QUANTUM HALL FERROMAGNETS

AKSHAY KUMAR

A DISSERTATION

PRESENTED TO THE FACULTY

OF PRINCETON UNIVERSITY

IN CANDIDACY FOR THE DEGREE

OF DOCTOR OF PHILOSOPHY

RECOMMENDED FOR ACCEPTANCE

BY THE DEPARTMENT OF

PHYSICS

ADVISER: SHIVAJI L. SONDHI

APRIL 2016

c© Copyright by Akshay Kumar, 2016.

All rights reserved.

Abstract

We study several quantum phases that are related to the quantum Hall effect. Our initial

focus is on a pair of quantum Hall ferromagnets where the quantum Hall ordering oc-

curs simultaneously with a spontaneous breaking of an internal symmetry associated with

a semiconductor valley index. In our first example – AlAs heterostructures – we study

domain wall structure, role of random-field disorder and dipole moment physics.

Then in the second example – Si(111) – we show that symmetry breaking near several

integer filling fractions involves a combination of selection by thermal fluctuations known

as “order by disorder” and a selection by the energetics of Skyrme lattices induced by

moving away from the commensurate fillings, a mechanism we term “order by doping”.

We also study ground state of such systems near filling factor one in the absence of valley

Zeeman energy. We show that even though the lowest energy charged excitations are charge

one skyrmions, the lowest energy skyrmion lattice has charge >1 per unit cell.

We then broaden our discussion to include lattice systems having multiple Chern num-

ber bands. We find analogs of quantum Hall ferromagnets in the menagerie of fractional

Chern insulator phases. Unlike in the AlAs system, here the domain walls come naturally

with gapped electronic excitations.

We close with a result involving only topology: we show that ABC stacked multilayer

graphene placed on boron nitride substrate has flat bands with non-zero local Berry curva-

ture but zero Chern number. This allows access to an interaction dominated system with a

non-trivial quantum distance metric but without the extra complication of a non-zero Chern

number.

iii

Acknowledgements

First of all, I would like to thank my adviser, Shivaji Sondhi, for giving an engineering stu-

dent a chance to explore theoretical physics. I am grateful to have had Shivaji as a teacher

and counselor. He has always been incredibly patient with me. I have also been fortunate

to have closely collaborated with many extremely bright physicists. Sid Parameswaran’s

boundless energy and Rahul Nandkishore’s speed of project execution always amazed me.

Working with Rahul Roy was simultaneously frustrating and exciting! I also thank my

undergraduate adviser Sankalpa Ghosh for motivating me to pursue a PhD.

I would also like to thank Ravindra Bhatt for reviewing this dissertation, as well as

Simone Giombi and Waseem Bakr for serving on my defense committee. I thank Thomas

Gregor for taking me on as an experimental project student. Moreover, I am indebted to

the support staff of the Physics Department for greatly simplifying my life as a graduate

student. They made it painless for me to attend conferences and summer schools, where

I had the chance to meet folks outside of the Princeton bubble; Boulder will always be a

favorite of mine.

Life at Princeton has not only been about research. I thank all of the wonderful friends

I have met here. Dining out at random restaurants with Vedika and Chaney was always fun.

I am grateful to them for proofreading parts of my thesis, and I owe it to Vedika for making

Dresden tolerable! Discussions with Liangsheng and Bin about life outside of physics were

always enlightening. I would like to especially thank my large Indian group of friends for

the movies, sports and board games on weekends.

Lastly I thank my parents for believing in me at all stages of my life. This dissertation

would not have been possible without their unending encouragement.

iv

Publications associated with this dissertation

1. Akshay Kumar and Rahul Nandkishore. Flat bands with Berry curvature in multi-

layer graphene. Phys. Rev. B 87, 241108(R) (2013)

2. Akshay Kumar, S.A.Parameswaran and S. L. Sondhi. Microscopic theory of a quan-

tum Hall Ising nematic: Domain walls and disorder. Phys. Rev. B 88, 045133 (2013)

3. Akshay Kumar, Rahul Roy and S. L. Sondhi. Generalizing quantum Hall ferromag-

netism to fractional Chern bands. Phys. Rev. B 90, 245106 (2014)

4. Akshay Kumar, S.A.Parameswaran and S. L. Sondhi. Order by disorder and by dop-

ing in quantum Hall valley ferromagnets. Phys. Rev. B 93, 014442 (2016)

Materials from this dissertation have been presented at the following places:

1. APS March Meeting 2014, Denver, CO

2. Seminar Series, Max Planck Institute for the Physics of Complex Systems

(MPIPKS), Dresden, Germany

3. Schlumberger-Doll Research Center, Cambridge, MA

v

To my parents.

vi

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

1 Introduction 1

1.1 Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Chern Insulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Quantum Hall Ferromagnet . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 Quantum Hall Valley Ferromagnet . . . . . . . . . . . . . . . . . . . . . . 12

1.5 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Microscopic Theory of a Quantum Hall Ising Nematic: Domain Walls and

Disorder 18

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Overview: Phases, Transitions and Transport . . . . . . . . . . . . . . . . 22

2.3 Microscopic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3.1 Hartree-Fock Formalism . . . . . . . . . . . . . . . . . . . . . . . 26

2.3.2 Estimates of Tc . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3.3 Properties of Sharp Domain Walls . . . . . . . . . . . . . . . . . . 31

2.3.4 Does the Dipole Moment Matter? . . . . . . . . . . . . . . . . . . 36

vii

2.3.5 Domain Wall Texturing . . . . . . . . . . . . . . . . . . . . . . . . 38

2.4 Disorder in the Microscopic Theory . . . . . . . . . . . . . . . . . . . . . 41

2.4.1 Random Fields from Impurity Potential Scattering . . . . . . . . . 41

2.4.2 Estimating Disorder Strength from Sample Mobility . . . . . . . . 43

2.5 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3 Order by Disorder and by Doping in Quantum Hall Valley Ferromagnets 49

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2 Silicon(111) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2.1 Effective Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 52

3.2.2 ν = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.2.3 ν = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.2.4 ν = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4 Silicon(110) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.5 Group-theoretic analysis of symmetry breaking . . . . . . . . . . . . . . . 59

