Download On Exotic Orders in Stongly Correlated Systems

On Exotic Orders in Strongly Correlated Systems
F. J. Burnell
A DISSERTATION
PRESENTED TO THE FACULTY
OF PRINCETON UNIVERSITY
IN CANDIDACY FOR THE DEGREE
OF DOCTOR OF PHILOSOPHY
RECOMMENDED FOR ACCEPTANCE
BY THE DEPRATMENT OF
PHYSICS
Advisor: S. L. Sondhi
November, 2009
c Copyright 2009 by F. J. Burnell.
All rights reserved.
Abstract
The key to understanding interacting many-body systems at low temperatures is often to identify their dominant ordering pattern. Though some types of order seem
extremely robust and prevalent in nature, those which lie at the frontier of our understanding are often the result of subtle competition between interactions, and hence
can be delicate and difficult to identify. This competition can lead to intricate phase
diagrams, as varying parameters favors different orders. They often also exhibit surprising emergent properties, including deconfined gauge fields or topological order not
present in the microscopic model. Understanding the nature and scope of these novel
orders is one of the key tools in mastering the vast practical and intellectual potential
of low-temperature correlated materials.
In this thesis, we explore intricate orders, both classical and quantum, which
emerge in several strongly correlated systems at low temperature. In Chapter 2, we
investigate magnetic ordering in classical antiferromagnets on a class of dilutions of
the face-centered cubic lattice. Exploiting the fact that the magnetic Hamiltonian
is that of a particular stacking of triangular planes, we find a 120 degree order for
Heisenberg and XY magnets, and an entropically driven 3-sublattice order in the Ising
case. In Chapter 3, we propose a (topologically ordered) gapless spin liquid state in
the highly frustrated pyrochlore antiferromagnet. We find the energetically favored
symmetric spin liquid state using a large Nf expansion, and present arguments for
its stability to fluctuations about the infinite Nf mean-field state. Chapter 4 explains
iii
and generalizes an interesting relationship between the algebraic structure of these
states and the representation theory of Lie groups. Chapter 5 turns to a classical
ordering with a fractal, devil’s staircase structure found in 1-d systems with infiniteranged interactions. The fate of these states in dipolar bosonic gases, where quantum
effects come into play, is investigated – along with an interesting generalization of the
classical problem. Finally, in Chapter 6 we consider how to stabilize states exhibiting
the topological order of fractional quantum Hall states in 3 dimensional crystals,
finding a parameter régime in doped graphite in which such states appear energetically
favorable.
iv
Acknowledgements
Looking back over the contents of this thesis, and indeed my life during the past
6 years, I can’t resist evoking one of my favorite quotes from J.R.R. Tolkien, who
reminds us: “Not all who wander are lost”. Indeed, the most rewarding part of
assembling what follows was to see, in retrospect, the paths connecting the different
regions of my field that I have passed through and realize that the journey has brought
me, quite coherently, to where I wanted to go. Of course, it didn’t always feel that
way at the time, and much of what I would like to acknowledge is the guidance and
support of those who helped me find both a road which could take me in the right
direction, and the faith in my destination to follow it.
I am forever grateful to my advisor, Shivaji Sondhi, for his insightful mentorship
and consistent good humor. At the time that he took me on, two years into my
PhD, my knowledge of condensed matter physics did not extend much beyond the
observation of one of my colleagues, in preparing for our general exams, that it seemed
mostly to be about taking Fourier transforms. In the four years that we have worked
together Shivaji has slowly filled in the vast unknown territory beyond Bloch waves,
and left me with a road map by which to navigate the varied terrain of correlated
many-body systems. Along the way he has introduced me to several parts of its
landscape which I find inspiring and beautiful – places that have convinced me that
I have wandered in the right direction.
From the outset Shivaji has been an astute manager, whose advice in everything
v
from the most technical aspects of research to how to seek out questions best suited to
my interests has been invaluable to me. He has scrutinized everything from conceptual
ideas to the wording of manuscripts with exacting care – taking the time to teach me
many skills beyond those necessary to complete the task at hand. Always ready with
unforeseen questions, he has repeatedly challenged me to understand things more
deeply, and to recognize honestly the boundaries of my own understanding. But
above all, he has been a constant reminder of what drew me to academic life in the
first place: a genuine intellectual, who finds fascinating questions in a multitude of
areas of life, and takes sincere pleasure in seeking their answers.
It has been a great experience to find myself part of a community of CMT students,
post-docs, and faculty during my time at Princeton. I would like especially to thank
Andrei Bernevig, for initiating the work that lead to the last chapter of this thesis, and
for his consistent faith in me; and Meera Parish, who has been a valuable collaborator,
notably in the contents of Chapter 5. In the past year I have also greatly enjoyed
collaborating with Steve Simon, though the results have not found their way into
this thesis. For his helpful insights at several junctures, as well as for reading the
many pages which follow, I am grateful to David Huse. I have greatly appreciated the
numerous exchanges of questions and ideas that I have had with Chris Laumann, who
has also been my go-to person in times of computational difficulty and often a source
of perspective. Charles Mathy has been a much-appreciated source of information,
especially in the field of cold atoms, as well as of energy and enthusiasm. And I
am grateful to Siddharth Parameswaran, not only for many interesting conversations
both within physics and without, but also for the generous favors he has done for me
as a friend – not the least of which has been to print and hole-punch all 230+ pages
of this thesis!
Outside of the physics department, I am also blessed with wonderful parents,
whose presence and support over the years has meant more to me than I can possibly
vi
fit into words. I am grateful to both of them for keeping me grounded through the
ups and downs of graduate school. To my mother, for the numerous ways in which
she has helped me differentiate between the directions I wanted to go, and the paths
I might follow because they happened to lie directly ahead. And to my father, for
reminding me at crucial moments along the way that good science is really about
enjoying yourself. I am also grateful to both of my siblings: to my elder sister Tara,
for the constant support and advice which was never more than a phone call away,
and to Kaitlyn, for being such a good and patient listener on subjects ranging from
mathematics to life choices.
There have been many other people who have shaped the course of my time at
Princeton, and whose names also belong in these pages. Beth, for many shared
adventures and for always remembering the good things in life; Anita, for making the
difference between student housing and home for many years; and Andrew, for always
having something unexpected to offer. Akash and Katie, for being caring room-mates
and faithful friends. My many friends at the 2D co-op, for countless memorable dinner
conversations which added so much color to campus life. And Robin, for everything
from lending me his computer, to editing job applications, to being the person I could
always call.
As in most of my travels in life, I set off to graduate school thinking I was on a
private mission to arrive at a particular series of destinations, and learn from these a
precise set of skills and facts. Only to find, in retrospect, that the most valuable part
of the experience has not been arriving at these planned destinations, but the many
unforeseen things I have learned from and shared with the people I have met along
the way. It has been a long, but overwhelmingly worthwhile, journey.
vii
Contents
Abstract
iii
Acknowledgements
v
Contents
viii
List of Figures
xiii
List of Tables
xv
1 Introduction
1
2 Classical Frustrated Magnetism on the diluted FCC lattice
11
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.2
Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.3
Antiferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.3.1
Eigenspectrum of Interaction Matrix . . . . . . . . . . . . . .
21
2.3.2
XY, Heisenberg and N > 3 cases . . . . . . . . . . . . . . . .
23
Ising case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.4.1
Zero temperature entropy . . . . . . . . . . . . . . . . . . . .
26
2.4.2
Order by disorder . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.4.3
LGW analysis . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
2.4.4
Monte Carlo results . . . . . . . . . . . . . . . . . . . . . . . .
30
2.4
viii
2.5
Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Quantum Frustrated Magnetism on the Pyrochlore lattice
31
34
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
3.2
The Large-N Heisenberg Model: Spinons and Gauge Fields . . . . . .
38
3.3
Mean-Field Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
3.3.1
Saddle Points of the Nearest Neighbor Heisenberg Model . . .
41
3.3.2
The Monopole Flux State . . . . . . . . . . . . . . . . . . . .
44
3.3.3
Low Energy Expansions of the Spinon Dispersion . . . . . . .
47
3.4
Stability of the Mean-Field solution: the role of the PSG . . . . . . .
50
3.5
The PSG of the monopole flux state . . . . . . . . . . . . . . . . . .
54
3.6
3.7
3.8
3.5.1
Symmetries of the pyrochlore lattice . . . . . . . . . . . . . .
55
3.5.2
The PSG . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
PSG invariance and the fate of the monopole flux state . . . . . . . .
58
3.6.1
Time reversal and Parity . . . . . . . . . . . . . . . . . . . . .
63
3.6.2
PSG symmetry and perturbation theory in the long wavelength
limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
Fluctuations about mean-field . . . . . . . . . . . . . . . . . . . . . .
67
3.7.1
Perturbation Theory on the lattice . . . . . . . . . . . . . . .
69
Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
4 Frustrated hopping problems on lattices derived from Lie Algebras 79
4.1
4.2
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
4.1.1
Lie Algebras and Hamiltonians . . . . . . . . . . . . . . . . .
80
4.1.2
Dirac-like continuum limits
. . . . . . . . . . . . . . . . . . .
85
4.1.3
What follows . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
Lattice construction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
4.2.1
87
Properties of our lattices.
. . . . . . . . . . . . . . . . . . . .
ix
4.3
4.4
4.5
4.6
4.2.2
Lattices in d = 2 . . . . . . . . . . . . . . . . . . . . . . . . .
90
4.2.3
Lattices in d = 3 . . . . . . . . . . . . . . . . . . . . . . . . .
92
Flux Hamiltonians in d = 2. . . . . . . . . . . . . . . . . . . . . . . .
97
4.3.1
SU(3) and kagomé . . . . . . . . . . . . . . . . . . . . . . . .
97
4.3.2
Flux Hamiltonians for Sp(4)=SO(5) . . . . . . . . . . . . . . 101
Flux Hamiltonians in d = 3 . . . . . . . . . . . . . . . . . . . . . . . 107
4.4.1
Flux on the SO(6) spinor lattice. . . . . . . . . . . . . . . . . 107
4.4.2
Flux on the SO(6) vector lattice. . . . . . . . . . . . . . . . . 113
4.4.3
Sp(6)
4.4.4
SO(7) spinor . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.5.1
Relevance to mean-field solutions of the Heisenberg model . . 121
4.5.2
Hamiltonians beyond the linear approximation . . . . . . . . . 124
4.5.3
Extensions to Higher Dimensions . . . . . . . . . . . . . . . . 125
Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . 126
5 The devil’s staircase in 1-dimensional dipolar Bose gases in optical
lattices
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.1.1
5.2
Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . 130
The ultra-cold dipolar Bose Gas . . . . . . . . . . . . . . . . . . . . 131
5.2.1
5.3
128
Bosons in 1D optical lattices . . . . . . . . . . . . . . . . . . . 131
Classical solutions and the devil’s staircase . . . . . . . . . . . . . . . 135
5.3.1
Commensurate Ground States . . . . . . . . . . . . . . . . . 136
5.3.2
The Devil’s staircase . . . . . . . . . . . . . . . . . . . . . . . 141
5.3.3
Structure of the q-Soliton State . . . . . . . . . . . . . . . . . 141
5.3.4
Energetics of the qSS . . . . . . . . . . . . . . . . . . . . . . . 143
5.3.5
Proof of devil’s staircase structure . . . . . . . . . . . . . . . 144
x
5.4
Away from the classical limit: Mott-Hubbard transitions in the strong
coupling expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.4.1
5.4.2
5.5
5.6
Strong Coupling Expansion
. . . . . . . . . . . . . . . . . . 147
Bosonization treatment of the phase transitions . . . . . . . . 152
Effects of the Trapping potential . . . . . . . . . . . . . . . . . . . . . 155
5.5.1
t = 0 physics and the LDA . . . . . . . . . . . . . . . . . . . . 156
5.5.2
Spatial Profiles of Atoms in a harmonic trap at finite t . . . . 158
5.5.3
Density profiles . . . . . . . . . . . . . . . . . . . . . . . . . . 161
Departures from Convexity . . . . . . . . . . . . . . . . . . . . . . . . 164
5.6.1
Effective convexity and thresholds for double-occupancy formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
5.7
5.6.2
Arguments for monotonicity of Uc with filling fraction . . . . . 176
5.6.3
Structure of the doubly-occupied régime . . . . . . . . . . . . 178
Interesting phenomena in the non-convex régime
. . . . . . . . . . . 180
5.7.1
A new staircase
. . . . . . . . . . . . . . . . . . . . . . . . . 181
5.7.2
Supersolids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
5.8
Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
5.9
Supplementary Material . . . . . . . . . . . . . . . . . . . . . . . . . 185
5.9.1
General Perturbation Theory . . . . . . . . . . . . . . . . . . 185
5.9.2
Bounds on the volume of Devil’s staircase lost at small t . . . 192
5.9.3
Creation of double occupancies in CGS states: calculation of
dipolar and quadrupolar interactions . . . . . . . . . . . . . . 196
5.9.4
Lattice-scale arguments for adding charge as double occupancies 199
6 Mechanisms for generating fractional quantum Hall states in 3-d
materials
204
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
6.2
Experiments on 3D compounds in high magnetic fields . . . . . . . . 209
xi
6.3
Candidate States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
6.3.1
Staged Quantum Hall Liquids . . . . . . . . . . . . . . . . . . 213
6.3.2
Staged Wigner Crystals . . . . . . . . . . . . . . . . . . . . . 215
6.3.3
Miniband States
. . . . . . . . . . . . . . . . . . . . . . . . . 219
6.4
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
6.5
Supplementary Material: Monte Carlo methods for quantum Hall States224
6.6
Supplementary Material: the Ewald Method . . . . . . . . . . . . . . 224
References
235
xii
List of Figures
2.1
The triangular lattice . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
2.2
Stacking vectors for the tunneled FCC lattice . . . . . . . . . . . . .
16
2.3
TS-FCC lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.4
Equivalence of TS-FCC and stacked triangular lattices . . . . . . . .
19
2.5
TS-HCP Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.6
Maximally flippable Ising configuration on the honeycomb lattice . .
27
2.7
Correlation of sublattice magnetizations in the semi-stacked triangular
Ising antiferromagnet. . . . . . . . . . . . . . . . . . . . . . . . . . .
32
3.1
Bond Order in The Monopole Phase
. . . . . . . . . . . . . . . . . .
44
3.2
Spectrum of the monopole flux state . . . . . . . . . . . . . . . . . .
46
3.3
Constantenergy surface of the monopole flux state . . . . . . . . . . .
47
4.1
The kagomé lattice and SU(3) . . . . . . . . . . . . . . . . . . . . . .
83
4.2
Planar pyrochlore and SO(5) . . . . . . . . . . . . . . . . . . . . . .
91
4.3
Planar pyrochlore and SP (4) . . . . . . . . . . . . . . . . . . . . . .
93
4.4
The octachlore lattice of SP (6) . . . . . . . . . . . . . . . . . . . . .
94
4.5
The 3-d checkerboard SO(7) spinor lattice . . . . . . . . . . . . . . .
95
4.6
The pyrochlore SO(6) spinor lattice . . . . . . . . . . . . . . . . . . .
96
4.7
Flux assignment on the kagomé lattice . . . . . . . . . . . . . . . . .
98
4.8
The flux on the SO(5) lattice . . . . . . . . . . . . . . . . . . . . . . 102
xiii
4.9
Possible flux assignments to the pyrochlore. . . . . . . . . . . . . . . 109
4.10 Flux assignment on the octachlore . . . . . . . . . . . . . . . . . . . . 114
4.11 Flux assignment on the octachlore II . . . . . . . . . . . . . . . . . . 115
4.12 Two possible flux assignments for on the SO(7) lattice . . . . . . . . 120
5.1
Perturbative calculation of the Mott to SF phase boundary . . . . . . 151
5.2
Local density approximation for filling profile in a finite trap . . . . . 157
5.3
Density profiles in a finite trap calculated using simulated annealing . 159
5.4
Density profiles in a harmonic trap at finite hopping I . . . . . . . . . 162
5.5
Density profiles in a harmonic trap at finite hopping II . . . . . . . . 163
5.6
Thresholds of stability to double occupancy in the classical system . . 167
5.7
Super-solidity in the doubly occupied régime . . . . . . . . . . . . . . 183
5.8
Dipolar interaction of double occupancies in certain odd-denominator
filling fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
6.1
Energetics of quantum Hall liquids . . . . . . . . . . . . . . . . . . . 214
6.2
Energetics of Wigner crystals . . . . . . . . . . . . . . . . . . . . . . 218
6.3
Energetics of the miniband and quantum Hall liquid states . . . . . . 219
6.4
Approximate phase portrait of doped graphite at high magnetic field
xiv
221
List of Tables
3.1
Mean-field energies for projected and unprojected ground states of the
mean field ansätze on the pyrochlore . . . . . . . . . . . . . . . . . .
44
3.2
PSG of Spinons under Proper Rotations . . . . . . . . . . . . . . . .
58
3.3
PSG action of screw rotations . . . . . . . . . . . . . . . . . . . . . .
58
3.4
T-breaking in the monopole state . . . . . . . . . . . . . . . . . . . .
64
3.5
Effect of Point Group rotations on low-energy eigenstates . . . . . . .
66
3.6
Effect of 2-fold glide rotations on low-energy eigenstates . . . . . . . .
66
4.1
Hopping amplitudes along different-length roots for mean-field solutions on several Lie lattices . . . . . . . . . . . . . . . . . . . . . . . . 124
5.1
Experimental values of dipolar interaction strength for polar molecules 134
5.2
Commensurate states . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
xv
Chapter 1
Introduction
It is interesting to ask what inspires research in various areas of physics. Since the
first forays into quantum mechanics in the 1920’s, the perplexing and counter-intuitive
phenomena of this microscopic world have captured the fancy of several generations
of physicists. Unveiling its mysteries has enabled many of the major technological
innovations of the twentieth century, from medical diagnostics such as MRI to the
unprecedented destructive power of the atomic bomb. It has also fundamentally
revolutionized our understanding of the world in which we live – shaking the foundations of long-held beliefs about the nature of space-time, energy, and determinism in
physics.
In this light, there are two perspectives from which to view the study of condensed
matter physics. First, it can be described as the study of materials with novel properties which may find use in our everyday lives. Over the past half-century our deepening understanding of many-body systems has enabled a multitude of technological
innovations, including the transistors, lasers, fiber optics, and liquid crystals which
are now an integral part of how we process, store, transmit and display information
on a daily basis. Second, many condensed matter systems exhibit emergent phenomena, germane to understanding systems at remote energy scales, in a régime where
1
2
a fantastic range of precise experimental probes can be brought into play. Thus the
famous Higgs particle soon to be under investigation at the LHC is the mathematical
cousin of the condensate of paired electrons in a super-conductor; ultra-high energy
relativistic electrons have mathematical analogues in room-temperature graphene;
and the topological field theories first proposed in the 1980’s as a tool for understanding quantum gravity are physically manifested in the low-energy description of
the quantum Hall effect.
The interplay between practical and intellectual motivations often uncovers theoretically interesting – but completely unanticipated– phenomena. For example, it
is unlikely that the link between magnetism and high-temperature super-conductivity,
even today only partially understood, would have been foreseen on theoretical grounds.
Nonetheless almost 3 decades of intensive experimental research, motivated chiefly by
the tremendous potential for applications of high-temperature superconductors, suggest a fundamental connection. Condensed matter physics is thus both a window
into many of the beautiful mathematical phenomena manifest in nature, with the
possibility of realizing these in the laboratory, and a constant source of new puzzles
when our elegant descriptions turn out not describe the experiments, or when experiments themselves point to qualitatively novel behavior. The marriage of these
two has birthed scientific advances both of tremendous practical importance, and of
intrinsic intellectual interest.
But what makes a material interesting, in this context? Perhaps the most honest
answer to this question is that its low-temperature behavior cannot be described by
physics similar to that which we use to describe the relatively few states of matter
commonly encountered at room temperatures – band insulators, Fermi liquids, and
simple magnetic states such as ferromagnets. Instead, it exhibits states of matter
which challenge these paradigms – often through exotic orders or competing interactions of a quantum mechanical nature.
3
Indeed, a survey of the major areas of condensed matter physics suggests that
quantum mechanics can become important at low temperatures in two rather different ways. First, it can determine the way a material orders at low temperatures. The
quantum statistics of fermions and bosons, for example, drives them to form Fermi
liquids and Bose condensates, respectively – neither of which is a phase of classical
matter. Second, the quantum non-commutativity familiar from Heisenberg’s uncertainty principle changes the nature of competing interactions: typically competing
terms in the Hamiltonian are not simultaneously diagonal, rendering the interplay
between them much more complex than in the classical context.
Quantum mechanical order is the essence of most of the currently understood novel
low-temperature phases of matter. The beautiful phenomenon of Bose condensation
is an excellent example of this. The Bose-Einstein condensate admits a simple description – a macroscopic number of bosons occupy a single quantum state – but one
which cannot arise unless the quantum statistics of the bosons is taken into account.
Interestingly, this is one case in which theory preceded experiment by many decades:
though superfluidity was discovered in Helium in the 1930’s, it was not until the advent of trapped atomic gases in the past decade that weakly-interacting Bose gases
were realized in the lab. The precise control afforded by modern optics enabled atoms
to be trapped and cooled below the Bose-Einstein condensation temperature, realizing one of the most dramatic predictions of quantum statistical mechanics [1, 2]– and
in the process, produced the most tunable existing experimental tool for engineering
interacting many-body systems.
Another example of novel order is the quantum Hall effect. If electrons in 2 dimensions are placed in a strong magnetic field, quantum mechanics dictates that when the
number of electrons is equal to the number of flux quanta of the magnetic field which
pierce the area of the sample, a special kind of insulating state with quantized Hall
resistance occurs. These insulating states are topological, in the sense that though
4
the bulk system is an insulator, its boundary has a conducting mode whose nature
is robust to all perturbations which do not open a gap in the bulk of the sample.
The integer quantum Hall phases are only the tip of the iceberg, however: since their
discovery in 1985 [3], a plethora of states with quantized resistivity have been discovered at fractional fillings [4]. In addition, other phases with similar topological
properties have been proposed in various physical contexts [5, 6, 7, 8, 9]. In all of
these systems we can understand the exotic properties of the system as due to the
fact that quantum mechanics is, like in the case of the Bose condensate, an integral
part of the low-temperature order.
Topologically ordered phases highlight an interesting feature frequently found in
correlated many-body systems – emergent phenomena which resemble physics expected at radically different energy scales. The relativistic field theories first conceived
to study the physics of fundamental particles appear as good effective descriptions
of collective behavior of materials in several contexts. For example the Dirac equation, conceived for highly relativistic electrons, proves to be the correct low-energy
description of charge carriers in graphene.1
Dirac fermions coupled to emergent
gauge fields, which describe the strong interactions in these systems, have also been
proposed to describe certain quantum magnets. Topological phases typically admit
an effective description in Chern-Simons theory. These surprising parallels with highenergy physics are one of the striking features of quantum mechanical order.
The experimental realization of weakly interacting Bose condensates in cold atoms
also demonstrated a second way in which quantum mechanics can come to play an
essential role in the physical properties of a system. From the starting point of a
weakly interacting gas, interesting variations can be introduced in the experimental
set-up: lasers can be used to generate an artificial lattice, and contact interactions
between the particles can be tuned to different strengths. As the strength of the
1
The interested reader can thus create her very own Dirac particles with a pencil and a little
Scotch tape.
5
interactions are increased, the system undergoes a transition from a superfluid phase
to a Mott insulating state in which each boson is effectively localized at a single
lattice site [10]. This dramatic change results from a competition between kinetic
terms which favor a superfluid, and interactions which prefer the Mott insulator. As
these two terms do not commute, the ordering patterns of these two states is very
different.
Competing interactions come into their own in systems where the Hamiltonian
contains non-commuting terms, neither of which alone is sufficient to understand the
dynamics. The most famous and longest-standing such problem is the enigma of hightemperature superconductivity. In the cuprates– the compounds in which high Tc was
discovered– strong interactions favor states with a single electron per copper atom
(Mott insulators). This invalidates the usual approach to superconductivity, in which
interactions are treated as perturbations about the non-interacting limit. However,
perturbative treatments about the strongly interacting limit are also of limited use.
In the superconducting phase these interactions alone do not fix a unique electronic
configuration. For example, in hole-doped samples some of the copper atoms will have
no associated electron; these holes are free to propagate in the lattice without paying
the cost of doubly occupying a site. One can envision resolving this degeneracy
using perturbation theory about the strongly interacting limit – but this predicts
fundamentally different physics than that observed in the cuprates. In fact, Nagaoka
[11] showed that doping a Mott insulator with a single hole leads to ferromagnetic
interactions, rather than the experimentally observed anti-ferromagnetism near the
onset of superconductivity. Instead, understanding the cuprates seems to require an
approach which can account for both the strong on-site inter-electron repulsion and
the kinetic energy of freely propagating holes. The former term is easily represented
in position space; the latter, in momentum. As these two bases do not commute,
however, the challenge of adequately representing both has remained an open question
6
for almost 30 years.
A similar spirit underlies the role of quantum mechanics in magnetism. In this
case the non- commuting terms are the values of spin along different axes. In geometries where it is energetically favorable to align all spins along a particular axis,
the quantum-mechanical nature of the spins enters as small fluctuations about this
order. However, in frustrated magnets this is not the case, and we expect that these
quantum fluctuations play a key role in determining the order itself. In sufficiently
frustrated systems, there can be a macroscopic number of degenerate classical solutions (in which each spin points in a fixed direction), and only quantum fluctuations
can resolve this degeneracy at zero temperature. We then find ourselves presented
with a conundrum similar to that of the cuprates: a strongly interacting system with
a large degeneracy of ground states and no simple state about which perturbation
theory is sufficient to pinpoint the characteristics of the phase.
If quantum mechanics is the source of exoticism in materials, what tools can we
use to understand its consequences? The notion that quantum mechanics engenders
competing, non-commuting interactions is very general; hence we virtually automatically venture into the realm of problems to which there is no obvious solution. A
great deal of progress has been made by considering situations in which one set of
mutually commuting terms dominates over the others, which can then be treated
perturbatively. Thus, for example, Landau Fermi liquid theory explains the general
behavior of metals, and can be used to deduce the existence of a phase transition towards BCS superconductivity [12]. Similarly, the behavior of most low-temperature
magnets is well-described by perturbing about a magnetically ordered solution, giving a spin-wave expansion which correctly predicts the absence of true long-ranged
order in dimension less than 3. In one spatial dimension, we may draw on powerful
non-perturbative techniques to give insight into these questions. The machinery of
bosonization – an approximate mapping between a large class of problems describing
7
interacting fermions on a lattice to the problem of non-interacting bosons – gives
insight into the nature of strongly interacting fluids in one dimension. We may go
even further than this and study the many integrable models in one dimension, for
which exact solutions are known.
However, at the intermediate couplings relevant to many important systems –
not the least of which is the cuprates – the perturbative approach is often flagrantly
inapplicable. Attempts to generalize non-perturbative methods from one to higher
spatial dimensions have also had little success. This leaves a wide range of intriguing
physical questions which require a different approach. One possible resort is to search
for new kinds of local order parameters, and try to establish that their orders are
energetically favored. Thus we could begin by supposing that at low temperatures
the system will break some symmetry, and that the symmetry-broken phase is well
described by controlled fluctuations about the new, symmetry-broken vacuum. Such
symmetry-breaking generally occurs because some bosonic field acquires a vacuum
expectation value; however, insightful choices of this order parameter can predict
a plethora of novel symmetry-broken states. The ensuing mean-field descriptions
have been useful tools in understanding phenomena such as superconductivity and
superfluidity; this approach has also proven fruitful in studying various types of spinorder in frustrated magnets. Hence one active frontier of condensed matter physics
is the search for new (exotic) types of symmetry-breaking orders described by local
order parameters.
More recently another, non-symmetry breaking, type of order has appeared on
the scene [13]. This order is not associated with a symmetry-breaking local order
parameter, but rather by the global topological properties of the phase, such as the
protected conducting edge modes of the quantum Hall states. The nature of this
order is fundamentally different from that of the symmetry-broken orders described
above; however, many topological phases are nonetheless best understood by starting
8
with approximate mean-field descriptions. These are constructed so as to include the
topological features of the phase in the mean-field state – most notably, emergent
gauge fields at low energies. Unfortunately the task of including fluctuations in these
gauge fields is delicate, making these states relatively difficult to describe accurately.
Nonetheless their novel properties – notably fractionalization of charge and, in 2
dimensions, possibly statistics – make them one of the most captivating manifestations
of the complex behavior engendered by mixing quantum mechanics and interacting
many-body systems.
As with the symmetry-breaking orders described above, an insightful approximate (mean-field) approach allows us to understand or predict these interesting new
phases of matter, in parameter régimes where perturbative approaches must fail. In
both cases, where the microscopic Hamiltonian cannot be treated directly with exact
methods, physical insight into the natures of the low-temperature ordering – whether
super-fluid, magnetic, or topological – is a crucial tool in understanding these complex
systems.
With this general overview of the context of this thesis, we now turn to its contents.
The themes of searching for novel quantum many-body states, and identifying the
associated low-temperature orders, will appear in many guises in the coming chapters.
We will begin in Chapter 2 with a study of symmetry-breaking order, in the context
of frustrated magnets. In the example discussed here the order is in fact a classical
ordering of the orientations of spin moments; however the techniques used to describe
quantum states with broken symmetry are very similar. The question for classical
magnetism is how best to resolve a seeming macroscopic degeneracy of ground states;
we study this problem for several different spin models (Heisenberg, XY, and Ising)
on a set of dilutions of the FCC lattice. In the first two cases energetics specify a
classical ordering pattern for the spins; in the third, the order is fixed only by entropic
considerations.
9
From here, we move in Chapter 3 to a study of a more exotic, topologically
ordered phase. The example we will consider is a so-called spin liquid state, in which
the magnetic order is topological in nature. We propose such a state on the highly
frustrated pyrochlore lattice, and explore its properties in a mean-field treatment, as
well as the delicate arguments for its stability. In so doing we will confront many of
the difficulties with this inexact route to describing topologically ordered states.
In studying this spin-liquid state on the pyrochlore lattice, we find an interesting
connection between the lattice geometry and the internal properties of the mean-field
Hamiltonian, many of which arise because it is an element of a Lie algebra related
to the lattice in a specific way. In Chapter 4 we take an amusing digression and
explore generalizations of this behavior on other lattices, describing several ‘spinliquid’ candidate states with interesting symmetry properties.
In Chapter 5, we consider a rather different type of system – namely, bosons in
one-dimensional optical lattices. Here quantum mechanics appears as a competition
between potential terms which seek to localize bosons on a particular site, and kinetic
terms which favor a superfluid state of indefinite local boson density. The resolution
of this is that the system finds itself in one of two ordered phases – a Mott phase,
in which order appears in the number of bosons on a given site in the lattice, and a
superfluid phase, in which optimizing kinetic energy results in phase coherence (and
hence superfluid order) in the bosons. We investigate the results of this competition
in detail for the case where the bosons have an infinite-ranged repulsive dipolar interaction, finding a series of phase transitions between Mott and super-fluid phases. By
considering a parameter regime where bosons may doubly-occupy sites in the lattice,
we will also find ‘super-solid’ phases in which the two orders co-exist.
Chapter 6 addresses a rather more pragmatic question – namely, when can the
exotic topological orders of fractional quantum Hall states exist in 3-dimensional systems? This question can also be framed in terms of a competition between kinetic
10
and potential terms – in this case, the kinetic terms favor electrons de-localized along
the direction of the magnetic field, while the potential terms favor the strong correlations in the planes perpendicular to this field which lead to the topologically ordered
fractional quantum Hall states. We find a parameter régime, potentially experimentally relevant to anisotropic crystals such as graphite, in which the potential terms
dominate and fractional states may arise.
In summary, this thesis explores several of the ways in which novel orders emerge
in the collective behavior of systems of many interacting particles at low temperatures.
More often than not, the resulting orders have inherently quantum-mechanical properties, and manifest phenomena such as quantum electrodynamics and topological
properties once thought to be germane only to describing the physics of fundamental
particles at ultra- high energies. In all cases we find that the collective behavior is
strikingly different from the simple dynamics of its constituent particles – and therein
lies the challenge and the magic of condensed matter physics. As P. W. Anderson so
aptly put it, “more is different”.
Chapter 2
Classical Frustrated Magnetism on
the diluted FCC lattice
2.1
Introduction
In this Chapter, we will consider one of the most frequently-asked questions in classical
magnetism: namely, we will consider various magnetic interactions in a novel lattice
geometry introduced by Torquato and Stillinger [14], and ask what types of magnetic
ordering occur. The question of magnetic order in materials with this structure
interesting principally since the structures are geometrically frustrated – because of
the lattice geometry, it is impossible to simultaneously minimize all interactions in
the magnetic Hamiltonian. The result is similar in spirit to the scenario of adding
extra, non-commuting interactions to the Hamiltonian (the effect of which, in the
magnetic context, is also typically called frustration). The classical ground states
of frustrated systems are thus weakly split in energy from other configurations – in
extreme cases, leading to a macroscopic degeneracy of classical ground states. Because
frustration weakens the classical order, fluctuations may play a significant role in the
low-temperature physics. It is thus precisely where classical ordering is weak that the
11
12
phenomena of quantum magnetism, to which we will turn in later chapters, comes
into its own.
Ordering in the presence of frustration is thus a subtle question. Understanding
the nature and dynamics of the classical solutions is an important point of departure
for addressing this issue. Specifically, if the spin per site S is large, the classical order
is the leading term in the 1/S expansion typically used to probe the quantum ground
states. Even at small spin, at finite temperatures thermal fluctuations about the
classical ground states often determine the ordering pattern. If the classical ground
state is unique, these fluctuations typically induce small corrections to the expected
ordering; where there are several degenerate classical ground states the fluctuations
break this degeneracy and select a particular magnetic order, in a phenomenon known
as ‘order by disorder’. Hence understanding the nature and degeneracy of classical
ground states in a frustrated system is a key step in unravelling its delicate order, at
least for the situations mentioned above where a controlled expansion in the fluctuations is valid. The question of how to proceed when such expansions fail will be left
to the next chapter.
Geometrical frustration is, as the name suggests, a phenomenon which depends
strongly on the geometry of the lattice under consideration. Here we will tackle the
issue of nearest-neighbor magnetic interactions on a family of dilutions of the face centered cubic (FCC) lattice and its relatives introduced by Torquato and Stillinger [14].
These lattices, which we shall term Torquato-Stillinger (TS) packings, are the lowestdensity known structurally stable periodic arrangements of spheres in 3 dimensions.
They are constructed by diluting a close-packed lattice to leave a regular structure
with a local coordination number of 7 and a spherical packing density of
√
2π
.
9
Their
close relation to the FCC structure suggests that materials with such packings should
be realizeable. Since the FCC lattice itself is known to produce highly geometrically
frustrated magnetic interactions [15, 16], it is interesting to ask what effect these
13
dilutions have on the magnetic properties of these crystal structures.
In this Chapter we examine the possible magnetic phenomena in TS packings.
Specifically we determine the ground states, low temperature ordering and the nature of the phase transition to the paramagnetic state in a large subclass of the TS
packings for nearest neighbor antiferromagnetic interactions for classical Ising, XY,
and Heisenberg spins and as well as for O(N) spins with N ≥ 4. In this task we
will be greatly aided by the simple observation that this subclass of TS packings is
topologically equivalent to a set of stacked triangular lattices with half of the stacking bonds removed. As stacked triangular lattices have been studied intensively (see
Refs. [17], [18] and references therein), we will be able to carry over various results
from that work. The results presented here were first published in Ref. [19].
In the following we first review the TS construction of their packings in Sect. 2.2,
and single out a dilution of the FCC packing (lattice) as exemplifying the stacked
triangular structure that we will focus on. We explain how this lattice turns out to
be identical, from the point of view of the magnetic Hamiltonians we consider, to an
infinite family of TS packings – and to the semi-stacked triangular lattices mentioned
above. Next we present results for antiferromagnetism on the TS-FCC packing and
its relatives. We first consider XY and Heisenberg interactions in Sect. 2.3.2; these
phases order according to the usual 120 degree state on the triangular lattice. We
then turned to the more frustrated Ising case in Sect. 2.4, where we find states which
order by disorder into a pattern which is again similar to that of the stacked triangular
antiferromagnet, with the crucial difference that the zero-temperature entropy scales
as L2/3 . We end in Sect. 4.6 with brief comments on the TS packings not studied in
this Chapter.
14
2.2
Lattices
General TS packings are constructed by removing one third of the sites from a close
packing of spheres. To describe them more precisely recall first that any three dimensional close packing of spheres can be obtained by stacking two dimensionally close
packed triangular lattices of spheres according to a prescribed stacking pattern. In
a given triangular plane, the interior of triangular plaquettes host depressions into
which further spheres may be placed. Of these we may select either all upward pointing triangles or all downward pointing triangles in which to place the spheres of the
next triangular layer. The standard description labels the original sites as, say, C
whereupon the two inequivalent depressions that host the second layer are labeled A
and B. All layers consist of spheres occupying one of these three sets of sites with
the rule that there is no repetition between adjacent layers—this gives rise to the
2N Barlow packings for N layers of spheres. As is well known, of these the repeated
sequences ABC yield the FCC lattice and AB or AC yield the hcp structure [20].
An equivalent description can be given in terms of two stacking vectors which allow
us to translate one triangular layer into a neighboring one. As they can be chosen
independently at each step, we recover the previous counting. For concreteness, let
us orient one of the triangular layers as shown in Fig. 2.1. Now the stacking vectors
are readily seen to be
Vα
Vβ
r
√
3a
2
a
ŷ + a
ẑ
= − x̂ +
2
6
3
r
√
3a
2
=
ŷ + a
ẑ
3
3
(2.1)
where a is the diameter of the spheres. Now, for example, the allowed configurations
of three planes can be written as CAC (or Vα , Vβ ), CBC (Vβ , Vα ), CAB (Vα , Vα ),
CBA (Vβ , Vβ ).
To dilute a sphere packing into a TS packing, one vertex of each triangle in the
triangular layers is removed, leaving stacked honeycomb layers. Now at each step
15
y
x
Figure 2.1: The triangular lattice
there are 6 choices—a choice between two of the A, B, or C sites followed by a choice
of which of the three equivalent sublattices of the triangular layer to dilute. All
choices lead to stable structures [14].
Equivalently, we may begin with one honeycomb layer and construct the rest of
the structure by displacing it successively by stacking vectors drawn now from a set
of six vectors. With our choice of orientation these are
r
√
3a
2
a
x̂ −
ŷ + a
ẑ
Vβ1 =
2
6
3
r
√
3a
2
Vβ2 =
ŷ + a
ẑ
3
3
r
√
a
3a
2
ŷ + a
ẑ
Vβ3 = − x̂ −
2
6
3
r
√
a
3a
2
Vα1 = − x̂ +
ŷ + a
ẑ
2
6
3
r
√
3a
2
Vα2 = −
ŷ + a
ẑ
3
3
r
√
3a
2
a
Vα3 =
x̂ +
ŷ + a
ẑ
2
6
3
(2.2)
Now, starting as above with a C plane, Vα1 through Vα3 generate A planes, while
Vβ1 through Vβ3 yield B planes. The projections of these vectors in the honeycomb
planes are shown in Fig 2.2. Observe that the projections of the Vβi are inverses of
the projections of the Vαi .
16
V
β2
V
V
α
α
1
3
V
V
β
β1
3
V
α
2
Figure 2.2: Projections of the 6 stacking vectors in the honeycomb planes. Note that
all Vαi result in a CA stacking, and all Vβi in a CB stacking.
Three comments are in order. First, the TS packings contain tunnels through the
parent Barlow packings. The stacking vectors can also be visualized as giving the
direction of the tunnels (in other words, the offset between the centers of the missing
spheres in adjacent planes). Second, in the close packed case, irrespective of the
stacking pattern, rotations by 2π/3 radians about either the vertex or the center of a
triangle are symmetries of the structure. After removing the centers of each hexagon,
however, such rotations map some occupied sites to unoccupied sites and vice versa,
breaking the symmetry. Third, all TS packings have a local coordination number of
7—3 nearest neighbors in a single honeycomb layer and 2 each in the layer above and
below.
Clearly, there are 6N TS packings for N honeycomb layers. In this Chapter we
focus on a subset of them which is 2N in number. We begin with one member of
this subset which is defined by the single stacking vector Vβ1 . This particular choice
known as the tunneled FCC lattice, is discussed extensively in Ref. [14]; we will review
its structure briefly here.
17
Written conventionally, this packing is a triclinic lattice with a two site unit cell.
The primitive lattice vectors are
a2
a3
√
3, 0)
√
3
3
= a( , −
, 0)
2 √2 r
3
1
2
= a( , −
,
)
2
6
3
a1 = a(0,
(2.3)
The two atoms of the unit cell are at positions
x1 = a(0, 0, 0)
x2 = a(1, 0, 0)
(2.4)
The resulting lattice, shown in Fig. 2.3(a), consists of the honeycomb lattice in the
xy plane, with nearest neighbors separated by a distance a. The honeycomb layers
are stacked in the z direction according to the FCC pattern, with the same stacking
vector Vβ1 between every honeycomb plane.
We are primarily interested in nearest neighbor antiferromagnetism. For nearest
neighbor interactions the TS-FCC lattice has an elegant reinterpretation that is extremely useful. As shown in Fig. 2.3(b) the honeycomb planes stack in such a way as
to create folded sheets of stacked triangular lattices. The folded sheets run along two
pairs of parallel edges in the hexagon. The remaining pair of edges bond neighboring
triangular sheets. This is made clear in Fig. 2.3(c) where we straighten out the triangular sheets and draw the topologically equivalent semi-stacked triangular lattice
or SSTL. Unlike the case of the stacked triangular lattice (STL), in which each site
has a nearest neighbor in the sheets above and below it, the stacking bonds in the
SSTL alternately join sites in one sheet to the sheets above and below. The lattice
co-ordination number is thus 7 as it should be.
The TS-FCC lattice is one of an infinite subclass that share the same topology
as the SSTL: any TS packing defined by stacking vectors that belong to one of the
18
(a)
(b)
(c)
Figure 2.3: (a) The TS-FCC lattice as a set of stacked honeycomb lattices. Bonds in
the honeycomb lattice (xy plane) are shown as bold red lines; bonds joining different
honeycomb layers are light blue lines. The two colorings of the sites differentiate the
2 sublattices. (b) A rotated view that exhibits the alternate decomposition as a set of
semi-stacked folded triangular planes. The planes are seen almost edge on and consist
of sites from both sublattices. The lighter red (darker blue) sites are connected to
dark blue (light red) sites in the folded triangular plane to the left (right). (c) The
topologically equivalent stacked triangular lattice, with unfolded triangular planes
now redrawn in the xy plane.
19
sets {Vαi , Vβi } is equivalent to triangular sheets stacked in this way. Thus there are
3 · 2N such packings for N layers—up to the overall factor of 3 for choice of sublattice
diluted, this is the same as the number of the parent Barlow packings.
Figure 2.4: A schematic of the formation of triangular planes. One layer of the
parent triangular lattice is shown, with black (white) circles representing occupied
(vacant) sites. The arrows show the projection of the stacking vector Vα2 in the
honeycomb plane. In a Barlow packing, the center of every upward facing triangle
is an occupied site in the next layer, and the center of each triangle would be the
apex of a tetrahedron. Both solid and dotted lines are nearest neighbor bonds for the
Barlow packing. In the equivalent TS packing shown here, only the centers of the four
triangles lying immediately below occupied sites are occupied in the next layer. The
solid lines show nearest neighbor bonds in the TS packing. The darkened diagonal
edges of the hexagon still form bases of triangles completed by the occupied sites in
the next layer; the horizontal edges of the hexagon do not, and lie in the direction of
stacking of the triangular sheets.
To see how this comes about, let us begin with stacking one plane above a reference
plane with say Vα2 and consider a given hexagon in the reference plane. Let us label
the three sets of parallel bonds on the hexagon by the indices on the stacking vector
projections orthogonal to them. As we see in Fig 2.4, two of the three sets of parallel
20
bonds on the hexagon are now also bonds on triangles while one set—set 1—of parallel
bonds is not. The same set is singled out when we use stacking vector Vβ2 instead.
It follows then that if we use a sequence of Vα2 and Vβ2 to stack, we will get a
sequence of honeycomb planes where the 2 and 3 bonds participate in triangles and it
is easy to convince oneself that this will lead to the claimed topology. More precisely,
the 2 and 3 bonds will lie in (folded) triangular planes connected by 1 (stacking)
bonds. Conversely, if we decide to switch from the 1 stacking vectors to the 2 or 3
stacking vectors at some stage we will interfere with this topology. Hence the result.
Figure 2.5: The TS-HCP lattice, showing stacking structure. Honeycomb planes are
stacked according to an alternating ABAB pattern. Bonds in the A planes are shown
here in green, and the B planes in red. Bonds joining different honeycomb planes are
shown in blue. In contrast to the TS-FCC case, the tunnels formed by vacant sites
zig-zag between layers, giving the structure a 2 sublattice chirality [14].
We have already discussed the TS-FCC lattice obtained by repeated stacking with
the vector Vα2 . As another example we display, in Fig 2.5, the TS-HCP structure
constructed using the repeated sequence Vβ2 , Vα2 . We emphasize that both of these
have the topology of the SSTL in Fig. 2.3(c).
In the balance of this Chapter we will be concerned with O(N) symmetric spins
placed on the sites of the TS-FCC lattice and other members of its class, interacting
via nearest-neighbor interactions alone. For these problems it will be sufficient to
consider such spins placed on the SSTL which is what we will do in the remaining.
21
This is a great simplification since it allows us to treat in one go an infinite family
of lattices with unit cells of arbitrarily large size. We will not treat the problem of
translating the results back to the original coordinates in the general case except for
the case of the TS-FCC lattice which we discuss in our concluding remarks.
2.3
Antiferromagnetism
We now turn to nearest neighbor antiferromagnetism on the TS-FCC lattice and its
equivalents. As noted above, we will study the equivalent problems on the SSTL.
Specifically, we wish to elucidate the nature of ordering in the Hamiltonians,
H=
X
Jij Sia Sja ,
(2.5)
ij
where
P
a
Sia Sia = 1, a ∈ {1, · · · , N}, i, j run over the sites of the SSTL, and Jij = J
when i, j are nearest neighbors and zero otherwise. We begin by collecting some
results on the eigenspectrum of the nearest neighbor interaction (adjacency) matrix
which will come in handy in our subsequent analysis.
2.3.1
Eigenspectrum of Interaction Matrix
We wish to find the eigenvectors and eigenvalues of the adjacency matrix, Jij ψj = ǫψi .
The SSTL differs from the STL in that translational symmetry is broken along two
of the triangular lattice vectors as well as along the stacking direction. Consequently,
it has a two site unit cell with sites of type 1 connected only to the triangular plane
above, while sites of type 2 are connected only to the triangular plane below, as
shown in Fig. 2.3(c). For convenience we switch to a co-ordinate system in which the
triangular planes lie in the x − y plane, and stacking bonds in the z direction. With
22
this choice the lattice vectors are
a1 = a(1, 0, 0)
√
a2 = a(1, − 3, 0)
√
1
3
a3 = a( , −
, 1)
2
2
(2.6)
and the 2-site unit cell now has sites at
u0 = (0, 0, 0)
√
1
3
u1 = a( , −
, 0) .
2
2
(2.7)
Readers are warned not to mistake this choice of axes for the SSTL for the choice of
axes used earlier to discuss the TS-FCC lattice (Fig. 2.3(a)); equally, the triangular
planes in the SSTL are not the triangular planes we began with in our discussion of
the parent Barlow packings.
The 2-site unit cell leads to eigenvectors that we parameterize in the form
ψi ≡ ψ(r, α) = eik·r uα (k) ,
where r ≡ {x, y, z} is the actual location of the site of type α = 1, 2. The residual
problem requires diagonalization of the 2 × 2 reduction of the adjacency matrix to
momentum space
kx
cos kx I + [2 cos cos
2
√
3ky
+ cos kz ] σx + sin kz σy
2
With these choices, the eigenvalues are
ǫ(k)/J =
cos kx
(2.8)

