procedure, we can infer a
1D quantum theory of the edge. Consider a quantum Hall droplet with filling factor
ν, confined in a finite region by a potential well. The electric field from the confining
31
potential generates a persistent current along the edge,
j = σxy ẑ × E
(1.26)
because of the nonzero Hall conductance; this implies that near the edge electrons
drift with velocity v = cE/B; we assert (without attempting a proof) that this must
also be the velocity of the edge excitations. If we pick x as a coordinate along the
edge, we may write a 1D density ρ(x) = nh(x) where n = ν/2πℓ2B is the bulk density
(see Fig. 1.6.) Since the edge waves are gapless, and propagate unidirectionally21
with velocity v, they should be described by a chiral wave equation
(∂t − v∂x )ρ = 0
(1.27)
The Hamiltonian is simply the energy, which we can compute classically as the work
done in displacing the charge a distance h against the electric field:
H=
Z
1
dx eρ(x)h(x)E =
2
Z
v
dxπ ρ(x)2
ν
(1.28)
We can rewrite (1.27) and (1.28) in momentum space as
ρ̇k = ivkρk
vX
H = 2π
ρk ρ−k
ν
(1.29)
k>0
where ρk =
√1
L
R
dxeikx ρ(x) and L is the length of the edge. From these, we infer that
if we take as generalized coordinates qk = ρk (k > 0), then the corresponding canonical
momenta are pk =
2πi
ρ ,
νk −k
and Hamilton’s equations are satisified. Quantizing the
theory is then a simple matter of imposing canonical commutation relations, [qk , pk′ ] =
21
We choose B to point in the −ẑ direction so that v is in the positive sense.
32
iδkk′ , whence
[ρk , ρ′k ] =
where k, k ′ =
2πm
L
ν
kδk+k′ ;
2π
[H, ρk ] = vρk
(1.30)
with m an integer. These commutation relations form a U(1) Kac-
Moody algebra, and are precisely those obtained by bosonization of an interacting
theory of chiral fermions; in other words, they describe a chiral Luttinger liquid [85].
Using standard techniques from bosonization, we can show that the operator that
2π
creates an electron is ψ(x) ∼ ei ν
edge (for the case ν =
1
)
m
Rx
dx′ ρ(x′ )
and that the electron propagator along the
is:
G(x, t) = hT ψ † (x, t)ψ(0)i ∼
1
(x − vt)m
(1.31)
This corresponds to a Luttinger parameter [85] m, which is the final piece of information we need to specify the edge theory.
1.5.6
Hierarchies
The Laughlin states are excellent variational ground states for filling factor 1/m;
however, it experiments show a host of other fractions which are not simple Laughlin
fillings. How are we to understand these new fractions?
The solution lies in any of a number of ‘hierarchy constructions’ that in effect
reduce the problem non-Laughlin fillings to an already solved problem. One approach
– pioneered by Haldane [86] and Halperin [91] – is to arrange affairs so that the
quasiparticles of an existing quantum Hall state themselves form a Laughlin liquid.
Another idea, due to Jain [103], is to consider the fractional quantum Hall effect as
the integer effect of a new ‘composite’ fermion. Finally, there is a third construction
due to Wen and Zee [247] which we shall not consider here. For a succinct comparison
of the advantages and shortcomings of the Jain and Haldane-Halperin approaches, we
33
defer to [113]. We note that while they are perfectly good variational states, not all
members of a given hierarchy may be realized since other phases, such as quasiparticle
Wigner crystals, may have a lower energy.
Haldane-Halperin Hierarchy
We start with the Haldane-Halperin hierarchy construction, which proceeds iteratively
at as follows: fractional quantum Hall states at a given level of the hierarchy are
obtained by forming Laughlin states of the quasielectrons or quasiholes of the previous
level; the uppermost level of the hierarchy consists of the Laughlin states.
Let us begin with a parent Laughlin state with ν =
1
,
m
and discuss how to con-
struct the next level of the hierarchy22 . As we have shown previously, on the disk the
flux and filling factor at commensuration are related by NΦ = m(N − 1). If we in
addition add Nqp quasiparticles, this is replaced by
NΦ = m(N − 1) + αNqp
(1.32)
with α = −1(+1) for quasielectrons (quasiholes). If we now require that the added
quasiparticles are themselves in a Laughlin
1
p
state, we must have a similar condition
N = p(Nqp − 1)
(1.33)
with p even. This requires some explanation. The evenness of p is because the
quasiparticles are nominally bosonic, and so they can only form even-denominator
Laughlin states. The replacement of NΦ by N is because the number of independent
single-quasiparticle states is N rather than NΦ , which is the number of available
electronic states. From (1.32) and (1.33), it follows that in the thermodynamic limit
22
Our discussion closely parallels that of Haldane in [192].
34
the first-level hierarchy state has filling
ν=
p
N
=
NΦ
mp + α
(1.34)
To determine the charge carried by quasiparticle of the hierarchy state, consider
adding a single electron to the system, so that we take N → N 0 + 1. We then have
0
NΦ = m(N 0 + 1 − 1) + αNqp
0
= m(N 0 − 1) + α(Nqp
+ αm)
≡ m(N 0 − 1) + αNqp
(1.35)
0
where we have defined a modified quasiparticle number Nqp = Nqp
+ αm, which
follows from the fact that an electron is equivalent to αm Laughlin quasiparticles.
We then find
0
N 0 + 1 = p(Nqp
− 1) + 1
0
= p(Nqp
+ αm − 1) − α(mp − α)
= p(Nqp − 1) − α(mp − α)
(1.36)
which corresponds to a state with (mp − α) quasiparticle excitations of the daughter
1
i.e. the reciprocal
incompressible fluid. It follows that the latter have charge ± mp−α
of the denominator of the filling fraction.
By iterating this procedure, one can construct an infinite hierarchy of incompressible states, at filling factors given by appropriately terminating the continued fraction
ν=
1
m+
α1
α2
p1 − p −...
2
35
.
(1.37)
If the underlying system is fermionic – as is the case in a 2DEG – this always leads to
an odd denominator. Whether any of these states are stabilized in the lowest Landau
level once again depends sensitively on the interactions.
Composite Fermions and the Jain Hierarchy
To motivate the Jain approach, it is instructive to rewrite the Laughlin wave function
in the following suggestive form: with m = 2k + 1, we have
Ψm
N ;Laughlin [z]
N
N
Y
Y
P
m − 14 r |zr |2
=
(zi − zj )2k Ψ1N [z]
(zi − zj ) e
=
(1.38)
i<j
i<j
which is a Jastrow factor multiplying a Slater determinant corresponding to a filled
lowest Landau level. Jain’s insight was to realize that the last term could be replaced
by other Slater determinants, corresponding to p filled Landau levels, ΨpN [z, z ∗ ]; since
the latter obviously involve terms from higher Landau levels, the resulting wavefunction must be projected into the lowest Landau level. When this is done, we obtain a
trial wavefunction that describes quantum Hall states at filling ν =
"N
#
Y
(k,p)
ΨN ;Jain [z] = P
(zi − zj )2k ΨpN [z, z ∗ ]
1
:
2k+p−1
(1.39)
i<j
A very useful understanding of the Jain hierarchy can be given within composite
fermion Chern-Simons theory (which is the subject of the next section). Here, we
perform a statistical gauge transformation that attaches 2k flux quanta to each electron, whose formal implementation introduces a Chern-Simons term into the action.
The resulting particles obey fermionic statistics, and are termed composite fermions.
Since composite fermions carry flux, the magnetic field seen by any one of them is the
sum of the external, fixed magnetic field and the statistical pseudomagnetic field due
to all the others. Thus, when the composite fermions are at uniform density they can36
cel part of the external magnetic field. As charged particles in the residual magnetic
field, they give rise to an auxiliary Landau problem, with its own pseudo-Landau
levels, sometimes termed “Λ” levels to emphasize their fictitious nature. When an
integer number p of these are filled, the result is the Jain state23 with ν =
1.6
p
.
2pk+1
Half-Filled Landau Levels
In the previous section, we showed two different ways to construct an infinite hierarchy of incompressible states at odd-denominator fillings. While somewhat involved,
they allow us to explain many of the observed quantized Hall plateaus. However,
experiments see two distinct behaviors when a Landau level is half filled: at ν = 21 ,
there is much evidence in favor of a compressible, Fermi liquid-like state, whereas at
ν=
5
2
– corresponding to a half-filled n = 1 Landau level above filled lowest Landau
levels for each spin polarization – there is a clear plateau in σxy . Evidently, neither
the Haldane-Halperin hierarchy nor Jain’s construction can explain the latter24 ; as
for the Fermi liquid-like state, this appears to be another beast entirely.
In this section, we shall show that both the compressible state at ν =
the incompressible ν =
5
2
1
2
and
state may be understood in a unified fashion in terms of
composite fermions, albeit in a manner that sets the incompressible state apart from
Jain’s hierarchy. To do so, we shall have to introduce two new pieces of physics: the
composite fermion Chern-Simons theory and paired quantum Hall states of composite
fermions.
23
Within the field theoretic approach, a careful treatment of the fluctuations of the Chern-Simons
field may be necessary to recover the wavefunction. For an example at ν = 31 , see [146].
24
The attentive reader might worry that the possibility left unmentioned might solve the problem;
she can rest assured that the Wen-Zee approach is also left wanting at even denominators.
37
1.6.1
Composite Fermions and the Halperin-Lee-Read Theory
We shall begin by discussing the Halperin-Lee-Read (HLR) Chern-Simons theory of
the half-filled Landau level. Our treatment shall be necessarily, even criminally, brief,
and we refer the reader interested in further details to both HLR’s original work [92]
and the excellent pedagogical review by Simon [215].
There are many different ways of motivating the Chern-Simons approach. Perhaps the conceptually most straightforward route is to begin in the first-quantized
Hamiltonian formalism, and observe that the unitary transformation,
Ψ → Ψ̃ ≡ ei2k
P
i<j
Im log(zi −zj )
Ψ
(1.40)
on the many-body wavefunction maintains the antisymmetry of the wavefunction, and
hence the statistics of the underlying particles are unchanged. We further observe
that the Hamiltonian is changed by this transformation: the vector potential A(rj ) →
A(rj ) + a(rj ), where the ‘statistical’ gauge field a is transverse, ∇j · a(rj ) = 0 and
satisfies
∇ × a(rj ) ≡ b(rj ) = −2kΦ0
X
i6=j
δ(ri − rj ).
(1.41)
The physical content of this statement is that each composite fermion – as the gaugetransformed electrons are known – sees a 2k-flux tube at the coordinates of each of
the others. In other words, we have attached a pair of fluxes to each electron in order
to convert it to a composite fermion. The composite fermions thus move in a gauge
field that is the sum of the statistical and background (i.e., external) contributions25 .
25
Our argument, with the minimal modification that 2k is replaced by 2k + 1, applies equally well
to the composite boson approach; indeed part of our discussion is adopted from [113] which focuses
on the latter.
38
With these preliminaries, we are ready to make the leap to a field theory. Consider a system of fermions, interacting with an external magnetic field A, and a
statistical Chern-Simons gauge field. This may be described by the path integral,
R
Z = D ψ̄DψDaµ e−S , where
β
2
e
~2 †
S =
dτ d r ψ (i~c∂0 − ea0 − µ)ψ +
−i∂i − (ai + Ai ) ψ ∗
2m
~c
0
Z
1
1
2 ′ †
′
† ′
′
ij
d r ψ (r)ψ(r)V (r − r )ψ (r )ψ(r )
(1.42)
a0 ε ∂i aj +
−
2kΦ0
2
Z
2
The above action is written in Coulomb gauge, where we impose the transversality
condition26 ∇ · a = 0. Here, V is be the electron-electron interaction and µ is the
chemical potential. We have chosen not to impose an external A0 . As usual, Φ0 =
hc/e is the quantum of flux.
The equations of motion can be obtained by varying the action with respect to ψ
and aµ . For ψ we obtain the standard equations for a nonrelativistic fermion moving
in the combined field aµ + Aµ , with a density-density interaction V . The equations
of motion for the Chern-Simons field are given by
b ≡ ∇ × a = −2kΦ0 ρ(r)
εij ej ≡ εij (∂0 aj − ∂j a0 ) = −
2kΦ0
j(r)
c
(1.43)
where
26
ρ(r) = ψ † (r)ψ(r)
e
~ †
†
ψ
∇ψ
−
ψ∇ψ
− ∗ (a + A)ψ † (r)ψ(r)
and j(r) =
r
∗
2m i
mc
(1.44)
In two dimensions, this means the natural second term in the gauge action, a × ∂0 a vanishes;
a manifestly gauge-invariant action can be obtained by undoing the gauge-fixing. See [113] for a
discussion in the composite boson context.
39
are the density and current density of the composite fermions, and we have defined
the ‘statistical electric field’ e . Note that varying the action with respect to a does
not, strictly speaking, lead to an equation for ej as written, since by our gauge fixing
εij ∂0 aj = 0 so that e = −∇a0 . Nevertheless we give the definition of e that is gaugeinvariant which we can always obtain from a gauge-invariant form of S where we undo
the gauge fixing in the path integral.
The composite fermion density is obviously equal to the density of the original
electrons; we observe in addition that since ψ is always coupled to the combined field
aµ + Aµ , we may write
δS
δS0
δS0
ie
jµ ≡
=
=
c
δAµ
δAµ
δaµ
(1.45)
where S0 is the action without the Chern-Simons term, thereby relating the electron
and composite fermion current densities. The first of the Chern-Simons equations
implements the flux-attachment transformation and is therefore equivalent to (1.41);
the second simply ensures that the flux attachment is preserved under time evolution,
and can be seen more or less as a consequence of the continuity equation relating ρ
and j.
At half-filling (ν = 21 ), we observe that hρi = n =
B
.
2Φ0
The mean-field solution of
the first Chern-Simons equation yields hbi = −2kΦ0 n, so that for k = 1, the meanfield statistical magnetic field exactly cancels the external magnetic field, hbi + B = 0,
and the composite fermions see zero field on average. Barring various instabilities –
which in channels other than the particle-particle channel are precluded by various
considerations, for rotationally invariant systems [210] – the ground state of a system
of fermions in zero field is to form a Fermi sea. This in essence is the heart of
composite fermion theory: Jain’s states can then be understood as arising when the
Landau diamagnetism of the ν =
1
2k
composite fermion metal leads to the formation
of a state with p filled pseudo-Landau levels. The Fermi liquid-like state has been
40
verified both in numerics [204] and by surface acoustic wave absorption [249] and
magnetic focusing [81] experiments.
For now, let us focus on the case of half-filling (k = 1), although our results apply
for any even-denominator filling ν =
1
.
2k
Thanks to the Chern-Simons constraint, we
are free to trade the quartic density-density interaction for a quadratic interaction
between fluxes that modifies the propagator for the gauge field. Naı̈vely it seems
that we can solve the theory exactly by integrating out the fermions. However, the
fermionic excitations are gapless, and as a result this procedure is not controlled
and can lead to various non-analytic and/or non-local terms in the resulting effective action27 . We can go one step beyond the saddle-point solution and incorporate
Gaussian fluctuations of the Chern-Simons field, as originally done by HLR. As we
show below, the resulting “random-phase approximation” (RPA) response is that of
a compressible phase with a Hall resistance tied to ρxy =
2e2
,
h
and a longitudinal re-
sistance that arises from scattering off both charged impurities as well as the random
flux configuration produced due to the rearrangement of the electron density in a
disordered external potential [92].
We provide a telegraphic account of how one computes the composite fermion
conductivity [215]. While computing the full electromagnetic response in the RPA
is somewhat involved [92, 215], the determination of the conductivity tensor – which
is ultimately the most significant observable in the quantum Hall context – is fairly
straightforward. We begin by observing that the equation for the Chern-Simons
electric field can be written as
e = 2kΦ0 (ẑ × j) ≡ −ρCS j
27
An example of the law of conservation of difficulty, or the ‘no free lunch’ theorem.
41
(1.46)
where
ρCS =


2kh  0 1 


e2
−1 0
(1.47)
is the Chern-Simons resistivity tensor. On the other hand, we have already shown
that the composite fermions interact with the combined gauge field aµ + Aµ ; the
resulting response equation takes the form
j = ρ−1
CF (e + E)
(1.48)
Here ρCF is the resistivity of composite fermions in the effective magnetic field hbi +
B. Of course, the Chern-Simons field is a purely internal quantity, and therefore
cannot be measured by a physical voltmeter; the conductivity tensor measured in
any experiment is the response of the current to the external field E. To determine
the measured conductivity σ, we must therefore eliminate e using the Chern-Simons
relation, to find a resistivity addition rule: j = σE, with
σ −1 = ρ = ρCF + ρCS
(1.49)
Using (1.49), we can compute the response of various composite fermion states, both
compressible and incompressible.
For the case of half-filling, the effective field vanishes and therefore ρCF is purely
diagonal,


ρCF = 
ρCF
xx
0

0 

ρCF
yy
(1.50)
leading to the the following measured conductivity:
σ=
1
CF
ρCF
xx ρyy
+

CF
 ρyy

2h 2
e2
42
2h
e2
− 2h
e2
ρCF
xx



(1.51)
We see immediately that is not the response of a quantized Hall phase, which is
appropriate: we do not expect a plateau from a compressible state. Note that within
the composite fermion Chern-Simons theory, the Hall resistance is quantized, although
there is dissipation i.e. nonzero longitudinal resistance
28
.
If we perform the composite fermion construction away from half-filling, the statistical and external fields no longer cancel exactly and hbi is nonzero. If, in this
weakened field, the composite fermions form integer quantum Hall states with p filled
pseudo-Landau levels, we have
ρCF =