3.5.1 Four-Valley Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.5.2 Six-Valley Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.6 Valley Skyrmion Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.6.1 Analytics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.6.2 Numerical Minimization . . . . . . . . . . . . . . . . . . . . . . . 68

3.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4 Generalizing Quantum Hall Ferromagnetism to Fractional Chern Bands 73

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.2 A Special flat C=2 band at 1/2 filling . . . . . . . . . . . . . . . . . . . . 75

4.3 Other flat C=2 bands at 1/2 filling . . . . . . . . . . . . . . . . . . . . . . 79

viii

4.4 Generalization to higher Chern bands . . . . . . . . . . . . . . . . . . . . 83

4.5 Fractional states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5 Flat bands with local Berry curvature in multilayer graphene 86

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.2 ABC stacked graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.3 Effect of BN substrate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.4 Details of Bandstructure Calculation . . . . . . . . . . . . . . . . . . . . . 95

5.5 Chern number from adatoms . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6 Conclusion 99

A Theta Functions 101

A.1 Basic Theta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

A.2 Modified Theta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Bibliography 102

ix

List of Tables

4.1 Analogies between gas and lattice systems. . . . . . . . . . . . . . . . . 73

x

List of Figures

2.1 (a) Model band structure used in this chapter, appropriate to describing

AlAs wide quantum wells. (b) Different phases as determined by com-

paring Imry-Ma domain size ξIM to sample dimensions LS . Top: For

ξIM � LS we find the QHRFPM. Bottom: For ξIM � LS the system

is dominated by the properties of a single domain, and is better modeled as

a QHIN. At intermediate scales, LS ∼ ξIM there is a crossover. . . . . . . . 20

2.2 Phase diagram as function of temperature (T ) and disorder strength

(W ), showing behavior of conductivity. The phases and critical points

are defined in the introduction. . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3 Valley symmetry-breaking field permits transport to probe the energy

scales of the QHIN/QHRFPM. (Inset) Domain structure as function of

disorder strength and valley splitting; dashed line shows a representative

path in ∆v leading to a transport signature similar to that in the main figure.

∆∗v is the valley splitting for which the system is single-domain dominated. 26

2.4 Mean-field and NLσM estimates of Tc. Dashed line shows the anisotropy

(λ2 ≈ 5.5) appropriate to AlAs. . . . . . . . . . . . . . . . . . . . . . . . . 31

2.5 (a) Surface tension and (b) dipole moment of a sharp DW as a function of

the effective mass anisotropy. Dashed line shows the anisotropy (λ2 ≈ 5.5)

appropriate to AlAs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

xi

2.6 Domain-wall texturing from Hartee-Fock Theory. (Top) Contour plot of

the average in-plane valley pseudospin 〈Sx〉 per unit magnetic length along

the domain wall, as a function of the mass anisotropy λ2 and the valley

Zeeman field gradient g, with the latter on a logarithmic scale. The dashed

line marks the anisotropy λ2 ≈ 5.5 relevant to AlAs; note that there is still

some texturing in this limit. (Bottom) Cut along dashed line, with g on a

linear scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.1 Valley ordering in Si(111) QH states. (Inset) Model Fermi surface. El-

lipses denote constant-energy lines in k-space. (Main figure) Schematic

global phase diagram, showing how the G = [SU(2)]3 o D3 symmetry

is broken to H0, HT at zero and finite temperature. The order parameter

spaces are O = G/HT for T > 0, and O = HT/H0 at T = 0. For ν = 1, 2,

D3 symmetry breaks continuously at Tc, but this becomes first-order around

ν = 3. Near ν = 2, 3 order by doping yields to thermal order-by-disorder

at T ∼ T ∗E-S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2 Possible valley-ordered states at ν = 1, 2, 3, including representatives

of Class I and II states for ν = 2, 3. Unfilled and fully-filled valleys are

shown as empty and filled ellipses; valleys partially-filled due to a particu-

lar choice of SU(2) vector within the two-valley subspace are shaded with

different colors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.3 Model Fermi surface and possible valley-ordered states for Si(110)

quantum wells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.4 Unit cell Γ of a skyrmion lattice with L = 1/√

sin γ. . . . . . . . . . . . . 68

xii

4.1 (a) Lower band Chern flux distribution over the Brilliouin zone for the

single-particle Hamiltonian Ho with m = −1.8. (b) Low energy many-

body spectrum for 8 fermions on a 4 × 4 lattice for the case of the single-

particle part of Hamiltonian chosen as Ho with m = −1.8 and V = 3U .

(Energies are resolved using total many-body momenta (Kx, Ky) which

are in units of 1/a.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.2 Ising ordered ground state for Hproj. at half filling. . . . . . . . . . . . . 77

4.3 2 species of domain wall considered in the main text. . . . . . . . . . . . 80

4.4 (a) Lower band Chern flux distribution over the Brilliouin zone for the

single-particle Hamiltonian H ′o with m = −1.8. (b) Low energy many-

body spectrum for 8 fermions on a 4 × 4 lattice for the case of the single-

particle part of Hamiltonian chosen as H ′o with m = −1.8 and V = 3U .

(Energies are resolved using total many-body momenta (Kx, Ky) which

are in units of 1/a.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.5 Spectra of the Hamiltonian in Equation 4.3 with m = −1 and choice

of orientation of domain wall as shown in Fig. 4.3(b). (a) t = 0. (b)

t = 0.3. (Momentum along domain wall is in units of 1/√

2a.) Similar

results are obtained for the other choice of V (~i). . . . . . . . . . . . . . . . 85

5.1 Low energy band structure for N layer ABC stacked graphene in the

presence of a vertical electric field. The band structure is plotted in the

vicinity of the ~K point, assuming that the potential difference between the

top and bottom layers ∆ = 0.167t0 ≈ 50meV . . . . . . . . . . . . . . . . 90

5.2 Bandwidth Λ of the lowest conduction band for N layer chiral

graphene. For N > 5, the bandwidth comes mainly from umklapp

scattering at the zone boundary. . . . . . . . . . . . . . . . . . . . . . . . . 92

5.3 Dispersions of the three lowest conduction bands for N = 7 along the

high symmetry directions. . . . . . . . . . . . . . . . . . . . . . . . . . . 93

xiii

5.4 Contour plot of Berry curvature in the lowest flat conduction band for

N = 7. The red/yellow regions come mainly from the K valley and have

positive curvature, while the blue regions come mainly from the K ′ valley

and have negative curvature. The Berry curvature integrated over the band

is zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

xiv

Chapter 1

Introduction

Condensed matter physicists study complex phenomena arising in materials having strong

interactions between Avogadro number of constituent particles like electrons. Even though

all materials have the same fundamental constituents, they can exhibit different forms.