2
± sin kz +
kx
2 cos cos
2
√
3ky
+ cos kz
2
!2 1/2

We will be especially interested in the minima of this dispersion relation as they yield
the soft modes that will dominate the ordering. Analysis of the possible minima of
23
ǫ(k)/J reveals that ǫmin /J = −2.5, and is attained for two inequivalent points in the
Brillouin zone. We will, however, find it convenient to choose two such points outside
the first Brillouin zone of the lattice as they facilitate comparison with the existing
analysis of the stacked triangular lattice. Accordingly, we will choose the pair:
 
1
4πi
ψ1 (r, α) = e 3 x eiπz  
1
 
1
4πi
ψ2 (r, α) = e− 3 x e−iπz  
(2.9)
1
Evidently, ψ2 (r, α) = ψ 1 (r, α).
2.3.2
XY, Heisenberg and N > 3 cases
For N ≥ 2, which includes the XY and Heisenberg cases typically of maximum interest, the ground states of the full lattice are simply the well known coplanar, three
sublattice ground states of the triangular antiferromagnet, stacked antiferromagnetically between the different layers. The reader will recall that the ground states of
the triangular antiferromagnet exhibit all spins confined to a plane in spin space with
three different orientations on the three sublattices making angles of 120 degrees with
each other. There is a single global rotational degree of freedom which carries over
into the TS-FCC lattice. The set of ground states is thus identical to those of the
STL.
As these states thus involve breaking a continuous global symmetry in three dimensions, we expect a single phase transition between the paramagnetic phase at
high temperatures and the 120 degree state at low temperatures. For the STL this
transition has been discussed extensively in the literature [18, 21, 22]. We will see
that the results on the nature of the transition do not change in our case although
the details will of course be sensitive to the altered microscopics.
24
Landau-Ginzburg-Wilson functional
We will now follow the standard route of constructing the Landau-Ginzburg-Wilson
(LGW) functional that controls the probability distribution of the soft modes from a
symmetry analysis. We will find that the LGW functional for the TS-FCC lattice is
essentially identical to that of the STL up to sixth order in the fields and thus should
be expected to lead to phase transitions in the same universality class as the latter
lattice.
We begin by writing (soft spin) configurations with energies near the two minima
(2.9) in the form:
Φa (r, α) = φa1 (r)ψ1 (r, α) + φa2 (r)ψ2 (r, α)
a
≡ φa1 (r)ψ1 (r, α) + φ1 (r)ψ 1 (r, α)
(2.10)
where a is the O(N) vector index and on the second line we have built in the real
valuedness of the fields.
The reader can check that, of the various symmetry operations on the underlying
lattice, there are two that give independent non-trivial actions that need to be considered in writing the LGW functional. These can be chosen to be a translation by
two steps in the x-direction,
Tx2 [φa1 (r)] = φa1 (r + 2ax̂)
= e
2πi
3
φa1 (r)
(2.11)
and inversion,
I[φa1 (r)] = φa1 (−r)
a
= φ1 (r)
(2.12)
In addition, we must consider the O(N) symmetry of the microscopic Hamiltonian.
25
Together these symmetries constrain the form of the LGW Hamiltonian to fourth
order in the fields to be:
H=
a
[ r + c⊥ (qx2 + qy2 ) + cz qz2 ] φa1 φ1
a
b
b
+ u4 (φa1 φ1 )2 + v4 (φa1 φa1 )(φ1 φ1 )
(2.13)
where we have summed over repeated indices. It is straightforward to confirm that
this Hamiltonian, when minimized, gives rise to the coplanar state we deduce from
the microscopic analysis. As H has exactly the same form as for the STL and thus
has been studied extensively, we will now review the known results on its phase
transitions.
Rernormalization group results on phase transitions
Renormalization group analyses of this Hamiltonian have been performed in the literature both in the large N and d = 4 − ǫ dimensional expansions [23]. An extensive
review of these and other analytic and numerical results can be found in Refs. [21]
and [18].
This work has shown that there are four contending fixed points, whose stability
varies with N. For N > Nc there is a single stable “chiral” fixed point, with v4 6= 0,
which controls a phase transition in a different universality class than that of the
ferromagnetic O(N) model.
Depending on the initial parameters, the flow may either lead to a second order
transition at this novel fixed point, or be unstable, signalling a first order transition.
A simulation would be needed to settle this question for the SSTL. For N < Nc there
are no stable fixed points, and the transition is necessarily first order.
The most reliable estimate of Nc comes from the Monte Carlo Renormalization
Group calculations of Ref. [24]. These results suggest that 4 < Nc < 8, and the cases
of maximum physical interest lie in the subcritical regime where the transition is first
26
order. This contradicts the results of many earlier numerical studies, which seemed
to indicate a second order transition about the chiral fixed point. The apparent
discrepancy stems from the presence of an attractive basin in the flow about complex
fixed points lying close to the real plane, which causes the transition to appear second
order for small system sizes [18].
2.4
Ising case
Thus far our analysis of the TS-FCC lattice has closely paralleled the analysis of
the STL. But now for the Ising case, a new and interesting feature enters which
distinguishes the two lattices. As is well known, a single triangular Ising layer exhibits
a macroscopic number of ground states [25, 26]. In the STL the ground states of the
stacked lattice are as many since they consist of single layer ground states repeated
antiferromagnetically (although the number can be boosted somewhat by picking
antiperiodic boundary conditions in the stacking direction). For a three dimensional
system, this is a submacroscopic number of states and thus the entropy per site
vanishes as T → 0. For the SSTL we find instead that the number of ground states
is again macroscopic and now there is a non-zero entropy per site as T → 0.
Despite this difference, the nature of the ordering at low temperatures in both
systems—driven by the order by disorder mechanism—turns out to be the same.
This is indicated by the coincidence of their LGW functionals (up to coefficients) and
we are also able to give numerical and analytic evidence to the same end.
2.4.1
Zero temperature entropy
Let us first consider a lower bound on the zero temperature entropy. We begin
with the “maximally flippable configuration” in a single triangular plane shown in
Fig 2.6. In this configuration, spins on two out of three sublattices are flippable,
27
in that they can be individually flipped without leaving the ground state manifold.
This configuration has as many flippable spins as can be packed into a ground state.
Observe that sites on one of the two sublattices are independently flippable and thus
generate 2N/3 ground states that bound the entropy of an isolated plane from below
by (log 2)/3 per site.
Figure 2.6: Maximally flippable configuration. Ising spins are shown on each site.
Frustrated bonds are (bold) red, unfrustrated bonds (light) blue.
Now consider stacking this configuration antiferromagnetically. In a given plane,
half of the sites are married to sites in the layer below, and the other half above. It
follows then that we may flip half the sites on one sublattice along with their partners
above and the other half with their partners below. This leads to a lower bound on
the ground state entropy
S/N > (log 2)/6
(2.14)
where N is the total number of sites in the system. The scaling with N establishes
the macroscopic character of the ground state entropy. In contrast, for the STL there
are only N 2/3 ground states. A simple upper bound
Su /N < 0.3383 . . .
is obtained by considering the entropy of decoupled triangular layers [25].
(2.15)
28
We remark that a binary alloy that forms in the TS-FCC family of structures would
thus be expected to exhibit a macroscopic zero temperature entropy, contributing to
its stabilization.
2.4.2
Order by disorder
The next question to consider is whether the ground state manifold breaks any symmetries, i.e. whether the unweighted average over all the ground states yields long
range order in the correlation functions.
What kind of order might one expect? As this order has to be selected entropically,
i.e. by the preponderance of a family of configurations in the ground state average,
we expect it to correspond to the configuration that has the greatest number of
nearby configurations reached by local moves. The stacked maximally flippable (MF)
configuration considered in our entropy lower bound meets this criterion—it is also
the three dimensional configuration with maximal flippability. To see this, observe
first that the constraint of inter-planar spin partnering is absolute in the ground state
manifold: no spin may be flipped independently of its partner. Spin configurations
which are stacked (the same in every layer) automatically partner flippable spins
to flippable spins, and thus the stacking bonds impose no additional constraints on
flippability. As the MF state maximizes the number of flippable spins in each plane,
stacking this state gives the maximum possible number of flippable spins for the
SSTL.
We should note however, that the spin distribution in the maximally flippable
configuration is not directly observable; instead, it must be dressed by the fluctuations
that select it. Two options emerge naturally. The first involves a three sublattice
structure with magnetizations (c, −c, 0) wherein one of the two sublattices of flippable
spins does all the flipping and thus exhibits a vanishing magnetization while the other
two sublattices exhibit equal and opposite magnetizations. The other exhibits a three
29
sublattice structure but now with two equivalent sublattices. The magnetizations
(d, −d′ /2 − d′ /2) reflect more completely the symmetries of the maximally flippable
configuration. The selection between these two is a matter of detail. The reader
should note that both options give rise to six symmetry equivalent states.
Unfortunately, direct demonstration that one of these options is realized is not
straightforward and we will not definitively answer this question in this Chapter
although we believe that symmetry breaking in the (c, −c, 0) is realized at T = 0.
Instead we will, in the next section, approach the existence and structure of the
ordered phase from the paramagnetic phase at high temperatures by constructing
the appropriate LGW functional.
But before we do that let us briefly comment on the difference between what we
have discussed here and the corresponding analysis of the STL Ising antiferromagnet.
On the STL, the ground state manifold exhibits long range order in the stacking
direction but only algebraic order in the planes—in the latter directions it exhibits
the known correlations of a single triangular layer [27]. This algebraic order is again
present at the wavevectors of the maximally flippable state (Fig 2.6). In the STL,
switching on a small T > 0 converts this to true long range order. The mechanism
is “order by disorder” which can be visualized as the entropic dominance of three
dimensional configurations in which flippable spins in the MF configurations in the
planes stack with a set of mobile solitonic defects [28, 29, 30]. In this setting it is by
now clear that a single low temperature phase in the (c, −c, 0) pattern is separated
from the paramagnet [31, 32]. The major qualitative difference between the STL and
the SSTL is then that in the latter fluctuations in the stacking direction are present
already at T = 0 and so we expect that (eventually) the low temperature ordering
can be understood by an analysis of the ground states alone.
30
2.4.3
LGW analysis
We now add another ingredient to our analysis of the Ising problem by applying the
LGW and Renormalization group analysis to this case. This yields
HI =
[ r + c⊥ (qx2 + qy2 ) + cz qz2 ] φa1 φ1
(2.16)
6
+ u4 (φ1 φ1 )2 + u6 (φ1 φ1 )3 + v6 (φ61 + φ1 )
where we have now kept terms to sixth order in the fields. This is necessary for
the second of these terms is the first one that breaks a U(1)/XY symmetry that is
present up to fourth order down to a Z6 (clock) symmetry. Consequently, there is a
discrete set of six symmetry equivalent states at low temperatures and we reproduce
a key feature of the Ising problem. The two possible signs of v6 correspond to the
two magnetization patterns discussed above. This term is dangerously irrelevant: it
is irrelevant at the critical fixed point that controls the transition into the broken
symmetry phase, but to get the correct low-temperature physics it cannot be set to
zero. Since it is irrelevant at the critical point, the transition is in the universality
class of the three dimensional XY model. It is worth noting that a finite stack will
exhibit a Kosterlitz-Thouless transition [33]. All these results parallel those for the
STL [34].
2.4.4
Monte Carlo results
The remaining challenge is to distinguish between the two ordering alternatives or
equivalently, to fix the sign of v6 . As this is sensitive to microscopics, we have chosen
to simulate the system to investigate this question.
In the simulations we used a simple spin-flip Monte Carlo algorithm. The algorithm allows 2 types of moves: a single spin flip, or a double spin flip which reverses
a pair of partnered spins in adjacent layers. While this is sufficient for our purposes
near the transition, at low temperatures it fails to be ergodic. The nature of the
31
problem is clearest at zero temperature, where only the double spin flip is allowed.
Hence a spin s1 in a given triangular plane may be flipped only if its partner s2 in
the adjacent plane is also flippable. As only 2 of the 6 in-plane neighbors of s1 are
partnered with in-plane neighbors of s2 , at T = 0 configurations exist in which this
pair can be flipped only after flipping spins in all other layers of the system. Hence
at low temperatures a more complicated, cluster-type method must be used.
We turn now to the data for systems of size 6 × 6 × 6 and 12 × 12 × 12 for the
temperature ranges where our algorithm is ergodic as evidenced by the decay in single
spin autocorrelations to zero. The system dimensions are chosen to be N triangular
planes of N 2 sites each with periodic boundary conditions in all directions.
For these systems we proceed as follows: In each configuration we compute and
order the three sublattice magnetizations as M1 > M2 > M3 . We then compute
the matrix of correlations hMi Mj i averaged over the run. The results are shown in
Fig 2.7. As the reader will note these correlators should, in the (c, −c, 0) state, exhibit
the values c2 , −c2 and 0 in the infinite volume limit. Our computations are consistent
with that and clearly inconsistent with the competing (d, −d′ /2 − d′ /2) state. This
includes details such as the multiplicity of the values observed, and their evolution
between the two system sizes.
2.5
Concluding Remarks
To summarize, we have studied nearest neighbor O(N) antiferromagnets on an infinite subset of the new family of packings introduced by Torquato and Stillinger and
established the nature of the ordering at low temperatures and the nature of the
phase transitions.
We have done our analysis in the equivalent representation of the SSTL. This has
the great advantage that we have dealt with the entire family of lattices at once—
32
1
0.8
0.6
<Mi Mj>
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
0.5
0.6
0.7
0.8
0.9
1
kB T/J
1.1
1.2
1.3
1.4
1.5
Figure 2.7: Correlations of sublattice magnetizations as functions of temperature,
indicating the phase transition from 3 sublattice order to a paramagnetic state. Two
lattice sizes are shown: a 6 × 6 × 6 lattice in (light) blue, and a 12 × 12 × 12 lattice
in (bold) red.
33
most of which have sizeable unit cells stemming from long periods in the stacking
direction. However, for a specific realization, it will be necessary to translate the
ordering back into the actual geometry of the lattice. For example, for the TS-FCC
lattice the ordering wavevectors ± 4π
x̂ common to all O(N) cases, translate into the
3
vectors
2π
±
a
√
√ !
3 6
0,
,
9 9
in the choices made in Equations (2.3) and (2.4).
One statistical mechanical remark may be interesting to readers. By this somewhat circular route we have discovered that the SSTL preserves the ordering of the
STL for the Ising problem, while exhibiting a greatly increased ground state entropy.
This analysis indicates that further dilution of the stacking bonds will further boost
the zero temperature entropy while still preserving the nature of the ordered phase
at asymptotically low temperatures.
Finally, we have taken a preliminary look at TS packings which are not in the
TS-FCC class. They appear, generically, to be more frustrated than the ones studied
in this Chapter and thus are an interesting topic for future work.
Chapter 3
Quantum Frustrated Magnetism
on the Pyrochlore lattice
3.1
Introduction
The study of classical magnets is essentially the study of various types of spatial
magnetic order, and the mechanisms which can generate them. However, quantum
systems often exhibit ‘exotic’ orders, of a type that classical systems do not – superconductivity and the quantum Hall effect being two examples which have had an
especially large impact on the field of strongly correlated systems. An important
component of quantum magnetism is the search for such exotic orders – of a type
not realizable by classical spin models – in magnetic systems. In this chapter, we
turn to the question of quantum magnetism, and propose a spin liquid state on the
highly geometrically frustrated pyrochlore lattice. Specifically, we consider S = 1/2
Heisenberg models, and attempt to construct a zero temperature phase that breaks
no symmetries of the problem— a fully symmetric quantum spin liquid.
Candidate magnetic states with exotic quantum mechanical ordering are often
referred to as quantum spin liquids, since they do not ‘crystallize’ into a state with
34
35
long range Néel order down to zero temperature. Interest in such states dates back
to Anderson’s introduction of the resonating valence bond (RVB) state [35] and then
to his suggestion[36], upon the discovery of the cuprate superconductors, that their
behavior was traceable to a parent spin liquid state.1
More recently, progress in
constructing actual models that realize spin liquid behavior2 has sparked a resurgent
interest in this field. This has also lead to an understanding of how these states
exemplify ordering beyond the broken symmetry paradigm: a large class of spin
liquids give rise to low energy gauge fields but not order parameters. That such
“topological phases”3 also underlie a fascinating approach to quantum computation
[39] only multiplies their interest.
Geometrically frustrated magnets are a natural place to look for states with such
intrinsically quantum mechanical order. Indeed, Anderson’s 1972 paper identified a
small value of the spin and geometric frustration as two sources of quantum fluctuations that could favor a spin liquid. The pyrochlore lattice is a natural object of study
in this context. It is highly frustrated and frequently realized as a sublattice of the
spinel or pyrochlore compounds. Potentially, it could host a spin liquid in d = 3 for
small values of the spin (though there is not, to date, a good experimental S = 1/2
antiferromagnet on the pyrochlore lattice). Much work has gone into studying its
magnetic properties in various contexts. Most notably, it is known to lack long range
order with nearest neighbor interacting classical spins [40] but instead to exhibit an
emergent gauge field and dipolar correlations as T → 0. Attempts to work about
the classical limit, in the spin wave (1/S) expansion have lead to some insight into
1
The history of the study of liquid-like behavior in quantum magnets dates back much further
than this, however. Bethe’s solution for the 1D spin chain revealed a perplexing difference between
the spin-1/2 and spin 1 cases, which can be understood in terms of valence bond effects. Indeed,
the existence of ’spinons’ was first identified in such systems.
2
For an introduction to this area see the recent Les Houches lectures by Misguich [37] and the
older review article Ref. [38].
3
Here we use the term “topological phases” in the looser sense of any phase with emergent gauge
fields. Strictly speaking the term should be reserved for cases where the low energy gauge theory is
a purely topological gauge theory.
36
the quantum “order by disorder” selection mechanism in this limit. Though the fate
of the 1/S expansion is not settled [41], there is little reason to think that it can be
informative when it comes to small values of spin, especially the S = 1/2 case that
interests us here. This is so partly because the selection mechanism at large S is weak
and leads to somewhat ornate states but also for the well-understood reason that it
misses out on tunneling processes that are sensitive to the Berry phases entering the
exact path integral [42, 43].
Given this unusual resistance to classical order, various authors have attempted
to directly tackle the S = 1/2 problem. Harris, Berlinsky and Bruder (HBB) [44]
initiated a cluster treatment in which the pyrochlore lattice is first decoupled into,
say, its up tetrahedra and then perturbatively reconnected. Subsequently Tsunetsugu [45] worked out a more complete treatment along the same lines and found a
dimerized state with a four sublattice structure. The criticism that this work predicts
symmetry breaking that is put in at the first step has attracted a potential rebuttal
in the work of Berg et al [46] with the “Contractor Renormalization” or CORE technique. An alternative perspective on this physics was provided in Ref. [47] where
it was shown that an SU(N) deformation produces a quantum dimer model whose
physics is very reminiscent of the HBB scenario. Unfortunately, the N = 2 limit is
manifestly problematic so it has not been possible to declare victory in this work. Yet
another attack on the problem [48] used an alternative large N theory—equivalent to
Schwinger boson mean field theory —and found a delicate energetics at small values
of spin (or boson density) which nevertheless strongly indicated that the spin 1/2
problem must break some symmetry.4
With this set of predictions of symmetry breaking as background, we bring another
approximate large N technique—that of “slave fermions” [49, 50]—to bear on the
4
Ref. [48] shows that at asymptotically small boson densities the system must break some symmetry. The minor caveat is that this does not rule out a different solution intervening right near
S = 1/2.
37
pyrochlore problem with a view to examining whether it produces a symmetric, spin
liquid, alternative. To this end we enumerate various translationally invariant mean
field solutions of which the lowest energy non-dimerized solution is one we call a
“monopole flux” state; upon Gutzwiller projection it also improves upon the fully
dimerized states. While this state does not break lattice symmetries in the manner
of the HBB scenario, it is not a spin liquid in the sense of breaking no symmetries at
all. Instead it is a chiral spin liquid [51, 52] and breaks parity (P) and time reversal
(T) symmetries. It also exhibits spinons in its mean field spectrum. We describe the
unusual mean field spectrum—which yields a Fermi surface consisting of four lines
intersecting at a point—and its low energy limit in some detail.
The stability of the mean field structure to fluctuations is the next question of
interest. We make progress in that direction by enumerating the projective symmetry
group (PSG) [53] of the state and showing that it forbids any terms that would
destabilize the mean field Fermi surface. We discuss how these symmetries restrict
the form of the continuum field theory describing the low-energy excitations about
the Fermi surface, and give arguments for the qualitative form of this action.
This chapter is organized as follows. We begin with a brief overview of the largeN/mean field slave fermion treatment of the Heisenberg model in Section 3.2. In
Section 3.3 we apply this technique to generate several mean field ansätze on the
pyrochlore lattice. We identify the lowest energy state, or monopole flux state, and
discuss its interesting properties. Section 3.4 reviews in general terms how the PSG
protects a mean field state against developing symmetry-breaking terms. Details
of the PSG for the monopole flux state can be found in Section 3.5. The PSG
derived arguments for the stability of the monopole flux state are given in Section
3.6, where we derive the general form of the symmetry permitted perturbations to the
Hamiltonian. We then attempt to address the question of fluctuations more directly,
by analyzing the fluctuations in the gauge theory problem. Section 3.7 sketches
38
both lattice and continuum approaches to this problem, though our results here are
inconclusive. We conclude in Section 3.8.
3.2
The Large-N Heisenberg Model: Spinons and
Gauge Fields
In this section we briefly review the large N fermionic approach to the S = 1/2 SU(2)
Heisenberg model which began as a mean field theory introduced by Baskaran, Zou
and Anderson [49] and was shortly thereafter systematized via a generalization to
SU(N) by Affleck and Marston [50].
In this approach, we first replace the bosonic spin operators of the Heisenberg
Hamiltonian
H=J
X
<ij>
Si · Sj
(3.1)
with bilinears in fermionic “spinon” operators:
Si =
1X †
c σαβ ciβ .
2 α,β iα
(3.2)
The resulting Hamiltonian conserves the number of fermions at each site and the
starting spin Hamiltonian is recovered if we limit ourselves to physical states with
exactly 1 particle per site. Up to a constant in the subspace of physical states, it can
be re-written in the suggestive form,
H=−
J X XX †
c cjα c†jβ ciβ
2 <ij> α β iα
(3.3)
A mean field theory arises upon performing the Hubbard-Stratonovich decoupling
H=−
XX
(c†iα cjα χij + h.c.) +
α <ij>
2 X
|χij |2
J <ij>
and locally minimizing the classical field χij to obtain self-consistency.
(3.4)
39
In order to understand the nature of fluctuations about such mean field solutions
it is conceptually convenient to consider the path integral defined by the equivalent
Lagrangian:
L =
X
c†i,α (i∂t + µ)ci,α +
i,α
+
X
i,α
"
X X
<ij>
α
φi (c†i,α ci,α − 1)
(c†iα cjα χij + h.c.) −
2
|χij |2
J
#
(3.5)
where φ is a Lagrange multiplier field enforcing the single occupancy constraint
P †
α ciα ciα = 1.
The above Lagrangian (3.5) is invariant under the local gauge transformations
c†i → c†i e−iθi
χij → χij ei(θi −θj )
φ → φ + ∂θ/∂t
(3.6)
which arise from the local constraints in the fermionic formulation. It follows that
we have reformulated the Heisenberg model as a problem of fermions that live on the
sites of the original lattice coupled to a U(1) gauge field and an amplitude field (the
phase and amplitude of χij ) that both live on the links of the lattice. In other words,
we may write χij = ρij eiaij , where aij → aij + θi − θj under the gauge transformation
(3.6). The mean field theory consists of searching for a saddle point with frozen link
fields.
As the Lagrangian (3.5) does not directly constrain the phase of the χij , it describes a strongly coupled gauge theory where the assumption of a weakly fluctuating
gauge field invoked in the mean field theory is, prima facie, suspect. To circumvent this barrier, Affleck and Marston [50] proposed a large N framework which
introduces a weak-coupling limit for the model (3.5) by extending the SU(2) spin
symmetry group of the Heisenberg model to SU(N) with N even. The result is a
theory of many spin flavors whose coupling strength scales as J → J/N. In the limit
40
that N → ∞, the corresponding mean field theory is exact; for sufficiently large but
finite N one hopes that a perturbative expansion gives accurate results. The validity of the qualitative features deduced at large N in the starting SU(2) problem is,
of course, hard to establish by such considerations and requires direct numerical or
experimental confirmation.
To effect the large N generalization, we replace the 2 spinon operators c↑ and c↓
with N spinon operators cα . The single occupancy constraint is now modified to
N
X
c†iα ciα =
α=1
N
2
(3.7)
and the large-N Hamiltonian has the form
H = −J/N
= −
XX
α,β <ij>
XX
α
c†jα ciα c†iβ cjβ
(c†iα cjα χij + h.c.) +
<ij>
N X
|χij |2
J <ij>
(3.8)
In the infinite N limit, the action is constrained to its saddle point and the mean
field solution becomes exact. Further, to lowest order in
1
N
the allowed fluctuations
involve moving single spinons, so that as N → ∞ we need only impose the constraint
(3.7) on average.
Away from N = ∞ the link fields, especially the gauge field, can fluctuate again
although now with a controllably small coupling. While the fate of the coupled
fermion-gauge system still needs investigation, the presence of a small parameter is a
great aid in the analysis, as in the recent work on algebraic spin liquids [54].
Finally, we note that the starting SU(2) problem is special, in that it is naturally
formulated as an SU(2) gauge theory [55, 56]. This can have the consequence that
the N = 2 descendant of the large N state, if stable, may exhibit a weakly fluctuating
SU(2) gauge field instead of the U(1) field that arises in the above description. We
will comment on this in the context of this Chapter at the end.
41
3.3
Mean-Field Analysis
3.3.1
Saddle Points of the Nearest Neighbor Heisenberg Model
We begin by enumerating mean field (MF) states which preserve translation invariance
on the pyrochlore. A mean field solution consists of a choice of link fields which
minimize the mean field energy functional for the Lagrangian (3.5)


X1
X
E(χ) = N 
|χij |2 +
(ε(k) − µ)
J
k
(3.9)
hiji
where ε(k) is the energy of a spinon of momentum k in the fixed background χij ,
and the chemical potential µ is chosen so that the constraint of 1 particle per site is
satisfied on average.
As discussed in the introduction, previous work on the Heisenberg model on the
pyrochlore lattice has led to ground states with broken symmetries. In this work
we are particularly interested in constructing a natural state on the pyrochlore that
breaks as few symmetries as possible. To this end, we begin our search with especially
symmetric ansätze for which ρij ≡ ρ is independent of i and j, and the flux Φ△ =
P
P4
△ aij through each face of the tetrahedron is the same. The net flux
i=1 Φ△
through each tetrahedron must be an integer multiple of 2π, since each edge borders
two faces such that its net contribution to the flux is 0 (mod 2π). This gives the
following 3 candidate spin liquid states:
1. Uniform: Φ△ = 0
2. π Flux: Φ△ = π
3. Monopole: Φ△ = π/2. Every triangular face of the tetrahedron has a π/2
outwards flux – equivalent to a monopole of strength 2π placed at the center of
each tetrahedron.
42
At infinite N a dimerized state is always the global minimum of (3.9) [57]; thus we
also consider
4. Dimerized: χij = χ0 on a set of bonds that constitute any dimer covering of the
lattice but zero otherwise.
The states (1-3) above are analogues of the uniform, π flux, and chiral states
studied previously on the square lattice [50, 52]. Of the above states, (1) and (2)
break no symmetries of the problem; the third preserves lattice symmetries but breaks
P and T.
The states (1) and (2) are in fact particle-hole conjugates: a particle-hole transformation maps c†i cj + c†j ci → −c†j ci − c†i cj , changing the sign of χ on each bond and
adding π flux to each triangular plaquette. At N = 2 this can be effectuated by an
SU(2) gauge transformation, so that the states (1) and (2) describe the same state
after Gutzwiller projection.
The mean field energies of these states are listed in the first column of Table I.
Consistent with Rokhsar’s general considerations [58, 57] the fully dimerized state is
lowest energy and the monopole flux state has the lowest energy of the non-dimerized
states. The mean field states with N set equal to 2 do not satisfy the single occupancy
constraint. While, in principle, perturbation theory in 1/N can greatly improve the
wavefunction in this regard this is a complex business (to which we return in Sections
IV and V) ill-suited to actual energetics. Instead, the somewhat ad hoc procedure of
(Gutzwiller) projecting the mean field wave function onto the Hilbert space of singly
occupied sites is typically employed to improve matters. This leads to resonances
and long range correlations that can substantially lower the mean field ground state
energy, particularly for spin-liquid type states.
Expectation values in the Gutzwiller projection of a state can be carried out using
a Monte Carlo approach, as described in Ref. [59]. A description of the numerical
method specialized to our problem is given in [60]. The second column in Table 3.1
43
shows the numerically evaluated energies of the 4 mean field states with Gutzwiller
projection. We see that the monopole flux state now emerges as the lowest energy
state of our quartet5 . Encouraged by this, and also because the state has various
elegant properties, we will focus in the remainder of this work on the properties of
the monopole flux state. Note however, that we have failed to preserve all symmetries
of the Hamiltonian even in this approach—we are forced to break T and P and thus
end up with a chiral spin liquid. We give a fuller description of the symmetries of the
state below.
Finally, we note that larger unit cells can be consistent with translationally invariant states.6 Such states have an integral multiple of π/2 flux through each triangular
plaquette, but also non-trivial flux through the hexagonal plaquettes in the kagomé
planes, as for the mean field states on the kagomé studied in Ref. [62]. By the same
arguments as employed for a single tetrahedron, we find that the flux through the
hexagons must have values 0 or π (mod 2π) to preserve the translational symmetry of the lattice. [A flux of π/2 per hexagonal plaquette necessarily breaks lattice
translations]. However, as noted in Table 3.1, we find that these states also have
higher energies than the monopole flux state both at mean field and upon Gutzwiller
projection.
The above results do not guarantee that the monopole flux state is the true ground
state of the nearest-neighbor pyrochlore antiferromagnet. Indeed, numerical studies
of the planar pyrochlore find energies of −.55J per unit cell for valence bond crystals
for a configuration compatible with the full pyrochlore lattice[63], and higher spin
interactions may be necessary to stabilize the monopole flux state, as is the case for
the π-flux state on the square lattice [50, 64]. Since such terms generically exist, our
focus here is on describing the energetically optimal spin liquid state, rather than on
5
Ref. [61] also performed Gutzwiller projection on these states and we find good numerical
agreement with their results
6
We are grateful to Michael Hermele for emphasizing this point.
44
Uniform
π Flux
Monopole
Dimer
(π, π)
(π/2, π)
EM F (unprojected)
−0.3333J
−0.3333J
−0.3550J
−.375J
−0.3333J
−0.3491J
EM F (projected)
−.3752 ± 0.0004
−.3752 ± 0.0004
−.4473 ± 0.0009
−.375J
−0.3751 ± 0.0008
−0.4356 ± 0.0003
Table 3.1: Mean-field energies for projected and unprojected ground states of the
mean field ansätze considered. The quoted mean field energies are the energy of
(3.4) plus the omitted constant − J4 per site required to make a correspondence with
(3.1). The states (π, π) and (π/2, π) are variants of the uniform and monopole state,
respectively, with flux π per hexagonal plaquette. The projected wave functions were
evaluated on a lattice of 5 × 5 × 5 unit cells, or 500 sites for configurations with a 4
site unit cell, and 1000 sites for configurations with an 8 site unit cell.
the exact ground state of the nearest-neighbor Heisenberg antiferromagnet.
3.3.2
The Monopole Flux State
3
2
1
4
4
2
3
Figure 3.1: Link field orientations in the monopole state with χ = ±iρ0 . Hopping
along the direction of an arrow induces a phase of π/2; hopping against the arrows,
a phase of −π/2. The flux on each triangular face is π/2 inwards. With this flux
assignment the monopole flux state breaks T and P , but is invariant under lattice
translations and rotations.
The monopole flux state exhibits a flux of π/2 per triangular face. To write down
the mean field Hamiltonian explicitly we must pick a gauge. We choose χij = ρ0 eiaij ,
with aij = ±π/2. The phase of ±i that a spinon picks up when hopping from site i
to j can be represented as an arrow on the corresponding edge, which points from i
45
to j (j to i) if the resulting phase is +(−)i. The orientation of the link fields, shown
in Figure 3.1, gives an inward flux of π/2 per plaquette.
The necessity of picking a gauge for the mean field solution causes, as usual, various
symmetries to be implemented projectively. For example, the assignment shown in
Figure 3.1 is not invariant under lattice rotations. However, the background link
fields after rotation can be gauge transformed to the original state, as expected from
the manifestly rotation invariant assignment of fluxes. We discuss these and other
symmetries in more detail in Sections 3.4 and 3.6; here we merely note that P and T
are the only symmetries broken by the monopole flux state.
The Hamiltonian for spinons in the gauge choice shown in

y
z
0
sin( kx +k
) sin( ky +k
)
4
4


kx +ky
z
X † 
0
sin( kx −k
)
sin( 4 )
4
H = −ρ0
Ψmkα 
sin( ky +kz ) sin( kx −kz )
0
mk,α

4
4

y
x
z
) sin( kz −k
) sin( ky −k
)
sin( kx +k
4
4
4
Fig. 3.1 is

z
sin( kx +k
)
4

y 
sin( kz −k
)
4
 Ψmkα.
x 
sin( ky −k
)

4

0
(3.10)
where Ψ is a 4-component vector, with Ψimkα = ciαmk . Here the index i labels the 4
sites in the tetrahedral unit cell.
Figure 3.2(b) shows a plot of the energy eigenvalues of (3.10) along the highsymmetry lines of the Brillouin zone. At half filling, the Fermi “surface” consists of
the lines k(±1, ±1, ±1) which join the point (0, 0, 0) to the center of the hexagonal
faces of the Brillouin zone of the cubic FCC lattice, line (L − Γ) in Fig. 3.2(b). Each
Fermi line has a pair of zero energy eigenstates.
Figures 3.3(a) shows a surface of constant energy E ≈ 0 near the Fermi surface. At
E = 0, the 4 bands intersect only at the origin and the constant energy surface is given
by the 4 lines described above. Surfaces of constant energy E ≈ 0, E 6= 0 consist of 4
cylinders enclosing the (1, 1, 1) directions, which are the surfaces of constant energy
for particle-like (E > 0) or hole-like (E < 0) excitations about the Fermi line. About
46
kz
L
U
Γ
ky
X
K
W
kx
(a)

3
1
0
-1
-

3
G
X
W
U
L
G
(b)
Figure 3.2: Spectrum of the monopole flux state. (b) shows the spectrum of the
monopole flux state. Note the Fermi line √13 (k, k, k). (a) shows the contour in the
Brillouin zone along which the spectrum is plotted.
47
the origin all 4 bands have energy linear in k, and another, diamond-shaped constantenergy surface appears. These surfaces intersect at the band crossings along the x, y,
and z axes7 .
-1.5
-1
-0.5
0
y
0.4
1
1.5
-1.5
-1
-0.5
0
0.5
1
1.5
(a)
x
-0.6
-0.4
-0.2
0
0.6
0.5
0.2
0
z
-0.2
-0.4
-0.6
0.6
0.4
0.2
0
y
0.2
0.4
0.6
-0.6
-0.4
-0.2
x
(b)
Figure 3.3: Constant energy surfaces of the monopole flux state, for E/J = 0.5.
Decreasing E/J makes the cylinders thinner. (a) A view of the cube of side length
π surrounding the origin. Eight cylinders surrounding the eight Fermi lines emanate
from the origin; at the origin a diamond-line shape (the low-energy spectrum of the
remaining two bands) can also be seen. This shape repeats at the cube’s corners
(±π, ±π). (b) A close-up view of the region surrounding the origin. Altering the
ratio E/J shrinks the entire structure, but does not change its shape.
3.3.3
Low Energy Expansions of the Spinon Dispersion
The low-energy structure of the monopole flux state can be divided into two regions:
R1 , the set of 4 Fermi lines sufficiently far from the origin, and R0 , the area near the
origin.
In R1 , only 2 of the four bands lie near the Fermi surface, and the low-energy
theory is effectively two-dimensional. Linearizing the Hamiltonian about one of the
7
This Fermi surface does not display fermion doubling in the naive sense; all four bands cross at
only one point in the Brillouin zone. This does not violate the result of Ref. [65], which assumes
that levels are degenerate only at a finite set of points.
48
Fermi lines gives:
H[ε, θ] = Ψ†1α [k, ε, θ](ε cos θτ1 + ε sin θτ2 )Ψ1α [k, ε, θ],
(3.11)
with energies ±ε, independent of θ. Here we have used the local coordinate system
r
2
u
v
u
v
(q1 , q2 , q3 ) = (k +
u, k − √ − √ , k − √ + √ ),
(3.12)
3
6
2
6
2
√
√
with θ(u, v) = tan−1 (v/u) and ǫ(u, v) = ρ0 u2 + v 2 /(2 2). Curiously, at mean
field the low energy spectrum is independent of the position k along the Fermi line,
depending only on the momentum component in the kagomé planes perpendicular
to the vector mli . Thus the linearized theory away from the origin consists of a
continuum of flavors of Dirac fermions confined to the kagomé planes orthogonal to
this line.
In R0 all 4 bands have energies vanishing linearly as k → 0, and the low-energy
Hamiltonian is given by:

 0
 k +k
x
y
X †α 

H = −ρ0
Ψk  4
 ky +kz
k,α
 4

kx +kz
4
kx +ky
4
ky +kz
4
0
kx −kz
4
kx −kz
4
0
kz −ky
4
ky −kx
4

kx +kz
4 
kz −ky 
4 

 Ψkα .
ky −kx 
4 
0
with energy eigenvalues
v
s X
u X
1
u1
2
ε = ±t
k ±
3
ki2 kj2 ,
8 i i
8
(3.13)

(3.14)
(i<j)
This dispersion relation also gives massless spinons; however, the theory is no longer
one of Dirac fermions.
In addition to four bands touching at the origin, the linearized Hamiltonian (3.13)
has 2 zero eigenvalues on each Fermi line. Restricting the spinors to the corresponding
low-energy subspace again yields the expression (3.11). Thus Eq. (3.13) captures the
49
principal features of the low-energy behavior not only in the vicinity of the origin,
but throughout the entire Brillouin zone.
The linearized Hamiltonian has several interesting features. First, we may express
it in terms of three matrices as follows:
H=
ρ0
(αx kx + αy ky + αz kz )
4
(3.15)
The α matrices are reminiscent of Dirac γ matrices, albeit with a tetrahedrally invariant, rather than rotationally invariant, algebraic structure. They do not comprise
a Clifford algebra, but obey the anti-commutation relations
{αi , αj } = 2δij +
√
6|εijk |Wk
{Wi , Wj } = 2δij
(3.16)
Further, in a 3+1 dimensional Dirac theory there are two matrices (γ0 and γ5 )
which anti-commute with all γi . In this sense our mean field Hamiltonian more
resembles a 2+1 dimensional Dirac theory: there is a unique matrix α0 such that
{α0 , αi } = 0, i = 1..3, given by


0
cos(ky ) cos(kx )
cos(ky ) cos(kz )
cos(kx ) cos(kz ) 



− cos(ky ) cos(kx )

0
cos(k
)
cos(k
)
−
cos(k
)
cos(k
)
1
x
z
y
z


(0)
α =√ 


N
−
cos(k
)
cos(k
)
−
cos(k
)
cos(k
)
0
cos(k
)
cos(k
)

y
z
x
z
y
x 


− cos(kx ) cos(kz ) cos(ky ) cos(kz ) − cos(ky ) cos(kx )
0
(3.17)
where N is a normalization factor such that (α(0) )2 = 1. In the continuum limit this
reduces to