h  0 1 


pe2
−1 0
(1.52)
which leads to a measured conductivity
σ=


1 e2  0 −1 


2k + 1p h
1 0
as appropriate to the Jain hierarchy state at filling ν =
(1.53)
p
.
2pk+1
As promised, we have
recovered the results of Jain’s variational approach within a field theoretic framework.
Fluctuations of the Chern-Simons gauge field should not significantly modify the
qualitative features of results away from half filling, since the states with p filled
pseudo-Landau levels are all gapped. However, this is emphatically not the case for
the gapless composite Fermi liquid at ν = 12 . Here, the inclusion of gauge fluctuations
beyond the RPA leads to various singularities. When the RPA corrections are taken
into account, the propagator of the gauge field is renormalized by the particle-hole
excitations of the composite fermions, and acquires both infrared and ultraviolet
28
This result, which is a natural consequence of the vanishing of the composite fermion Hall conCF
ductance σxy
in the HLR approach, is somewhat controversial; in particular, particle-hole symmetry
– which applies in the lowest Landau level approximation – requires that at half-filling the electron
Hall conductance, rather than the Hall resistance, be quantized. See [136] for a discussion of this
point.
43
divergences. While the latter can be explained away as unphysical, the former lead
to singularities in various physical quantities when the RPA-improved propagator is
used in computations that go beyond the RPA, such as the fermion self-energy and
corrections to the fermion-gauge vertex. In particular, the composite fermion spectral
function vanishes logarithmically for Coulomb interactions, and even faster for shortranged interactions between the bare electrons [92, 223]. Various careful studies [115,
116, 117, 118, 127, 171] show that while the low-frequency, long-wavelength response
does not deviate significantly from the RPA result, other quantities– for instance, the
2kF susceptibility – may acquire divergences or nonanalyticities. Such singularities
make the composite fermion approach a far from controlled theory. Ultimately, the
most serious objection – which also applies also to the composite boson LandauGinzburg theories – is on very physical grounds: namely, that the Chern-Simons
approach fails to properly build in the the lowest Landau level structure. Frequently
quoted in this context is the “effective mass problem”, which in its most obvious
form is that the gaps predicted for incompressible states by the Chern-Simons-RPA
method have an unphysical dependence on the band mass of the electrons, rather
than being set purely by the interaction scale e2 /εℓB as appropriate to a fractional
quantum Hall state constrained to the lowest Landau level [215].
There have been various attempts to construct a more tractable theory for ν = 12 .
These include both phenomenological Fermi liquid approaches [223, 217, 215], as well
as pioneering work by Read [199] and by Pasquier and Haldane [189] for the bosonic
case at ν = 1 and the Hamiltonian theory of Murthy and Shankar [167]. In one
way or another, each of the latter three approaches attempts to build a theory for
composite fermions that lives entirely within the lowest Landau level and thus avoids
the singularities inherent in the HLR approach. While they have some success, they
are fairly cumbersome to work with, and so we continue to use the Chern-Simons
approach with all the necessary caveats. In the end, any predictions that we make will
44
have the flavor of phenomenological guesswork: they will have to find their vindication
in numerical data or experimental fact.
1.6.2
Paired States of Composite Fermions
We have successfully explained the compressible state at filling factor ν =
1
2
as a
Fermi liquid of composite fermions. How are we to understand the state at ν = 25 ,
which is incompressible and has a quantized Hall conductance?
One solution, which has the merit of hewing closely to the historical development
of the subject, is to simply guess a wavefunction; this was done by Moore and Read ,
where they used the conformal block approach to produce a wavefunction for an evendenominator incompressible state [164]. Rather than take this route, we shall sacrifice
historical accuracy for the sake of physical clarity. To motivate our approach, let us
return to the computation of the response of the composite Fermi liquid. There,
we observed that the measured resistivity was the sum of the composite fermion
resistivity, which is computed in zero magnetic field, and a Chern-Simons term which
was purely off-diagonal. The dissipation inherent in the former resulted in a state
without quantized Hall response. Were it not for this, we would have produced a state
which had an even-denominator Hall conductance. It is natural, therefore, to ask how
we can arrange affairs so that the compressible state acquires a bulk gap – necessary
for the precise quantizaton of σxy and dissipationless transport. In other words, how
can we gap a system of fermions in zero field so that transport is dissipationless?
The answer is to destroy the composite fermion Fermi surface by forming a superconductor via the Bardeen-Cooper-Schrieffer (BCS) mechanism. The resulting
superconducting state29 will have a Meissner effect – expulsion of flux from the bulk
– which, thanks to the flux-charge equivalence of Chern-Simons theory implies a
charge gap. Since superconductors cannot support electric fields in the bulk, we must
29
Much of what we say about the superconducting properties applies equally to the composite
boson theory for odd denominators.
45
have e+E = 0, which leads to the necessary quantized Hall conductance30 . Moreover,
vortices of the superconducting order parameter carry electric charge thanks to the
Chern-Simons term; owing to the halving of the quantum of flux in the paired state
and the half-integer Hall conductance, their charge is quantized in units of 4e .
This identification of the incompressible state in a half-filled Landau level as a
paired quantum Hall state of composite fermions [164, 82] has a number of close
connections to the original approach based on conformal field theory (CFT), which
we summarize briefly. First we note that in the simplest case of spinless fermions, the
pairing necessarily occurs in the p-wave channel [200]. The resulting paired state can
be in one of two phases: a weak-pairing phase (corresponding to the ‘BCS’ picture
of long-ranged pairs in position space) and a strong-pairing phase (the ‘BEC’ or
‘molecular’ limit where the pairs are tightly bound.) In each case, we can compute
first-quantized electronic wavefunctions by projecting the BCS wavefunction onto a
sector with a definite number of composite fermions, and multiplying the result by the
(zi − zj )2k factor required by the composite fermion flux attachment transformation.
In the weak-pairing phase, we recover (asymptotically) the ‘Pfaffian’ wavefunction
of Moore and Read, which at ν =
Ψ2k
MR [z] = Pf
1
2k
has the form
1 P
1 Y
2
(zi − zj )2k e− 4 j |zj |
zi − zj i<j
(1.54)
where the Pffafian of a 2L × 2L antisymmetric matrix M is
L
Y
1 X
Pf[M] = L
signσ
Mσ(2k−1),σ(2k) .
2 L! σ∈S
k=1
(1.55)
2L
By solving the Bogoliubov-de Gennes (BdG) equations for a vortex in the paired
state – corresponding to a quantum Hall quasiparticle– we can show that it supports
30
Throughout, we assume that all the action takes place in the uppermost, partially filled Landau
level; thus at ν = 52 , we are taking for granted that the underlying filled Landau levels will provide
the deficit 2e2 /h needed to obtain the correct Hall conductance.
46
a zero-energy fermionic bound state, known as a Majorana zero mode [200, 101]. As
a result, the vortices (quasiparticles) acquire non-Abelian statistics: the low-energy
Hilbert space with nonzero quasiparticle number is finite-dimensional, and braiding
the vortices leads to a unitary evolution within in this low-energy subspace31 – thus
rederiving a striking prediction of Moore and Read. Other topological properties –
such as the existence of neutral gapless chiral edge modes and the counting of ground
state degeneracies – can be computed using the BdG equations and shown to match
the CFT predictions.
In the strong-pairing phase, the wavefunction is no longer of the Pfaffian form,
owing to the tightly bound position space pairs. Asymptotically we simply find a
wavefunction describing the resulting bosonic molecules, which describes an Abelian
phase. In going from weak to strong pairing, the system undergoes a topological
phase transition, as they correspond to different topological orders and hence different
phases of matter.
We thus have a unified picture of the two distinct sets of phenomena at even
denominators, summarized as follows: at such fillings, the electrons form a composite fermion Fermi liquid state. At ν =
1
,
2
the inter-electron repulsion is sufficient
to render pairing ineffectual in destroying the Fermi surface, and the compressible
state survives. At ν = 25 , the modified form of the electronic wavefunctions in the
n = 1 Landau level softens the repulsion, and the Fermi surface is unstable to pairing,
leading to an incompressible state. This picture has received some support from numerical studies; it naturally carries with it the presumption that the pairing strength
is tunable. We shall make implicit use of this ability in Chapter 2 when we show
31
This has been proposed as a platform for ‘topological’ quantum computation. For a review, see
[170] .
47
that in the weak-coupling32 , ‘BCS limit’ the pairing energetics enforce quasiparticle
attraction, leading to a Type I quantum Hall liquid.
It is possible to show that other paired states – including those with spin or other
internal degrees of freedom – can be captured within the composite fermion-BCS
approach. We shall not discuss these further, but direct the interested reader to [200]
for details.
As a final comment before we close this section, we point out that the fact that
half-filled Landau levels can support both a compressible Fermi-liquid like phase and
an incompressible paired state suggests that there may be interesting avenues for
exploration, in heterostructures engineered to have spatially varying interaction parameters that support pairing in some regions of the sample and suppress it in others.
As the gapless state has no natural length scale, it should support pairing correlations
with power-law decay33 . This leads to a quantum Hall version of the superconducting
proximity effect, and closely related analogs of Andreev reflection and the Josephson
effect, which may serve as yet another probe of the fractional quantum Hall regime
[184]. No comparable statements can be made for odd denominator quantum Hall
states, essentially because there is no competing Fermi-liquid-like state at such fillings.
1.7
Landau-Ginzburg Theories of the Quantum
Hall Effect
We have argued that even-denominator quantum Hall states are usefully described
as ‘superconductors’of composite fermions. We mentioned, but did not explicitly
demonstrate, that the presence of a Chern-Simons term for the statistical gauge
32
There is a subtle and for our purposes not particularly important distinction between weak
coupling, which refers to the pairing energy scale, and weak pairing, which is a statement about the
size of the pair wavefunction. For a discussion of this point see [200].
33
These should survive the inclusion of interactions, as indeed is the case in the simpler example
of a repulsive Fermi liquid in Chapter 9.
48
field allowed us to translate various properties of the superconducting phase into
the language of the quantum Hall effect. This approach is not restricted to halffilled Landau levels; we can similarly model odd-denominator quantum Hall states as
superconductors with Chern-Simons electrodynamics, except that we no longer have a
natural interpretation of the superconductor as a condensate of Cooper pairs. Instead,
we map the original problem of electrons in a magnetic field to one of composite bosons
in zero field. The condensed phase of the composite bosons then corresponds to the
quantum Hall state. It is to the Chern-Simons Landau-Ginzburg theories that result
from this analysis that we now turn.
1.7.1
Composite Boson Chern-Simons theory
We give a brief introduction to the Chern-Simons Landau-Ginzburg theory introduced by Zhang, Hansson and Kivelson (ZHK) [258] to describe odd-denominator
fractions, and by extension the Haldane-Halperin hierarchy construction. The ZHK
case predates the composite fermion approach to the half-filled Landau level, and
rests on the idea of statistical transmutation – a flux attachment transformation that
cancels part or all of the external field, similarly to the composite fermion approach,
but which leads to a bosonized description of the quantum Hall problem.
We return to the unitary transformation (1.40-1.41) , but this time attach an odd
number of flux quanta to each electron, so that
Ψ → Ψ̃ ≡ ei(2k+1)
P
i<j
Im log |zi −zj |
∇ × a(rj ) ≡ b(rj ) = −(2k + 1)Φ0
Ψ
X
i6=j
δ(ri − rj ).
(1.56)
Unlike in the composite fermion case where an even number of fluxes were attached,
here the symmetry of the many-body wavefunction has changed: Ψ̃ is now symmetric
in all its coordinates. The gauge-transformed electrons obey Bose statistics, and
49
we refer to them as composite bosons. The flux attachment transformation can be
captured as before within a Chern-Simons field theory, this time involving a bosonic
field ϕ but otherwise closely resembling (1.42)
β
2
e
~2 S =
dτ d r ϕ (i~c∂0 − ea0 − µ)ϕ +
−i∂i − (ai + Ai ) ϕ
∗
2m
~c
0
Z
1
1
2 ′
2
′
′ 2
µνλ
−
(1.57)
d r |ϕ(r)| V (r − r )|ϕ(r )| .
ε aµ ∂ν aλ +
2(2k + 1)Φ0
2
Z
2
∗
In writing this action, we have chosen to undo the transverse gauge-fixing in (1.42) so
that the action is manifestly gauge-invariant34 . The equations of motion that follow
are very similar to the composite fermion case, except that the boson order parameter
doubles as the density,
b ≡ ∇ × a = −(2k + 1)Φ0 |ϕ(r)|2
εij ej ≡ εij (∂0 aj − ∂j a0 ) = −
(2k + 1)Φ0
j(r)
c
(1.58)
and j is the composite boson current, which is identical to (1.44) but with ψ replaced
by ϕ.
Similarly to the composite fermion case, the composite bosons experience an effective field which is the sum of the statistical and external contributions. Thus, at
commensurate fillings ν =
1
,
2k+1
the effective field vanishes and we have a theory
of interacting bosons in zero field. In order to capture the phenomenology of the
Hall effect, we replace the long-range Coulomb interaction with a contact repulsion
between the bosons, so that the bosonic part of the action takes on the familiar
34
Except under transformations that are do not equal the identity on the boundary. This lack of
invariance under “large” gauge transformations is a standard feature of a Chern-Simons theory, that
ensures that the Chern-Simons coupling is not renormalized in any perturbative, momentum-shell
renormalization procedure, which will always produce gauge-invariant corrections [208].
50
Landau-Ginzburg form [258],
β
2
e
~2 S =
dτ d r ϕ (i~c∂0 − ea0 − µ)ϕ +
−i∂i − (ai + Ai ) ϕ
∗
2m
~c
0
1
λ
2
+ |ϕ(r)|2 − ρ −
εµνλ aµ ∂ν aλ .
(1.59)
2
2(2k + 1)Φ0
Z
2
∗
while the gauge field obeys Chern-Simons electrodynamics.
At commensuration, the boson field is condensed, the system is in a superconducting phase and exhibits a Meissner effect; this translates to incompressibility of
the quantum Hall system through the Chern-Simons constraints. Exactly as in the
case of a paired composite fermion state, the total electric field e + E = 0 in the
condensed phase, so that the measured response is appropriate to a Hall plateau at
filling ν =
1
.
2k+1
The vortex excitations now carry an electric charge q =
e
,
2k+1
as
appropriate to Laughlin quasiparticles. As we shall have much more to say about
the vortex excitations in Chapter 3, we defer a detailed discussion until then. For
further discussion on how to derive other aspects of fractional quantum Hall effect
phenomenology from the effective field theory – including potential pitfalls of this
approach – we refer the reader one of the reviews on the subject [113, 257].
1.7.2
A Landau-Ginzburg Theory for Paired Quantum Hall
States
We have already argued that an incompressible state can be obtained in a half-filled
Landau level by forming a paired state of composite fermions. Response in such paired
states can be calculated by standard techniques from the theory of superconductivity
[69], and verifies our claim that the low-frequency, long-wavelength response is appropriate to that of an even-denominator quantum Hall plateau. Working within the
Bogoliubov-de Gennes formulation, we can extract additional information such as the
51
wavefunction of the N-electron ground state, the statistics – sometimes non-Abelian
– of quasiparticles, and the existence of neutral edge modes.
In this section, we sketch a derivation of a Landau-Ginzburg free energy for a
paired quantum Hall state, which we shall put to use in Chapter 2 to study the
energetics of quasiparticles and the structure of the quantum Hall plateau around
even-denominator fillings. While our derivation is formally valid only near the transition – really, a crossover – temperature Tc , for phenomenological purposes we can
use it down to T = 0. Our Landau-Ginzburg theory does not take into account
the non-Abelian statistics of the quasiparticles; this would necessitate an additional,
non-Abelian Chern-Simons gauge field for the nontrivial braiding statistics [73, 72].
While this is an important and fascinating aspect of the theory of paired Hall states,
it is somewhat peripheral to our interests and therefore we do not discuss it further.
We restrict ourselves to the spinless case and accordingly begin with the gaugefixed composite fermion action, (1.42), but with an additional external vector potential A0 . We add to this a phenomenological p-wave pairing interaction ‘by hand’ [69].
This takes the form
Sp = g
q
q † ψ −iωn + ηp , −k +
eiθk e−iθk′ ψ † iωn , k +
2
2
n,′ n,p,
X
(1.60)
k,k′ ,q
q
q ′
′
ψ iωn + ηp , k +
×ψ −iω + ηp , −k +
2
2
where ωn =
2π(n+1)
β
and ηp =
n′
2πp
β
′
(1.61)
are fermionic and bosonic Matsubara frequencies.
With this term added, the derivation proceeds as follows:
1. First, we use the Chern-Simons constraint to rewrite the density-density interaction V as an interaction between Chern-Simons fluxes. Once this is done, Sp
is the only term in the action that is quartic in the fermion operators.
52
2. We perform a Hubbard-Stratonovich decoupling of Sp in terms of a pair field
∆, so that we obtain quadratic terms of the form ∆ψ † ψ † , ∆∗ ψψ, and
∆2
g
at the
cost of an additional functional integral over ∆.
3. Next, we perform the quadratic integral over the fermionic fields, integrating
them out in favor of ∆. The resulting functional integral takes the form
∆2
1
a0 ∇ × a + S0 [∆, aµ + Aµ ] +
Z[Aµ ] =
D∆Daµ exp −
2kΦ0
g
Z
2
d rdτ
′
′
+
(∇ × a(r) − 2kΦ0 ρ) V (r − r ) (∇ × a(r ) − 2kΦ0 ρ)
(2kΦ0 )2
Z
(1.62)
where ρ̄ is a uniform positive background density chosen to render the Coulomb
contribution nonsingular, and S0 [∆, aµ + Aµ ] is the result of integrating out the
fermions.
4. We now work near T = Tc , and follow the usual arguments in the derivation of
the Landau-Ginzburg action to focus on the zero Matsubara frequency component of S, which suffices to obtain a free energy. We can expand the result in the
standard Landau-Ginzburg form in terms of a suitably rescaled order parame
2
2
e
(a + A) Ψ + c2 (|Ψ|2 − |Ψ0 |2 ) +
ter Ψ ∝ ∆35 [191] , i.e. F ∼ c1 −i∇ + ~c
χ0 e(a0 + A0 )2 , where the last contribution is from the response to the external
field a0 + A0 , and we use the fact that the static compressibility of a superconductor is essentially the same as the underlying Fermi liquid, χ0 =
m∗
.
2π~2
5. Finally, we integrate out a0 ; this imposes the Chern-Simons constraint on the
flux, and results in a contribution of the form
1
(4kΦ0 )2 χ0
+
A0 ∇×a
.
2kΦ0
The latter
term provides the correct Hall response to an external electric field E = −∇A0 ,
35
In general the order parameter can have more components, but our problem is simplified by
working in the spinless case with a coupling that is attractive for only one of the two independent
p-wave chiralities.
53
while the former is an effective Maxwell energy that traces its origin to the
compressibility of the underlying Fermi liquid normal state. The final result,
with appropriate additional rescalings and redefinitions, is a free energy of the
form (2.1), whose properties are discussed in detail in Chapter 2.
Of course, we have still to perform a functional integral over a as well as one over
the Hubbard-Stratonovich field ∆; however, we expect that while this may change
details of various estimates, the qualitative features elaborated in Chapter 2 are robust
to such corrections.
The approach we have taken to analyzing the composite fermion pairing problem
has been entirely phenomenological: we asserted that pairing occured, and inserted
an ad hoc pairing interaction Sp . Ideally, we would like to derive the pairing physics
directly from the Coulomb interaction, but this is fraught with difficulties [35]. Recent
progress on understanding the pairing transition within the renormalization group
approach, whose preliminary results appear consistent with the results of Chapter 2,
may shed additional light on these issues [154].
1.7.3
Off-Diagonal Long Range Order in the lowest Landau
level
The central appeal of the effective field theory approach to the fractional quantum Hall
effect is that it allows us to understand the topologically ordered quantum Hall phase
as arising from the condensation of a bosonic order parameter. This simplifcation is
achieved by means of the nonlocal flux attachment transformation. Unfortunately, by
its very nature of partitioning the magnetic field into fluxes attached to charges, the
flux attachment approach discards any information about the structure of Landau
levels; the resulting theory is thus purely phenomenological, and various parameters
have to be taken as physical inputs. For instance, since the effective field theory has to
54
simultaneously (a) reproduce Kohn’s theorem on the cyclotron resonance frequency
and (b) give the correct scale (∼
e2
)
εℓB
to collective modes in the lowest Landau level,
the question of an appropriate choice for the effective mass remains a thorny one.
It is natural to ask, in light of such issues, if we can construct a version of the
composite boson/paired composite fermion construction that builds in Landau level
physics at the outset. In important early work in this direction, Girvin and MacDonald [78] produced a lowest-Landau level operator – equivalent to the bosonic order
parameter ϕ of the Zhang-Hansson-Kivelson theory – that exhibited ‘algebraic offdiagonal long-range order (ODLRO)’ , i.e. its two-point correlations exhibits powerlaw rather than exponential decay with distance. This was later extended by Read
[198] to construct an operator that exhibits true off-diagonal long-range order – the
two-point correlator tends to a constant at infinite separation. We briefly describe
how the ‘Read operator’ is constructed for the Laughlin fractions; similar arguments
can be made for paired Hall states.
1
m
which we shall deQ
note by |0L; Ni; this has the usual (unnormalized) coordinate representation i<j (zi −
P
1 P
2
zj )m e− 4 k |zk | . It will prove useful to define ψ(z) ≡ ∞
n=0 an ϕn (z), where an the
We begin with the N-particle Laughlin ground state at ν =
second-quantized field operator for the nth single-particle lowest Landau level basis
state ϕn (z) (see Section 1.5.1 ), as well as the Laughlin quasihole operator, which in
Q
first quantization is simply U(z) = N
i=1 (zi − z).
Read observed that the two-point correlator
h0L ; N|Ũ † (z)m ψ(z)ψ † (z ′ )Ũ (z ′ )m |0L ; Ni = ρ−1 h0L ; N + 1|ρ(z)ρ(z ′ )|0L; N + 1i
→ ρ0
(1.63)
as |z − z ′ | → ∞, with |z|, |z ′ | > N. Here, we have defined a normalized version of the
quasihole operator Ũ (z)m |αi ≡ (hα||U(z)|2 |αi)−1/2 U(z)m |αi.
55
The result (1.63) follows more or less immediately from the observation that the
simultaneous addition of m fluxes and one electron to an N-electron Laughlin state
must have nonzero overlap with the N +1 electron Laughlin state, and the fact that the
Laughlin states have liquid-like density correlations. Note that since the expectation
value of the Read operator hOR (z)i ≡ hψ † Ũ m i vanishes in a Laughlin state while its
square-expectation is nonzero, the Laughlin state is not a pure state. An example
of the latter, in which ψ † Ũ m also has an expectation value, can be constructed by
superposing Laughlin states of various particle numbers36
|0L ; θi =
∞
X
N =1
αN e−iN θ |0L ; Ni
(1.64)
where αN is real and squares to a binomial distribution on N with mean N̄ ≫ 1 and
variance of order N̄ , and θ is an arbitrary parameter. For this state, and arbitrary z,
it is easily verified that
hOR (z)iθ → ρ0 1/2 eiθ
(1.65)
as N̄ → ∞. Thus the Read operator serves as an order parameter for the fractional
quantum Hall state. In exact analogy with a BCS superconductor, the fixed-particlenumber Laughlin wavefunction follows as
|0L; Ni =
Z
2
†
m −|z 2 |/4
d z ψ (z)U(z) e
N
|0i
(1.66)
While not quite local as it involves a flux insertion, the Read operator can be uniquely
associated with a single point, and therefore it can be used to construct a lowestLandau-level field theory for the fractional quantum Hall states [198, 194]. However,
the Chern-Simons Landau-Ginzburg formalism is significantly easier to work with;
36
This should be familiar to readers well-versed in the theory of superconductivity; this is exactly
how one constructs states in which the Cooper pair operator has a definite expectation value.
56
we therefore adopt this in Part II of this thesis, under the assumption that the longwavelength predictions of both approaches are morally the same.
1.8
Type I and Type II Quantum Hall Liquids
Superconductors famously come in two varieties, which differ in their response to external magnetic fields: Type I superconductors phase separate into superconducting
and normal regions, with flux concentrated in the latter, while Type II superconductors form an Abrikosov lattice of vortices, each carrying a single flux quantum.
The analogy between superconductors and fractional quantum Hall phases suggests
that there is a similar distinction between Type I and Type II quantum Hall liquids,
manifested in dramatically different patterns of charge organization upon doping with
quasiparticles. While Type II quantum Hall liquids exhibit the Wigner crystallization
of fractionally charged quasiparticles traditionally assumed to occur about commensurate fillings in the clean limit, their Type I cousins exhibit either phase separation
or for sufficiently long-ranged interactions, its frustrated mesoscopic analogs. Surprisingly, this quite general dichotomy was pointed out only recently, when it was argued
– as recounted in Chapter 2 – that Type I behavior occurs in paired quantum Hall
states, in the ‘BCS limit’, i.e. when the pairing scale is weak [185]. While the focus
of that work was the Pfaffian phase seen in the vicinity of filling factor ν = 5/2, the
results generalize implicitly to all paired states.
In Part II of this thesis, we show how to obtain Type I behavior at several other
fractions, both within effective field theory as well as more microscopically in terms
of Hamiltonians projected into the lowest Landau level. In both approaches, we rely
crucially on the existence of special points in the space of parameters, at which the
interaction between quasiparticles 37 vanishes, and the underlying quantum Hall fluid
is simultaneously gapped. Adding a weak attractive perturbation then renders the
37
More precisely, quasiholes but not quasielectrons, as elaborated in Chapter 3.
57
quasiparticles attractive without closing the gap required for their existence, which
leads immediately to Type I behavior. Further modifications – such as the introduction of interactions of varying range and competing signs – can frustrate phase
separation, leading to a variety of mesoscale phases upon doping.
The distinction between Type I and Type II liquids thus adds an additional layer of
complexity to the characterization of various fractional quantum Hall phases. While
they share topological quantum numbers – such as ground state degeneracies on
nontrivial manifolds, and charge and statistics of fractionalized excitations – they have
significantly different quasiparticle energetics, reflected in the structure of their Hall
plateaus. Other aspects of their phenomenology, such as their response to disorder,
may further distinguish the two regimes.
1.9
ν = 1 is a Fraction Too: Quantum Hall Ferromagnets
So far, our discussion of the Hall effect has not included spin, or other internal degrees
of freedom. What happens when we add these to the problem?
In the case of spin, if we consider the lowest Landau level of a two-dimensional
electron gas in free space, the answer is: not much. This is because the Zeeman energy
gµB B which characterizes the gap between the different spin polarization states, is
exactly equal to the cyclotron gap ~ωc , for g = 2 as appropriate to free space. The
lowest Landau level has spins aligned with the magnetic field; every other spin-up
Landau level corresponding to the nth oscillator level is degenerate with the spindown Landau level of the (n − 1)th oscillator level. The gap to spin excitations is
the same as the gap to inter-level transitions, and so once we’re in a regime where
58
the lowest Landau level approximation may be made, the spin degrees of freedom are
‘frozen out’, and therefore don’t significantly change the physics at ν = 138 .
In real materials, however, two things conspire to alter this situation. First of all,
the effective mass in these systems is much smaller than the physical electron mass
(m∗ /m ≈ 0.068 for the conduction band of GaAs), and second, spin-orbit scattering
reduces the effective g factor (g ≈ 0.4 in GaAs.) The first effect increases the cyclotron
gap, whereas the second reduces the Zeeman splitting; the net result is that the ratio of
the two energy scales is reduced from 1 to about 1/70. This means that at sufficiently
low temperatures, the kinetic energy is quenched and the system may be considered
confined to the lowest Landau level, but the spin degrees of freedom remain free to
fluctuate. In fact, it is reasonable to ignore the Zeeman splitting at leading order,
and consider the two spin states to be degenerate.
The first and immediate conclusion is that without interactions, there can be
no ν = 1 quantized Hall state seen in any realistic experimental conditions. The
problem is once again massively degenerate – we have enough electrons to fill one
Landau level, but are presented with two degenerate levels – and therefore our only
recourse is to interactions as a means of resolving the degeneracy to produce a gapped
(i.e., incompressible) state. In this sense, ν = 1 is very similar to the case of fractional
filling, although we can for the most part treat the interactions within Hartree-Fock
theory and use Slater determinant trial states, unlike the case of the more ‘fractional’
fractions.
It remains to ask what ground state results, once interactions are included. When
the latter are purely repulsive – a reasonable assumption in the systems of interest –
we argue that the system must be ferromagnetically ordered at ν = 1. This is because
z
a spin-polarized state (with Stot
=
N~
)
2
has a wavefunction totally symmetric in its
spin indices; the exclusion principle demands that the spatial part of the wavefunction
38
At higher fillings, the two-dimensional electron gas in free space is an example of Landau level
coincidence, which can also lead to a quantum Hall ferromagnet!
59
is antisymmetric in every pair of electron coordinates, and thus the charge density has
a node when any two electrons approach each other, which minimizes the interaction
energy – essentially the same effect that leads to Hund’s rule in atomic physics. At
ν = 1 the ground state can be written as a simple Slater determinant
39
; in second-
quantized form, we have
|Ψi =
Y
X
c†X,↓ |0i
(1.67)
where c†X,σ is the creation operator for a single-particle state in the lowest Landau level
at guiding center coordinate X and spin σ. Our choice of polarization is appropriate
to the case of GaAs, where the Zeeman term while small, is nevertheless nonzero and
negative, thereby providing a weak symmetry breaking that picks a down-spin ground
state.
The ν = 1 state in GaAs is thus an itinerant quantum ferromagnet. In many ways,
this is the simplest itinerant ferromagnet: the usual competition between the increase
in kinetic energy and the decrease in interaction energy in a polarized state is rendered
moot as the kinetic energy at ν = 1 is quenched by the quantizing magnetic field.
Thus, the exchange gain prevails, leading to a polarized state. We note in passing
that there is an interesting intermediate example, of a three dimensional electron gas
in high fields and small Zeeman coupling where the kinetic energy is only partially
quenched; the question of itinerant magnetism in such a system to our knowledge
remains an open problem.
1.9.1
Spin Waves
In the absence of spin, the only neutral collective modes at ν = 1 correspond to
quasielectron-quasihole pairs, labeled by their momentum q. It is easy to show [76]
39
This is the case even for the case of short-ranged interactions, which is often useful as a first
approximation to the problem. We shall, however, discuss the Coulomb case as it is not too much
more complicated.
60
that these necessarily have a gap set by the cyclotron frequency, ∆(q) → ~ωc as
q → 0. When spin is included, however, there is a new branch of neutral excitations:
spin waves (or magnons); these also involve quasielectron-quasiholes pairs, but now
with opposite spin. Since (as g → 0) a flipped spin is degenerate with the original
one in the noninteracting problem, it is clear that the characteristic energy scale of
the spin wave excitations is set, not by the cyclotron frequency but by the interaction
scale40
e2
.
εℓB
The dispersion of these modes can be shown to be [112, 76]
∆sw (q) = gµB B +
r
2 2 π e2
k ℓB
−k 2 ℓ2B /4
1−e
I0
2 εℓB
4
(1.68)
where I0 is the modified Bessel function. That this dispersion is gapless and quadratic
as q → 0 is unsurprising, since we require such an excitation branch by Goldstone’s
theorem.
1.9.2
Skyrmions
What is the lowest-energy charged excitation in the quantum Hall ferromagnet?
Naively, we would expect that we should simply remove a down spin or add a up
spin, without disturbing the remainder. An estimate for the gap to creating such a
quasiparticle-quasihole pair at infinite separation is to simply take the k → ∞ limit
p 2
of (1.68), giving a gap ∆p-h = gµB B + π2 εℓe B .
We can, however, do better by taking advantage of the ability to produce ‘spin
textures’, topologically nontrivial configurations of the ferromagnetic order, while
varying the charge density. This rests on the observation that the exchange gain for
any given electron is essentially a local contribution from its interaction with those
within a few magnetic lengths of it. Thus, if while adding an up spin (removing
a down spin) electron we simultaneously rotate the other spins around the added
40
This can be most easily understood by noting that a flipping a spin involves a loss of exchange
energy.
61
(removed) electron, we pay a far lower exchange energy. We can then slowly relax the
spin configuration back to the down spin background over several magnetic lengths in
a circularly symmetric fashion. A moment’s thought suffices to realize that since the
spin should interpolate smoothly between ‘up’ near the center and ‘down’ far away,
there is necessarily a rotation in the local spin orientation as we encircle the added
(removed) electron, leading to a topologically stable configuration of the ferromagnetic
order parameter, commonly referred to as a skyrmion (see Fig. 1.7). While a skyrmion
enjoys a significantly lower exchange contribution to the energy, it has an increased
Zeeman cost; the competition between this and the Hartree energy of the nonuniform
charge distribution sets the size and the energy gap of the resulting excitation. We
can estimate the size of a skyrmion (λ) and the cost of a skyrmion-antiskyrmion
pair (∆sk ) using the nonlinear sigma model of the next subsection. To logarithmic
accuracy at small Zeeman coupling we find
3
9π 2 ℓB
(g| log g|)−1
=
28 εa
"
#
r
1/6 1/3
1 π e2
3π 18
εa
∆sk (g) =
1+
(g| log g|)1/3
2 2 εℓB
4
π
ℓB
λ
ℓB
(1.69)
where a = ~2 /m∗ e2 is the Bohr radius [218]. The cost of a skyrmion-anti skyrmion
pairs is thus – in the limit of vanishing Zeeman coupling – one-half the cost of the
simple spin-flip pair; below, we verify that the skyrmions carry an electrical charge.
This continues to hold for small but nonzero vaues of g, so that in this limit the
lowest-energy charged excitations of the SU(2) symmetric quantum Hall ferromagnet are charged skyrmions. The associated spin textures have various observable
consequences for tranport and magnetic resonance experiments.
62
Figure 1.7: Skyrmion spin configuration.
1.9.3
Low-energy Dynamics
An elegant treatment of the dynamics of the quantum Hall ferromagnet may be
derived [218] within the Chern-Simons Landau-Ginzburg approach with appropriate
modifications to include the spin index [137]. Without elaboration, we simply present
the low-energy effective Lagrangian that results from this analysis41 :
ρs
Leff = αA[n(r)] · ∂t n(r) + (∇n(r))2 + g ρ̄µB n(r) · B
2
Z
1
−
d2 r ′ V (r − r′ )q(r)q(r′)
2
(1.70)
with the constraint [n(r)]2 = 1. Here, α and ρs are couplings that depend on the
interaction scale
e2
εℓB
and ρ̄ is the average density. A is the vector potential of a
unit monopole at the origin of the spin-space Bloch sphere, i.e. ∇ × A = n, and is
chosen to give the precessional dynamics to n required for the quantum-mechanical
equations of motion for a spin. The first three terms of Leff are the standard terms in
41
In writing this Lagrangian we have followed the authors of [218] in neglecting a Hopf term,
which is the transcription of the Chern-Simons term required in all long-wavelength theories of the
quantum Hall effect to the sigma-model description; while important to obtain fermionic statistics
for the skyrmionic quasiparticles, for our purposes this is not essential.
63
the nonlinear sigma model treatment of a ferromagnet [71]; the new ingredient in the
quantum Hall ferromagnet is the final term, which represents an interaction between
topological or Pontryagin densities
q(r) =
1 ij abc a b
ε ε n ∂i n ∂j nc .
8π
(1.71)
The spatial integral of q gives the Pontryagin index (Chern number) of n(r) thought
of as a map from the plane (suitably compactified by including the point at infinity)
to the spin sphere. This is the topological invariant that underlies the stability of
a skyrmionic configuration to smooth deformations of the order paramter. To see
that q is also the density of electric charge of a skyrmion configuration – and thus
explain the inclusion of the Coulomb interaction term in (1.70) – consider the following
argument, from [76]. An electron described by position coordinate xµ , moving in a
static background spin configuration nν can be described by the Lagrangian
e
~
e
L0 = − ẋµ Aµ + ṅµ Aµ [n] = − ẋµ (Aµ + aµB )
c
2
c
(1.72)
where the first term is the usual coupling to the electromagnetic gauge field, and
the second is a contribution coming from the Berry phase from the changing local
field ṅµ = ẋν ∂ν nµ . This defines a Berry vector potential for transport in the spin
background, aµS ≡ − Φ20 ∂µ nν Aν [n]. In writing L0 in this form, we have assumed that
the exchange coupling is strong enough to force the electron spin to follow the local
orientation, and the
~
2
factor is appropriate to a spin- 12 particle. The additional Berry
potential produces a pseudo-magnetic field bS , which is easily verified to be simply
bS (r) = −Φ0 q(r) where q is the Pontryagin density defined previously.
If we adiabatically deform the spin configuration n, the electronic degrees of freedom see this as an added Berry flux. Since the Berry potential couples to the electrons
in identical fashion as the physical electric field, it follows by the same argument as
64
for the charge of a Laughlin quasiparticle that the adiabatic deformation produces a
change in the charge density
δρ(r) =
σxy
bS (r) = −νeq(r)
c
(1.73)
The integral of the Pontryagin index over all space vanishes, unless the spin configuration n has nonzero topological index (skyrmion number). Thus, skyrmions in a
quantum hall ferromagnet at filling factor ν carry νe units of electric charge, which
is identical to the charge of the Laughlin quasiparticle. The spin of the skyrmion is
a somewhat more delicate issue; for a discussion, see [172].
An alternative approach which allows a direct evaluation of terms in the ferromagnetic energy functional was developed by [163]. In its essence, the method involves
explicitly computing the energy for long-wavelength fluctuations about the ferromagnetic ground state using the algebra [79] of operators projected into the lowest Landau
level. The relation between the topological and electrical charges can also be derived
microscopically in this fashion. The results are consistent with [218] and are readily
extended to cases where the symmetry of the low-energy theory is not immediately
obvious, such as those in Part III of this thesis.
1.9.4
Other Examples
We focused above on the electron spin, as it is an illustrative example and the best
studied to date. However, there are various other internal degrees of freedom –
such as the semiconductor valley pseudospin, layer index in double quantum wells,
and Landau level index when different Landau levels are brought into coincidence
in tilted fields, to name a few. The symmetry of the resulting ferromagnet depends
on details of the interaction, which can introduce various anisotropies; for instance,
in Chapter 4 we study an example where owing to an anisotropic effective mass
65
tensor the ferromagnet has a strong easy-axis (Ising) anisotropy, while in bilayer
systems the tendency to favor equal fillings in both layers leads to an easy-plane
(XY) system [163]. Each symmetry class has distinctive features; we defer discussion
of the easy-axis and easy-plane cases to later in this thesis and to the cited reference,
respectively. A general classification of quantum Hall ferromagnets into different
pseudospin anisotropy categories based on the symmetries of their interactions may
be found in [110].
Quantum Hall ferromagnetism is not restricted to integer Landau levels with interactions, but can be generalized to other fillings, for instance ν =
the question of whether such ferromagnetic behavior occurs at ν =
5
2
1
3
[218]. Indeed,
is a central issue
in determining whether it is in fact a non-Abelian quantum Hall state [53].
1.10
Antiferromagnetic Analogs and AKLT States
It is clear that the fractional quantum Hall states are extraordinary from the conventional Landau-Ginzburg-Wilson perspective of broken-symmetry phases of matter:
they break no symmetries; they exhibit fractionalization of quantum numbers; they
have a nontrivial ground state degeneracy on a torus; and so on. This complex of
phenomena was soon recognized as characteristic of a new kind of order emergent in
the low-energy description of strongly correlated quantum matter, commonly termed
topological order. Unlike traditional broken-symmetry phases where the natural lowenergy description is a sigma model in terms of a local parameter, topological phases
are described by an emergent gauge symmetry; while on occasion a particular gauge
may be found in which a description in terms of a local order parameter obtains, such
examples are fortuitous exceptions to the general rule that no such description is
possible. There are by now many different theoretical examples of topological phases,
66
although the quantum Hall states remain the only ones on firm experimental footing;
this is because their topological order manifests itself in transport and is thus readily
measurable.
Quantum antiferromagnets in low dimensions have proven to be an abundant
source of strongly-correlated phases, since they naturally have strong fluctuations
and, in addition, can be readily frustrated by competing interactions or geometry.
Topological phases are no exception, as evidenced by the multitude of topologically
ordered ‘spin liquid’ states proposed on various lattices in d = 2 and 3. Many, if
not all, of these phases can be captured within mean-field constructions where the
spins are decomposed into fermionic spinons or Schwinger bosons42 , and standard
techniques from the study of Fermi or Bose gases can then be applied to the new
variables; since the fractionalization into the emergent degrees of freedom artificially
enlarges the Hilbert space, an emergent gauge field is introduced to constrain calculations to the physical subspace [245]. For a review of experimental and theoretical
developments in the study of spin liquids, we direct the reader to [139]; there are also
recent numerical results [152, 253] that have received much attention. We shall not,
however, discuss these further in this thesis.
Instead, we focus on a somewhat simpler set of antiferromagnetic phases, that are
not topologically ordered – they have no nontrivial groundstate degeneracies and host
no fractionalized excitations. However, they do not break any lattice or rotational
symmetries, their ground states lack order owing to quantum fluctuations and (in
some cases) geometric frustration, and they are the exact ground states of local – indeed, nearest-neighbor – Hamiltonians. These are the quantum paramagnetic valence
bond solid states originally proposed by Affleck, Kennedy, Lieb and Tasaki (AKLT)
as ground states for one-dimensional spin-1 chains [8, 9]. The motivation of those authors was to construct a rigorous example of an integer-spin system that was in accord
42
In this sense, the fractionalization is manifest at the outset. This is closely related to the parton
construction of quantum Hall phases [104, 242, 244].
67
with Haldane’s conjecture [87] that half-integer spin chains support gapless spinon
excitations and power law-correlated ground states, while integer chains are gapped
and exhibit exponential correlations. The essential insight of the AKLT approach is
to build a wavefunction that incorporates quantum fluctuations from the ground up
through the following steps: (a) expand the Hilbert space by decomposing every spin
into spin- 21 constituents (b) build a quantum-disordered state by placing these into
pairwise singlets along bonds; and (c) impose a constraint, of symmetrization on each
site, to project back to the physical degrees of freedom.
While they are not topologically ordered, the fact that AKLT states build in ‘good’
correlations at the outset and have model Hamiltonians places them on conceptually
similar footing with Laughlin’s trial wavefunctions for the fractional quantum Hall
effect. These similarities were crystallized in work by Arovas, Auerbach, and Haldane,
who showed how to construct families of AKLT states on arbitrary lattices through
the Schwinger boson approach43 [17]; the construction constrains the spin on a site
to be a half-integer multiple of the coordination number. These states were shown
to be exact ground states of Hamiltonians that, in the original variables, could be
written in terms of projectors onto states of definite total angular momentum of pairs
of spins. Finally, it was noted that the ground state correlations could be calculated
in terms of a finite-temperature classical model on the same lattice, by working in
the basis of coherent states of spin in which the wavefunction has a Jastrow form –
closely paralleling Laughlin’s plasma analogy.
Are the AKLT states really quantum disordered? This is a less trivial question
than it might seem at first glance – recall that the Laughlin wavefunctions eventually
evolve crystal-like correlations for large enough m; a similar complication could in
principle occur in the AKLT approach. Here, the fact that the classical model is at
finite temperature comes to our rescue, since it would necessarily have to break a
43
While a two-dimensional example on the honeycomb lattice was already known to AKLT, Arovas
et. al. were the first to systematically give a prescription for all lattices.
68
continuous symmetry to order, which is precluded in d = 1 and d = 2 by the MerminWagner theorem
44
[153]. The three-dimensional case is not so straightforward, and
here the question must be settled by explicit study of the ground state correlations,
which is the subject of Part IV of this thesis. As we shall have much more to say
about AKLT states in Part IV, we eschew a more technical discussion here.
We close by noting that in d = 1 the spin-1 AKLT state is the exact ground state
for the Heisenberg model with an additional, nearest-neighbor biquadratic term. By
varying the strength of the biquadratic term, the ground state can be studied numerically; these show that the AKLT state is adiabatically connected to the pure
Heisenberg model, and therefore captures universal properties of the phase. Whether
this remains the case in higher dimensions remains, to our knowledge, an open question that warrants further study.
1.11
This Thesis
We are now ready to give a summary of the results in this thesis, some of which we
have alluded to in the preceding pages. The remainder consists of eight chapters,
broadly organized into four parts. The overarching theme, such as there is one, is
the search for useful characterizations of order in quantum Hall systems. The hope is
that either by identifying order parameter-like variables in which the quantum Hall
effect can be understood in more conventional terms – as exemplified, for instance,
by the Chern-Simons Landau-Ginzburg approach – or by studying states, like quantum Hall ferromagnets, that break internal symmetries we can gain insight into the
quantum Hall regime by drawing on our understanding of superconductivity or ferromagnetism. We also attempt to do the reverse, and apply the model Hamiltonian-trial
wavefunction-classical analogy approach so successful in the study of the fractional
44
Note that this does not rule out the Wigner crystal transition in the classical plasma corresponding to the Laughlin state, since the presence of long-range interactions invalidates the assumptions
of [153].
69
quantum Hall states to the study of lattice spin models, in the hope of discovering
new correlated states of quantum matter. In the following, we present a few attempts
in both these directions.
By drawing on the analogy, in Part II, between the quantum Hall effect and superconductivity, we propose that the dichotomy between Type I and Type II superconductors has a close parallel in the classification of fractional quantum Hall phases.
We discuss an initial example of paired Hall states at weak coupling (Chapter 2),
before generalizing the Type I construction to several other fractions (Chapter 3).
In Part III, we move to a discussion of valley-ordered quantum Hall states. Here,
we study the interplay between the broken-symmetry phenomena associated with
valley ordering and the topological order inherent in quantum Hall systems, in the
hope that they have measurable consequences in experiments. We study two examples
with different symmetries: Ising-nematic order in AlAs (Chapter 4) and the breaking
of an SU(4) symmetry in bilayer graphene (Chapter 5). In the former example,
we identify two new phases, the quantum Hall Ising nematic and the quantum Hall
random-field paramagnet, and discuss various experimental signatures of both; this
work is directly relevant to recent transport measurements in alumnium arsenide
(AlAs) heterostructures. In the bilayer graphene example, we explain the sequence
of broken-symmetry Hall plateaus, identify a new kind of excitation – a charge-2e
skyrmion – and discuss experimental scenarios in which these might be observed.
Part IV ranges further afield, and discusses antiferromagnetic analogs of the fractional quantum Hall effect. Our motivation is to add to the list of provably quantumdisordered ground states of spin systems in dimensions higher than two. Using the
AKLT approach, we construct three-dimensional examples on the diamond and pyrochlore lattices (Chapter 6), before generalizing the construction to Cayley trees
and regular random graphs, which are formally infinite-dimensional. We show that
the latter support a new class of AKLT states with spin glass correlations (Chapter
70
7), that are conceptually intermediate between the quantum paramagnetic and Néel
states on Euclidean lattices.
The final part of the thesis (Part V) contains a pair of miscellaneous results.
The first of these argues that breaking continuous symmetries on random graphs
does not give rise to Goldstone modes (Chapter 8). The second is devoted to a
renormalization-group treatment of superconducting correlations in an interacting
Fermi liquid (Chapter 9). While on the face of it these diverge from the general
theme of this thesis, the former result is central to our study of AKLT spin glasses in
Chapter 7, while the latter work informed much of our of investigation, currently in
progress, of proximity effects between the composite Fermi liquid and its descendant
paired quantum Hall states [184].
71
Part II
Type I Quantum Hall Liquids
72
Chapter 2
The Weakly Coupled Pfaffian as a
Type I Quantum Hall Liquid
2.1
Introduction
As we have seen, there is a deep and precise relation between superconductivity and
the quantum Hall effect, which can be formally implemented by replacing the physical
Maxwell gauge field by the statistical Chern-Simons gauge field [78, 258, 198]. The
perfect conductivity of the superconductor then corresponds to the quantization of
the Hall conductance, the Meissner effect to incompressibility, and quantized vortices
to fractionally charged quasiparticles. In some cases, this relation goes further, in
that the Hall fluid is a condensate of fermion pairs. Specifically, the quantized Hall
phase seen in the proximity of the half-filled second Landau level, ν = 5/2, is likely
associated with the Pfaffian or Moore-Read state [164]. The ideal Pfaffian state has
a natural interpretation as a weakly-paired state of composite fermions [200, 82] with
p + ip symmetry, i.e. it bears an analogous relationship to the metallic “composite
Fermi liquid” state of the half-filled Landau level [92] as the corresponding BCS state
does to an ordinary metal.
73
In this chapter, we explore an important and hitherto little-studied aspect of
the quantum-Hall/superconductor relationship, particularly relevant to the Pfaffian
phase, namely, that quantum Hall states, like superconductors, generically exhibit two
length scales: a screening length λ that characterizes the decay of density deviations
and a coherence length ξ [147] that characterizes variations of the superfluid order.
Further, the ratio of these scales crucially influences the structure of vortices and and
thence the response of the ideal quantum Hall states to global density changes much
as they influence the response of superconductors to applied magnetic fields.
Accordingly we propose that quantum Hall fluids should come in two classes:
a) Type II quantum Hall fluids where (roughly) λ ≥ ξ and density deviations are
accommodated by the introduction of single quasiparticles/vortices and b) Type I
quantum Hall fluids where λ < ξ and quasiparticles are unstable to agglomeration and
form multi-particle bound states or if the interactions are sufficiently short-ranged,
phase separate entirely. Intuitively, Type I behavior arises at ξ ≫ λ as two vortices of
size ξ are able to reduce their joint energy by an amount of order of their individual
creation energies Ev ∼ (ξ)0 by merging and thus reducing the region over which the
order parameter is suppressed while only paying an interaction which is parametrically
smaller, e.g. a Coulombic cost of order e2 /ξ.
Examples of Type II quantum Hall liquids abound—indeed, it has been implicitly
assumed that all quantum Hall fluids are of that kind and that they exhibit a single
p
length scale, which ultimately is the magnetic length (ℓB = ~c/eB). We will present
and analyze a first example of a Type I quantum Hall liquid.
This liquid is the weak coupling limit of the Pfaffian state. On general grounds, the
identification of the Pfaffian phase as a paired state of composite fermions carries with
it a natural presumption that the pairing strength is tunable and that, in particular,
it is possible to reach a weak coupling regime where, as in BCS superconductors, ξ
is parametrically larger than the zero-temperature charge screening length, λ(T =
74
ρ − ρPf
CF Metal
Crossover region
Type II
Type I
Quasiparticle
crystal
s
plet
Dro
s
Stripe
Bubbles
0
տ
Ideal Pfaffian
ր
λ/ξ →
Figure 2.1: Schematic phase diagram of the Pfaffian
We show the Pfaffian phases with fixed Coulomb interactions as a function of density
deviation from ν = 5/2 (pseudomagnetic field) and the Landau-Ginzburg parameter,
λ/ξ. Typical configurations of the different inhomogeneous phases are shown, with red
representing the Pfaffian and white the metallic phase. For short-ranged interactions,
the microemulsion phases in the Type I region are replaced by a phase-separated
intermediate state, with the Pfaffian fraction decreasing continuously to zero as the
boundary to the composite fermion metal is approached.
0) ∼ ℓB 1 . Indeed, numerical studies suggest that such a weak coupling regime does
exist for suitable microscopic interactions [203, 160]. In the balance of this chapter
we analyze the properties of the weakly coupled Pfaffian and show that instead of the
various vortex/quasiparticle glass or crystal phases typically invoked for the quantum
Hall plateaux, it exhibits patterns of charge organization associated with frustrated
phase separation.
2
ne
This result follows from the London equation ej = − mc
2 (a + A) and the compressibility of the
m
composite Fermi liquid δρ = 2π~2 ea0 in conjunction with the Chern-Simons equations b + B = κδρ
and j/c = −ẑ × (∇a0 ), where n is the density, a0 the scalar potential and the other quantities are
defined in the text.
1
75
2.2
Landau-Ginzburg theory
For concreteness, we begin with a half-filled Landau level of spinless electrons. As
is usual in the fermionic Chern-Simons approach, two quanta of statistical flux are
bound to each electron so that if we treat the Chern-Simons gauge field at mean-field
level, exactly at ν = 5/2 the electrons form a Fermi sea of composite fermions in zero
net field [92].
As outlined in Chapter 1, we follow the phenomenological approach of previous
studies [69], and add an explicit p-wave attraction between the composite fermions.
At T = 0, a further BCS mean-field decoupling of this attraction produces a p + ip
superconducting state with coherence length ξ and a gap to creating Bogoliubov
quasiparticles (‘Bogolons’) ∆ = ~vF /ξ. Due to the Chern-Simons electrodynamics of
the statistical gauge field, vortices carry charge e/4 which , as we will see shortly, is
localized on a length scale λ, which plays the role of the London penetration depth
in superconductors. The Bogolons are neutral fermionic excitations of the quantum
Hall fluid [164, 22, 34, 162].
For weak attraction, ∆ ≪ εF or equivalently ξ ≫ λ 2 . Moreover, in this limit, the
vortex creation energy, Ev ≫ ∆. (A standard estimate – which we will improve below
– yields Ev ∼ εF .) Thus, for weak coupling there exists a finite temperature regime
where vortices are sparse, nv ∼ e−Ev /T and can be neglected on length scales ≪
ℓB e+Ev /2T . There is a well-defined crossover temperature TcM F ∼ ∆/2 corresponding
to the mean-field transition, at which significant pairing onsets.
Near TcM F , we can follow the procedure of Section 1.7.2 and derive a LandauGinzburg description from the fermionic Chern-Simons path integral by integrating
out the fermions in favor of a complex order parameter Ψ. In units in which |Ψ| = 1
in the uniform Pfaffian phase, and the condensation energy density is ε0 /ξ 2 (where
2
In a given Landau level both ∆ and εF will depend on the Coulomb energy, e2 /ℓB . Their ratio
can be tuned by adjusting pseudopotentials [203, 160].
76
ε0 ∼ εF ), this takes the form
FLG
=
ε0
"
#
2
Z
∇
2
1
2
2
dr dr [∇ × a]2
− a Ψ + 2 1 − |Ψ|
+λ
i
2ξ
x
Z
1
λC
′
+
drdr ∇ × a − 8πδρ r
∇ × a − 8πδρ r′
′
2
|r − r |
Z
1
e
+
dr A0,ext [∇ × a]
ε0
8π
Z
(2.1)
Here, the first term is the usual expression for the energy of the condensed phase in
powers of Ψ. In the second term, which is formally equivalent to a Maxwell term,
λ2 is largely proportional to the inverse compressibility of the the metallic composite
Fermi lqiuid. The gauge field a(r) ≡
8π
(a(r) +
κ
Aext ) is the sum of the Chern-Simons
and external gauge fields, which satisfy ∇ × a = κρ(r), ∇ × Aext = −κρPf = B
respectively, where κ−1 = e/2hc is the Chern-Simons coupling, and ρPf is the density
of the half-filled Landau level of the ideal Pfaffian state. The third term, which has
no direct analogue in the theory of superconductivity, is the Coulomb interaction,
with δρ = ρ̄ − ρPf the difference between the density set by the positive background
and the commensurate density. We have introduced an exponent x which takes value
x = 1 when the Coulomb interaction is unscreened, but can take the value x = 3
(corresponding to dipolar interactions) when there is a metallic gate present. (For
x = 1, λC ∼ e2 /εF , while for the dipolar case, λ3C ∼ e2 d2 /εF where d is the distance to
the metallic gate.) The final term ensures that the uniform superconducting (Pfaffian)
state has the correct Hall response. ξ and λ represent the Landau-Ginzburg coherence
length and penetration depth of the Chern-Simons superconductor.
Two comments are in order. First, the formal derivation of the Landau-Ginzburg
theory is valid only near TcM F and at lower temperatures non-local “Pippard” effects will need to be included as is the case for Type I superconductors 3 . However,
3
At T = 0 a self-consistent treatment of nonlocal effects yields an actual penetration depth
∼ (λ2 ξ)1/3 [230], ensuring a sensible metallic (ξ → ∞) limit of the Pfaffian.
77
for a first pass at the problem, we will ignore the subtleties involved and use the
Landau-Ginzburg theory down to T = 0 with its parameters considered to be phenomenological constants. Second, as FLG is formally obtained by integrating out the
composite fermions, the full partition function would be obtained by performing a
further path-integral over the Hubbard Stratonovich field, Ψ, and the Chern-Simons
gauge field, a. For present purposes, we will treat this problem in saddle-point approximation, where we focus on the configuration of Ψ and a which minimizes FLG . This
will be adequate for the purposes of understanding vortex structure and interaction
which is our focus in the remaining.
2.3
Vortex structure
We turn now to the vortices. First, let us consider the structure of the vortices in
the absence of a long ranged interaction, (λC = 0). Now FLG has exactly the form
of the conventional free energy of a superconductor, with screening length λ. In the
extreme Type I limit, λ ≪ ξ, the vortex has a core with magnetic flux spread roughly
over a region of size λ while the order parameter is suppressed over a much larger
region of size ξ. Translated into quantum Hall language, the charge density of the
quasiparticle is confined primarily to a region of size λ which is much smaller than
the region over which the Pfaffian order is disrupted.
This structure can be captured by a simple variational ansatz (for a vortex of
vorticity N) with a single parameter, L, that has been shown [255] to be accurate in
78
the limit ξ/λ → ∞:



γ(r/L)N
for r < L
Ψ(r) = e
√

 1 − γ ′ K0 ( 2r/ξ) for r > L


 (r/L2 ) for r < L
a(r) = N θ̂

 (1/r) for r > L
iN θ
(2.2)
where, for continuity, γ ′ = (1 − γ)/K0 (x), and for continuity of the derivative, γ =
√
xK1 (x)/[NK0 (x) + xK1 (x)], with x = ( 2L/ξ). Here the flux (charge) is uniformly
distributed inside a disk of radius L, and is zero outside. The K0 Bessel function is the
solution of the linearized Landau-Ginzburg equations, so the ansatz has the correct
asymptotic form at long distances from the vortex core but also (less obviously)
immediately outside it. For a given value of λ/ξ, we determine L by numerically
optimizing the free energy of this profile. In the extreme Type I limit for a single
vortex (N = 1), ξ/λ ≫ 1, L ∼ λ log(ξ/λ) and Ev ∼ εF / log(ξ/λ).
Turning now to vortices in the Coulomb problem, the general features are similar
and hence we can gain quantitative guidance by adopting the same variational ansatz
as in Eq. 2.2. However, in this case the ansatz does not have the proper form at long
distances; from general considerations, as first discussed in the context of abelian
quantum Hall states [219], it follows that in the presence of Coulomb interactions the
vortex has power-law tails in the charge and current density, which decay like r −3 and
r −2 respectively. These do not, however, affect the basic length scales in the problem
significantly.
There are, of course, features of the vortex structure that cannot be captured
by our Landau-Ginzburg theory, most notably the existence of a bound Majorana
zero mode and thus the non-abelian statistics [200, 101]. While of great fundamental
interest [170], these subtleties are irrelevant for present purposes.
79
2.4
Vortex Binding and Optimal Droplets
The structure of the vortex in a Type I quantum Hall fluid, with the charge density
confined on a distance much shorter than that over which the order parameter is
suppressed, implies that individual vortices are unstable to aggregation. To see this,
consider two vortices separated by a distance ξ. As they share a common region of
suppressed pairing (|Ψ| < 1) they gain an energy of order ε0 while paying a Coulomb
energy of order ε0 (λC /ξ) which is parametrically smaller.
In the short-range case (λC = 0) the Landau-Ginzburg theory is formally identical
to that of a superconductor, where vortex aggregation continues indefinitely when
√
λ/ξ < 1/ 2 [230] and thus any number of vortices phase separate macroscopically.
The situation is different with Coulomb interactions where the cost of macroscopic
phase separation is superextensive. Instead, when the concentration of vortices is
small, they aggregate into droplets of vorticity Nc and size ξ. We can obtain an
estimate of Nc by minimizing an expression for the energy density of vortices (ε0 +
ε0 N 2 λC /ξ)δρ/N, where the expression in the bracket is an energy of one droplet with
q
vorticity N. In the extreme Type I limit, this leads to an estimate Nc ∼ λξC . This
agrees with a variational calculation of the optimal droplet using (2.2). Note that
a bubble with vorticity Nc and charge eNc /4 is, itself, a new sort of quasiparticle.
If Nc is odd/even, this quasiparticle will have/not have a Majorana zero mode and
corresponding non-Abelian/Abelian braiding statistics.
2.5
Phase diagram near ν = 5/2
As we move away from the ideal Pfaffian filling factor, ν = 5/2, we introduce a pseudomagnetic field corresponding to the difference between the average density ρ̄ and
the commensurate density ρPf . For short ranged interactions (λC = 0) the quantum
Hall system behaves exactly like the superconductor and we get phase separation
80
between Pfaffian and metallic regions. Specifically, the additional charge coagulates
in a “normal” region with density δρc =
1
ξλ
which is the transcription of the thermo-
dynamic critical field Hc (more precisely Bc ) to this setting. For ρ̄ − ρPf > δρc , the
entire system is metallic.
However, in the presence of Coulomb interactions, the phase diagram of the Type
I Pfaffian is richer than that of its superconducting cousin (see Fig. 2.1.) Now the
tendency towards phase separation is frustrated—which is a problem much studied
in recent years in several contexts [105, 221, 179]. At small deviations from the magic
density, optimal bubbles (discussed above) form a triangular lattice bubble crystal.
For larger values of |ρ̄ − ρPf |, the bubble crystal typically gives way to a stripe phase.
At still larger values, one typically finds an anti-bubble or droplet phase, which in this
case consists of puddles of Pfaffian phase embedded in the majority metallic phase.
The character of these various “microemulsion phases”, and the structure of phase
transitions between them will be explored in future publications. Note that phases
where the Pfaffian percolates will manifestly exhibit the quantum Hall effect. In the
remaining phases the issue turns on the balance between the Pfaffian version of the
proximity effect [184] and the effects of quantum and gauge fluctuations. The reader
should contrast this complexity with the case of the Type II Pfaffian where the density
deviation is always accomodated via a triangular lattice quasi-particle Wigner crystal
formed on the background of the uniform Pfaffian state—analogous to the Abrikosov
lattice in a Type II superconductor.
2.6
Experiments
Recent experiments on the 5/2 state are consistent with Pfaffian order. We believe
that these are further consistent with either Type I or Type II behavior. Importantly,
the fundamental unit of charge is e/4 even in the Type I regime, despite the binding
81
of multiple vortices/quasiparticles in the bulk, and the experiments to date have
probed exactly this quantization. These include [54, 193] which measure the charge
tunneled across point contacts, [250] which measure the charge contained in a region
via edge-state interferometry, and scanning probe measurements [236] which are based
on the existence of disorder-induced compressible puddles. A detailed analysis of the
experiments is outside the scope of the present article; we defer this to future work.
2.7
Concluding Remarks
Our identification of the weakly coupled Pfaffian as a Type I quantum Hall liquid is
not particularly sensitive to various approximations, such as the neglect of fluctuations
about various saddle points, that we have employed in our treatment—essentially it
depends on the existence of a regime where ξ is asymptotically larger than λ ∼ ℓB .
The features we obtain in that limit are macroscopic or semi-macroscopic and thus
should be robust as well. That said, it is worth drawing attention to a subtlety that
we ignored in the main text. The metallic phase is not really immune to the pseudomagnetic field—it has “quantum oscillations” as a function of density (flux) reflecting
the existence of various integer (and even fractional) quantum Hall states at special
densities. Sufficiently close to ν = 5/2 and in the strongly Type I limit, the relevant
states will be fairly weak and likely to modify the properties of the bubbles only at
extremely low temperatures. Of course, our considerations are sensitive to the inclusion of disorder which will destroy true long range order in the various microemulsion
phases and at sufficient strength, the underlying Pfaffian state as well.
In closing, we note that the nature of the phases produced by frustrated phase
separation does depend critically on the range of the frustrating interactions [221], i.e.
x in Eq. 2.1. For x > 3, the interactions are short-ranged, and can be lumped with the
compressibility term, yielding a renormalized value of λ. However, in the interesting
82
dipolar case, x = 3, macroscopic phase separation is only marginally forbidden. Here,
the typical density in a puddle is δρ ≈ δρc , and the size of the bubbles defines an
emergent length scale that grows exponentially as the size of the dipole (distance to
the gate d) decreases. This suggests that the search for Type I quantum Hall liquids
would be greatly advanced by investigating gated 2DEGs.
83
Chapter 3
More Type I Quantum Hall
Liquids
In this chapter, we construct bosonic Chern-Simons Landau-Ginzburg theories with
Type I behavior, and use these to examine the properties of isolated quasiparticles
and their interactions. We then provide a prescription for constructing microscopic
Hamiltonians for Type I quantum Hall liquids in the lowest Landau level, before
proceeding to a discussion of experimental realizations of the same. We close with
an account of signatures of Type I behavior, focusing in particular on how it differs
from the Type II case.
3.1
Chern-Simons Landau-Ginzburg Theory
We begin by recalling how Type I and Type II behavior arise in the familiar context
of superconductivity. A superconductor is described by a Landau-Ginzburg theory
where the order parameter couples to the electromagnetic field, and has two characteristic length scales: a coherence length characterizing the healing of superfluid
order, and a screening length that measures the penetration of magnetic fields into
the superconductor. Their ratio is controlled by a single parameter, the dimensionless
84
√
quartic coupling λ of the Landau free energy. At a characteristic value of λ (1/ 2 in
the usual convention), these two scales become equal. At this ‘self-dual’ point, something remarkable happens: the equations of motion governing the minimum energy
solutions of arbitrary vorticity reduce to a first order system [33]. This reduction has
the consequence that vortices/antivortices do not interact at this special point. It
follows straightforwardly that decreasing (increasing) λ from its self-dual value gives
rise to an effective attraction (repulsion) between them, leading to Type I (Type II)
behavior.
A very similar development can be given for quantum Hall fluids in the framework
of the Chern-Simons Landau-Ginzburg theory of the simplest quantum Hall fluids
[258], which we now sketch. As discussed in Chapter 1, near filling factor ν = 1/k
the corresponding action is formally derived by transforming the original microscopic
electronic degrees of freedom to a bosonic field ϕ by binding an odd number (k) of
magnetic flux quanta to each electron, and at the cost of introducing an additional
fictitious Chern-Simons gauge field a. The resulting theory has a Lagrangian density
L = ϕ∗ (iD0 ) ϕ −
2
εµνρ
|Dϕ|2
2
aµ ∂ν aρ
−
λ
|ϕ(r)|
−
ρ
+
2m∗
2kΦ0
(3.1)
where Φ0 = hc/e is the flux quantum, Dµ = ∂µ − i(a + A)µ is the covariant derivative,
Aµ are the components of the [fixed] external electromagnetic field, ρ the commensurate density, and λ is a phenomenological short-range repulsion between the bosons.
The final term is the Chern-Simons term for a, and is to be contrasted with the usual
B 2 term in the Landau-Ginzburg functional of a superconductor. A and ρ are related
by ∇ × A ≡ B = kΦ0 ρ.
The Chern-Simons Landau-Ginzburg equations of motion are easily derived from
L. Variations in ϕ are described by a gauged nonlinear Schrödinger equation, while
the equations for a are pure constraint: they bind (a) magnetic flux to charges and
85
(b) electric fields to currents [258]. As a result, b ≡ ∇ × a = −kΦ0 |ϕ|2 so that the
composite bosons see an effective pseudomagnetic field Beff = b + B = kΦ0 (ρ − |ϕ|2 ).
The ground state solution exactly at ν = 1/k—corresponding to the commensu√
rate Laughlin state—is a uniform configuration with ϕ = ρeiθ and a = −A and thus
Beff = 0. For repulsive interactions (λ > 0) the linear modes/quadratic fluctuations
about this solution are gapped. The Chern-Simons Landau-Ginzburg equations also
admit nonlinear vortex and antivortex solutions carrying quantized Chern-Simons
flux. As a consequence of the charge-flux relationship, they carry fractional electric
charge ±e/k, corresponding to Laughlin quasiholes and quasielectrons, respectively.
For nonlinear stability and incompressibility it is necessary that the energy for creating a widely separated vortex anti-vortex pair remain positive—which holds for all
λ ≥ 0 [226].
Away from the commensurate filling ν = 1/k, Beff is necessarily nonzero and the
ground state must involve a nonzero density of vortices/antivortices. The spatial
organization of these objects again depends crucially on the sign of their interaction.
Generically, working this out requires the simultaneous solution of all three ChernSimons Landau-Ginzburg equations for multi-vortex configurations but at λ = λsd ≡
kΦ0 /2m∗ —the self-dual point—the problem simplifies greatly to solving a pair of
first-order equations,
D̄ϕ ≡ (D1 + iD2 ) ϕ = 0
b = −kΦ0 |ϕ|2
(3.2)
for positive vorticity solutions or quasiholes. The general solution for ϕ is a configuration of ni -fold vortices at coordinates Ri , whose core scale is set by the magnetic
length ℓB = (~c/eB)1/2 . The non-trivial piece of analysis now requires a solution of
86
the equation for the amplitude:
X
|ϕ(r)|2
ℓ2B 2
∇ log |ϕ(r)|2 =
− 1 + 2π
ni δ(r − Ri )
2
ρ
i
(3.3)
after which the phase of the configuration can be reconstructed [102]. While (3.3) still
requires a numerical solution, we only need a very general property of its solutions,
P
that the energy of the solutions depends only upon the net vorticity Nv = i ni ,
E=
BΦ0
Nv ,
2m∗
(3.4)
and is independent of how it is distributed spatially. Thus, exactly at the self-dual
point quasiholes do not interact.
For negative vorticity solutions – multi-quasielectron configurations, no comparable simplification occurs, in contrast to the case of a superconductor. Quasielectrons
of (3.1) thus always repel, however this interaction is short ranged on the scale of
ℓB . Finally, at the self-dual point, numerical analysis yields a gap [226] to making
quasielectron-quasihole pairs. Putting all of this together we find the self-dual ground
state is gapped and stable, while its quasiholes do not interact and the quasielectrons
have a short ranged repulsion.
It follows that if we vary λ from its self-dual value, the quasiholes attract for
λ < λsd and repel λ > λsd , while quasielectrons repel in both cases. Thus we obtain
Type I behavior for one sign of doping in the simplest Chern-Simons Landau-Ginzburg
theory. But we can do better. While the restriction to a local scalar self-interaction
is natural in a superconductor, the more general non-local density-density interaction
1
δL =
2
Z
dr′ (|ϕ(r)|2 − ρ)v(r − r′ )(|ϕ(r′ )|2 − ρ)
87
(3.5)
is just as natural in the context of the Hall effect. Let us perturb the self-dual model
with a term of this form where we pick v(r) to be a) attractive, b) weak enough to
not close the gap to making a quasiparticle-quasihole pair and c) with a range ≫ ℓB .
Now we can get the quasihole and quasielectrons to attract at long distances which
is sufficient for both to phase separate at finite densities and thus to Type I behavior
for both signs of doping.
3.2
Paired quantum Hall states
A different Landau-Ginzburg derivation of Type I quantum Hall behavior can be given
for paired quantum Hall states as sketched in the particular context of the Pfaffian
state in Chapters 1 and 2, although the basic ideas generalize trivially to other paired
quantum Hall states. Here we begin with the fermionic Chern-Simons rewriting of
electron dynamics and look to solve the coupled Chern-Simons Bogoliubov-de Gennes
equations that describe the pairing of composite fermions. At weak coupling we
can integrate out the composite fermions to derive a Landau-Ginzburg free energy
with an effective Maxwell term for the Chern-Simons gauge field arising from the
compressibility of the composite fermion liquid. This closely resembles the LandauGinzburg theory for superconductors in that now there are explicitly two length
scales—the pairing length and the screening length. Type I behavior then follows at
weak coupling (the BCS limit) as the coherence length greatly exceeds the penetration
depth.
3.3
Microscopic Models
We now turn to a large class of microscopic models for fractional quantum Hall
effect states which realize the key properties of the self-dual point of the ChernSimons Landau-Ginzburg theory and thus can be perturbed to yield Type I quantum
88
Hall fluids in exactly the same fashion. These are not new models—they have been
constructed historically to render various desirable wavefunctions exact ground states
starting with the work of [233, 86].
An illustrative example of how this works is the ν = 1/3 state, in many ways the
prototypical fractional quantum Hall state. At this filling the ideal Laughlin state is
the (essentially) unique 1 ground state of the model “hard core” or “pseudo-potential”
Hamiltonian [233, 86]
H1/3 =
X
i<j
∇2 δ (2) (ri − rj ).
(3.6)
All available evidence is consistent with the proposition that at exactly ν = 1/3 the
ground state is separated by from collective modes and from quasielectron-quasihole
pairs by a gap that remains non-zero in the thermodynamic limit. Further, all states
with a given number of quasiholes are degenerate, or in other words, the quasiholes
do not interact; quasielectrons on the other hand have a weak short-range repulsion.
These features of the model Hamiltonian clearly parallel those of the self-dual point
of the Chern-Simons Landau-Ginzburg theory. Therefore, we may follow the strategy
adopted previously: by perturbing about H1/3 with a weak, long ranged interaction,
we can make the quasiparticles either attract or repel without closing the gap and
destabilizing the ground state. We have thus replicated our perturbative construction
of Type I (and II) quantum Hall liquids at ν = 1/3 within a lowest Landau level
treatment.
The 1/3 Laughlin state is just one example of a much larger (indeed, infinite) class
of microscopic wavefunctions inspired by conformal field theory which are gapped and
exact ground states of short-range Hamiltonians. It is believed that these criteria are
satisfied by all states belonging to the so-called Read-Rezayi sequence [201], and their
particle-hole conjugates. The Read-Rezayi states have filling ν = k/(km+2), where k
is a nonzero positive integer, and m is odd for quantum Hall states of fermions; their
1
The caveat allows for topological degeneracies on closed manifolds, in this case 3g for genus g.
89
wavefunctions obey a generalized Pauli principle, and they can be obtained as the
densest zero energy states of k + 1 body model Hamiltonians, which we shall denote
(k,m) 2
by HRR
. The k = 1 case corresponds to the Laughlin states, while k = 2, m = 1
corresponds to the Moore-Read (Pfaffian) state. Particle-hole conjugates of the RR
wavefunctions describe quantum Hall states at filling factor ν = 1 − k/(km + 2). The
(k,m)
corresponding model Hamiltonians are obtained by conjugating HRR , and the roles
of quasiholes and quasielectrons are reversed. Our results apply, mutatis mutandis,
to these as well.
All such model Hamiltonians have noninteracting quasiholes and weakly repulsive
quasielectrons and therefore correspond to the boundary between Type I and Type
(k,m)
II behavior. Small perturbations of the HRR
break the degeneracy of the quasi-
hole sector, and modify the energies of the (nondegenerate) quasielectron states. Of
course, the underlying quantum Hall state survives, as it is protected by a gap; the
change is only seen in the energetics of excitations, and not in the topological properties of the phase. While generic perturbations lead to interaction profiles that are
non-monotonic, there exist those that lead to interactions that are purely attractive
(Type I) or purely repulsive (Type II)
3
for both quasiholes and quasielectrons, al-
though arbitrarily weak perturbations may not suffice for the latter. It is therefore
possible to produce a Hamiltonian whose excitation spectrum is characteristic of a
Type I quantum Hall liquid, but whose topological properties are those of the underlying Read-Rezayi state. Note that while the quantum numbers of the fractionalized
quasiholes of Type I and Type II quantum Hall liquids are identical, in the former
case individual quasiholes are not stable against agglomeration.
2
These Hamiltonians annihilate not only the ground state but also quasihole and edge excitations.
However, the ground state and states with various different quasihole content can be distinguished
by their angular momenta. The densest zero-energy state is the RR state with no quasiholes, which
(k,m)
we refer to as the ground state. Properly speaking we require only that HRR are gapped with
respect to collective modes and neutral quasielectron-quasihole pairs.
3
Two-body perturbations should suffice for this purpose, even in cases where k + 1 body interactions are needed to stabilize the ground state.
90
3.4
Type I plateau formation
The transport response of a Type II quantum Hall fluid exhibits plateaux in two
distinct senses. First, in a clean system it exhibits plateaux in chemical potential;
this is a restatement of the existence of a gap to making charged excitations. Second,
in a system with weak disorder it exhibits a plateau with filling factor. This is
understood as arising from the localization of doped quasiparticles by a combination
of interactions and disorder with the latter dominating at long wavelengths. For a
Type I quantum Hall liquid the formation of plateaux with chemical potential follows
again from the existence of a charge gap. In the presence of disorder there is again
plateaux formation. Now disorder destroys phase separation and instead pins the
quasiparticles in clumps as first discussed in an early paper [120]. As a consequence,
Type I fluids also give rise to plateaux—the canonical property of quantum Hall
fluids. Also, as the charge of the minimum deconfined charged excitation is the same
for Type II and Type I fluids derived from the same parent state, Type I plateaux
will also exhibit a combination of activated and variable range hopping transport at
low temperatures.
3.5
Experimental Realizations
In this chapter we have been primarily concerned with a point of principle—the
general existence of Type I quantum Hall fluids—and have mostly discussed Hamiltonians somewhat far from those typically found in experimental systems. We now
turn briefly to the prospects for experimental realizations of Type I physics. Here
there are two broad remarks we wish to make. The first is that apart from potential
cold atom realizations, experimental systems involve repulsive Coulomb (1/r) or, in
the presence of a nearby conducting plane, dipolar (1/r 2 ) interactions. For both of
these, any tendency to phase separation stemming from short distance physics will
91
necessarily be frustrated by the long ranged part of the interactions with the frustration being much weaker for dipolar interactions. This frustration will then lead
to microemulsion phases, such as bubble and stripe phases, discussed at length in
a general context elsewhere. [These phases in turn will be disrupted at long wavelengths by disorder.] The second remark is that paired quantum Hall states appear
at this point to be the most likely candidates for realizing Type I behavior—for these,
the weak coupling limit should lead to Type I behavior automatically as discussed in
more detail in Chapter 2.
The prototypical paired state, the Pfaffian, can be tuned to weak coupling either
by changing quantum well thickness [181], or using graphene samples with a screening
plane [182]. Another example is a bilayer system, with each layer at ν = 1/2. For
large layer separation d ≫ ℓB , the ground state is simply two decoupled composite
Fermi liquids and hence compressible, but pairing of composite fermions between
layers becomes increasingly favored as d is decreased; for d → 0 the ground state is
an interlayer paired quantum Hall liquid [161]. At intermediate d & ℓB , the pairing
gap will be small reflecting weak coupling and the resulting paired state will be Type
I.
3.6
Concluding Remarks
In this chapter, we have shown that the existence of special points where the quasiparticles are non-interacting and yet the quantum Hall state is protected by a gap, allows
the construction of Type I and Type II quantum Hall fluids by simple perturbation.
Conversely, this identifies these special points as poised on the boundary of Type I/II
behavior. For the Chern-Simons Landau-Ginzburg theory this is in turn tied to the
self-dual reduction of the equations of motion and it is interesting to ask whether this
has a meaning for model Hamiltonians such as (3.6). Finally, an analogous set of
92
Hamiltonians have been studied for T and P symmetric topologically ordered states,
e.g. Kitaev’s toric code [119], various Rokhsar-Kivelson points [205] and the LevinWen hamiltonians [141], where again the quasiparticles are non-interacting. It is also
interesting to ask what Type I doping will look like in these cases.
93
Part III
Valley Ordered Quantum Hall
States
94
Chapter 4
Nematic Valley Ordering in
Quantum Hall Systems
4.1
Introduction
A remarkably diverse set of phases, exhibiting the quantum Hall effect, are observed
in sufficiently clean two-dimensional electron systems subjected to a high magnetic
field [50]. Of these, a particularly interesting subset occurs in multi-component quantum hall systems, where in addition to the orbital degree of freedom within a Landau
level, electrons have low energy “internal” degrees of freedom, such as spin or a
“pseudo-spin” associated with a valley or layer index. Quantum Hall states in such
systems, in addition to the topological order characteristic of all quantum Hall states,
feature broken global spin/pseudospin symmetries [218, 163] – a phenomenon termed
quantum Hall ferromagnetism. The entangling of the charge and spin/pseudospin
degrees of freedom leads to novel phenomena in ferromagnetic quantum Hall states,
including charged skyrmions [218], finite temperature phase transitions [163], and
Josephson-like effects [163, 50].
95
In the cases studied to date, the global symmetry is an internal symmetry that acts
on spin/pseudospin. In this chapter we study a situation where the global symmetry
acts simultaneously on the internal index and on the spatial degrees of freedom.
This occurs naturally in a multi-valley system where different valleys are related
by a discrete rotation, so that valley (pseudospin) and rotational symmetries are
intertwined. An example of such a system which is central to our discussion in this
chapter is the AlAs heterostructure [213, 211, 180], where two valleys with ellipsoidal
Fermi surfaces are present, as illustrated in Fig. 4.2).
This linking of pseudospin and space in this system has two significant consequences at appropriate filling factors such as ν = 1. First, in the clean limit the onset
of pseudo-spin ferromagnetism, which occurs via a finite temperature Ising transition,
is necessarily accompanied by the breaking of a rotational symmetry that corresponds
to nematic order, with attendant anisotropies in physical quantities. We shall call the
resulting phase a quantum Hall Ising nematic (QHIN). (The Ising-type pseudospin
ferromagnetism is consistent with the general classification of anisotropies in quantum Hall ferromagnets [110]). Second, any spatial disorder, e.g. random potentials
or strains, necessarily induces a random field acting on the pseudospins which thus
destroys the long ranged nematic order in the thermodynamic limit. Interestingly,
though, the resulting state still exhibits the quantum Hall effect at weak disorder so
we refer to itas the quantum Hall random field paramagnet (QHRFPM).
Although for concreteness we shall focus on the simple case of the ν = 1 state in
AlAs-heterostructures, our findings are readily extended to other values of ν and a
variety of multi-valley systems.
96
4.2
Symmetries
The only exact symmetries of quantum Hall systems are the discrete translational and
point group symmetries of the underlying crystalline heterostructures. However, in
many circumstances, there are additional approximate symmetries, some of which are
continuous. To the extent that spin-orbit coupling can be ignored, there is an approximate U(1) spin-rotation symmetry about the direction of the magnetic field. Since
q
~c
, and the Fermi wave-length, λF , are long compared
the magnetic length, ℓB = eB
to the lattice constant, the effective mass approximation is always quite accurate, so
it is possible to treat the translation symmetry as continuous. If the electrons occupy
only a valley or valleys centered on the Γ point in the Brillouin zone, the effective
mass approximation also elevates a Cn point-group symmetry to a continuous U(1)
rotational symmetry. Terms which break this symmetry explicitly down to the discrete subgroup come from corrections to the effective mass approximation, and so are
smaller in proportion to (a/λF )2 , where a is the lattice constant of the semiconductor.
All three of these approximate symmetries hold in GaAs heterostructures.
However, once there are multiple valleys centered on distinct symmetry-related
Bloch wave-vectors, the effective mass tensor for each valley is, generically,
anisotropic.
Thus, already in the effective mass approximation, individual val-
leys do not exhibit full rotational invariance; there are only the original discrete
set of rotations which are associated with a simultaneous interchange of valleys.
These discrete symmetries are unbroken for weak interactions in zero magnetic
field. However, we show that in the presence of a strong magnetic field they are
spontaneously broken at certain filling factors.
Specifically, in the two valley case considered explicitly here (Fig.4.2) the Hamiltonian has an approximate Z2 × U(1) invariance: The Z2 represents the operation of
a π/2 rotation combined with valley interchange. The U(1) reflects an approximate
conservation of the valley index, which is violated only by the exponentially small
97
Coulomb matrix elements, Viv , which involve the intervalley scattering of a pair of
electrons. The quantum Hall ferromagnet should thus exhibit a finite-temperature Z2
or Ising symmetry breaking phase transition, accompanied by a spontaneous breaking
of the rotational symmetry from C4 to C2 , i.e. to Ising-nematic ordering. It is important to note that although Viv breaks the approximate U(1) symmetry, the Ising
symmetry is exact.
We should note that there is a well understood counter-example to our general
argument concerning the lack of continuous symmetries in multi-valley systems which
is realized at a (110) surface in silicon (Si). Here the two dimensional electron gas
occupies two valleys centered at k = ±Q/2, Q being shorter than the smallest reciprocal lattice vector. In this case, the only rotational symmetry is a symmetry under
rotation by π. Yet, to the extent the intervalley scattering, Viv , can be neglected, this
problem was shown by Rasolt et al [197] to have an SU(2) pseudo-spin symmetry.
This derives from the fact that, in this case, the effective mass tensors in the two
valleys are identical. While in the case of interest to us there is only a discrete Z2
symmetry, due to the effective mass anisotropy in each valley, in the limit of small
anisotropy there is a reference SU(2) symmetry which is only weakly broken. For
clarity, and without loss of generality, we will in places consider this analytically
tractable limit, although in reality, the mass anisotropy in AlAs is not small.
4.3
Ising anisotropy
The single-particle Hamiltonian in each of the valleys, labeled by the index κ = 1, 2,
P
(p −Kκ,i +eAi /c)2
, where K1 = (K0 , 0) and K2 = (0, K0 ) are the
is given by Hκ = i=x,y i 2m
κ,i
positions of the two valleys in the Brillouin zone. We work in Landau gauge, A =
(0, −Bx), in which eigenstates can be labeled by their momentum py that translates
into the guiding center position X = py ℓ2B . The lowest Landau level eigenfunctions
98
ky
2
1
1
kx
2
Figure 4.1: Model band structure for AlAs quantum wells
Ellipses represent lines of constant energy in k-space. There are two non-equivalent
anisotropic valleys, 1 and 2.
in the two valleys are given by
2
eipy y uκ 1/4 − uκ (x−X)
2ℓ2
B
,
e
ψκ,X (x, y) = p
Ly ℓB π
(4.1)
where λ2 = (m1,x /m1,y ) = (m2,y /m2,x ) is the mass anisotropy in terms of which
u1 = 1/u2 = λ.
Proceeding to the effects of the Coulomb interactions, we notice that the terms in
the Hamiltonian that involve inter-valley-scattering processes require large momentum transfer, of order π/a, and therefore they are small in proportion to a/ℓB . In
accord with that, we write the Hamiltonian as follows,
H = H0 + Hiv , H0 =
1 X
V (q)ρκκ (q)ρκ′ κ′ (−q),
2S κ,κ′
99
(4.2)
where S = Lx Ly is the system’s area, ρκκ is the density component within valley
κ, V (q) =
2πe2
εq
is the matrix element of the Coulomb interaction, and Hiv denotes
inter-valley scattering terms1 , which we neglect for now.
To account for the spatial structure of Landau level eigenfunctions, we follow the
standard procedure of projecting the density operators onto the lowest Landau level
(see, e.g., [163]):
−
ρκκ (q) = Fκκ (q)ρ̄κκ (q), Fκκ (q) = e
„
2
q2
qx
+uκ 4y
4uκ
«
,
(4.3)
where the magnetic translation operator is given by,
ρ̄κκ (q) =
X
X̄
eiqx X̄ c†κ,X+ cκ,X− , X± = X̄ ±
qy
.
2
In the limit of vanishing mass anisotropy, λ → 1, the Hamiltonian H0 is SU(2)symmetric, so at filling factor ν = 1 there is a family of degenerate fully pseudo-spin
polarized ground states, favored by the exchange interactions.
Ψα,β =
Y †
(αc1,X + βc†2,X )|0i, |α|2 + |β 2| = 1.
(4.4)
X
In this notation, the components of the nematic order parameter are given by nx =
αβ ∗ + α∗ β, ny = iαβ ∗ − iα∗ β, nz = |α|2 − |β|2, where n2 = 1. We can use the states
(4.4) to obtain a variational estimate of the energy per electron of the system for
different (uniform) values of the order parameter which should be reliable at least for
λ near 1. The result is
E0 = −∆0 (D1 +
D2 n2z ),
1
1
∆0 =
2
r
π e2
2 εℓB
(4.5)
Here we notice that Hiv has a contribution Viv that breaks U(1) symmetry, as well as a contribution that preserves the number of electrons within each valley; the former is exponentially small
in a/ℓB , while the latter, although only algebraically small, has negligible value [183].
100
where
D1 = (C1 + C2 )/2, D2 = (C1 − C2 )/2,
r
p
2λ
2 K( 1 − 1/λ2)
√
,
, C2 =
C1 =
π
1 + λ2
λ
(4.6)
(4.7)
K being the complete elliptic integral of the first kind. Clearly, when λ 6= 1, the
SU(2) symmetry is broken down to Z2 × U(1) and the resulting QHIN indeed has an
Ising (easy-axis) symmetry.
For the experimentally relevant case, λ2 ≈ 5, κ ≈ 10, the anisotropy reaches a
relatively large value of 5 K at B = 10 T. Let us also note, for subsequent use, that the
Ising symmetry can be explicitly broken in experiments by the convenient application
of a uniaxial strain [213], which then acts as a valley Zeeman field.
4.4
Thermal properties
In order to understand the behavior of the system more generally, and in particular
to describe the properties of domain walls and excitations, we need to account for
spatially varying order parameter configurations. The classical energy functional for
smooth configurations of the order parameter can be obtained approximately for
|λ − 1| ≪ 1 by the method of [163]:
ρs
E[n(r)] =
2
where α ≈
3 ∆0
(λ−1)2 .
32 2πℓ2B
Z
α
d r(∇n) −
2
2
2
Z
d2 rn2z ,
(4.8)
The symmetric part of the stiffness coefficient in Eq.(4.8) is
2
√e
16 2πεℓB
+ O(λ − 1). In writing Eq.(4.8), we have neglected anisotropic
R
ρ R
stiffness terms of the form, ρ2A d2 rnz ((∂x n)2 − (∂y n)2 ), 2A′ d2 r[3(∇nz )2 − (∇n)2 ].
given by, ρs =
While these terms are also quadratic in the gradient expansion, in the limit |λ−1| ≪ 1,
the first term is at most cubic [183] in λ − 1, such that ρA ≈ o((λ − 1)2 ), while the
101
second term is quadratic, ρA′ = o((λ − 1)), and so they are much smaller than the
gradient term we have kept.
The nematic ordering temperature can readily be estimated from Eq.(4.8), which
is precisely the continuum limit of the 2D Heisenberg ferromagnet with weak Ising
anisotropy. Consequently, Tc vanishes for α = 0, but only logarithmically, due to the
exponential growth of correlations in the Heisenberg model,
kB Tc ∼ 4πρs log−1 [ρs /αℓ2B ].
(4.9)
Since, in reality, the anisotropy is not small, a more robust estimate is just Tc ∼ ρs .
This puts it in the range of several Kelvin, well above typical temperatures at which
quantum Hall experiments are carried out, which range from a few tens to a few
hundred milli-Kelvins [213].
4.5
Domain walls and quasiparticles
The topological defects of an Ising ferromagnet are domain walls, in this case domain
walls across which the valley polarization changes sign. We obtain a domain wall
solution by minimizing the classical energy in Eq.(4.8) to obtain the length scale L0 =
p ρs
√
which characterizes the domain-wall width, and the surface tension, J ∼ ρs α,
α
its creation energy per unit length. The domain wall solution obtained in this way
spontaneously breaks the approximate U(1) symmetry as the energy is independent
of the choice of the axis of rotation of n in the plane perpendicular to nz . Naturally,
since the domain wall is a one dimensional object, thermal or quantum fluctuations
restore the symmetry, but at T = 0, and in the absence of explicit symmetry breaking
perturbations, what remains is a gapless “almost Goldstone mode” and power-law
correlations along the domain wall. A small gap in the spectrum and an exponential
102
fall-off of correlations beyond a distance ξiv are induced when the effects of the weak
intervalley scattering terms, Viv , are included.
In AlAs, the anisotropy is λ2 ≈ 5, so L0 is only 30% greater than ℓB , which
indicates that our treatment should be supplemented by microscopic calculations
that can better handle a strong Ising anisotropy [186]. A variational ansatz for a
domain-wall can be constructed of the same form as in Eq. 4.4 by treating α and β
as (complex) functions of X, with asymptotic forms (α, β) → (1, 0) as X → −∞ and
(α, β) → (0, 1) as X → ∞. In the limit of large λ, the optimal such state consists
of a discontinuous jump between these two limiting values across the domain wall,
so that the domain wall width is simply equal to ℓB . When inter-valley scattering
is absent, such a wall, being a boundary between two different quantum Hall liquids
(one with ν1 = 0, ν2 = 1, the other with ν1 = 1, ν2 = 0), supports two counterpropagating chiral gapless modes— one with pseudo-spin “up” and the other with
pseudo-spin “down.” Coulomb interactions between the two modes turn this into a
type of Luttinger liquid. This connects smoothly to the description obtained above
in the limit of weak anisotropy, and indeed the Luttinger liquid action can be derived
explicitly from a σ-model description [59, 158] by augmenting the classical energy in
Eq. (4.8) with an appropriate quantum dynamics.
The other excitations of interest are charged quasiparticles and it is well known
that in the SU(2) limit at λ = 1 they are pseudospin skyrmions of divergent size [218].
However, the smallness of L0 at λ2 ≈ 5 alluded to above implies that for the experimentally relevant case the quasiparticles will be highly, if not completely, valley
polarized.
103
4.6
Properties of the clean system
For T < Tc , where the pseudospin component nz has a nonzero expectation value,
C4 rotation symmetry is spontaneously broken to C2 . Thus, non-zero values of any
non-trivial traceless symmetric tensor can also be used as an order parameter.
Ideally, thermodynamic quantities, for instance the difference in the valley occupancies, provide the conceptually simplest measures of the broken symmetry. However, such quantities are not easily measured in practice. Following our remark above,
we should just as well be able to use the experimentally accessible transport anisotropy
ratio
N=
σxx − σyy
6= 0
σxx + σyy
(4.10)
as a measure of nematic order. However, at T = 0, where σaa = 0, N is ill-defined.
This problem can be resolved by measuring σaa at finite temperature, T > 0, and
then taking the limit T → 0. (Alternatively, one could imagine working at finite
frequency, and then taking the limit as the frequency tends to 0.) However, in practice,
the conductivity is strongly affected by the presence of even weak disorder, so any
practical discussion of the resistive anisotropy must be preceded by an analysis of the
effects of disorder.
4.7
Length scales from weak disorder
By analogy with the random-field Ising model [30], we know [10] that even an arbitrarily weak random valley-Zeeman field destroys the ordering of the QHIN, leading
to formation of “Imry-Ma” domains of opposite valley-polarization. In AlAs such
disorder can stem from random strains, which lead to position-dependent relative
shifts of the energies of the two valleys. While the average strain (i.e. the average
pseudo-magnetic field) can be externally controlled [213], fluctuations of the strain
104
ξIM
LS
ξIM ≫ LS
Figure 4.2: Phases of nematic ordering in dirty and clean systems
.
Top: when the sample size, LS ≫ ξIM , we have many domains of opposite nematic
orientation in the sample, characteristic of the QHRFPM. Bottom: for LS ≪ ξIM ,
the sample is characterized by a single nematic orientation, i.e. it is in the QHIN
phase.
are inevitable. Random fluctuations of the electric potential ( which we denote Ud to
distinguish it from the interaction, V ) also give rise to a random valley field.
The coupling of random strain and potential disorder to the QHIN order parameter
is
1
Est =
2
Z
d2 r h(r)nz (r),
(4.11)
where the random field h(r) = [hst (r) + hpot (r)] with hst (r) ∝
∂u(r)
∂x
− ∂u(r)
, u(r) being
∂y
the displacement of point r of the crystal, and2
hpot (r) =
2
(mx − my )ℓ2B
2π~2
"
∂Ud
∂x
2
−
∂Ud
∂y
2 #
.
(4.12)
The derivation of this term is somewhat involved, so we defer the details to Appendix A.
105
On the basis of this analysis, we expect that the random valley-Zeeman field is
smooth, with a typical correlation length ℓdis ≫ ℓB . For weak disorder, the Imry-Ma
domain size is set by the mean-squared strength of the random field (assumed to have
R
zero mean and to be short-range correlated) W ≡ d2 rhh(r)h(0)i and the domain
wall energy per unit length, J (defined above) as ξIM ∝ exp[ A(J )2 /W ] where A is
a number of order 1. Because of the exponential dependence on disorder, it is possible
for ξIM to vary, depending on sample details, from microscopic to macroscopic length
scales.
Disorder also leads to scattering between the valleys although this is again suppressed due to the mismatch between the reciprocal lattice vector and the length
scale of the dominant potential fluctuations. There is thus a second emergent length
scale, ξiv , which is the length scale beyond which conservation of valley pseudo-spin
density breaks down. However, this length scale approaches a finite value in the
limit of vanishing disorder due to intervalley Coulomb scattering, discussed above.
Different regimes of physics are possible depending on the ratio of ξIM /ξiv . Finally,
especially when the filling factor deviates slightly from ν = 1, there is a length scale,
ξQP , which characterizes the quasi-particle localization length. Because the magnetic
field quenches the quasiparticle kinetic energy, even for extremely weak disorder, we
expect ξQP ∼ ℓB is relatively short.
4.8
Intrinsic resistive anisotropy
In a quantum Hall state at low temperatures, dissipative transport is usually due to
hopping of quasiparticles between localized states, accompanied by energy transfer
to other degrees of freedom [212]. Typically, transport is of variable-range-hopping
(VRH) type, such that the optimal hop is determined by the competition between the
energy offset of the two states and their overlap. We will now apply these ideas to our
106
system when its transport primarily involves hopping of electrons between localized
states within one of the valleys. This requires a) either that a uniaxial strain be
applied to substantially eliminate domain walls and achieve valley polarization in the
proximity of ν = 1 or that the sample be smaller than ξIM and b) that ξiv be large
compared to ξQP . For each valley the localization length is anisotropic, owing to the
mass anisotropy, which results in the anisotropy of the corresponding contribution to
the VRH conductivity.
The contribution to the resistive anisotropy from quasiparticles in valley 1, N1 , can
be computed as follows: First, we transform the anisotropic VRH problem into the
√
√
isotropic one by rescaling coordinates, x = x̃/ λ, y = λỹ. In the new coordinates
the effective mass tensor is isotropic, which, given the uncorrelated nature of the
potential, implies that the VRH problem is isotropic 3 , and therefore σ̃xx = σ̃yy . Since
the ratio of the conductivities in the original coordinates is given by
N1 =
1 − λ2
,
1 + λ2
σxx
σyy
=
1 σ̃xx
,
λ2 σ̃yy
(4.13)
which is negative for λ > 1, as expected, i.e. it is more difficult for particles to
move in the direction of larger mass. Clearly, the resistive anisotropy produced by
quasiparticles in valley 2 is N2 = −N1 .
At ν = 1 localized states in both valleys are present, and due to combined particlehole/valley-reversal symmetry of the state (in the absence of Landau-level mixing),
the density of localized states should be same: The resistivity is thus expected to
be isotropic! However, for ν 6= 1, particle hole symmetry is broken. Consider the
case in which nz = +1, which corresponds to filling valley 1 states. Then, at slightly
filling factor, ν = 1 − δν with 1 ≫ δν > 0, the density of localized states for valley
κ = 1 exceeds that for valley κ = 2. Due to the exponential sensitivity of the VRH
3
Under the assumption that the energy transfer in VRH is due to phonons, and the electronphonon coupling is scalar.
107
conductivity to the density of states, this implies that the contribution of valley 1 to
the total conductivity dominates, leading to an anisotropy of the total conductivity
N ≈ N1 . Conversely, for ν = 1 + δν, N ≈ N2 = −N1 ! It is worth noting that
the scaling argument presented above for VRH regime is likely more general, and
also applicable to the regime of thermally activated transport, which is relevant at
intermediate temperatures.
4.9
Domain walls and the QHRFPM
We now move away from the above limit to where domain walls are a significant
contributor to the transport—to systems much bigger than ξIM and at weak uniaxial
strain. Now, dissipative transport is complicated by the existence of multiple emergent length scales. Transport within a nematic domain proceeds by variable range
hopping and/or thermal activation of quasiparticles. For length scales larger than
ξIM , it is likely to be dominated by transport along domain walls, which will have
insulating character or metallic character depending on whether viewed at distances
large or small compared to ξiv .
A key question is whether the quantum Hall effect survives the formation of domains. This is trickiest when no net valley Zeeman field is applied where in the
thermodynamic limit the domain walls form a percolating network. In the limit
ξiv → ∞, the associated edge channels are conducting, and the domain wall network
can be expected to be well described by two copies of the Chalker-Coddington network
model [43] at criticality. This implies a critical metallic longitudinal DC conductivity
of order e2 /h and the absence of quantum Hall effect. However, at length scales longer
than ξiv (or temperatures less than Tiv ∼ 1/ξiv ) the domain wall states are localized,
which implies a phase that exhibits the quantum Hall effect without Ising/Nematic
order—the QHRFPM. Needless to say, in the absence of substantial amount of short
108
ranged disorder (which can produce a relatively small value of ξiv ) the topological
(quantum Hall) order in the QHRFPM is likely to be fragile. (In some ways similar
results were obtained by Rapsch et al. [196], who considered an SU(2)-symmetric
disordered quantum Hall ferromagnet, where magnetic order is destroyed by forming
a spin glass without destroying the quantum Hall effect. )
In the presence of a uniform valley Zeeman field h̄ the existence of the quantum
Hall effect is much more robust. Even weak fields can restore a substantial degree of
valley density polarization as domains aligned with this field grow while those aligned
opposite to it shrink. Consequently, the domain walls no longer percolate but rather
are separated by a finite distance that grows with increasing h̄.
While we have yet to construct a detailed theory of the transport in this regime,
it is clear that the characteristic energy scale characterizing the dissipative transport
will rise rapidly from Tiv for h̄ = 0, to the clean-limit gap ∆0 ∼ ρs for substantial
values of h̄. It is also important to note that this equilibrium response will come
embedded in a matrix of dynamical phenomena characteristic of the random field
Ising model that can be translated straightforwardly to the case of the Ising-nematic,
as has been discussed in another context in [40]. In particular, the macroscopic
nematicity induced by the application of h̄ will be metastable for long times, even
upon setting hhi = 0—thus giving rise to hysteresis.
4.10
Experiments
Turning briefly to experiments we note that an anomalously strong strain induced
enhancement of the apparent activation energy at ν = 1 has been observed[213] in
AlAs, where it was tentatively attributed to the occurrence of valley skyrmions. As
we noted earlier, in view of our estimate of a large Ising anisotropy skyrmions of the
requisite size (about 15 flipped pseudospins) are implausible. We would like to suggest
109
that it is more plausible that these remarkable observations are associated with the
growth of QHIN domains. In support of this idea, we have estimated the domain
size from the long ranged part of the potential disorder alone and find that it should
be order the distance to the dopant layer, ξdis ≈ 50 nm which is thus much smaller
than the system size. However, we currently lack a plausible estimate of ξiv which
is sensitive to the short ranged part of the disorder and which is needed to round
out this explanation. Direct measurements of resistive anisotropies, and of hyseretic
effects characteristic of the random field Ising model [40] could directly confirm this
proposal.
4.11
Related Work
We note that there is a sizeable body of existing work on Ising quantum Hall ferromagnets produced at level crossings of different orbital LLs, which is typically achieved
by applying tilted magnetic fields. These systems exhibit enhanced dissipation at
coincidence [52, 166, 231]which is the analog of a dissipation peak at zero valley Zeeman field in our language. Qualitatively, our results are consistent with this earlier
work. Where we differ is in our contention that the domain walls do not, even at zero
valley Zeeman field, produce dissipation at T = 0—in the previous work [111, 44] this
was not explicitly addressed, in part as the focus was on accounting for the unexpected dissipation. The reader will also note that the QHIN studied here differs from
the “nematic quantum Hall metal” (NQHM) phase which has been observed [145]
in ultra-clean GaAs-AlGaAs heterostructures for fields at which the n > 1st Landau
level is nearly half filled. Unlike our system, the NQHM is a metallic state which does
not exhibit the quantum Hall effect, but has a strongly anisotropic resistivity tensor.
110
4.12
Concluding Remarks
The distinctive feature of our system is the breaking of a global symmetry that combines spatial and internal degrees of freedom. This physics and its attendant consequences will generalize immediately to other ferromagnetic fillings in the present
system and then to other experimentally established examples of multi-valley systems
such as monolayer and bilayer graphene4 [260, 62], where two valleys are present, and
Si (111) [58], where, depending on the parallel field, either 4 or 6 degenerate valleys
can be present. Potentially, our ideas could apply farther afield in the case of 3D Bismuth (Bi), where three electron pockets related by 2π/3, 4π/3 rotations are present.
Recently, high-field anomalies in transport and thermodynamic properties of Bi were
found [143, 21], which may indicate spontaneous breaking of the Z3 valley symmetry
driven by the magnetic field, reminiscent of quantum Hall ferromagnetism.
4
Indeed, the latter is the subject of the following chapter, although we consider a situation where
the full SU (2) symmetry is preserved.
111
Chapter 5
Charge 2e skyrmions in bilayer
graphene
5.1
Introduction
In the previous chapter, we analyzed in some detail an example of a valley quantum
Hall ferromagnet with a low-energy Ising degree of freedom. Owing to the underlying
nematic pattern of symmetry breaking arising from the valley anisotropy, we were
able to predict anisotropies in transport as sharp experimental signatures of quantum
Hall ferromagnetism. Additionally, the system could be driven into a polarized state
by applying valley Zeeman field by straining the sample, allowing a systematic study
of transport properties in the nematic state.
We now study another valley ferromagnet, that we argue should exist in bilayer
graphene. Although in this case the low-energy theory does not have Ising anisotropy,
it shares both of the above experimentally appealing features with AlAs. In bilayer
graphene the electronic eigenfunctions belonging to the two valleys reside in different
layers. As a result, the valley populations can be tuned by applying a transverse
electric field – which thus acts as a valley Zeeman field – and furthermore, since the
112
surface is accessible textures in the valley degree of freedom can be studied by means
of scanning probes – providing an experimental window on the valley ferromagnet.
Unlike the Ising case, however, the enhanced symmetry supports charged skyrmions
which are the dominant low-energy degrees of freedom and which will be the central
focus of this chapter.
Before discussing the bilayer case, it is perhaps worth briefly reviewing the example
of the monolayer. There, the fourfold valley and spin degeneracy of the zero-energy
Landau level is lifted by Coulomb interactions, giving rise to new spin and/or valley
polarized incompressible quantum Hall states [260, 3, 107]. The effective Hamiltonian
that describes the Coulomb interactions within single Landau level is approximately
SU(4) symmetric [176, 80] with respect to the rotations in the combined spin/valley
space. The splitting of the Landau levels corresponds to a spontaneous breaking
of the SU(4) symmetry, and the ground state is thus a quantum Hall ferromagnet.
The precise order in which the spin and valley degeneracies are lifted is determined
by the interplay between the Zeeman interaction and the valley anisotropy [11, 2] –
both of which are much smaller than the Coulomb interaction. The spin- and valleypolarized quantum Hall states in the monolayer were predicted to feature unusual
edge states [1, 64], as well as spin and valley skyrmions [254], which are smooth
topologically nontrivial textures of the ferromagnetic order parameter that carry the
electron charge e [218].
Bilayer graphene also features a Landau level at zero energy, which in addition
to the spin and valley degeneracies already present in a single layer has a twofold
orbital degeneracy: in each valley there are two zero-energy states (a = 0, 1), with
wave functions corresponding to the ground state and the first excited state of the
magnetic oscillator [150]. Taking into account all the degeneracies, the zeroth Landau
level in the bilayer is eight-fold degenerate; Coulomb interactions are expected to
lift this eight-fold degeneracy [23]. In this chapter, we consider interaction-induced
113
Ã
E, B
B̃
d
A
B
a
Figure 5.1: Bilayer graphene lattice.
Electrons can hop from layer to layer between the à and B (black) sites. The lowenergy degrees of freedom reside on the A (red) and B̃ (blue) sites.
quantum Hall states at even filling factors, and analyze the properties that arise due
to the orbital isospin1 . We show that these quantum Hall states exhibit interesting
collective and topological excitations, and in particular predict that pairs of charge e
excitations bind into skyrmions that carry charge 2e.
Such binding of charges is surprising, because the Coulomb interactions between
electrons are purely repulsive. A similar binding phenomenon was predicted for spin
quantum Hall ferromagnets with small Zeeman interaction [144, 173]. The weak
pairing of skyrmions considered in those examples, however, can occur only when the
Zeeman interaction is extremely small; in contrast, charge 2e skyrmions in bilayer
graphene can be thought of as robust tightly bound pairs, which exist in for a wide
range of the effective Zeeman interaction.
We also analyze the dependence of skyrmion energy and size on the effective valley
Zeeman splitting, which can be tuned [42, 178] by creating a potential difference
between the top and bottom layers, ∆v = eEd, where d = 0.34 nm is the separation
between the layers (see Fig. 5.1). Slightly away from even filling factors, |∆ν| =
1
We use the term isospin for the orbital index, to contrast with the valley pseudospin and the
electron spin.
114
|ν − 2M| ≪ 1, there is a finite density of charge-2e skyrmions in the system. At small
density, the skyrmions form a triangular lattice, while above a critical density, ∆ν∗ ,
they form a bipartite square lattice [36, 6].
5.2
Landau levels of Bilayer Graphene
We begin our discussion by recalling the Landau level spectrum in bilayer
graphene [150].
The low-energy excitations near the K, K ′ point are described
by the Schrödinger equation εψK,K ′ = HK,K ′ ψK,K ′ , with the Hamiltonian
HK,K ′ = −