Solids and liquids are the most familiar examples of phases of matter. Exotic phases in-

clude the superconducting, ferromagnetic and anti-ferromagnetic phases. Physicists are

concerned with answering a few basic questions: What kinds of phases are possible for a

given material? How can we develop a material having the desired properties? What are

the properties which help us distinguish the hundreds of different kinds of matter? In this

thesis, we will focus on the last question.

The traditional way of classifying phases of matter uses the Ginzburg-Landau theory

of spontaneous symmetry breaking. Symmetry breaking is the phenomenon of a system’s

ground state not having the full symmetry of the Hamiltonian describing the system. A

crystal is an example of a broken translational symmetry state. In the past few decades,

a new way of differentiating between phases has been developed without using symmetry

breaking. They can be distinguished by “topological” properties. Topologically ordered

states are characterized by the presence of some special properties: lack of a local or-

der parameter, a ground state degeneracy dependent on the topology of space, fractionally

1

charged quasi-particles obeying fractional exchange statistics in the bulk, robust gapless

boundary excitations, and robust fundamental properties such as the quantized value of

Hall conductance. Quantum Hall fluids are examples of topological phases [40]. States

with different topological orders can not change into each other without a phase transition.

In the traditional description of topologically ordered phases, the presence of a global

symmetry is not a requirement. Recently, symmetry protected topological (SPT) phases

have also been discovered [24]. These are defined to have no topological order in the bulk,

but their distinctions are protected by a global symmetry. While the gapless boundary

excitations in intrinsically topologically ordered phases are robust against any local pertur-

bations, those in SPT order are robust only against local perturbations that do not break the

symmetry. Topological insulators [49] comprised of non-interacting fermions are examples

of SPT phases.

The interplay between symmetry and topology can give rise to new phases of matter:

symmetry enriched topological (SET) phases and quantum Hall ferromagnets. SET order

[54, 79, 35] refers to phases that have the same topological order but are distinct in the pres-

ence of a symmetry. Systems with the same topological order and the same symmetry can

be in different SET phases with different symmetry fractionalization on the quasi-particles.

For the purposes of this thesis, we will be interested in situations in which intrinsic topolog-

ical ordering is accompanied by the breaking of internal symmetries—such as the global

symmetries associated with the electron spin, valley or layer pseudospin. The resulting

broken-symmetry state, termed a quantum Hall ferromagnet, possesses—in addition to the

topological order common to all quantum Hall states—a distinctive set of phenomena re-

lating to the low-energy pseudospin degrees of freedom. These include charged skyrmions

and finite-temperature phase transitions, to name a few. In this thesis, we study various

kinds of quantum Hall ferromagnetic phases and their analogs in lattice systems.

We begin with an introduction to the quantum Hall effect in two-dimensional electron

gases (2DEGs) placed in high magnetic fields (Sec. 1.1). First, we consider the case of

2

spinless electrons and later take the internal degrees of freedom into account (Sec. 1.3).

In Sec. 1.2 we also give a short introduction to topological band theory and discuss Chern

insulators. Finally in Sec. 1.4, we discuss quantum Hall valley ferromagnets in detail and

end this Chapter with an outline for the rest of the thesis (Sec. 1.5).

1.1 Quantum Hall Effect

Consider an electron with massm and charge−e, moving in the xy-plane in the absence of

any magnetic field. Because of translational invariance along both x and y directions, the

eigenfunctions are plane waves and the energy eigenvalues form a continuous spectrum.

What happens when a magnetic field ~B = Bz is turned on ? The Hamiltonian for this

system is given by H = (~p+ e ~A/c)2/2m where ~A is the vector potential. If we choose the

Landau gauge Ax = 0, Ay = Bx, translational invariance is broken in the x direction. The

new eigenstates are indexed by the y momentum and a discrete index n, which we shall

henceforth call the Landau level (LL) index,

ψn,ky =1

(2nn!π1/2lB)1/2

1

L1/2y

eikyyHn(x+ kyl2B)e−(x+kyl2B)2/2l2B (1.1)

where Lx and Ly are the sample dimensions, Hn is the nth Hermite polynomial and lB =

( ~ceB

)1/2 is the magnetic length. The corresponding eigenvalues are En = ~ωc(n + 12)

where ωc = eBmc

is the cyclotron frequency. Since the energies do not depend on ky, a large

degeneracy– equal to the number of flux quantum threading the sample –is associated with

every Landau level [40]. To summarize, a non-zero magnetic field reorganizes a continuum

of energy levels into a discrete spectrum of highly degenerate Landau levels.

A two dimensional electron gas (2DEG) can be realized in metal-oxide-semiconductor

field effect transistors (MOSFET) and semiconductor heterostructures. For example, en-

ergy bands in a AlAs/GaAs heterostructures can be used to build a quantum well confining

the transverse motion of electrons. Different sub-bands arise in the electronic band struc-

3

ture. For the purposes of this thesis, we will assume that the spacing between the sub-bands

is much larger than the temperatures at which experiments are performed and also ignore

the spread of the electron wavefunction in the transverse direction. Now we will describe

the historical experiments performed on such samples and give explanations for the obser-

vations.

Consider a gas of spinless electrons in a semi-infinite plane. The motion of electrons is

confined in the y direction and a current I is flowing in the x direction. Acording to classical

arguments [9] the Hall resistance is given as RH = VHI

= Bne

, where VH is the voltage

developed in the y direction and n is the number density of electrons. The longitudinal

resistance is independent of B. The same results can also be obtained through an argument

based on Lorentz covariance [40].