α(0)
0

−1
√

= 3
√
−1
 3

−1
√
3
√1
3
√1
3
0
√1
3
−1
√
3
0
√1
3
−1
√
3

√1
3

−1 
√
3

√1 
3
0

(3.18)
50
α(0) acts as a spectrum inverting operator on H, interchanging hole states at energy
−E(k) with particle states at energy E(k).
We point out that many of the interesting features of the low-energy spectrum of
the monopole flux state can be generalized to a class of lattices whose geometry is
related to certain representations of Lie groups [66]. Indeed, the four sites in the tetrahedral unit cell can be viewed as the four weights in the fundamental representation
of SU(4); hopping on the pyrochlore is then analogous to acting with the appropriate
raising and lowering operators. This perspective gives an explicit connection between
the hopping Hamiltonian (3.13) and the ladder operators in the fundamental representation of SU(4). Analogous hopping problems can be studied for various other Lie
group representations, as outlined in detail in the next Chapter.
To summarize, the monopole flux state is a spin liquid which preserves all symmetries of the full Hamiltonian except P and T . At mean field level it has gapless
spinons along a 1-dimensional Fermi surface of 4 lines which intersect at the origin.
Though strictly at N = ∞ it has higher energy than the dimerized state, Gutzwiller
projection suggests that for N = 2 this is no longer the case, and the monopole flux
state is the lowest energy simple mean field ansatz after projection. We now turn our
attention to what can be said about the stability of this rather unusual mean field
state.
3.4
Stability of the Mean-Field solution: the role
of the PSG
Next, we would like to address the question of whether the mean field solutions
described above maintain their basic properties at finite N and whether this holds
all the way to N = 2. This is a difficult problem, whose complete solution is not
51
available even for the longer studied cases of the algebraic spin liquids in d = 2
8
However, following that work the general idea would be to try and understand if the
state is truly stable at large enough N while leaving the question of stability at small
N to detailed numerical investigation.
There are several questions here. First, is the mean field solution locally stable?
Second, is it the global minimum? Third, assuming the answer thus far is in the
affirmative, is the expansion about the mean field solution well behaved? Ideally, this
would mean convergent, but it would be sufficient to know that it does not destroy
the qualitative features of the gapless spinon dispersion at mean field. For example,
in the case of the algebraic spin liquids in d = 2 the spinons interact and acquire
anomalous dimensions away from N = ∞ but they remain gapless in the vicinity
of a discrete set of points [54]. Finally, what is the spectrum of collective (gauge)
excitations that arise in this expansion?
Based on the experience with spin liquids in d = 2, answering the first two questions in the affirmative is likely to require the addition of more terms to the Hamiltonian although it may be possible to choose them so that they become trivial at
N = 2 [50]. We have not investigated this in detail but there does not appear to be
an obstacle to doing this.
The third and fourth questions require detailed consideration of the symmetry
properties and the detailed dynamics of the expansion which is that of a lattice gauge
theory with matter and gauge fields in some fashion. In this work we will carry out
the first part of this program which goes under the study of the “Projective Symmetry
Group” (PSG) discussed in detail by Wen [53]. In this section we review the concept
of the PSG and its implications for perturbative expansions. We also show that at
N = ∞, or in mean field theory, the PSG already helps us understand the stability
of particular mean field solutions; to our knowledge this particular aspect has not
8
See Refs. [54, 67] and references therein.
52
appeared in the literature before.
Turning first to the PSG, observe that though the original Hamiltonian formulated
in terms of spin operators is invariant under the full space group of the pyrochlore
lattice, the actual mean field Hamiltonian of the monopole flux state is not: many
of the symmetry transformations map the mean field Hamiltonian into different but
gauge equivalent Hamiltonians. Thus, when working in the gauge theory formulation
of the problem, the actual symmetry transformations of the mean field Hamiltonian
have the form:
ci → gs (s(ci ))
(3.19)
where s is an element of the space group, and g is a gauge transformation. As the
full Hamiltonian is gauge invariant, Eq. (3.19) is simply an alternative formulation of
the lattice symmetries. Hence as emphasized by Ref. [53], these projective symmetry
operators are exactly analogous to lattice symmetries in the original spin problem.
Indeed, the correct choice of gauge transformation ensures that both HM F and H are
invariant under the PSG, so that the family
Hλ = HM F + λ(H − HM F )
(3.20)
is also invariant and perturbative corrections in H − HM F cannot break the PSG
symmetry.
Before discussing the implications of PSG symmetry for the monopole flux state,
we would like to briefly underline how the PSG constrains the mean field theory at
infinite N which is a much simpler but still instructive exercise.
Ignoring the dimerization instability, the monopole flux state is a mean field minimum for nearest-neighbor couplings. The PSG is the symmetry group of the corresponding mean field Hamiltonian. We may now ask what happens to the PSG if
further neighbor couplings are included in the Hamiltonian: in particular do they lead
to terms in the new mean field Hamiltonians that modify the PSG found earlier?
53
At N = ∞ this is a problem of minimizing the expectation value of the sum of
the quadratic Hamiltonian in Eq.(3.4) and the new generic terms
′
δH = −
′
XX
α
(c†iα cjα χij
(ij)
N X
+ h.c.) +
|χij |2
Jij
(3.21)
(ij)
wherein the primed sum runs over non-nearest neighbor bonds and the Jij are much
smaller than the nearest neighbor J. We will now show that, generically, the result
of the new minimization for the perturbed problem preserves the PSG for the nearest
neighbor problem. While we use the language of perturbing about the monopole flux
state, the argument is general.
With the addition of the perturbation, the functional that we need to minimize
over the full set of {χij } is:
′
EM F = hH
(F )
+ δH
(F )
iH+δH
N X
N X
+
|χij |2 +
|χij |2 .
J <ij>
Jij
(3.22)
(ij)
(0)
where the superscript F denotes the fermionic part of the Hamiltonian. Let χij
denote the values of the link fields when δH ≡ 0, i.e. in the monopole flux state. For
small Jij we expect the new minimum to lie not far from the old one, whence the link
(0)
fields will be close to the values χij . Consequently we will compute the expectation
value required in the above equation in perturbation theory in δH about H. (If such
an expansion fails to have any radius of convergence then we are already parked at a
phase transition and no stability argument is possible.)
This expansion,
EM F ({χij }) = E0 + h0|δH (F )|0i +
X |h0|δH (F )|ni|2
+···
E0 − En
NX
N X
+
|χij |2 +
|χij |2 ,
J
Jij
n>0
hiji
(3.23)
(ij)
where the numerical indices refer to the ground and excited states of the unperturbed
(0)
Hamiltonian H (F ) ({χij }), has three properties that we need. First, the linear term
54
takes the explicit form,
−
XX
α
(ij)
hc†iα cjαiδχij + h.c.
(3.24)
(0)
where δχij is χij for the new bonds and the deviation from χij for the nearest neighbor
bonds. This implies that new minimization likes to turn on exactly those χij that
transform as the expectation values hc†iα cjαi. If these are, in fact, what get turned on,
then the new mean field Hamiltonian will indeed inherit the PSG of the starting one.
The second property that we need can be established by considering a decomposition
of δH into a piece that commutes with the PSG generators and another piece that
does not. It is straightforward to see that terms from quadratic order and beyond
must give rise to a potential which is even in powers of the non-PSG conserving piece
of δH. Finally, at sufficiently small Jij the potential for the χij must be stable due to
the explicit factors of 1/Jij . Together these properties imply that the new minimum
must be in the “direction” selected by the linear term and hence will exhibit the same
PSG as before.
3.5
The PSG of the monopole flux state
We will now describe the PSG of the monopole flux state, and its implications for
stability at the mean field level. In brief, the space group of the pyrochlore lattice
is F d3̄m, which contains 24 symmorphic and 24 non-symmorphic elements. For our
purposes it is most convenient to divide these elements into the 24 proper elements
composed of rotations and translations, and 24 improper elements involving a reflection or inversion. The rotations comprise a proper subgroup of F d3̄m, while the
improper elements are generated by the product of the inversion operator (inversion
is taken about one of the lattice sites) with rotations.
The PSG of the monopole flux state has the following general structure:
• Translations : FCC translations, combined with the identity gauge transforma-
55
tion.
•P 0 space group elements : These elements are symmetries when combined with
appropriate gauge transformations, which induce a π phase shift at some the sites in
the unit cell.
•P i space group elements : These elements are symmetries when combined with
an appropriate gauge transformation, as above, and a time reversal transformation.
•Charge conjugation C : The charge conjugation operator maps ci → c†i .
3.5.1
Symmetries of the pyrochlore lattice
We will begin by describing these symmetries in detail. The space group F d3̄m
of the pyrochlore lattice consists of the 24 element tetrahedral point group 4̄3m,
and a further 24 non-symmorphic elements, involving a combination of rotations or
reflections with translation by half of a lattice vector. We will briefly describe the
actions of these symmetry operations here. All vectors are expressed in the basis of
the standard cubic FCC unit cell.
The actions of the tetrahedral point group P 0 fix the position of one tetrahedron’s
center (at e.g. (a/8, a/8, a/8)). Its elements are:
1. The identity
2. 8C3 : there are four 3-fold axes, one passing through each vertex and the center
of the opposite face. Rotations about this axis permute the 3 vertices not on the
axis. These we label C1 ..C4 , C12 ..C42 , where Ci , Ci2 fix site i of the tetrahedral
unit cell.
3. 3C2 : There are 3 2-fold axes, parallel to the x, y, and z axes. Each axis bisects
a pair of edges on the tetrahedron; the ensuing rotation exchanges pairs of
vertices. These we label Cx , Cy , Cz .
56
4. 6σd : A plane of reflection passes through each edge, and out the center of the
opposite face. These planes lie on the diagonals with respect to the FCC cubic
unit cell, and hence are called diagonal reflections.
5. 6S4 : An improper rotation of degree 4 about the axis bisecting 2 edges (parallel
to the x, y, or z axis) is also a symmetry. The tetrahedron is rotated by π/2
about e.g. (1/8, 1/8, z) and reflected through the plane z = 1/8. This operation
squared produces one of the 2-fold rotations, so each axis contributes 2 group
elements.
The remaining non-symmorphic elements, which we distinguish from their symmorphic counterparts by adding a˜, are:
1. 6C̃4 : There are three 4-fold screw axes: (3/8, 1/8, z), (3/8, y, 1/8, and (x, 3/8, 1/8).
The symmetry rotates the lattice by π/4 about such an axis, and translates by
1/4 of the side length of the FCC cubic unit cell along the axis. Each axis
accounts for 2 elements of the quotient group, as C42 = tC2 , with t an FCC
translation and C2 one of the 2-fold rotations of the point group. These we
label C̃x , C̃x2 , C̃y , C̃y2 , C̃z , C̃z2 .
2. 6C̃2 : Along each of the 6 edges of the tetrahedron (the FCC basis vectors) there
is a 2-fold screw axis. The lattice is translated along the edge of a tetrahedron,
then rotated by π about this edge. These we label C̃ij , where C̃ij has a screw
axis along the line joining sites i and j.
3. 3σ̃h : The x, y and z planes of the cubic unit cell each contain a horizontal glide
plane. The lattice is translated along an FCC vector in the plane, e.g. by
(1/4, 1/4, 0), and then reflected through the plane – in our example, through
z = 0.
4. i The lattice is inverted about the origin.
57
5. 8S̃6 : The products of the 8 C3 rotations with the inversion give 8 improper
3-fold rotations. (These are not in the point group because they map a single
tetrahedron onto its neighbor.)
For our purposes these 48 elements divide into 24 C elements involving pure rotations and translations, and 24 S elements involving improper rotations, reflections,
or inversions. The S elements are not symmetries of the monopole flux state, as they
map monopoles to anti-monopoles; to construct the appropriate symmetry elements
they must be combined with a time reversal transformation. Since all such elements
can be expressed as a product of a C element with the inversion, this is simply a
consequence of the fact that while P and T are separately broken in the monopole
flux state, the combination PT is still a symmetry.
3.5.2
The PSG
We now list the PSG transformation rules for the symmetry operations described
above, shown in 3.5.2 and 3.5.2. Throughout, we use the gauge illustrated in Fig.
3.1, in which all bonds are either ingoing or outgoing from site ‘1′ ; starting from a
different gauge will permute the gauge transformations listed here (note that this has
no effect on which bonds are allowed or disallowed by the PSG, however). These
tables only show the mapping between the site labels 1...4; it is important to bear in
mind the effect of the translations in the case of the non-symmorphic elements, which
reverse the directions between sites by interchanging up and down triangles. To this
end we also include a table of momentum transformations under these operations.
Since S elements are products of C elements and inversion, it is sufficient to consider the 24 rotation operations, together with the operator iT . Group multiplication
in the PSG is valid modulo the global gauge transformation ci → −ci , which clearly
does not alter the Hamiltonian. Thus only the relative phases of the 4 sites in the
tetrahedral unit cell are relevant to the PSG transformation.
58
Table 3.2: Action of
1 C1 C12 C2
c1 c1 c1
c4
c2 c3 c4 −c2
c3 c4 c2 −c1
c4 c2 c3
c3
kx kz ky −kz
ky kx kz
kx
kz ky kx −ky
c1
c2
c3
c4
kx
ky
kz
3.6
C̃12
c2
c1
c3
−c4
ky
kx
−kz
C̃13
c3
−c2
c1
c4
kz
−ky
kx
PSG Point Group Rotations on
C22
C3
C32
C4
C42
−c3
c2
−c4
c3
−c2
−c2
c4
c1
−c1
c3
c4
−c3 −c3
c2
c1
c1
−c1
c2
−c4 −c4
ky
kz −ky −kz −ky
−kz −kx −kz −kx kz
−kx −ky kx
ky −kx
Table 3.3:
C̃14 C̃23
c4
−c1
c2
c3
−c3
c2
c1
c4
−kx −kx
kz −kz
ky −ky
PSG
C̃24
−c1
c4
c3
c2
−kz
−ky
−kx
action
C̃34
−c1
c2
c4
c3
−ky
−kx
−kz
spinon operators.
Cx
Cy
Cz
−c4 −c3
c2
−c3
c4
−c1
c2
c1
c4
c1
−c2 −c3
kx −kx −kx
−ky ky −ky
−kz −kz kz
of screw rotations
C̃x
C̃y
C̃z
C̃x3
c2
c4
c3
c3
−c4
c1
c4
c1
c1
c2
−c2
c4
c3
−c3
c1
−c2
kx
kx
kz −kz
−kz kz
ky
ky
ky −ky −kx kx
C̃y3
c2
c3
−c4
c1
−ky
kx
kz
C̃z3
c4
−c3
c1
c2
ky
−kx
kz
PSG invariance and the fate of the monopole
flux state
To deduce what restrictions PSG invariance imposes on the spectrum, we begin with
a generic 4 × 4 quadratic Hamiltonian
H (2) =
X
Jij c†i cj
(3.25)
ij
The bonds Jij connect arbitrary sites in the lattice, but respect the lattice symmetries.
In what follows, we will use the PSG to restrict the possible quadratic terms, and show
that all terms allowed by symmetry vanish at the Fermi surface. Hence the Fermi
surface of the monopole flux state is unaffected by PSG-preserving perturbations to
the Hamiltonian. For simplicity we will drop the superscript
(2)
in the remainder of
59
this section to simplify the notation.
Though the inversion P and time reversal T are broken in the mean field state, the
combination P T leaves both the full and mean field Hamiltonians invariant. Terms
invariant under this transformation have the form:
(J ′ + iJ ′′ )c†x cx+δ + (J ′ − iJ ′′ )c†x cx−δ
(3.26)
and the Hamiltonian is real in momentum space. Further, invariance under charge
conjugation forces all spatial bonds to be purely imaginary: under C,
(J ′ + iJ ′′ )c†x cx+δ + (J ′ − iJ ′′ )c†x+δ cx
→ (−J ′ + iJ ′′ )c†x cx+δ + (−J ′ − iJ ′′ )c†x+δ cx
(3.27)
so that J ′ = 0 if C symmetry is unbroken. In momentum space, if we write the
Hamiltonian as ψH (k)ψ, Eqs. (3.26) and (3.27) imply that H (k) is real and an odd
function of k. We may express elements of the matrix H (k) as a superposition
Hab (k) =
X
R
JR;ab sin [k · (R + rab )]
(3.28)
where R is an FCC lattice vector, the indices a, b label sites within the unit cell, and
JR;ab is the coupling between sites a and b separated by the lattice vector R, and
the vector rab in the unit cell. This is the general form for a function periodic in the
Brillouin zone.
Diagonal Terms Let H11 ...H44 be the diagonal elements of H . To restrict the form
of H11 , we consider the action of all PSG operations that map site 1 in the tetrahedral
unit cell onto itself. These are (see Appendix 3.5 for labels and actions of the PSG
60
elements) {C1 , C12 , C̃23 , C̃24 , C̃34 }, which transform H11 in the following way:
C
H11 (kx , ky , kz ) →1 H11 (kz , kx , ky )
C2
→1 H11 (ky , kz , kx )
C̃23
→ H11 (−kx , −kz , −ky )
C̃24
→ H11 (−kz , −ky , −kx )
C̃34
→ H11 (−ky , −kx , −kz )
(3.29)
which allows us to express H11 in a form where its symmetries are manifest as:
H11 (kx , ky , kz ) =
1
[H11 (kx , ky , kz ) + H11 (kz , kx , ky )
6
+H11 (ky , kz , kx ) − H11 (kx , kz , ky )
−H11 (kz , ky , kx ) − H11 (ky , kx , kz )]
(3.30)
Similarly we can relate H22 , H33 , and H44 to H11 by considering operations which
interchange site 1 with sites 2, 3, and 4 respectively. These imply:
H22 (kx , ky , kz ) = H11 (ky , kx , −kz )
H33 (kx , ky , kz ) = H11 (kz , −ky , kx )
H44 (kx , ky , kz ) = H11 (−kx , kz , ky )
(3.31)
While multiple transformations map between each pair of diagonal elements, the
group structure and invariance of H11 under PSG transformations ensures that these
mappings all yield the same result.
The reader should note that Eqs. (3.30) and (3.31) ensure that along the Fermi
lines k = ±(1, ±1, ±1) all allowed diagonal terms vanish.
It is worth digressing to make one more comment on the diagonal terms. Using the
symmetrized form of H11 in Eq. (3.30) above, we can rewrite the term in Eq. (3.28)
61
with a fixed R and a = b = 1 as
JR;11 sin(k · R) =
1
JR;11 [sin(kx Rx + ky Ry + kz Rz )
6
+ sin(kz Rx + kx Ry + ky Rz )
+ sin(ky Rx + kz Ry + kx Rz )
− sin(kx Rx + kz Ry + ky Rz )
− sin(kz Rx + ky Ry + kx Rz )
− sin(ky Rx + kx Ry + kz Rz )]
(3.32)
Expressions of the form (3.32) vanishes if any two coefficients are equal; non-vanishing
terms occur only for a sum of at least three FCC translations. Physically this corresponds to a hopping between a site and its translate some three lattice vectors
distant.
Off-Diagonal Terms As H is real in momentum space, Hab = Hba . To restrict the
form of H12 , consider the action of all PSG elements which either map sites 1 and 2
to themselves, or interchange them. These are {C̃34 , C̃12 , Cz }, which transform H12
according to
C̃
34
H12 (kx , ky , kz ) →
−H12 (−ky , −kx , −kz )
C̃12
→ H21 (ky , kx , −kz )
C
→z −H21 (−kx , −ky , kz )
(3.33)
Again, transformations mapping sites 1 and 2 onto other sites in the unit cell can be
used to deduce the form of the remaining off-diagonal elements. Hence
C
H12 (kx , ky , kz ) →1 H13 (kz , kx , ky )
C2
→1 H14 (ky , kz , kx )
C2
→4 −H23 (−ky , kz , −kx )
C
→2 −H24 (−kz , kx , −ky )
Cy
→ −H34 (−kx , ky , −kz )
(3.34)
62
This gives off-diagonal entries:


0
H12 (kx , ky , kz )
H12 (kz , kx , ky )
H12 (ky , kz , kx ) 



H12 (kx , ky , kz )
0
H12 (ky , −kz , kx ) H12 (kz , −kx , ky )





H (k , k , k ) H (k , −k , k )
0
H
(k
,
−k
,
k
)
 12 z x y
12 y
z
x
12 x
y
z 


H12 (ky , kz , kx ) H12 (kz , −kx , ky ) H12 (kx , −ky , kz )
0
(3.35)
where again we can make the symmetries manifest by writing
H12 (kx , ky , kz ) =
1
[H12 (kx , ky , kz ) + H12 (ky , kx , kz )
4
+H12 (ky , kx , −kz ) + H12 (kx , ky , −kz )] .
(3.36)
Again, it is useful to focus on the contribution to H12 from bonds with a given R
which can now be seen to come with the factor:
kx + ky
)
2
kx + ky
+ sin(ky Rx + kx Ry +
)] .
2
cos(kz Rz )[sin(kx Rx + ky Ry +
(3.37)
Eq. (3.37) shows that H12 (kx , ky , kz ) vanishes along the lines (k, −k, −k), (−k, k, −k).
Of course, this can also be seen directly from Eq. (3.36).
Now we may consider the fate of the monopole flux state’s exotic Fermi surface.
Since PSG rotations map between different Fermi lines, it is sufficient to consider
possible alterations to the spectrum on Fermi line (k, k, k). The most general form
that H can have about the line (k, k, k) is:


0
H12 (k, k, k) H12 (k, k, k) H12 (k, k, k)




H12 (k, k, k)
0
0
0


H=

H (k, k, k)

0
0
0
 12



H12 (k, k, k)
0
0
0
(3.38)
which has two zero eigenvalues. Thus terms allowed by symmetry add neither a chemical potential nor a gap to any part of the Fermi lines, and preserve the characteristic
63
structure of the monopole flux state, with 2 low energy states about each Fermi line,
and 4 low energy states about the origin.
Note that nothing prevents the Fermi velocity vF from being modified as a function
of the momentum along the line. Indeed, (3.38) implies that the general form of vF
is:
X
vF (k)
√
=
JR;12 (Rx + Ry + 1/2) cos(2kRz ) cos 2k(Ry − Rx )
2
R
JR;11
+
(Rx sin(2kRx ) sin 2k(Rz − Ry )
3
+Ry sin(2kRy ) sin 2k(Rx − Rz ) + Rz sin(2kRz ) sin 2k(Ry − Rx ))(3.39)
.
3.6.1
Time reversal and Parity
One striking feature of HM F is that it is odd under both T and P , reminiscent of the
chiral spin state first described in Ref. [52]. Though T is naively broken, some care
must be taken to show that the apparent T breaking is physical and that |ψi, T |ψi
are gauge inequivalent states[67]. Readers familiar with this subtlety from discussions of T breaking on the square lattice, should note that the pyrochlore lattice is
not bipartite and hence naive time reversal is no longer equivalent to particle-hole
conjugation. But most directly, as explained in Ref. [52], the operator
Eijk = Si · (Sj × Sk )
(3.40)
where the spins i, j, and k lie in a triangular plaquette, is odd under T and P . Hence
if hEijk i|ψi 6= 0, the state |ψi breaks time reversal.
At mean field level,
−i
hE123 i = hχ12 χ23 χ31 i − hχ13 χ32 χ21 i
2
(3.41)
and states with an imaginary flux through triangular plaquettes are T -breaking. For
the monopole flux state, we have confirmed numerically that this T -breaking in each
plaquette is robust to Gutzwiller projection; the results are shown in Table 3.4.
64
Lattice size
3×3×3
5×5×5
hE△ i
0.039
0.043
Table 3.4: Expectation values of the T -breaking operator hE△ i for triangular faces of
the tetrahedra.
We also note the curiosity that at infinite N, the spectrum-preserving nature
of T and P allows us to construct additional symmetries which are not, however,
symmetries of the full H. Particle-hole symmetry at each k allows us to construct
the following 2 discrete symmetries of HM F :
T̃ : |ψ(x, t)i → α(0) |ψ(x, −t)i
P̃ : |ψ(x, t)i → α(0) |ψ(−x, t)i
(3.42)
where α0 was defined in Eq. (3.17). Both of these commute with the non-interacting
Hamiltonian: since α0−1 Hα0 = −H, we have
hψ|T̃ †H T̃ |ψi = −hψ|H ∗ |ψi
= hψ|H|ψi
hψ|P̃ † H P̃ |ψi = −hψ|H T |ψi
= hψ|H|ψi
(3.43)
The matrix structure of α0 is such that T̃ and P̃ are not symmetries of the full
Hamiltonian, however, and will not be robust to perturbative corrections about mean
field.
65
3.6.2
PSG symmetry and perturbation theory in the long
wavelength limit
We have established that invariance under the PSG transformations and charge conjugation forbid both mass and chemical potential terms on the Fermi lines. Here
we explore how these PSG symmetries are realized as symmetries of the linearized
low-energy theory away from the origin, and hence see in that setting why they are
protected perturbatively.
Consider the linearized theory about the Fermi line ml1 = (k, k, k). A general
Hamiltonian in the 2 × 2 space of low-energy states can be expressed as:
H(k) = ψ1† (k)h(k)ψ1 (k)
(3.44)
h(k) = µ(k) + m(k)σ3 + ε(k, v)(cos(θ)σ1 + sin(θ)σ2 )
where k is the component of the momentum k along the line, and (v, θ) are the
magnitude and angle respectively of the momentum perpendicular to the line. Here
ψ11 (k) = (0, ω 2, 1, ω) · mcα (k)
ψ12 (k) = (0, ω, 1, ω 2) · mcα (k)
(3.45)
with ω = e2πi/3 . The states (3.45) are eigenstates of the rotation operator C1 which
rotates about the (1, 1, 1) direction, with C1 ψ1j = ω j ψ1j .
Under charge conjugation,
†
ψ11 (k) → ψ12
(−k)
†
ψ12 (k) → ψ11
(−k)
(3.46)
The corresponding symmetry operator in the continuum theory is
C : ψ1 (v, θ) → σ1 [ψ1† (v, π − θ)]T
(3.47)
with the Fermi surface points at k and −k interchanged. This implies m(−k) = m(k),
and µ(−k) = −µ(k).
66
ψ11
ψ12
ψ21
ψ22
ψ31
ψ32
ψ41
ψ42
ψ11
ψ12
ψ21
ψ22
ψ31
ψ32
ψ41
ψ42
Table 3.5: Effect of Point
C1
C2
C3
2
ωψ11
ω ψ31 −ωψ41
ω 2 ψ12
ωψ32 −ω 2 ψ42
ψ41
ωψ21
ψ11
2
ψ42
ω ψ22
ψ12
−ωψ21
ψ41
ωψ31
−ω 2 ψ22
ψ42
ω 2 ψ32
2
−ω ψ31 ωψ11 −ω 2 ψ21
−ωψ32 ω 2ψ12 −ωψ22
Group rotations
C4
Cx
2
−ω ψ21 −ψ41
−ωψ22 −ψ42
ωψ31
−ψ31
2
ω ψ32 −ψ32
−ψ11
ψ21
−ψ12
ψ22
ωψ41
ψ11
ω 2 ψ42
ψ12
on low-energy eigenstates.
Cy
Cz
ωψ31
ωψ21
ω 2 ψ32
ω 2 ψ22
−ω 2 ψ41 −ω 2 ψ11
−ωψ41 −ωψ12
−ω 2 ψ11 ω 2 ψ41
−ωψ12
ωψ42
ωψ21
−ωψ31
ω 2 ψ22 −ω 2 ψ32
Table 3.6: Effect of 2-fold
C̃12
C̃13
C̃14
2
ωψ22
ω ψ32 −ωψ42
ω 2 ψ21
ωψ31 −ω 2 ψ41
ψ12
ωψ22
ψ22
2
ψ11
ω ψ21
ψ21
−ωψ32
ψ12
ωψ32
−ω 2 ψ31
ψ11
ω 2 ψ31
2
−ω ψ42 ωψ42 −ω 2 ψ12
−ωψ41 ω 2ψ41 −ωψ11
glide rotations on low-energy eigenstates.
C̃23
C̃24
C̃34
2
−ω ψ12 −ψ12
ωψ12
−ωψ11 −ψ11 ω 2 ψ11
ωψ32
−ψ42 −ω 2 ψ22
2
ω ψ31 −ψ41 −ωψ22
−ψ22
ψ32 −ω 2 ψ42
−ψ21
ψ31
−ωψ41
ωψ42
ψ22
ωψ32
2
ω ψ41
ψ21
ω 2 ψ31
Tables 3.6.2 and 3.6.2 list the action of the rotation elements of the PSG on these
low-energy states. Since (3.45) is PT invariant, we need only consider the rotation
elements; the improper elements will act on these states with the same effect. Proper
rotations map clockwise rotating states to clockwise states, and counter-clockwise to
counter-clockwise. The improper rotations, conversely, map counter-clockwise rotating states into clockwise rotating states, and vice versa.
An analysis of the PSG transformations in Tables 3.6.2 and 3.6.2 reveals that the
glide rotations (C̃ij ) map clockwise rotating states to counter-clockwise states while
reversing the direction of the corresponding Fermi line: ψi1 (kli ) → ψi2 (−kli ). This
transformation leaves the mean field Hamiltonian invariant. In the 2 × 2 basis, this
is because
T : |ψ(v, θ)i → σ1 |ψ(v, π − θ)i
(3.48)
67
is a symmetry of the mean field Hamiltonian. Note that the momentum transformation can be realized in 3 dimensions by a π rotation about the line x = y, and hence
should also send k → −k, though there is no way to deduce this from the form of the
mean field eigenstates. The symmetry transformation (3.48) reverses the sign of the
mass term, but not of the chemical potential, implying that m(−k) = −m(k) and
µ(k) = µ(−k). Hence we conclude that in the continuum theory about a given Fermi
line, the symmetries C and T prevent a gap or chemical potential from arising.
One might ask why we have not considered mass gaps of the form mσ1 or mσ2 ;
both of these choices turn out to violate either (3.47) or (3.48). Indeed, both choices
explicitly break the rotational symmetry of the spectrum about the Fermi line.
3.7
Fluctuations about mean-field
The symmetry arguments presented in the previous sections suggest a story similar
to that which emerges for the π flux state on the square lattice, in which the gapless
points are protected by symmetry, but strong interactions renormalize the dimensions of the fermion and gauge fields to a non-Gaussian fixed point. The important
difference is that, unlike for the square lattice case, where the long-wavelength action
is the well-studied problem Dirac fermions in 2 + 1 dimensions, the long-wavelength
Hamiltonian (3.13) on the pyrochlore does not have relativistic – or even rotational –
invariance. Hence calculating renormalized quantities, even at 1-loop order, presents
a major technical challenge which we will not confront here. Instead, we will present
some simple arguments for the form of the allowed corrections.
Before proceeding, let us sketch the steps required to carry out the full calculation
of the first-order corrections to the theory. Since we are working in a 1/N, rather
than weak coupling, expansion, it is not sufficient to calculate all diagrams to oneloop order. In particular, the one-loop contribution to the photon propagator is of
68
order 0 in 1/N, since the N flavors of fermions possible in the loop compensate for the
√
suppression by 1/ N at each vertex. Thus integrating out fermions induces a kinetic
term for the photon at 0th order in 1/N, given by the series of diagrams involving
insertions of this fermion loop. The resulting photon propagator in turn feeds into
corrections to the fermion propagator and vertex at order 1/N. Ultimately it is these
corrections that we are interested in, as they will indicate possible instabilities in the
Fermi surface of the monopole flux state. Their form, however, will rather obviously
depend on the photon propagator at one loop. Hence the first important step in
understanding the effect of fluctuations about mean-field is to deduce the form of the
photon’s kinetic energy.
Here we will focus on a lattice derivation of the gauge-field propagator, which
illustrates several interesting features of the perturbation expansion. First, we show
how beginning with 12 links in each unit cell on the lattice, we arrive at 2 propagating
gauge modes, as expected in a continuum theory in 3 dimensions. Second, we will
argue that integrating out the high-energy fermions, which is equivalent to adding the
effect of short loops on the lattice to the gauge field action, generates a rotationally
invariant photon propagator. Since we only include short loops, it is also useful to
consider the restrictions that symmetries impose on the photon propagator, and ask
what kinds of terms are allowed in principle. We will show that a topological term, of
the form E · B, is also allowed in the gauge-field action. From this starting point one
could, theoretically, compute corrections to the fermion propagator as well, though
we leave this arduous task for future work.
Before carrying out this lattice derivation, let us mention one caveat in the above
treatment. On the lattice we integrate out only high-energy fermions; the usual
procedure in the continuum is to integrate out all fermions, down to the four Fermi
lines. As stated above, the partial integration leads to a photon propagator with
full rotation invariance. However, we could also integrate out all fermions sufficiently
69
far from the origin that we may use the effective low-energy theory about each line.
In this case it is easy to see that the low-energy modes near each line induce gauge
kinetic terms which break its full rotation invariance to a discrete subgroup. This in
turn may alter the structure of infra-red divergences in higher-loop corrections to the
fermion propagator. Hence the results we describe here, while useful in elucidating
aspects of the qualitative form of the full Lagrangian, may not accurately represent
all of the important structure in the gauge kinetic terms.
3.7.1
Perturbation Theory on the lattice
Our derivation of the lattice gauge kinetic action will proceed in two steps. We begin
here by asking what equal-time gauge invariant operators may arise at quadratic order
in the gauge fields from integrating out fermions on the pyrochlore lattice. We then
consider terms involving time derivatives of the gauge fields in Sect. 3.7.1.
Gauge invariant terms with no time derivatives are closed spatial loops (Wilson
lines) on the lattice. Such loops automatically respect lattice symmetries, and hence
constitute the full set of time-independent, gauge invariant terms allowed by symmetry. We imagine coarse-graining the theory in space, in which case shorter loops
(corresponding to integrating out higher-energy fermions) are the first terms generated. Here we consider the effect of the simplest plaquette contributions – those of the
triangular and hexagonal plaquettes. We will be interested in terms in the effective
action quadratic terms in the gauge field –all others being irrelevant at long wavelengths. Where appropriate, we obtain these by expanding the Wilson loop operators
to quadratic order.
This approach is useful not only in giving insight into the form of the gauge field
propagator, but also in addressing the following conundrum. The unit cell of the
pyrochlore lattice has 12 distinct edges, and hence 12 possible distinct gauge fields.
In our continuum theory, we expect a single photon mode. We will see that 6 of
70
the extra gauge fields acquire infinite mass in the continuum limit, and hence do not
appear in the long-wavelength theory. Another 4 modes are pure gauge, and have
flat band dispersions. This leaves 2 physical gauge invariant modes, as we expect.
Let us now evaluate the contribution of the triangular plaquette terms to the
effective action. We label the 12 gauge fields in the unit cell axy , axz , ayz , bxy , bxz , byz
on the up tetrahedra, and cxy , cxz , cyz , dxy , dxz , dyz on the down tetrahedra. Here axy
and cxy lie on the x+y-directed edges, while bxy and dxy lie on the x−y-directed edges.
The net linear contribution of the 8 triangular plaquettes in the unit cell vanishes.
Expanding the exponentials of the Wilson lines to quadratic order gives the effective
action:
(axy − byz − axz )2 + (axy − bxz − ayz )2
+ (bxy − axz − ayz )2 + (bxy + byz − bxz )2
+ (cxy − cyz − dxz )2 + (cxy − cxz − byz )2
+ (dxy + cyz − cxz )2 + (dxy − dxz + dyz )2
(3.49)
This equation can be expressed as an 12 × 12 matrix which has two 6 × 6 diagonal
blocks. Diagonalizing each of these blocks gives three gapped and three gapless modes.
The gapless modes will become the gauge degrees of freedom; they are conveniently
expressed in terms of the six fields Ax , Ay , Az , Bx , By , Bz , where
axy = Ax + Ay
bxy = Ax − Ay
cxy = Bx + By
dxy = Bx − By
(3.50)
and so on.
At this juncture, we have an apparent excess of gauge fields. To amend this, and
obtain a kinetic action for the propagating modes, we must also include Wilson lines
71
from the hexagonal plaquettes. There are 4 distinct types hexagonal plaquettes, one
in each of the 4 kagomé planes. Each hexagonal plaquette contains one a, b- copy and
one c, d copy of each of the 3 possible triangular edges associated with this kagomé
plane. This gives a contribution of the form:
X
~
r
[−bxz (~r) + cxy (~r − ẑ + x̂) + ayz (~r + x̂ + ŷ)
−dxz (~r + x̂ + ŷ) − axy (~r + ŷ + ẑ) − cyz (~r)]2
(3.51)
for each kagomé plane. Substituting in the expressions for the 6 low-energy gauge
modes, this gives:
X
~
r
[Ax (~r) − Ax (~r + ŷ + ẑ) + Ay (~r + x̂ + ŷ) − Ay (~r + ŷ + ẑ) − Az (~r) + Az (~r + x̂ + ŷ)
+Bx (~r − ẑ + x̂) − Bx (~r + x̂ + ŷ) + By (~r − ẑ + x̂) − By (~r) − Bz (~r) + Bz (~r + x̂ + ŷ)]2
(3.52)
Each field Ai , Bi appears along two edges; the second edge is translated relative to
the first by the FCC vector in this triangular plane which does not involve xi . (For
example, Ax appears along the two edges x̂ + ŷ and x̂ − ẑ, with a relative translation
of 2(ŷ + ẑ). The other four kagomé planes give similar contributions, but we must
be careful to keep the same origin ~r in both up and down tetrahedra. After Fourier
transforming, the net contribution of these terms is:
L1 (k)L1 (−k) + L2 (k)L2 (−k) + L3 (k)L3 (−k) + L4 (k)L4 (−k)
(3.53)
where
L1 (k) = a4 + b1 + a2 eik1 − b4 eik2 − a1 eik2 − b2 e−ik4
L2 (k) = −a5 + b1 + a3 eik1 + b5 eik3 − a1 eik3 − b3 eik5
L3 (k) = a6 + b2 e−ik4 + a3 eik2 − b6 ei(k3 −k4 ) − a2 eik3 − b3 ei(−k4 −k6 )
L4 (k) = a4 − b5 + a6 e−ik5 − b4 eik6 + a5 eik6 − b6 e−ik4
(3.54)
72
where all gauge fields have momentum k. Because only three of the 4 hexagons can
be chosen such that two of the triangles surrounding each lie on the same pair of
tetrahedra, the gauge fields bi have an extra translation e−ik4 in L3 .
After substituting in the appropriate values for the 6 physical fields, which are in
this case:
1
1
a5 = 12 (Az − Ay ) a3 = (Ax + Az ) a4 = (Ax − Az )
2
2
1
1
1
b1 = 2 (By + Bx ) b2 = (Bz + By ) b3 = (Bx + Bz )
2
2
1
1
1
b4 = 2 (Bx − Bz ) b5 = (Bz − By ) b6 = (By − Bx )
2
2
(3.55)
and using
k1 = kx + ky
k2 = kz + ky
k3 = kx + kz
k4 = kx − kz
k5 = kz − ky
k6 = ky − kx
(3.56)
we find that the resulting matrix has 4 flat gauge bands. This leaves two propagating
modes, as expected for a photon in 3 dimensions.
The eigenvalues corresponding to the propagating modes are:
λ(k) = 6 − 2 cos(kx ) cos(ky ) − 2 cos(kx ) cos(kz ) − 2 cos(ky ) cos(kz )
(3.57)
λ vanishes only at the origin of the Brillouin zone, and for small k has an expansion
λ(k) ≈ ~k 2
(3.58)
as one would expect for the spatial contribution to the action of the 3 + 1 dimensional
photon.
Gauge fields on arbitrary lattices
It is amusing to consider in more abstract terms how we arrived at the correct number
of propagating gauge degrees of freedom. Why did this counting work out, and is it
73
generally so on an arbitrary lattice? Here we present a very simple counting argument
that answers this in the affirmative.
First, consider gauge theory on a Bravais lattice in d dimensions. The unit cell of
the Bravais lattice has a single site and d lattice vectors. Each of the d links has a
different associated gauge field, and there is one gauge transformation corresponding
to a gauge rotation at the site. Hence we should find d − 1 propagating gauge modes,
as expected in d spatial dimensions.
If we add a second site to the unit cell, we create at least one new link per
unit cell. If we add only one new link, this adds one gauge field – but also one
gauge transformation, since the phase on each of the two sites in the unit cell can be
rotated independently. Hence we have preserved the number of propagating degrees
of freedom for the gauge field.
When adding a third site, we have two options. We may connect it to a single
other site in the unit cell, in which case as above we have added one site and one link.
Or, we may connect it to both of the other sites, in which case we have added one site,
two links, and one plaquette. As in the example of the pyrochlore described above,
the plaquette furnishes a constraint which eliminates one gauge degree of freedom.
There remain d + 2 gauge fields and 3 independent gauge transformations, giving the
correct d − 1 propagating gauge modes.
In fact, the above holds generally: when adding a new site to the unit cell, we
necessarily add at least one link and one gauge transformation. The net effect of
these, then, clearly does not alter the number of propagating gauge modes. We may
also add additional new links joining the new site to existing sites in the unit cell.
Each additional link will, however, add a plaquette: since each site in the unit cell
is connected to at least one other site before the additional link is added, afterwards
there will be one new closed loop in the set of all links. This will impose an additional
plaquette constraint, eliminating the extra gauge field from the new link.
74
Thus we see by simple counting that the result we found on the pyrochlore should
hold in general: irrespective of the choice of lattice, the number of independent gauge
transformations and constraints imposed by plaquettes ensures that the effective lattice action always has the correct number of propagating gauge modes.
Time Derivatives
The lattice treatment above addressed only the time-independent components of the
gauge field action. We now consider the contribution that time derivatives of the
gauge field can make to the effective action. To do so, we will argue on the basis of
symmetry, rather than calculation.
As explained in Sect. 3.6.1, the monopole state breaks both time reversal and
parity microscopically, but is invariant under their combined action. Hence we consider terms which preserve this combination, as well as the symmetries of the lattice
and, of course, gauge invariance.
It is convenient to fix the gauge A0 = 0, and consider only time-independent gauge
transformations. In this case gauge invariance does not impose any constraints on
terms quadratic in the photon field which involve time derivatives. For i, j ∈ {x, y, z},
under gauge transformations we have:
Ȧµ Aν → Ȧi (Aν − ∂ν Λ)
= Ȧµ Aν + Aµ ∂ν Λ̇
= Ȧµ Aν
(3.59)
and all such quadratic terms are allowed. Terms of the form Ȧµ Ȧν are also allowed
by gauge invariance.
This indicates that, after fixing the gauge A0 = 0 and restricting to time-independent
gauge transformations, the possible gauge-invariant terms quadratic in the photon
75
fields have the form:
Ȧµ
X
Aν
!
and
Ȧµ Ȧν
.
(3.60)
where can in principle denote any closed curve on the lattice – though in practice
we will consider only terms from short loops.
Next, what restrictions are imposed by P T invariance? We have:
T Ax,y,z
= −Ax,y,z
T A0
nP Ax,y,z
T
X
P A!0
Aν
= A0
= −Ax,y,z
T Ȧx,y,z
= Ȧx,y,z
T Ȧ0
= −Ȧ0
P Ȧx,y,z
(3.61)
= −Ȧx,y,z
= A0 P Ȧ0
= Ȧ0
!
X
P
P
= − ( Aν ) P ( Aν ) =
Aν
.
(3.62)
These simply state that time reversal acts by complex conjugation, changing links
with gauge field terms eiA to links with gauge field terms e−iA ≡ ei(−A) . If A is
on a spatial link, it is thus odd under T ; on a time-like link we should also change
the orientation of the link itself, rendering A0 T -even. P reverses the orientation of
the spatial links – a transformation under which Ax,y,z is odd, but A0 is even. Note
that a product of gauge fields around a closed curve is invariant under P , which
simultaneously reverses the orientation of the gauge field and the curve itself.
Of the possible terms of this form, the following are invariant under P and T
individually (here i, j denote spatial indices):
!
X
Ȧi A0
Ai
Ȧ0
Ȧ0 Ai
Ȧi Ȧj
Ȧ0 Ȧ0
(3.63)
The first line will lead to terms of the form ωpiA0 Ai in the continuum limit – which
are indeed expected in the usual photon propagator Fµν F µν , though in our choice
of gauge they vanish. In the second line, invariance under lattice rotations restricts
76
possible terms to Ȧ2i and Ȧ20 . The former is again the expected contribution to Fµν F µν ;
the latter vanishes in our choice of gauge.
Further, we must consider terms which violate P and T individually, but are
invariant under their combined action. Setting A0 = 0, the relevant gauge-invariant
terms are:
Ȧj
X
Ai
!
.
(3.64)
This is the well-known topological term in 3 + 1 dimensions, F ∗ F ≡ E · B. Though
we note its possible appearance, we do not expect it to play an important role in the
physics of the monopole state, except at the spatial boundaries of the material.
In summary, we have analyzed the form of the 1-loop photon self-energy by integrating out high-energy fermions on the lattice. We find that of the 12 naive gauge
fields on the lattice, only two correspond to propagating physical degrees of freedom,
as expected of a 3 + 1 dimensional gauge theory. The time-independent gauge field
action induced by terms about triangular and hexagonal plaquettes on the pyrochlore
is then, in the continuum limit, simply the rotation-invariant gauge field propagator
Fij F ij . To this, we add terms involving time derivatives which are allowed by symmetry. The symmetry analysis suggests that the gauge field action then has two
contributions: a T and P -independent contribution from Fµν F µν , and a T and P
breaking topological term of the form E · B.
To carry this analysis further requires deducing both the contributions to the
gauge-field propagator which stem from the rotational-symmetry breaking effects of
long-wavelength fermions, and the resulting 1-loop correction to the fermionic propagator. This task we leave for future contemplation.
77
3.8
Concluding Remarks
We have discussed an interesting mean field (large N) solution to the Heisenberg
model on the pyrochlore lattice. This is a P and T breaking state in which all
triangular plaquettes have an outward flux of π/2. After Gutzwiller projection, this
state has lower energy than all other mean field states considered, including the
simplest dimerized state. Its low-energy physics is rather striking, with a spinon Fermi
surface of lines of nodes preserving the discrete rotational symmetries of the lattice.
The symmetries of the Hamiltonian suggest that this Fermi surface is perturbatively
stable and thus should characterize a stable spin liquid phase, at least at sufficiently
large N.
However, our analysis of stability is thus far based only on symmetries and does not
rule out dynamical instabilities. The study of such instabilities requires adding back in
the gauge fluctuations that are suppressed at N = ∞ and studying the coupled system
consisting of spinons and gauge fields. In principle this can be done entirely within
the linearized theory, which correctly describes the entire Fermi surface - though in
practice the required computations are technically challenging. A second question is
whether non-perturbative effects might destabilize the monopole state. In the well
studied case of two dimensional algebraic spin liquids it took a while to understand
that this coupled system could, in fact, support a gapless phase at sufficiently large
N despite the compactness of the gauge fields. In the present problem there is also
the specific feature that at N = 2, as the background flux per plaquette is U(1),
such an analysis should incorporate fluctuations of an SU(2) gauge field [53]. We
expect that a similar analysis holds here, and that PSG arguments are sufficient to
ensure that the Fermi lines are dynamically stable. Encouraged by this, in the next
Chapter, we will construct other mean-field like Hamiltonians with similarly peculiar
Fermi surfaces, and explore their special relationship to Lie Algebras.
The applicability of our ideas to the strict case of N=2 will ultimately have to
78
be tested either in actual experiments looking for T and P breaking and for the
distinctive spinon spectrum of nodal lines, or in numerical work looking for the same
features, e.g. computations of the correlations of the triple product operator in Eq.
(3.40). Spinon Fermi lines could be detected directly through neutron scattering, or
indirectly through measurements of the specific heat, which for non-interacting Fermi
lines in 3 dimensions is quadratic in T .
Finally, we note that our initial motivation in this study was to see if we could
construct a fully symmetric spin liquid on the pyrochlore lattice for S = 1/2 in
contradiction with previous studies using other techniques. We have not succeeded in
that goal and, as the technique in this Chapter has produced a pattern of symmetry
breaking distinct from the ones considered previously, the fate of the S = 1/2 nearestneighbor Heisenberg antiferromagnet on the pyrochlore lattice remains undeciphered.
Chapter 4
Frustrated hopping problems on
lattices derived from Lie Algebras
4.1
Introduction
In the previous Chapter, we studied in detail the physics of the monopole flux state,
which arose as an energetically favorable mean-field solution to a certain large-N
expansion of the Heisenberg model on the pyrochlore lattice. While the study of
such spin-liquid type states is interesting from the point of view of understanding
when truly quantum phases of matter can be realized in real physical systems, the
monopole flux state has several features which are interesting in their own right at the
mean-field level. First of all, the Fermi surface of 4 intersecting lines, which preserves
all rotational symmetries of the crystal but breaks SO(3) rotational invariance, is
quite striking. Second, the algebraic structure which gives rise to the mean-field
energy spectrum – consisting of a set of Dirac-like matrices which anti-commute with
each other to give a set of anti-commuting matrices – begs the question of whether
generalizations to more complex such algebras are possible.
In this Chapter, we will explicate the origins of these interesting algebraic struc-
79
80
tures, and construct a family of similar models on a special set of lattices. These
lattices have a special geometrical relationship to Lie algebras, which will allow us to
make an explicit connection between certain quadratic (free-fermion) Hamiltonians
in a background gauge flux, including the one discussed in the previous Chapter, and
elements of a matrix representation of the Lie Algebra in question. This relationship
explains the two unusual features of the monopole flux state outlined above. We note
that this Chapter closely follows the presentation of Ref. [66].
4.1.1
Lie Algebras and Hamiltonians
As explained above, our approach will be to study the properties of certain Hamiltonians which describe fermions hopping on a lattice in the presence of a background
gauge field. As we have seen, Hamiltonians of this type arise as mean-field Hamiltonians for nearest-neighbor spin-1/2 Heisenberg models. When these represent valid
mean-field solutions, the phases which result are generally spin-liquid like in that they
preserve the rotational and translational symmetries of the lattice. For the cases we
study, in general they will also share with the monopole state the characteristic that
they break time reversal at the microscopic scale.
There is, of course, a large set of possible mean-field Hamiltonians and we next
need to describe the restrictions that generate the family that we study. This brings us
to the second context in which our Hamiltonians can be situated and which intrigues
us most: the deep ties our Hamiltonians have to Lie algebras. These ties are twofold:
ideas from Lie algebras are central to generating the very lattices the fermions move
on and the Hamiltonians can be written as direct sums of pieces that are Lie algebra
elements in specified representations. Consequently we find that properties of Lie
algebras also control some striking features of the resulting spectra. Sometimes we
fully understand these connections and sometimes we do not, though in all cases we
will share what we know with the reader.
81
Let us first summarize how the unit cell and the underlying lattice have direct
group theoretic significance. A discussion of the group theory used here can be found
in [68, 69].
Consider the unit cell. Recall that the generators of a (semi-simple) Lie algebras
can be partitioned into a maximally commuting Cartan subalgebra Hi : [i = 1, . . . r] =
H whose eigenvalues label the weights, and a set of ladder operators Eα and their
adjoints Eα† = E−α that act on the states to raise (lower) the weights by α. The vectors
α are called the roots. The states within any irreducible representation (multiplet)
of a Lie algebra may therefore be visualized as a collection of points in a space of
dimension r, called the rank. The coordinates of the points are the simultaneous
eigenvalues of H. The roots that help us move around these points are also vectors
in the same space. For example, in the case of the rank-2 group SU(3), (whose
commuting quantum numbers are traditionally called isospin and hypercharge in the
physics literature) the fundamental (quark) representation is an inverted triangle,
the anti-quark is a triangle, and the eight-dimensional adjoint representation is a
hexagon with two null weights at the center. The six nonzero roots correspond to the
six corners of the hexagon.
The unit cells of the lattices we consider correspond to certain special representations called minuscule representations.
Starting with any one state in a minuscule
representation or multiplet, we can obtain all the others by acting with the Weyl
group, the group of reflections about the hyper-planes normal to the roots. All
weights of a minuscule representation are on the same footing and in particular all
have the same length. Thus the quark and anti-quark are minuscule while the adjoint
representation, with weights of both zero and non-zero length, is not.
This unit cell is now used to decorate a lattice, which is a subset of the root
lattice LR . Recall that roots, like weights, also live in r dimensions. So it is possible
to choose r roots, called simple roots, as a basis. A basic result of group theory is that
82
every root is an integer linear combination of the simple roots. There are of course
a finite number of them in any algebra of rank r. The root lattice LR is the infinite
lattice formed with the same basis with but with any integer set of coefficients. For
SU(3), the six roots form a hexagon, while the root lattice is the infinite hexagonal
lattice.
The lattice on which our fermions move is L2R , the subset of LR whose points
have even integer coefficients, decorated by a basis corresponding to a minuscule representation.
When applied to the quark representation of SU(3), this yields the kagomé lattice,
as shown in Fig. 4.1(b). The inverted triangle (Fig. 4.1(a)) is the fundamental quark
representation which forms the unit cell that decorates the root lattice L2R , which
is a hexagonal lattice with twice the lattice spacing as the root lattice LR . The
origin of coordinates is at the point named q in the figure. If you stare at the figure
hard enough, you can also see it as a hexagonal lattice decorated by the conjugate
representation, the anti-quark, whose weights are the negatives of the quark. The
center of one such unit cell is labeled q̄ in the figure. The quark and anti-quark
unit cells are corner sharing. These features are common to all our models and
exist because the unit cell and the lattice are constructed from weights and roots in
a particular way. A proof of this will be given in Section 4.2.1. Let us now turn
to the construction of the Hamiltonians on the above set of lattices. Gauge fields
will enter our models in the form of purely imaginary hopping amplitudes which
can be ±i. This restriction means that on any triangular face the flux can only be
±π/2.1 In other words the background gauge field is an Ising-like variable, and time
reversal symmetry is broken. Also, we will require that the gauge fields exhibit the
periodicity of the Bravais lattice L2R — this has the gauge invariant content that there
is no net flux passing through the lattice. It is worth noting here that generically
1
Evidently, the flux is the gauge invariant variable. Our choice of gauge fields is convenient for
the purposes of this Chapter.
83
in problems of this kind we must view symmetries as projective, i.e., the underlying
group operations will have to be accompanied by additional gauge transformations
to make the symmetry manifest [53], say the way Lorentz transformations have to
be accompanied by gauge transformations in relativistic field theories to establish
Lorentz covariance. The classification scheme relevant to our problem appears to be
that of Color Groups, in which each face of the crystal is colored black or white, which
we may read as ±π/2 of flux [70, 71].
q
q
µ
µ
2
1
a2
a1
µ
3
(a)
(b)
Figure 4.1: The kagomé lattice and SU(3): The unit cell (a) is the quark triplet, with
weights µ1 ...µ3 as indicated. The kagomé lattice (b) is formed by decorating L2R (the
subset of the SU(3) root lattice with even integer coefficients) with this unit cell. L2R
is a hexagonal lattice with basis vectors a1 and a2 which are twice the simple roots
of SU(3). The origin of coordinates is marked q. The lattice may also be viewed as
being decorated by anti-quark unit cells, one of which is centered at q̄
.
We are now in a position to specify the Hamiltonians of interest. Since the flux
added to this lattice will be translationally invariant on L2R , we may go to momentum
space to solve for the dispersion relation. Evidently H(k) will be a matrix that acts
on the states of the minuscule representation since they constitute the unit cell.
With our choices of background gauge fields we then arrive at manifestly hermitian
84
Hamiltonians of the form
H(k) =
X
Cα (k)(Eα + Eα† )
(4.1)
α⊂Σ+
where the coefficients Cα are real, satisfy
Cα (−k) = −Cα (k)
(4.2)
and Σ+ are the positive roots. The roots may be divided into positive (Σ+ ) and
negative (Σ− ) roots by drawing a plane through the origin which does not contain
any roots. This choice is basis-dependent. For any choice of basis we may choose an
ordering of the basis vectors so that the positive roots are those whose first nonzero
component is positive.
Since the ladder operators move us around the multiplet, it is reasonable to consider H of the form (4.1). However we must bear in mind that this is not the most
general possibility on this lattice. For example the model only allows hops between
sites that differ by a single root while there are minuscule representations where the
states differ by more than a single root. Other restrictions implied by this form of H
will be discussed later.
Observe that we have obtained an unusual connection between the lattice and the
hopping Hamiltonian: the former is obtained by decorating the root lattice of a Lie
algebra with the weights of one of its representations, and the hopping Hamiltonian (in
momentum space) is an element of the same Lie algebra in the same representation!
It is worth emphasizing that this connection does not involve the symmetry group of
the problem. For example, the Lie algebra SU(3) that shows up on the kagomé lattice
does not generate the actual symmetries of the hopping problem or even correspond
to any symmetries of the starting SU(2) Heisenberg model on that lattice.
One final point on the construction of our lattices and Hamiltonians: Although our
lattice construction works in all dimensions, we shall limit ourselves to lattices in d = 2
and d = 3, both because these are the dimensions we could encounter in the lab and
85
because the rules for attaching flux break down in higher dimensions wherein areas
cannot be oriented unambiguously. In this latter aspect the flux Hamiltonians that
give rise to Dirac problems are special as they involve π flux, which does not carry an
orientation, and thus their generalization to arbitrary dimensions is straightforward.
4.1.2
Dirac-like continuum limits
The discretized Laplace operator on lattices typically looks like a hopping problem
with zero background flux. It is an interesting fact that when Kogut and Susskind
[72] set out to discretize the Dirac operator on cubic lattices, they were naturally
led to introduce a background flux of π per plaquette. Indeed, the Affleck-Marston
work ended up “rediscovering” this earlier construction in the case of two spatial
dimensions.
The flux Hamiltonians we consider evidently have a family resemblance to the π
flux Hamiltonians which they generalize. It is also the case that their low energy
limits exhibit a family resemblance to the Dirac theory which is the third context in
which they appear to be interesting.
As a first step in explaining what we mean let us observe that zero energy plays
a special role in our entire set of Hamiltonians. Normally, this comes about via a
particle-hole symmetry, where at every momentum k states at energy E are accompanied by states at −E. In our problems, the choice of background flux ensures that
H(−k) = −H(k). Consequently for every level that is negative and hence occupied
at k, there is one at −k that is empty and unoccupied so that the combination of the
two bands is particle-hole symmetric and E = 0 is again special. We should observe
that this is very useful in the original context of the mean-field theory of various
Heisenberg models for this ensures that the half-filled band exhibits a Fermi energy,
EF = 0.
In the π flux phase, the structure about E = 0 is that of the Dirac theory. In
86
our examples we find generalizations that we term Dirac-like or pseudo-Dirac. By
Dirac-like, we mean a generalization in three respects. First, although the spectrum
is linear in momentum for small momentum, it possesses only discrete and not full
rotational invariance. Second, the square of the Hamiltonian H is not proportional to
the unit matrix, though a higher degree polynomial is. While a result of this sort is
inevitable for any finite size matrix, the fact that the polynomial often contains just
even powers leads to a result that is sufficiently reminiscent of the Dirac case. The
third generalization we encounter is that in addition to isolated Dirac (or Dirac-like)
points, we often find entire lines and even planes of zero energy.
The zeros are interesting in and of themselves in the Lie algebraic setting: in
some cases, the lines of zeros are along the direction of the weights while in other
cases the planes of zeros are simply related to the roots and so on. In some cases we
can understand the locus of zeros without explicit computation by appealing to ideas
from group theory, while in many cases we could neither anticipate nor explain the
zeros.
There is, inevitably, a matrix structure that goes with the low energy Dirac-like
theory and generalizes the gamma matrices but it does not appear to be immediately
interesting in and of itself although we will exhibit it in one especially interesting
case.
4.1.3
What follows
In Section 4.2 we explain the construction of the lattice in detail, and describe the
examples we will study in the remainder of the Chapter. We discuss the hopping
problem for the d = 2 lattices in Section 4.3, studying several possible background
fluxes. In Section 4.4 we carry out the same analysis for lattices in d = 3. Here
the reader can find in some detail a discussion of the pyrochlore lattice, which we
understand the best in terms of group theory and which also displays interesting
87
mathematical structures. In Section 4.5 we comment on the utility of our Hamiltonians as mean-field solutions of the Heisenberg model on our various lattices and on
two extensions of our analysis. Section 4.6 contains concluding remarks and discusses
open problems.
4.2
Lattice construction
Recall our recipe for generating the lattices:
• Generate L2R , the even sector of the root lattice, that is to say, even integer
combinations of the simple roots.
• Decorate each lattice point with a minuscule representation.
As noted in the Introduction, although our lattice construction works in all dimensions, we shall limit ourselves to lattices in d = 2 and d = 3, due to their physical
relevance and more absolutely because the rules for attaching flux break down in
higher dimensions wherein areas cannot be oriented unambiguously. Hence we study
rank 2 and 3 groups only. Here is the list of candidates.
• d = 2: SU(3) and SO(5)=Sp(4). The exceptional group G2 does not have minuscule representations. The group SO(4) factors into two independent SU(2)
factors and will only be discussed very briefly.
• d = 3: SU(4)=SO(6), SO(7), and Sp(6). There are no exceptional groups of
rank 3.
The minuscule representations in each case will be listed as we go along.
4.2.1
Properties of our lattices.
The lattices we manufacture by the rules listed above have some interesting features
that will be established in this section. Before we do so in general, let us pause to
88
examine a simple example, the rank-2 group SU(3). This exercise will help us better
motivate and understand the general case.
The only minuscule representations are the quark and antiquark. Let us begin
with the quark representation.
The weights, numbered 1, 2, 3 in Fig. 4.1(a), are
1
1
µ1 = (1, √ ),
2
3
1
1
µ2 = (−1, √ ),
2
3
1
µ3 = (0, − √ )
3
(4.3)
and point to the vertices of an equilateral triangle.
One significant feature of these weights that we will invoke is that