†2



1  0 π 
 0 π 
,
 + ĥw , ĥw = ξv3 

2m π 2 0
†
π 0
(5.1)
where π = px −ipy . For the K valley the upper (lower) component of the wave function
corresponds to the amplitude on sublattice A(B̃) (see Fig. 5.1), which belongs to the
bottom (top) layer; for the K ′ valley the order of components is reversed. The
effective mass m can be expressed in terms of the ÃB interlayer hopping amplitude
γ1 ≈ 0.39 eV and the Fermi velocity in the monolayer vF ≈ 106 m/s, m = γ1 /2vF2 ;
ξ = +1(−1) for K ′ (K) valley. The trigonal warping term ĥw originates from the weak
√
direct AB̃ hopping, γ3 ≪ γ1 [150], with an effective velocity given by v3 = ( 3/2)γ3 a,
a = 0.142 nm.
To analyze the Landau level spectrum, we work in Landau gauge, Ay = Bx,
Ax = 0, in which the eigenstates can be classified according to the value of the wave
vector ky , ψK,K ′ (x, y) = eiky y ψK,K ′ (x). The wave vector ky translates into the guiding
p
center position, X = ky ℓ2B , where ℓB = ~c/eB is the magnetic length. For simplicity
we shall choose units below such that ℓB = 1.
115
First, let us ignore the small trigonal warping term. Then the effective 1d Hamiltonian for ψK,K ′ (x) takes the following form,