However real-life experiments do not agree with these results. The basic experimental

observations [62, 129] are as follows: Instead of showing a linear dependence on B, the

Hall resistance trace has a series of plateaus. Further the longitudinal resistance is approx-

imately 0 within the plateaus and peaks at the steps between the plateaus. The quantized

values of Hall conductance are m e2

hwhere m is either an integer or belongs to a special list

of rational fractions (more on this later). The former case is known as the integer quantum

Hall effect (IQHE) and the latter as the fractional quantum Hall effect (FQHE). The IQHE

is observed when the filling factor ν (ratio of number of electrons to number of flux quan-

tum threading the sample) is close to an integer and the FQHE is observed when ν is close

to certain special rational fractions. This quantization is universal and is independent of mi-

croscopic details of the semiconductor material, but properties like the width of a plateau

are non-universal. How did our earlier arguments break down ? Disorder present in real-

life samples breaks translational invariance and thus those arguments do not go through.

First we explain how disorder leads to the IQHE in a non-interacting electron gas and then

discuss FQHE in an interacting gas.

4

For non-interacting electrons in the absence of a magnetic field, all the states are local-

ized in one and two dimensions for arbitrarily small disorder [12]. However the localization

properties change in the presence of magnetic field. Let us use the case of no disorder and

B 6= 0 as the starting point of our discussion. In the plot of density of states versus en-

ergy, there are δ function peaks at LL energies and all eigenstates are delocalized. Turning

on a random potential leads to a broadening of the spectrum around the LL energies and

delocalized states only at the center of band. This kind of spectrum can be explained us-

ing a semi-classical model of electron dynamics in a smooth random potential [40]. This

explains the existence of plateaus of Hall conductance at integral multiples of e2

h. When

the magnetic field is varied and the Fermi energy crosses the LL centers, the Hall conduc-

tivity increases because the delocalized states get filled. The vanishing of the longitudinal

resistance can be explained by a finite energy gap to creating particle-hole excitations in

the bulk.

For the case of fractional ν, let us ignore disorder at first. For simplicity we consider

only the fractions ν < 1 here. In the absence of interactions the ground state manifold

has a massive degeneracy. Thus interactions are needed to pick a ground state(s). The trial

wavefunction approach has been very successful for solving this many-body problem. The

idea is to guess a good wavefunction for the incompressible many-body ground state built

out of the states in the lowest Landau level. Laughlin’s wavefunction works very well for

ν = 13

and 15

[70]. Jain’s construction- which involves building trial wavefunctions from

filled pseudo-Landau levels of composite fermions [57]- explains plateaus at ν = p2pk+1

where p and k are integers. Moreover the gapped quasiparticle and quasihole excitations

carry fractional charge [40]. Now, let us introduce disorder. Just like in the integer filling

case, it localizes the quasiparticles and leads to plateaus in the conductivity trace. One last

point worth mentioning about the FQH state is that it has topological degeneracy [136]. All

these properties come together to make it a topologically ordered state.

5

Finally, what explains the universality of the quantized values of Hall conductance ?

The amazing precision and robustness of the quantization can be explained using Laugh-

lin’s gauge argument [60]. The Hall conductance for a non-interacting system at integral

filling factor can also be written as a topological invariant [90], known as the Chern num-

ber. Since it is a discrete index it can not be changed by making small perturbations in

the random potential. Thus various configurations of disorder lead to plateaus at the same

quantized values. A similar result holds for the fractional filling cases. We will discuss

Chern numbers in more detail in the next section.

1.2 Chern Insulator

In this section, we focus on non-interacting particles moving in a perfectly periodic po-

tential. An insulator has an energy gap separating the occupied valence band states from

the empty conduction band states. A notion of topological equivalence between different

insulating states can be defined: two insulators are said to be topologically equivalent if one

can be continuously deformed into another without closing the band gap. A trivial insulator

is one which is topologically equivalent to the atomic insulator. All insulating states are

not trivial and this leads to the concept of topological insulators. Perhaps it’s most famous

example is the IQH state. This is an instance of a Z valued classification in terms of the

first Chern number (more on this later).

However, a Chern insulator is just one of the several possible classes of topological

insulators. The notion of topological band theory can be generalized to make a periodic

table of topological insulators [49]. Ten symmetry classes are specified by the presence

or absence of time-reversal symmetry, particle-hole symmetry and chiral symmetry. A

Chern insulator is a 2D topological insulator, governed by a Hamiltonian of no particular

symmetry. Insulators in different dimensions and of different symmetries can be classified

6

in different ways. For the purposes of this thesis we will focus only on d = 2 class A (no

symmetry) insulators.

The classification of the d = 2 class A insulator can be obtained by homotopy the-

ory [6]. Consider a two band model for an insulator. There are two eigenstates for every

crystal momentum ~k in the toroidal brilliouin zone. Pick a particular ~k and choose it’s

two orthonormalized eigenvectors as the basis vectors. Now the eigenstates at any ~k can

be obtained by acting with a U(2) transformation on the basis vectors. Also, each eigen-

state is defined only up to a global ~k dependent phase. Hence all the information about

the eigenstates is encoded in U~k which belongs to the set U(2)/U(1) × U(1). Moreover all

information about the band structure is encoded in a mapping: ~k → U~k. Thus, information

about the state’s topology is present in the homotopy classes of this mapping. If the map-

ping can be continuously deformed to a unit transformation, then the insulator is trivial. In

our example the mappings can be classified according to a Z valued winding number, also

known as the first Chern number. This calculation can also be generalized to the case of

more than one conduction and valence band.

The Chern invariant can formally be written in terms of the Berry phase [14] associated

with the Bloch state |~k〉 of a particular band. When a Bloch wave function is transported in

a closed loop in the Brillouin zone it acquires a phase given by the line integral of the Berry

potential ~A = i〈~k|~∇~k|~k〉. Using Stokes’ theorem, it can be rewritten as a surface integral

of Berry flux F = ∇~k× ~A. The Chern invariant is the total Berry flux through the Brillouin

zone.

C =1

2π

∫d2~k F (~k) (1.2)

It can take only integer values [13].

How does the IQH state fit into this picture ? We can think of Landau levels as also

forming a band structure. The magnetic translation operators do not commute with one

another in general [39], but they commute if a unit cell with hceB

area is used. So Bloch’s

theorem can still be used to label the degenerate states of a LL by 2D crystal momentum.