a i = j
µi · µj =


b i 6= j
(4.4)
where a = 1/3 and b = −1/6. In other words, there are just two possible values for
µi · µj , between a weight and itself (a) and between a weight and any other (b).
The nice thing about SU(3) is that in the quark representation, the weights form
a simplex where every corner is equidistant from every other, the difference between
any two corners (any side of the simplex) is a root, and these are the only roots. Note
that this implies six roots for SU(3), since each edge of the triangle can be traversed
in two directions. So the root system of SU(3) is
Σ(SU(3)) = µi − µj ≡ αij
[i 6= j : 1, 2 or 3]
(4.5)
We may choose the simple roots to be α12 and α23 and in terms of them the lattice
L2R is defined as the set of points
2R = 2mα12 + 2nα23
(4.6)
≡ ma1 + na2
We can now put the expanded root lattice together with the minuscule representation: in Fig. 4.1(b) we have placed one triangle at the origin (labeled q) and made
89
copies at every lattice point in L2R . As the reader can see, we end up with the kagomé
lattice.
Let us note that the two features, Eqs. (4.4,4.5), generalize in the obvious manner
for all SU(N), where the weights now point to the vertices of an N−simplex.
Now we turn to some general features of our lattices valid for all the groups we
will study, not dependent on the special features of SU(N) alluded to above. To see
what they might be, look at Fig. 4.1(b), and observe the following features:
1. The lattice can also be viewed as L2R decorated by the conjugate (anti-quark)
representation with reversed weights. The original representation and the conjugate share corners, and every site is shared in this manner.
2. If the particle can hop to any point labeled i from a point labeled j in the same
unit cell by moving a displacement dij , it can keep moving an extra dij to reach
a point labeled j in the adjacent unit cell. In other words the edges of the unit
cell and the conjugate unit cell that meet at a shared corner are continuations
of each other with no change in direction.
We will now furnish the proofs of these results in the general case.
Theorem I: The original lattice 2R + µi can be rewritten as 2R + 2µ1 − µi .
In other words, if at a new origin displaced from the old one by 2µ1 , we place the
conjugate representation and make copies of it using any element of L2R , we get the
old lattice. The result is just as valid if we use any other weight 2µj in place of 2µ1 .
Proof:
2R + 2µ1 − µi = 2R + 2µ1 − 2µi + µi = 2R′ + µi
(4.7)
where we have used the fact that 2µ1 − 2µi , being an even integer multiple of weight
differences, is then an even integer multiple of roots, which in turn is a translation
within L2R .
90
Note that the choice of origin at 2µ1 is arbitrary: the choice 2µ2 differs by 2µ2 −2µ1 ,
an even integer multiple of roots, and hence a translation within L2R . Theorem II If the particle can hop to any point labeled i from a point labeled
j in the same unit cell by moving a distance dij , it can keep moving an extra dij to
reach a point labeled j in the adjacent unit cell.
Proof: Since dij is a difference of weights it is some integer combination of simple
roots. Moving an extra distance dij , corresponds to a total displacement by an even
integer combination of simple roots, which is a symmetry of L2R . It follows that if
we start at a point labeled j we must end up a point also labeled j. 4.2.2
Lattices in d = 2
We have already discussed SU(3) in the last section. Now we will deal with SO(5)
and Sp(4). These two Lie algebras are mathematically equivalent up to cosmetic
differences which will be displayed.
SO(5)
We begin with the more familiar group SO(5) which preserves the norm
x2 =
5
X
x2i
(4.8)
i=1
and has a defining representation of 5 × 5 orthogonal matrices.
We choose as Cartan generators H1 = L12 and H2 = L34 which generate rotations
in the 12 and 34 planes. In terms of the coordinates
x±
I =
x1 ± ix2
√
2
x±
II =
x3 ± ix4
√
2
x0 = x5
(4.9)
we may write the invariant in this spherical basis as
2
x =
x20
+2
II
X
a=I
x−a xa .
(4.10)
91
The vector x itself serves as a 5-dimensional representation. The components x±
I
and x±
II are eigenstates of H1 , H2 with eigenvalues H = (±1, 0) and (0, ±1) while
x0 = x5 does not respond to either rotation and has eigenvalues (0, 0). The vector
representation is not minuscule since the weights are of unequal length.
The only minuscule representation is the 4-component spinor, with weights
1 1
µ = ± ,±
.
2 2
(4.11)
These form a square as in Fig. 4.2(a).
a
q
2
µ
µ
3
1
q
µ4
a1
µ2
(a)
(b)
Figure 4.2: Planar pyrochlore and SO(5): The unit cell (a) of the spinor representation is the square of edge unity. The resulting square lattice, with hoppings along
all roots, is shown in (b). The lattice vectors a1 , a2 are a basis for the subset of the
SO(5) root lattice with even integer coefficients.
The eight roots for SO(5) are given by
Σ(SO(5)) = ±ei ; ±ei ± ej
i 6= j = 1, 2
(4.12)
where ei is a unit vector in direction i. The short roots connect states along the
coordinate axes while the long ones go diagonally.
The same spinor also forms a representation of SO(4). However SO(4) has only
the four long roots and one can see that the weights ±( 12 , 12 ) do not talk to the pair
±( 12 , − 21 ), which means the representation is reducible. We do not discuss it here.
92
The simple roots are e1 − e2 and e2 ; our root lattice is 2R = 2m(e1 − e2 ) +
2ne2 = 2m′ e1 + 2n′ e2 , a square lattice of sides 2. A site on the decorated lattice is
2R + (± 21 , ± 21 ).
Since the spinor is self-conjugate, the original squares share corners with identical
squares, and the resulting lattice is the square lattice. When links along the long
roots are included as in Fig. 4.2(b), the structure is known variously as the square
lattice with crossings (SLWC), checkerboard lattice, or planar pyrochlore. Note also
that if you can hop from site j to site i on one unit cell, you can hop once more by
the same amount to hit site j in the next unit cell.
Sp(4)
The weights of the 4-dimensional minuscule representation of Sp(4) are
µ = (±1, 0), (0, ±1).
(4.13)
Since the group is the collection of 4 × 4 symplectic matrices, this is also called the
defining representation. (The same terminology applies, say to the 6-dimensional
vector representation of SO(6), which is the group of 6 × 6 orthogonal matrices.)
The roots of Sp(4) are
Σ(Sp(4)) = ±ei ± ej and ± 2ei
i 6= j : 1, 2
(4.14)
Now the long roots connect points parallel to the axes and short roots in diagonal
directions, as shown in Fig. 4.3(a). Note that this just the rotated and rescaled
version of SO(5).
This concludes the enumeration of lattices in d = 2.
4.2.3
Lattices in d = 3
With the warm up from d = 2 we can proceed rapidly through d = 3 where the
candidates are SU(4)=SO(6), Sp(6), and SO(7).
93
q
q
µ
2
µ
µ
4
1
µ
3
Figure 4.3: Planar pyrochlore and SP (4): The unit cell (a) for the defining representation of Sp(4) is the square rotated by 45 degrees. The entire lattice (b) is the
rotated version of the SO(5) spinor.
Sp(6)
The only minuscule representation for Sp(6) is the defining 6-dimensional one. The
weights are
µ = (±1, 0, 0), (0, ±1, 0), (0, 0, ±1)
(4.15)
which form an octahedron which is self-conjugate. The decorated lattice is most
readily visualized as corner sharing octahedra, which we will refer to as the octachlore
lattice, depicted in Fig. 4.4. The roots are
Σ(Sp(6)) = ±ei ± ej and ± 2ei
i 6= j : 1, 2 or 3
(4.16)
The short roots allow you to hop along the edges of the octahedron while the long
roots take you straight across the unit cell to the antipodal point. This is just the
d = 3 version of Sp(4).
SO(7)
Next we turn to SO(7). Only the spinor is minuscule. It has self-conjugate weights
1 1 1
µ = (± , ± , ± )
2 2 2
(4.17)
94
Figure 4.4: The octachlore lattice of SP (6): blue octagons are copies of the defining
representation of Sp(6); red octagons are in the conjugate representation. Note that
we have not drawn the bonds along the long roots.
which lie at the corners of a unit cube. The roots
Σ(SO(7)) = ±ei ± ej and ± ei
i 6= j : 1, 2 or 3
(4.18)
allow hops along edges, and diagonally across faces, but not along the body-diagonal.
The root lattice L2R is cubic with edges of size 2. The decorated lattice is made
of corner sharing unit cubes, with face diagonal hopping on every second cube as
one proceeds along any of the three cubic axes (Fig. 4.5). This is one possible 3
dimensional variant of the checkerboard lattice discussed in Sections 4.2.2 and 4.2.2;
we shall refer to it as the 3 dimensional checkerboard lattice. The other, which we
will discuss in the next section, is the pyrochlore.
In higher dimensions the SO(2N + 1) spinor leads to the N-dimensional checkerboard lattice: an N dimensional cubic lattice with links on all face diagonals.
SO(6)
We obtain SO(6) from SO(7) if we drop the short roots ±ei :
Σ(SO(6)) = ±ei ± ej i 6= j : 1, 2 or 3.
(4.19)
Consider the spinor representation of SO(7). Without the short roots ±ei , we
can only flip the signs of the components of each weight two at a time. This means
95
Figure 4.5: The 3-d checkerboard SO(7) spinor lattice: blue and red cubes show the
spinor representation and its conjugate (identical in this case).
the 8-dimensional spinor of SO(7) breaks down into two irreducible representations
with four weights each. The first has an even number of negative weights
1 1 1
µ1 = ( , , ),
2 2 2
1 1 1
µ2 = (− , − , ),
2 2 2
1 1 1
µ3 = ( , − , − ),
2 2 2
1 1 1
µ4 = (− , , − ) (4.20)
2 2 2
and the other has these weights reversed and hence an odd number of negative weights.
This is general: the irreducible spinor of SO(2N + 1) becomes two irreducible representations of SO(2N), called left and right handed spinors.
If you join the 4 points in either spinor multiplet, you will see the tetrahedra that
form the weights of the SU(4) quark and antiquark representations. In other words
the right and left handed spinors of SO(6) are the quark and anti-quark of SU(4).
(This is why we will not study SU(4) separately.) The tetrahedra form the familiar
pyrochlore lattice shown in Fig. 4.6.
The extensions of this construction to higher dimensions yield corner sharing
simplices that form the natural generalizations of the kagomé and pyrochlore lattices
[14].
96
Figure 4.6: The pyrochlore SO(6) spinor lattice: Blue tetrahedra are in the fundamental representation of SU(4), or the right-handed spinor representation of SO(6);
red tetrahedra are in the conjugate representation.
Since for SU(4) the difference of any two weights of the quark is a root and there
are no others, the roots of SO(6) may just as well be written as
Σ(SO(6)) = µi − µj
[i 6= j : 1, 2, 3, or 4]
(4.21)
a result we will invoke later.
The third minuscule representation of SO(6) is the 6-dimensional defining representation with weights
µ = (±1, 0, 0),
(0, ±1, 0),
(0, 0, ±1)
(4.22)
just as in Sp(6). The decorated lattice again is made of corner-sharing octahedra, as
shown in Fig. 4.4. However, without the long roots ±2ei , one can hop only along the
edges but cannot jump directly from a point to its antipodal point.
This concludes the enumeration of lattices.
Structure factor
Before we turn on the flux let us note that the structure factors for these lattices have
a simple group theoretic interpretation. When we do a sum over all sites (indexed by
97
l) we find
S(k) =
ik·rl
l∈L2R +µ e
P
≡ SL2R (k)χ(k)
=
P
ik·2R
L2R e
ik·µj
j∈µ e
P
=
P
L2R
eik·2R Treik·H
(4.23)
where Hi = Hi , the ith element of the Cartan subalgebra, whose j th eigenvalue is µij ,
and χ(k) = Treik·H is just the character of the representation.
4.3
Flux Hamiltonians in d = 2.
We will turn on fluxes by attaching arrows to each bond. The sense of the arrow will
remain fixed as we move along bonds in any one direction. The hopping amplitude
will be ±i if we go along (against) the arrow. Time reversal reverses every arrow,
sending H → −H. Since time reversal also reverses k, this means that in cases we
study,
H(−k) = −H(k).
(4.24)
If all hopping amplitudes are pure imaginary, the flux through each triangle is
±π/2. It is important to remember that the arrows themselves do not stand for
physical quantities. For example a tetrahedron with uniform flux π/2 coming out of
each face is invariant under all symmetries of the tetrahedron although the arrows
will look different if we say rotate the figure. This will, of course, be built into a PSG
(Projective Symmetry Group) analysis [53].
4.3.1
SU (3) and kagomé
Let us begin with the first case, the kagomé lattice, now with the flux shown in
Fig.4.7. Note that as we go counter-clockwise around the (quark) triangles 1-2-3-1,
we get a product (i)(+i)(−i) = +i. This is so for every quark triangle. The antiquark
triangles will have the opposite flux.
98
3
2
1
µ2
2
µ1
µ3
3
Figure 4.7: Flux assignment on the kagomé lattice. There is a phase factor of ±i as
we move along (against) the arrow.
Consider hops from the site numbered 1 in the central unit cell shown, to the sites
numbered 2 to its left (on the same cell) and right (on the cell to the right). Hopping
along (against) the arrows brings a factor of ±i. We note that this state was one of
several mean-field solutions found by [73] for uniform hopping on the kagomé lattice.
In terms of creation and destruction operators c and c† we have, in obvious notation, a contribution to H :
H1,2 (k) = c†2 c1 ieik·(µ2 −µ1 ) + h.c − (1 ↔ 2)
= (c†2 c1 + c†1 c2 ) sin(k · α12 )
(4.25)
where we have dropped an overall factor of 2, suppressed the k dependence of the
operators, and as before,
α12 = µ1 − µ2
(4.26)
Upon noting that c†1 c2 = Eα12 , the generator corresponding to the root α12 , we see
that when all hops are included we get what we shall the canonical form
H(k) =
X
Σ+
sin(k · α) Eα + Eα†
(4.27)
where the sum is over Σ+ , the positive roots, which have the form µi − µj with i < j.
In this Chapter we will often modify the canonical form in two ways: replace
sin(k · α) by k · α and attach various signs sα = ±1 in front of each term so H
99
assumes the more general form
H(k, sα ) =
X
Σ+
sα sin(k · α)(Eα + Eα† ).
(4.28)
The role of these signs is to modify the relative phase of the hopping amplitudes on
various bonds, and therefore the fluxes.
Putting in explicit values, we get for the canonical case


√
1
0
sin x
sin( 2 (x + 3y))


√


1
H(k) = 
sin x
0
− sin( 2 (x − 3y))


√
√
1
1
0
sin( 2 (x + 3y)) − sin( 2 (x − 3y))
(4.29)
where x and y stand for kx and ky respectively.
The determinant of this matrix
√ √ 1
1
|H| = −2 sin x sin
x − 3y
x + 3y
sin
(4.30)
2
2
√
shows zeros along the lines x = 0 and x = ± 3y, which are precisely the directions
of the weights (or their negatives)! The determinant also has a simple form in terms
of the three positive roots:
|H| = −2 sin(k · α12 ) sin(k · α13 ) sin(k · α23 )
(4.31)
We do not know how this generalizes for SU(N). But we do know how to understand the lines of zeros as follows.
The Hamiltonian has the form
X
i<j
sin(k · (µi − µj ))(Eαij + Eα† ij )
(4.32)
Suppose we set k = µ1 . (All points on the simplex are the same and we pick one that
is easier to analyze.) The simplex has the property that



a i = j
µi · µj =


b i 6= j
(4.33)
100
It follows that the argument of any sine in which µ1 does not appear will vanish
and the ones where it does will have the same value a − b. The resulting matrix,
proportional to sin(a − b), has non-zero entries only in the first row or column. This
leaves a 2 × 2 submatrix of zeros which kills the determinant:
3
X
sin(a − b) (Eα1j + Eα† 1j ) = 0
(4.34)
j=2
This will happen for all SU(N) because each raising or lowering operator in the
fundamental representation has only one non-zero entry. (Geometrically, this is equivalent to the statement that the N-simplex has no parallel edges). The components of
the Hamiltonian which do not vanish for k = µ1 are
sin(a − b)
N
X
†
(E1j + E1j
)
(4.35)
j=2
But all of these are hoppings to or from site 1, so that all non-zero entries lie in either
the first row or the first column of the matrix. The resultant (N − 1) × (N − 1) null
submatrix ensures that the determinant is zero at this particular value of k.
Let us now replace sin x by x, since none of the key features are lost and the
algebra is more manageable, especially in the problem of diagonalization. So we will
set

0


H(k) =  x

1
2
x+
√
3y
The determinant of this matrix is
x
1
2
0
− 12
− 12 x −
√
3y
0
x+
√

3y
√ 

x − 3y 

√
√
1
|H| = − x(x − 3y)(x + 3y)
2
√
which shows zeros along the same lines x = 0 and x = ± 3y as before.
(4.36)
(4.37)
If we change the sign of the term multiplying any of the generators, we get the
same lines of zeros. This is expected since we simply reverse the flux penetrating
101
every triangle, which inverts the spectrum. If we reverse the signs of any two of
them, it makes no difference even to the flux. The reader can check the lines of zeros
are not altered by any change of sign.
As explained earlier, this is a problem where H(k) = −H(−k), and the pair of
points at ±k together produce E → −E symmetry. With the Fermi energy at zero
we are at half-filling, the relevant filling for mean-field solutions of the Heisenberg
model.
The energies themselves are fairly complicated and not displayed here. It turns
out two of them never vanish away from the origin and one of them produces all the
zeros: if we move around the unit circle, it vanishes six times. Linearizing near these
zeros will produce one-dimensional Dirac fermions that will control the low-energy
physics. ( Near the origin all six Dirac excitations will get mixed up.) If all this were
part of a mean-field calculation, we would be looking at this Dirac field minimally
coupled to a gauge field if we wanted to consider fluctuations. One could ask if the
mean-field solution remains stable in their presence. These questions require a PSG
analysis of the type carried out for the monopole flux state in the previous chapter.
4.3.2
Flux Hamiltonians for Sp(4)=SO(5)
Recall that the groups Sp(4) and SO(5) are the same. The roots of one are the
rotated and rescaled versions of the other and no new physics will come from looking
at both. We will only work with SO(5) since it may be more familiar to the reader.
Before writing down the hopping matrix we need to define the basis. The states
are numbered 1 through 4 in Fig. 4.8 with 1 = ( 21 , 21 ) etc. We use a tensor product
102
of two Pauli matrices, σ and τ , to operate on the two labels. We take as generators
Ee1
=
1
σ
2 +
†
⊗ I = E−e
1
Ee2
=
1
σ
2 3
†
⊗ τ+ = E−e
2
Ee1 ±e2
= ∓ [Ee1 , E±e2 ] =
± 21 σ+
(4.38)
⊗ τ±
E−e1 ∓e2 = Ee†1 ±e2
The canonical Hamiltonian
In this basis we choose the canonical form
P
H(k) =
(k · α)(Eα + Eα† )
 Σ+
y
x
 0

 y
0
y−x

= 
 x
y−x 0


x+y
x
−y

x+y 


x

 .
−y 


0
(4.39)
The orientation of the arrows corresponding to this H are shown in Fig. 4.8. In other
words, rather than write down some arrows and deduce the hopping matrix from
these, we are writing down a canonical matrix in the Lie algebra and asking what
hopping elements it implies.
3
1
4
2
Figure 4.8: Flux assignment for the canonical SO(5) Hamiltonian. There is a phase
factor of ±i as we move along (against) the arrow. The flux in each triangle alternates
as we go counterclockwise in the case depicted.
If we look at the flux in each triangle we find that it alternates: triangles sharing
a face diagonal, such as triangles 132 and 342 in Fig. 4.8, have the same flux, while
103
triangles not sharing a face diagonal (triangles 132 and 142) have opposite flux. The
determinant here is
|H| = 4(x4 − x2 y 2 + y 4)
(4.40)
It has permutation symmetry but not full rotational symmetry. It has no zeros
anywhere away from the origin. We do not know a simple way based on group theory
to understand this.
The characteristic polynomial is
E 4 − 4E 2 (x2 + y 2 ) + 4(x4 − x2 y 2 + y 4)
(4.41)
resulting in the particle-hole symmetric spectrum:
√ q
√
E = ± 2 x2 + y 2 ± 3xy
The reason behind the symmetry E → −E is the matrix


0
0
1 
 0


 0

0
1
0


G=
 ,
 0

−1
0
0




−1 0
0
0
(4.42)
(4.43)
with G2 = −I. Since
G · H · G−1 = −H
(4.44)
it follows that H and −H have the same spectrum.
The anatomy of the operator G is interesting. Suppose we wanted to manufacture
an operator that reversed the sign of H by conjugation. We could accomplish this
by a parity operation that exchanges each weight with its negative- this should flip
every hopping term. However there are many ways to flip the weights since we can
take each state to its parity reversed state, times any unimodular phase factor, which
must be a sign if we want the hoppings amplitude to be ±i. Suppose we picked a G′
104
with all positive signs:

We would find
 0

 0

′
G =
 0


1

G′ HG′−1
 0

 −y

=
 x


x+y
0
0
0
1
1
0
0
0

1 

0 


0 


0
−y
x
0
y−x
y−x
0
x
y
(4.45)

x+y 


x



y


0
(4.46)
This is clearly not −H. However, if we go back to the lattice and ask what the
corresponding hopping amplitudes are we will find that all the fluxes are reversed,
though some arrows are reversed and some are not. Since −H also has all fluxes
reversed (reversing every bond will reverse the product over every triangle) the two
must be gauge equivalent. It turns out that appending minus signs to states 3 and 4
is one way to flip the arrows that needed to be flipped. The operator G is the product
of G′ and a diagonal matrix that multiplies 3 and 4 by minus signs. This is to be
expected in a gauge theory, where the symmetry is projectively realized [53].
Since such a procedure will work for any self-conjugate representation, we will not
explicitly construct the operator in future occasions.
We can understand now why the spectrum of H had the full set of lattice symmetries even though the flux alternated. Under any of the lattice symmetry operations,
we either left the flux alone or reversed it. Neither affects the determinant since these
operations at worst exchange E → −E, which has no effect on the spectrum. This
feature will be seen again when we consider other groups.
Since H is 4 × 4, and the characteristic equation is even in H, it satisfies an
equation of the form
H 2 − f (k) · I
2
= g(k) · I
(4.47)
105
where f and g are scalar functions. Eq. (4.47) reduces to the Dirac form if g(k) = 0
and f is constant.
The non-canonical Hamiltonian
Let us now change the sign in front of any of the terms in Eq. 4.39. It turns out that
the determinant is sensitive only the relative sign of the two long roots that reach
diagonally across the square. Here is what we get when we flip the coefficient of the
E corresponding to the root e1 − e2 :

 0

 y

Huni (k) = 
 x


x+y
y
x

x+y 


0
−(y − x) x


−(y − x) 0
−y 


x
−y
0
(4.48)
If we compute the flux now, we find it is uniform in all triangles, hence the
subscript in Huni . This in turn means that Huni will be invariant (up to gauge
transformations) under symmetry operations of the lattice.
The characteristic polynomial is
E 4 − 4E 2 x2 − 4E 2 y 2 + 4x2 y 2
(4.49)
p
√ q
E = ± 2 x2 + y 2 ± x4 + x2 y 2 + y 4
(4.50)
resulting in the spectrum
which has E → −E symmetry because G once again anticommutes with H. The
spectrum has zeros along the lines x = 0 and y = 0. Note that along these directions,
αij · ex = 1 or 0. As in the kagomé case, the lines of zeros are axes of symmetry
of the unit cell. All edges not orthogonal to these axes have equal projections onto
them (up to sign). With signs as in Huni , this results in a pair of identical rows in
the Hamiltonian – and thus lines of zeros along the short roots for both the linearized
and lattice versions of H.
106
The Hamiltonian with unequal hopping
In the above we have considered a more general case
H(k, sα ) =
X
Σ+
sα sin(k · α)(Eα + Eα† )
(4.51)
where sα = ±1 is a possible sign. In this problem there were essentially just two
choices, the ones with uniform and alternating fluxes, determined by the relative sign
of the two long hops.
In a problem like SO(5) = Sp(4), where there are roots (i.e. bonds) of two different
lengths, we could also play with the relative strengths of the hopping across long and
short bonds. There is no obvious inspiration from group theory on how to choose from
the continuum of possibilities, though only some choices will give SU(N) mean-field
solutions. The only consolation is that only two different lengths are allowed for the
roots of any semi-simple Lie algebra and among the cases we study this happens only
for SO(2N + 1) and Sp(2N).
We just mention one extreme case where long hops are set equal to zero:

 0

 y

Hshort(k) = 
 x


0
y
x
0
0
0
0
x
−y
Now the determinant is
|Hshort | = (x2 + y 2)2
0



x 


−y 


0
(4.52)
(4.53)
which describes two Dirac points at the origin. Indeed, this is just the flux phase on
the square lattice originally described by [50]. The unit cell is of course twice as big
as it needs to be, so that the two Dirac points of the traditional unit cell have both
come to the origin.
107
More generally, with a magnitude c for the long hops and relative signs all positive,
the eigenvalues are
q
√
E = ± (1 + c2 )(x2 + y 2 ) ± 2cxy c2 + 2
(4.54)
and the spectrum has one Fermi point at the origin. For relative signs chosen as in
Huni , the eigenvalues are
q
p
E = ± (1 + c2 )(x2 + y 2 ) ± 2c x4 + y 4 + c2 x2 y 2
At the special values c = 0,
√
(4.55)
2, these give a Dirac spectrum. For all c 6= 1, the
Fermi surface is a single point at the origin. Values of c corresponding to mean-field
solutions are given in Section 4.5.1.
4.4
Flux Hamiltonians in d = 3
Luckily we have to consider just three groups: SO(6)=SU(4), SP (6), and SO(7).
The only minuscule representation of the latter is the spinor. There are three minuscule representations for SO(6): two spinors (quark and antiquark of SU(4)) and the
six-dimensional vector representation. SP (6) has one minuscule representation– the
defining one.
4.4.1
Flux on the SO(6) spinor lattice.
The weights forming the tetrahedron are
1 1 1
µ1 = ( , , ),
2 2 2
1 1 1
µ2 = (− , − , ),
2 2 2
1 1 1
µ3 = ( , − , − ),
2 2 2
1 1 1
µ4 = (− , , − ) (4.56)
2 2 2
The positive roots are, in terms of orthogonal unit vectors,
e1 ± ej
j > i = 1, 2, 3.
(4.57)
108
However, as pointed out in Eq. (4.21), it is more convenient to note that since this
is also an SU(4) quark representation, we could write them in terms of the weights
(4.56) as
+
αij
= µi − µj
j > i.
(4.58)
In view of what we saw in d = 2 we are going to admit the more general case
H(k, sα ) =
X
Σ+
sα sin(k · α)(Eα + Eα† )
(4.59)
where sα = ±1 is a possible sign in front of each term.
The canonical Hamiltonian
If we pick all signs positive, (which means the arrow always goes from a site with a
lower index to one with a higher index) we obtain the flux assignment in Fig. 4.9(a).
This gives the canonical Hamiltonian

 0

 x+y

H=
 y+z


x+z
with determinant
x+y
y+z
0
z−x
z−x
0
z−y
x−y

x+z 

z−y 


x−y 


0
|H| = 4x4 − 4x2 y 2 − 4x2 z 2 + 4y 2 z 2
(4.60)
(4.61)
Note that it lacks the discrete symmetries of the lattice. This is to be expected since
the flux on each face is not the same.
Consider its zeros. It vanishes along the weight directions, k ∝ µi. This is to be
expected since the right-handed spinor is also the SU(4) quark representation and
we have seen that for SU(N), because the weights form a simplex, when k ∝ µi
109
only terms corresponding to roots involving µi remain (and that all have the same
coefficient in front). The rest vanish, so that H has just one nonzero row or column.
But we find in addition that there are entire planes along which there are zeros. For
example for any linear combination k = aµ1 + bµ2 or k = aµ1 + bµ4 , the determinant
vanishes. However it does not vanish for k = aµ1 + bµ3 unless a = 0 or b = 0. This
variation is to be expected since the flux is not symmetric on the tetrahedron.
Once again if we can use more powerful group theoretic methods to know when
determinants of certain elements of the Lie algebra of the type of Eq. (4.59) will
vanish, we will be able to anticipate this result rather than just observe it.
3
3
4
4
2
1
2
2
2
1
4
3
3
4
(a)
(b)
Figure 4.9: Possible flux assignments to the pyrochlore. (a) Flux assignment breaking
rotational symmetries. (b) Flux assignment preserving rotational symmetries.
The uniform monopole case
If we flip the sign of the 24 and 42 matrix elements (corresponding to the root µ2 −µ4 =
α24 ) we obtain the flux assignment of Fig. 4.9(b). The Hamiltonian is

Hmono
 0

 x+y

=
 y+z


x+z
x+y
y+z
0
z−x
z−x
0
−(z − y)
x−y
x+z



−(z − y) 



x−y


0
(4.62)
110
with determinant
|Hmono | = 4x4 + 4y 4 + 4z 4 − 4x2 y 2 − 4x2 z 2 − 4y 2z 2
(4.63)
which has the discrete symmetries of the lattice. The subscript on Hmono reflects the
fact that the flux is the same on all faces of the tetrahedron and comes from a unit
monopole at its center. Evidently, this is the Hamiltonian for the monopole flux spin
liquid state discussed in the previous chapter.
The reader may well ask how many more such signs are we going to play with.
Luckily we are done.
To understand this, we need to transcribe the Hamiltonian to the corresponding
factors of ±i on the edges of the tetrahedron. As mentioned above, the case with all
sα = 1 has a factor of i if we move from a corner to another with larger index (and a
−i if we move the other way). It is readily verified that the two faces not involving
the bond 24 have an outward flux of 21 π (a factor i around the triangular faces) and
the other two the reverse. Clearly the choice of matrix elements violates the discrete
symmetries of the tetrahedron.
On the other hand if we flip the coefficient of the α24 term, we get an arrangement
with all outward fluxes equal to π/2 and we are led to Hmono , with a tetrahedrally
symmetric determinant.
Other choices of sign will only yield one of two options: the flux is uniform (could
be ± 21 π) and the determinant is symmetric, or the flux assignment breaks the symmetry with two positive and two negative faces. Different choices for the latter will
correspond to determinants in which the asymmetric roles of x, y and z are interchanged.
We will now elaborate further on the case Hmono , which describes uniform flux, as
it has various nice properties, albeit at the cost of some redundancy with the previous
Chapter. To be consistent with the notation used there, we will briefly revert to the
custom of referring to momentum components as kx or ky rather than simply x or y.
111
The energy levels of Hmono are
v
s X
u X
u
2
E(k) = ±t2
ki2 kj2
ki ± 2 3
i
(4.64)
(i<j)
The spectrum has E → −E symmetry since there is a matrix G obeying G2 = I that
anticommutes with Hmono :
Hmono G = −GHmono
where

 0

1 
 −1
G= √ 
3
 −1

−1
1
1
0
1
−1
0
1
−1
(4.65)

1 

−1 


1 


0
(4.66)
That a matrix G which ensures E → −E symmetry should occur is less obvious
than in the SO(5) case, since tetrahedron is not inversion symmetric (self-conjugate).
The inversion operation maps the tetrahedron formed by the right-handed spinor
representation of SO(6) to that formed by the left-handed representation. Hence G
is not a simple geometric operation on sites in the unit cell.
We are tempted to cast Hmono in Dirac form
Hmono = αx kx + αy ky + αz kz
(4.67)
since G seems to be like the matrix β which anticommutes with the three α’s in
the Dirac Hamiltonian. However the resemblance to the Dirac case is not complete
2
because α’s do not form a Clifford algebra and Hmono
is not a multiple of the unit
matrix.
What one finds is
[αi , αj ]+
= 2δij +
√
3|εijk |Wk
(4.68)
[Wi , Wj ]+ = 2δij .
In other words, the anticommutator of the α’s is proportional to the unit matrix plus
some amount of W ’s, and the W ’s obey a Pauli algebra. Thus if we square H(α),
112
move the stuff proportional to the unit matrix to the left hand side and square again,
we will end up with a multiple of the unit matrix. Indeed this is so:
(H 2 − 2k 2 )2 = 12(kx2 ky2 + kx2 kz2 + ky2 kz2 )
(4.69)
where the subscript on H and the identity I have been suppressed.
That we should end up with the form encountered in the d = 2 SO(5) case of Eq.
(4.47)
(H 2 − f (k) · I)2 = g(k)I
(4.70)
is due to the same reasons: the characteristic polynomial P (H) is even and of fourth
order in H, i.e. quadratic in H 2 . It can therefore be cast in the form Eq. (4.70). To
get all details of f and g we would of course need to actually evaluate P (H):
P (H) = H 4 − 4H 2 (kx2 + ky2 + kz2 ) + 4(kx4 + ky4 + kz4 − kx2 ky2 − ky2 kz2 − kz2 kx2 )) = 0. (4.71)
The anticommutator algebra in Eqs. (4.68) stems from the fact that the Hamiltonian lives in the Lie algebra of the right handed spinors of SO(6), which is also the
quark of SU(4).
Recall that it is possible to write the generators of SO(N) in the spinor case in
terms of the Dirac γ- matrices: σµν , which generates rotations in the µ − ν plane,
may be expressed as σµν = 2i γµ γν . Although the γ matrices are 8 × 8, bilinears in
them like σµν form reducible representations with two 4 × 4 blocks, these being the
quark and antiquark of SU(4). The two blocks are eigenstates of γ7 = iγ1 · · · γ6 with
eigenvalue ±1. If we want the quark we can work with these 8 × 8 matrices and focus
on just the top left hand corner. In this block γ7 is just a number equal to 1.
Consider the following operator
H = iγ1 (γ6 − γ4 )kx + iγ3 (γ2 + γ6 )ky + iγ5 (γ4 + γ2 )kz = HR ⊕ HL
(4.72)
where HR and HL are 4 × 4 blocks corresponding to right and left handed spinors, or
quark and antiquark representations.
113
With a judicious choice of basis for the γ matrices its upper left-hand corner, HR
is just our Hamiltonian αx kx + αy ky + αz kz . Thus if we do not stray from this block
we can view the α’s as bilinears of γ matrices. Not so obvious is the fact that the
W ’s which come from two powers of α are also bilinears in γ.
The closure under anticommutation of the σµν or the α’s and W ’s is a special
property of SO(6). In general, if you multiply two of them you will get something
quartic in the γ’s even after some of them reduce to quadratic terms upon invoking
γ 2 = I. The quartic ones can be rewritten as γ7 times a quadratic, upon inserting
the square of the “missing” two γ matrices. In the sector with γ7 = 1, these are just
quadratic in the γ’s.
4.4.2
Flux on the SO(6) vector lattice.
In the defining vector representation the generators are represented as follows in terms
of canonical creation and destruction operators c and c† :
= c†i ci − c†−i c−i
Hi
i = 1, 2, 3
Eei ∓ej = c†i c±j − c†∓j c−i
(4.73)
i<j≤3
with generators of negative roots defined as the adjoints of the positive ones above.
The canonical Hamiltonian
In this basis the usual sum over positive roots with all coefficients positive yields the
matrix