 0
HK,K ′ = −~ωc 
2
a†X
a2X
0


,
aX = i(∂x + (x − X)),
(5.2)
where ωc = eB/mc is the cyclotron energy. The Hamiltonian (5.2) has two zero
a
iXy
modes with the following wave functions, ψK,K
(0, ϕa,X (x)),
′ (x, y) = e
a = 0, 1,
where ϕa,X (x) denotes the a-th excited level of the magnetic oscillator. Below we shall
denote the annihilation operators of the zero modes by ca,κ,X , κ = K, K ′. Including
the trigonal warping term ĥw does not change the energies of the zero modes [150].
The corrections to the wave functions of the zero modes due to ĥw are proportional to
p
(v3 /vF ) × γ1 /~ωc [6]; for experimentally relevant field values, B ∼ 20 T, the square
root term is of the order one, so the corrections are negligible due to the smallness of
the ratio v3 /vF ≪ 1.
5.3
Splitting of the zero-energy Landau level
As our first step, we explain the manner in which Coulomb interactions lift the eightfold degeneracy of the zeroth Landau level. Before presenting the technical aspects
of our calculation, we briefly summarize the arguments that lead to the order of
splittings in Fig. 5.3. The effective Coulomb interaction Hamiltonian for the zeroth
Landau level in the bilayer is approximately SU(4) symmetric in the valley-spin space.
However, the symmetry in the orbital isospin space is broken at the outset due to the
different orbital wave functions of the two states [23]. This results in the following
picture of how the degeneracy is lifted: at even filling factors (ν = 2M filled sublevels) M pairs of orbital states with the same valley and spin are filled, while the
states at odd filling factors, ν = 2M + 1 are obtained from the ν = 2M quantum Hall
116
state by filling one of the remaining states with orbital isospin a = 0. This sequence
of splittings is due to two facts: (i) exchange energy within the Landau level with
isospin a = 0 is higher than that for the Landau level with isospin a = 1; (ii) there
is exchange energy between filled a = 0 and a = 1 Landau levels with the same spin
and valley, which makes the energy of the state where a = 0, 1 Landau levels with the
same spin and valley are filled (e.g. 0K ↑ and 1K ↑) lower than the energy of a state
polarized in the orbital space along a = 0 direction (e.g., 0K ↑, 0K ′ ↑ Landau levels
are filled). The order in which valley and spin degeneracies get lifted is determined
by the competition between the symmetry-breaking terms: the Zeeman interaction,
Ez = gµB B, and the effective valley Zeeman interaction ∆v . In the experiment ∆v
is typically small, and it can be tuned by gates [178]. We assume that ∆v is tuned
to be smaller than Ez . Furthermore, we assume that ∆v is non-zero and favors the
K valley 2 . In the following, we shall be especially interested in the states at filling
factors ν = −2, +2, marked in Fig. 5.3. Since these two states are related by the
particle-hole symmetry, we shall focus on the ν = −2 state.
We now proceed to a more quantitative analysis. Neglecting Landau level mixing allows us to project the Coulomb Hamiltonian onto the subspace spanned by
eigenfunctions corresponding to the zero-energy Landau level,
Hint =
1 X
V (q)ρκ (q)ρκ′ (−q),
2S
′
(5.3)
qκκ
where ρ(q) are the density operators restricted to the zeroth LL, S = Lx Ly is the
sample volume, κ, κ′ are valley indices, and the matrix element is given by V (q) =
2πe2
.
εq
2
When ∆v is extremely small, the ordering in the valley space is determined by the charging
energy, which favors states where the charge is distributed equally between the layers. We shall
assume that ∆v , while still small, is sufficient to avoid this situation.
117
spin
valley
1K ′ ↓
0K ′ ↓
orbital
ν = +2
1K ↓
0K ↓
E=0
1K ′ ↑
0K ′ ↑
ν = −2
1K ↑
0K ↑
Figure 5.2: Schematic of the splitting of the zeroth Landau level
We show the order in which we fill different broken-symmetry states in bilayer
graphene. Note that the energy levels are shown schematically, and the splittings
between them are not to scale.
The projected density components are given by
ρκ (q) =
X
ab
Fab (q)ρ̄ab
κ , ρ̄κ (q) =
a,b
were X± = X̄ ±
qy
,
2
and F00 (q) = e−q
X
X̄
2 /4
exp iqx X̄ c†a,κ,X− cb,κ,X+ .
, F11 (q) = (1 − q 2 /2)e−q
2 /4
(5.4)
are the usual
+iqx −q
form-factors for the lowest LL and the first excited LL, and F01 (q) = − qy√
e
2
F10 (q) =
qy −iqx −q 2 /4
√
e
2
2 /4
,
are the form-factors corresponding to the density components
which mix the two orbital states.
For the purpose of determining the order of splitting, it suffices to compare the
energies of trial states which are Slater determinants corresponding to filled Landau
levels. At filling factor |ν| ≤ 3 (ν + 4 filled sub-levels) our trial states may be chosen
118
as
|Ψν i =
ν+4
YY
i=1 X
d†i,X |Ωi, d†i,X =
X
i
Ūa,κ,s
c†a,κ,s,X ,
(5.5)
a,κ,s
where Ū is a unitary matrix, s is the electron spin, and Ω corresponds to an empty
zeroth Landau level3 . The problem then reduces to determining the choice of the
unitary matrix U such that the energy is optimized.
To find the ground state for ν = −2, we compare the energies of two states: (i) two
a = 0 Landau levels with different valley and/or spin indices are filled, for example,
d†1 = c†0,K,↑, d†2 = c†0,K ′,↑ , and (ii) a = 0 and a = 1 Landau levels with the same valley
and spin indices are filled, d†1 = c†0,K,↑ , d†2 = c†1,K,↑. The energy of the first state is
twice the exchange energy of a non-degenerate lowest LL,
1
hHint i1 = −2N∆0 , ∆0 =
2
r
π e2
2 εℓB
(5.6)
where N is the total number of states in one non-degenerate Landau level (i.e. when
we fix spin, valley and orbital indices). Averaging the Coulomb interaction over the
N∆0 . Thus the energy of the second state is
second state, we obtain hHint i2 = − 11
4
lower than the energy (5.6) of the first state, and the spin- and valley-polarized state
|ψ0 i is the ground state at ν = −2. The state at ν = +2 can be obtained from |ψ0 i
by charge conjugation.
5.4
Excitations
As discussed in the section above, the ν = −2 quantum Hall state is valley polarized,
and thus corresponds to a valley quantum Hall ferromagnet 4 . There are two types
of excitations this supports: ‘single-particle’ excitations, and topological defects of
3
This is filling factor ν = −4 in our notation.
It is also spin-polarized, but owing to the high g-factor (∼ 2) in graphene spin symmetry is
broken explicitly by the magnetic field, rather than spontaneously by the interactions.
4
119
the valley pseudospin order parameter, known as skyrmions. These are topologically
nontrivial smooth textures of the order parameter [218], which carry charge, a fact
which may be qualtitatively understood as arising from the entangled dynamics of
the charge and spin/valley degrees of freedom [218]. The question of which of these
is the lowest-energy excitation and thus determines the physics about commensurate
fillings is a matter of detail, which depends sensitively on the Zeeman energy and the
pseudospin stiffness, and it is to this issue that we now turn.
The lowest-energy electron-hole pair at ν = −2 is obtained by moving an electron
with orbital isospin a = 1 from the filled Landau level into one of the empty Landau
′
= 4∆0
levels. The energy of such excitation, Eeh = 27 ∆0 , is lower than the energy Eeh
of a particle-hole excitation that is obtained by removing an electron with isospin
a = 0.
Can skyrmions exist as energetically stable excitations in bilayer graphene?
Skyrmions of charge e are energetically unfavorable because they involve flipping
valley isospin (or spin) for either a = 0 or a = 1 states in some region, and in that
region the filled a = 0 and a = 1 states would have different valley isospin (spin),
which leads to a loss of the exchange energy ∆0 per flipped valley isospin.
Another possibility is skyrmions of charge 2e, which can be created by making
two identical valley textures for a = 0 and a = 1 orbital states. Such textures
are described by a unit vector n, with nz = −1(+1) corresponding to filling K(K ′ )
states. On an intuitive level, we expect such textures to be energetically favorable:
since a = 0 and a = 1 orbital states rotate simultaneously, no exchange between 0
and 1 states is lost. Below, we find the energy of the 2e skyrmion, and, by comparing
it to the energies of the single-particle excitations, establish that such skyrmions are
indeed energetically favorable.
Before we proceed to the quantitative analysis of charge 2e skyrmions, let us
compare excitations at the even and odd filling factors. For simplicity, let us consider
120
the excitations of the state ν = −3, which corresponds to the filled 0K ↑ Landau
level. The lowest energy electron-hole pair is obtained by removing an electron from
the 0K ↑ level and putting it in the 1K ↑ level; owing to the exchange between 0K ↑
odd
and 1K ↑ states, the energy of such excitation, Eeh
= ∆0 , is lower than the energy
odd
Ẽeh
= 2∆0 of an excitation where the excited electron resides in a Landau level with
a different valley and/or spin index. The existence of orbital skyrmions at ν = −3
is unlikely, because such skyrmions correspond to filling a = 1 states in some region,
which leads to a loss of the exchange energy equal to ∆0 /4 per flipped orbital isospin.
5.4.1
Skyrmion energy
Now we briefly describe the calculation of the energy of charge 2e skyrmion. As a
first step, we derive an effective Hamiltonian describing valley textures of the order
parameter,
|ψi = e−iÔ |ψ0 i.
(5.7)
In our analysis, we follow the microscopic approach developed in [163]; however, the
dynamics of the order parameter in the bilayer graphene is richer than that in the case
of SU(2) and SU(4)-symmetric quantum Hall ferromagnet, owing to the presence of
the orbital degree of freedom. We parametrize the rotation operator Ô as follows,
Ô =
X
µ
Ωµab (q)Ŝab
(−q),
(5.8)
q,a,b,µ
µ
Ŝab
(−q) =
X
X̄
eiqx X̄
µ
τκκ
′ †
c
cb,κ′ ,X+ ,
2 a,κ,X−
(5.9)
where τ are the Pauli matrices. The rotation (5.8) is described by four complex
parameters,
ua = Ωxaa + iΩyaa , a = 0, 1, v = Ωx10 + iΩy10 , w = Ωx01 + iΩy01 .
121
(5.10)
The parameters u0 (u1 ) correspond to rotations that involve 0K and 0K ′ (1K and
1K ′ ) states, while v, w parametrize rotations which transform 0K into 1K ′ , and 1K
into 0K ′ , and vice versa. To simplify calculations, we assume that the rotations
are small (|ua | ≪ 1, |v| ≪ 1, |w| ≪ 1). Then we can expand the texture energy
†
E = hψ0 |eiÔ He−iÔ |ψ0 i − hψ0 |H|ψ0 i in series in the powers of Ô. This procedure
yields an effective action for general textures that involve ua , v, w [6]. As a next
step, we restrict our attention to the low-energy excitations, where a = 0 and a = 1
states are rotated simultaneously in the valley space, which corresponds to setting
u0 = u1 = u. Integrating out v, w variables [6], we obtain the energy of the low-energy
textures. For what follows, we rewrite the resulting energy functional in terms of the
O(3) order parameter n = (−uy , ux , 0),
ρs
Est =
2
Z
d2 r (∂µ n)2 ,
ρs =
25
∆0 .
64π
(5.11)
Although we assumed above that n deviates slightly from n = (0, 0, −1), the rotational invariance in valley space ensures that (5.11) is valid for any slowly varying
configuration of the order parameter.
†
Next, we evaluate the charge density of the texture, δρ = hψ0 |eiÔ ρ̂e−iÔ |ψ0 i −
hψ0 |ρ̂|ψ0 i, where ρ̂ is the density operator. We find that the charge density is twice
the Pontryagin index density,
δρ(r) = 2ep(r), p(r) = −
1
εµν (n[∂µ n × ∂ν n]).
8π
(5.12)
This relation differs from the usual SU(2) quantum Hall ferromagnet case [218, 163]
by a factor of 2, which corresponds to the fact that the texture rotates states in both
a = 0, 1 LLs.
122
Apart from the stiffness term (5.11), there are two other contributions to the
texture energy: the valley Zeeman term and the long-range Coulomb interaction,
Hz = ∆v n0
Z
2
d r nz , Hcoul
1
=
2
Z
d2 rd2 r′
δρ(r)δρ(r′ )
,
ε|r − r′ |
(5.13)
where n0 = 1/2πℓ2B is the Landau level electron density.
The simplest topologically nontrivial texture of the order parameter n has topological charge 1 and an electric charge ±2e. This is to be contrasted with the usual
skyrmions [218], which carry charge ±e. In the limit of vanishing ∆v , the Coulomb
repulsion forces skyrmions to be infinitely large. Then the skyrmion energy is determined solely by the stiffness term,
Esk = 4πρs =
25
∆0 .
16
(5.14)
The energy of the skyrmion-antiskyrmion pair, 2Esk = 25∆0 /8, is lower than the
energy of two electron-hole pairs, which equals 7∆0 . Therefore, in the limit ∆v → 0,
pairs of electron (hole) excitations bind into charge 2e skyrmions (antiskyrmions).
At finite ∆v the skyrmion size ls is determined by the competition between the
effective valley Zeeman and Coulomb energies [218]. Optimizing the skyrmion energy
with respect to ls , we find with logarithmic precision,
ls
≈
ℓB
9π 2
32
1/3
˜ −1/3 | log ∆
˜ v |−1/3 ,
∆
v
(5.15)
˜ v = ∆v /(e2 /εℓB ). The skyrmion energy is increased compared to the case
where ∆
˜ v) =
∆v = 0, Esk (∆
25
∆
16 0
˜ 1/3
˜ 1/3 , where A =
+ A∆0 ∆
v | log ∆v |
34/3 π 5/6
.
211/6
When the filling factor is tuned away from ν = −2, the system contains a finite
density of skyrmions; the long-range interaction between them stabilizes a variety of
skyrme crystal phases that can be studied numerically [49].
123
5.5
Experiments
The rapid progress of experimental studies on graphene-based materials has led to
several simultaneous investigations of the quantum Hall regime in bilayer graphene.
Two different groups [62, 261] have observed the splitting of the eightfold degeneracy of the zero-energy Landau level. The measured activation gaps suggest that
these states originate from interaction effects; [7] provides a lucid summary of the
experimental situation.
As noted previously, an important and experimentally relevant aspect of this work
is that since the zeroth Landau level states in different valleys reside in different layers,
any valley texture of the order parameter directly translates into a spatial modulation
of the charge density in both layers. This in turn results in a spatial modulation of the
local density of states (LDOS), which can be probed directly by scanning tunneling
microscopy (STM), owing to the fact that the surface of bilayer graphene is exposed.
Thus STM provides a way to study the properties of a single charge 2e skyrmion, as
well as the properties of a lattice of such skyrmions. We note that STM has already
been employed to study electronic properties of monolayer graphene with atomic
spatial resolution [142, 259]. LDOS features on the scale of ∼ 1 meV, can be resolved
by this method, and this should be sufficient to observe the LDOS modulation in the
quantum Hall ferromagnet, whose energy scale is set by the Coulomb interaction, of
the order of 10 meV at B = 10 T [1] . STM can be used to study the skyrmion size
(5.15) as a function of the valley Zeeman interaction, as well as the properties of the
Skyrme lattice as a function of filling factor and valley Zeeman interaction. Finally,
we note that in samples with long-range disorder, the charge-2e skyrmions will lead to
an even-odd asymmetry in charging spectra of individual disorder-induced quantum
dots, which can be studied in scanning single-electron transistor experiments [148].
124
5.6
Concluding Remarks
We have presented a second example of a valley quantum Hall ferromagnet, with
physics rather distinct from the Ising case of the previous chapter. While there is
much experimental evidence for skyrmions in the historically best-known quantum
Hall ferromagnet, the ν = 1 state in gallium arsenide quantum wells, a systematic
study of their properties is hindered by two significant obstacles: first, the 2DEGs
are buried deep in semiconductor heterostructures, and are hence inaccessible to
surface probes; second, the Zeeman coupling while small – allowing for the existence
of skyrmions – is not tunable. The combination of working with bilayer graphene and
focusing on a valley degree of freedom overcomes both obstacles, and should enable
a detailed experimental catalog of the properties of quantum Hall ferromagnets, that
can then inform other examples less suited to such study.
125
Part IV
Antiferromagnetic Analogs of the
Hall Effect
126
Chapter 6
Order and Disorder in AKLT
Antiferromagnets
6.1
Introduction
Quantum antiferromagnets have been a fertile field of research for a half century,
exhibiting a great richness and variety of physical phenomena. In more recent decades,
starting with Anderson’s introduction of the RVB state [12, 61] and accelerating with
the discovery of the cuprate superconductors [13], much attention has focused on
antiferromagnets that allow for disordered ground states due to a mix of frustration
and quantum fluctuations 1 . In an important step, Affleck et. al [8, 9] showed how
to construct models that build in a great deal of both these effects by using local
projectors - models for which (essentially unique) ground states can be determined
analytically. These AKLT models have spins given by S =
z
M,
2
where M is any
integer, and z the lattice coordination number. The associated ground states have
the added feature that their wavefunctions can be written in Jastrow (pair product)
form. A general feature of such wavefunctions is that the ground-state probability
1
For a review, see [207]
127
densities can be viewed as Boltzmann weights corresponding to a local, indeed nearest
neighbor, Hamiltonian for classical spins on the same lattice. Using this unusual
quantum-classical equivalence we can understand many properties of the states via
Monte Carlo simulations of the associated classical model.
In d = 1 and d = 2, the AKLT states are disordered
2
for any spin due to the
Hohenberg-Mermin-Wagner theorem. In particular, the d = 1 case is the celebrated
AKLT chain which realizes the S = 1 Haldane phase. In this chapter, we study AKLT
states in d = 3, which are relatively less well-understood than their one and two
dimensional counterparts. Moreover, in three dimensions, the Hohenberg-MerminWagner theorem no longer applies and therefore whether an AKLT state of a given
spin is disordered or instead exhibits long range order is no longer automatic. Instead
a computation is now required to settle this question and it is this issue that we
address in this chapter by a combination of mean-field arguments and Monte Carlo
simulation. Specifically, we discuss the AKLT states on the simple cubic and diamond
lattices, where there is no (geometrical) frustration, as well on the highly frustrated
pyrochlore lattice, where the attendant complications lead to a macroscopic ground
state degeneracy of the associated classical model. Of course, all the models we study
have frustration from competing interactions.
On the cubic lattice we find that all AKLT states starting with the “minimal”
(smallest spin) S = 3 state are ordered with the standard two sublattice Néel pattern.
The diamond lattice has a small coordination number and thus larger fluctuations
and we find that on it the the minimal S = 2 state is disordered while all higher
spin states are ordered with the two sublattice Néel pattern. On the pyrochlore
lattice, the geometrical frustration of the lattice plays a significant role. In mean field
theory for the companion classical model we find a macroscopic number of solutions
corresponding to as many energy minima. While the mean field estimate for the
2
Note that in [8], the authors rule out the existence of Néel order for lattices of coordination
number z = 3.
128
critical spin (transition temperature) already indicates that the minimal S = 3 model
on the pyrochlore lattice is disordered, the large number of competing states indicate
that the true boundary between disorder and some form of order lies at much larger
values of spin. Indeed, a basic simulation leads to a conservative bound in which
disordered ground states persist up to S = 15. Given the unphysical complexity
of the AKLT Hamiltonians at such large spins we do not pursue a more precise
determination of this boundary in this work. Indeed, readers may take as the main
fruits of our work the identification of the S = 2 AKLT model on the diamond lattice
and the S = 3 AKLT model on the pyrochlore lattice as (not too common) instances
of three dimensional spin Hamiltonians with quantum disordered ground states.
It is worth noting that models with quantum disordered ground states are currently objects of intense interest in the context of topological order and more specifically, in the context of topological quantum computing. We note that our disordered
models do not yield topologically ordered states; they do not posess a topological
degeneracy or host fractionalized excitations. The disordered states herein are described either as fully symmetric valence bond solids or, in the long wavelength sense,
as quantum paramagnets. To understand why this is the case, it is instructive to
recall how a closely related strategy works to produce topologically ordered states in
S = 1/2 models, including instances in d = 3. This strategy, initiated by Chayes,
Chayes and Kivelson [46], and brought to fruition in work by Raman, Moessner and
Sondhi (RMS) [195] works with spin- 21 analogs of the AKLT models called Klein models [122]. Unlike AKLT models, Klein models have many ground states—indeed they
select the macroscopically many nearest neighbor valence bond coverings of a lattice.
This selection of a degenerate manifold underlies the emergence of topological order.
More precisely, the work of RMS showed that Klein models could be controllably perturbed on a family of lattices in order to select a topologically ordered (RVB) state
in this ground state manifold. In this fashion they could construct SU (2) symmetric
129
models with Z2 topological order in d = 2 but also models with Z2 and U(1) order in
d = 3 3.
The rest of this chapter is organized as follows: In Section II, we present a brief
summary of the AKLT construction. We then proceed in Section III to review the
mean field analysis of the AKLT states [16]. We then specialize in Section IV to
bipartite lattices, and compute the transition temperature for the simple cubic and
diamond lattices using Monte Carlo simulations. We determine that while the simple
cubic lattice exhibits Néel order for all choices of M (and thus S), the diamond
lattice allows a quantum disordered state in the M = 1 (S = 2) case. We then go
on to discuss the AKLT states on the frustrated 3D pyrochlore lattice, and discuss
the mean field analysis and classical ground states in this case. We find that the
pyrochlore lattice admits quantum disordered states for many values of M; while
the exact value of Mc was not determined, we find evidence from simulations that it
exceeds 5, corresponding to S = 15.
6.2
AKLT States: A Brief Review
The central idea of the AKLT approach [8], is to use the idea of quantum singlets
to construct correlated quantum-disordered wavefunctions, which are eigenstates of
local projection operators. One can then produce many-body Hamiltonians using
projectors that extinguish the state, thereby rendering the parent wavefunction an
exact ground state, typically with a gap to low-lying excitations. A general member
of the family of valence bond solid (AKLT) states can be written compactly in terms
of Schwinger bosons [17]:
|Ψ(L ; M)i =
3
Y
b†i↑ b†j↓
hiji
−
b†i↓ b†j↑
M
|0i .
See also the closely related work on an XXZ model in d = 3, in [96].
130
(6.1)
This assigns M singlet creation operators to each link hiji of a lattice L . The total
boson occupancy per site is given by zM, where z is the lattice coordination number,
and the resultant spin on each site is given by S = 12 zM. Thus, given any lattice, the
above construction defines a family of AKLT states with S = 21 zM, where M is any
integer. The maximum possible spin on any link is then Sijmax = 2S −M, and therefore
|Ψ(L; M)i is extinguished by any Hamiltonian constructed out of projectors PJ (ij)
onto link spin J, provided 2S − M + 1 ≤ J ≤ 2S. The projectors, which transform
as SU(2) singlets, may be written as polynomials in the Heisenberg coupling Si · Sj
of order 2S. Explicitly, one has
2S
Y
Si · Sj + S(S + 1) − 21 J ′ (J ′ + 1)
.
PJ (ij) =
1
J(J + 1) − 12 J ′ (J ′ + 1)
2
J ′ =0
(6.2)
(J ′ 6=J )
The AKLT states have a convenient representation in terms of SU(2) coherent
states, as first shown in [17]. In terms of the Schwinger bosons, the normalized spin-S
coherent state is given by |n̂i = (p !)−1/2 (zµ b†µ )p | 0 i, where p = 2S, with z = (u , v)
a CP 1 spinor, with u = cos(θ/2) and v = sin(θ/2) eiϕ . The unit vector n̂ is given by
na = z † σ a z, where œ are the Pauli matrices. In the coherent state representation,
Q
the general AKLT state wavefunction is the pair product Ψ = hiji (ui vj − vi uj )M .
Following [17], we may write |Ψ|2 ≡ exp(−Hcl /T ) as the Boltzmann weight for a
classical O(3) model with Hamiltonian
X 1 − n̂i · n̂j ,
Hcl = −
ln
2
(6.3)
hiji
at temperature T = 1/M. All equal time quantum correlations in the state |Ψi may
then be expressed as classical, finite temperature correlations of the Hamiltonian Hcl .
The consequences of this exact quantum-to-classical equivalence, which is a general
feature of Jastrow (pair product) wavefunctions, were noted in [17]. This represen131
tation is also useful in establishing exact results, such as the existence of a unique
infinite-volume ground state on the honeycomb lattice [114].
On one and two-dimensional lattices,the Hohenberg-Mermin-Wagner theorem precludes long-ranged order4 at any finite value of the discrete quantum parameter M.
Thus, while the S = 2 Heisenberg model on the square lattice is rigorously known to
have a Néel ordered ground state [174], the S = 2 AKLT Hamiltonian, which includes
up to biquartic terms, has a featureless quantum disordered ground state, called a
‘quantum paramagnet’. In three dimensions, we expect Néel order for large M, corresponding to low temperatures in the classical model. If the Néel temperature for
Hcl on a given lattice satisfies Tc > 1, then Mc < 1, and all the allowed AKLT states
on that lattice exhibit long-ranged order.
The issue of whether or not the AKLT states can be in the quantum disordered
phase on a given lattice can be investigated by a combination of mean-field calculations and classical Monte Carlo simulations, which we present below.
6.3
Mean Field Theory
A mean field analysis of the classical model of eqn. 6.3 on bipartite lattices was
described in Refs. [17, 16]. In the general case, we may begin with the Hamiltonian
of eqn. 6.3, and we write n̂i = mi + δn̂i , with hn̂i i = mi . Expanding Hcl to order δn̂i ,
P
we obtain the mean field Hamiltonian H̃ MF = E0 − i hi · n̂i , where the mean field
hi is given by
X′
hi = −
j
mj
,
1 − mi · mj
4
(6.4)
A rigorous proof of this theorem for classical spin systems may be found in [153]. While it does
not address this particular interaction, the point of view we take is that if one chooses an ordered
ground state and examines the effect of fluctuations, the result is that the long-wavelength modes
destroy order, thereby precluding symmetry-breaking; while we do not have an explicit proof of
this statement, the physical motivations seem reasonable, and it is in this sense that we invoke the
theorem.
132
where the prime restricts the sum on j to nearest neighbors of site i. Self-consistency
then requires
R
dn̂i n̂i exp hi · n̂i /T
mi = hn̂i i = R
,
dn̂i exp hi · n̂i /T
which yields mi = mi hi |hi |, with local magnetization
mi = coth
6.4
hi
T
−
T
hi
.
(6.5)
(6.6)
Unfrustrated Lattices: Simple Cubic and Diamond
6.4.1
Mean Field Transition
On an unfrustrated, bipartite, lattice a sublattice rotation n̂i → ηi n̂i , with ηi = ±1 on
the A (B) sublattice, results in a ferromagnetic interaction, and if we posit a uniform
local magnetization m we obtain the mean field
h=
zm
,
1 + m2
(6.7)
where z is the lattice coordination number. This results in a mean field transition
temperature TcMF =
1
z,
3
i.e. McMF = 3z −1 . All AKLT states on bipartite lattices
in more than two space dimensions will exhibit two sublattice Néel order, provided
M > Mc . According to the mean field analysis, long ranged order should pertain for
z ≥ 3, which would be satisfied by almost any three-dimensional structure. However,
mean field theory ignores fluctuations, hence it overestimates Tc and underestimates
Mc . Therefore the possibility remains that a quantum disordered AKLT state may
exist in a three-dimensional lattice. We examine two cases, the simple cubic lattice
(z = 6) and the diamond lattice (z = 4). We shall address this issue via classical
133
Monte Carlo simulations of our model on both lattices. We note that, of these,
the diamond lattice is the stronger candidate as it is more weakly coordinated, and
McMF =
3
4
is sufficiently close to threshold that fluctuations are likely to drive the true
Mc to be greater than unity.
6.4.2
Monte-Carlo Simulations
The classical Hamiltonian Hcl of eqn. 6.3 consists of nearest neighbor interactions
v(ϑij ) where ϑij = cos−1 (n̂i ·n̂j ) is the relative angle between spins on neighboring sites
i and j, and v(ϑ) = − ln sin2 21 ϑ . This interaction strongly suppresses ferromagnetic
alignment, with a logarithmically infinite barrier, but has a smooth quadratic minimum v(ϑ) ≈ 41 (ϑ − π)2 when ϑ ≈ π. We have simulated the equivalent ferromagnetic
model, with interaction v(ϑ) = − ln cos2 12 ϑ .
We used a multithread Monte Carlo approach, in which simultaneous simulations
with independent initial configurations were used to produce M independent Markov
chains each with N configurations [37], which were then written to a file. For every
independent thread, we performed checkerboard sweeps of the lattice using a standard
Metropolis Monte Carlo technique [190]. In each Monte Carlo step, we produced a
vector δn̂, with length distributed according to a Gaussian and pointing in a random
direction, which was used to generate a new spin unit vector
n̂′i =
n̂i + δn̂
|n̂i + δn̂|
(6.8)
The standard deviation of the Gaussian was adjusted by hand until a significant
fraction of proposed moves were accepted (we left it at σ = 0.5.)
The number of Monte Carlo steps per site (MCS) and the number of independent
threads were adjusted to count roughly the same number of autocorrelation times
134
for each sample size 5 . For each chain, we obtained the average value of the Binder
cumulant, and averaged this across chains to get a single number for each temperature. We estimated the error from the standard deviation of the M independent
thread averages. This is free of the usual complications of correlated samples inherent in estimating the error from a single chain, and frees us of the need to compute
autocorrelation times to weight our error estimate.
Plots were made of the Binder cumulant [29], defined to be
2 2
(M )
B = 1 − 2 ,
3 M2
where M =
P
i
(6.9)
n̂i is the total magnetization.
For any system of Heisenberg spins in the thermodynamic limit, the Binder cumulant has value
2
3
in the low-temperature (ordered) phase and value
4
9
in the high-
temperature phase. These are easily seen by assuming a gaussian distribution for |M|
at high temperature, and using the result that all the expectation values of powers
of M · M are equal in the ordered phase. For a finite system, the limiting values
continue to be close to these estimates, but the interpolating behavior is different for
each system size; the primary utility from our point of view is that finite-size scaling
analysis of B reveals that it has a fixed point at the transition temperature [29]. We
may therefore determine Tc by plotting the Binder cumulant for a series of different
lattice sizes, and determining the points where the curves cross.
Before simulating our modified interaction, we checked our code by determining
the (known) transition temperatures for the standard Heisenberg model on the simple
cubic lattice [190] and the Ising model on both the diamond and the simple cubic
5
System configurations were recorded after each lattice sweep, so 1 MCS is the natural unit of time
along the Markov chains. A precise determination of the autocorrelation time was not performed,
but plots of the error estimate were made for blocks of increasing length and initial position along the
chain, which allowed us to check the convergence of physical quantities; the final block, consisting
of the latter half of the chain, was used to perform averages in each thread.
135
Disordered
Néel
Disordered Sc = 1.83
Sc = 2.38
Néel
S = 2M
S = 3M
3
6
9
12
2
15
4
6
8
10
Figure 6.1: Transition temperature for classical companion models
Top: Binder cumulant plots for valence-bond states on the cubic and diamond lattices.
The T -axis scale is chosen using a rough estimate of Tc so as to provide approximately
the same window in natural units T /Tc for both cases. In each case, the total number
of spins being simulated is 2 · L3 . We perform a fit of the data (weighted by the error
bars) to a parabola and estimate Tc from the intersection of the best-fit lines. We
can be reasonably confident that the curves have an intersection from the fact that
they change order on either side of the crossing, and become separated by more than
a standard deviation as we move away from the crossing. We obtain Tc ≈ 1.66 for
the cubic lattice and Tc ≈ 0.85 for diamond.
Bottom: Phase diagrams for AKLT models inferred from classical Monte Carlo simulations of |ΨAKLT |2 . Red shading denotes disordered phases, blue denotes Néel order,
and the ticks on the axis correspond to values of M for which an AKLT state can be
constructed.
136
lattices [67], as well as comparing the high-temperature susceptibility from simulations
to the predictions of the high-temperature expansion [222]. All these agreed well with
the expected values, at least to the accuracy we need to determine whether Tc is less
than or greater than 1. Recall that if Tc < 1, then Mc > 1, which means that the
minimal AKLT state, with M = 1, is on the disordered side of the phase transition.
Using our Monte Carlo simulations, we obtain estimates of Tc for the families
of AKLT states on the simple cubic and diamond lattices (see Fig. 6.1). Although
our simulation techniques were not particularly sophisticated, they were sufficient to
pin down Tc to a reasonable degree of accuracy, and certainly enough to determine
whether Tc > 1. Our simulations allow us to estimate that TcSC ≈ 1.66 on the simple
cubic lattice, and that TcD ≈ 0.85 for the diamond structure. Therefore, we conclude
that while all the simple cubic AKLT states are Néel-ordered, the minimal (S = 2)
AKLT state in diamond is a featureless quantum disordered state.
6.5
Frustrated Lattice: The Pyrochlore
The pyrochlore is a lattice of corner-sharing tetrahedra and can be constructed from
the the diamond lattice by placing a site at the midpoint of each bond, resulting in a
quadripartite structure (Fig. 6.2). The pyrochlore lattice is highly frustrated from the
perspective of of collinear antiferromagnetism; the canonical nearest-neighbor classical
Heisenberg antiferromagnet on this lattice has an extensive ground-state degeneracy
and remains a quantum paramagnet at all temperatures [159].
Our problem has a different form for the interaction and hence the results for the
nearest neighbor problem, which build on the high degree of degeneracy for a single
tetrahedron, do not apply. Indeed, as we discuss below, the logarithmic form of the
interaction energy leads to the selection of a unique single-tetrahedron ground state
up to global rotations. However, the full lattice still exhibits a substantial ground
137
Figure 6.2: The quadripartite pyrochlore lattice.
The lattice is formed from corner-sharing tetrahedra. Alternatively, we can view it as
being constituted from four interpenetrating FCC lattices, corresponding to the four
different colors of sites.
state degeneracy on account of its open architecture indicating an anomalously low
transition temperature which we roughly bound from above by T ≈ 0.2. We now
turn to the details of these assertions.
6.5.1
Single-Tetrahedron Ground States
Numerical minimization on a single tetrahedron finds the lowest-energy configuration
to be one where each pair of spins make an angle ϑij = cos−1 (n̂i · n̂j ) = cos−1 − 13 .
This means that the spins are pointing either towards or away from the corners of
a regular tetrahedron in three-dimensional spin space. We proceed to search for soft
modes, by expanding the energy to quadratic order and studying the resulting normal
modes. We find that there is a pair of soft modes corresponding to global rotations,
and another where three spins rotate about the axis defined by the fourth. The latter
mode leads to a degeneracy of ground states of the full lattice as discussed below.
138
We note that for the Heisenberg antiferromagnet with interaction n̂i · n̂j , the
2
P
single tetrahedron Hamiltonian is HΓ = MΓ , where MΓ = i∈Γ n̂i is a sum of the
spin vectors over all sites in the tetrahedron Γ. The ground state manifold MΓ = 0
is then five-dimensional, since one can choose any two vectors n̂A and n̂B , then take
n̂C = −n̂A and n̂D = −n̂B . The four freedoms associated with choosing n̂A and n̂B are
then augmented by another freedom to rotate the C and D spins about the direction
n̂A + n̂B . A large-N analysis[100] finds that the O(N) pyrochlore antiferromagnet is
paramagnetic down to T = 0.
6.5.2
Ground States on the Full Lattice
There are many ways in which we can construct degenerate states on the lattice
that simultaneously satisfy the minimum-energy constraint on every tetrahedron. We
begin by describing the simplest such states which form a discrete family. To this end,
label the four spins defined by the single-tetrahedron constraint (with a fixed joint
orientation) as A, B, C and D. If we use only these four orientations for each spin, we
have the constraint that none of them can occur twice on the same tetrahedron; this
translates to the statement that spins on neighboring links must be different. This is
the same constraint as for ground states of the antiferromagnetic 4-state Potts model.
We therefore conclude that one family of ground states of the classical Hamiltonian
on the pyrochlore lattice are in a one-to-one correspondence with the ground states of
the 4-state Potts antiferromagnet on the pyrochlore lattice. Readers familiar with the
lore on the kagomé problem[99] will recognize the resemblance to the planar ground
states there which are in correspondence to ground states of the 3-state Potts model.
As in the kagomé problem, from this set of ground states others can be constructed
by identifying sets of spins which can be locally rotated by an arbitrary amount at
zero energy cost. These are sets of spins, say of type B, C and D which are connected
139
to other spins solely by spins of type A. Clearly one can rotate this set by an angle
about the A axis at zero energy cost.
While we have not parametrized the full, continuous, space of ground states an
extensive lower bound on the degeneracy of the ‘Potts submanifold’ of ground states
can be obtained as follows. First, we note that the number of allowed configurations
of the 3-state Potts model on a kagomé lattice with M sites is given [99, 25] by
gk ≈ (1.20872)(2M/3) . Next we partition the pyrochlore into four sublattices, so that
the sites that lie on a single tetrahedron are each on different sublattices. Choose one
sublattice, and fix the spins on that sublattice to be one of the four types (say A.)
Now, looking down through the tetrahedra, one sees alternating layers of triangular
and kagomé planes; the kagomé planes are made up of B, C, and D spins, while
the triangular planes are made up of A spins. In each kagomé plane, we have M
spins, whose configurations are those of the 3-state Potts model. If we now let Nk
be the number of kagomé planes, we must have that M · Nk = 43 N, where N is the
total number of spins in the system. We then have for the number of states in this
restricted submanifold
grestricted = 4 · gkNk ≈ 4 · (1.20872)N/2
(6.10)
where the factor of 4 stems from the fact that we can choose any one of the four spins
to be fixed in the triangular planes. Since we’ve restricted ourselves to considering a
certain submanifold of the ground states in the above argument, it is clear that we
have obtained a lower bound for the degeneracy of the Potts submanifold.
6.5.3
Bounds on Tc
Each of the ground states identified above can serve as a basis for a mean-field treatment of the system and all of them yield the same TcMF . This vast set of “soft modes”
140
is, of course, a signature that the true Tc ≪ TcMF . Thus we may begin with a calculation of TcMF which can serve as an upper bound on the true Tc .
Consider a spin at site i in the pyrochlore lattice. Expanding in small fluctuations
about any ground state, we have the same mean-field Hamiltonian as in the general
mean field Ansatz of section 6.3, with the mean field given by eqn. 6.4. In a mean-field
treatment, each of the neighbor spins n̂j is to be replaced by its average mj = hn̂j i =
mêj in the particular ground state that we are considering. In any ground state, we
note that the angle between any pair of nearest neighbors is ϑij = cos−1 − 13 . In
P
addition, the spins on a tetrahedron add to zero, which allows us to write j ′ mj =
−mi . If we further recall that each spin lies on exactly two tetrahedra, we obtain the
following expression for the mean field acting at site i:
X′
hi = −
j
mj
2m
=
2 êi
1 − mi · mj
1 + m3
(6.11)
Note that we have only made use of the local structure of the ground state, and so
our treatment here is relevant for the transition into any state in the ground state
manifold. Substituting the mean field in eq. 6.11 into the self-consistency condition
(eqn.6.6), we find, in a manner similar to the bipartite case, that the mean-field
estimate of the transition temperature is TcMF = 23 . From this alone we conclude that
the M = 1 state on the pyrochlore lattice is quantum disordered.
There is little question that the actual Tc is much lower than the mean-field
estimate and therefore Mc is much higher than
3
,
2
allowing many more quantum
disordered states. As is familiar from other highly frustrated magnets, where the
ground-state manifold encompasses a vastly degenerate set of states the transition
will be driven by the ‘order-by-disorder’ mechanism wherein a particular state or
subset of states is favored by entropic effects at low temperatures. This is a weak
effect and hence Tc is typically a small fraction of TcMF .
141
While we have not performed an extensive Monte Carlo analysis on the pyrochlore
lattice owing to the complexity of the ground state and the difficulty of defining a
simple order parameter, prior experience with pyrochlore antiferromagnets suggests
that the transition temperature is sufficiently small that the corresponding AKLT
Hamiltonians are rather complicated functions of the spins; thus there is little reason
in the current context to locate the transition or the nature of the ordered phase with
greater precision.
However, we note that the same set of ground states arises in the the classical
Heisenberg model with nearest neighbor bilinear and biquadratic interactions with
the latter chosen to disfavor collinearity. This is a physically plausible model and
warrants further investigation, which we shall defer to future work.
6.6
Concluding Remarks
To summarize, we have studied AKLT states on two unfrustrated and one frustrated
lattice in d = 3 by a combination of mean-field theory and Monte Carlo simulations
for the associated classical models. We find that the simple cubic lattice is Néel
ordered at all values of the singlet parameter M and spin S; the diamond lattice,
on the other hand, is quantum-disordered for M = 1 (S = 2), and Néel ordered for
M > 1. On the pyrochlore lattice we find that the M = 1 (S = 3) model is definitely
disordered and the boundary between disorder and order very likely lies above M = 5.
While quantum-disordered ground states in low (i.e. one and two dimensions)
have often been discussed, three dimensions has historically been the province of long
range order. Hence our disordered models on the diamond and pyrochlore lattices
significantly expand the set of possibilities for quantum ground states of models with
Heisenberg symmetry in d = 3.
142
In recent work, one of us has generalized the AKLT construction to SU(N) spins
[16]. In the near future, we intend to investigate the SU(4) simplex state on the
pyrochlore lattice introduced in this work by methods similar to the ones used in this
chapter.
143
Chapter 7
AKLT Models with Quantum Spin
Glass Ground States
7.1
Introduction
The initial construction of the AKLT models was motivated by the search for quantum
disordered states in low dimensions. This works only too well, as we have seen: in
d = 1 and d = 2 the mapping to finite temperature classical models ensures, by the
Mermin-Wagner theorem, that all cases lead to quantum paramagnetic or valence
bond solid ground states. In d > 2 this is no longer true and a computation was
needed to decide which models order and which do not. In the previous chapter, we