7

This produces a series of flat bands and, upon the introduction of a periodic potential with

the same lattice periodicity, they gain dispersion [49]. Thus Chern insulators are just lattice

analogs of quantum hall states. Historically, the Haldane model on a honeycomb lattice

was the first example of a tight binding model that gives rise to robust quantization of Hall

conductivity in the absence of a net external magnetic field [45]. It is a model of graphene

having nearest neighbor and next-nearest neighbor hopping and subjected to zero net flux

through a unit cell. Simpler models on a square lattice have also been proposed [13].

But what is the physical consequence of all this ? The Hall conductivity σxy for an

insulator can be calculated by computing the expectation value of the current density to

first order in perturbation theory in an external electric field [125, 39].

σxy =ie2

hLxLym2

∫d2k〈1~k|px|2~k〉〈2~k|py|1~k〉 − 〈2~k|px|1~k〉〈1~k|py|2~k〉

(E1 − E2)2= C

e2

h(1.3)

where |1~k〉, |2~k〉 are the eigenstates of the bulk hamiltonian and E1, E2 are the correspond-

ing eigenvalues. Hence σxy is insensitive to smooth changes in the parameters in the Hamil-

tonian. When the bulk bands have non-trivial topology, the surface of an insulator shows

robust metallic behavior. These conducting states are similar to the edge states seen at the

interface between integer quantum Hall state and vacuum. The chiral edge states can be

seen by solving the Haldane model in a semi-infinite geometry.

Now we can ask the next logical question: Can there be a fractional quantum Hall liquid

for interacting electrons hopping on a lattice ? We will only be interested in matching the

physics of the Chern band to that in the lowest LL. Now a LL is flat and thus, at a fractional

filling, interactions pick a ground state. But, in general, a Chern band has dispersion, which

leads to kinetic energy also playing a role in choosing the ground state. Hence in order

to mimic the FQH scenario, an obvious condition involving a hierarchy of energy scales

should be satisfied: band dispersion � interaction strength � band gap. Considerable

effort has gone in engineering nearly flat Chern bands on hexagonal, kagome and checker-

8

board lattices [123, 122, 87]. Evidence for FQH states at 1/3, 1/5 fillings has been reported

in finite-size studies of short-ranged interactions projected to these bands [87, 113, 107].

They find three signatures of the FQH state: a gap to particle-hole excitations, a many-body

Chern number close to the filling fraction (in units of e2

h), and a topological degeneracy.

But this is not the full story. There are two more criteria for deciding a good host for

fractionalized phases. They involve both the topology and the geometry of a Chern band.

The first criterion is that the Chern band should have near-uniform Berry curvature because

this would ensure that the algebra of the long wavelength density operators projected to the

Chern band is the same as the Girvin-McDonald-Platzman algebra [40] that is obeyed by

similar operators in the lowest LL [93]. The second criterion imposes constraints on the

Fubini-Study metric tensor [7] constructed for the Chern band [108].

The connection to the FQHE was made more explicit in Refs. [97, 72]. These ref-

erences give a mapping from Landau gauge eigenfunctions to hybrid Wannier functions,

which can be used to translate model wave functions and Hamiltonians from the lowest LL

to Chern bands. Also the adiabatic continuity between the model Hamiltonians written for

a Chern band, and more realistic interactions has been verified [75]. A few candidates have

been suggested for experimentally realizing a fractional Chern insulator: optical lattices

with short-range interactions [27] and also with dipolar interactions [143].

Another possibility that can be realized in lattice models is a band with higher Chern

number [133]. This provides a promising arena for new collective states of matter and also

leads to interesting possibilities. For instance, the most favorable situation that selects frac-

tional Chern insulators is not necessarily the one that mimics Landau levels. Neupert et al

[89] find that giving width to the bands can sometimes stabilize a fractionalized topological

phase in a bigger region of parameter space. Moreover bands with higher Chern number

have no direct analog in the continuum. To explore new physics beyond single Landau lev-

els, it is natural to consider topological flat band models with higher Chern numbers. Such

models without long range hopping have been proposed in [134, 142, 127]. The existence

9

of a number of bulk insulating states has been established at fractional filling in such flat

bands [135, 139].

1.3 Quantum Hall Ferromagnet

Until this point we have assumed that the internal degrees of freedom of the electrons are

frozen out. In this section, we lift this assumption and consider the quantum Hall effect

in multicomponent systems. Let us begin by taking the spin of an electron into account.

In the case of the lowest Landau level of a two-dimensional electron gas in free space,

nothing interesting happens. This is because the Zeeman energy which characterizes the

gap between the different spin polarization states, is exactly equal to the cyclotron gap for

g = 2 as appropriate to free space. The gap to spin excitations is the same as the gap to

inter-level transitions. So the spin degrees of freedom are frozen out, and therefore do not

significantly change the physics at ν = 1.

However in the limit of negligible Zeeman coupling, the degeneracy of each LL gets

doubled. Hence ν = 1 is like a fraction and the quasiparticle gap arises because of the

many-body interaction. Coulomb interactions choose a spin polarized ground state [40], so

we have an itinerant quantum ferromagnet. This is essentially the answer given by Hund’s

rule in atomic physics. This state has a quantized Hall coefficient and a broken global

internal symmetry (SU(2) spin symmetry in this example). This phenomenon is termed

quantum Hall ferromagnetism (QHFM).

Such a scenario is made possible in GaAs by two things. First of all, the effective

mass in these systems is much smaller than the physical electron mass (m∗/m ≈ 0.068),

and second, spin-orbit scattering reduces the effective g factor (g ≈ 0.4) The first effect

increases the cyclotron gap, whereas the second reduces the Zeeman splitting. The net

result is that the ratio of the two energy scales is reduced from 1 to about 1/70. Thus, at low

10

temperatures, the kinetic energy is quenched and the system may be considered confined

to the lowest Landau level, but the spin degrees of freedom remain free to fluctuate.