H=






0
0
x−y
x+y
x−z
x+z



0
0
−x − y y − x
−x − z z − x 


x − y −x − y 0
0
y−z
y+z 


x+y y−x
0
0
−y − z z − y 



x − z −x − z y − z
−y − z 0
0


x+z z−x
y+z
z−y
0
0
(4.74)
114
where the rows and columns are numbered as follows: (1, 0, 0, ) ≡ 1, (−1, 0, 0) ≡
−1, (0, 1, 0) ≡ 2, ...(0, 0, −1) ≡ −3, the components being just the eigenvalues of
H1 , H2 and H3 . The site labels and corresponding factors of ±i are shown in Fig.
4.10. Note that the flux alternates from one face to the next.
3
−3
2
−2
−1
−1
1
−2
2
−3
3
Figure 4.10: Flux assignment to the octachlore in accordance with signs of the group
generators of the vector representation of SO(6).
The determinant vanishes identically because there are two zeros at every k. If
we pull them out we find
|H| = 48(x2 y 2 + y 2 z 2 + z 2 x2 ) ∗ 0 ∗ 0
(4.75)
that is to say, the product of the nonzero energies is 48(x2 y 2 + y 2 z 2 + z 2 x2 ).
Why does H have all the discrete symmetries when the flux alternates? The
answer is that any rotation is equivalent to a change of the sign of the overall flux,
which in turn corresponds to time-reversal, and does not affect the determinant in a
problem with E → −E symmetry.
Extra zero-energy bands occur when any two coordinates vanish, i.e., along the
axes, which corresponds to the direction of the weights. We can understand this to
the extent we could understand the SU(N) and SO(5) cases. If
H=
X
i<j
h
i
(k · (ei ± ej )) Eei ±ej + Ee†i ±ej
(4.76)
115
it follows that if we set k = e1 say, only roots of the form e1 ± ej will survive and
that too with the same coefficient. The matrix will have only two non-zero rows and
columns – for the sites at ±e1 in the unit cell. With the flux assignment of (4.74),
one row is exactly the negative of the other, resulting in two extra zero energy bands
in both the linearized and the lattice Hamiltonian.
Near any line of zeros we can define a 2 dimensional Dirac field, except near the
origin when they all collide and modify each other.
Again there is a matrix G which anticommutes with the Hamiltonian, and acts
upon the unit cell as the inversion. Its existence results from the fact that the unit
cell is inversion symmetric, while the directions of all fluxes are reversed by inversion.
The non-canonical Hamiltonians
We could append signs for each term, but this gives spectra which break the lattice
symmetries. We have not looked deeply into what kind of zeros result in that case.
We did however note the following. Suppose we start with a hopping problem on
an octahedron with uniform flux in every face as in Fig. 4.11.
3
−3
2
−2
−1
−1
1
−2
2
−3
3
Figure 4.11: Flux assignment to the octachlore preserving rotational symmetries.
When we extracted the H(k) for that problem we found it could not be written
in terms of generators of SO(6). It is important to understand why we have this
problem here but did not when we considered the spinor of SO(6) = quark of SU(4).
116
There each root or generator connected only two states. If we did not like the sign of
the matrix element given by group theory we just put a negative sign in front of that
generator using sα . But here, each root connects two pairs of points, corresponding
to parallel edges of the octahedron. For example
Ee1 +e2 = c†1 c−2 − c†2 c−1
(4.77)
connects points labeled (−2, 1) and (−1, 2) in Fig. 4.11 with opposite matrix elements.
If we do not like the relative sign, we cannot do anything about it. This is exactly
what happens in the case of the octahedron with uniform flux in every face. To
describe it, we would have to use the generators of the much larger group SU(6). But
that is not the game we are playing: we want to work within a group, SO(6) being
the operative one here.
This could have happened to the SO(5) spinor, whose short roots connected opposite sides of the square with same sign for the horizontal roots ±e1 and opposite
signs for the vertical roots ±e2 . Luckily this choice of signs corresponded to the case
of interest.
4.4.3
Sp(6)
The defining representation of Sp(6) is the same octahedron as in SO(6) with the
same weights. The generators can be written in terms of creation and destruction
operators as
Hi
= c†i ci − c†−i c−i
Eei ∓ej = c†i c±j ∓ c†∓j c−i
E2ei
= 2c†i c−i
i = 1, 2, 3
i<j≤3
with negative roots being given by adjoints of the above.
(4.78)
117
The non-canonical Hamiltonian
It is interesting to consider first the canonical H with all signs positive and only
the short roots. (As noted before when we have two different root lengths, we have
the freedom to chose the scale of each type of term. Keeping only short roots is an
extreme case.) We find








Hshort (k) = 






0
0
0
0
x−y x+y
x+y y−x
x−z x+z
x+z z−x
The determinant has the value
x−y x+y x−z x+z



x+y y−x x+z z−x 


0
0
y−z y+z 


0
0
y+z z−y 



y−z y+z 0
0


y+z z−y 0
0
|Hshort| = −32(x2 + y 2 )(x2 + z 2 )(y 2 + z 2 )
(4.79)
(4.80)
that is to say, zeros along the weights. The logic is the same as in SO(6) since the
long roots that distinguish between them have been suppressed. Note however that
matrix elements are different now: the two pairs of states connected by a generator
do not always have opposite matrix elements. Thus in this case along the axes there
are 2, rather than 4, zero energy bands.
What is surprising is that the energies do not change for any choice of signs !
A non-canonical Hamiltonian with unequal coefficients
Consider the following matrix involving the long roots:
118








H“x+z”, 1 
2 





2x
0
x−y x+y x−z
−x − z
2x
0
x + y y − x −x − z z − x
x−y
x+y
0
2y
y−z
y+z
x+y
y−x
2y
0
y+z
z−y
x−z
−x − z y − z y + z 0
−x − z z − x
2z
y + z z − y 2z
0















(4.81)
where the subscripts remind us of two ways in which it differs from the canonical
form: the x + z term has a minus sign relative to the canonical form, and the hopping
matrix element for the long roots is half as big as the canonical one. As for the latter
point, consider the term 2x. It indeed equals k · 2e1 , but the generator E2e1 = 2c†1 c−1
has another two in it. So this term should have been 4x. But with the choice of sign
and hopping (4.81) we get
|H“x+z”, 1 | = −16(x + y)2(x + z)2 (y + z)2
2
(4.82)
which has zeros along planes x + y = 0 etc.
When we put in the canonical strength (4x etc.) we did not find any interesting
spectra for many choices of sign that we tried.
4.4.4
SO(7) spinor
Recall that the only minuscule representation of SO(7) is the spinor and that the
lattice we associate with it is cubic, with face diagonals but no body diagonals (Fig.
4.12(a)). The generators in this representation can be expressed in the direct product
space of three Pauli matrices:
E±e1
=
1
σ
2 3
E±e3
=
1
σ
2 ±
E±e2
⊗1⊗1
Ee1 ±e2 = ± 21 1 ⊗ τ+ ⊗ α±
Ee1 ±e3 = ± 21 σ+ ⊗ τ3 ⊗ α±
=
1
σ
2 3
⊗ τ3 ⊗ α±
⊗ τ± ⊗ 1
Ee2 ±e3 = ± 21 σ+ ⊗ τ± ⊗ 1
(4.83)
119
where the labels 1, 2, and 3 correspond to the directions of the co- ordinate axes.
The matrix
G = σ2 ⊗ τ1 ⊗ α2
(4.84)
acts as an inversion operator on the unit cell and anti-commutes with all of the
symmetric generators Eα + E−α so that the spectrum has symmetry under E → −E
no matter what the signs.
The canonical Hamiltonian
If we ask what hopping amplitudes are associated with the canonical case we find
that each square plaquette has π flux.
The generators Ei±j fix the flux through the triangular plaquettes to be ±π/2.
The form of these generators dictates that two opposing pairs of triangular faces will
have diagonals with the same orientation; the third will have diagonals with opposite
orientations. It turns out that in this case a uniform flux through the triangular
plaquettes is impossible.
If we choose all signs to be positive, then three of the cube’s faces have flux π/2
outwards through all triangular plaquettes, and the remaining three to have flux
−π/2.
The zeros of energy can be found from
|H| =
1
(x2 + y 2 + z 2 − 2(xz − yz − xy))2 (x − y + z)4
256
(4.85)
Here not all cubic symmetries are preserved, but permutations of the x, −y, and z
axes (corresponding to rotations of the cube about the (1, −1, 1) body diagonal) map
positive fluxes to positive fluxes and vice versa, so that some of the cubic symmetries
are preserved.
120
The non-canonical Hamiltonian with alternating flux
As in the octahedral case, the other interesting case is the alternating flux pattern
shown in Fig. 4.12(b), in which rotations by π/4 about the x, y, and z axes reverse
the signs of all fluxes. In this case, we append minus signs to the x + z, x − y, and
y − z terms. The energies are remarkably simple:
E=±
1√
3(x ± y ± z)
2
(4.86)
which vanish along the planes x = ±y ± z. For example, if the sites of the unit cell
are labeled 1 − 8 as shown in Fig. 4.12(b), momenta in the plane x = −y − z obey
k · α1j = −k · α8j , producing a pair of linearly dependent rows in the Hamiltonian.
In the alternating flux case the cubic symmetries are preserved since a π/4 rotation
reverses the signs of all fluxes, which is gauge equivalent to reversing the signs of all
hoppings and thus the sign of H. Invariance under E → −E thus ensures that this
is a symmetry.
These two possibilities are the only ones preserving the permutation symmetry of
the x, y and z axes.
2
2
1
1
4
4
3
3
6
6
5
8
5
8
7
7
(a)
(b)
Figure 4.12: Two possible flux assignments for on the SO(7) lattice. Each square
plaquette has flux π, while the triangular plaquettes on each face may be chosen to
have equal (a) or alternating (b) flux.
121
4.5
4.5.1
Comments
Relevance to mean-field solutions of the Heisenberg
model
We will now revisit the Hamiltonians of Sections 4.3 and 4.4 with a view to asking
whether they are, in fact, mean-field solutions of the SU(N) Heisenberg model. Since
the mean-field approach was outlined in the previous Chapter we will not repeat
this description here, but focus on the general results which can be inferred about
the Hamiltonians presented here. A few of these have been discussed previously in
the literature. The kagomé Hamiltonian of Section 4.3.1 was identified as a meanfield solution by [73]. Several authors have discussed SU(N) mean-field states on
the square lattice with second neighbor hopping [52, 74, 67, 75], though these have
focused on the gapped chiral spin state quite unlike that of Section 4.3.2; we believe
that previous mean-field studies on the checkerboard lattice [76] have been restricted
to dimerized states.
We wish to extend these results and establish that many of the Hamiltonians
discussed above govern mean-field solutions. To show this, we must argue that the
mean-field equations admit solutions in which the hoppings tα are purely imaginary
with the correct relative signs, and hoppings along roots of equal length are equal in
magnitude.
The mean field equations can be written:
Jij (R)
tij (R) =
V
Z
X
φ∗i,α (k)φj,α (k)eik·(rij +R)
(4.87)
k∈1BZ E (k)<0
α
where Jij (R) is the spin-spin coupling for the original Heisenberg Hamiltonian, and
tij (R) is the hopping matrix element between these sites for the mean-field Hamiltonian. Here i and j label positions in the unit cell, and R is a lattice translation
122
vector. The φ(k) are obtained from the eigenfunctions according to
†
φi,α (k) = h0|cik ψk,α
|0i
(4.88)
where α is a band index.
We now show that if the spectrum is invariant under k → −k, then as H(−k) =
−H(k), Eq. (4.87) ensures tij = −t∗ij . Taking the complex conjugate of Eq. (4.87),
and using the fact that H(k)|ψ(−k)i = −ǫ(k)|ψ(−k)i gives:
t∗ij
Jij
=
V
Z
X
φi,α (k)φ∗j,α(k)eik·rij
(4.89)
k∈1BZ E (k)>0
α
where we have dropped the dependence on R, which is 0 for sites in the same unit
cell. Using the completeness relation
X
φ∗i,α (k)φj,α(k) = δij
(4.90)
α
and the fact that the eigenvectors of H(k) are real up to a common phase, this gives
for i 6= j:
t∗ij
Jij
= −
V
= −tij .
Z
X
φ∗i,α (k)φj,α(k)eik·rij
k∈1BZ E (k)<0
α
(4.91)
A similar argument shows that if the spectrum preserves lattice symmetries, the
magnitude of tij must be the same on all roots αij of a given length, with the relative
signs fixed by the action of the corresponding gauge transformations on the φi . In
cases with two different length roots, the mean-field equations specify a particular
relative hopping strength.
Many of the flux configurations discussed in Sections 4.3 and 4.4 are therefore selfconsistent mean-field Hamiltonians. Consider first the lattices in which all links have
the same length (and are related by lattice symmetries): the kagomé, pyrochlore, and
octachlore. In these cases (4.87) gives equal hopping amplitudes on all links provided
123
that the spectrum does not break the lattice symmetries. The kagomé Hamiltonian
has been discussed by [73]; it is somewhat exceptional among the Hamiltonians we
consider in that its spectrum is not inversion symmetric at fixed k. The monopole
Hamiltonian of the pyrochlore lattice (Section 4.4.1), as well as SO(6) (Section 4.4.2)
and short-root SP (6) (Section 4.4.3) Hamiltonians on the octachlore lattice have
spectra that preserve lattice symmetries and thus should be mean-field states by the
argument above. We have verified numerically that the monopole, SO(6) vector and
short-root SP (6) Hamiltonians are viable mean-field solutions.
The lattices constructed from the spinor representation of SO(2N + 1), as well
as the canonical lattice of SP (2N), have roots of two different lengths. In all three
examples discussed here, Hamiltonians with hopping only along the short roots are
mean-field solutions; for SO(5) this gives the flux state of [50] and for SO(7) the 3-d
version thereof. If we are interested in Hamiltonians with non-zero hopping along the
long roots, we must find the ratio tl /ts consistent with Eq. (4.87).
A summary of the allowed values of tl /ts is given in Table 4.5.1. For SP (6)
generic values of tl /ts do not give symmetric spectra, and we find no mean-field
Hamiltonians with tl > 0. The alternating flux Hamiltonian of SO(7) and both flux
assignments discussed in Section 4.3.2 for the SO(5) spinor lattice give symmetrypreserving spectra whose mean-field tl /ts can be calculated numerically. The ratio
tl /ts at mean-field depends on the relative magnitudes of the spin-spin coupling Jl /Js
in the original Heisenberg Hamiltonian; solutions with tl > 0 exist only in a restricted
range of Jl /Js , as shown in Table 4.5.1.
Needless to say, this analysis does not preclude the existence of other flux assignments leading to a lower mean-field energy. Indeed states with lines and planes of
zeros, as many of our examples have, are often energetically disfavored at mean-field
[73] due to the large phase space near E = 0 relative to gapped or mostly gapped
states. Also one must bear in mind that dimerized mean-field states of lower energy
124
Jl /Js
1.14
1
.9
.8
.7
.6
SO(5) (alt)
tl /ts
0, 1.46
0
0
0
0
0
SO(5) (uni)
tl /ts
0, 1.10
0, 1.21
0, 1.36
0, 1.73
0, 4.02
0
SO(7) (alt)
tl /ts
0, 1.0
0, 1.0
0, 1.1
0, 1.2
0, 1.3
0, 1.6
SP (6)
tl /ts
0
0
0
0
0
0
Table 4.1: Relative strengths of hopping along the long (tl ) and short (ts ) roots as
determined by the mean-field equations for the SO(5), SO(7), and SP (6) hopping
problems. For SP (6) we find only the tl = 0 solution at mean-field level. For SO(5)
we find consistent mean-field solutions with |tl | > 0 for 5 > Jl /Js > .68 in the
uniform flux case, and 1.3 > Jl /Js > 1 in the alternating flux case. For SO(7) we
find consistent mean-field solutions with |tl | > 0 for 3.5 > Jl /Js > 0.5.
inevitably exist [57, 77]. However, as pointed out in the previous Chapter, corrections to the mean-field solution for N < ∞ often alter the relative stability of various
mean-field states, so we should not take this issue too seriously. Among the meanfield solutions discussed here, perhaps the most interesting example is the monopole
Hamiltonian of the pyrochlore lattice. In Chapter 3, we showed that it corresponds
to the lowest energy symmetry preserving mean-field solution to the SU(N) Heisenberg model, and has lower energy than the dimerized mean-field ground states after
Gutzwiller projection is used to enforce the constraint of single occupancy [78].
4.5.2
Hamiltonians beyond the linear approximation
This work has focused on the linearized versions of lattice Hamiltonians, in which we
have replaced
sin(k · r) → k · r.
(4.92)
However, many of the interesting properties of the spectra are unaffected by this
substitution.
First, surfaces of zero energy which are related to symmetries of the unit cell will
125
not be affected. Recall that we find several zero-energy surfaces along directions of
the unit cell for which k · rij takes on values ±c, 0 for some constant c. If k · rij is
replaced by sin(k · rij ), the only effect on the Hamiltonian at these points is to change
the value of the constant c; hence the zero eigenvalues remain. As discussed above,
this yields lines of nodes along the weight vectors for the SU(N) lattice model for any
N. The zero-energy manifolds of the alternating flux S0(5) and SO(7) Hamiltonians,
and the SO(6) and SP (6) Hamiltonians discussed above for the octachlore lattice,
are also preserved under (4.92). In other words, all zero-energy surfaces listed in
Sections 4.3 and 4.4 which reflect symmetries of the lattice unit cell are unaffected
by the substitution (4.92).
Second, the symmetry of the spectrum will remain. Symmetries in the spectra
occur when flux is assigned in a way that preserves the lattice symmetries, and the
substitution (4.92) cannot alter the symmetry properties of the state.
Finally, it is interesting to note that on the lattices with inversion-symmetric unit
cells (namely lattices related to representations of SO(N) or SP (2N)), the operator
G which anti-commutes with the linearized Hamiltonian also anti-commutes with the
lattice Hamiltonian. This happens because G in these cases is simply the inversion
operator multiplied by an appropriate gauge transformation, and inversion maps every edge to another edge associated with the same symmetric generator of the Lie
group representation. Thus G anti-commutes separately with all of the symmetric
P
generators – and hence also with the lattice Hamiltonian Σ+ (Eα + Eα† ) sin(k · rα ).
4.5.3
Extensions to Higher Dimensions
We have already noted that the generalization of our Hamiltonians to d ≥ 4 is problematic. For completeness we note here that the lattice construction described in
Section 4.2 can be applied to the appropriate representations of the Lie groups discussed above in arbitrary dimension. Assigning a hopping of ±i to each directed edge
126
will result in a Hamiltonian related to the group generators by Eq. (4.28), for which
H(k) = −H(−k). Additionally, for all of the cubic lattices (SO(N) spinor, SO(2N)
vector, and the defining representation of Sp(2N) ) a matrix G can be found which
anti-commutes with the Hamiltonian, leading to a time-reversal invariant spectrum.
In general, however, the symmetry operations of the resulting unit cell make it
impossible to assign flux in such a way that the lattice symmetries are unbroken. The
notable exception is the SO(2N + 1) spinor case with only the short roots, where all
fluxes are π ≡ −π. This gives the N dimensional Dirac Hamiltonian.
4.6
Conclusions and Outlook
In this Chapter we constructed a class of lattices inspired by the root and weight
systems of Lie algebras. The lattices had as their unit cells minuscule representations
of the standard Lie groups which decorated the underlying lattice L2R , elements of
the root lattice with even coefficients. We observed that the lattices could equally
well be viewed as decorations of L2R by the conjugate representation, which shared
corners with the original one. While construction works for any rank r we stuck to
r = 2, 3 since these were experimentally accessible and because these allowed an
unambiguous assignment of flux on the triangular faces of the unit cells. Remarkably,
they also correspond in many cases to known lattices like the pyrochlore, kagomé or
checkerboard. Even our octachlore lattice is a motif in the perovskite structure.
We find this last aspect enticing, for it hints that it may be possible to relate more
physics on these lattices to the underlying Lie algebras. Indeed, as we were finishing
up this work we came across recent work by Arovas [79] who constructs generalized
AKLT models on the kagomé and pyrochlore lattices which naturally involve local
degrees of freedom that live in the fundamental representations of SU(3) and SU(4)
respectively.
127
However, our own work makes a different connection. We considered hopping
Hamiltonians which, when written in momentum space, were elements of the Lie
algebra, linear in the momentum k for small k, and obeyed H(k) = −H(−k).2 By
varying the signs in front of each generator we could alter the fluxes in the faces of
the unit cell. We found Dirac or Dirac-like spectra at points, lines and even sheets.
The locus of the zeros had strong ties to the directions of the weights or roots. We
could anticipate and thus understand some of them using ideas from Lie algebras but
often were just able to draw attention to them. It seems very likely that an assault
using ideas from Lie algebras can yield further understanding. To begin with one
must employ a more systematic way to represent weights and roots in dual bases:
simple weights for the former and simple roots for the latter. One should also use
color groups to classify symmetries of this problem where the triangular faces of the
unit cell are colored with flux ±π/2. Of all the properties associated with H, the
determinant seems most likely to yield to group theoretic methods. It has uniformly
proven to be a much simpler and more symmetric function of the momenta than
individual eigenvalues.
The spectra often had E → −E symmetry. For self-conjugate representations we
could fully understand this feature and indeed use our understanding to construct an
operator G that anticommuted with the Hamiltonian and explained this feature.
While such hopping problems typically arise as lattice regulators for continuum
theories or as mean-field theories for quantum spin models, in this Chapter we have
studied them in their own right. While we did observe that most of them are candidates for interesting mean-field theories of quantum Heisenberg models on the same
lattices, a fuller investigation of the fluctuations would be required to establish their
value in that setting.
2
To belabor this point, we have an entire unit cell represented by a quark state, while Arovas has
a quark state at each site of the unit cell.
Chapter 5
The devil’s staircase in
1-dimensional dipolar Bose gases in
optical lattices
5.1
Introduction
When commensuration effects compete with long-ranged interactions, a startling richness of phases can arise. This interplay between long-ranged forces and lattice effects
has been studied in a variety of models – from the Ising model [80] to interfaces
between crystal surfaces [81]. In this chapter, we explore a new context in which
such physics leads to rich mathematical structure – that of the cold dipolar Bose
gas. Recent advances in laser cooling and trapping have opened the possibility of
creating cold Bose gases in optical lattice potentials. If these bosons have sufficiently
strong dipolar interactions, the phase portrait is controlled by the interplay between
the infinite-ranged dipolar repulsion1 , and commensuration effects due to the optical
lattice – giving a potential experimental realization of the diverse phases expected
1
The dipolar interaction can be made repulsive everywhere on the line by polarizing the dipoles
with an external field
128
129
in such a system. Furthermore, considering which parameters can be easily tuned in
such experiments naturally opens questions about the ground states of these systems
in new regions of parameter space.
The driving force behind the strikingly rich phase diagram in these systems is that
long ranged interactions on a lattice can stabilize exceptionally intricate ground-state
structures in the classical, or strong coupling, limit. A particularly elegant example of
this is the case of classical particles in a one-dimensional (1D) lattice interacting via
an infinite-ranged convex potential studied by Pokrovsky and Uimin, and Hubbard
(PUH) [80, 82, 83]. Here, it can be shown that the ground state filling fraction as a
function of chemical potential µ is a complete devil’s staircase [84], in which every
rational filling fraction between 0 and 1 is stable over a finite interval in µ, and the
total measure of all such intervals exhausts the full range of µ.
This devil’s staircase of PUH has dramatic consequences for the physics of quasi
1D cold atomic gases. Building on the existing understanding of this classical limit,
we consider two perturbations of the devil’s staircase that arise naturally in the experimental setting of cold atomic gases. The first of these is the introduction of a
quantum kinetic energy, due to the finite depth of the optical lattice, which now renders the problem sensitive to particle statistics. We focus on the bosonic case, as this
case is most likely to be realized experimentally. Tuning away from the classical limit
has the well understood effect of initiating a competition between the crystalline,
Mott phase that exists at zero hopping and the superfluid (Luttinger liquid in d = 1)
that must exist at all fillings at sufficiently large hopping. The phase diagram we
find is thus an extremely complex variant of the usual Hubbard model story, with an
infinite number of Mott-Hubbard lobes corresponding to commensurate phases.
The second perturbation involves tuning the onsite interaction independently of
the dipolar potential. This allows for a controlled departure from convexity, and generates new classical states comparable in complexity to the PUH states considered
130
previously. This introduces doubly occupied sites in the classical limit. While describing the resulting phase diagram in complete and rigorous detail is beyond the
scope of this project, we give an account of the “staircase” structure of the initial
instability and of regions of the phase diagram where the classical states exhibit superlattices of added charge built on underlying PUH states. At least some of these
regions exhibit devil’s staircases of their own. Finally, upon introducing hopping we
are led to an infinite set of “supersolids”—which in this context are phases that are
both Luttinger liquids and break discrete translational symmetries.
5.1.1
Chapter Outline
We begin in Section 5.2 with an overview of the model and a sketch of the relevant
experimental parameters. 5.3 gives a detailed description of the intriguing infinite
family of classical solutions to this model, described in [80, 83], and discuss the fractal
structure known as the devil’s staircase which constitutes the classical phase portrait.
Then in Section 5.4, we use a strong coupling perturbation theory similar to that
previously studied in the Bose-Hubbard model [85], and in extended Bose-Hubbard
models with nearest-neighbor interactions [86, 87], to calculate the boundaries of
the Mott-Hubbard lobes where these commensurate classical states liquify. We also
review how bosonization techniques admit a Luttinger liquid description of these
phase transitions. This approach proves helpful in understanding the expected density
profiles of atoms in a parabolic trapping potential – a question we address for both
quantum and classical models in the convex limit in Section 5.5.
In Section 5.6, we switch gears and evaluate the classical phase diagram as U0 is
tuned away from convexity. This reveals a mind-bogglingly complex array of possible
phases, only some of which we are able to describe here. However, one class of such
phases, in which the ground states resemble a two-component version of the HUP
ground states, are of particular interest. Section 5.7 discusses in more detail the
131
classical structure of these phases, as well as their behavior at finite hopping – where
we find super-solid like phases. We have relegated several detailed calculations to
Section 5.9, which contains supplementary material.
5.2
The ultra-cold dipolar Bose Gas
The unprecedented control over experimental parameters in trapped ultracold atomic
gases has substantially widened the range of possible exotic phases of matter that can
be explored. Optical lattices can be used to simulate simple lattice models and/or
vary the system dimensionality, while the interatomic interactions can be varied via
a magnetically-tunable Feshbach resonance [88]. Thus far, the focus in these systems
has largely been on contact interactions, since these generally provide a good description of atom-atom scattering in the low energy limit. However, with the realization of
atomic gases with strong magnetic dipole moments [89] and the prospect of working
with molecules that exhibit electric dipole moments [90, 91], there is now substantial interest in examining the physics of long-ranged interactions in these systems2
. In this section we will present the model, motivated by these new experimental
possibilities, which we will discuss in the remaining sections of this chapter.
5.2.1
Bosons in 1D optical lattices
We begin by discussing the basic physics of dipolar molecules in an optical lattice.
Experimentally, the scenario is as follows: atoms, or in some cases even molecules,
can be trapped using spatially varying electric fields. In optical traps, this frequency
can be tuned to be close to an atomic transition. In this case the atoms are highly
polarizable at this frequency, and hence feel a relatively strong trapping potential. A
lattice potential can then be made for the gas of atoms, by using counter-propagating
2
For a review of the current understanding of dipolar interactions, and the parameter régime
currently accessible to experiments, see [92, 93]
132
laser beams to create a periodic pattern of maxima and minima of the electric field’s
intensity. The depth of the lattice is tuned by the intensity of the laser light; the
lattice spacing is determined by the wavelength of the standing light-waves.
To explore 1D physics, in practice a 3D optical lattice is generated, but with
a lattice depth and spacing much greater in the first two dimensions than in the
third. This effectively creates an array of 1D tubes, with a relatively weaker periodic
potential and shorter inter-particle spacing along each tube. Typically, the whole
ensemble of tubes is also confined in a long parabolic trap, to ensure that the particles
remain inside the optical lattice system.
Here we will ignore the complications due to interactions between tubes. To
simplify the discussion, we will also begin by omitting the parabolic trapping potential
– we will return to its effects in Section 5.5. Hence the Hamiltonian we study is:
X 1
UX
n
n
+
ni (ni − 1)
i
j
r3
2 i
i<j ij
X
X †
−µ
ni − t
ci ci+1 + h.c.
H = V0
i
(5.1)
i
This model describes bosons in a deep 1D optical lattice with hopping amplitude t, onsite interaction energy U, and an infinite-ranged dipole-dipole interaction V0 /r 3 . Since
the hopping potential t is controlled by the depth of the optical lattice, it can be tuned
over a wide range of values. In the cases where the bosons are atoms, e.g. 52 Cr [89], the
on-site interaction U may also be easily tuned using Feshbach resonances. We set the
dipolar interaction to be maximally repulsive by aligning the dipoles perpendicular to
the 1D chain. We also note that though dipolar interactions will couple the different
√
tubes, aligning the dipoles at an angle such that cos(θ) = 1/ 3 cancels this interaction
in one of the two remaining dimensions. Hence to access the 1D régime, one would
ideally work with a single 2D array of optical tubes.
Though our focus here is on the realm of possibly, rather than currently, attainable
states of the dipolar Bose gas, it is instructive to ask what range of parameters can
133
be attained in current experiments. The critical ratios are t/V0 and U0 /V0 .
In cold atomic gases, there are essentially two ways to generate dipolar interactions: by cooling atoms, such as
52
Cr, with large magnetic dipole moments, or by
creating molecules with electric dipole moments – typically by generating heteronuclear molecules which have an intrinsic dipole moment, for example, with
41
K
87
Rb.
Techniques for cooling and trapping atoms with large magnetic dipole moments have
already been developed [94]; cooling dipolar molecules is significantly more difficult,
though significant progress towards creating cold molecules in their 2-body ground
states has been achieved, most notably in fermionic systems [95, 96]. Since the experimentally attainable electric dipole moments are much larger than their magnetic
counterparts, it is interesting to consider the parameter r’egimes accessible in both
types of experiments.
Current ultracold bosonic molecules have electric dipole moments of the order of
d ≈ 1 Debye; for
52
Cr and
87
Rb the magnetic dipole moments are 6µB and 1µB ,
respectively. Using these values, the dipolar interaction strength at a distance of n
lattice spacings is
d2 (3.338 × 10−30 Cm)2
d2
1 (dD)2
=
n3 4πǫ0 a3
4π8.854 × 10−12 C 2 /(Nm2 )a3 n3
1
1.0 × 10−22 Nm
≈
3
n a3
1 µ0 (µµB )2
(4π × 10−7 NA−2 µ2 (9.27 × 10−24 Am2 )2 1
=
=
n3 4πa3
4πa3
n3
µ2
≈
8.6 × 10−27 Nm
3
3
na
El
Vdip
=
M ag
Vdip
(5.2)
where a is the optical lattice spacing in nm, d = 0.6 is the electric dipole moment
measured in Debye, and µ = 6.0 for
52
Cr and 1.0 for
87
Rb is the magnetic dipole
moment in Bohr magnetons.
To find the hopping parameter requires calculating the overlap integrals of Wannier functions as a function of the depth of the optical lattice. The result is that the
134
hopping t is given by [97]:
4Er
t= √
π
A
Er
3/4
√
e−2 A/Er
(5.3)
where A is the intensity of the laser beam, and Er is the recoil energy of a single
atom, Er = h2 /(8ma2 ). Current experiments can attain lattice spacings of order
a = 500nm, at laser intensities on the order of 10 − 20 recoils. This gives a range
of hopping parameters listed in Table 5.2.1. We see that, assuming that the electric
dipole moments of the molecules do not significantly alter the facility with which
they can be trapped (a reasonable assumption since the frequency of the molecular
transitions is significantly lower than that of the atomic transitions to which the laser
light is tuned3 ), fractions t/V0 10−2 are well within the range of experiments on polar
molecules. For magnetic dipole moments, the numerical value of t/V0 is larger by a
factor of 300 (for
52
Cr) to 6000 (for
87
Rb), and hence we do not expect to see any
Mott physics associated with these states in currently realistic trapping potentials.
a(nm)
A/Er
5
10
15
20
300
400
500
0.184 0.25 0.306
0.048 0.065 0.081
0.016 0.021 0.026
0.006 0.008 0.010
Table 5.1: Experimentally attainable values of the hopping t/V0 as a function of
the lattice constant and laser intensity in the approximate range of experimentally
realizable values. Here we use parameters relevant to polar molecules – an electric
dipole moment of 0.6 Debye, and the mass of 41 K 87 Rb. The corresponding values for
52
Cr are larger by a factor of ≈ 300.
The effect of deforming the on-site repulsion U away from convexity is rather more
difficult to probe experimentally. To attain this régime, Feshbach resonances must be
used to tune the scattering length to be relatively large and negative, to compensate
3
Recall that the optical trap is created using laser light that is tuned to be close to an atomic
transition, so that the atoms are highly polarizable at this frequency.
135
for the strong dipole-dipole interaction felt by two bosons confined to the same lattice
site. Calculating the overlap of the Wannier wave functions in a deep optical lattice,
where the particles are effectively localized at a single lattice site, gives
√
3/4
8πas
A
as
≈ 5 × 10−29 Nm
U=
Er
a
Er
a
(5.4)
where as is the scattering length of the bosons, tuned by Feshbach resonances, and
we take
A
Er
= 20, a = 500nm. To this we must add the effective dipolar interaction,
whose approximate order of magnitude we obtain by noting that two bosons confined
to the same site are a separation of no more than a/2 apart – giving a dipolar energy
of order 2 × 10−30 Nm. Hence a priori one would expect that on-site interactions
are easily tuned away from convexity by making the scattering length as sufficiently
negative. In an optical lattice a reasonable limit [88] is aS < a, so some tuning away
from convexity may be possible4 .
5.3
Classical solutions and the devil’s staircase
The notion that longer-ranged interactions can stabilize fractional fillings in the Mott
state is well-known in the study of extended Hubbard models. With only on-site interactions, the Hubbard model itself has commensurate states only at integer fillings;
extended Hubbard models with repulsion between neighboring sites can stabilize such
states at half filling, and so on. The HUP case is simply the extreme limit of the
extended Hubbard model: convex interactions favor arrangements in which particles
are spread out as homogeneously as possible given the filling fraction. In the case of
infinite-ranged convex interactions, this yields a constraint that must be satisfied at
all distance scales, leading to a pattern that is unique, up to global lattice transfor4
We are not aware of existing calculations which account for both scattering and dipolar interactions in an optical lattice; hence this estimate is obtained by assuming that the relevant parameter
is the effective scattering length due to both scattering and dipolar interactions, which we take to
be positive.
136
mations, for every filling fraction. Hence the classical configurations are devoid of all
local degeneracies, and commensurate structure exists at all length scales.
In our model Hamiltonian (5.1), if U is sufficiently large the potential is everywhere
convex – meaning that
V (x) ≤ λV (x − 1 + λ) + (1 − λ)V (x + λ) for 0 ≤ λ ≤ 1 .
The t = 0 (classical) ground states of (5.1) in this case are therefore those of a
PUH Hamiltonian: for every rational filling fraction ν = p/q, the ground state is
periodic with period q [80]. Each such ground state is unique up to global lattice
translations [98]. We denote these states commensurate ground states (CGS). Adding
or removing a single particle from a CGS in the infinite volume limit produces a qsoliton state (qSS) containing q fractionally charged solitons of charge 1/q. We now
review the nature and energetics of the CGS and qSS states.
5.3.1
Commensurate Ground States
We now review the detailed construction of the CGS classical ground states. To do so,
it is convenient to characterize a state by the set of all of its inter-particle distances.
Call particles lth neighbors if there are l − 1 occupied sites lying between them (their
separation will generally exceed this). Any static configuration of particles on the
(1)
(2)
(N )
lattice is characterized by the set (S) of sets Sl ≡ {rl , rl , ...rl
} of lth neighbor
distances between a particle and its lth neighbor to the right. (On a finite-sized chain
(i)
it is simplest to consider periodic boundary conditions, in which case rl
exists for
all i and l < N.) The energy of such a configuration is
E=
XX
l
(i)
V (rl )
(5.5)
i
where V (r) is the (convex) potential. Here the second sum is a sum over individual
particles; the first sum is a sum over all neighbors of each particle. A solution which
137
minimizes the inner sum for each l also minimizes the energy; hence it suffices to find
the set Sl for which
El =
X
(i)
V (rl )
(5.6)
i
is minimal, and establish that a configuration for which Sl has this form for all l does
indeed exist.
Energetics of convex interactions
The first step in constructing the HUP ground states is to establish which Sl , for
a given l, minimizes the energy (5.6). For convex interactions, the optimal Sl is a
maximally compact set – e.g. the one for which all rl are as close as possible to the
same value. This is commensurate with the intuition that convex interactions favor
maximally uniform distributions of particles, and we will prove presently why this is
so.
For a given l and filling fraction ν, what is the maximally compact Sl ? In the
(i)
absence of a lattice, maximal compactness would imply rl ≡ rl for all i; however on
a lattice we must also allow Sl = {rl , rl + 1}. To see why, recall that the sum of all
distances between first neighbors must be the total length of the lattice – or, more
generally, that for a lattice of length L we have
X
(i)
rl = lL.
(5.7)
i
The average separation between lth neighbors is thus:
r̄l =
1 X (i)
r = l/ν
N i l
(5.8)
where ν = N/L. For rational fillings ν = p/q, this gives:
r̄l = lq/p
(5.9)
Thus either p|l and the maximally compact set is Sl = {r¯l } = pl q, or r̄l is not an
(i)
integer, and hence Sl must have at least two elements – rl ∈ {rl , rl + 1}. Hubbard
138
[80] showed by construction that for all rational ν there exists a solution in which
Sl = {rl , rl + 1}, and hence that these are ground states.
Before outlining the details of this construction, let us show why maximally compact sets Sl correspond to minimal energy solutions. For convex potentials V (r)
V (rl + x + y) + V (rl − x) > V (rl ) + V (rl + y)
(5.10)
In other words, given 2 pairs of points with the same average, the average of any
convex function V will be smaller over the pair which is closest together. This is
simply a reiteration of the familiar definition of convexity for continuous functions,
stating that the line joining any 2 points on the graph of V (x) lies above the graph.
Now, given two distinct Sl with the same average r̄l , the convexity of the potential
(5.10) ensures that the set with the narrowest distribution of its elements will have
the minimum energy. More precisely, let Sl∗ be the maximally compact set at level l,
containing either one or 2 elements, and let
X
rl = Mrl + N(rl + 1) .
(5.11)
rl ∈Sl∗
Any other set Sl with the same mean must by definition contain at least one element
(1)
rl
(1)
such that either rl
(1)
> rl + 1 or rl
< rl . (If there is only one element in Sl∗ then
of course both limits are given by rl ). Let us assume that the former case holds, and
(1)
that there exists an rl
= rl + 1 + x for some integer x. As both sets have the same
average, this means that
X
(1)
rl ∈Sl ,rl 6=rl
rl = Mrl + (N − 1)(rl + 1) − x
(5.12)
In other words, the mean of the remaining elements must be shifted correspondingly
(2)
to the left. Picking one of the shifted elements rl
(1)
if Sl∗ contains only rl ), create Sl
≤ rl (where strict inequality holds
(1)
by shifting rl
(1)
→ rl
(2)
− 1, rl
(2)
→ rl
+ 1 and
leaving all other elements of Sl unchanged. Then
(1)
(1)
(2)
(1)
(2)
E(Sl ) − E(Sl ) = V (rl ) + V (rl ) − V (rl − 1) − V (rl + 1 > 0
(5.13)
139
(1)
where the inequality follows from the convexity of the potential, since for rl
(1)
the elements of Sl
(2)
> rl
are more narrowly distributed about the mean than the elements
(1)
of Sl . This construction can be repeated so long as some rl
> rl + 1 exists – in other
words, so long as Sl 6= Sl∗ ; at each step in the construction the new configuration
has lower energy. Since this holds for all l, the optimal solution is characterized by
S ≡ {Sl }l with each Sl maximally compact.
Constructing the CGS
Hubbard [80] gives a construction which, for any rational filling p/q, constructs a
configuration of particles on the lattice for which every set of lth neighbor distances
is maximally compact. A thorough discussion of this solution is given in [98]; here
we present an overview.
Hubbard’s solutions are periodic, and can be expressed in the form:
n1 n2 ...nj
(5.14)
where ni represents the spacing between the ith and (i + 1)st occupied sites. Given a
continued fraction {n0 , n1 , ..., nk }, let
pi
= 1/(n0 + 1/(n1 + ... + 1/ni ))
qi
(5.15)
be the numerator and denominator of the related continued fraction {n0 , n1 , ...ni }, i <
k. Now consider
X0 = n0
Y0 = n0 + 1
(5.16)
with
Xi = (Xi−1 )ni −1 Yi−1
Yi = (Xi−1 )ni Yi−1
(5.17)
140
Filling CGS particle soliton
1/q
q
q−1
2/5
23
22
3/7
223
222
5/12 23223
22322
hole soliton
q+1
33
323
32323
Table 5.2: Occupancy patterns in a few CGS
for all i > 1. The Hubbard solution is given by Xk . A few examples of these CGS
states and their solitons are given in Table 5.3.1.
One can show by induction that this solution has the correct filling fraction. The
numerator and denominator of the full continued fraction can be expressed in terms
of numerators and denominators of shorter continued fractions, via [99]
pk = nk pk−1 + pk−2
qk = nk qk−1 + qk−2
(5.18)
Using (5.16), we have X0 = n0 which contains one particle per n0 sites; Y¯0 = n0 + 1
contains one particle per n0 + 1 sites. Equation (5.17) states that if Xi−1 contains
pi−1 particles in qi−1 lattice sites, and Yi−1 contains pi−1 + pi−2 particles in qi−1 + qi−2
lattice sites (with p−1 = 0, q−1 = 1), then Xi contains ni pi−1 + pi−2 = pi particles in
ni qi−1 + qi−2 = qi lattice sites. Yi contains (ni + 1)pi−1 + pi−2 = pi + pi−1 particles in
(ni + 1)qi−1 + qi−2 = qi + qi−1 sites. Thus at every step in the process the solution
Xi has filling fraction
pi
qi
corresponding to the truncation of the continued fraction
expansion at ni .
Further, it is easy to convince oneself by inspection that these solutions do indeed
satisfy Hubbard’s criterion that Sl = {rl , rl + 1}5 .
5
A proof of this fact can be found in [98]
141
5.3.2
The Devil’s staircase
We have seen how Hubbard’s construction gives the ground states of any infiniteranged convex potential at rational filling fractions. We now address the issue of
these states’ stability. Bak and Bruinsma [84] showed that the range of µ over which
each CGS is stable is given by
∞
X
n=1
At a filling fraction of
p
q
nq
nq
2nq
+
−
(nq + 1)3 (nq − 1)3 (nq)3
.
(5.19)
the result is independent of p, and falls off sharply as a
function of q. At t = 0 these intervals cover the entire range of µ pertinent to fillings
less than unity, giving the devil’s staircase structure.
To understand Eq. (5.19, one must consider the effect of adding or removing a
single particle from the CGS. We shall see that at filling fraction
p
q
this produces q
fractionally charged solitons of charge 1/q, each of which distorts the periodic ground
state by altering the distance between one pair of adjacent particles by 1. (For every
commensurate state, there is a unique distortion of this type which minimizes the
potential energy). This results in a q-soliton state (qSS).
5.3.3
Structure of the q-Soliton State
To understand why charge fractionalizes, consider the possible Sp after one charge is
added or removed from a chain of length L containing N particles. At rational filling
N/L = p/q, for l 6= 0 mod p, we have
rlij ∈ {rl , rl + 1}
(5.20)
for all lth neighbors i and j in the HUP CGS. For any value of l, the CGS must satisfy
Nrl + Nrl +1 = N
Nrl rl + Nrl +1 (rl + 1) = lL
(5.21)
142
where Nrl , Nrl +1 are the number of lth neighbor pairs separated by distance rl and
rl + 1, respectively. After adding or removing a particle, the most energetically
favorable lattice configuration requires Sl be maximally compact for every l. For
l < p, this requires that the set of possible radii {rl , rl + 1} remain unchanged; for
l = p, however, we must now assume Sp = {q, q ± 1}. These new pth neighbor
separations constitute the solitons of the qSS. After adding a particle,
Nr′l + Nr′l +1 = N + 1
Nr′l rl + Nr′l +1 (rl + 1) = lL
(5.22)
giving the net change in the distribution of lth neighbor distances:
Nr′l − Nrl = ±(rl + 1)
Nrl+1 − Nr′l +1 = ±rl
(5.23)
where +, − correspond to adding or removing a particle, respectively. Substituting
rp = q (one charge removed) or rp = q − 1 (one charge added) into (5.23) shows that
the qSS contains exactly q solitons. Thus each soliton has a positive (or negative, in
the case of holes) charge of 1/q with respect to the parent lattice.
In a finite system, the qSS is also a HUP state with rational filling, obtained
by distributing the q solitons as evenly as possible on the chain in order to satisfy
Hubbard’s criterion at all l. The denominator of such a state will be of the order
of the number of lattice sites. It is useful to conceptualize filling fractions whose
denominators are comparable to the system size as being qSS states of a related CGS
of smaller denominator.
Energetically speaking, in a finite system one must also account for the repulsion
between solitons; for most of what follows, we will drop these terms and consider the
infinite volume limit in which solitons are infinitely far apart. In this limit the HUP
qSS consists of q free solitons in the ‘background’ lattice of the parent HUP state.
143
5.3.4
Energetics of the qSS
We now turn to the question of the energetics: over what range of chemical potential
is each CGS solution stable against the formation of solitons? Eq. (5.23) above shows
that the energy costs of adding and removing particles are, respectively,
E+ = −µ +
+
X
l=nq
E− = µ +
+
X
l6=0(mod p)
nqV (nq − 1) − (nq − 1)V (nq) + ...
X
l6=0(mod p)
X
l=nq
[(rl + 1)V (rl ) − rl V (rl + 1)]
[rl V (rl + 1) − (rl + 1)V (rl )]
nqV (nq + 1) − (nq + 1)V (nq) + ...
(5.24)
where we ignore soliton repulsion terms present in a finite system. Setting E± = 0
in Eq. 5.24, we obtain the values µL (p/q) and µR (p/q) of the chemical potential at
the left and right extremities of the plateau. The width of the plateau (Eq. 5.19) is
given by the difference between these. The sum over l 6= 0(modp) contributes equal
amounts to µL and µR , since lth neighbors separated by rl and rl + 1 both exist in
the initial solution, and creating solitons simply adjusts their relative frequencies.
However, the solitons replace some rp by rp ± 1 – giving different contributions to
particle and hole like solitons. This gives the width of the plateau quoted in Eq.
(5.19):
µR (p/q) − µL(p/q) =
X
n
nq [V (nq − 1) + V (nq + 1) − 2V (nq)]
(5.25)
which, by the assumption of convexity, is strictly positive. Since rnp = nq, the range
of stability depends only on q.
On the infinite-length chain, the function ν(µ) has a complete Devil’s staircase
structure. This means firstly that the function is monotonic and contains no finite
jump discontinuities [98]. Secondly, the set of all such intervals for rational fillings
144
p/q with q > 1 completely covers the interval 0 ≤ µ ≤ E− (ν = 1), and in the infinite
chain limit all ground states are periodic HUP states at rational fillings.
5.3.5
Proof of devil’s staircase structure
Here we examine a few of the details required to show the devil’s staircase structure.
The fact that ν(µ) has no jump discontinuities essentially follows from the fact that
between any pair of rationals on the real line, there is another rational. We thus
focus on the second claim – that all ground states between ν = 0 and ν = 1 are
commensurate (except possibly on a set of measure 0). To do so, it suffices to show
that the intervals of stability are disjoint, and that the sum of their lengths is the
length of the relevant interval in µ.
To show that the ranges of µ over which each rational filling is stable must be
disjoint, we first point out that as the potential V is a convex function of the interparticle spacing, the energy E is a convex function of the filling fraction. Indeed,
were this not so, it would be energetically favorable at some filling fraction to break
a system of length L at filling
p
q
into(say) two subsystems of filling
p
q
+ x and
p
q
−x –
which, as we know from Hubbard’s solution, does not occur. This is because if ǫ(ρ)
is the energy per unit length of the state at filling ρ, then:
p
p
p
2ǫ( ) < ǫ( + x) + ǫ( − x)
q
q
q
(5.26)
and the energy is a convex function of filling fraction.
For any p′ /q ′ < p/q, choose l such that p′ /q ′ = p/q − (l + 1)/L, so that the state
at filling (N − l)/L contains
p′
L+1
q′
particles. The upper boundary of stability of the
state p′ /q ′ then occurs at the chemical potential where the energy of this state is 0.
Let E(N/L) denote the lattice contribution to the energy of the state at filling N/L,
145
such that the total energy of this state is µN + E(N/L). We have:
′
N −l
N −l−1
p
= E
−E
µR
q′
L
L
p
N
N −1
= E
−E
µL
q
L
L
′
p
p
N
N −l−1
N −1
N −l
µL
− µR
= E
+E
−E
−E
q
q′
L
L
L
L
(5.27)
which is positive-definite by the convexity of E as a function of filling. Similarly, for
p′
q′
> pq , we choose p′ /q ′ = p/q + (l + 1)/L
′
p
N +l+1
N
N +1
N +l
p
− µR
=E
+E
−E
−E
µL
q′
q
L
L
L
L
(5.28)
which is again non-negative by convexity. Thus the intervals of stability of different
rational fillings do not overlap.
It remains to show that the sum of the intervals of stability of all rational fillings
completely covers the range 0 ≤ µ ≤ µ(1)L. According to (5.24), the chemical
potential µ(1)L at which the ν = 1 state becomes unstable is:
∞
X
µ(1)L =
(n + 1)V (n) − nV (n + 1)
n=1
∞
X
= 2
V (n)
(5.29)
n=1
To compute
P
x∈Q (µR (x)
− µL (x), we must add together the length of the interval
for all possible denominators q, multiplied by the number of reduced fractions with
this denominator. The function which gives this multiplicity is the Totient function
φ(q), which counts the number of integers less than q which are relatively prime to q.
Hence the combined length of all intervals is
X p
p
− µL
I ≡
µR
q
q
=
p/q
∞
X
q=2
φ(q)
X
n
nq [V (nq − 1) + V (nq + 1) − 2V (nq)]
(5.30)
146
Collecting the contributions to V (r) for each r gives