showed showed via Monte Carlo simulations and mean-field arguments that AKLT
models on the diamond and pyrochlore lattices exhibit quantum-disordered ground
states for small spin sizes while on the cubic lattice all models exhibit Néel order.
In this chapter we take the exploration of higher dimensional AKLT models in a
different direction—we study them on locally tree-like lattices of fixed connectivity z,
which are known to physicists as Bethe lattices. Here we shall consider two physically
distinct systems. The first is the Bethe lattice constructed as the limit of a family of
144
Figure 7.1: Different realizations of the Bethe lattice.
(Left) Cayley tree and (Right) Regular random graph, both with coordination z = 4.
Cayley trees. This construction yields a system with a finite surface to volume ratio
and without loops. The second is a typical member of the ensemble of random graphs
of fixed connectivity. These graphs are locally tree-like in the thermodynamic limit;
however they also have long loops of logarithmically divergent size. These loops of
both even and odd lengths introduce topological frustration into the system. The
two constructions of locally tree-like lattices yield different physics. Representative
examples of both cases for a small system size are shown in Fig. 7.1.
For the infinite Cayley tree, we exhibit an exact solution using the quantumclassical correspondence. Specifically, we use a generalized transfer matrix technique
to obtain exact solutions for various statistical quantities in the ground state of the
tree. We note that the AKLT model on the Bethe lattice has been studied before
directly within the quantum formalism [60]; we suspect that readers will find our
solution simpler. We find one quantum disordered state (M = 1 on the z = 3 tree)
and two that are critical (M = 2 on z = 3 and M = 1 on z = 4), in that the
correlation functions decay exponentially at precisely the rate required to balance
145
the exponential growth of the graph. All other cases exhibit Néel order. We address
the question of whether the bulk excitations are gapless in cases when the AKLT
wavefunction has critical or Néel correlations. We find, perhaps surprisingly, that the
system is always gapped to local excitations, and that the only gapless excitation is
a global one connecting the different broken-symmetry ground states. We connect
this to related work in [133] (reproduced in Appendix 8 of this thesis), in which we
conjecture that this is a generic feature of symmetry breaking quantum models on
the Bethe lattice, related to the spectrum of the graph Laplacian.
On random graphs of fixed connectivity, Néel ordering in the companion classical
model is frustrated by the presence of the long loops. To study this case, we appeal to
the cavity techniques familiar from the theory of classical disordered systems. These
have been applied recently to a variety of discrete statistical mechanical problems on
random graphs and there is much evidence that the (approximate) techniques are on
solid ground. From this analysis, we conclude that for z ≤ 10 there are disordered
states at small spin and spin glass ground states at larger spin as well as a couple of
cases where the state is critical. For z > 10 all AKLT models have ground states with
spin glass order. By spin glass order, we mean states with fixed but randomly oriented
local magnetizations and that the set of such states is larger than those connected
by global rotations alone. We argue that the spectrum of local excitations above the
pure states in this set is again gapped.
Of our various results we would especially like to flag these last mentioned. The
nature of quantum glass phases is a subject of much interest – especially as to how
much of the elaborate framework of the classical subject may be lifted into the quantum world. The AKLT construction provides a direct line of approach to this problem
and does so using Hamiltonians without random couplings but from graph disorder
alone.
146
This chapter is organized as follows: in Section 7.2, we introduce the AKLT model
on an arbitrary graph via the Schwinger boson formalism. We proceed to construct
a companion classical model that captures the structure of the ground state wavefunction by introducing a basis of SU(2) coherent states. In Section 7.3, we develop
transfer-matrix technology to solve the companion classical model on the (bipartite)
Bethe lattice exactly, and obtain the transition temperature and correlation functions
in the paramagnetic and Néel-ordered phases. In Section 7.4 we investigate the energy gap using a variational ansatz for the excited states. Finally, in Section 7.5 we
consider the extension of this analysis to the spin glass transition expected on regular
random graphs and consider some of the quantum consequences of the classical glassy
phase.
7.2
AKLT and arbitrary graphs
As is evident from (6.1), we may define the AKLT states on an arbitrary graph; if the
graph has fixed connectivity z, then the resulting model has the same spin on each
site. On graphs with a boundary, this is not automatic, since the boundary sites will
have fewer neighbors z ′ . There are several ways to deal with this boundary effect.
The first is to work with a system with a lower spin on the boundary: in that case
S ′ = 21 z ′ M. The quantum state of this non-homogeneous system is unique. Another
option is to add (z − z ′ ) additional Schwinger bosons of either flavor to the edge sites;
there is not a unique way in which to do this, leading to a multitude of degenerate
ground states classified by the state of the boundary spins. When translated to the
companion classical model, the latter option can be viewed as connecting each of
the boundary spins to (z − z ′ ) fixed spins, each with an orientation specified by the
behavior of an independent spin-1/2 degree of freedom; thus, the different degenerate
states of the homogeneous AKLT model on a graph with boundary can be understood
147
by choosing different fixed boundary conditions for spins in an additional, outer ring
of leaf spins. Finally, one can opt to get rid of the boundary by, for instance, taking
periodic boundary conditions on a Euclidean lattice. As we will discuss in Sec. 7.5,
when generalized to tree-like graphs, this approach leads to a spin glass phase in
the companion model and thus provides a non-trivial new quantum spin glass to the
AKLT phase diagram.
7.3
Transfer Matrix Solution of the Classical
Problem on Trees
The classical Hamiltonian that describes the properties of the ground-state wavefunction of an AKLT model with singlet index M, written in the basis of SU(2) coherent
states is given by
βHcl = −M
X
hi,ji
1 − n̂i · n̂j
log
2
(7.1)
Before we proceed, we note that on bipartite graphs we can perform a gauge
transformation by flipping every spin at odd depth to obtain a ferromagnetic model.
This gives us
βHcl = −M
X
hi,ji
1 − n̂i · n̂j
log
2
(7.2)
As usual in the treatment of tree models, we consider the statistical state ψ 0 (n̂0 )
(marginal distribution) of a cavity spin n̂0 at the root of a branch of the tree. This
unnormalized distribution can be found in terms of the cavity states of its z − 1
148
neighbors by summation:
0
ψ (n̂0 ) =
=
Z
Dn̂1 · · · Dn̂z−1 T (n̂0 , n̂1 )ψ 1 (n̂1 ) · · · T (n̂0 , n̂z−1 )ψ z−1(n̂z−1 )
Z
Dn̂′1 · · · n̂′z−1 M(n̂0 ; n̂′1 , · · · n̂′z−1 )
Z
× Dn̂1 · · · Dn̂z−1 T (n̂′1 , n̂1 )ψ 1 (n̂1 ) · · · T (n̂′z−1 , n̂z−1 )ψ z−1 (n̂z−1 )(7.3)
where
β log
T (n̂0 , n̂1 ) = e
h 1−n̂
i ·n̂j
2
i
=
1 − n̂i · n̂j
2
β
(7.4)
is the transfer matrix of the AKLT model and
M(n̂0 ; n̂′1 , · · · n̂′z−1 ) = δ(n̂0 − n̂′1 ) · · · δ(n̂0 − n̂′z−1 )
(7.5)
is the merge matrix.
The merge matrix M defines a multilinear map from the z − 1 state spaces of the
neighbor spins to the state space of the root. This lifts naturally to the appropriate
complexified tensor product spaces1 and thus we will find it natural to write the merge
and transfer operations abstractly using Dirac notation:
M =
=
Z
Z
Dn̂0 Dn̂′1 · · · Dn̂′z−1 δ(n̂0 − n̂′1 ) · · · δ(n̂0 − n̂′z−1 )|n̂0 ihn̂′1 | · · · hn̂′z−1 |
Dn̂ |n̂ihn̂| · · · hn̂|
(7.6)
1
A few comments on the nature of the classical statistical state space of a vector spin are in order,
as we have so cavalierly complexified and tensored it into a much more quantum mechanical looking
system. Physical cavity distributions ψ(n̂) must be real, normalizable, nonnegative functions on
the sphere. By its construction as a marginalization (summing out) procedure, equation (7.8) must
produce such a physical output given physical inputs, even though we have extended it over C. A
more important subtlety arises in the normalization of states – the standard L2 norm associated
with the Dirac inner product is not necessarily 1 for a properly normalized probability distribution.
Since the probabilistic L1 norm is incompatible with the Hilbert space structure, it is much simpler
to work with unnormalized vectors and keep in mind that a probabilistic interpretation only applies
in the standard basis.
149
and,
T =
Z
Dn̂Dn̂′ T (n̂, n̂′ ) |n̂ihn̂′ |.
(7.7)
Thus, equation (7.3) becomes
|ψ 0 i = M T |ψ 1 i ⊗ · · · ⊗ T |ψ z−1 i .
(7.8)
We now focus on the stability of the paramagnetic state against Néel ordering.
Hence, we have to use boundary conditions that are consistent with this kind of
ordering. As discussed at the end of Section 7.2, one can either use free boundary
spins with lower S, or connect the boundary spins to some fixed additional spins: in
the latter case, the additional spins must all have the same orientation to allow the
Néel ordering. This remark is particularly important because, as we will see later
in Section 7.5, on a random regular graph the boundary conditions on any given
tree-like subregion are fixed self-consistently, and in general are not consistent with
Néel ordering, leading to a disordered spin glass state. Assuming uniform boundary
conditions, we obtain the unique state at depth d − 1 by merging the z − 1 states at
level d using the T and M operators:
|d − 1i = M(T |di)⊗(z−1) .
(7.9)
The natural basis to work in is that of states with definite angular momentum,
i.e. states |l mi, which are eigenstates of the angular momentum operators L2 , Lz .
In the coordinate basis, these are simply the spherical harmonics, and as shown in
Appendix B, they are eigenstates of the transfer matrix with eigenvalue λl .
It remains for us to understand exactly how the merge operation acts in the
angular momentum basis. If we insert resolutions of the identity in the angular
150
momentum basis into (7.6), we obtain
M =
=
=
Z
Z
Dn̂ |n̂ihn̂| · · · hn̂|
Dn̂
X X
l0 ,m0 l1 ,m1
X X
l0 ,m0 l1 ,m1
×
Z
···
···
X
lz−1 ,mz−1
X
lz−1 ,mz−1
|l0 , m0 ihl0 , m0 |n̂i (hn̂|l1 , m1 ihl1 , m1 | ⊗ · · ·
· · · ⊗ hn̂|lz−1 , mz−1 ihlz−1, mz−1 |)
|l0 , m0 i (hl1 , m1 | ⊗ · · · ⊗ hlz−1 , mz−1 |)
m
Dn̂ Yl0m0 ∗ (n̂)Yl1m1 (n̂) · · · Ylz−1z−1 (n̂)
(7.10)
For the case z = 3, the integral in (7.10) is simply the Clebsch-Gordan coefficient
that characterizes the fusion of two SU(2) spins. For higher values of z, this is the
appropriate generalization of the Clebsch-Gordan coefficient describing the fusion of
(z − 1) SU(2) spins. Thus we see that the merge operation, when written in the
angular momentum basis, has a natural interpretation as the fusion rules for the
O(3) symmetry group.
The paramagnetic state - here represented in Fourier space by the |00i state
- is always a fixed point: it is an eigenstate of the T -matrix, and the fusion of
any number of |00i states is again a |00i state. We proceed via linear stability
analysis: we introduce a perturbation into a state that is not uniformly weighted
on the sphere, and see if this grows or shrinks under the iteration procedure. We
note that we can decompose any such state into spherical harmonics, and so we write
P
|di = |00i + ε ∞
l6=0,m clm |l mi into (7.9) to obtain
|d − 1i = M
λ0 |00i + ε
= λz−1
0 |00i
+ε(z − 1)λ0q−2
151
!⊗(z−1)
X
λl clm |l mi
X
λl clm |l mi + O(ε2 )
l6=0,m
l6=0,m
(7.11)
where we have used the fact that fusing any number of |00i states with an |l mi state
results in an |l mi state. We renormalize to leading order and find that the iterated
state is, to linear order
|d − 1i = |00i + ε(z − 1)
X λl
clm |l mi + O(ε2 )
λ0
l6=0,m
(7.12)
The perturbation is irrelevant (shrinks under iteration) if the coefficient of the
linear term is less than 1, and relevant if it is greater than 1. The critical point is
reached when
λl
1
=
λ0
z−1
(7.13)
for any l. Using the temperature dependent expression (B.4) for the λl , one can show
that the dipole instability (l = 1) is the first one encountered as the temperature is
lowered, and therefore sets the transition temperature.
Using the results of Appendix B (replacing β by the singlet parameter M), we
obtain
Mc =
2
z−2
(7.14)
We see that for z = 2, 3, 4, Mc = ∞, 2, 1, while for all other values, Mc < 1. Since M
must be a positive integer, we see that for the chain (z = 2) all values of M correspond
to quantum-disordered states (which follows from the Mermin-Wagner theorem and
the original AKLT result [9]) whereas for z = 3, the M = 1 state is disordered while
the M = 2 state is critical, and finally for z = 4, the M = 1 state is critical. Bethe
lattices of higher connectivity will always have ordered AKLT ground states for any
value of M. See Fig. 7.5.
Finally, we consider the correlation function hn̂0 · n̂d i within the paramagnetic
phase. This is given by considering the response of hn̂0 i to a field on n̂d – which is the
same as asking how the dipole l = 1 perturbation propagates along a chain of length
152
d in a background of trivial l = 0 cavity states. This immediately implies
hn̂0 · n̂d i ∝
λ1
λ0
d
=
M
M +2
d
(7.15)
Notice that this implies that the naive correlation length never diverges – as usual
with tree models, phase transitions occur when the correlation decays at the same rate
as the growth of the lattice. For a slightly more detailed calculation, see Appendix C.
As an aside, we note that we can use the same generalized transfer matrix technique to obtain the transition temperature for the Heisenberg model, a result first
obtained by Fisher [66] using a different method. This serves as a test of the technique
proposed here.
7.4
Variational Bounds on the Gap
We now perform a variational computation of the gap to excitations in the critical
model, similar to the Single-Mode Approximation (SMA) discussed in [17]. The
central idea of the SMA is to construct an excitation orthogonal to the ground state
by acting on it with a local operator, and then to reduce the energy of this excitation
by delocalizing it, thereby decreasing its kinetic energy. A variational bound on the
energy gap is given by
0 ≤ ∆ ≤ ∆SM A =
hΨSM A |H − E0 |ΨSM A i
hΨSM A |ΨSM A i
(7.16)
This approach is designed to optimize the energy due to the off-diagonal matrix
elements in the excited sector, which will be proportional to the usual graph Laplacian
for a nearest neighbor model. On the Bethe lattice, the spectrum of the Laplacian is
unusual. As argued in [133] and Appendix 8, there is necessarily a gap to hopping excitations on tree-like graphs despite the existence of symmetry related ground states.
153
Thus, in some sense the SMA calculation is doomed to failure as it will never be able
to close this gap. Nonetheless, it is interesting to see how this plays out in an exact
model.
We consider a rooted Cayley tree, and in order to restrict ourselves to studying
excitations confined to the bulk as the size of the tree grows we suppress excitations
far from the center using an infrared regulator λ. We therefore study variational
wavefunctions of the form
|ΨSMA i = |λi =
N
X
i=1
uλ (i)Siz |Ψi
(7.17)
where uλ (i) is a function only of the depth ρi of site i referenced to the root of the
Cayley tree. The SMA gap is
∆SM A
hλ|H − E0 |λi
= lim lim
λ→0 N →∞
hλ|λi
(7.18)
After some algebra, we may write (with the understanding that we always take N to
infinity before taking λ to zero):
∆SM A = lim
N →∞
λ→0
"P
uλ (i)uλ (j)h[Siz , [H, Sjz ]]iΨ
P
z z
i,j uλ (i)uλ (j)hSi Sj iΨ
i,j
#
(7.19)
Asserting homogeneity of the graph, and noting that the Hamiltonian is a function
only of Si · Sj where i and j are nearest neighbors, we may re-express this as
∆SM A = lim
N →∞
λ→0
#
P
i,j uλ (i)Aij uλ (j)
f×P
|i−j| u (j)
λ
i,j uλ (i)γ
"
(7.20)
where f = 41 h[Siz − Sjz , [H, Siz − Sjz ]]iΨ , and we denote by A the adjacency matrix of
the Cayley tree; the correlations in the ground state are always exponential, and go
as γ |i−j| , where γ →
1
z−1
from below as we approach criticality. We have therefore
154
reduced the problem of the SMA on the Bethe lattice to understanding (i) the spectrum of the graph Laplacian (the adjacency matrix up to a sign) and (ii) the behavior
of the ground-state correlations.
2
Our choice for a variational ansatz is to take uλ(i) = e−λρi . This is motivated by
the fact that the number of sites at a given distance from the center grows exponentially, and therefore in order to remain in the bulk of the tree, we need to cut off the
wavefunction faster than exponentially2 . We perform the summations by converting
the sum over sites into a sum over depths, approximating the sums by integrals and
using steepest-descent. We find that, at criticality, the gap is nonvanishing:
∆G
SM A
#
"
√
z−2 z−1
f
√
∼ z
8 z − 1 + 4 z−1
log (z−1)
(7.21)
Excitations constrained to live in the bulk are therefore always gapped, even at criti√
cality. The factor of z − 2 z − 1 is precisely the spectral gap for bulk excitations on
a Cayley tree, as discussed in Appendix 8. A state where uλ (i) is independent of position must be gapless in the broken-symmetry phase of the model, since it connects
the different broken-symmetry ground states. We cannot recover this state through
a correctly regulated calculation in the chosen order of limits, however.
There is a straightforward physical argument for this gap. By our choice of vari√
ational ansatz, we cut off the excitation at some depth D ∼ 1/ λ. On a Euclidean
lattice of dimension d, this costs a surface energy ∝ D d−1 which is normalized by the
weight of the wavefunction ∝ D d . As D → ∞ (λ → 0), the SMA gap therefore vanishes as D −1 . On tree-like lattices, both the surface area and the bulk normalization
scale as (z − 1)D ; the boundary is always a finite fraction of the bulk. Thus, the ratio
remains finite as D → ∞ and the gap cannot close [98, 133].
2
In principle, an exponential regulator with a sufficiently fast decay constant also works. The
calculation proceeds in a similar fashion but the interpretation is more complicated and no more
enlightening.
155
7.5
AKLT model on regular random graphs
We now consider the same model on a regular random graph [106] of connectivity z.
The ensemble of these graphs is constructed by assigning uniform probability to all
possible graphs of N vertices, such that each vertex is connected to exactly z links.
There are several reasons why statistical models defined on this ensemble of graphs
are interesting:
1. A central property of this ensemble [106] is that typical lattices are locally
tree-like; their loops have a length diverging logarithmically with the size N
of the system; this implies that one can develop a method to solve statistical
models on these graphs based on the same recurrence equations that are exact
on trees. This is known as the cavity method [156].
2. Despite being locally tree-like, they do not have any boundary, all sites playing
statistically the same role (in the same way as periodic boundary conditions
impose translation invariance on a finite cubic lattice). Moreover, the freeenergy of regular random graph models is self-averaging with respect to their
random character in the thermodynamic limit. In other words for large enough
N a single sample is a good representative of the ensemble average.
3. Typical graphs are characterized by many large loops of even and odd length;
this strongly frustrates the antiferromagnetic ordering, which gives way to a
spin glass phase instead3 .
In the following we will be particularly interested in the implications of the last point
for the quantum problem.
The reasoning outlined in section 7.2 clearly applies to the random graph model,
whose AKLT ground state is therefore described by a classical Hamiltonian of the
3
See [156] for a general discussion and [126] for the explicit computation of the phase diagram of
a classical Ising antiferromagnet.
156
form (7.1), where the pairs hi, ji are connected by a link of the random graph. The
main difference between the tree model and the random graph model is that the
recurrence equation (7.3) now does not hold for the full graph; it only holds for a
tree-like subregion of the graph, and has to be initialized using the boundary values
of the ψ i (n̂i ) that are determined by the summation over the rest of the graph. In
other words, the recurrence on the subregion is initialized from random self-consistent
boundary conditions, determined by the rest of the system: these boundary conditions
are not consistent with Néel ordering, which is therefore frustrated, as discussed in
Section 7.3 above. However, since the tree-like subregions grow in size when N → ∞,
equation (7.3) must be iterated a very large number of times. One can classify the
different phases of the system by studying its fixed points [156, 155].
To calculate the stability of the paramagnetic solution against spin glass ordering
we observe that the spin glass transition is signaled, as usual, by the divergence of
the classical spin glass susceptibility [31]:
χSG =
1 X
[hn̂i · n̂j i]2 .
N ij
(7.22)
The details are discussed in Appendix C; the result is that χSG is finite if for all l
λl
1
≤√
.
λ0
z−1
(7.23)
Once again the instability originates in the l = 1 sector and occurs at
MSG = √
2
z−1−1
(7.24)
We see that MSG = ∞ for z = 2, MSG = 4.828 for z = 3 (hence the system is a spin
glass for M ≥ 5), MSG = 2 for z = 5 and MSG = 1 for z = 10 (in these cases the
system is critical), and it is smaller than 1 for any z > 10. See Fig. 7.5.
157
Figure 7.2: Phases of AKLT models on trees
Phase diagrams for AKLT models with singlet parameter M on tree-like lattices with
coordination z. On the Cayley tree the transition is from paramagnetic (PM) to
Néel-ordered (AF) phase at the solid blue line with no spin glass. On regular random
graphs the transition is from the paramagnetic to spin glass ordered (SG) phase at
the dashed green line — there is no antiferromagnet. The models with Bethe lattice
critical correlations are labeled with large dots.
With the Néel-ordered phase suppressed, the quantum paramagnet extends further in the z − M plane than on the Cayley tree models. Nonetheless, it is clear that
the paramagnetic solution develops an instability to spin glass ordering at large M.
What are the properties of this low temperature phase? Most of the detailed work on
classical spin glasses has focussed on discrete models. The AKLT mapping provides
a classical vector model with weakly divergent interactions whose glass phase has
not yet been studied. Nonetheless, most of the qualitative features of the classical
multiple-valley picture should still hold and these provide an intriguing scenario for
the quantum system. The classical Gibbs measure decomposes into a collection of
clustering pure states α = 1 . . . N with essentially disjoint support. In each of these
states, the ψαi (n̂i ) – and thus the local magnetizations – are macroscopically different.
This strongly suggests that the quantum AKLT ground state |Ψi itself is a super-
158
position over a collection of macroscopically distinct degenerate ground states |Ψα i
each of which corresponds to one of the classical clustering states.
While we believe that the above picture holds in general, a rigorous derivation is
problematic. In the following we will attempt to justify it in more detail and point
out some of the subtleties that must be dealt with. First, there does not yet exist
a detailed study of the classical vector model4 – for the purposes of this chapter, we
shall assume that, modulo the O(3) global symmetry, the qualitative behavior is that
of the better studied Ising antiferromagnet on a regular random graph [126].
By analogy to this model, two different phases exist: at high temperatures, the
stable phase is a paramagnet where ψ i (n̂i ) = ψ(n̂) = 1 is the same for all sites, and
it is the unique fixed point of Eq. (7.3). At low temperatures, the stable phase is a
spin glass, characterized by the existence of many non-symmetry related pure states
labeled by α = 1, . . . , N within which connected spatial correlators vanish. This is
related to the existence of many different fixed points of Eq. (7.3), and reflects the
decomposition of the thermodynamic Gibbs measure P = exp(−βHcl )/Z as follows:
P ({n̂}) =
N
X
wα
α=1
Z
dg Pα ({g · n̂})
(7.25)
where the Pα are representative classical pure state measures. By integrating over
g, the O(3) of global rotations of spin space, we account for the continuous family
of symmetry related pure states associated to each representative state α. One can
access these representative pure states by adding a uniform infinitesimal field to the
classical model, but as the quantum AKLT state is a singlet, we prefer to work without
explicitly breaking this symmetry [31].
The Ising spin glass models with two-body interactions which have been studied,
such as the Sherrington-Kirkpatrick model [157] and the random graph antiferro4
Although there is a replica symmetric treatment of a related model with the additional complication of random interactions. See [48].
159
magnet [126], are characterized by a continuous spin glass transition5 with a finite
collection of pure states throughout the spin glass phase6 . We shall therefore assume
that this is true of our collection of representative pure states. Indeed, all we will
need is that N grows at most polynomially in N as the thermodynamic limit is taken.
The pure state decomposition (7.25) has striking consequences for the structure
of the low-energy states of the quantum AKLT Hamiltonian. To wit, we argue that
|Ψi =
N
X
√
α=1
wα
Z
dg g |Ψα i
(7.26)
up to exponentially small corrections in the thermodynamic limit, where the |Ψα i
can be interpreted as a collection of symmetry breaking degenerate quantum ground
states whose correlations correspond to the classical pure states Pα .
We argue this in three parts. First, we assume the existence of a collection of
quantum states |Ψα i such that
|h{n̂i}|Ψα i|2 = Pα ({n̂i })
(7.27)
and show that the quantum state (7.26) reproduces the observables of the classical
decomposition of (7.25). Second, we will show that up to exponentially small corrections each of the |Ψα i are themselves orthogonal ground states. Finally, we address
the issue of the existence of such states given the classical distributions Pα .
5
Other mean field spin glass models, such as the p-spin model, can have a number of states
scaling exponentially in system size. These models show a discontinuous spin glass transition. However, antiferromagnetic models with two-body interactions usually do not show this phenomenology,
therefore we will not investigate this transition here.
6
See [157], and in particular the reprint on page 226, for a more detailed discussion of this delicate
statement.
160
The first part follows the argument of [32] but we rephrase it in terms of density
matrices. Consider the density matrix of the proposed state (7.26):
ρ = |ΨihΨ| =
X√
wα wβ
α,β
Z
dg
′
Z
dg g ′ |Ψα ihΨβ |g †
(7.28)
Given any local operator Ô depending only on spins7 , its expectation value in state
|Ψi is given by
hÔi = Tr ρÔ
Z
Z
h
i
X√
′
dg Tr g ′ |Ψα ihΨβ |g †Ô
=
wα wβ dg
α,β
=
X√
α,β
wα wβ
Z
dg
′
Z
dg hΨβ |g † Ôg ′|Ψα i
(7.29)
We now argue that the interference term is negligible. That is,
hΨβ |g † Ôg ′ |Ψα i = δαβ δg g′ hΨα |g †Ôg|Ψα i
(7.30)
up to exponentially small corrections in the thermodynamic limit. This follows from
the observation that |Ψα i and |Ψβ i have macroscopically distinct magnetization patterns that completely break the O(3) symmetry. In particular, the classical configurations {n̂i } on which the wavefunction |Ψα i is concentrated have extremely small
weight in any other wavefunction β 6= α – this remains true even with arbitrary global
rotations allowed between them. If α = β but g and g ′ differ, then the configurations
with weight are again macroscopically distinct by virtue of the net global rotation
g −1 g ′ between them. The fact that the observables are local and have bounded matrix
elements does not modify these assertions. Since N is finite for our antiferromagnetic
model, the finite sum over the exponentially small corrections does not modify the
7
Such observables are diagonal in the coherent state basis and therefore their correlations follow
from the classical measure.
161
result:
hÔi =
X
wα
α
Z
dg hΨα |g †Ôg|Ψα i
(7.31)
Inserting a complete set of coherent states and using (7.27) reproduces the classical
distribution (7.25). The probability of finding the quantum system in a state α (with
respect to the state |Ψi) is the same as that of the classical problem: both are given
by wα .
Furthermore, the |Ψα i must have exponentially small energy with respect to the
quantum Hamiltonian. Since hHi = 0 in the AKLT state, using equation (7.31) and
the rotational invariance of H, we have that
0 = hHi =
X
α
wα hΨα |H|Ψαi
(7.32)
up to exponentially small corrections. Since each of the terms in the sum is nonnegative, it follows that
hΨα |H|Ψαi . O(e−N )
(7.33)
Thus, the |Ψα i are a collection of nearly orthogonal, nearly zero energy states, each
of which generates a further continuous collection of such degenerate states under the
action of O(3).
Finaly, we turn to the slightly thorny question of whether states satisfying (7.27)
exist. The problem is that the coherent state basis is overcomplete for any fixed spin
size S = zM/2 and we cannot necessarily find quantum states which have given exp
pansions in this basis. That is, a priori we cannot simply set h{n̂i }|Ψα i = Pα ({n̂i })
and know we have a well-defined quantum state for spins of size S. In the large
spin limit, there is no problem as the coherent states become a complete, rather than
162
overcomplete, basis. This coincides with the zero temperature limit of the classical
companion model and thus the pure states |Ψα i may be identified with the (many
degenerate) minima of the energy function (7.1). At finite M, we cannot find such
finely localized states in the coherent state representation – the most localized state
has solid angular scale ∼ 1/M – but the finite temperature fluctuations around the
classical minima will also smear the Pα at a similar scale.
On the other hand, we already used above that the states have disjoint support, up
to exponentially small corrections in N. For a given classical configuration {n̂i }, only
one state contributes to h{n̂i }|Ψi significantly. Thus, in a given region of classical
configuration space, h{n̂i }|Ψi coincides with one of the h{n̂i }|Ψα i, and conversely,
each of the h{n̂i }|Ψα i can be seen as the restriction of the full h{n̂i }|Ψi to that
region. Since the 1/M smoothing is local in configuration space, it is safe to assume
that the states h{n̂i }|Ψα i are as smooth as the original h{n̂i }|Ψi, therefore |Ψα i can
be defined without ambiguity. Based on the above arguments, we think it likely that
at least an approximate pure state decomposition of the form proposed above can be
found even at finite M.
In summary, we obtain the following picture for the low energy spectrum in the
spin glass phase: the non-clustering paramagnetic AKLT ground state |Ψi can be
decomposed in a superposition of several almost degenerate states, whose energies are
of order exp(−N). These states enjoy the clustering property (vanishing of connected
correlations) and are characterized by amorphous order (the local magnetizations are
different in each state). It would be nice to check this scenario explicitly by means of
exact diagonalization.
It would be very interesting to obtain more detailed information on the spectrum
of such spin glass Hamiltonians. For instance, a natural question is whether there
is an energy gap between the degenerate low-lying spin glass states and the excited
states. Indeed, we expect Goldstone (or Halperin-Saslow) modes [93] associated with
163
twisting of the amorphously magnetized states |Ψα i. While it is difficult to explicitly
construct a coarse-graining procedure to produce an effective theory of such modes
on a tree-like graph, one usually expects that such a theory applies in sufficiently high
dimensions [84]. Insofar as the effective theory is an elastic hydrodynamics living on a
tree-like graph, the corresponding modes should remain gapped [133]. This suggests
that the low energy spectrum is indeed gapped in any given pure state sector.
7.6
Concluding Remarks
In this chapter, we extend the study of AKLT models to locally tree-like graphs
of fixed connectivity by exploiting the quantum-classical mapping of the associated
wavefunctions. On the infinite Cayley tree, we recover the results obtained in [60] .
We find that the Bethe lattice possesses the peculiar property that it is possible to
choose parameters (for z = 3, 4) so that the corresponding AKLT state is critical. A
variational calculation of the gap is unable to produce gaplessness, which is consistent
with the arguments of Chapter 8 that this is a general feature of locally tree-like
graphs: essentially, one cannot deform a uniform excitation into long-wavelength
rotations of the order parameter, without jumping a gap in the Laplacian spectrum.
Turning to regular random graphs, we find that the companion classical model
is unstable to spin glass ordering within a cavity analysis. This is a general feature
of classical antiferromagnetic models on such graphs, but has striking consequences
given that the pecularities of such mean-field-like glasses should directly transfer to
the quantum ground state of the AKLT model. This provides an alternative route
to the study of quantum glassy order in tree-like models (see for example [135, 124,
123, 227, 39, 109]). We argue that there are now many (nearly) degenerate quantum
ground states with macroscopically distinct magnetization patterns, but that there
164
remains a gap to Halperin-Saslow waves for geometric reasons analogous to the simpler
case of the antiferromagnet.
There are several avenues for future research. One obvious direction is to study
the classical vector spin glass and the corresponding classical measure. In a different
vein, we observe that the AKLT construction applies at a very special point in the
space of quantum Hamiltonians. To what extent do the features of the quantum
AKLT glass extend to regions proximate to this exactly solvable point? Ideally, the
AKLT glass would capture the essential features of a broader range of quantum spin
glasses, playing a role reminiscent of that played by the S = 1 AKLT chain in relation
to the Haldane phase.
165
Part V
Miscellany and Marginalia
166
Chapter 8
There are no Goldstone bosons on
the Bethe lattice
8.1
Introduction
The study of statistical mechanical systems on the Bethe lattice dates back to Bethe’s
early work on binary alloys in the 1930s[27]. Formally, the Bethe lattice is a regular graph with no loops in the infinite size limit; its tree-like local structure makes
it amenable to real space self-consistent treatments for systems with short ranged
interactions without entirely losing the notion of distance, as happens if instead we
resort to infinite ranged interactions. Specifically, for ferromagnetic, and hence unfrustrated, models it makes the Bethe-Peierls approximation exact. For frustrated
systems, such as spin glasses, the simplification achieved is more problematic and has
been the subject of much ongoing work[156]. Altogether, there is a large set of results
on various aspects of classical statistical mechanics on the Bethe lattice.
Quantum statistical mechanics on the Bethe lattice has received less attention
by comparison. One early set of results concern the Hubbard model on the Bethe
lattice in the infinite coordination limit [75]. More recently, purely bosonic models
167
on the Bethe lattice have been the subject of investigation with the introduction of
the quantum cavity method. [135, 209, 125] This has led to computational results
on the quantum Ising model, the quantum Ising spin glass and on the Bose-Hubbard
model. In addition, we note variational work on the ground state of the quantum
Ising model [168] and on a spherical model of a spin glass [124].
In this chapter, we study the quantum unfrustrated (ferromagnetic) problem on
the Bethe lattice analytically. For technical reasons, we do this nominally in the
form of the spherical model for a scalar field but it is essentially also the large Nf
limit of the nearest neighbor O(Nf ) quantum rotor model on the Bethe lattice. We
find some surprising results from the perspective of quantum phase transitions on
Euclidean lattices. First, we find that the critical point is marked by a single global
mode (or Nf modes in the O(Nf ) interpretation) descending in energy to become
degenerate with the ground state while all other states remain a finite distance away
in energy. Second, no local operators couple to this mode in the thermodynamic
limit so that all local correlation functions exhibit a gap at criticality. Alternatively,
once the thermodynamic limit has been taken, the response to a field on any finite
subvolume is bounded even if that volume is taken to infinity. Third, this state of
affairs persists into the broken symmetry phase in that while there are 2 (or O(Nf )
worth of) degenerate broken symmetry states as required by the global symmetry,
there are no Goldstone bosons and all local correlators remain gapped. We believe
our results are robust to moving away from the spherical/Nf = ∞ limit. Indeed, the
Nf = 2 problem is the particle-hole symmetric transition in the Bose-Hubbard model
studied using the cavity method by Semerjian et al [209] whose results imply that
a macroscopic superfluid density develops at the quantum phase transition (QPT)
without closing the excitation gap in the single site correlation function.
These results should surprise the reader. On the face of it, the lack of Goldstone behavior confounds the standard intuition regarding broken symmetry phases.
168
Aficionados of classical models on the Bethe lattice may counter, rightly, that their
phase transitions are accompanied by exponential rather than power law correlations,
as this is sufficient for the classical susceptibility to diverge. However, if one computes
the quantum susceptibility – by which we mean the response to an infinitesimal field
applied over a subvolume whose size is taken to infinity after the thermodynamic
limit – it remains finite, because the matrix elements in question do not couple to the
global mode. The analogous susceptibility of the classical model diverges to signal
the phase transition.
We turn now to a brief discussion of what we mean by the Bethe lattice as it
plays an important role in the following. The traditional, and simplest, way to define the Bethe lattice is through a sequence of Cayley trees. The Bethe lattice then
corresponds to the interior of a large tree where each site is connected to exactly
z neighbors. This construction makes clear that the number of sites within a fixed
distance of a given site scales exponentially with distance and thus connects, qualitatively, with the infinite dimensional limit of hypercubic lattices that is also often
invoked in the context of making self-consistent treatments exact. Unfortunately, the
boundary of a finite Cayley tree is always a finite fraction of the bulk whence the
choice of boundary conditions can complicate the thermodynamic limit[57]. Indeed,
the difference between a ferromagnet and a glass can be phrased solely in terms of
boundary conditions. To get around this problem, one can proceed differently and
consider instead the ensemble of z−regular graphs with N nodes. This consists of all
graphs where each site is connected to exactly z nearest neighbors. Picking a graph
at random from this ensemble in the N → ∞ limit yields an alternative definition of
the Bethe lattice which is known to be perfectly satisfactory for the classical ferromagnetic problem[108] and that is what we shall use in the following. At finite N,
members of the ensemble contain loops of characteristic size log N and the graphs are
169
not entirely homogeneous. However, homogeneity is restored in the limit N → ∞ as
the loop size diverges.
With this definition in hand, we can comment on the feature of the Bethe lattice
that brings about the surprising features we report: the boundary of any subvolume
of the lattice is always a finite fraction of the whole whence (arbitrarily) localized
excitations cannot be made low energy. In more formal terms, the Bethe lattice
has what is known to graph theorists as a positive Cheeger constant h – that is
the minimal ratio of boundary to bulk of its nontrivial subgraphs. A theorem – the
Cheeger bound – guarantees that the Laplacian has a gap of at least h2 /2 between
the uniform Perron-Frobenius ground state and the rest of the spectrum. Moreover,
this property is shared by a large collection of so-called expander graphs[98], of much
interest in quantum and classical information theory, and we expect our qualitative
results to be shared by models on any such graph.
In the balance of this chapter, we first introduce and solve the classical spherical
model on the Bethe lattice which introduces most of the ingredients needed in a simpler setting. This classical model on the Bethe lattice has been solved previously[41]
by a different technique. We next introduce and solve the quantum spherical model
which exhibits the features we described above. (We note that the quantum spherical
model on Euclidean lattices has been solved by Vojta[237].) We end with a discussion
and comments on the generality of our results.
8.2
The Classical Spherical Model
The classical spherical model on a z-regular graph G with N nodes is defined by the
Hamiltonian
H=
1X
ϕi Lij ϕj
2 ij
170
(8.1)
subject to the global constraint that
by
P
i
ϕ2i = N. Here, the graph Laplacian is given
Lij = −Aij + zδij
(8.2)
A is the adjacency matrix of G, and the overall shift of z ensures that H is positive
semidefinite. Alternatively, we can view this choice of the Hamiltonian as the one
which gives a system of coupled oscillators with nonnegative frequencies.
8.2.1
Phase transition
The partition function of the model (at inverse temperature β = 1/T ) is given by the
expression
Z(β) =
Z Y
dϕi e−βH[ϕ] δ
i
X
i
ϕ2i − N
!
(8.3)
Representing the delta function by a Lagrange multiplier λ and performing the Gaussian integral over the ϕi , we obtain the effective action
Z(β) =
Z
dλe−N [ 2N Tr log{βL−2iλ1}+iλ]
1
(8.4)
As N → ∞, this integral may be performed by steepest descent. The effective
propagator for the ϕ field is given by (defining −µ/T = 2iλ):
hϕi ϕj i = T (L + µ1)−1
ij
and the constraint that
(8.5)
P 2
hϕi i = N gives the self-consistency condition
i
N = Tr[T (L + µ1)−1 ]
171
(8.6)
We may rewrite this in terms of the eigenvalues εα of the graph Laplacian
1 X
1
1
=
T
N α εα + µ(T )
(8.7)
where we have explicitly written µ(T ) to emphasize that self-consistency forces µ to
depend on temperature. Throughout, we use indices α, β to label the modes of the
Laplacian L with energy εα and eigenvector uαi and i, j to refer to sites of the lattice.
The spectrum of z-regular random graph Laplacians and the related problem of
hopping on the Bethe lattice have been extensively studied [151, 47]. The spectrum
√
√
consists of a continuum of states in the interval [z − 2 z − 1, z + 2 z − 1], with the
density of states
z
ρc (ε) =
2π
p
4(z − 1) − (ε − z)2
z 2 − (ε − z)2
(8.8)
However, the above density of states does not include the uniform Perron-Frobenius
√
eigenvector, which has amplitude 1/ N on each site and energy 0. Thus, the full
density of states is
ρ(ε) =
1
δ(ε) + ρc (ε)
N
(8.9)
The existence of this spectral gap, and the uniqueness of the low-lying state is guaranteed by the fact that the z-regular random graphs have a positive Cheeger constant.
We now return to the self-consistency equation Eq. (8.7) and rewrite it in terms
of the density of states Eq. (8.9):
1
1
=
+
T
Nµ
√
z+2
Z z−1
√
z−2
dε
ρc (ε)
ε+µ
(8.10)
(z−1)
The usual argument for Bose-Einstein condensation now follows: as T → 0, we must
decrease µ in order to satisfy this sum rule. At high temperatures, the thermodynamic
limit can be taken straightforwardly without paying espercial attention to the uniform
172
state, and one can satisfy the self-consistency equation. However, since µ cannot
decrease below the lowest eigenvalue of the Laplacian - as this would render the
steepest descent calculation unstable - the smallest that µ can be is zero. This occurs
at a critical temperature Tc given by