The projected spin density and charge density operators do not commute within the

lowest LL [40]. So when spin is rotated, charge gets moved. Hence spin textures carry

charge. We can ask what is the lowest-energy charged excitation in the quantum Hall fer-

romagnet? The answer isn’t the the naive excitation made by simply removing a down

spin or adding an up spin. For small enough Zeeman energy, the lowest energy charged

excitations are topologically nontrivial spin configurations called skyrmions [120]. While

a skyrmion enjoys a significantly lower exchange contribution to the energy, it has an in-

creased Zeeman cost; the competition between this and the Hartree energy of the nonuni-

form charge distribution sets the size and the energy gap of the resulting excitation. The

cost of a skyrmion-anti skyrmion pairs is thus – in the limit of vanishing Zeeman coupling

– one-half the cost of the simple spin-flip pair. An elegant treatment of the dynamics of

the quantum Hall ferromagnet may be derived within the Chern-Simons Landau-Ginzburg

approach [120].

Various other “pseudospin” degrees of freedom are also possible: the layer index in

double quantum wells, semiconductor valley pseudospin, and the Landau level index when

different Landau levels are brought into coincidence in tilted fields. In the case of two

degrees of freedom, the “pseudospin” can be mapped to a fictional spin 1/2 degree of free-

dom. The symmetry of the ferromagnetic ground state at ν = 1 depends on the details

of the interaction. For example in the case of a bilayer system, interactions between elec-

trons in the same layer are stronger than the interactions between electrons in different

layers. This leads to a tendency to fill both the layers equally and hence leads to “easy

plane anisotropy”. Again spin textures carry charge and this leads to topologically stable

merons being the lowest energy charged excitations [82]. Moreover, broken symmetry can

persist to nonzero temperatures even as quantum Hall order is lost [23].

11

QHFM is not restricted to integer Landau levels with interactions, but can be general-

ized to other fillings, for instance ν = 13

[120]. A general classification of quantum Hall

ferromagnets into different pseudospin anisotropy categories based on the symmetries of

their interactions may be found in [59]. In the following section, we study quantum Hall

valley ferromagnets in detail.

1.4 Quantum Hall Valley Ferromagnet

In many semiconductors, electrons can occupy multiple degenerate energy band minima,

or ‘valleys’ in momentum space. Both valley locations and dispersion relations of electrons

occupying them are determined by the symmetries of the lattice. Silicon and Germanium

are standard examples of such a multi-valley semiconductors [9]. Here we focus on two-

dimensional electron gases (2DEG’s) confined to Si quantum wells. The valley degeneracy

of Si depends on the orientation of the interface, as this choice can break the crystal sym-

metries responsible for the exact valley degeneracy in bulk Si. The first example of valley

QHFM was Si(110) in the presence of a strong interface potential [106, 105, 17, 137]. Al-

though in bulk Si the valleys indeed have substantial anisotropy oriented along different

axes, the two valleys that survive in the low-energy dispersion upon projection into the

(110) plane have identical anisotropies; therefore, the symmetry here is again SU(2). If we

consider the spin to be frozen, this is very similar to the spinful case of GaAs. Thus similar

phenomena emerge, such as low-energy skyrmionic ‘valley textures’ and gapless neutral

Goldstone modes associated with the breaking of the continuous valley pseudo-spin sym-

metry. Corrections— beyond the effective mass approximation —which break the SU(2)

symmetry group into a smaller group, are discussed in [104].

Another example of multi-valley QHFM is graphene [144]. Here, the Dirac disper-

sion is identical and to good approximation isotropic in the two valleys. When Zeeman

and spin-orbit interactions are neglected, it’s Landau levels are fourfold degenerate. If

12

we restrict ourselves to the lowest Landau level, this system has an approximate SU(4)

isospin symmetry. (When the Zeeman and spin-orbit interactions are taken into account,

the symmetry group is reduced to SU(2), at-least for ν < 1 [3] where the short range

interactions do not play much of a role. But for ν > 1, interactions lead to a breaking

down of SU(2) valley symmetry to either Z2 or U(1) [3].) Quantum Hall Ferromagnetic

phases in mono-layer graphene have been clearly observed in experiments [144, 43, 44].

Valley Skyrmionic excitations and collective modes associated with such phases have been

studied in [114, 141, 31, 4, 91]. Lastly the multi-component fractional quantum Hall effect

has also been observed in high-mobility graphene devices fabricated on hexagonal boron

nitride substrates [29] and attempts [3, 119] have been made at explaining the experimental

findings.

In cases of bilayer graphene, the emergent symmetry is approximately SU(8). It sup-

ports a variety of quantum Hall ferromagnetic ground states where the spins and/or valley

pseudospins and/or orbital pseudospins collectively align in space [69]. It has also been

shown that at even filling factors, electric charge is injected into this system in the form of

charge 2e Skyrmions [2]. This is a rare example of binding of charges in a system with

purely repulsive interactions.

Until now we have looked at cases where the global symmetry is an internal symmetry

that acts on spin/pseudospin. There are situations where the global symmetry acts simulta-

neously on the internal index and on the spatial degrees of freedom. This occurs naturally

in a multi-valley system where different valleys are related by a discrete rotation so that val-

ley pseudospin and rotational symmetries are intertwined. Examples of such systems are

Si(111) 2DEG and AlAs heterostructures. Preference for one valley over the others should

automatically distinguish some spatial directions, as long as the valleys are inequivalent.

This in turn leads to anisotropies in experimental measurements.

In the case of a Si(111) interface [8], effective mass theory predicts a six-fold degener-

acy [121]. (See Chapter 3 for details.) This valley degeneracy is quite robust. It cannot be

13

lifted by changing the width of the confining well or by an interface potential. Considering

this degeneracy to be exact is surely an idealization. In a more realistic situation, the six-

fold valley degeneracy can be lifted due to wafer miscuts and strains arising from lattice

mismatch. While the valley splitting due to the former mechanism is negligible compared

to the cyclotron gap [78], the latter can be more significant [128, 110]. This problem has

been largely solved by working with 2DEGs on a H-terminated Si(111) surface [34, 65].

Since different valleys are related by a discrete rotation so that valley and rotational sym-

metries are intertwined, we have a multivalley system where the symmetry that is broken

is a global symmetry that acts simultaneously on the internal index and the spatial degrees

of freedom. Later in this thesis, we will study the interplay between broken symmetry and

topological order in the context of the QH states observed in 2DEGs confined in Si(111)

quantum wells.