X X
X
X

I=
φ(q)(r + 1) +
φ(q)(r − 1) − 2
φ(q)r  V (r) .
r
q|r+1
q|r−1
(5.31)
q|r
Using the identity6
X
φ(q) = r
(5.32)
q|r
gives
I =
X
r
= 2
(r + 1)2 + (r − 1)2 − 2r 2 V (r)
X
V (r)
(5.33)
r
which is precisely the boundary of stability of the ν = 1 state, establishing the desired
result.
5.4
Away from the classical limit: Mott-Hubbard
transitions in the strong coupling expansion
The physics of the Mott transition in one dimension has been studied in the context of
classical phase boundaries, as the commensurate-incommensurate transition, as well
as in quantum mechanics via the Hubbard and extended Hubbard models. Basically,
as quantum or thermal fluctuations in particles’ positions increase, commensurate
order is destroyed by a condensation of solitonic defects. In the quantum mechanical
case, these defects cost potential energy, but are favored by kinetic terms, driving
a transition at some finite hopping on the lattice. This physical mechanism is also
responsible for a commensurate- incommensurate transition in systems with infiniteranged convex interactions. However, the fractal structure of the classical ground
states leads to a phase portrait considerably more complex than in cases studied
previously. Here we will examine the structure of this phase portrait.
6
Found on Wikipedia!!
147
The qualitative behavior of our system in the convex regime is reminiscent of the
Bose-Hubbard model, with commensurate Mott lobes ceding to superfluid states as
t increases. It is convenient to treat a state with large q as a state with smaller q
at a nearby filling in which a crystal of dilute solitons has formed. Hopping tends
to liquify dilute crystals of solitons: at large separation the inter-soliton repulsion is
smaller than the kinetic energy gained from delocalization. (The latter grows as 1/r 2 ,
and the former as
1
,
r3
at large separations). The delocalized solitons destroy long-
ranged spatial order, creating a Luttinger liquid with full translational symmetry.
Hence, as t increases, the system undergoes a transition from the Mott insulating
CGS to a Luttinger liquid state, with larger q states liquifying at smaller t.
5.4.1
Strong Coupling Expansion
To find the position of the phase boundary, we generalize the method of Ref. [85]
and compare the energies of the CGS and its adjacent qSS to third order in t using
standard time-independent perturbation theory. This approach assumes that the
phase transition is continuous, so that for a given t > 0, values of µ for which
EqSS (µ) = ECGS (µ) constitute the phase boundary. This assumption is well-founded,
since the soliton repulsion ensures that the energy cost of creating multiple solitons
is larger that of a single soliton, thus favoring a second-order transition.
In a finite system one must account for the repulsion between solitons; here we
drop these terms and consider the infinite volume limit. In the limit that the solitons are sufficiently well separated that we may neglect their interactions, the qSS
is highly degenerate and can be expressed in terms of a band of solitonic momentum eigenstates. Here we consider only the bottom of the band, which lies at zero
momentum.
To find the transition, we compare the energies (calculated up to third order in
t) of the CGS (which is the t = 0 ground state) and the qSS. The calculation of
148
the energies is carried out using standard time-independent third order perturbation
theory. The energy corrections are calculated in terms of the t = 0 ground and excited
states of the CGS and qSS.
Perturbation theory in the CGS
To calculate these energy corrections, we must consider excitations about the classical CGS generated by perturbative hopping, and their matrix elements with the
unperturbed state. The zeroth order CGS is given by the HUP solution |ψ(CGS)i,
which is non-degenerate (up to global translations). Hopping creates excited states
of the form:
(0)
|ψex
(x)i = b†x bx+1 |ψ(CGS)i
At filling
p
q
(5.34)
there are exactly 2p distinct such hoppings which must be considered:
one in each direction for each occupied site in the HUP unit cell.
To calculate the energy corrections, we must calculate matrix elements of the
ground and excited states with H1 , and the energy differences δEi between the corresponding classical ground and excited states. Note that the first order wave function
contains only terms which can be transformed into the classical ground state by only
one hopping – in other words, only the excited states of the form (5.34) and (5.37)
have non-zero contributions. The details of the perturbative approach are outlined
in Sect. 5.9.1; here we will only state the results.
The first and third order corrections to the CGS energy are zero, because the CGS
is non-degenerate and hence any odd number of hoppings produces a state orthogonal
to the ground state. At ν = p/q, the second-order correction is given by
(2)
ECGS = −2
(0)
where ∆Ei = Ei
(0)
− E0
p
Nt2 X 1
p i=1 ∆Ei
(5.35)
is the difference in potential energies between the ground
and the excited state formed by hopping from the ith occupied site in the ground state
149
configuration. As the ground state is periodic, it suffices to calculate these energies
for the p distinct particles in the repeated pattern.
Perturbation theory in the qSS
We now repeat this analysis for incommensurate fillings, which have solitonic ground
states. The qSS is given at zeroth order by the HUP qSS, consisting of q HUP solitons
sufficiently far apart that soliton-soliton interactions can be ignored. The energy of
the qSS is then simply q times the energy of a single soliton on an infinite lattice.
The qSS ground states are highly degenerate in this limit, since all translates of each
soliton have equal energy. We therefore work in the basis in which the hopping term
is diagonal, namely:
(0)
|ψqSS (k)i =
X
x
eikx |ψqSS (x)i
(5.36)
where the sum runs over occupied lattice sites, and |ψqSS (x)i is the state containing
one soliton beginning at lattice site x. The soliton hops by q sites when a single boson
on one of its edges is hopped by one site; hence the position x is an integer multiple of
qa. The perturbation theory is essentially the same for qSS states containing solitons
and anti-solitons; functionally the difference between these is in the energy gaps to
the local excitations.
Local excitations, or defects, of the qSS are again generated by hopping one particle away from its preferred position. These can be expressed in the form:
|ψr (k)i =
X
x
(0)
eikx b(x + r)b† (x + r ± 1)|ψqSS (k)i
(5.37)
The state ψr (k) describes correlated propagation of a soliton and a defect r lattice
sites away7 . Note that for each r the defect has 2 possible orientations, depending
on whether the hopping has been towards or away from the defect.
7
In the following the word soliton applies strictly to the HUP solitons described in section (5.3.1);
lattice distortions due to other hoppings we will call defects. The two are indeed different as solitons
carry global topological charge, while defects – which constitute local re-arrangements of the charge–
do not.
150
Computing the resulting matrix elements gives the energy corrections:
(1)
EqSS = −2qt cos(kqa)
(2)
EqSS
(3)
EqSS
N/q
X
X
t2
t2
−q
= 2q cos(2kqa)
∆Er1 ,−1
∆Eri ,α
i=1 α=±1
cos(2kqa)
cos(2kqa)
3
−
= −2qt cos(kqa)
∆Er1,−1 ∆Er1,−1 +q (∆Er1,−1 )2
N/2 X
1
1
3
−2qt cos(kqa)
−
2
(∆E
)
∆Eri ∆Eri+1
r
i
i=1
(5.38)
Here k is the soliton momentum, a is the lattice constant, and q is the denominator of
the commensurate filling fraction, which appears here because one hopping displaces
the soliton by q lattice spacings. The subscripts on Eri ,±1 indicate the distance
between particle i and the soliton, and the direction of the hopping relative to the
soliton. The special distance r1 describes an excited state in which an anti-soliton
is sandwiched between 2 solitons (for a solitonic qSS), or vice versa (for a qSS with
anti-solitons). These states contribute extra terms to the energy corrections of the
qSS because of the ambiguity as to which soliton is associated with the ground state
qSS.
Fig. 5.1 shows the results of the perturbative calculation for selected filling fractions. The t = 0 axis corresponds to the classical limit, in which the CGS states
comprise a complete devil’s staircase: every value of µ corresponds to a rationally
filled ground state, except for a set of measure 0. The figure shows the resulting Mott
lobes: inside each lobe the CGS is stable and the system is in a Mott insulating state.
The Mott gap vanishes on the boundary of the lobe; outside of this region solitons
proliferate and the system is in a Luttinger liquid phase.
For any t > 0, only a finite number of insulating states exist; the rest are liquid
states with a superfluid of condensed solitons. The function ν(µ) is no longer a devil’s
staircase, but rather a piecewise smooth function, with plateaux of constant density
separated by liquid phases whose density varies continuously with µ. The size of the
151
U/V
0
0.4
3/7
2/5
0.35
µ/V
0
0.3
0.25
1/3
0.2
0.15
2/7
0.1
0.05
0
0
1/4
1/5
1/6
0.005
0.01
0.015
0.02
0.025
0.03
0.035
t/V
0
Figure 5.1: Perturbative calculation of the Mott to SF phase boundary in the (t, µ)
plane, shown here for U = 20. Here the strength of all couplings is measured relative
to that of the dipolar interaction strength V0 . Each lobe encloses a Mott insulating
region in which the filling is fixed; the region outside the lobes is a superfluid of
solitons. The 1/3, 1/5, 1/6, 2/7, 2/7 and 3/7 -filled lobes are shown here. Every
commensurate state of the complete Devil’s staircase has a Mott lobe, but the range
of hoppings over which a state exists falls off sharply with its denominator.
152
commensurate region decreases sharply with q: states of higher q have both smaller
ranges of stability in the classical limit, and larger energy corrections relative to the
CGS, so that the volume of the corresponding Mott lobe scales approximately as
1/q 5 .8
The total volume occupied by liquid states can be estimated from the first-
order approximation to the Mott lobe boundaries; we find that for small but fixed t
the volume of the liquid region scales as approximately t2/5 . (Details of the calculation
are given in Sect. 5.9.2). At sufficiently large t we expect all insulating states to be
unstable, and the particle density to vary smoothly with µ.
5.4.2
Bosonization treatment of the phase transitions
The qualitative nature of the Mott transition can be deduced from existing knowledge
of 1D commensurate-incommensurate phase transitions, which we summarize here.
We approach the phase transition from the Luttinger liquid side, where bosonization
can be used to treat the kinetic term and dipolar interactions exactly. The lattice
potential can be added perturbatively; when the coupling of the density to the lattice
becomes relevant, this signals the Mott transition.
In bosonized form, our system is described by a Hamiltonian that is a sum of a
Luttinger piece, accounting for the kinetic term and dipolar interactions
9
, and a
sine-Gordon term which emulates the lattice in the continuum limit. We also include
a parameter δ, which parametrizes the deviation from commensurate filling. Close to
filling pq , the bosonized Hamiltonian is thus:
Z
h
i
1
u
2
2
H=
dx uK(πΠ(x)) + (▽φ(x)) + g cos(2qφ(x) − δx)
2π
K
Here φ is related to the density of the physical bosons by
X
1
ρ(x) = ρ0 − ▽ φ(x)
ei2n(πρ0 x−φ(x))
π
n
8
(5.39)
(5.40)
This scaling uses a first-order in t approximation to the boundaries of the Mott lobe.
Though the potential is infinite ranged, it falls off quickly enough that the qualitative description
is identical to that for short-ranged interactions [100]. This is true for any long-ranged potential
with a finite Fourier transform at 0 momentum.
9
153
and Π is the momentum conjugate to φ. u and K are the usual Luttinger parameters,
expressible in terms of the hopping and interaction terms of the original system. The
sine-Gordon term adds a periodic lattice potential
Z
2π
dxρ(x)V cos( x)
a
(5.41)
to the model; g parameterizes the strength of the coupling to this lattice.
Near commensurate densities ρ0 =
1
qa
+ δ, the expression (5.40) for the density
shows that the slowly varying modes are precisely those for which
n=q
(5.42)
where n is the resonant harmonic from the density (5.40). The resulting expression
for the coupling (5.41) to the lattice is
Z
V
dx cos(2qφ(x) − δ) .
For densities
p
q
we obtain the same expression, since the state at filling
(5.43)
p
q
consists
of several charge density wave instabilities at wave vector q, whose relative positions
also become pinned inside the Mott region.
The Hamiltonian (5.39) is well studied in both the context of the Mott transition
[100] and the Frenkel-Kontorowa model of surface interfaces [81]. The upper and
lower Mott lobes join in a cusp which is not accurately described by the perturbation
theory– since ‘small’ hopping implies that the ratio t/δE of the hopping relative to
the energy gap to the solitonic states must be small, the range of t over which the
perturbative treatment is valid decreases with q and never encompasses this point.
The bosonized treatment reveals that crossing the edge of the Mott-Hubbard lobe
induces one of two different types of phase transitions. At the cusp joining the upper
and lower Mott lobes, a constant-density phase transition, with δ = 0 in Eq. (5.43)
of the Kosterlitz-Thouless type occurs. For δ 6= 0 the transition is well described by
a simple two-band model with quasi-particles that are gapped in the commensurate
√
phase, and with a density increasing as µ − µc near the transition on the liquid side.
154
Derivation of the Luttinger description near the phase transition
To derive these results, the Hamiltonian (5.39) must be treated using a perturbation
series in g. This is easy to do for δ = 0. One finds that for K < Kc , the cosine
term becomes relevant at low energies, and locks the state into a crystalline phase
commensurate with the lattice. For K > KC , so-called domain walls (in our case,
lattice solitons and anti-solitons) proliferate, and the phase becomes a liquid. In the
renormalized Hamiltonian, this predicts a transition from the liquid to the solid phase
at t ∼ 1/q 4 at filling pq , consistent with the perturbative results of Section 5.4.1. At the
transition, K jumps discontinuously from the universal value Kc = 1/2 (approaching
from the liquid state) to 0 in the gapped Mott phase. This is the standard signature
of a Kosterlitz-Thouless transition.
A phase transition with δ = 0 occurs only for the value of µ for which the model is
particle-hole symmetric. For non-zero δ, the situation is similar, but at constant µ the
renormalization of the doping flows to strong coupling before the phase transition;
hence the transition is not accurately described by perturbation theory about the
Luttinger Hamiltonian. In this case the transition is best studied using the LutherEmery solution– that is, re-scaling the problem and exploiting the fact that it can be
mapped onto a system of weakly interacting fermions (in our case actually the lattice
solitons) with an upper and lower energy band separated by a gap.
To do this, we first scale the fields in Eq. (5.39) according to φ̃ = qφ + δx/2, Π̃ =
Π/q. Then
1
H=
2π
Z
u
2
2
˜
˜
˜
dx uK̃(π Π(x)) + (▽φ(x) − δx/2) + cos(2φ(x))
K̃
where K̃ = Kq 2 . The C-IC transition occurs at a value of K =
1
,
2q 2
(5.44)
or in this picture
K̃ = 1/2. Here we will drop all˜, and work in the scaled system.
Second, we translate the Luttinger Hamiltonian (5.44) back into the language of
155
spinless fermions:
H =
X
k
vF (c†Rk cRk − c†Lk cLk ) + ∆(c†Rk cLk + c†Lk cRk )
+g2 (ρR (k)ρL (−k) + H.C) + g4 [ρR (k)ρR (−k) + ρL (k)ρL (−k)] (5.45)
This is convenient because the potential cos 2φ due to commensuration effects has
been mapped to the quadratic term c†R cL + h.c.. Thus at K̃ = 1, along the LutherEmery line where the system is particle-hole symmetric, the interaction terms g2 and
g4 vanish and we have:
H0 = vF k(c†Rk cRk − c†Lk cLk ) + ∆(c†R cL + h.c.)
(5.46)
p
This simply describes free massive fermions, of energies ǫ = ± vF2 k 2 + ∆2 , with the
mass gap ∆ is set by the coupling of the cosine term. For K̃ 6= 1, Schulz [81] showed
that a parametrization can be chosen such that the g4 term vanishes, and the coupling
of the remaining interaction is proportional to the doping away from commensuration,
which vanishes at the phase transition. Thus the non-interacting model gives a good
description of the physics very near the phase transition.
In practice, the quasi-particles in (5.46) may be identified with Hubbard solitons
(or distortions of the commensurate lattice) which become gapless and proliferate
at the transition. In the solid phase, the solitons are gapped, and do not occur at
sufficiently low temperatures. At the phase boundary the chemical potential crosses
the bottom of the upper band, and solitons proliferate. Near the transition on the
√
liquid side, this predicts d ∼ µ − µc with µc the chemical potential at the phase
transition.
5.5
Effects of the Trapping potential
In practice, any experimental realization of the HUP model will involve a finite sized
atom trap, generally of length not more than a few hundred lattice sites. In addition,
156
the trapping potential is not generally flat, but rather is well approximated as a
harmonic potential. Here we will try to address the possible effects of this trap on
the system.
We can estimate the effects of a harmonic trapping potential (present in current
cold-atom experiments) using the local density approximation – this assumes that
the trapping potential is slowly varying enough that it can be simply incorporated
into the chemical potential, resulting in a spatially-dependent µ. Trajectories along
the 1D chain then correspond to cuts in Fig. 5.1 at fixed t/V0 . Thus, at t = 0 we
expect different commensurate fillings at different points along the trapped chain,
with the most stable states (fillings 1/2 and 1/3) occupying the largest regions within
the trap. However, commensurate states with a period q greater than the lengthscale
over which the trapping potential varies cannot exist in the trapped system since they
violate the local density approximation. The situation is improved slightly when t
is nonzero but small, where small islands of Mott states at various fillings p/q are
separated by regions of superfluid. These fluid regions can interpolate continuously in
density between the two commensurate states. We will discuss the resulting density
profiles for various chemical potentials. In doing so, we ignore the CDW correlations
of the liquid states, which near filling p/q also occur at length scale q; hence this
treatment is a modest improvement of the LDA.
5.5.1
t = 0 physics and the LDA
The simplest approach to determining the density profile in a finite harmonic trap is
to use an effective Local Density Approximation (LDA) treatment. In other words,
given an approximation to the function ν(µ) as calculated by [84], one can generate a
profile of the filling ν(x) as a function of position, x, in the trap, treating the trapping
potential as a spatially varying chemical potential. In this approach, we neglect the
fact that each filling p/q requires at least q lattice sites to be realized, and that the
157
range of x over which ν = p/q may well encompass less than q sites.
µ 0 =.2
0.37
0.33
0.36
0.32
0.35
µ
µ
0.34
0.31
0.3
µ 0 =0.3
0.34
0
100
200
0.33
300
0
100
x
µ 0 =0.4
0.43
300
200
300
µ 0 =0.5
0.5
0.49
0.42
0.48
µ
µ
0.41
0.4
0.47
0.46
0.39
0.38
200
x
0.45
0
100
200
x
300
0.44
0
100
x
Figure 5.2: Local density approximation (based on the HUP solution for infinite µ)
for a trapping potential V (x) = (x − x0 )2 /(L/4)a2 − µ0 (where L is the total number
of sites in the trap) shown for various values of µ0 . We expect the large plateaux of
small denominator states to remain as features in the actual finite-volume solution,
but that most smaller plateaux will disappear into regions of ambiguous density.
Figure 5.5.1 shows sample LDA profiles for various values of the trap minimum µ0 ,
taken from a numerical computation of ν(µ) cut off at q = 200. For some parameter
choices, most of the trap lies in a chemical potential range that is strongly locked at a
commensurate filling with low denominator, and the LDA gives a plausible rendering
of the density profile over much of the trap. However the figure also shows clearly
that for some regions of the trap the LDA is a very poor approximation, as it predicts
158
fillings ν = p/q with q much too large to fit into the number of lattice sites over which
the state is stable.
In the classical limit, another approach is to simulate the density profile numerically. We have used simulated annealing to generate profiles; Figure 5.5.1 shows
numerically generated profiles for the same trapping potentials as in Figure 5.5.1.
Note that in regions of transition between different commensurate states, the exact
density is somewhat ambiguous, and can vary depending on what technique is used
to calculate it.
Though neither of the techniques discussed above proves particularly illuminating
in the transition areas between plateaux of relatively small q states, they illustrate
some practical features of the HUP system in a trap. First, the spatial homogeneity
of the density depends strongly on the depth of the trap. Since the range in µ over
which the lowest q states (ν = 1/2, 1/3...) are stable is much larger than the ranges of
stability of higher denominator states, a trap with a modest curvature can be ‘locked’
everywhere at a density of 1/2. As µ0 is shifted, however, portions of the trap fall
outside the range of half-filling, and since all of the nearby states have much larger
denominators, their ranges of stability will be correspondingly much smaller. In a
trap of only a few hundred sites, for the most part these states will not be stable over
a wide enough range to produce a strong scattering peak. Hence it is the strongly
locked regions which are the most visible experimentally, and these, fortunately, are
well -described both by LDA and simulated annealing techniques.
5.5.2
Spatial Profiles of Atoms in a harmonic trap at finite t
We have seen that the LDA gives a poor description of the expected profiles for highdenominator filling fractions in a finite trap. However, in practice such states would
not in any case exist in any currently attainable experiment. Hence in practice we
expect density profiles which consist of phase separated regions of density-locked Mott
159
µ0=−0.2
0.34
µ0=−0.3
0.38
0.3
0.36
ν
0.37
ν
0.32
0.28
0.35
0.26
0.34
0.24
0
100
200
0.33
300
µ0=−0.4
0.55
0
100
300
200
300
µ0=−0.5
0.55
0.5
200
0.5
ν
ν
0.45
0.45
0.4
0.4
0.35
0.3
0
100
200
300
0.35
0
100
Figure 5.3: Density profiles calculated using simulated annealing for the same potentials as in Figure 5.5.1. The multiple lines show multiple runs of the simulation,
indicating that the states at low-denominator filling fractions are very robust, while
states at higher values of q (q > 4) are energetically delicate and require simulations
over longer time periods than those undertaken here.
160
states interspersed with compressible fluid states whose density varies continuously
with the trapping potential.
The qualitative properties of the quasiparticle fluid, at least near the transition,
are well described by the Hamiltonian (5.46). To describe the density of the Luttinger
liquid regions, we make a quantitative mapping between the 2-band model (5.46), and
the perturbative description of the quasi-particle fluid valid for small t.
If we ignore the weak interactions between quasi-particles, this effective description has 2 free parameters: the effective Fermi velocity in the upper band, and the
band gap. By calculating these parameters using the perturbative hopping model, a
quantitative matching between the Luther-Emery solution described in Sect. 5.4.2
and our system can be achieved. We will use this model to generate profiles of particle
densities in the trap at finite filling.
The band gap ∆ is given simply by the difference in energies of the IC states with
one extra particle and one extra hole. At t = 0 this is just the devil’s staircase pattern
calculated by [84]; at finite t it can be extracted from the perturbative calculation of
Sect. 5.4.1.
To match the Fermi velocities, we match the true free soliton Hamiltonian:
H (t) = s† (k) [∆ + µ + t(1 − cos(qak))] s(k)
(5.47)
onto the effective Hamiltonian
H (S) = s† (k)
p
(vk)2 + ∆2 s(k)
(5.48)
where v is the Fermi velocity in the original 2-band model.
Both H (t) and H (s) are quadratic in k near the bottom of the upper band. The
effective model describing the dynamics of particles in the upper band only is obtained
in both cases by linearizing the spectrum about kF in the upper band. This gives:
H˜(t) = s† (k) [∆ + µ + t(1 − cos(aqkF )) + kaqt sin(aqkF )] s(k)
p
H˜(s) = v 2 kF k/ (vkF )2 + ∆2
(5.49)
161
Of course µ is such that the constant terms in the first equation cancel, and both
energies are linear in k. Hence we can match:
v 2 kF /
p
(vkF )2 + ∆2 = qat sin(qaKF )
(5.50)
To lowest order in kF (valid at low quasiparticle densities) this gives:
v 2 = (qa)2 ∆t
(5.51)
We may use this result to calculate the density of solitons in the IC region near
the phase transition. In particular, if we ignore quasiparticle interactions, we have
Z µ
L
ǫ
√
N =
dǫ
2
2πv ∆ ǫ − ∆2
L p 2
=
µ − ∆2
2πv
(5.52)
where in the non-interacting model ∆ = µc is the chemical potential at the transition,
which can be estimated using the perturbative calculation of the phase diagram.
(Interactions will renormalize the gap in principle, though in our case we calculate
the gap perturbatively, and this result should be accurate for sufficiently small t.)
Substituting in the value of v obtained in equation (5.51), and accounting for the
fact that each soliton has a charge density of
1
q
relative to the background charge
density of the lattice, we obtain the charge density near the C-IC transition:
ρ(q) =
2πq
1
√
2
µc t
p
µ2 − µ2c
(5.53)
This gives the desired expression for the density of quasi-particles as a function of the
parameters of the original HUP Hamiltonian.
5.5.3
Density profiles
We now use the results of the previous section to estimate the density profiles in a
realistic trap, by using a finite t LDA. This scheme first identifies the commensurate
162
µ0=0.2
0.35
µ0=0.3
0.4
0.3
0.25
ν
ν
0.2
0.15
0.2
0.1
0.05
0
0
100
0
300
100
x
µ =0.4
µ =0.5
0.4
0.4
ν
0.45
0.35
200
300
200
300
0
0.5
0.45
0.3
0
x
0
0.5
ν
200
0.35
0
100
200
x
300
0.3
0
100
x
Figure 5.4: Density profiles in a harmonic trap at t/V = 0.001. Stable plateaux
can be seen at 1/2, 2/5, 1/3, and 1/4 filling, separated by Luttinger liquid regions of
continuously varying density.
163
µ0=0.25
0.35
µ0=0.35
0.35
0.25
0.25
0.2
0.2
ν
0.3
ν
0.3
0.15
0.15
0.1
0.1
0.05
0.05
0
0
100
200
0
300
100
x
µ =0.45
µ =0.55
0
0.5
0
x
200
300
200
300
0
0.6
0.5
0.4
0.4
0.3
ν
ν
0.3
0.2
0.2
0.1
0.1
0
0
0
100
200
x
300
−0.1
0
100
x
Figure 5.5: Density profiles in a harmonic trap at the experimentally realizable value
t/V = 0.02. Here only the 1/2 and 1/3 filled commensurate states are stable.
164
regions of the trap; since for the values of t considered here all but a few very lowdenominator fractions are unstable, the problem of patterns too short to fit into the
allotted number of lattice sites does not arise. At the edges of each commensurate
region we use (5.53) to predict the local density profile; a polynomial interpolation is
used to join the various liquid regions.
Figure 5.5.3 shows typical charge density profiles in a trap t/V = 0.001. Commensurate plateaux at filling fractions 1/2, 2/5, 1/3, and 1/4 can be stabilized, depending
on the chemical potential at the bottom of the trap. Between these plateaux we see
regions of Luttinger liquid.
Figure 5.5.3 shows density profiles for t/V = 0.02, the value in principle attainable
by experiments on polar molecules.
Though we do not claim to predict accurately the density profile deep in the liquid
region, where both soliton interactions and lattice commensuration effects have a
strong impact on the solution, Figure 5.5.3 gives an accurate representation of the
commensurate regions and their immediate vicinity. Experimentally speaking, the
most interesting features of these profiles are the spatially separated commensurate
plateaux, which even at t/V = 0.02 can cover a significant fraction of the trap’s
volume, and hence should be experimentally visible in the structure factor.
5.6
Departures from Convexity
The PUH CGS are the classical ground states so long as the potential is everywhere
convex. Since the on-site potential U is tunable experimentally, it is interesting to
ask what happens to these states as U is lowered away from convexity and double
occupancies begin to form. Of course, as U is lowered still further, triple and higher
occupancies will also form. However, as the barrier to triple occupancies is 3U,
instabilities towards triple occupancy at a given filling will set in at approximately
165
one third the value of U for instabilities to double occupancy. For U in the range of
relevant for interesting physics about the 1/2-filled state which we will describe below,
for example, triple occupancies will not be favored at any filling fraction. Hence we
will not analyze triply occupied states here—the doubly occupied régime has enough
challenges of its own.
As U0 is deformed away from convexity, a series of thresholds exists, at values
of U0 decreasing monotonically with the density of bosons. At each (commensurate)
density, we consider two thresholds. The lower threshold is where double occupancies
start to form spontaneously in the CGS, and a new non-HUP classical ground state
takes over at this commensurate filling. While our primary interest is in ground state
transitions, much insight is gained by also computing a second, upper threshold at a
(qSS)
given filling. Past this threshold, U = Uc
, a particle added to the state goes in as
a double occupancy instead of fractionalizing into q solitons. We will see that these
two thresholds allow us to understand many striking features of the phase diagram.
(qSS)
Fig. 5.6 plots Uc
(CGS)
(blue) and Uc
(qSS)
half-filling. The values of Uc
(red) for states with q ≤ 15 in the vicinity of
shown there are obtained by numerical minimization
(CGS)
in the sector with one added charge at the specified filling. The values of Uc
are obtained by numerical minimization over configurations at the specified filling
that contain exactly one double occupancy. The latter is the correct answer for all
ν = p/q for which p and q are not both odd. In such cases the double occupancy and
its surrounding charge rearrangements give rise to even moments starting with the
equivalent of a quadrupole. Consequently, double occupancies repel at all distances
and enter via a continuous transition at the computed threshold. However, when p
and q are both odd, double occupancies have a dipole-like moment, causing them
to attract at long distances. The transition in this case is first order, and the true
(CGS)
Uc
lies above our numerically determined value. We ignore this gap due to
first-order effects here, as we do not expect it to be very large. Indeed, a relatively
166
straightforward calculation shows that the minimum in the dipolar potential between
two such defects (see Sect. 5.9.3) places the two defects at least a distance q apart;
hence this energy gap decreases at least as
1
.
q3
This is a small perturbation in states
of filling p/q where p and q are both large.
This section outlines the energetic arguments for the locations of these thresholds
and the nature of the new ground states, and discusses the interesting features of
Fig. 5.6. We will end by exploring what can be said about the phases deep in the
non-convex regions – which will lead to an interesting new series of states which will
form the subject of the last two sections of this chapter.
5.6.1
Effective convexity and thresholds for double-occupancy
formation
We begin by understanding where each CGS becomes unstable to forming double
(CGS)
occupancies. Let Uc
(ν) be the thresholds at which the PUH ground states give
way to ones with at least one double occupancy—these are marked as the red points in
(CGS)
Fig. (5.6). Observe that these thresholds increase monotonically with ν, Uc
(CGS)
Uc
(ν ′ ) >
(ν) for ν ′ > ν.
To understand this monotonicity, along with the approximate locations of these
thresholds, we considering the implications of convexity at a given filling. Sufficient
conditions for convexity [80] are that, for all x,
1
15
(V (0) + V (2x)) ≥ V (x) or U ≥ V0
2
8
1
x3
(5.54)
For convexity to hold everywhere, (5.54) must hold for x = 1; below this threshold
double occupancies may occur. However, when perturbing about a given convex
solution at fixed µ, solutions will be stable approximately until U violates (5.54) for
x = rm , the minimal inter-particle distance. This implies that states with lower filling
fractions are more stable against double occupancies, as the potential gain in lattice
167
(CGS)
(qSS)
Figure 5.6: Numerically calculated values of Uc
(red) and Uc
(blue), for a
selection of filling fractions ν. Quoted values of U are measured relative to V0 . A
(CGS)
(qSS)
segment of the y-axis between Uc
(1/2) and Uc
(1/2) has been removed for
better resolution of the rest of the phase diagram. The green and yellow boxes,
bordered by black dots, indicate the approximate regions where the ground states
consist of double occupancies in the 1/3 and 1/2-filled states respectively. Between
the shaded regions the ground states consist of double occupancies on other, higherdenominator states.
168
energy from doubly occupying a site is smaller. This is in contrast to the stability of
the commensurate states as t increases, where the denominator of the filling fraction
determines stability.
For example, consider the states ν < 1/2. In reality, these states contain no pairs
of particles separated by x = 1, so the convex solutions should be stable until U0
violates (5.54) for x = 2 – i.e. for U0 ≥ .235. More generally, we expect the Hubbard
solutions to remain the ground states so long as U0 satisfies Eq. (5.54) for x ≥ rm ,
the minimum inter-particle spacing.
Arguments for the approximate locations of the transitions
For fillings ν = 1/q we can understand this more rigorously by considering the energetics of nearest neighbors. (The potential 1/r 3 falls off quickly enough that nearestneighbor interactions dominate the energetics in this case). Since all particles in the
initial CGS have the same separation, forming a double occupancy in the CGS results
in replacing
(q + 1)V (q) → (q + 2)V (q + 1) + U0
.
(5.55)
Removing a particle entirely changes the potential energy by (q + 1)V (q) − qV (q + 1).
The double occupancy, however, adds 2 NN distances of q + 1 to the lattice: one of
these was originally a NN separation of q; the other was a NNN separation of 2q –
which we omit here as it is a higher-order term.
Equation (5.55) shows that the scale at which double occupancies become energetically favorable in the CGS is determined by q: if ( 5.54) holds for x = q, then
double occupancies will not form. Indeed, writing
q=
q
q
(q + 1) + (1 −
)0
q+1
q+1
(5.56)
169
we see that for V convex,
V (q) = V (
q
q
q
(q + 1) + 0) ≤
V (q + 1) + (1 −
)V (0)
q+1
q+1
q+1
or (q + 1)V (q) ≤ qV (q + 1) + U0
(5.57)
Hence if V is convex at the length scale set by q, double occupancies cannot be present
in the classical ground state.
(qSS)
Similarly, we can gain an intuitive grasp for the energetics of Uc
by considering
only nearest-neighbor interactions, and repeating the above analysis. If a soliton is
formed in the qSS, the q solitons have replaced
(q − 1)V (q) → qV (q − 1)
(5.58)
whereas inserting a particle as a double occupancy induces 2 nearest neighbor distances of q. The net NN difference in lattice energy between adding a particle as
solitons and adding a particle as a DO is thus
δE = (q + 1)V (q) + U0 − qV (q − 1)
(5.59)
In this case, DO’s will not form if Eq. (5.54) is satisfied for x = q − 1.
Thus, we expect that the CGS and qSS will become unstable to double occupancies
at approximately
U < Uc (CGS) ≈
U<
Uc (qSS) ≈
15V0 1
8 q3
15V0
1
8 (q − 1)3
(5.60)
In short, states of higher density will become unstable to double occupancies at larger
(qSS)
values of U, and at a given filling there is a finite gap between Uc
(CGS)
and Uc
.
What about more general filling fractions? Repeating the above analysis, we see
that for fillings p/q, p > 1, nearest neighbor distances are r, r +1. Double occupancies
in the CGS replace (r+1)V (r) by (r+2)V (r+1). Likewise the energy cost of a soliton
170
is (r +1)V (r)−rV (r +1), while inserting an extra particle as a double occupancy adds
2V (r + 1).10 Hence the nearest neighbor energy cost of forming a double occupancy
in the CGS and qSS is:
CGS δE = (r + 2)V (r + 1) − (r + 1)V (r) + U0
qSS δE = (r + 2)V (r + 1) − (r + 1)V (r) + U0
(5.61)
Hence in general, we expect the approximate value of U0 at which these transitions
occur to be set by the smallest nearest-neighbor distance in the CGS. However, the
(qSS)
scale of the difference between Uc
(CGS)
and Uc
is determined by further neighbor
interactions – in fact, not until pth neighbors are included in the energetic calculations
is the set of allowed separations different in the qSS and CGS. Hence this splitting
will be much smaller in this case than for the 1/q -filled state with q = r in Eq. (5.61).
We now begin to understand the basic features of Figure 5.6. To a first approximation (obtained by considering only NN distances), we have:
1
1
)
UcqSS ( ) = UcCGS (
q
(q − 1)
and, for all ν with
1
q
<ν<
(5.62)
1
,
q−1
1
UcCGS (ν) = UcqSS (ν) = UcqSS ( )
q
(5.63)
That is, the value of U0 at which double occupancies first appear is approximately
decreasing with the filling. There is a large gap between its values for CGS and qSS
states at filling fractions 1q ; the jump for fractions pq , p > 1 is much smaller.
Form of the doubly occupied ground states and computing the exact loci
of transitions
To calculate the exact loci of the transitions requires finding the exact occupation
pattern in the doubly occupied states, and computing its energy relative to that of
10
Since V (r) > V (r + 1) in the case of interest, we would expect this to be a lower bound. In fact
the double occupancy always polarizes the qSS such that its two NN distances are exactly r + 1.
171
states without double occupancy. The second part of this task is easy to do numerically, if not analytically; here we discuss the energetically optimal configurations with
double occupancy in both qSS and CGS states.
qSS states:
We begin with the somewhat simpler qSS states. To find the optimal configuration,
we first search for the optimal position in the repeating CGS pattern at which to add
a double occupancy. Intuitively, this will be at the occupied site with the lowest local
charge density. Second, we ask whether adding a charge at this site creates further
distortions of the CGS state. Though we will not prove that this is the globally
optimal configuration, the resulting configuration will give the least distortions of the
CGS configuration for a given local charge density, and hence should be the ground