Tc = 

√
z+2
Z z−1
√
z−2
(z−1)
dε
−1
ρc (ε) 

ε 
=
z(z − 2)
z−1
(8.11)
This result for the transition temperature was previously derived in [41]. Clearly,
Tc = 0 for z = 2, and there is no finite-temperature transition on the chain, as
expected. However, Tc is finite for z > 2, and below this temperature, we cannot satisfy the self-consistent equation in the naive thermodynamic limit discussed
above. Rather, we must keep track of the uniform mode in (8.10) and take µ =
−1
T
T
+ O(N −2 ) for T < Tc .
1 − Tc
N
For comparison, in the O(Nf ) model, the Lagrange multiplier λ would become a
field λi , which at finite N on the random graph need not be homogeneous at the large
Nf saddle point. However, the infinite Bethe lattice is homogeneous and so in the
thermodynamic limit we expect to recover the same saddle point as in the simpler
spherical model.
8.2.2
Ordered phase
How does the system behave in the low temperature phase? Consider the correlation
function computed in the eigenbasis of L, namely
hϕα ϕβ i =
173
T δαβ
εα + µ
(8.12)
Below the critical point, the amplitude of the lowest eigenmode is given by
T
hϕ0 ϕ0 i = = N
µ
T
1−
Tc
(8.13)
The thermal occupation of the uniform state is macroscopic, indicative of long range
order in the system. In light of this, we revisit the steepest descent calculation, but
this time treat the lowest mode separately. First, integrate out the N − 1 higher
modes
Z(β) =
Z
2
dλdϕ0 eiλ(ϕ0 −N )
P
Z Y
− 12
(βεα −2iλ)ϕ2α
×
dϕα e α6=0
α6=0
=
Z
" „
#
«
ϕ2
1 P
−N iλ 1− N0 + 2N
log{βεα −2iλ}
dϕ0 dλ e
α6=0
and now perform the λ integral by steepest descent, giving the self-consistency equation (again µ/T = −2iλ)
ϕ2
1= 0 +T
N
Z
√
z+2 z−1
√
z−2 z−1
dε
ρc (ε)
ε+µ
(8.14)
√
For T near Tc , µ ≪ z − 2 z − 1 so we expand in powers of µ and solve:
1
µ≈
bT
where b =
R z+2√z−1
√
z−2 z−1
ϕ20
T
−1
+
N
Tc
(8.15)
dε ρcε(ε)
2 .
Using this solution and approximating the steepest descent value of the integrand
to quadratic order in µ(ϕ0 ), we find that
Z(β) ≈
Z
dϕ0 e−N Vef f (ϕ0 )
174
(8.16)
where the effective potential for ϕ0 is (dropping constants):
1
Vef f (ϕ0 ) =
2bT 2
"
ϕ
√0
N
2 4 #
1 ϕ0
T
√
−1 +
Tc
2
N
(8.17)
Thus, we see that the effective potential has Landau-Ginzburg form, and predicts
√
a mean field symmetry breaking for T < Tc such that hϕ0 i ∼ N. That is, hϕi i ∼
√
hϕ0 i/ N ∼ O(1).
8.2.3
Spatial correlations
We now consider the spatial behavior of the Green’s function Gij = hϕi ϕj i. As this is
ultimately a Gaussian theory, the only interesting correlator is the two-point function.
Since we do not know the exact eigenvectors of the graph Laplacian at finite N, we
cannot obtain Gij from Gαβ directly. Instead, we work directly in the thermodynamic
limit of the Bethe lattice and assume that Gij = G(|i − j|) is translation invariant.
In the disordered phase, Gij satisfies the equation of motion
(Lii′ + µδii′ )Gi′ j = −δij
(8.18)
Using the ansatz Gij = Ae−|i−j|/ξ , one finds
Gij =
2(z − 1)
p
(z − 2)(z + µ) + z (z + µ)2 − 4(z − 1)
!|i−j|
p
(z + µ) − (z + µ)2 − 4(z − 1)
×
2(z − 1)
(8.19)
which, as µ → 0+ at the phase transition, reduces to
Gij =
z−1
(z − 1)−|i−j|
z(z − 2)
175
(8.20)
This agrees with the well-known correlation length ξ = 1/ ln(z − 1) of the Bethe
lattice at criticality and corresponds to a diverging global susceptibility.
Since the uniform mode has vanishing weight in the disordered phase, it makes no
contribution to the Green’s function for T > Tc . In the low temperature phase, the
uniform mode has macroscopic occupation corresponding to a non-zero value of hϕi i.
In this case, the expression (8.12), calculated at µ = 0, corresponds to the connected
correlation function Gcij = hϕi ϕj i − hϕi ihϕj i throughout the low temperature phase.
That the correlation length sticks at its critical value and the susceptibility remains
divergent throughout the condensed phase is the Bethe lattice analogue of the classical
Goldstone theorem, which arises quite naturally when this model is viewed from the
large Nf point of view.
8.3
The Quantum Spherical Model
The quantum spherical model is given by the Hamiltonian [237]
1 X 2 1X
ϕi Lij ϕj
π +
H= g
2 i i
2 ij
(8.21)
with canonical commutation relations [ϕi , πj ] = iδij , and the mean spherical constraint:
X
i
hϕ2i i = N
(8.22)
As discussed in [237], the partition function can be rewritten in functional integral
form as
Z(g, β) =
+
Z
dλ
Y
i
( Z
dϕi (τ ) exp −
0
β
dτ
"
1 X
ϕ̇i (τ )2
2g i
!#)
X
1X
ϕi (τ )Lij ϕj (τ ) + iλ
ϕ2i (τ ) − N
2 ij
i
176
where τ is an imaginary time parameter. Transforming to frequency space and expanding in eigenvectors of L,
Z(g, β) =
Z
dλ
dϕα (ω)
(8.23)
α
− 21
× e
Y
P
ω,α
h “ 2
”
i
ϕα (ω) β ωg +εα −2iλ ϕα (−ω)−iλN
where we sum over discrete frequencies ωn =
2πn
.
β
The self-consistency equation is now given by
1=
1
T X
N ω,α ω2 + ε + µ
α
g
(8.24)
Performing the frequency summation, we obtain
√
g
1 X
1 p
√
1=
coth
β g(εα + µ)
N α 2 εα + µ
2
(8.25)
This result was computed directly within the Hamiltonian formalism in [237]. Note
that as g → 0, we recover the classical result Eq. (8.7).
8.3.1
Quantum phase transition
At T = 0, the self-consistency equation reduces to
1
1
1 X
√
√ =
g
N α 2 εα + µ
=
1
√ +
2N µ
√
z+2
Z z−1
√
z−2 z−1
177
ρc (ε)
dε √
2 ε+µ
(8.26)
Following a similar argument as in the classical case, we find that the system has a
quantum critical point at the critical coupling gc , defined by
√
z+2
Z z−1
gc−1/2 =
√
z−2
ρc (ε)
dε √
2 ε
(8.27)
(z−1)
This integral is convergent, and can be done numerically to find the precise value of
gc . Again, for g < gc we must be careful regarding the thermodynamic limit and take
√
√ −1
p
g
1 − g/gc
µ=
2N
(8.28)
in order to determine the macroscopic occupation of the zero mode. As expected,
the quantum spherical model undergoes a Bose condensation transition at the critical
coupling gc .
8.3.2
Correlations
The imaginary time Green’s function at zero temperature is most easily computed by
taking a limit from finite temperature. From the partition function Eq. (8.23), we
find
Gαβ (τ )
=
X
ω
=
T δαβ
eiωτ
+ εα + µ
dω
eiωτ
2π ωg2 + εα + µ
√
√
g
e− g(εα +µ)|τ |
δαβ √
2 εα + µ
−→ δαβ
T →0
ω2
g
Z
(8.29)
Each mode α decays exponentially in imaginary time at a rate corresponding to the
gap to exciting that mode. Thus, the spectral response of the time-ordered propagator
p
in real frequency space is precisely a delta function at ω = g(ε + µ).
178
Now we are in a position to understand the peculiarity of the quantum condensation transition on the Bethe lattice. The single site Green’s function is given by
transforming Gαβ back to position space using the eigenmodes uαi of the graph Laplacian:
√
g
1
√
Gii (ω) =
√ δ(ω − gµ)
N2 µ
√
X
p
g
α α
δ(ω − g(εα + µ))
+
ui ui √
2 εα + µ
α>0
(8.30)
Without detailed knowledge of the eigenvectors of the Laplacian but merely its density
of states, this formula already allows us to sketch the spectral response of local single
particle excitations as in Fig. 8.3.2. As expected, the quantum phase transition is
signaled by the continuous closing of a spectral gap to a uniform mode. The unusual
feature is that this state is isolated from the remainder of the spectrum and its weight
vanishes as 1/N in the disordered phase. Meanwhile, in the ordered phase, the weight
√
p
g
1
√ = 1 − g/gc
N2 µ
(8.31)
is finite, reflecting the condensation into the uniform state.
The general spectral features of correlations sketched above may be supplemented
in detail by assuming homogeneity of the infinite Bethe lattice and exploiting our
knowledge of the classical spherical model. The frequency resolution of the zero
temperature Green’s function Gij (ω) is given by the mode sum
Gij (ω) =
X
α
uαi uαj
ω2
g
+ εα + µ
(8.32)
which has the same form as the mode sum for a classical model at chemical potential
µ+
ω2
.
g
Thus, we use the classical Green’s function in Eq. (8.19) and make this
substitution for µ. The formal expression is rather unenlightening but there is much
179
Figure 8.1: Support of Gij (ω) at zero temperature for z = 3.
The isolated (blue) line indicates the contribution of the uniform mode which vanishes
as 1/N above gc (dashed) but is finite below gc signaling the long-range order (solid).
Obtained from numerical inversion of Eq. (8.26).
information in the pole structure (see Fig. 8.3.2). The most important feature is a
pair of vertical square root branch cuts on the imaginary axis corresponding to the
bulk spectrum. These never pinch off the real axis as g is varied; rather, the closest
branch point sets the dominant decay rate, which agrees exactly with the previous
discussion.
At finite N the branch cuts break into lines of poles and in addition one should
recall the contribution of the uniform mode. This provides a pair of isolated simple
√
poles at ±i gµ with residue 1/N in the disordered phase. As g → gc+ , these poles
pinch off the real axis corresponding to the closing of the gap and the phase transition.
In the condensed phase, these poles have non-vanishing residue indicating the long
range order in imaginary time.
Finally, we note that the static susceptibility χ of the quantum model to a global
field (applied before the thermodynamic limit) is simply given by the zero frequency
lattice sum of Gij (ω), which is equivalent to summing Eq. (19) over the lattice. This
susceptibility diverges throughout the broken symmetry phase, as usual. However,
180
Figure 8.2: Analytic structure of Gij (ω) in the complex plane.
Wiggly lines (green) indicate branch cuts. The two crosses (red) indicate the poles
due to the zero mode with residue 1/N. The following formulae for cut locations are
given at µ = 0 (criticality): The hyperbolic
given by Re [ω]2 − Im [ω]2 =
q cuts are
√
−zg. The vertical cuts extend between ±i g(z ± 2 z − 1). These cuts control the
slowest decay since we can deform the contour past the uniform mode pole in the
thermodynamic limit.
taking a large subvolume and computing its susceptibility after the thermodynamic
limit gives a convergent result, reflecting the absence of Goldstone bosons in the
spectrum.
8.4
Concluding Remarks
There are two salient qualitative features of the spherical model condensation transition on the Bethe lattice that we have derived above: (1) despite its vanishing weight
in the response to local excitations, there is a global mode closing the many-body
gap in the disordered phase. Alternatively, this can be viewed as a ‘midgap’ mode
driving the transition, where by midgap we mean in the middle of the gap to local
excitations. (2) It is via macroscopic occupation of this mode that long range order
develops in the symmetry-broken phase. However, local excitations remain gapped.
181
In fact, not only are there no Goldstone bosons, the spectral response of any operator
with bounded support will remain gapped.
We believe these features are quite general to unfrustrated transitions on the Bethe
lattice and expander graphs. For one thing, as already noted in the introduction, our
results are entirely consistent with those of [209] on the Bose-Hubbard model wherein
the particle-hole symmetric transition is the O(Nf ) model at Nf = 2. This indicates
that the features we have found at Nf = ∞ are robust throughout the range of O(Nf )
models and not merely artifacts of the spherical model limit. We believe this claim will
be susceptible to a proof in the 1/Nf expansion, but defer such a calculation for the
time being. Furthermore, the results should apply to sufficiently weakly disordered
ferromagnetic models, for which the disorder does not close the mode gap.
More intuitively, these results should be stable to the inclusion of self-interactions
of the scalar field beyond the self-consistent Gaussian theory treated here. First,
at the critical point, weak interactions cannot close the gap between the soft global
modes and other massive modes; indeed, they typically cause the low energy density of
states to decrease even further. Second, the usual heuristic argument for a Goldstone
mode is seen to lead to a gap as follows: consider the simplest case of an O(2) = U(1)
broken symmetry in the broken phase. In the usual fashion we derive an effective
Lagrangian for the phase fluctuations in the symmetry-broken phase by freezing out
modulus fluctuations in |hϕi|:
Leff ≈
X 2 1
X
1
|hϕi|2
θi Lij θj
θ˙i + |hϕi|2
2g
2
i
ij
(8.33)
This is a quadratic action for a massless free field θ, which would lead to gapless behavior of local excitations were the Laplacian gapless. However, the Laplacian spectrum
on expander graphs is gapped to all modulated excitations. While the global rotation
is still, necessarily, of zero energy, it no longer follows that there exist local excitations
182
of arbitrarily low energy. This corresponds to a breakdown of our intuitive understanding of generalized rigidity. Quite generally then, we conjecture that there is an
‘anti-Goldstone’ theorem that applies to symmetry breaking transitions on expander
graphs.
We should also note that the fully connected models for which mean field theory is
exact exhibit similarly gapped behavior, even after the usual rescaling for extensivity.
Consider the fully connected Heisenberg ferromagnet whose Hamiltonian is H =
P
2
−J/N ij Si · Sj = −J/NStot
. Trivially, the gap to the first excited state is J and
does not vanish in the thermodynamic limit. The fully connected graph does not have
bounded coordination so is not technically an expander graph, however the surface
to volume ratio is, nonetheless, very large.
Clearly there are many subtle issues that may invalidate the usual physics lore
when models are considered on non-Euclidean graphs. As quantum models are studied more extensively in the context of quantum complexity theory and on random
networks, an enhanced understanding of universal symmetry breaking properties will
potentially play an important role in their analysis.
183
Chapter 9
The Superconducting
Susceptibility of a Fermi Liquid
9.1
Introduction
Thinking within the framework of the renormalization group (RG) is always an insightful way to approach the low energy, long wavelength behavior of physical systems, but it does not always provide a significant computational advantage over
more straightforward, perturbative, methods. The exceptions are cases that involve
marginal and near marginal couplings where making sense of perturbative divergences
is far easier within the framework of the RG. Indeed, the historical development of
RG methods had mostly to do with exactly these cases leading up to the discovery
of the epsilon expansion by Wilson and Fisher [252, 68].
A subclass of these canonical RG applications consists of problems with marginally
relevant or irrelevant couplings. Celebrated examples of the former are the Kondo
problem[14, 251] and QCD[83] while the textbook example of the latter is the critical behavior of ferromagnetic models (equivalently, of vector models in the language
of field theory) in four dimensions[262]. In all of these cases, non-trivial logarithms
184
appear in physical quantities either in the ultraviolet (marginally relevant) or the infrared (marginally irrelevant), and reflect the extremely slow variation of the coupling
at issue. An instructive example, with close parallels to our concerns in this chapter, is
the critical behavior of the specific heat in four dimensional O(N) ferromagnets/vector
models. Here the non-interacting gaussian theory exhibits a logarithmic divergence
C ∼ − log |t| with t =
T − Tc
.
Tc
As the quartic coupling is marginally irrelevant, i.e., it dies logarithmically in the
infrared and the theory flows to the non-interacting limit, one may then suppose that
the divergent susceptibility of free-field theory will remain intact even upon inclusion
of the interactions. However, as is long established, interactions lead to the more
complex logarithmic dependence
4−N
C ∼ [− log |t|] N+8
which diverges as (− log |t|)1/3 in the Ising case (N = 1) but vanishes as (− log |t|)−1
when N → ∞. Observe that the logarithms appear generically with fractional powers
and a variable sign. This complexity is not easily unraveled perturbatively and instead
it is much better to resort to the machinery of the RG, specifically to the derivation
and solution of the RG (differential) equations obeyed by the correlation functions,
which we shall simply call Callan-Symanzik equations.
Let us turn now to the problem considered in this chapter. The RG treatment of
interacting fermions identifies the Landau Fermi liquid as a fixed point of a momentum
shell RG characterized by set of exactly marginal couplings consisting of the Landau
F function. In addition however, there is a marginally irrelevant coupling which is the
185
repulsive BCS (Cooper channel) coupling.1 Absent this marginal flow the fermions
have a logarithmically divergent susceptibility to superconductivity 2 . We wish to
ask what happens to this divergence when the marginally irrelevant flow is taken into
account. We show here using the RG that this divergence goes away and the uniform,
zero frequency superconducting susceptibilty of the Fermi liquid is therefore finite.
It is useful at this point to clarify what we mean by the Fermi liquid. As the
reader is no doubt aware, the actual RG flow for an interacting fermion system with
typical repulsive interactions leads inevitably to a superconducting instability via
the Kohn-Luttinger effect wherein screening produces an effective interaction that is
attractive in a higher angular momentum channel, as discussed for example in [210].
What we have in mind therefore is the RG flow for a system where the dominant bare
couplings in the BCS channel are repulsive and renormalize to smaller values for a
large range of scales (temperature or energy) while the growing attractive couplings
are still small—operationally this is what one means by a Fermi liquid. We can,
however, formalize this understanding by working with Hamiltonians which contain
only the Landau couplings and the reduced BCS couplings that are left in a naive
application of the RG to the interacting fermion problem. In this approximation,
the Kohn-Luttinger scale has truly vanished but we are still left with a marginally
irrelevant operator about the Fermi liquid Hamiltonian whose flow can be studied.
Our central result is possibly not new—certainly it is almost present in the large
lore on superconductivity and superconducting fluctuations[129] and we would be
delighted to hear from readers who can point us to an explicit, relevant citation. It
does not, however, appear to be widely known and at first blush comes as a surprise, as
condensed matter theorists are conditioned to think of the Fermi liquid as exhibiting
correlations morally identical to that of the Fermi gas up to the effects of the Landau
1
If the BCS coupling is attractive, it is marginally relevant and we have the physics of the
superconducting instability.
2
For a discussion of susceptibilities, see for instance [97].
186
parameters. Ex post facto the intuition we would offer is that the repulsive interaction
goes away just slowly enough at long distances that the usual buildup of the divergence
of the superconducting susceptibility is undone. Possibly there is a deformation of
our present problem where this exact cancellation can be modified to yield a more
complex residual of the kind cited for four dimensional ferromagnetism above.
In any event, we view our work as a contribution to the RG analysis of interacting
fermions and as such we trust readers will find it interesting as well. In the process, we
show how to derive Callan-Symanzik equations to one loop for interacting fermions
for composite operators made from the fundamental Fermi fields—which may be of
interest to readers even beyond the specifics of our application to the computation of
the superconducting susceptibility.
In the following we begin with a quick summary, in Section II, of the RG for
interacting fermions that leads to the Fermi liquid fixed point(s) and the flow in
its vicinity. Next we derive the relevant Callan-Symanzik equations (Section III),
solve them (Section IV) and end with a brief discussion (Sections V and VI) and an
Appendix that contains some technical details. Note that while our results certainly
apply in both two and three dimensions, to avoid unnecessarily complications we work
with the pedagogically simpler case of d = 2 throughout this chapter.
9.2
Review of Results from the RG
We begin the technical part of our discussion with a summary of results from the
renormalization group approach as applied to interacting fermion systems. We shall
provide a telegraphic review, referring the reader interested in further details to more
pedagogical discussions, such as [210]. Readers familiar with the technology and
results can skip ahead to Section III.
187
We focus on the following following action, written for a system of electrons with
a circular Fermi surface in d = 2 and spin directions α =↑, ↓ :
S =
X Z
−∞
α=↑,↓
+
∞
XZ
µ,ν
αβ
dω
2π
{ki }
{ωi }
Z
2π
0
dθ
2π
Z
Λ
−Λ
dk
ψ̄α (ωθk) (iω − vF k) ψα (ωθk)
2π
uµναβ (1, 2, 3, 4)ψ̄µ(1)ψ̄ν (2)ψα (3)ψβ (4)
≡ S0 + SI
(9.1)
where ψµ (i) = ψµ (ωi θi ki ) is the fermion/Grassman field, ω is the frequency, k =
|K| − KF is the radial component of momentum measured relative to the Fermi
momentum KF , θ denotes the angle on the Fermi surface, vF is the Fermi velocity,
and Λ << KF is the ultraviolet cutoff.
In the second term SI , the measure is
Z
{ki }
{ωi }
3
Y
=
Z
∞
−∞
i=1
K4 =K1 +K2 −K3
Z
Z 2π
dθi Λ dki
×
0
2π
−Λ
2π
dωi
2π
Θ(Λ − |k4 |).
(9.2)
The renormalization group transformation involves three steps: (i) integrating out
all momenta between Λ/s and Λ, and correcting terms in the action as needed in the
process; (ii) rescaling frequencies and the momenta as per (ω, k) → s (ω, k ) so that
the cutoff in k is once again at ±Λ; and finally (iii) rescaling fields ψ → Zψ ψ to keep
the free-field action S0 invariant. Thus S0 is a fixed point of this RG and the possible
interactions can be classified as relevant, irrelevant, or marginal with respect to this
transformation and fixed point.
At tree level, it is easily shown that interactions with six or more fields are irrelevant, as is the ω or k dependence of quartic couplings. While this is very much like
ϕ44 , the ϕ4 theory in four dimensions, there the coupling is just a number, u(0, 0, 0, 0),
188
describing collision of particles at zero external momentum, here the quartic couplings
can depend on the angles on the Fermi circle. Given that momentum is conserved,
we need to pick only three of the angles independently, say θ1 , θ2 and θ3 . However,
the fact that the momenta come not from the plane but a very thin annulus leads to
additional constraints [210].
Consider the left half of Figure 9.1 where all momenta lie on the Fermi circle.
Given 1 and 2 , it is clear that 3 and 4 have to equal them pairwise. That is θ1 = θ3
and θ2 = θ4 or the exchanged version. Now consider the generic case of momenta
that lie in a very thin shell |k| < Λ, rather than right on the Fermi circle. It is not
surprising that now θ1 ≃ θ3 and θ2 ≃ θ4 with deviations of order Λ/KF . We ignore
the dependence of u on such tiny angular differences when Λ/KF → 0, and we ignore
any k -dependence (since it is irrelevant). Thus we define u with all k = 0 and θ1 = θ3
and θ2 = θ4 :
u(θ1 , θ2 , θ3 , θ4 ) = u(θ1 , θ2 , −θ1 , −θ2 ) = u(θ1 , θ2 )
= u(θ1 − θ2 ) ≡ F (θ).
(9.3)
It is important to bear in mind that even though we evaluate u at k = 0 for
angles (θ1 = θ3 ) and (θ2 = θ4 ), we do not imply that only forward scattering is
allowed or that all momenta lie on the Fermi surface: rather we allow all momenta
in the measure defined by Eqn. 9.2 but ignore the dependence of u on k and the tiny
differences (θ1 − θ3 ) and (θ2 − θ4 ). (It is like saying that in ϕ44 , u = u(0, 0, 0, 0) does
not mean the external legs are limited to zero, only that u is the same for all values
of external momenta.) Indeed a small amount of non-forward scattering is not only
allowed at any nonzero cut-off, it is essential to produce a nonzero compressibility in
the “q” limit.
189
4
3
3
2
1
1
2 = −1
4 = −3
KF
KF
F
V
Figure 9.1: Kinematics of the couplings F and V .
In summary, in studying the flow of the four point coupling, we will choose the
legs to be on the Fermi surface (since k dependence is irrelevant), i.e, we study the
flow of F .
Kinematics allows one more coupling function besides F . Consider the right half
of Figure 9.1 with all momenta on the Fermi circle, but with θ1 = −θ2 so that the
incoming momenta add up to zero. Now the outgoing pair of momenta can point in
any direction, as long as they add up to zero, which lets them point in any pair of
mutually opposite directions. This leads to a function V in the BCS channel:
u(θ1 , θ2 , θ3 , θ4 ) = u(θ1 , −θ1 , θ3 , −θ3 ) = u(θ1 , θ3 )
= u(θ1 − θ3 ) ≡ V (θ)
(9.4)
Once again, even though u will be assumed to depend on just the directions θ1 and θ3
(via their difference), for any Λ > 0, scattering between states of the total momentum
P ≃ Λ will be kinematically permitted.
The fate of the couplings F and V , marginal at tree level, is determined by one
loop diagrams. The one-loop analysis of [210] shows that F is strictly marginal while
V flows. It is also shown there that the one-loop flow is exact in the limit Λ/KF → 0.
190
In the remainder, we will assume F = 0 and that V is repulsive and angle independent. While this simplifies the discussion, the first and last restrictions are inessential.
Inclusion of F will not affect the physics of the Cooper channel for kinematic reasons,
and while the angular dependence of V can affect details of various calculations, it
cannot alter the fact that singular logarithms arise in the same. As long as the system
remains rotationally invariant, any angular dependence in V can be accounted for by
expanding in different angular momentum channels, and deriving flow equations for
each channel. Each channel will have its own Callan-Symanzik equation, and the
remainder of the arguments follow from this. On the other hand, it should be clear
from the preceding discussion that restricting V to be repulsive is necessary, since
attraction leads to the BCS instability, and of course the Fermi liquid fixed point
cannot capture the physics of superconductivity.
Thus,
SI =
XZ
µ,ν
αβ
{ki }
{ωi }
g0
εµν εαβ ψ̄µ (1)ψ̄ν (2)ψα (3)ψβ (4)Θ(λ − |P |)
4
(9.5)
where εµν is the antisymmetric tensor. The additional condition Θ(λ − |P |) ensures
that the total momentum of the Cooper pair is less than λ which itself is assumed to
obey λ << Λ. This will not be a limitation in our study of superconductivity since
only the interaction at P ≃ 0 will come into play.
We will use the standard parametrization of s:
s = eσ so that
dσ =
dΛ
ds
=− .
s
Λ
(9.6)
We will typically choose to rescale by an amount close to unity, in which case
dσ =
ds
= ds since s → 1.
s
191
(9.7)
Note that the cutoff is being changed infinitesimally at each RG step; as the RG
transformations are infinitesimal, this permits us to derive differential equations that
capture the content of the flows.
It is shown in [210] that
β(g) ≡
dg
= −ag 2
dσ
(9.8)
where a is a positive constant. The solution to this equation is
g(σ) =
g0
.
1 + aσg0
(9.9)
For repulsive g0 we see a logarithmically vanishing g(σ) ≃ 1/σ. One may expect that