Recent experimental [41, 112, 111, 115, 92, 42, 96] and theoretical work [1] has also

focused on AlAs heterostructures. AlAs has two valleys with ellipsoidal Fermi surfaces.

(See Chapter 2 for details.) While Si has all valley minima inside the Brillouin zone, AlAs

has valey minima at the edge of Brillouin zone. Here, the linking of pseudospin and space

has significant consequences at “ferromagnetic” filling factors, such as ν = 1. Here, the or-

der parameter is an Ising variable. In the absence of disorder, pseudospin ferromagnetism

onsets via an Ising-type finite-temperature transition and is necessarily accompanied by

broken rotational symmetry, corresponding to nematic order. The resulting state at T = 0,

dubbed the quantum Hall Ising nematic (QHIN), has an intrinsic resistive anisotropy for

dissipative transport near the center of the corresponding quantum Hall plateau. Also the

QHIN phase is unstable to quenched random spatialfields. Disorder thus destroys the long-

range nematic order, giving rise to a paramagnetic phase. Provided that there is (arbitrarily

weak) intervalley scattering, this continues to exhibit the QHE at weak disorder and low

temperatures, and is hence termed the quantum Hall random-field paramagnet (QHRFPM).

Transport in this phase is dominated by excitations hosted by domain walls between dif-

14

ferent orientations of the nematic order parameter. The existence of the two phases was

originally established within a long-wavelength nonlinear sigma model (NLσM) field the-

ory. We provide a microscopic analysis of this system in Chapter 2.

Another example of Ising-type valley QHFM is trilayer graphene. At first sight, it

would appear to exhibit a higher symmetry group similar to its mono-layer cousin; however,

the inclusion of ‘trigonal warping’ effects into the band structure [73] could break this down

to an Ising symmetry.

Lastly Ref. [124] discusses a model where there are two different orientations for the

principal axes of the effective-mass tensor in the various valleys, and the magnetic field is

applied along a direction that is symmetric with respect to these orientations. They study

the system as a function of the electron density, magnetic field strength, the effective-mass

anisotropy, the electronic g factor, and the number of degenerate valleys. Depending on the

parameters, they find that the ground state may contain spin-density waves or valley-density

waves.

Finally, a far more speculative example of valley QHFM is a 3D system like Bismuth

which has three degenerate valleys with different orientations of ellipsoidal Fermi surfaces.

Transport experiments in Bismuth [148, 10, 74, 94] have demonstrated orientational sym-

metry breaking in the presence of a magnetic field that is not too far from the quantum limit,

which could be consistent with some valley-ordering scenarios. However the situation in

3D is much less clear, as the ability of a magnetic field to enhance the effect of correlation

is greatly diminished.

1.5 Thesis Outline

The remainder of this thesis consists of five chapters. In Chapter 2, we study the the

interplay between spontaneously broken valley symmetry and spatial disorder in the AlAs

multivalley semiconductor in the quantum Hall regime. We provide a detailed microscopic

15

analysis of the quantum Hall Ising nematic phase, which allows us to (i) estimate its Ising

ordering temperature; (ii) study its domain-wall excitations, which play a central role in

determining its properties; and (iii) analyze its response to quenched disorder from impurity

scattering.

In Chapter 3, we examine the Si(111) multi-valley quantum Hall system and show that

it exhibits an exceptionally rich interplay of broken symmetries and quantum Hall ordering

already near integer fillings in the range 0-6. We show that the symmetry breaking near

filling fractions 2, 3 and 4 involves a combination of selection by thermal fluctuations

known as “order by disorder” and a selection by the energetics of Skyrme lattices induced

by moving away from the commensurate fillings, a mechanism we term “order by doping”.

We also study it’s ground state near filling fraction one in the absence of valley Zeeman

energy. We show that a non-trivial, complex analytic and quasi-periodic valley texture with

charge one does not exist. Thus even though the lowest energy charged excitations are

charge one skyrmions, the lowest energy skyrmion lattice has charge >1 per unit cell.

In Chapter 4, we study the interplay between quantum Hall ordering and spontaneous

sublattice symmetry breaking in multiple Chern number bands at fractional fillings. Pri-

marily, we study fermions with repulsive interactions near half filling in a family of square

lattice models with flat C = 2 bands and a wide band gap. By perturbing about the particu-

larly transparent limit of two decoupled C = 1 bands and by exact diagonalization studies

of small systems in the more general case, we show that the system generically breaks sub-

lattice symmetry with a transition temperature Tc>0 and additionally exhibits a quantized

Hall conductance in the limit of zero temperature. We also discuss generalizations to other

fillings and higher Chern numbers.

In Chapter 5, we demonstrate that ABC stacked multilayer graphene placed on boron

nitride substrate has flat bands with non-zero local Berry curvature but zero Chern num-

ber. The flatness of the bands suggests that many body effects will dominate the physics,

while the local Berry curvature of the bands endows the system with a nontrivial quantum

16

geometry. The quantum geometry effects manifest themselves through the quantum dis-

tance (Fubini-Study) metric, rather than the more conventional Chern number. Multilayer

graphene on BN thus provides a platform for investigating the effect of interactions in a

system with a non-trivial quantum distance metric, without the complication of non-zero

Chern numbers. Finally in Chapter 6, we conclude by summarizing our work and giving

directions for future research.

17

Chapter 2

Microscopic Theory of a Quantum Hall

Ising Nematic: Domain Walls and

Disorder

2.1 Introduction

Recent experimental [115, 34, 92, 42, 96, 65] and theoretical work [1] has focused on

quantum hall ferromagnets in which the symmetry in question is between the different val-

leys (i.e., conduction band minima) of a semiconductor. In previous work [1], it was noted

that a generic feature of such multivalley systems is that the point-group symmetries act

simultaneously on the internal valley pseudospin index and on the spatial degrees of free-

dom. This linking of pseudospin and space has significant consequences at “ferromagnetic”

filling factors, such as ν = 1:

(i) in the absence of disorder, pseudospin ferromagnetism onsets via an Ising-type finite-

temperature transition and is necessarily accompanied by broken rotational symmetry,

corresponding to nematic order. The resulting state at T = 0, dubbed the quantum

18

Hall Ising nematic (QHIN), has an intrinsic resistive anisotropy for dissipative trans-

port near the center of the corresponding quantum Hall plateau.