state.
First, how do we identify the locus of lowest local charge density? It is useful to
consider a few simple examples:
• For states of the form 1/q, there is only one site per unit cell – hence no freedom
in where to place the double occupancy. Using the notation of Section 5.3.1, we
see that states of filling 2/q have occupancy patterns ... r r + 1 r r + 1..., and
again both sites in the unit cell have the same local charge density.
• At filling 3/q, the occupancy pattern is either ... r r r + 1 r r r + 1..., or
... r r + 1 r + 1 r r + 1 r + 1.... In the former case, the DO cannot sit between
the two NN distances of r; in the latter, it must sit between the two NN distances
of r + 1.
• In the state 5/13 = ...2 3 2 3 3..., the DO must sit between the two NN distances
of 3 – which we will henceforth denote as the 5th site in the unit cell, using the
convention that we count from left of the first interval shown.
• In the state 5/12 = ...2 3 2 2 3..., the DO must sit on the first or second site in the
172
unit cell (these being equivalent, up to reflection of the entire pattern). In this
case, considering only nearest neighbors would suggest that it could sit anywhere
but the 4th site. However, occupying sites 3 and 5 leads to second neighbor
distances of 4, 5, instead of 5, 5 for sites 1 and 2, making these energetically
preferable.
This suggests the following simple algorithm: begin by considering the sum of the
two nearest-neighbor distances s1 (i) = ri,l + ri,r to the left and right of each site in the
pattern. If one of the s1 (i) is less than the rest, place the DO here. If not, generate the
set of sums of right and left second neighbor distances: s2 (i) = ri,l +ri−1,l +ri,r +ri+1,r .
If this set has a unique largest element at site i, place the DO there. If not, proceed
to the set of sums of right and left third neighbor distances, and so on. One can
proceed in this way up to p − 1st neighbor distances; sites for which sk (i) = sk (j)p−1
k=1
have identical local charge density – as in the example above at ν = 5/12– and hence
are related by reflections and translations of the unit cell.
Having found the optimal site at which to add the DO, we must then ask whether
inserting charge here results in a further re-arrangement of the background charge
of the lattice. Basically, dipolar ‘charges’ on nearby sites are repelled by the extra
charge at the DO. Charge will be forced away from the doubly occupied site when the
potential energy gain in doing so is greater than the cost of compressing the state by
the corresponding amount in the remainder of the lattice. When such compression
is favorable, it will result in particle-like solitons forming near the DO and being
repelled to infinite distance. In practice, this means that displacing one charge, say
at the right of the DO, one lattice site outwards will push all charges to its right one
site outwards as well. This creates a single soliton at ∞, as we have lengthened one
pth neighbor distance from q to q + 1.
It is straightforward to calculate the relevant energies. The lattice energy associ-
173
ated with a single soliton given by Eq. (5.24):
∆E1 = 1/q
X
p6=0(mod q)
X
[(rp + 1)V (rp ) − rp V (rp + 1)]+
nqV (nq−1)−(nq−1)V (nq)+... .
n
We must compare this to the energy gained by creating an extra hole between the
DO, and site i, a distance d to its right (say). This energy has two contributions: one
infinite sum for the change in interaction of site i and all particles to its right with
the double occupancy, and one double infinite sum for the change in interaction of
site i and all particles to its right with all occupied sites to the left of the DO. This
gives:
∆E2 =
∞
X
k=i
+
1
1
−
3
(xk − xi + d + 1)
(xk − xi + d)3
i−1 X
∞
X
j=−∞ k=i
1
2
−
3
(xk + xj − xi + d + 1)
(xk + xj − xi + d)3
(5.64)
where xk here denotes the position of the k th particle on the chain, relative to an
arbitrary origin. The sum over j here includes the doubly occupied site; the first line
of Eq. (5.64) counts only the extra interaction due to the second particle occupying
this site.
The important point here is that the sum of the second line of Eq. (5.64) and the
soliton energy is positive-definite by convexity of 1/r 3. Further, their sum depends
only on which particle in the unit cell lies at site i, rather than the value of d directly.
The gain in energy from moving away from the DO (first line of Eq. (5.64), conversely,
falls of rapidly with d. Hence if d is sufficiently large, the repulsion of the DO will be
too weak to push the charge outwards. Further, once we have found a site for which
∆E1 + ∆E2 > 0, no charge further from the DO than this site will be displaced.
It is thus straightforward to compute ∆E1 + ∆E2 for the sites close to the DO to
determine which charges will be pushed outwards. The algorithm begins with sites
closest to the DO; if ∆E1 + ∆E2 < 0, the charge in question and all charges further
from the DO are pushed outwards by one site, creating a single soliton infinitely far
174
away. One then moves outwards to the next closest charges, and repeats the process
until a site is reached for which ∆E1 + ∆E2 > 0.
CGS States
Forming double occupancies in the CGS is qualitatively different, since we think
of re-arranging the existing dipoles, rather than adding a new one. In this case
it is simplest to think of forming the DO by pinching charge inwards towards an
unoccupied site on which the DO will form.
The pinching operation involves picking a site xd on which the DO will be formed,
and moving some number of particles (initially at positions {Ri }) to the left of xd
rightwards, and some number of particles to the right of xd (initially at positions
{Li }) leftwards.
For example, the 1/3 state introduces {Li } = {1}, and {Ri } = {2, 5}, relative to
xd . A DO can be formed in the 1/3-filled state as follows:
1
: 1001001001001
3
Pinch L1 , R1 : 1 0 0 0 2 0 0 0 0 1 0 0 1
Pinch R2 : 1 0 0 0 2 0 0 0 1 0 0 0 1
(5.65)
Similarly, for the 2/5 state, {Li } = {1, 4}, and {Ri } = {1, 4}
2
: 1010010100101001
5
Pinch L1 , R1 : 1 0 1 0 0 0 2 0 0 0 1 0 1 0 0 1
Pinch R2 : 1 0 0 1 0 0 2 0 0 1 0 0 1 0 0 1
(5.66)
Hence again, we must first find the optimal site onto which to pinch the charge,
and then ask whether other charges will move after the DO has formed. The lowest
energy final charge configurations are as symmetric as possible: if the local charge
density is lower on one side of the DO, this inhomogeneous distribution will exert a
force on the excess charge of the doubly occupied site. Thus we find the optimal site
by searching for holes which sit at symmetry centers of the configuration.
175
This approach reveals an important even-odd type effect in forming DO in the
CGS. If q − p is odd, every hole in the unit cell has an equal number of holes in the
pattern to its right and to its left, and we will show that it is possible to choose a hole
about which occupied sites are distributed symmetrically. Pinching in these states
will not alter this symmetry, and hence two DO’s repel. If q − p is even, however,
such a site does not exist and the final charge distribution is not symmetric. This
leads to a dipole-like moment for each double occupancy, causing these to attract at
long distances.
As a simple example of how this works, consider the states at 2/5 and 3/5 filling.
The occupancy patterns are:
ν = 2/5 :
...1010010100...
ν = 3/5 :
...1011010110...
(5.67)
The important difference between the two is that at ν = 2/5, there is an odd number of
holes in the unit cell – and hence somewhere in the unit cell there is a pair of occupied
sites separated by an odd number of holes. The distribution of charge required by
convexity is such that the hole at the center of this interval is a center of symmetry
of the charge distribution. The double occupancy can then be formed by pinching
particles onto this center of symmetry. The ensuing charge distribution is therefore
symmetric. At ν = 3/5, the center of symmetry of the pattern occurs on an occupied
site, and hence there is no way to pinch the charge completely symmetrically. The
result is a configuration which has a dipole-like moment due to this charge asymmetry.
Indeed, in general HUP solutions will have a center of symmetry, since this is the
most homogeneous way to distribute charge. For general p/q with p − q odd, the unit
cell contains an odd number of holes, and this center of symmetry consequently lies
on an unoccupied site. For p − q even, conversely, it lies on an occupied site. Thus
176
the HUP solutions also result in the even-odd effect observed in the phase transitions
to doubly occupied states.
This pinching construction gives a simple algorithm for constructing DO in the
CGS: first we find a hole which sits at a center of symmetry, and pinch the two
nearest bosons onto this site. This can only create attractive interactions from the
remaining charges towards the DO. We then iterate through the neighboring particles
and compute the energy of moving each one inwards. (Here one must bear in mind
that for less dense CGS states, it may be energetically favorable for particles to move
inwards by several lattice sites). If this energy is negative, we move the particles
inwards and proceed to the next nearest occupied sites. If the energy is positive, no
further re-distribution of charge will occur.
Hence for both CGS and qSS states, a simple algorithm exists to find the optimal
distribution of occupied sites, given that a double occupancy will form. This allows
us to compute the exact difference in potential energy between configurations with
and without a single double occupancy – and hence calculate the threshold values
of U0 at which double occupancies begin to form. The result is the classical phase
portrait Fig. 5.6.
5.6.2
Arguments for monotonicity of Uc with filling fraction
From our above considerations, at fixed filling, the CGS is always more stable against
forming double occupancies than the qSS. Indeed, a simple estimate suggests that
the gap separating these instabilities is approximately the difference in energies for
adding and removing a particle to the state– which is equal to the interval in µ over
which the state is stable at infinite U. We now use this to understand an intriguing
(CGS)
property of Fig. 5.6 – namely, the monotonicity of Uc
with ν.
The monotonicity can be understood as a simple consequence of the fact that
(qSS)
Uc
(CGS)
(ν) > Uc
(ν). Consider a filling ν ′ > ν. As ν ′ can be constructed by adding
177
(CGS)
charge to ν, we conclude that Uc
(qSS)
(ν ′ ) > Uc
(ν); the latter corresponds to the
threshold at which adding an extra filling ν ′ − ν in the form of the solitons of ν loses
out to adding it in the form of double occupancies on top of the PUH state at ν.
As filling factors only slightly greater than ν involve a dilute addition of charges, we
(CGS)
further conclude that limν ′ →ν + Uc
(qSS)
(ν ′ ) = Uc
(ν).
(CGS)
A somewhat more involved argument shows that limν ′ →ν − Uc
(CGS)
(ν ′ ) = Uc
(ν) .
We have:
lim Uc(qSS) (ν ′ ) = Uc(CGS) (ν)
(5.68)
ν ′ →ν −
since the state at filling ν − δ is the state at filling ν with a density δ of hole-like
solitons. As this density decreases a single particle added as a double occupancy
will induce a charge configuration increasingly similar to that of forming a double
occupancy in the p/q-filled state (which can be thought of as forming q hole-like
solitons by removing a single particle, then re-inserting this particle as a double
occupancy and letting the charge settle into its optimal distribution.) Further, the
(qSS)
gap between Uc
(CGS)
(ν ′ ) and UC
(ν ′ ) vanishes as the denominator of the state ν ′
(CGS)
goes to infinity, implying continuity of UC
from the left.
In short, our considerations so far imply intricate behavior for the location of
(CGS)
the initial ground state instability, namely that Uc
(ν) is a monotone increasing
function of ν on the set of rationals with a discontinuity at each rational value of ν:
lim Uc(CGS) (ν ′ ) > Uc(CGS) (ν) = ′lim− Uc(CGS) (ν ′ ).
ν ′ →ν +
ν →ν
(CGS)
The first relation is a strict inequality, as the scale is set by the gap between Uc
(qSS)
and Uc
(ν), which is finite. The second is equality, because the gap is set by
(qSS)
limδ→0 Uc
(ν)
(CGS)
(ν + δ) − Uc
(ν + δ), which is 0.
178
5.6.3
Structure of the doubly-occupied régime
The previous sections have outlined some rather stringent constraints on the thresh(qSS)
olds Uc
(CGS)
and Uc
, which give significant insights into the nature of the phase
portrait of the classical system as U0 is decreased: monotonicity locates the transitions of higher denominator states relative to those of lower denominator states. Thus
we can understand the coarse features of this instability by considering first the most
stable states (of denominator q ≤ q0 , for some q0 small enough to allow the exact
thresholds to be computed), and deducing the expected behavior at fillings close to
these.
We now turn to the question of what can be said about the phases with U < Uc . At
any given filling, tracking the evolution of the ground state with decreasing U0 after
double occupancies have been introduced is a problem of considerable complexity.
Here we use the ideas developed thus far to identify a family of regions in the (ν, U0 )
plane where simpler descriptions emerge—these are indicated, in two simple cases,
by the shaded regions on the figure. The basic idea is that once the ν = p/q qSS
becomes unstable to double occupancy, any particles added to the ν = p/q state
will be added as double occupancies, since these repel less strongly than solitons (see
(qSS)
below). Hence at first sight we expect, in the region Uc
(CGS)
(p/q) > U > Uc
(p/q),
states of filling ν > p/q to consist of double occupancies in the ν = p/q state.
At rational fillings the double occupancies will arrange themselves in a crystal thus
generating a commensuration distinct from that of the underlying p/q state—we will
refer to these as doubly-commensurate states.
(qSS)
A detailed discussion of why, for U0 < Uc
(ν), increasing the filling fraction
can only form new double occupancies, and never new solitons, is presented in Sect.
5.9.4. It is useful, however, to understand the simple physical reason underlying this:
adding particles as solitons results in a charge distribution that is maximally spread
out in space; inserting them as double occupancies produces a maximally localized
179
charge distribution. If the change in density is infinitesimal, the solitons or double
(qSS)
occupancies will be infinitely far apart, and thus by definition if U0 < Uc
(ν)
these extra particles will enter the ground state as double occupancies. If the change
in density is finite, we must add some number of particles per unit length on the
lattice. Since the potential 1/r 3 falls of rapidly in space, the repulsion between two
added charges in a given distance is smallest when the extra charges are as localized as
possible –that is, when both form double occupancies. Hence as the density increases,
so does the energetic payoff of forming DO, rather than solitons.
Our discussion so far suggests that for all δν > 0, states at filling ν + δν with
(CGS)
Uc
(qSS)
(ν) < U0 < Uc
(ν) consist of the doubly commensurate states described
above. However, as the density of added charge increases, the parent state itself
becomes less stable to forming extra double occupancies. This can lead to transitions
in which the structure of the CGS collapses to a crystal of double occupancies over a
background of significantly smaller filling.
We have carried out a simple analysis of the location of this instability for the 1/2
and 1/3 plus double occupancy regions in Fig. (5.6) at selected fillings and these are
marked by the black dots in the figure. We do not know of a general algorithm to
compute this threshold at arbitrary filling, but the principal is easy to illustrate for
the 1/2-filled state. Consider the following two states at filling 2/3 = 1/2 + 1/6:
... 2 0 1 0 1 0 2 0 1 0 1 0 ...
... 2 0 0 2 0 0 2 0 0 2 0 0 ...
(qSS)
The first state will be stable for U0 < Uc
(5.69)
(1/2), but sufficiently large. However,
the energy gained by pinching this state to form the second state is greater than the
energy gained by simply forming a DO in the 1/2-filled state, as the two particles
that are pinched are moving away from sites with charge 2, rather than charge 1. The
black dots in the figure are calculated by calculating the difference in energies of such
simple configurations.
180
It is interesting to notice that the black dots do not behave monotonically with
filling fraction. To understand why, consider the following two states at filling 3/4:
... 2 0 1 0 2 0 1 0 2 0 1 0 ...
... 2 0 0 2 0 0 2 0 2 0 0 2 0 0 2 0 ...
(5.70)
Because of the greater density in the first state, the best we can do by pinching is
ensure that 2/3 of the double occupancies sit at least 3 lattice sites apart. Hence once
at least half the sites are doubly occupied, the potential gain due to forming extra
double occupancies is actually less than for smaller fillings.
In summary, Fig. 5.6 gives an accurate picture of the locations of the initial phase
transition due to deforming U0 away from convexity for arbitrary fillings. We have
constructed the general form of the resulting states with double occupancy. As U0
is decreased even further, our analysis suggests that there are further transitions to
states with a higher density of double occupancies, whose structure we have not systematically understood. While we have found sizeable regions which can be described
as simple descendants of the 1/2 and 1/3 states, we are not at present able to estimate
the sizes of analogous regions for higher denominator fractions. Of course, to have a
full solution of this would be equivalent to tracking the evolution of each ν as U is
(CGS)
decreased from Uc
5.7
.
Interesting phenomena in the non-convex régime
In Section 5.6, we have expended considerable effort understanding the various phase
boundaries in the classical system as U0 is decreased away from the convex limit.
We now focus on a particularly interesting region of this phase diagram – namely,
that in which the ground states are derived by adding double occupancies to a parent
CGS configuration. We will first discuss the classical limit of these phases, showing
that these contain a re-scaled version of the devil’s staircase. Focusing on the devil’s
181
staircase near 1/2 filling, we then consider the effect of adding hopping to the mix,
and find a phase diagram with super-solid like regions in which a Luttinger liquid of
double occupancies co-exists with a commensurate 1/2-filled background.
5.7.1
A new staircase
The discussion of Sect. 5.6 has led us to the doubly-commensurate states in the (ν, U0 )
phase diagram: we remind the reader that such states are constructed by periodically
doubly occupying some fraction of the sites in a CGS. We now discuss the ground
state configurations of these double occupancies, and show that in at least some cases
these doubly commensurate states can form a devil’s staircase of their own.
To explore such states, we will work in the parameter range where increasing
density effectively adds double occupancies to a parent CGS state, without forming
DO in the parent state itself. We have argued in Sect. 5.6.3 that at least for the
1/2 and 1/3 filled states, this description is apt over sizeable regions of the phase
diagram. Though we have not calculated the lower thresholds for higher denominator
states, we expect that a similar description holds for arbitrary parent fillings ν and
(CGS)
Uc
(qSs)
< U0 < Uc
, at least over a modest range of densities.
First consider states constructed from double occupancies in the 1/2-filled state,
which exist in the region shaded in yellow in Fig. (5.6). The energetics of such
states can be divided into a) the constant interaction of the parent 1/2-filled PUH
configuration with itself, b) the constant interaction of the added charges, irrespective
of their location, with the parent 1/2 filled configuration and c) the interaction of the
added charges with themselves. This last part involves an interaction between the
added charges which is convex again and thus leads to PUH configurations sitting on
a lattice with a doubled lattice constant. The energy cost of adding a single double
occupancy is U + Vd , Hence the doubly-occupied sites comprise a Devil’s staircase
with µ → µ + U + Vd , and the widths of all intervals decreased by a factor of 8.
182
Here Vd =
1
8
P∞
1
n=1 n3
is the interaction energy of each double occupancy with the
underlying 1/2-filled state. At fixed U, this staircase is complete over the range
of fillings for which increasing the particle density infinitesimally does not induce
‘excess’ double occupancies to form in the half-filled background lattice. In the case
of the 1/2-filled state, for U sufficiently close to the upper cutoff this gives a complete
staircase on 1/2 ≤ ν ≤ 1.
Similar structures exist for all 1/q-filled states in the appropriate range of U.
As mentioned before, we do not, at present, understand the situation for doubly
commensurate descendants of general rational fillings.
5.7.2
Supersolids
Thus far our considerations away from the convex limit have been purely classical.
But we can equally consider states obtained from these modified classical states upon
the introduction of hopping. Specifically, let us consider the fate of the doubly commensurate descendants of the PUH 1/q states considered above.
In a manner entirely analogous to the problem with which we began this paper, the
superlattice of added charges can melt via the motion of its solitons as t is increased
resulting in a phase transition between the doubly commensurate state and a “supersolid” like phase in which the background 1/q-filled CGS coexists with a Luttinger
liquid. This is, in a sense the d = 1 version of the supersolid in higher dimensions,
but it is worth noting that the d = 1 version in our problem exhibits a more divergent
CDW susceptibility than superfluid susceptibility as T → 0.
To get a more quantitative account of these new phases, we may repeat the strong
coupling treatment above. Fig. (5.7) shows the phase portrait at intermediate values
of U near ν = 1/2. The black line traces the infinite-U Mott lobe, over which the
background 1/2-filled state is stable against forming solitons. The red line shows the
threshold at which it is energetically favorable to add a single double occupancy to the
183
1/2 filled state. The blue curves show the positions of the Mott lobes for the doubly
commensurate states. The presence of double occupancies stabilizes the 1/2-filled
state against proliferation of solitons, so that the background remains commensurate
at least within the infinite-U 1/2-filled Mott lobe, shown in black. This 1/2-filled
super-solid phase has also been shown to exist in an extended Bose-Hubbard model
with second-neighbor repulsion [101]; these numerical results are consistent with the
phase portrait shown here for small t/V .
U = V0
1.8
1.6
5/6
µ/V
0
1.4
3/4
2/3
1.2
1
0.8
0.6
0.05
0.1
0.15
t/V
0.2
0.25
0.3
0
Figure 5.7: Phase portrait in the vicinity of doubly commensurate states about 1/2
filling at U = V0 . Again, µ, t, and U in the figure are measured relative to V0 . The
black lines indicate the boundaries of the Mott lobe at U = 20. The red curve shows
the chemical potential at which it becomes energetically favorable to add particles to
the half-filled state as double occupancies, for U = V0 . The blue curves show some
Mott lobes of the doubly-occupied staircase region; the region between these and the
black lines (which delineate the region of stability of the 1/2-filled state at infinite U)
is a super-solid state.
184
Perturbation theory in the super-solid régime
The phase diagram in Figure 5.7 is obtained using 4th order perturbation theory in
t. As the calculation of the energy corrections is rather involved, we have included
the details in Sect. 5.9.1. Here we will discuss the form of these corrections, and the
qualitative features of the phase diagram.
The calculation is similar to that described in the convex case in Sect 5.4, with
one key difference: all ground states must have holes at every second site. Adding
a charge to a doubly commensurate crystal produces an extra double occupancy –
which in turn produces solitons in the doubly occupied sites. That is, as we do not
allow triple occupancies to form, and it is energetically unfavorable to place extra
particles on the unoccupied sites, we may only add particles by forming solitons in
the doubly-occupied superstructure. At filling 1/2 + p/2q, for example, the relevant
ground state is given by the qSS of the p/q filled state, with all distances stretched
by a factor of 2. Hence only even-order corrections to the energy are non-vanishing:
an odd number of hoppings produces a state with at least one particle on a site that
is vacant in the parent half-filled state.
As discussed in the Sect. 5.9, we can therefore describe the corrections in terms
of an ‘effective’ hopping Hamiltonian, H1ef f , which describes the hopping between
occupied sites. H1ef f contains both a ‘trivial’ correction, in which a boson hops off a
given site and back on, and a ‘hopping’ term in which a boson hops from one occupied
site to another. The second-order correction to the energy in this case is:
E (2) = −hψ0 |H1ef f |ψ0 i
(5.71)
The trivial part of H1 gives approximately the same correction for commensurate
and solitonized states (except for small differences in the energy denominators). The
hopping term, on the other hand, has a first-order matrix element with solitonized
states, but not with the CGS. Thus the situation is analogous to that of qSS solitons in
185
the convex limit: solitonized states have a negative energy correction at leading order
in perturbation theory which their commensurate brethren do not. Thus as t increases
the double occupancies undergo a commensurate-incommensurate transition, as seen
in Fig. 5.7.
5.8
Concluding Remarks
The infinite range of the dipolar interaction does produce, as promised, intricate
phase diagrams for the one dimensional dipolar bosonic gas. Particularly striking
are the singular staircase functions that showed up in our analysis in three different
settings: in the ν(µ) curve for the exactly dipolar classical problem, in the function
(CGS)
U0c
which marks the instability of the PUH states when the onsite U is tuned
down and in the ν(µ) curves in selected regions of the (ν, U ) plane. The other main
set of results pertain to the presence of a large, indeed, infinite number of transitions
between Mott crystals and Luttinger liquids or supersolids. The challenge of observing
some of this physics in cold atomic gases is not trivial—the major obstacles are getting
a reasonable simulacrum of a one dimensional gas of infinite extent. On the positive
side, the control parameters we study here are eminently tunable.
5.9
5.9.1
Supplementary Material
General Perturbation Theory
Here we review the perturbation theory pertinent to treating the hopping terms in
the general Hamiltonian, and in particular hopping in the super-solid state.
(0)
(0)
Let H0 be the unperturbed Hamiltonian, with eigenstates |ψi i of energy Ei .
The perturbation δH1 induces corrections to the wave function and energies; the ith
(i)
(i)
order corrections are denoted |ψj i, Ej . When the lower indices are omitted, we will
186
be referring to corrections to the ground state energy and wave-function (which is all
we’re interested in here.)
Perturbation theory can be summarized by the following equations:
|ψi = |ψ (0) i + δ|ψ (1) i + δ 2 |ψ (2) i + δ 3 |ψ (3) i + ...
E = E (0) + δE (1) + δ 2 E (2) + δ 3 E (3) + ...
(H0 + δH1 )|ψi = E|ψi
(5.72)
(0)
We stipulate that hψ0 |ψ (i) i = δi0 , and find the recursion relation for the energy is:
!
n−1
X
(0)
(0)
(Ei − E0 )hψi |ψ (n) i =
E (n−j) hψi |ψ (j) i − hψi |H1 |ψ (n−1) i
(5.73)
j=1
where |ψi i label excited states of the unperturbed Hamitlonian.
We can explicitly write out the first few terms of these series in terms of the
unperturbed wave functions and energies. We drop all superscripts, which are 0.
E (1) = hψ0 |H1 |ψ0 i
X
1
|ψi ihψi |H1 |ψ0 i
|ψ (1) i = −
E
−
E
i
0
i
X
1
E (2) = −
|hψ0 |H1 |ψi i|2
E
−
E
i
0
i
X
1
1
|ψj i
hψj |H1 |ψi ihψi |H1 |ψ0 i
(Ej − E0 )
Ei − E0
j
i
1
−
hψ0 |H1|ψ0 ihψj |H1 |ψ0 i
Ej − E0
X hψ0 |H1 |ψ0 i
=
−
|hψ0 |H1 |ψj i|2
2
(E
−
E
)
j
0
j
|ψ (2) i =
E (3)
X
1
+
hψ0 |H1 |ψj ihψj |H1 |ψi ihψi |H1 |ψ0 i
(Ei − E0 )(Ej − E0 )
(5.74)
The matrix elements hψi |H1 |ψj i are all ±t or 0, except for certain hoppings in the
vicinity of the soliton, which have the form t(1 + cos(kq)).
We are interested in corrections higher than 3rd order to investigate possible supersolid states, which consist of a background of filling 1/n, and double occupancies at
187
filling p/(nq). The non-vanishing energy corrections occur at multiples of n, as other
orders cannot map the ground state back to itself.
Here we consider the simplest case, n = 2. Since the first order correction to E
vanishes, we may write ψ (2) in the form
|ψ (2) i =
where H1ef f =
P
X
j
1
|ψj ihψj |H1ef f |ψ0 i
(Ej − E0 )
1
i Ei −E0 H1 |ψi ihψi |H1
(5.75)
is the effective hopping Hamiltonian. Here |ψi i
has one occupied odd site. |ψj i differs from the ground state by 2 hoppings. There are
two such terms: |ψj0 i has occupation of even sites only, and |ψj1 i has two occupied
odd sites. It is useful to separate these two terms explicitly:
|ψ (2) i =
X
j0
X
1
1
|ψj0 ihψj0 |H1ef f |ψ0 i+
|ψj1 ihψj1 |H1ef f |ψ0 i (5.76)
(Ej0 − E0 )
(E
−
E
)
j1
0
j
1
The first piece looks like the first-order correction to a wave function on a re-scaled
lattice, with an effective hopping coefficient given by H1ef f . The second piece does
not contribute to the second-order energy correction. Hence in this notation, we also
have
E (2) = −hψ0 |H1ef f |ψ0 i
(5.77)
This is the leading-order energy correction. The third-order correction to the energy
vanishes; the wave-function correction is:
(3)
|ψ i =
X
i
−
1
|ψi i
Ei − E0
X
j
1
hψi |H1 |ψ0 ihψ0 |H1ef f |ψ0 i
Ei − E0
!
1
hψi |H1 |ψj ihψj |H1ef f |ψ0 i
Ej − E0
(5.78)
188
This gives the fourth order energy correction:
E
(4)
1
1
hψ0 |H1 |ψi ihψi |H1 |ψ0 ihψ0 |H1ef f |ψ0 i
=
Ei − E0
Ei − E0
i
!
X
1
−
|ψj ihψj |H1ef f |ψ0 i
Ej − E0
j
X
1
= hψ0 |H̃ ef f |ψ0 ihψ0 |H1ef f |ψ0 i −
|hψ0 |H ef f |ψi1 i|2
Ei1 − E0
i1
X
1
|hψ0 |H ef f |ψi0 i|2
−
Ei0 − E0
i
X
(5.79)
0
where H̃ ef f =
1
i (Ei −E0 )2 H1 |ψi ihψi |H1 .
P
The last term is the analogue, in our effective
theory, of a ‘second-order’ correction. The first two terms give an extra correction
due to the structure of the underlying lattice.
Matrix Elements and formulae used for numerics in the super-solid states
To evaluate the perturbative energy corrections, we must calculate the relevant matrix
elements. Here we outline this calculation order by order for the super-solid phases,
in which the results are somewhat complex. The corresponding calculations for the
convex régime are a straightforward adaptation of the results of [85], and are not
included here. To simplify the language, for the remainder of this section we will use
qSS and CGS to refer to states with and without solitons in the doubly occupied sites,
respectively.
Second Order correction
The second-order energy is given by
E (2) = −hψ0 |H1ef f |ψ0 i
(5.80)
hψ0 |H1 |ψi i
(5.81)
The matrix elements
are
√
2t when |ψi i involves hopping from a doubly occupied site (as there are two
√
identical particles to choose from), and t otherwise. (This is because bi |n0 i = n0 |n0 −
189
√
1i. A factor of 2 is also introduced by hopping onto an occupied site, since b†i |n0 −
√
1i = n0 |n0 i. Hence hopping a particle off of a doubly occupied site, and then back
on, incurs a factor of 2).
Hence, for the CGS,
E (2) = −2t2
"
X
i6=iDO
X
1
2
+
Ei − E0 i Ei − E0
DO
#
(5.82)
where Ei corresponds to hopping the ith particle to the right. (The overall factor of
2 then accounts for the left hoppings). This can be written conveniently in the form:
X ef f
E (2) = −2
ti,R
(5.83)
i
f
where tef
i,R =
t
Ei −E0
if site i − 1 is singly occupied, and
2
Ei −E0
if it is doubly occupied.
This accounts for the denominator when hopping a particle from site i − 1 one site
f
to the right, to site i. We may equally well define tef
i,L , which is
singly occupied, and
2
Ei −E0
1
Ei −E0
if site i + 1 is
if it is doubly occupied.
In the solitonized ground states, there is an additional contribution, in which the
particle at one end of the soliton hops two sites to the right (or left) effectively hopping
the soliton by 2q sites. This extra contribution is equivalent to the first order energy
f
gain of the solitonized state in the non-super-solid phase. The tef
for this is the same
i
as for this particle hopping out and returning to the same site. Thus for the qSS
X ef f
(2)
f
EqSS = −2
ti,R − 2tef
(5.84)
is cos(2kq)
i
where
f
tef
is
contains the appropriate energy denominator for the hopping from site at
the end of the soliton. The factor of 2 in front of the sum accounts for left and right
hoppings; in the second term, the particle can hop from the right end of the soliton
ending at site is to the left end of a (new) soliton beginning at is + 2, or vice versa (
reversing left and right). The difference between the CGS and qSS energy corrections
is thus:
δE
(2)
p
N h
i
X
2(p ± 1) X CGS
QSS
ef f
CGS
ti,R
= −2
ti,R − ti,R − 2tis cos(2kq) −
p
i=1
i=1
(5.85)
190
Here N is the number of particles on the chain. In practice the sum may be truncated
at finite N as the difference between CGS and qSS contributions falls off rapidly far
from a soliton. In this term, however, we have assumed that there is one particle
in the CGS for each particle in the qSS; we must correct for this by subtracting the
(average) energy of one particle in the CGS, given in the last term.
Fourth order correction
The fourth order energy correction is:
E (4) = hψ0 |H̃ ef f |ψ0 ihψ0 |H1ef f |ψ0 i −
−
X
i0
X
i1
1
|hψ0 |H ef f |ψi0 i|2
Ei0 − E0
1
|hψ0 |H ef f |ψi1 i|2
Ei1 − E0
(5.86)
The first term in the sum is a product of E (2) , calculated above, and Ẽ (2) , given by
"
#
X
X
1
2
+
(5.87)
Ẽ (2) = −2t4
2
2
(E
(E
−
E
)
i − E0 )
i
0
i
i6=i
DO
DO
Here again, the solitonized ground state has an extra contribution compared to the
CGS, of the form (5.84). Each term in this sum is uniquely labeled by choosing two
site indices, i and j, to be the target sites of intermediate hopping, and a direction
of hopping onto and off of each.
In the second term, |ψi i are states which are connected to |ψ0 i by hopping two
particles k and j onto odd sites. The choice of k and j uniquely fixes the intermediate
state |ψi1 i. Note, however, that for each choice of k and j there are two possible
hopping sequences: |ψ0 i → |ψk i → |ψi i → |ψj i → |ψ0 i, and |ψ0 i → |ψk i → |ψi i →
|ψk i → |ψ0 i, where |ψk i denotes the state in which particle k has hopped, but particle
j has not. These two possible paths differ only in the order in which the hopping
back to the ground state is performed.
191
Hence the second term of (5.86) can be expressed as:
X
1
1
−
ni nj
+
2
(E
(Ei+1,j+1 − E0 )(Ei+1 − E0 )(Ej+1 − E0 )
i+1,j+1 − E0 )(Ej+1 − E0 )
i6=j
X 1
1
2
−
ni
+ δni ,2
(Ei+1 − E0 )(Ej+1 − E0 )2
(Ei−1,i+1 − E0 )(Ei+1 − E0 )(Ei−1 − E0 )
i
(5.88)
Here i and j are sites which are occupied by ni and nj bosons in the ground state.
Ei+1 , Ej+1 are the energies of the intermediate states produced by hopping a boson
off of sites i and j (by hopping to the right; the term for hopping to the left appears
with a − sign). Ei+1,j+1 represents the energy of the intermediate state with particles
hopped from both sites i and j. Here again, in the case of solitonized states the
particle at the end of the soliton can hop either ‘out and back’, or ‘out and out’ – the
latter producing a translation of the soliton.
For both CGS and qSS states, the sum of the first two terms in Eq. (5.86) is small.
In particular, the cross-terms between hoppings of the soliton itself and hoppings of
the background lattice approximately cancel. This is as it should be – if they did
not, these terms would give an energy splitting between qSS and CGS states which
diverges in the thermodynamic limit!
The final contribution resembles a second-order contribution with a re-scaled lat(4)
(4)
tice constant and hopping terms which we will call Ea and Eb . The first of these
has the form:
Ea(4)
f
2
X |tef
i−2,R |
= −2
Ei − E0
i
where i labels the locus of the hopped particle in the excited state, and i − 2 its locus
in the ground state (prior to hopping). The factor of 2 accounts for hoppings to the
f
left and right. The tef
i−2,R is the effective hopping from site i − 2 to site i − 1 (which
gives the same energy denominator as a leftward hop from site i to site i − 1). As
in the non-super-solid case, there is one special intermediate configuration |ψi∗ i in
the solitonized case which permits two distinct hoppings which return to the ground
192
state, as the ground state in this case is a momentum eigenstate. The intermediary
configurations for the ‘forward’ and ‘backward’ hoppings are mirror images of each
other in this case. Hence for the solitonized ground state:
"
#
f
2
ef f 2
X |tef
|
|t
|
∗
i−2,R
i
Ea(4) = −2
+
∗
E
−
E
E
i
0
i − E0
i6=i
(5.89)
s
We exclude hopping to the right from is in the first sum, as it merely hops the soliton,
and has been included at second order. The special hopping from site i∗ gets a matrix
element of 4, as it also occurs once in the first sum.
The final contribution represents the effect of hopping both particles off of a doubly
occupied site and then back, and has the form
(4)
Eb
=
X
iDO
f 2
2t2 |tef
2t2
i,R |
ef f 2
ef f ef f
+2
4
|ti,R | + |ti,R ti,L |
Ei,RR − E0
Ei,RL − E0
(5.90)
where Ei is the energy of the intermediate configuration with both particles removed
from the original site. Here one factor of 2 is, as usual, to account for both left and
right hoppings. In the first term, both particles are hopped to the same site; the
√
factor of 4 = ( 2)4 arises because all hops move a particle either onto or off of a
doubly occupied site. In the second term, one particle is hopped to the right and one
to the left; consequently there is only a single factor of 2 due to double occupancies.
However, the return hop may be executed in the same order as the outward hop
f 2
ef f ef f
(|tef
i,R | term), or the opposite order (|ti,R ti,L |). The form of this correction is the
same for the CGS and for solitonized states.
5.9.2
Bounds on the volume of Devil’s staircase lost at small
t
Here we estimate the total volume of the liquid states in the phase diagram for small
but finite t.
193
First, let us review the situation at t = 0, where the total volume of liquid states
is 0. The range of stability of a Mott lobe with denominator q is given by
X
n
nq (V (nq + 1) + V (nq − 1) − 2V (nq))
(5.91)
To calculate the volume occupied by all rationally filled states, we sum over q. The
multiplicity of each q is the number of rationals between 0 and 1 with denominator
q, which is given by the Totient function φ(q). This gives the volume:
X
φ(q)
X
[qnV (qn + 1) + qnV (qn − 1) − 2qnV (qn)]
(5.92)
n
q
Switching the order of summation in the first term, and letting m = qn + 1, gives:


X X

φ(q) (m − 1)V (m)
(5.93)
m
q|m−1
The other terms in (5.92) can be treated similarly. The expression in square brackets
is just (m − 1), so that the total is
X
X
[(m − 1)2 + (m + 1)2 − 2m2 ]V (m) = 2
V (m)
m
(5.94)
m
which is in fact the threshold of stability (from below) of the integer-filled state.
Now let us calculate the amount of this volume lost to superfluid states at small
t. Since the form of the perturbative corrections is difficult to deduce exactly in q, we
work to linear order in t. Hence from each Mott lobe of denominator q, a swath in
chemical potential of length 2qt has been lost to superfluidity. Summing over q gives:
∆1 = 2t
X
qφ(q)
(5.95)
q≤q0
To avoid subtracting off an infinite correction, we sum only over those values of q for
which the Mott lobe has not, at this value of t, entirely disappeared. The threshold
value of q is given by:
2tq0 =
X
n
2
=
nq
nq
nq
+
−2
3
3
(nq + 1)
(nq − 1)
(nq)3
π q(2 + 3 cot(π/q)2 ) − 3π cot(π/q)(1 + cot(π/q)2 )
3
3q
(5.96)
194
At t = 0, q0 is infinite; hence for small t q0 is very large, and we may expand the
right-hand side to leading order in 1/q0 , giving:
t=
π4
15q05
(5.97)
To complete the calculation, we must also estimate the volume of the lost Mott
lobes:
∆2 =
XX
q>q0
n
nq
nq
nq
φ(q)
+
−
2
(nq + 1)3 (nq − 1)3
(nq)3
(5.98)
For q0 sufficiently large, we may use the bounds:
n
< φ(n) < n
ln(n)C
and take the leading order in q, supposing q0 is large:
X
nq
nq
nq
2π 4
+
−
2
≈
(nq + 1)3 (nq − 1)3
(nq)3
15q 4
n
(5.99)
(5.100)
Now we may use the bounds (5.99) to evaluate the sum over q:
X
q>q0
φ(q)
X 2π 4
2π 4
<
15q 4 q>q 15q 3
(5.101)
0
The last sum can be evaluated exactly in terms of di-Γ functions; to leading order in
1/q0 , we obtain:
4π 4
∆2 <
15q02
(5.102)
Calculating the lower bound is somewhat trickier; however, we may approximate it
as an integral. The integral may be evaluated for integer values of the constant C, so
we should round C up and then calculate. The result is:
1
2q02 ln(q0 )C
(5.103)
Note that here we do not expect the coefficient to be captured by the integral, only
the qualitative behavior in q0 . Hence we have, for some constant α,
4π 4
α
<
∆
<
2
q02 ln(q0 )C
15q02
(5.104)
195
It remains to calculate ∆1 . To do so, we will first prove the identity:
n
n
X
1X
n 3 3 n 2 1 n
kφ(k) =
kµ(k) ⌊ ⌋ + ⌊ ⌋ + ⌊ ⌋
3 k=1
k
2 k
2 k
k=1
(5.105)
where µ(k) is the Mobius function, which takes on values of +1 for prime numbers,
0 for perfect squares, and −1 otherwise. This can be shown by induction in n; the
base case is true since φ(1) = µ(1) = 1. For the inductive step, we must show:
n
n+1 3
n
1X
3 n+1 2 3 n 2
kµ(k) ⌊
⌋ − ⌊ ⌋3 + ⌊
⌋ − ⌊ ⌋
(n + 1)φ(n + 1) =
3 k=1
k
k
2
k
2 k
1 n
1 n+1
⌋ − ⌊ ⌋ + (n + 1)µ(n + 1)
(5.106)
+ ⌊
2
k
2 k
Now, ⌊ n+1
⌋a − ⌊ nk ⌋a vanishes except when k|n + 1, in which case it gives
k
a
a n+1
n+1
−1
−
k
k
(5.107)
Substituting this into the series above, we obtain:
" #
2
X
n+1
1
n+1
n+1 3 1
kµ(k) 3
−3
+1+3
− +
(n + 1)φ(n + 1) =
3
k
k
k
2 2
k|n+1,k<n+1
+(n + 1)µ(n + 1)
n
X
µ(k)
2
= (n + 1)
k
k|n
= (n + 1)φ(n + 1)
(5.108)
where the last equality is a basic identity of Totient functions. (Proved on Wikipedia).
Now we may use the fact that
n
n
n
−1<⌊ ⌋≤
k
k
k
(5.109)
to put bounds on the series. The upper bound is:
n
n
n
n3 X µ(k) n2 X µ(k) n X
+
+
µ(k)
3 k=1 k 2
2 k=1 k
6 k=1
(5.110)
while the lower bound is:
n
n
n
n3 X µ(k) n2 X µ(k) n X
µ(k)
−
+
3 k=1 k 2
2 k=1 k
6 k=1
(5.111)
196
To estimate these contributions, we use the relations:
n
X
µ(k)
6
+ O(1/n)
π2
=
k2
k=1
n
X
µ(k)
< log(n) + 1
2
k
k=1
n
X
µ(k) < n
(5.112)
k=1
Hence at large n, we have
n
X
3
kφ(k) = n
k=1
6
+ O(log(n)/n)
π2
(5.113)
Plugging this into the expression for ∆1 above, we have
∆1 = 2t
∆1
∆1
q0
X
qφ(q)
q=1
4tq03
1
π2
α log(q0 )
1
<
+
+O
q0
q0
3
4tq0
α log(q0 )
1
>
1−
+O
π2
q0
q0
(5.114)
Plugging in for t in terms of q0 , and keeping only the leading term, we obtain:
∆1 ≈
4π 2
15q02
(5.115)
Adding up both contributions, we obtain the following asymptotic form for the
volume of the liquid states in µ:
β
4π 2
α log(q0 )
4π 4 4π 2
α log(q0 )
1−
< ∆1 +∆2 <
1+
(5.116)
+
+
q02 ln(q0 )C 15q02
q0
15q02 15q02
q0
We can see that we expect the result to scale approximately as
5.9.3
1
,
q02
or as t2/5 .
Creation of double occupancies in CGS states: calculation of dipolar and quadrupolar interactions
Here we discuss the formation of double occupancies in the CGS. We will make use
of the fact that a double occupancy can always be formed by pinching about a hole
197
in the CGS – that is, by moving some number of particles on the right of that hole
leftwards, and some number of particles to the left of that hole rightwards.
Recall that by even states, we mean states in which either p or q is even; states
for which both are odd are odd states. In an even state, the pinching is symmetrical;
in an odd state it cannot be. This produces a quadrupolar interaction between DO
in even states, but a dipolar interaction for odd states.
The pinching operation described in Sect. 5.6.1 can be described by:
Ri → Ri − ai
Lj → Lj − bj
(5.117)
where Ri , Lj are the positions of the particles relative to the hole in the initial
configuration from which the double occupancy is formed. (For an odd state, Ri has
one more element than Li .)
Now let us consider interactions between double occupancies. The interaction
energy between two DO formed a distance d apart is given by the difference in energies
of the configuration before and after all distances Ri , Li have been altered, minus the
difference in energy when only a single DO has been formed. The relevant terms are
the interaction energies between particles that have been pinched to form the first
198
defect with particles that have been pinched to form the second:
X
i,j
−[1/(d − Ri − Lj + bi )3 + 1/(d − Ri − Lj + aj )3 ]
−1/(d − Ri − Lj + ai + bj )3 − 1/(d − Ri − Lj )3 ]
−[1/(d + Ri + Lj + bi )3 + 1/(d + Ri + Lj + aj )3
−1/(d + Ri + Lj + aj + bi )3 − 1/(d + Ri + Lj )3
+[1/(d + Ri − Rj + bj − bi )3 − 1/(d + Ri − Rj − bi )3
−1/(d + Ri − Rj + bj )3 + 1/(d + Ri − Rj )3 ]
+[1/(d + Li − Lj + aj − ai )3 − 1/(d + Li − Lj − ai )3
−1/(d + Li − Lj + aj )3 + 1/(d + Li − Lj )3 ]
(5.118)
For the even case, Ri = Li and ai = bi , giving
X
i,j
−[1/(d − Rij + ai )3 + 1/(d − Rij + aj )3 − 1/(d − Rij + ai + aj )3 − 1/(d − Rij )3 ]
−[1/(d + Rij + ai )3 + 1/(d + Rij + aj )3 − 1/(d + Rij )3 − 1/(d + Rij + aj + ai )3 ]
+2[1/(d + Rij − 2Lj + aj − ai )3 − 1/(d + Rij − 2Lj − ai )3
−1/(d + Rij − 2Lj + aj )3 + 1/(d + Rij − 2Lj )3 ]
(5.119)
where Rij = Li + Lj . Each term in square brackets is negative definite, by convexity.
A straightforward manipulation of the standard definition of convexity shows that for
a > b and any x, we have
V (x + a) + V (x + b) > V (x + c) + V (x + (a + b − c)) .
(5.120)
Equally, for 1/r 3 interactions, the last square bracketed term is necessarily less than
the sum of the first two (since these lie ’closer in’, if you will) and hence the overall
interaction is repulsive.
For the odd case, again all terms in square brackets are negative (by convexity).
However, as there are more Ri than Li , there are more terms in the last 2 lines than
199
in the first two. (One more term, to be precise). Since the last 2 lines are negative,
this results in an interaction that is repulsive at long distances.
At short distances, however, the interaction is dominated by the first line, which
gives a positive contribution (and hence the double occupancies repel at sufficiently
short distance scales). Certainly, when max(Ri + Lj ) ≈ d, we expect the potential to
be repulsive and the minimum therefore occurs at some d > d0 = max(Ri + Lj ). The
transition is thus to a density of double occupancies that sit at least d0 sites apart,
and the transition is to a density of double occupancies that is less than 1/d0.
Further, d0 should increase linearly in q. This is easiest to see by means of example,
but basically creating a DO will form hole-like solitons, by which I mean you can
change distances of q to q + 1, but not to q + 2. To arrive at such a configuration
involves a re-arrangement over q sites to the right of the DO (and something like q/2
to the left, seemingly).
As an example, we plot the 2 DO interaction potential as a function of separation
between the doubly occupied site of each DO for the 1/3 and 1/5 filled states in Fig.
5.8. The minima sit at 15 and 75 lattice spacings, respectively.
Hence as U0 is lowered, we expect even states to undergo a second-order phase
transition, as the interaction between DO is quadrupolar and hence repulsive. For
odd states the interaction is attractive, suggesting a first-order phase transition in
which a finite density of defects forms. However, this density of defects vanishes at
least as 1/q, and quite possibly more rapidly than this.
5.9.4
Lattice-scale arguments for adding charge as double
occupancies
Here we present a more complete argument that charges added as solitons have a
stronger mutual repulsion energy than charges added as double occupancies. At long
length scales we may employ a simple intuition based on electrostatics. That is, when
200
−7
4
x 10
3
VD
2
1
0
−1
50
60
70
80
rDO
90
100
110
120
20
30
40
rDO
50
60
70
80
−5
0
x 10
−0.5
−1
VD
−1.5
−2
−2.5
−3
−3.5
10
Figure 5.8: Two-DO interaction potential VD for the 1/3 (lower plot) and 1/5 (upper
plot) filled states. Here rDO is the distance between the two double occupancies in
units of the lattice constant. The minima of these potentials are at rDO ≈ 75 and 15,
respectively, for the 1/3 and 1/5 -filled states.
201
adding charges to the system we must compare both their self-energy (in this case,
the energy of adding a single charge) and their interaction. The interaction energy
between n charges q, spread over some finite interval of length L in the system, is
3
E(n) q Ln . This is always less than the interaction energy between mn charges of
3
charge q/m for any integer m > 1, which is E(m, n) mq Lmn
= m2 E(n). We work in
L
the régime where the self-energy favors doubly occupied states over solitons – hence
the charge will never fractionalize.
Here we outline a lattice argument to show that this intuition gives the correct
result even at short distances. We treat only the case ν ≥ 1/2, as the form of the
doubly occupied states is simpler to describe; however, we expect similar arguments
to hold for arbitrary filling fractions.
Consider adding a finite density of charges to a state with existing double-occupancies.
We break the energy cost of introducing the new particle into two pieces:
E = El + EI
(5.121)
where El is the self-energy, and EI is the interaction energy. The interaction energy
contains the repulsion between the extra charge and any existing double-occupancies.
If the extra charge breaks into solitons, EI also contains repulsion terms between
these solitons. Since we add a finite density of charge, we can consider that each
charge is effectively confined to a finite region of length r in the system.
Consider first the 1/2-filled state. If the charge fractionalizes, two solitons will be
formed. We place these solitons at r1 and r, respectively, from the double occupancy.
Here r1 and r measure the distance from the DO to the outer-most edge of each
soliton. One soliton sits as far from the DO as possible, as the repulsion between the
DO and each soliton is greater than the soliton-soliton repulsion, as a DO has integral
charge while the charge of a soliton is fractionalized. We may think of forming this
arrangement by excising a hole at r1 , pushing the rest of the pattern inwards, and
202
placing an extra particle at r. For r = 10, this looks like:
20101010101
20101101010
20101101011
(5.122)
Note that r1 is necessarily odd, and r here must be even. Then clearly we have:
(r−r1 −1)/2 X
1
1
1
EI (sol) = minr1
−
+ 3
3
3
(r1 + 2n)
(r1 + 2n + 1)
r
n=0
∞
X
1
1
+2
−
(r − r1 + 2n)3 (r − r + 1 + 2n + 1)3
n=0
1
1
+
−
(5.123)
(r − r1 + 2n + 1)3 (r − r1 + 2n + 2)3
The first line here is the interaction between the soliton and the DO; the second line
gives the soliton-soliton repulsion. (One pair of terms for each particle in the soliton).
The sums are all positive, since the 1/r 3 potential is monotonically decreasing. Hence,
the interaction energy from adding a particle as solitons is strictly greater than that
for adding it as a double occupancy:
EI (sol) >
1
= EI (DO) .
r3
(5.124)
Since r above is arbitrary, this argument also applies to the case of adding a
particle to a system with multiple double occupancies in the 1/2-filled state. Hence
in any finite system, we can induct on the number of particles to show that every
filling p/q > 1/2 consists only of double occupancies in the 1/2 filled state, and
never of soliton-like insertions ...0110.... In an infinite system, an infinite number of
particles must be added at once in order to change the filling fraction; in this case
the particles want to spread out homogeneously and hence we again confine them to
within some distance r of existing double occupancies. (Similar arguments show that
the repulsion between four solitons in a finite region is greater than the repulsion
between two double-occupancies, as the latter can spread farther apart).
203
Finally, we may generalize this argument to arbitrary fillings p/q > 1/2. Again
we form solitons by pushing a hole from some radius r1 to the farthest possible radius
r from the DO, and inserting an extra particle at r. This is illustrated below for the
2/3 filled state:
2011011011011011
2011011110110110
2011011101110111
(5.125)
We may hop some (but not all) of the particles back to their original positions to
form solitons, as shown in the third line. The energy of such a state clearly obeys
EI (sol) >
1
= EI (DO)
r3
(5.126)
and hence again, all additional particles will enter as double occupancies. This shows
that the doubly occupied state is locally stable at all length scales.
Chapter 6
Mechanisms for generating
fractional quantum Hall states in
3-d materials
6.1
Introduction
The fractional quantum Hall effect is understood as arising from strong correlation
effects between electrons in quasi-2 dimensional systems. Following several tantalizing threads of experimental evidence on Bismuth and graphite [102, 103], there is
reason to ask how and when such phenomena could be observed in 3D materials in
strong magnetic fields. The fundamental challenge in realizing fractional quantum
Hall physics in such systems is that kinetic terms in the third dimension tend to delocalize electrons between planes, thus destroying the intricate in-plane correlations
associated with the fractional states. In this Chapter we will discuss a new possibility for stabilizing fractional states in layered materials with relatively weak z-axis
hoppings. Specifically, we find that at sufficiently low densities, staging transitions,
in which layers alternately have enhanced or depleted electron density, into fractional
204
205
states are energetically favored. Staging naturally suppresses the z-axis kinetic energy, effectively pinning the staged electrons into a fixed layer. In this way we find
a new set of candidate wave functions for 3D materials which exhibit the striking
fractional conductance plateaux of 2D quantum Hall systems.
Let us begin with a brief review of the quantum Hall effect in 2D systems. When
a 2D electron gas is placed in a strong magnetic field, plateaux are observed in the
transverse resistivity when the ratio of electron density to magnetic flux (or filling
fraction) takes on certain values. These special values are either integers, (the integer
quantum Hall effect) or members of a particular set of fractions (the fractional quantum Hall effect). The integer quantum Hall effect (IQH) arises because, in a strong
magnetic field, electrons essentially occupy non-overlapping cyclotron orbits, whose
radii are determined by the strength of the magnetic field. This results in a series
of Landau levels, each of which contains a fixed number of electronic states. When
a Landau level is fully occupied, the system is an incompressible quantum fluid – an
integer quantum Hall state. Since there is only one way to occupy all states within a
Landau level, these integer states are fundamentally non-interacting.1 At fractional
fillings, however, there is a very large degeneracy within each Landau level. In 2D,
this leads to a series of incompressible states at certain rational filling fractions, stabilized by the Coulomb repulsion between electrons. The simplest of these are the
Laughlin [104] states which arise at filling fraction 1/(2n + 1) for integer n.
In a 3D material, the situation is complicated by the fact that electrons can gain
kinetic energy by dispersing between layers. At integer filling fractions in the quantum
Hall régime, one thus expects ground states in which the electrons propagate freely
along the field direction, while retaining the in-plane cyclotron motion appropriate to
the lowest Landau level. At fractional fillings, however, electrons appear to be faced
with a dilemma: they may either minimize Coulomb repulsion via strong correlations
1
In real systems, perturbatively including interactions typically results in each electron occupying
an admixture of states in different Landau levels; this is known as Landau level mixing.
206
between electrons in a small number of adjacent layers, or they may minimize their
kinetic energy by dispersing along the axis parallel to the magnetic field. As yet, there
is no known electronic state optimizing both.
2
Our objective here is to propose
states which minimizes Coulomb repulsion at the expense of the kinetic energy gained
by dispersing along the direction of the field. We will show that these states are
energetically favored over known competitors in a parameter régime relevant to some
layered crystals, such as graphite.
To motivate this approach, it is useful to review the status of the integer quantum
Hall effect in 3 dimensions. There are two known scenarios for the existence of 3D
states at integer filling in which the dispersion along the field direction is suppressed.
First, the band-structure can conspire to pin electrons to a given layer. Such an
effect has been proposed to occur in graphite [106]. The basic principle here is that
if the Landau gap is large relative to the z-axis hopping, the full non-interacting
3D spectrum can in certain cases have a gap. In this case it is known that the
Hall conductance remains quantized [107]. In graphite, band structure considerations
suggest that, at sufficiently strong magnetic field, the n = 0 and n = 1 Landau levels
of graphite are gapped and a single 3D IQHE plateau exists. The Bernal stacking of
the graphene layers, which strongly suppresses the z-axis hopping, is essential to the
existence of the 3D IQH in this scenario.
Second, integer 3D quantum Hall states can arise via interaction effects. In this
case, we begin with a state consisting of filled Landau levels in each plane and a filled
band in the z direction. It has been shown that adding interactions can lead to a 2kF
instability which opens a gap at the one dimensional Fermi level. The Fermi level is
then pinned in the many-body gap and the system exhibits a Quantum Hall effect.
Though both of the above scenarios describe integer states, they underline two
important features of 3D quantum Hall physics which we will exploit here. First, in
2
An interesting new mean-field approach to this question has, however, recently been proposed
by Levin and Fisher [105].
207
certain cases (most notably in graphite), the crystal structure conspires to naturally
suppress the hopping in one direction. This will prove to be important to the stability
of fractional states which effectively trap charge within a given plane. Second, chargedensity wave instabilities along the field direction are known to exist in 3D systems
under consideration here. Though the mechanism in that case is a higher-order effect
than the staging transitions we propose here, it makes the appearance of staging
transitions – which are simply longer-wavelength charge density waves along the field
direction – seem quite natural. This is encouraging, as we find that without the
staging transition, inter-layer Coulomb repulsion prevents the formation of any of the
strongly correlated states considered here at fractional filling.
To generate 3D fractional quantum Hall (FQH) states which are energetically
plausible, we will draw inspiration from several possiblities discussed in previous work
on semi-conducting quantum well bi-layer and multi-layer systems. These systems
differ from layered crystals in two important ways. First, in semiconducting multilayers the inter-layer spacing is typically not less than the order of the magnetic length.
In real materials, this ratio is much less than one at any realistic value of the magnetic
field. Second, multi-layered semiconductor systems naturally have relatively small
matrix elements for charge transfer between the layers; hence in isotropic materials
we expect that the ratio of the kinetic energy due to dispersion between layers to
interaction energy is much larger than the equivalent quantity in typical multi-layers.
Because of these differences, the electronic arrangements studied in semi-conducting
multi-layers are not directly applicable to layered crystals. Nonetheless they constitute a useful starting point from which to construct candidate variational wavefunctions; hence it is useful to briefly review them here. For a small number of layers,
Halperin [108] proposed a series of states which generalize the Laughlin states to include inter-layer correlations, thus minimizing the Coulomb repulsion of electrons in
different layers. It is easy to generalize both Laughlin’s and Halperin’s wave functions
208
to give a 3D FQH state [109, 110], possibly with correlations between layers. However, MacDonald and coauthors [109] considered the energetics of these 3D states,
and found that they are never the variational ground-states in an layered crystal.
Contrarily, they found that the magneto-plasmon gap closes. The phases become
unstable at this point, and a transition to a 3D crystal state occurs whenever the
distance between the layers is roughly smaller than half the magnetic length. MacDonald [111] also established that in multilayers, in the Hartree Fock approximation,
the gain in correlation energy from distributing electrons unequally between the layers can exceed the electrostatic cost of increased inter-layer Coulomb energy leading
to a staging transition. Ref. [112] proposed a third type of candidate state, the
spontaneous inter-layer coherent miniband state (SILC), consisting of a combination
of lowest Landau level (LLL) states in each layer which forms a band in kz . Unlike
the other states proposed for semi-conducting systems, this last favors potential over
kinetic energy, and hence is the chief competitor of 2D correlated states in layered
materials.
In this Chapter, we propose a new variational ground-state, a correlated 3D FQHstaged state which is energetically competitive in layered crystals where which the
in-plane and out-of plane inter-atomic distances are comparable. We compare four
~ field in the ultra-quantum limit
different potential ground states for materials in B
with strong Coulomb interactions and find that 3D FQH ground-states do not occur
in layered crystals unless they are accompanied, at very low densities, by staging
transitions similar to the ones observed in graphite intercalation compounds [113]
and predicted to occur in multi-quantum-well systems by MacDonald [111]. In the
parameter range relevant to graphite, we obtain the phase diagram as a function of
the electron filling fraction, the ratio of inter-layer separation to magnetic length,
and the magnitude of the c-axis hopping. At high carrier density and zero c-axis
hopping, a (staged) integer quantum Hall liquid is energetically favorable. As the
209
filling in occupied layers decreases below ν = 1, liquid states with correlations between
layers can develop, and become energetically favored because of their decreased interplanar Coulomb energy. At extremely low densities, staged crystal states tend to
be energetically favored. A SILC miniband state becomes favored over the staged
integer liquids, but not their fractional counterparts, at the c-axis hopping relevant
to graphite. Thus we find a parameter regime in which fractionally filled, staged
quantum Hall liquid states are the expected ground states of the layered system.
We begin this Chapter with a brief overview (Sect. 6.2) of the current status of
experiments on layered crystals in high magnetic fields. In Sect. 6.3, we will discuss
the various candidate states which we consider, and explain in brief how their energies
are calculated. Details of the calculations are given in Sect. 6.5 for the correlated
liquid states, and Sect. 6.6 for the crystalline states. In Section 6.4, we discuss the
phase portrait that this suggests for the system, and ascertain the parameter range
in which fractionally correlated states may exist in 3D materials. The key results of
this Chapter were first presented in Ref. [114].
6.2
Experiments on 3D compounds in high magnetic fields
Before presenting our theoretical model in detail, we give a brief overview of the
current experimental puzzles and challenges for 3D quantum Hall systems. We note
that much interesting progress has been made in the area of quantum Hall multilayers
[115, 116], in which, for example, systems with very large (order 200) numbers of 2D
quantum wells have been observed to display integer [117] effect. Here, however, we
focus on experiments in 3D crystals, which are our principal interest.
The integer Quantum Hall effect was first observed in 3 dimensions (3D) in the
Bechgaard salts [118, 119]. These organic superconductors have a quasi- 1D struc-
210
ture of weakly coupled chains; the integer quantum Hall effect was observed only in
strongly layered samples, well described by planes of chains, with very weak interplane coupling. These 3D Quantum Hall states can be explained as a series of 2D
states, weakly coupled so that the band-width in the z-direction be smaller than the
2D Landau gap. So far, 3D FQH states have not been observed in these compounds,
even in layered samples. On general grounds this is not unexpected due to their
low electron mobility and small many-body gap, which tend to destroy the range of
stability of the fractional plateaux.
These difficulties of large disorder and relatively small many-body gap are, however, ameliorated in other less strongly layered materials. Recent experimental interest in graphene, graphite and Bismuth has focused on the Dirac nature of carriers
and on the IQH effect. The Dirac dispersion in these materials gives electrons large
mean-free paths and large Landau gaps. As a result, large IQH plateaus are observed
in graphene, even at room temperature [120].
Recent experiments in Bismuth suggest the possibility of a 3D bulk strongly correlated electron state in the fractional filling regime [102]. The experiment reports
quasi-plateaus in ρxy at fields which are integer multiples of the field required to drive
the system in the quantum limit. These results are unexpected since Bismuth, unlike
graphite, does not have a layered structure. Though it is tempting to attribute these
plateaux to strong correlation effects, these effects can be understood by considering
CDW-like instabilities in the presence of interactions and strong spin-orbit coupling
[121].
The situation in graphite is more promising, however. Largely because of the
Bernal stacking of graphene planes, graphite naturally has a relatively small kinetic
energy scale along the c axis, which favors states with electrons pinned in-plane. The
carrier density in these samples can in principle be tuned to give low electron-like
or hole-like carrier densities in each graphene plane. Experiments on Kish graphite,
211
in which the conductivity is highly suppressed along the c-axis, have shown good
evidence of integer [103] quantum Hall plateaux, and more recently found suggestive
evidence of fractional states [122].
Hence, unlike in semi-conductor multi-layers, where the existence of quantum Hall
plateaux at fractional fillings is well-established, the status of fractional quantum Hall
physics in truly 3D materials is less certain. While the results in both graphite and
bismuth are suggestive, the complicated band structures of both of these materials
makes it difficult to draw definite conclusions from these results. In light of this, it is
especially interesting to ask in what parameter régimes one would theoretically expect
true fractional states, with strong inter-electron correlations leading to fractionalized
charge and statistics.
6.3
Candidate States
We set out to answer this question by constructing a new family of candidate 3D
states at fractional filling – the staged Laughlin and Halperin liquid states. Here we
will describe these, along with their most natural competitors (3D Wigner crystals
and non-interacting states with charge transport along the z axis).
Let us consider a layered crystal subject to a magnetic field parallel to one of its
crystallographic axes. The magnetic field quenches the kinetic energy of electrons in
lattice planes (layers) perpendicular to the magnetic field. A plausible scenario for this
situation, in which the separation between layers is much smaller than the magnetic
length, is that the states will stage, in the manner first proposed by MacDonald
for multi-layer quantum wells [111]. The relatively small inter-layer separations in
layered crystals favor staged states which consist of n layers of electron gas of density
σ e − δσ, and one layer of density σ e + nδσ, where σ e is the initial electron density
per layer, and δσ is the staging density. For crystalline states, staging increases
212
the distance between electrons which lie directly above one another, decreasing the
crystal’s Coulomb energy at small inter-layer separations; for liquid states staging
becomes advantageous when the gain in correlation energy by increasing the electron
density in the filled layers outweighs the Hartree cost of distributing charge unevenly
between the layers. Thus as the separation between layers decreases at fixed filling
per layer, staged states become increasingly energetically favorable.
We calculate the staging Hartree energy as described in [111]. We model the
system as a stack of planes, each of uniform charge density σi , where σi = −nδσ if
i|(n + 1) and δσ otherwise. This charge density is comprised of an immobile positive
background (ionic) charge density σ e in each plane, and a staged electronic charge
density −σ e +σi . n is called the staging number. The 3-d charge density is conveniently
described in Fourier space as a sum of Kroeneker δ terms
ρ(qz ) = ρ1 (qz ) + ρ2 (qz )
V X
ρ1 (qz ) = δσ
δ(qz , 2πj/d)
d j
V X
δ(qz , 2πj/(n + 1)d)
ρ2 (qz ) = −δσ
d j
(6.1)
where V is the total sample volume, and d is the lattice constant in the direction
of the magnetic field. This charge density configuration has a Hartree energy Es =
P 2π
e2
2
qz q 2 |ρ(qz )| , or:
ǫV
z
Es =
e2 (δν)2 d 2
(n + 2n)
ǫl ν0 12l
(6.2)
where ν0 ≡ 2πl2 σ e , and δν = 2πl2 δσ is the electronic filling staged from each layer.
One could also consider multiply staged states, in which three or more different staging
numbers n exist. However, we numerically found that the states of lowest energy are
those with charge distributions of the form in Eqn. (6.1), described by a single staging
number.
213
6.3.1
Staged Quantum Hall Liquids
The first class of candidate states that we consider are the staged liquid states. The
simplest such staged liquids consist of n layers depleted of electrons, and one layer
of electronic charge density σ e + nδσ (which we will call the occupied layer) in a
quantum Hall liquid state. The total energy of a staged liquid state is then:
E=
n
1
El (ν0 − δν) +
El (ν0 + nδν) + Es
n+1
n+1
(6.3)
where n is the staging number, El (ν) is the energy of the liquid at filling ν, Es is the
Hartree energy given by (6.2), ν0 is the mean filling factor, and 0 ≤ δν ≤ ν0 is the
amount of charge staged out of each layer.
As Laughlin states have good in-plane correlation energies, we consider first the
case where the occupied layers are in a Laughlin state. We will refer to Laughlin
states at filling fraction 1/m in the occupied layers as (0, m, 0) states. Since the
filling fraction in these layers is then fixed at a value of ν = 1/m for some odd integer
m, the quantities ν, ν0 and n are related by n + 1 =
ν
.
ν0
Using (6.3), the optimal
staged liquid state for a given ν0 and d/l can be found by optimizing over δν and n.
We find that optimal states in the regimes of interest are fully staged (δν = ν0 ), so
that the first term of the right-hand side of (6.3) vanishes.
At small d/l, the simple uncorrelated liquid state described above gives way to
liquids with inter-layer correlations [109], which we call Halperin liquid states. These
states have wave functions of the form
ψ=
Y
k;i,j
(k)
(zi
(k−1) m1
− zj
)
(k)
(zi
(k+1) m1
− zj
)
Y (k)
(k)
(zi − zj )m2
(6.4)
i<j
where k indexes the layer, and i, j index particles within a layer. We refer to states
of the form (6.4) as (m1 , m2 , m1 ) states. Fig. [6.1] shows the energies for several
of these states as a function of inter-layer separation and density, calculated using
Monte Carlo techniques (see Appendix).
214
−0.25
−0.3
−0.35
−0.4
E
E
010
E
030
E
050
E
070
E
111
E
131
E
−0.45
−0.5
−0.55
151
−0.6
−0.65
0
1
2
d/l
3
4
5
Figure 6.1: Energies of unstaged Laughlin (dashed lines) and Halperin (solid lines)
liquid states as a function of inter-layer separation over magnetic length (d/l). All
2
energies are reported as energies per electron, in units of eǫl , where l is the magnetic length. At large separation the energy increases with d/l as the correlations in
Eqn. (6.4) do not optimally minimize the intra-planar Coulomb interactions and their
energy grows (see text). The minimum energy occurs when correlations effectively
separate electrons from both their in-plane and out-of-plane neighbors.
215
When d/l is very small, one can imagine a low energy ground state of staged
Halperin liquid states, with n layers completely depleted of electronic charge for
every occupied layer at filling ν = 1/(2m1 + m2 ). In the layered system with a given
mean filling per layer ν0 , the staging number is fixed by n + 1 =
1
.
ν0 (2m1 +m2 )
This in
turn fixes the separation (n + 1)d between occupied layers, and the energy of such a
state is
E = El ((n + 1)d) + Es (d, n, ν)
(6.5)
where El ((n + 1)d) is the Coulomb energy of the state (6.4) at inter-layer separation
(n + 1)d, extrapolated from the Monte Carlo results of fig. 6.1, and Es is the Hartree
staging term. Fig. 6.1 shows an important property of the energies as a function of
d/l. The wave function (6.4) contains correlations which separate electrons in adjacent
layers, at the expense of decreasing their average separation in a given layer. Since
these correlations do not depend on d/l (the out of plane correlations in Eqn. (6.4)
(k)
(k−1)
depend on zi −zj
, i.e. only on the in-plane distance between particles in different
layers), as d/l increases the energy of the Halperin states also tends to increase. The
inter-planar Coulomb interaction becomes weaker, so that the energetic gain from
inter-layer correlations decreases, while the energetic cost of decreased separation inplane relative to the Laughlin state of the same filling remains constant. At large d/l
the Laughlin liquid becomes the ground state.
6.3.2
Staged Wigner Crystals
In two dimensions, it is well known that for a low density of carriers, a transition
occurs from a FQH state to a triangular lattice Wigner crystal of lattice constant
√ 1/2
. The Wigner crystal is energetically favored over the FQH liquid state
a = l ν3π
q
at fillings ν < 1/7 [123]. In multi-layer systems at small d (d/l < 0.9 2π
), inter-layer
ν
repulsions favor an in-plane lattice that is square as this permits larger separations
between sites in neighboring planes [109]. As d decreases, multiple phase transitions
216
between different stackings of squares in the vertical direction occur; reference [109]
identifies states in which electrons at the same in-plane co-ordinates are separated by
M = 2, 4 or 5 layers, as well as states with incommensurate stackings. In this work
we consider only true crystal states with well-defined spatial periodicity, and will not
include structures with incommensurate stackings.
Numerical studies [123] indicate that the total energy of these crystalline states is
well approximated by
E = E0 ν 1/2 + E2 ν 3/2 + E3 ν 5/2 .
(6.6)
Here E0 is the classical contribution, given by the energy of the Madelung sum, E2 is
the phonon contribution, and E3 is an extra higher moment contribution calculated
by a fit to a numerical evaluation of the Coulomb energy. For the low fillings at which
Wigner crystals are expected to occur, the total energy is well approximated by the
first two terms.
The classical contribution E0 can be calculated using the Ewald method, described
in detail in [124]. We slightly modify this method to account for the fact that the
background charge is localized in the planes. At each value of d, energies were tabulated for a variety of crystal stackings and stagings; the configurations of lowest
energy were selected. We estimate E3 by numerically extrapolating the results of
[109] as a function of nd/l to the values of interest here ((n + 1)d/l ≈ 0.2 for most
optimally staged crystal states).
We compute the contribution of lattice vibrations E2 by postulating a Lam-Girvin
[123] variational wave function of the form
Ψ = exp
1X
ξi Bij ξj
4 ij
!
Y
φR~ j (~rj )
(6.7)
j
~ j = (Xj , Yj ) is the position of the lattice site, ξj ≡ (xj − Xj ) + i(yj − Yj ) is
Here R
the deviation of the particle’s position from this site in two-dimensional complexified
217
coordinates, and φR~ (~r) are the lowest Landau level coherent states,
1
2
~ 2 2
~
φR~ ~r = √
e−(~r−R) /4l eiẑ·~r×R/2l
2πl2
(6.8)
The correlation matrix B is a variational parameter; minimizing the variational energy
gives
ωL(~k) − ωT (~k) iθ~
e k
ωL (~k) ωT (~k)
2
m∗ l2 X ~
ωL(k) + ωT (~k)
E2 = e
4
B~k =
(6.9)
~k
where ωL and ωT are the longitudinal and transverse phonon frequencies of the classical crystal, respectively, and E is the energy of the phonon modes. The phonon
frequencies are calculated using the method of [125]. For multiple-site unit cells, we
fix Bk as a single function independent of site indices within the unit cell. In this
case the energy is given by (6.9), with ωT , ωL replaced by the square roots of the
eigenvalues of the 2 × 2 matrix formed by averaging the second order correction to
the Coulomb potential over all sites in the unit cell.
Fig. 6.2 shows the energies E0 and E2 of the crystalline states over a range of interlayer separations. At very small d/a, the dominant interaction for a M-layer crystal
stacking is between an electron and its translates M layers above and below; the
associated energy increases roughly as
1
.
Md
To avoid this cost, the crystal undergoes
several transitions to stackings of higher M over the range of inter-layer separations
shown in fig. 6.2 [109]; at sufficiently small d/a even the M = 5 stacking does not
adequately separate charges from their closest vertical neighbors and the energy grows
rapidly. Though we computed the energy of all three crystal structures (2- 4- and
5- layer crystal stackings) at each value of d/a, only the lowest of these energies is
shown in the fig. 6.2.
At the very small values of d/l found in graphite and bismuth, none of these crystal
structures represent a stable ground state. We therefore allow for Wigner crystals with
218
−0.8
E0
−1
−1.2
0
0.2
0.4
d/a
0.6
0.8
1
0.2
0.4
d/a
0.6
0.8
1
4
3
E2
2
1
0
0
Figure 6.2: Calculated energies (per electron) of the unstaged crystalline states, as a
function of d/a where a is the crystal lattice constant. Energies are shown in units of
e2
. Top: Coulomb sum contribution (E0 ). As d/a decreases several phase transitions
ǫl
occur to crystal stacking patterns with a larger unit cell in the z direction, consistent
with the findings of [109]. Bottom: Phonon contribution (E2 ) for the crystal states of
lowest E0 . As d/a decreases the mean inter-electron distance decreases and E2 grows.
staged charge densities, requiring that charge depleted layers be completely emptied
of electrons (δσ = σ e ). The total energy for the staged Wigner crystal is given by
E = Ecrys + Es
(6.10)
where Ecrys is given by (6.6), using the values of E0 and E2 shown in fig. 6.2 at the
effective inter-layer separation (n + 1)d, and Es is given by (6.2) with δν = ν0 . At a
given inter-layer separation d and mean filling ν0 , we find the energetically optimal
Wigner crystal by choosing the staging number which minimizes (6.10).
219
−0.1
(0,m,0) liquid
(1,1,1) Liquid
Miniband, t=0.01
Miniband, t=0.033
Miniband, t=0.05
−0.2
−0.3
E
−0.4
−0.5
−0.6
0.05
0.1
0.15
ν0 0.2
0.25
0.3
Figure 6.3: A comparison of the energies of the (0, 1, 0) and (1, 1, 1) staged liquids
with those of SILC miniband states as a function of mean filling for d/l = 0.1.
2
Energies and t⊥ are shown in units of eǫl . At the physical value t⊥ = 0.033 · e2 /ǫl the
SILC miniband state is energetically favored at high densities, and has lower energy
than the staged integer-filled liquid state. The ν0 dependence of the liquid energies
can be understood from fig. 6.1 by noting that at fixed d/l and occupied filling ν,
(n + 1)d/l = νd/lν0 ).
6.3.3
Miniband States
The final candidate state we consider is the Spontaneous Inter-Layer Coherent State
(SILC) miniband of Hanna, Dı́az-Vélez, and MacDonald [112]. This state allows
electrons to disperse between layers, at the expense of creating states which do not
have the in-plane correlations associated with fractional statistics. Such states can
nonetheless occur at fractional fillings, by partially filling the band created by the
dispersion parallel to the field.
220
The SILC wave function is given by the Slater determinant state
|ψi =
Y
q,X
†
Cq,X
|0i
(6.11)
where the creation operator generates the state
†
h~r|Cq,X
|0i
Np
1 X ijqd
= p
e ψj,X (~r)
Np j=1
ψj,X (~r) = √
1
2πl2
(6.12)
2
2 /2l2
χ(z − jd) eiXy/l e−(x−X)
,
Here, X is the center of the Landau strip, and χ(z − jd) describes the vertical localization of the electron to the j th plane; Np is the number of planes (layers). The
product X is over all states in the lowest Landau level (LLL), but only the lowest
energy states in the momentum band are filled: −πν/d < q ≤ πν/d.
The energy of the SILC is given by [112]
2
Z 2
2t⊥ sin(πν)
νe2 X
d r −r2 /2l2 sin(πjν)
Ec
e
−
=
2
Ne
ǫl j
4πl
πjν
πν
(6.13)
where t⊥ is the hopping matrix element in the stacking direction. Fig. 6.3 compares
the energies of the SILC miniband states at d = 0.1 for several t⊥ values with the
energies of the staged liquid states. The energy of the SILC miniband state depends
2
strongly on the c-axis hopping matrix element t⊥ . For t⊥ = 0.01 eǫl , at the values of d
pertinent to graphite the SILC miniband state is never a ground state, as shown in
2
the fig. 6.3. At intermediate values ( t⊥ = 0.03 eǫl ), the SILC miniband state is the
2
ground state for sufficiently high densities. For t⊥ ≥ 0.05 eǫl , the SILC miniband state
is the ground state for all densities shown.
6.4
Results
We now compare the energies of the three types of states outlined before in a parameter regime experimentally attainable for graphite. We model graphite as a quantum
221
Figure 6.4: Approximate phase portrait for graphite in the range B ≈ 30T . SILC
miniband states are navy blue; the Laughlin liquids are colored dark blue ((0,3,0)
state), pale blue ((0,5,0) state), and turquoise ((0,7,0) state). The Halperin liquids
are colored orange ((1,1,1) state), red ((1,3,1) state), and brown ((1,5,1) state). Crystalline states are shown in green.
Hall multilayer of graphene sheets. Within each graphene plane, we assume that the
magnetic length is sufficiently large that the positive charge of the graphene crystal
is well approximated as a uniform surface charge σ e . Further, we take the interplane separation d to be fixed, with a value of approximately d = 3.4 Å. We consider
relatively strong magnetic fields (B ≈ 30 T).
In graphite, at B = 30 T, the relevant length scales is l = 47 Å, giving d/l ≈ 0.07.
The c-axis hopping in the LLL, as calculated from the c-axis bandwidth of the LLL
[106], is t⊥ ≈ 10 meV and the Landau gap is given by Egap = 30 meV. The energy
scale is set by
e2
ǫ0 l
= 0.31 eV. We assume that the mean filling fraction can be tuned
independently of the magnetic field.
222
The expected phase portrait for the system is shown in fig. 6.4. At constant d/l,
as the mean filling decreases (in real materials, these would be differently doped samples), the ground state shifts from a SILC miniband state through (1, 1, 1), (0, 3, 0),
(1, 3, 1), (0, 5, 0), (1, 5, 1), and (0, 7, 0) liquid states; at sufficiently low ν0 a phase
transition to a staged crystalline state (shown in green) occurs. At physically rele2
vant values of t⊥ ≈ 0.03 ǫe0 l , the SILC miniband state is the ground state only at high
densities where integer-filled Laughlin liquids would otherwise occur.
The structure of the phase portrait reflects the competition between Hartree and
Fock energy contributions. The correlation energy (the Fock term) decreases with
increasing density, and thus always favors maximal staging. As ν0 decreases at fixed
d, the correlation energy of staging to a given filling ν remains fixed, while the Hartree
energy due to staging increases approximately linearly in n+ 1 = ν/ν0 . Hence a series
of transitions to states of lower filling in the occupied layers occurs. Once ν < 1/7,
the crystalline states have lowest energy. At fixed ν, a transition occurs between a
Halperin state at higher ν0 (and hence smaller separation nd between occupied layers)
and a Laughlin liquid at lower ν0 (favored at larger nd). The effect of increasing d
at fixed ν0 is similar: it increases the effective cost of staging, and hence favors less
filled states; at fixed ν a phase transition between Halperin and Laughlin liquid states
occurs.
The crystalline states investigated in this paper have all assumed ‘pancake’ charge
distributions confined to individual lattice planes. However, c-axis hopping will allow
the electron charge distribution to spread out in the z-direction. For weak hopping,
the crystalline states should be stable throughout much of their phase diagram, but
eventually as t⊥ increases, states like the SILC are preferred. In the continuum,
one can define a family of variational three-dimensional crystalline states as generalizations of the two-dimensional quantum Wigner crystal of Maki and Zotos [126],
223
writing
h
i
Ψ = det ψR~ i ,Zi ~rj , zj ,
(6.14)
ψR,Z
~r, z = φR~ ~r ϕ(z − Z)
~
(6.15)
where ~rj is the electron position projected onto the (x, y) plane, and z is the c-axis
coordinate, and where
is a product of the in-plane lowest Landau level coherent state and a trial single
particle wavefunction describing the localization of the electron along the c-axis. An
obvious choice for ϕ(z) would be a harmonic oscillator wavefunction with a length
scale λ that is a variational parameter, determined by the local curvature of the
self-consistent crystalline potential. In this paper, we underestimate the stability of
our crystalline states; allowing the electrons to delocalize somewhat in the transverse
dimension will lower their energy. We will report on calculations based on these
states, and their obvious extensions to correlated Wigner crystal states described in
Eqn. 6.7, in a future publication.
We conclude that fractionally filled quantum Hall states cannot occur as stable
~ fields for which the magnetic length
ground-states in layered crystals under realistic B
is much larger than the c-axis lattice constant unless the density is sufficiently low
so that a staged FQH state becomes energetically favorable. In graphite, this corresponds to an electronic carrier density on the order of 1014 electrons/m2 in each
layer, or 3 × 1017 electrons/cm3 – approximately an order of magnitude less than the
typical low-temperature mobility in graphite. At even smaller densities, the FQH
staged ground-states give way to a staged 3D Wigner crystal with large unit cells in
the direction parallel to the magnetic field.
224
6.5
Supplementary Material: Monte Carlo methods for quantum Hall States
To use the Monte Carlo method initially used by Laughlin for FQH systems [104] in
a layered system, we chose periodic boundary conditions and compute the Coulomb
energy of each configuration as a Ewald sum over repeated copies of the fundamental
cell. Particle configurations are generated with a probability dictated by the squared
wave function; the energy is calculated by averaging the Coulomb potential over a
large number of configurations. The fundamental cell is a square in plane of side
p
length 2πl2 N2D /ν, where N2D is the total number of particles per layer. These
dimensions are chosen to enclose the maximum possible area over which the distribu-
tion of electrons in a droplet is essentially uniform, minimizing boundary effects. At
each step in the simulation, only particles which are found within the fundamental
cell are included in calculating the energy, though particles outside this region may
return at a later Monte Carlo step.
The vertical dimensions of the fundamental cell are determined by the minimum
number of layers which gives accurate energies. The height of the unit cell Nlayers d
must be at least several times the mean in-layer inter-particle spacing; otherwise
the computed energy will be artificially high due to particles Nlayers + 1 layers apart
which lie directly above each other. At very small d this limits the accuracy of the
simulations, as extremely large numbers of layers must be used to obtain reasonable
values.
6.6
Supplementary Material: the Ewald Method
Here we will outline the Ewald method, and explain the minor modifications to it
necessary to work with a neutralizing background charge confined to layers. This
225
section is largely based on the approach of Ref. [124], which also includes some of
the relevant proofs.
2
The basic idea of the Ewald method is to utilize a convergence factor e−sr to
compute lattice sums of electrostatic interactions efficiently. The convergence factor
renders the sums absolutely convergent, allowing several manipulations to be made
in order to express the sum as a rapidly converging series. After subtracting off the
effect of a neutralizing background charge, the limit s → 0 can be taken to yield a
finite result.
Lattice sums
The total electrostatic energy of the state is given by
E=
1 e2 X αi αj
2 4πǫ i6=j Rij
(6.16)
where the factor of 1/2 stems from the fact that each interaction is counted twice.
Here αj is the charge at the j th site in the unit cell, in units of e, and Rij is the
distance between sites. This gives an energy per particle
1 e2 X αi αj
E(ri ) =
2 4πǫ R 6=0 Rj
(6.17)
j
We begin by computing the above sum with an exponential convergence factor.
Consider, therefore, the sum
1
e2 X
1
2
E(rk ) = αk
αj e−sRi
2 4πǫ i,j
|Ri + rj |
(6.18)
where Ri are the lattice vectors of a Bravais lattice, and rj , j = 1..d are the positions
of the sites in the unit cell (relative to ri , which we will take to sit at the origin).
This gives the classical Coulomb energy per particle in the lattice. The parameter s
renders the sum absolutely convergent; the final answer will be obtained by taking
the limit s → 0.
226
We use the Laplace transform
1
1
=√
x
π
Z
2
dte−tx t−1/2
(6.19)
to re-write the expression above as
Z
e2 αk X
2
2
√
E=
αj dte−s(Ri ) e−t|rj +Ri | t−1/2
4πǫ 2 π i,j
(6.20)
We now divide the integration over t into two domains:
Z
∞
−sR2i −t|Ri +rj |2 −1/2
dte
e
t
=
0
Z
η2
−sR2i −t|Ri +rj |2 −1/2
dte
e
t
+
Z
∞
2
2
dte−sRi e−t|Ri +rj | t−1/2
η2
0
(6.21)
The parameter η 2 > 0 is arbitrary, and in practice is chosen to give the fastest possible
convergence of the series.
The second integral will lead to an absolutely convergent sum, and can be computed directly:
1
lim √
s→0
π
Z
∞
2
2
dte−sRi e−t|Ri +rj | t−1/2 =
η2
1
erfc(η|Ri + rj |)
|Ri + rj |
(6.22)
The first integral could be computed directly, but the resultant sum diverges as i →
∞, s → 0. This divergence is canceled by an equal and opposite divergence from the
lattice sum (where all charges are positive). To see that this is so, we re-express the
lattice sum in momentum space, using the Poisson summation formula
X
f (Ri + rj ) =
Ri
π 3/2 X
F̃ (2πKi )e2πiKi rj
V0 K
(6.23)
i
where Ri are lattice vectors, r is any element of a basis for the unit cell, and Ki
the corresponding reciprocal lattice vectors, with Ri Kj = δij . Here F̃ is the Fourier
transform of f .
To exploit this formula, we re-write the integrand as
2
2
2
2
e−sRi e−t|Ri +rj | t−1/2 = e−(s+t)|Ri +t/(s+t)rj | e−ts/(s+t)rj t−1/2
(6.24)
227
Choosing the lattice vectors to be Ri =
√
s + tâi , Ki =
√ 1 bˆi
s+t
with â, b̂ the basis
vectors of the original (in this case one-dimensional) lattice, one obtains:
Z η2
αk X
2
2
√
αj
dte−(s+t)|Ri +t/(s+t)rj | e−ts/(s+t)rj t−1/2
2 π ij
0
Z η2
t
παk X
2η
2 2
2
=
αj
dte−π ki /(s+t) e2πi s+t ki rj e−ts/(s+t)rj t−1/2 (t + s)−3/2 − αk2 √ (6.25)
2V0 ij
π
0
The last term comes from the fact that the Ri = 0, rj = 0 term is omitted from the
original position space sum, but is included in the identity (6.23). Hence we subtract
this contribution from the momentum space part.
Now, if Ki 6= 0, the resulting integral over t is finite and the limit s → 0 may be
interchanged with the integral [124]. The result is
αi X
1
2 2
2
αj e−π Ki /η e2πiKi rj
V0 K 6=0
πKj2
(6.26)
i
The Ki = 0 term must be treated separately. It gives
Z η2
παk X
2
αj
dte−ts/(s+t)rj t−1/2 (t + s)−3/2
Φ0 =
V0 j
0
Substituting u =
q
t
s+t
(6.27)
gives
Z η2 /(s+η2 )
2παk X
2
Φ0 =
αj
due−surj
V0 s j
0
(6.28)
In the limit s → 0, this gives:
Φ0
The divergent
1
s
s 2
2παk X
s
2
=
αj 1 − 2 − rj + O(s )
V0 s j
2η
3
X
2π
π
2π X
=
αk αj
−
−
αi αj rj2 + O(s2 )
2
V0 s V0 η
3V0 j
j
(6.29)
term is canceled by adding a neutralizing background charge. De-
pending on the geometrical distribution of the neutralizing charge, the second term
may or may not be canceled. (For example, in an ionic crystal where the neutralizing
228
charge is incorporated in the lattice sum, both of these terms vanish upon summing
over j.) The last term gives a finite correction proportional to the dipole moment of
the distribution of charge within the unit cell.
Thus the net energy of the k th particle in the unit cell is given by:


n
2
X
X
1
1 e 
Ec (k) =
αk αj 
erfc(η|Ri + rj |)
2 4πǫ j=1
|Ri + rj |
(6.30)
Ri +rj 6=0
1 X −π2 Ki2/η2 2πiKi rj 1
2η
2π
π
2π 2
+
r −√
e
e
+
−
−
V0
πKj2
V0 s V0 η 2 3V0 j
π
K 6=0
i
where n is the number of sites in the unit cell.
Stacked neutralizing background
We wish to consider graphite as a collection of stacked graphene planes, each of
which is in a quasi two dimensional quantum Hall state. As such, we wish to assume
that the neutralizing background charge which compensates for the electron lattice
is distributed uniformly within the graphene planes, but is confined to these planes.
To account for this, we must compute the sum
Z
∞
−s(r 2 +Zi2 )
X
e2 αk
2 e
E(k) =
ρ0 d r p
4πǫ
r 2 + Zi2
i=−∞
(6.31)
We also wish to consider staged states, in which there are n crystalline planes
of electronic charge density ρ − δ, and one liquid plane of electronic charge density
ρ0 + nδ. If we ignore in-plane correlations between the liquid and crystalline layers,
then this state can be described as the sum of a Wigner crystal and a layered state.
The layered state has a uniform in-plane charge density ρ(x, y, i) ≡ ρ0 (i) in the ith
layer, with n layers of a neutralizing background charge ρ0 and an n + 1st layer of
charge density −nδρ. Hence it is useful to consider a slightly more genera version of
(6.31), given by
E(k) =
∞ X
n
X
αk e2
i=−∞ j=0
4πǫ
ρ0 (j)
Z
2
2
e−s(r +Zi )
d2 r p
r 2 + (Zi + zi )2
(6.32)
229
where ρ0 (i) is the background charge in the ith layer. Here the exponential term is a
convergence factor, and the limit s → 0 must be taken to attain the final result. We
allow for an n + 1 site unit cell in the vertical direction, to account for staged states.
Following the usual Ewald method, we use the Laplace transform
1
1
=√
x
π
Z
2
dte−tx t−1/2
(6.33)
to re-write the expression above as
Z
∞ Z
e2 X ρ0 (j) X
2
2
2
2
2
√
E=
d r dte−s(r +Zi ) e−t(r +(Zi +zj ) ) t−1/2
4πǫ j
π i=−∞
(6.34)
Since the convergence factor ensures that the both integrals converge absolutely,
we may interchange the order of integration above without affecting the final result.
Performing the integral over r gives:
∞ Z
X
√ αk e2 X
2
2
ρ0 (j)
dte−t(Zi −zj ) e−sZi t−1/2 (s + t)−1
E= π
4πǫ j
i=−∞
(6.35)
Now we must divide the integration over t into two domains:
Z
∞
0
t
ts 2
dt
2
√
e−(s+t)(Zi + s+t zj ) e− s+t zj =
t(s + t)
Z
η2
t
ts
2
2
dte−(s+t)(Zi + s+t zj ) e− s+t zj t−1/2 (s + t)−1
0
+
Z
∞
t
ts
2
2
dte−(s+t)(Zi + s+t zj ) e− s+t zj t−1/2 (s + t)−1
η2
(6.36)
The second integral will lead to an absolutely convergent sum, and the limit s → 0
may be taken without further ado.
√
lim 2 π
s→0
Z
rdr
Z
∞
η2
t
ts 2
−(s+t)(Zi + s+t
zj )2 − s+t
zj −1/2
dte
e
t
(s + t)
−1
√
2 π −(Zi +zj )2 η2
=
e
− 2π|Zi + zj |erfc(η|Zi + zj |)
η
√
=
π
Z
∞
2
dte−t(Zi +zj ) t−3/2
η2
(6.37)
Again, it proves convenient to do the first integral in momentum space. Choosing
√
1 ˆ
bi with â, b̂ the basis vectors of the
the lattice vectors to be Ri = s + tâi , Ki = √s+t
230
original (in this case one-dimensional) lattice, one obtains:
Z
n
√ XX
π
ρ0 (j)
i
η2
t
2
ts
2
dte−(s+t)(Zi + s+t zj e− s+t zj t−1/2 (s + t)−1
0
j=0
n
XX
π
=
ρ0 (j)
(n + 1)h i j=0
Z
η2
dte−π
2 k 2 /(s+t)
i
e2πiki zj t−1/2 (t + s)−3/2
(6.38)
0
where (n + 1)h is the size of the unit cell in the stacked direction.
The divergences of (6.38) are in the k = 0 contribution, and can be treated as in
the lattice sum above. In the limit s → 0, we obtain
n
X
j=1
ρj
(n + 1)h
2π
π
2π
− 2+
|zj |2
s
η
3
(6.39)
The remaining series is convergent, and the limit s → 0 may be taken to obtain
X X
1
1
2 2
2
ρj
e−π kz /η e2πikz rj 2
(n + 1)h j
πkz
(6.40)
kz
We must also add the self-energy of the layered state. If the planar layers are
infinite in extent there is no difference between the field felt by a point charge in a
layer, and that felt by a uniform surface charge contained in a finite area. Hence the
sums and integrals are identical to the point-charge case considered above; the only
difference is that the charge αk of the k th lattice site must be replaced by the charge
per unit lattice area ρk A0 .
Combining these with the result for the lattice, and averaging over the sites in the
231
unit cell, we obtain the mean electrostatic energy per particle
( m
"
#
m
X
4πǫ
1
1
1X X
−π 2 |Ki |2 /η2 2πiKi rjk
Ec =
αk αj
erfc(η|Ri + rjk |) +
e
e
e2
2 k=1 j=1
|Ri + rjk |
πV0 |Ki |2
i
)
m
X
2π
π
2η
2π
−
|rjk |2 − √
+
αk αj
−
2
sV
V
η
3V
π
0
0
0
j=1
(
m
n
X
X
X 2 √π
−(Zi +zjk )2 η2
+
αk ρj
e
− 2π|Zi + zjk |erfc(η|Zi + zjk |)
η
j=1
i
k=1
"
#)
n
X
1
αk X
2π
π
2π
2
2
2
+
ρj
e−π kz /η e2πikz zjk 2 +
− 2−
|zjk |2
nh j=1
πk
s
η
3
z
kz
( n
√
n
X 2 π
1X X
−(Zi +zj )2 η2
+
A0 ρk ρj
e
− 2π|Zi + zj |erfc(η|Zi + zj |)
2 k=1 j=1
η
i
"
#)
n
X
X
2π
2π
A0 ρk
1
π
2 2
2
ρj
e−π kz /η e2πikz zj 2 +
(6.41)
+
− 2−
|zj |2
nh j=1
πkz
s
η
3
k
z
Here Zi are the lattice vectors for the repeating unit cell in the stacking direction,
and zi the vertical positions of the other stacked planes. m denotes the number of
sites in the unit cell, and n the number of layers after which the layered state repeats.
Charge neutrality ensures that the coefficients of the terms
2π
s
−
π
η2
cancel. The
overall coefficient of these terms is
m
m
m
n
n
n
1 X X αj X X
ρj
1 XX
ρj
αk +
αk
+
A0 ρk
2
V0
(n + 1)h 2
(n + 1)h
j=1
j=1
j=1
k=1
(6.42)
k=1
k=1
When summing over all sites in the unit cell (where this necessarily includes a sum
over all layers) the condition for charge neutrality is
m
X
k=1
and hence, using A0 =
V0
(n+1)h
αk = −
n
X
A0 ρk
(6.43)
k=1
as the area of the 2-d unit cell, the quantity (6.42)
vanishes.
Further simplifications can be made when we further restrict the form of the
charge distribution. For clarity, in the following we will restrict to the case where the
232
charge αk is the same at each site on the crystal, and assume that the crystal unit cell
contains only one site in each layer. (The first condition holds in all cases of interest
since each site of the Wigner crystal contains one electron; the second condition
is generally applicable since the known preferred structures in-plane are square or
triangular lattices. The generalization to multiple-site unit cells is straightforward.)
We postulate that the charge distribution has the form
α
− δρ in crystalline layers
A0
= nδρ in liquid layers
ρj = −
where A0 ≡
V0
h
(6.44)
is the area of the in-plane unit cell. The layering consists of n
crystalline layers followed by one liquid layer, and repeats after n + 1 layers.
Under these assumptions, the interaction energy of the crystal with the layered
state partially cancels the interaction energy of the layered state with itself. In particular, the sum over crystalline layers has the form:
"
#
X
n
X
X
1
α − (α + A0 δρ)
ρj
F (Zizjk ) +
G(kz , zjk )
2
j=0
i
kz
"
#
n
X
X
1 X
=
α
ρj
F (Zi zjk ) +
G(kz , zjk )
2 j=0
i
kz
"
#
n
X
X
X
1
ρj
F (Zizjk ) +
G(kz , zjk )
− A0 δρ
2
j=0
i
k
(6.45)
z
Substituting the charge distribution (6.44) into the last expression gives
"
#
n−1 X
X
X
1 2
− α − (A0 δρ)2
F (Zi zjk ) +
G(kz , zjk )
2
j=0
i
kz
"
#
X
X
nδρ
+
(α − A0 δρ)
F (Zi znk ) +
G(kz , znk )
2
i
k
z
(6.46)
This is the energy of the k th layer interacting with the entire layered state (including
liquid planes). Here zjk the distance of the j th layer to the k th layer within the unit
233
cell. The final missing contribution comes from the interaction of the liquid layers
with the entire layered sate. These further contribute, for each unit cell in the vertical
direction,
1
2
(
(−α − A0 δρ)nδρ
n−1
XX
i
+ A0 (nδρ)
F (Zi , zjn ) +
j=0
2
X
i
X
G(kz , zjn )
X
G(kz , 0)
kz
F (Zi , 0) +
kz
!
!)
!
n−1
X
X
X
1
= αnδρ
−
F (Zi , zjn ) −
G(kz , zjn )
2
j=0
i
kz
"
!
!#
n−1
n−1
X
X
X
X
A0 n
nF (Zi , 0) −
F (Zi, zjn ) +
nG(kz , 0) −
G(kz , zjn )
(δρ)2
+
2
i
j=0
j=0
k
z
(6.47)
Here the sum over zjn averages over all possible distances in the unit cell from the
liquid layer.
Combining the two contributions (6.46) and (6.47), and summing over all layers
234
in the unit cell, gives
( n−1
"
#
4πǫ
1
1
1 X 2 X
2
2
2
E =
α
erfc(η|Ri + rjk |) +
e−π |Ki| /η e2πiKi rjk
2
e2
2 j,k=0
|R
+
r
|
πV
|K
|
i
jk
0
i
i
)
2η
2πα2 X
−
|rjk |2 − |zjk |2 − α2 √
3V0 j
π
( n−1 n−1 "
X X X 2√π
1 α2
2 2
2
− A0 (δρ) )
e−(Zi +zjk ) η
− (
2 A0
η
j=0
i
k=0
−2π|Zi + zjk |erfc(η|Zi + zjk |))
#)
X
1
1
2
2
2
+
e−π kz /η e2πikz zjk 2
(n + 1)h k
πkz
z
−A0 n(δρ)2
n−1
X
j=0
1
+ A0 n2 (δρ)2
2
π 2
−
A (δρ)2
3V0 0
X
F (Zi , zjn ) +
i
"
X
X
G(kz , zjn )
kz
F (Zi, 0) +
i
X
G(kz , 0)
kz
( n−1
X
j,k=0
|zjk |2 − 2n
n−1
X
k=1
|zkn |2
!
#
)
(6.48)
where zjk is the distance between the j th and k th layers, and
F (Zi, zj ) =
X 2√π
i
G(kz , zj ) =
η
−(Zi +zj )2 η2
e
− 2π|Zi + zj |erfc(η|Zi + zjk |)
X
1
1
2 2
2
e−π kz /η e2πikz zj 2
(n + 1)h k
πkz
(6.49)
z
The total energy (6.48) conveniently separates into a lattice contribution (proportional to α2 ) and a stacking contribution (proportional to δ 2 . The latter is precisely
the staging energy calculated by [111].
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