such a marginally irrelevant coupling cannot affect the superconducting susceptibility
which diverges logarithmically in the noninteracting limit. The rest of this chapter
aims to show why this is not the case and what finally happens.
9.3
Calculation of the Beta Function and Anomalous Dimension
While we expect many of our readers to be familiar with the computation of the
RG functions, for the sake of completeness, and because the anomalous dimension
calculation may be unfamiliar to some, we briefly review the procedure here. Following Shankar [210], we compute corrections to the action from integrating out modes
between Λ − dΛ to Λ via the cumulant expansion. Essentially, we treat the coupling
as a perturbation δS, and use the result
2
1
2
heδS i = e[hδSi+ 2 h(δS) i−(hδSi) ]
192
(9.10)
Figure 9.2: One-loop diagrams for the beta function and anomalous dimension
of ψ̄ ψ̄
where all averages are over the modes being integrated out. In each term, we are to
select the number of ‘external’ legs, which will belong to the modes below the shell of
integration, and this determines which term in the action will be corrected by the term
obtained by integrating out the remaining fields. In order to determine the anomalous
dimension of the Cooper pair operator, the easiest method is to add a source term
of the form JO ψψ (and its complex conjugate) to the action and determine how it
gets renormalized at one loop; a moment’s thought will suffice to realize that this is
equivalent to determining γO .
The one-loop diagrams contributing to the β-function and the anomalous dimension of O are shown in Fig. 9.2.
Evaluating the first diagram, we find that δg = ag 2 dΛ
, or in other words that
Λ
= −ag 2 , since ds = −dΛ/Λ . From the second diagram, we find that
β(g) = dg
ds s→0
δJO
JO
= ag dΛ
; combining this with the tree level rescaling we find that JO → s1−ag JO ,
Λ
which gives us the result (cf. Eq. 9.23) that 2 + 2γO = 2 − 2ag, or in other words, that
γO = −ag. In these expressions, a is a positive constant, whose value is unimportant;
the significant point is that it is the same constant in both β and γO .
193
9.4
Derivation of the Callan-Symanzik Equations
So far, we have reviewed how different coupling functions in the Hamiltonian flow
under the RG. This allowed us to classify relevant and irrelevant perturbations, and
determine that the repulsive Cooper channel interaction, being marginally irrelevant,
does not lead to an instability: the system flows. In this section, we discuss how
to derive a set of differential equations that describe how the correlation functions
evolve under the RG flow. First, we discuss how to derive the equations for correlation
functions without composite operators, and then discuss the procedure for the Cooper
pair operator.
9.4.1
Fundamental Fermi field correlators
We begin with the correlation function 3 , defined with an action S and cutoff Λ:
hψ̄ω̄ θ̄k̄ ψωθk iS,Λ =
Z
D ψ̄Dψ Λ e−SΛ [ψ̄,ψ] ψ̄ω̄ θ̄k̄ ψωθk
Z
D ψ̄Dψ Λ e−SΛ [ψ̄,ψ]
(9.11)
where D ψ̄Dψ Λ indicates that the functional integral is over modes of all momenta
upto Λ. Here it is understood that the momenta k are well within the cutoff. Note
also that the momenta and frequencies corresponding to ψ̄ and ψ are different; in fact,
the result will have a delta function that forces these to be the same, but since the
delta function is dimensionful, it affects the scaling, so we maintain different momenta
explicitly through the calculation. Integrating out a shell dΛ of momenta between Λ
and Λ + dΛ = Λ − |dΛ| ≡ Λ/s ≡ Λ/(1 + dσ), we have [210]
hψ̄ω̄θ̄k̄ ψωθk iS,Λ =
Z
−S ef f [ψ̄< ,ψ< ] < <
ψ̄ω̄ θ̄k̄ ψωθk
D ψ̄ < Dψ < Λ e Λ/s
s
Z
−S ef f [ψ̄< ,ψ< ]
D ψ̄ < Dψ < Λ e Λ/s
s
3
We suppress spin indices for convenience.
194
(9.12)
ef f
where SΛ/s
is an action for a theory with cutoff Λ/s, with the appropriate corrections
to parameters, and the < superscript denotes ‘slow’ modes that lie within the reduced
cutoff. In order to obtain a theory with the same cutoff, we rescale momenta in the
−1/2
second equation. We will write ψ ′ (ω ′θ′ k ′ ) = Z̃ψ
ψ < (ω ′ /s, θ, k ′/s), where 0 < |k ′ | <
ef f
Λ. Further, we assume that action may be written as SΛ/s
[ψ̄ < , ψ < ] = SΛ [ψ̄ ′ , ψ ′ ] +
δSΛ [ψ̄ ′ , ψ ′ ], where δS is O(dσ). We obtain
hψ̄ω̄ θ̄k̄ ψωθk iS,Λ = Z̃ψ
Z
′
′
D ψ̄ ′ Dψ ′ Λ e−SΛ +δSΛ ψ̄sω̄
θ̄sk̄ ψsωθsk
Z
D ψ̄ ′ Dψ ′ Λ e−SΛ +δSΛ
= Z̃ψ hψ̄sω̄,θ̄,sk̄ ψsω,θ,sk iS+δS,Λ
(9.13)
where in the second step we dropped the primes as the fields are integrated over.
Since δS depends on the parameter s of the flow, we can write S + δS = Ss ; then,
we have
hψ̄ω̄ θ̄k̄ ψωθk iS,Λ = Z̃ψ hψ̄sω̄,θ̄,sk̄ ψsω,θ,sk iSs ,Λ
(9.14)
The left side of this equation is manifestly independent of s. Therefore, differentiating both sides with respect to s, and dividing through by Z̃ψ ,
d log Z̃ψ
d
hψ̄sω̄,θ̄,sk̄ ψsω,θ,sk iSs ,Λ + hψ̄sω̄,θ̄,sk̄ ψsω,θ,sk iSs ,Λ = 0
ds
ds
The second term can be rewritten as
4
4
We note that Z, the partition function, is independent of s, and implicitly use the fact that the
correlation function forces ω = ω̄, etc; in a moment, we will switch to a discussion of the Green’s
function, and this complication will be avoided.
195
d
hψ̄
k̄ ψsω,θ,sk iSs ,Λ
ds sω̄,θ̄,s
∂
∂
= ω
+k
+
∂ω
∂k
dg ∂
hψ̄sω̄,θ̄,sk̄ ψsω,θ,sk iSs ,Λ
ds ∂g
(9.15)
where we note that the only change in the action to one loop order is in the coupling
constants. Collecting terms, and taking the s → 1 limit, we arrive at the CallanSymanzik equation for the two-point correlation function,
∂
∂
∂
ω
+k
+ β(g) + 3 − 2γψ hψ̄ω̄ θ̄k̄ ψωθk iS,Λ = 0
∂ω
∂k
∂g
dg ds s→1
=
d log Z̃ sion 3 − 2γψ = ds ψ where β(g) =
dg
dσ
s→1
(9.16)
is the beta function, and we define the anomalous dimen.
Next, we observe that the Green’s function is related to the correlation function
as hψ̄ω̄θ̄k̄ ψωθk iS,Λ = G(ω, θ, k)δω,ω̄ δθ,θ̄ δk,k̄ . Since the δ functions each have dimension
−1, it is easily verified that the Green’s function satisfies the relation
∂
∂
∂
ω
+k
+ β(g) + 1 − 2γψ G(ω, θ, k) = 0
∂ω
∂k
∂g
(9.17)
(2,0)
Finally, we rewrite this as an equation for the amputated Green’s function, Γψ
≡
G−1 :
∂
∂
∂
(2,0)
+k
+ β(g) − 1 + 2γψ Γψ (ω, θ, k) = 0
ω
∂ω
∂k
∂g
(2,0)
We see that for the free theory, where β = γ = 0, Γψ
above relation, as expected.
196
(9.18)
∼ iω − vF k satisfies the
9.4.2
Cooper pair correlator
The central objects of our discussion are the Cooper pair operator and its two-point
correlation function. We define the composite operator that creates an s-wave Cooper
pair at frequency Ω and momentum P << Λ by
ŌΩ,P =
=
ψ̄ ψ̄
Z ∞
−∞
Ω,P
dω
2π
Z
Λ
−Λ
dk
2π
Z
2π
0
dθ
ψ̄
P ψ̄
P
2π ω+Ω,K+ 2 ,↑ −ω,−K+ 2 ,↓
(9.19)
where (k, θ) refer to K. (Since P << Λ, if K lies in the shell so will K +
P
.)
2
The
operator O is likewise made of two ψ’s.
Consider the expectation value
hŌΩ̄,P̄ OΩ,P i =
Z "Y
#
D ψ̄Dψ e−SΛ [ψ̄,ψ] ŌΩ̄,P̄ OΩ,P
Λ
Z "Y
Λ
#
(9.20)
D ψ̄Dψ e−SΛ [ψ̄,ψ]
In the first stage of the RG transformation, we must integrate out all fields at
momenta between Λ/s and Λ, so that the remaining action only has terms that are
at momenta < Λ/s. This is a complicated process for the composite operator since it
contains terms that must be integrated out. We can write the composite operator as
<,<
<,>
>,<
>,>
OΩ,P = OΩ,P
+ OΩ,P
+ OΩ,P
+ OΩ,P
(9.21)
where the <, > denote whether the two fields entering the operator are below or
<,<
above the cutoff respectively. The operator OΩ,P
is the descendant of the composite
>,<
<,>
operator in the theory with cutoff Λ/s; OΩ,P
and OΩ,P
are mixed terms, whose >
>,>
field must be integrated over; and OΩ,P
is composite operator made up entirely of
197
fast modes that will be integrated out. In integrating over the fast modes we may
functionally average over the free field action, since the deviations from the free action
will only produces terms of higher order in g.
When we take the product of composite operators, we note that only terms with
even numbers of ‘fast’ (>) modes will survive the functional average over fast modes.
<,< <,<
Of these, one is the term ŌΩ̄,
O , which is the descendant of the two-point funcP̄ Ω,P
<,> >,<
tion in the lowered-cutoff theory; then there are the mixed terms 5 , ŌΩ̄,
O
and
P̄ Ω,P
>,< <,>
>,> >,>
ŌΩ̄,
O ; and finally there is the piece made up entirely of fast modes, ŌΩ̄,
O .
P̄ Ω,P
P̄ Ω,P
The first term is analogous to ψ < ψ < in our earlier discussion of the fermion Green’s
function. The remaining three pieces have no analog for non-composite operators,
and we must determine how they alter the RG flow.
They clearly form additive corrections to what comes from the descendant term.
Just for computing this additive term we simplify things by invoking a familiar result
from computing the T = 0 Cooper bubble explicitly: namely, that the correlations
√
depend on (P, Ω) only via the combination Ω̃ = Ω2 + P 2. So we will simplify the
following discussion by choosing P = 0.
Now there are no mixed terms since |K + P2 | = | − K + P2 | (when P = 0) and both
fields in ŌΩ̄,0 and OΩ,0 are either above the cut-off or below it.
This leaves only the term consisting entirely of fast modes. It remains to integrate
over K, ω, i.e. we have (on converting to the Fermi-surface coordinates, performing
the trivial angular integral, and recalling that all stray factors of KF from the measure
are absorbed into the definitions of the fields)
>,> >,>
hŌΩ,0
OΩ,0 i
→ 2×
∼
5
Z
Λ
Λ/s
dk
2π
Z
∞
−∞
1
dω
2
2π ω + vF2 k 2
1
1
1−
2πvF
s
The other mixed possibilities are forbidden because only like spins may be contracted.
198
(9.22)
As was mentioned earlier, this is a term of order −dΛ/Λ = 1 − 1/s = dσ; we
show below that this leads to an inhomogeneous term in the C-S equation for the
two-point function. As an aside, we point out (but do not prove) that for any higher
correlation function of composite operators beyond the two-point function, and for
any other correlation functions with external legs and composite operator insertions,
this contribution is of a higher order in dσ, and will not lead to inhomogeneous
terms. For aficionados of the RG for the ϕ4 theory, this is the fermionic analog of
the statement that the specific heat is the only operator that is not multiplicatively
renormalizable, the signature of which is the appearance of inhomogeneous terms in
the Callan-Symanzik equation for the specific heat[262].
Finally, we turn to the more familiar term, which is the ‘descendant’ operator
in the cutoff theory. We still need to determine how this behaves under the next
two RG steps. It is sufficient to consider how the composite operator gets rescaled;
the two-point function scaling follows immediately. Before implementing the RG
transformation, we should rewrite the composite operator in terms of the momentum
relative to the Fermi momentum. Using the fact that K = (KF + k)K̂, where k ∈
[−Λ, Λ] at any stage of the RG, on expanding to linear order in the momentum P ,
we find the displacement from the Fermi surface is given by k± ≡ ±K + P2 − KF ≈
k ± P cos θkP , while the angular coordinates are given by θ+ = θk − 2KP F sin θkP , θ− =
π + θk +
P
2KF
sin θkP . Therefore,
ŌΩ,P
Λ
P
sin θkP
=
ψ̄↑ ω + Ω, k + P cos θkq , θk −
2KF
k,ω
P
×ψ̄↓ −ω, k − P cos θkP , π + θk +
sin θkP
2KF
Z
×Θ(Λ − |k+ |)Θ(Λ − |k− |)
199
where
RΛ
k,ω
≡
R∞
dω
−∞ 2π
R 2π
0
dθk
2π
RΛ
dk
.
−Λ 2π
We have absorbed a factor of
√
KF into the
definition of the fields, as is usual in the RG. The last two factors are to ensure that
the actual momenta in the fermion lines are consistently within the cutoff.
It follows that the descendant operator is
<,<
ŌΩ,P
Λ/s
P
sin θkP
=
ω + Ω, k + P cos θkP , θk −
2KF
k,ω
P
<
×ψ̄↓ −ω, k − P cos θkP , π + θk +
sin θkP
2KF
Z
ψ̄↑<
where we have used the step functions6 to set the cutoff to Λ/s. In this expression,
we should now redefine (k ′ = sk, ω ′ = sω) to shift the cutoff back to its full value,
and then use the definition of the field rescaling once again, to find7
<,<
ŌΩ,P
Λ
P sin θk′ P
= s Z̃ψ
ω + sΩ, k + sP cos θk′ P , θk′ −
2KF
k ′ ,ω ′
P sin θk′ P
×ψ̄↓′ −ω ′ , k ′ − sP cos θk′ P , π + θk′ +
2KF
−2
Z
ψ̄↑′
′
′
This is the tree-level rescaling of the composite operator; at one loop, the composite operator acquires an anomalous dimension, whose effects may be captured with
another renormalization parameter Z̃O (we lump the factor of s−2 with this as well).
Thus, we have the overall rescaling:
1/2
<,<
′
ŌΩ,P
= Z̃O Z̃ψ ŌsΩ,sP
(9.23)
where the ′ denotes the fact that we have related fields in two different theories, the
original and the one after the RG step.
Following the same steps used in deriving the C-S equation previously, and combining the scaling of the descendant operator (Eq. 9.23) with the inhomogeneous term
6
7
Note that if both k± lie within the cutoff, k itself must lie within the cutoff.
Note that the RG does not change angles, a fact that is used implicitly here.
200
(Eq. 9.22) contributed by the pieces of the composite operator lying above the cutoff,
we have the following equation obeyed by the flow of the two-point function hŌOi:
hŌΩ̄,P̄ OΩ,P iS,Λ
1
=
2πvF
1
1−
s
+ Z̃O Z̃ψ2 hŌsΩ̄,sP̄ OsΩ,sP iSs ,Λ
(9.24)
Once again, we may use the fact that the left hand side and thus the right hand
side of this expression are independent of s to perform the same manipulations as
before, and arrive at the expression
1
∂
d 0 =
Ō
OsΩ,sP
+ β(g) + 2 + 2γO hŌΩ̄,P̄ OΩ,P iS,Λ +
2πvF
∂g
ds s→1 sΩ̄,sP̄
S,Λ
(9.25)
where we have defined 2 + 2γO =
2 d log(Z̃O Z̃ψ
)
ds
s→1
We leave it as an exercise to the reader to show that the final term on the right
∂
∂
+ P ∂P
on the two point corhand side is equivalent to the action of the operator Ω ∂Ω
relation function; any discrepancies between the two vanish because of the symmetry
of the Fermi surface. Making this substitution, we find
∂
1
∂
∂
Ω
+P
+ β(g) + 2 + 2γO hŌΩ̄,P̄ OΩ,P iS,Λ = −
∂Ω
∂P
∂g
2πvF
(9.26)
As a final step, we note that once again we need to remove a trivial delta function
in going between the expectation value and the correlation function, hŌΩ̄,P̄ OΩ,P iS,Λ ≡
(0,2)
ΓO (Ω, q)δΩ̄,Ω δP̄,P . Thus, we finally arrive at the inhomogeneous Callan-Symanzik
equation for the two-point Cooper pair correlator,
∂
∂
∂
1
(0,2)
+P
+ β(g) + 2γO ΓO (Ω, q; g, Λ) = −
Ω
∂Ω
∂P
∂g
2πvF
201
(9.27)
9.5
Solution of the Cooper Pair Callan-Symanzik
Equation
The next step is to solve the Callan-Symanzik equations: given a bare coupling
constant g0 , we wish to determine the long-wavelength, low-frequency behavior of
(0,2)
ΓO . Since the dimensionless combination of the frequency, momentum, and cutoff
√
that enter the correlation functions must be of the form Ω2 + P 2 /Λ ≡ Ω̃/Λ, (where
a phenomenological velocity - that depends on the Fermi-liquid parameters such as m∗
∂
∂
∂
- has been set equal to unity.8 ) Using this, we argue that Ω ∂Ω
+ P ∂P
↔ −Λ ∂Λ
when
acting on the correlation functions; this gives the slightly more tractable equation
∂
1
∂
(0,2)
− β(g) − 2γO ΓO (Ω̃; g, Λ) =
Λ
∂Λ
∂g
2πvF
(9.28)
We may now solve this equation by the method of characteristics [262], and find
(0,2)
ΓO (Ω̃; g0 , Λ) = e−2
Ω̃
Λ dx
1
x
R
1
−
2πvF
γO (g(x))
Ω̃
Λ
(0,2)
ΓO
1; g
Ω̃
Λ
!
,1
!
dx −2 R1x dyy γO (g(y))
e
x
(9.29)
d
g(x) = −β(g(x)) and g(x = 0) ≡ g0 .
dx
(9.30)
Z
1
with
x
Using the results for the beta function and anomalous dimension calculated in
Sec. 9.3, we have
 2
g Ω̃
Λ
(0,2)
 Γ(0,2)
ΓO (Ω̃; g0 , Λ) = 
O
g0
8
1; g
Ω̃
Λ
!
,1
!
"
1
−
g
2πvF g02
Ω̃
Λ
!
− g0
#
(9.31)
Such phenomenological terms cannot in general be obtained within the RG.
202
where the flow of the coupling is given by
g
Ω̃
Λ
!
=
g0
1 − ag0 log Ω̃
Λ
(9.32)
which clearly reflects the fact that g is marginal: as we take Ω̃ → 0, g vanishes
logarithmically. Note that we must always have Ω̃ < Λ, so that the logarithm in the
denominator is positive and does not lead to any singularity as we take Ω̃ → 0.
Since g Ω̃
is marginal, in the limit of interest, the second term in Eq. 9.31
Λ
dominates, and we have
"
!
#
Ω̃
1
(0,2)
g
− g0
ΓO (Ω̃; g, Λ) ∼ −
2πvF g02
Λ
Ω̃ log
a
Λ
1
=
2πvF 1 + ag log Ω̃ 0
Λ
(9.33)
We see that for g0 strictly zero, the expression diverges logarithmically as Ω̃ → 0,
reflecting the singularity in the zero-frequency, zero-momentum pairing response of
the free Fermi gas at T = 0. However, for any finite g0 , we find that the response is
(0,2)
nondivergent: ΓO (Ω̃; g, Λ) ∼ 1/2πvF g0 as Ω̃ → 0. Observe that, nicely enough, this
answer itself diverges as g0 → 0.
9.6
Another route
We have outlined our derivation above at some length for we were interested in a
particular method of getting the answer, in which we follow the irrelevant coupling all
the way to zero while continuing to renormalize. For the generic marginal coupling,
this is the easiest way to go and even for fermions there are other problems, e.g.
involving gauge fields, where we expect this technique will be the way to go.
203
However, there is another route to our answer—as readers may guess by looking
at it. In this approach we renormalize until we get to an exactly solvable problem and
then we appeal to the exact results. In our problem, the action with the purely BCS
channel interaction corresponds to the reduced BCS Hamiltonian, which has infinite
range interactions and is thereforw exactly solvable by a saddle point computation
in the infinite volume limit. The same method shows that the RPA result for the
superconducting susceptibility for this problem is exact. This has precisely the form
(9.33) with Λ now being the scale at which we switch to the exact solution. Adding
in the additional operator renormalizations gathered en route will change the answer
but not the finiteness of the result or its behavior as g0 → 0. Indeed, in this approach
it is also straightforward to explicitly include the Landau couplings as the resulting
Hamiltonian is still exactly solvable [140].
The general procedure we have described in this section is also what is used in
the implementations of the the RG for interacting fermions known as the Functional
Renormalization Group (FRG), see for example Appendix B in [97]. The difference
is that as in such work generally relevant flows with multiple coupling constants
get stopped at some scale the resulting problem is not typically exactly solvable in a
controlled sense. However, that has to do with the ends to which the FRG is put—the
idea is the same.
9.7
Concluding Remarks
Our result (9.33) verifies the claim with which we began—namely that the superconducting susceptibility of the Fermi liquid is finite due to the intercession of the
marginally irrelevant BCS coupling that is present even for repulsive interactions.
This exact compensation of the leading singularity by the irrelevant flow is somewhat surprising from the RG perspective, certainly if you compare with the results
204
on ferromagnets in four dimensions that we reviewed in the introduction. Possibly
multi-band systems will lead to richer possibilities but that is a subject for future
work. It is also of interest to extend the RG approach taken here to the derivation
of the effects of the marginal flow on the electron Green’s function. As this requires
application of the RG at two loops, this will require going beyond the straightforward
momentum shell method used in this chapter.
205
Appendix A
Impurity-induced random Zeeman
field
We briefly summarize the argument that leads to a coupling between a local
anisotropy in the disorder potential Ud and the Ising-nematic order parameter n.
Since the form factors of the two valleys are different, we expect that the portion of
the disorder potential that is antisymmetric in valley indices will lead to a spatially
dependent single-particle splitting between valleys; in the limit when the cyclotron
gap diverges, i.e. when the LLL approximation is exact, this is the only contribution,
and we can argue from symmetry that the corresponding random field should take
the form (∂x2 − ∂y2 )Ud (at least in the small-anisotropy limit). Note, however, that
this term is a total derivative, and contributes significantly only at the boundary of a
domain. To go beyond this, we must relax the ωc → ∞ limit, and allow for the effects
of Landau-level mixing to first order in Ud ; since this allows for terms of order Ud2 /~ωc ,
the random field now receives contributions of the form ((∂x Ud )2 − (∂y Ud )2 )/~ωc ,
which is not simply a boundary term.
To derive the higher-order contribution to the single-particle valley splitting from
the LL mixing terms, we make a simplifying assumption: namely, we ignore inter206
actions. While the interactions may combine with the effects of disorder to modify
details of the calculation, we expect that their neglect does not change the qualitative
features of our results. The Hamiltonian for noninteracting electrons in AlAs is, in
the Landau basis
Hni =
X
n,X
(n~ωc − µ) c†n,κ,X cn,κ,X +
X
mn
Ud (−q)Fκκ
(q)eiqx X̄ c†m,κ,X+ cn,κ,X−
q,X̄,κ,n,m
(A.1)
where we have generalized the notation of Chapter 4 to include Landau level indices
n,m. The form factors are given by
nm
Fκκ
(q)
m!
=
n!
iq
√ x − qy
2uκ
r
uκ
2
n−m
n−m
Lm
qy2 uκ
qx2
+
2uκ
2
2
qx
e− 4uκ −
2u
qy
κ
4
(A.2)
nm
mn
for n ≥ m, with Fκκ
(q) = Fκκ
(−q)∗ , and Lαn is the generalized Laguerre polynomial.
Following the approach of Yang and Haldane [88], we compute a renormalized
effective potential within the lowest Landau level by including Landau level mixing
in perturbation theory. The resulting potential is
V̂LLL =
X
κ
Ud,eff
(−q)eiqx X̄ c†0,κ,X+ c0,κ,X−
(A.3)
q,X̄,κ
where, to first order in LL mixing,
κ
00
Ud,eff
(−q) = Ud (−q)Fκκ
(q)
X Ud (−q′ )Ud (q′ − q) q×q′
|Ud |3
0n
′
n0
′
i 2
Fκκ (q − q )Fκκ (q ) + O
+
e
n~ωc
~ωc
′
q ,n6=0
(A.4)
We are primarily interested in the valley symmetry-breaking contribution from
this term, so we consider only the portion antisymmetric in κ. Assuming that the
207
disorder potential is smooth on the scale of ℓB , we may expand in gradients of Ud ; to
quadratic order in qx , qy , only the n = 1 term in the sum contributes, and we find
1
2
UdSB (−q) = Ud,eff
(−q) − Ud,eff
(−q)
2
1
∂x − ∂y2 Ud −q
= − λ − λ−1
4
1
+
λ − λ−1 (∂x Ud )2 − (∂y Ud )2 −q
2~ωc
(A.5)
The leading piece vanishes except on domain boundaries, as discussed; thus, the
dominant valley splitting arising from impurities is due to the second term. In
the weak anisotropy limit, we may add this symmetry-breaking term to the full
Hamiltonian and then perform the gradient expansion of E[n]; the result is a term
R 2
d r hpot (r)nz (r), with hpot given by Eq. 4.12.
A.1
Characterizing the disorder potential from
the mobility
We may estimate the strength Ud of the smooth random potential from the measured
sample mobility µ and the distance d of the dopant atoms from the plane of the
2DEG, and using the results of the previous section, deduce the parameters of the
random Zeeman field h. Taking the dopants to be Poisson-distributed, and assuming
that the potential fluctuations are screened by electrons in the 2DEG, we can estimate
the fluctuations of the potential in the plane of the 2DEG to be [56]
h|Ud (q)|2 i = (U0 d)2 e−2qd
(A.6)
where U0 is determined by the screening length and should be proportional to the
impurity density.
208
The scattering rate due to this potential is
Wp,p′ =
2π
|Ud (p − p′ )|2 δ (Ep − Ep′ )
~
(A.7)
Estimating the transport time, assuming that it is dominated by the Fermi surface,
we find
1
m
=
τtr
2π~2
Z
π
−π
dθ′
2π
2π
~
2 θ ×2d
2
(1 − cos θ)U02 d2 e−2kF sin
(A.8)
where the (1−cos θ) comes from the Boltzmann equation and is needed since processes
that scatter electrons at small angles do not contribute significantly to dissipation and
hence have a smaller effect on 1/τtr . For kF d ≫ 1, we have
m
~2
1
2
=
(U
d)
0
τtr
π~2
8(mvF d)3
(A.9)
Using the fact that 1/τtr = e/(mµ) where µ is the mobility,
U0 ≈
where we take m =
√
8πe~3 kF3 d
µm2
1/2
(A.10)
mx my .
Finally, we note that the characteristic length scale of the disorder potential is
roughly the distance of the dopant plane from the 2DEG, allowing us to estimate
that |∇Ud | ∼ U0 /d. Using the results above, the characteristic value of the symmetry
breaking term USB is given by
(d)
∆SB ∼
4
1
lB
(m
−
m
)
U02
x
y
2
2
2π~
d
(A.11)
(d)
corresponding to a sigma model random field h ∼ ∆SB /ℓ2B . Since h is correlated
roughly over a distance d, we find that the characteristic width of the random field
distribution is W ∼ (hd)2 .
209
Appendix B
Transfer Matrix for the AKLT
Model
The transfer matrix may be thought of as a map between functions defined on the
sphere:
Z
(T f )(n̂) =
dn̂′
T (n̂, n̂′ )f (n̂′ )
4π
(B.1)
where n̂, n̂′ ∈ S N −1 . In our case, as in most cases of interest, the transfer matrix is
a rotational scalar and we can work in the angular momentum basis. The eigenvalue
must depend only on the L2 eigenvalue l and not on the Lz eigenvalue m. It therefore
suffices to solve the problem in the case m = 0.
We wish to solve the eigenvalue equation:
λl fl (n̂) =
Z
S2
Dn̂′
T (n̂, n̂′ )fl (n̂′ )
4π
(B.2)
Since the kernel T depends only on n̂ · n̂′ = cos θ, we work in polar coordinates with
the z-axis along n̂ and substitute x = cos θ to obtain
1
λl fl (1) =
2
Z
1
dx T (x)fl (x)
−1
210
(B.3)
A natural guess for the eigenfunctions is that they are Legendre polynomials. The
)β .Using standard identities,
transfer matrix in our case is T (x) = ( 1+x
2
λl,AKLT
[Γ(β + 1)]2
=
Γ(β − l + 1)Γ(β + l + 2)
(B.4)
A similar discussion for the Heisenberg model for arbitrary N and the case of SU(N)
and Sp(N) groups, may be found in [63].
211
Appendix C
Stability against spin glass ordering
on a regular random graph
We follow closely the derivation of [256], Appendix A. In the thermodynamic limit,
the spin glass susceptibility
χSG =
1 X
[hn̂i · n̂j i]2
N ij
(C.1)
can be rewritten, by taking the average over the random graphs and using translational invariance, as
χSG =
∞
X
d=0
Nd [hn̂0 · n̂d i]2
(C.2)
where Nd is the number of sites at distance d from a reference site. The sum is
convergent as long as
lim (Nd )1/d [hn̂0 · n̂d i]2/d ≤ 1
d→∞
212
(C.3)
Note that (Nd )1/d → z − 1 for large d. In the paramagnetic phase, hn̂0 · n̂d i is given
by the response of hn̂0 i (the root) to a small magnetic field coupled to n̂d , a leaf at
distance d, of a tree whose other nodes are in the paramagnetic state |00i. Hence we
get (repeated indices summed):
dhn̂i0 i
=
hn̂0 · n̂d i ∝
dhid
Z
dψ
dn̂0 n̂i0
0
(n̂0 )
dhid
(C.4)
Clearly the term that gives the exponential dependence on d is the variation of ψ 0 (n̂0 )
with respect to hd . Using the recursion relation (7.3) we can rewrite it as
dψ 0 (n̂0 )
=
dhd
Z
dn̂1 · · · dn̂d
dψ 0 (n̂0 ) dψ 1 (n̂1 )
dψ d (n̂d )
·
·
·
dψ 1 (n̂1 ) dψ 2 (n̂2 )
dhd
and the exponential dependence is related to the eigenvalue of the transfer matrix
dψd (n̂d )
dψd+1 (n̂d+1 )
= T (n̂d , n̂d+1 ). These can be obtained by repeating the analysis
of section 7.3. Indeed if we use the same ket notation and write the variation
P
δ|d + 1i = ε ∞
l6=0,m clm |l mi we can use the results of section 7.3 to obtain
δ|di = ε
X λl
clm |l mi + O(ε2 )
λ0
(C.5)
l6=0,m
The absence of the factor (z − 1) with respect to Eq. (7.12) is due to the fact that
here we are only varying one of the neighbors of a given spin, the neighbor on the
path linking the root to the given leaf at distance d. The relevant eigenvalues are
therefore λl /λ0 and we obtain the condition
(z − 1) max{(λl /λ0 )2 } ≤ 1
for the convergence of χSG .
213
(C.6)
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