(ii) as a quenched random field is a relevant perturbation to Ising order in d = 2, the QHIN

is unstable to spatial disorder—such as random potentials or strains—that gives rise

to such fields. Disorder thus destroys the long-range nematic order, giving rise to

a paramagnetic phase. Provided that there is (arbitrarily weak) intervalley scatter-

ing, this continues to exhibit the QHE at weak disorder and low temperatures, and

is hence termed the quantum Hall random-field paramagnet (QHRFPM). Transport

in this phase is dominated by excitations hosted by domain walls between different

orientations of the nematic order parameter, and is extremely sensitive to the appli-

cation of a symmetry-breaking ‘valley Zeeman’ field—for instance, due to uniaxial

strain—which can tune between percolating and disconnected domain walls.

Two aspects of this picture are particularly striking and should apply to a variety of valley

quantum Hall ferromagnets. The first is the role of valley anisotropy in establishing the

nature of the symmetry breaking. Systems with valleys that have identical anisotropies

(for instance, graphene), will exhibit an enhanced SU(2) valley pseudospin symmetry. It

is the valley anisotropy in the present situation that entangles rotations in space with those

in pseudospin space, and also reduces the order parameter to an Ising variable. Second,

we emphasize that the QHIN and the QHRFPM that naturally emerge in this situation both

exhibit quantum Hall behavior, but on parametrically different scales: the QHRFPM shows

quantized conductivity only at temperatures below the scale of domain wall-excitations,

typically dominated by weak interactions and/or disorder, and hence, much lower than the

intrinsic anisotropy scale characteristic of QH transport in the QHIN.

A specific example of experimental interest[115, 92, 42, 111] and our focus in this

chapter is the case of wide quantum wells in AlAs heterostructures. Here, two valleys with

ellipsoidal Fermi surfaces are present, as shown in Fig. 2.1. (Valley minima are at the edge

of Brillouin zone.) Owing to the anisotropic effective mass tensor in the two valleys, indi-19

kx

ky

1 1

2

2

�IM

LS

�IM � LS

Figure 2.1: (a) Model band structure used in this chapter, appropriate to describing AlAswide quantum wells. (b) Different phases as determined by comparing Imry-Ma domainsize ξIM to sample dimensions LS . Top: For ξIM � LS we find the QHRFPM. Bottom:For ξIM � LS the system is dominated by the properties of a single domain, and is bettermodeled as a QHIN. At intermediate scales, LS ∼ ξIM there is a crossover.

vidual electronic states no longer exhibit full rotational invariance. Only discrete rotations

of the axes, accompanied by a simultaneous interchange of the valleys remain as symme-

tries of the system. It is in this specific sense that the internal index is entangled with the

spatial symmetries.

The existence of the two phases was originally established within a long-wavelength

nonlinear sigma model (NLσM) field theory, which also provides a caricature of their prop-

erties and the above phase diagram in the weak-anisotropy limit. While it is expected that

this treatment captures qualitative features of valley Ising physics reasonably well, to make

a quantitative connection to experiments a microscopic understanding is essential. Here,

we provide such a microscopic analysis of the QHIN, focusing specifically on properties

of domain walls which as we have argued are central to this system.

20

A summary of the main results of this chapter, which also serves to outline its organiza-

tion, follows. We first place this work in context by providing a summary of the important

aspects of valley-nematic ordering in the quantum Hall effect in Sec. 2.2, focusing on

qualitative features of the phase diagram, the role of thermal fluctuations and quenched

disorder, and transport signatures of the QHIN/QHRFPM phases. We then proceed to our

technical results. First, we set up a Hartree-Fock formalism (Sec. 2.3.1), which we use to

obtain a mean-field estimate of the transition temperature out of the thermally disordered

phase (Sec. 2.3.2). We proceed to construct a solution of the HF equations corresponding

to a ‘sharp’ domain wall (Sec. 2.3.3), where the valley pseudospin changes its orienta-

tion abruptly at the wall; this is expected to be an accurate description of physical domain

boundaries in the ‘strongly Ising’ limit of large mass anisotropy. We determine the proper-

ties of the sharp wall as a function of the mass anisotropy, specifically its surface tension

and dipole moment, the latter a property which is not captured in the NLσM limit. We

clarify the effect of this dipole moment on critical behavior and domain wall energetics

(Sec. 2.3.4). We then relax the sharp-wall approximation and numerically solve the HF

equations to quantify the amount of ‘texturing’ in a soft domain wall as a function of the

anisotropy (Sec. 2.3.5) – we note that texturing is a prediction of the NLσM that remains

valid at high anisotropies. We next turn to an analysis of disorder within the microscopic

theory, where we first establish that anisotropies in the screened random impurity potential

act as a valley-selection mechanism, translating into a random field acting on the Ising or-

der parameter (Sec. 2.4.1), which we compute in Landau-level mixing perturbation theory.

We discuss how to estimate the strength of the disorder from the mobility, a measure that is

readily accessible to experiments (Sec. 2.4.2). Taken together, the domain wall parameters

and the random field studies yield estimates for the characteristic domain size due to the

disorder, allowing us to make partial contact with experiments (Sec. 2.5). All these results

are obtained for the microscopics of the AlAs heterostructures which were the original

motivation for our study of valley-nematic order.

21

2.2 Overview: Phases, Transitions and Transport

The temperature-disorder phase diagram of multivalley 2DEGs exhibiting Ising valley or-

dering can be sketched as follows (see Fig. 2.2). In the absence of disorder, there is a

finite temperature transition into an Ising nematic ordered phase, which exhibits transport

features of the QHE. While strictly speaking, the QHE is a zero-temperature phenomenon,

in a slight abuse of terminology we will nevertheless refer to the entire phase below Tc

in the zero-disorder limit as the QHIN. The quantization of the Hall conductivity and the

vanishing of the longitudinal conductivity are only exponentially accurate at finite temper-

ature, i.e. corrections are exponentially small. While there is a thermodynamic transition

associated with the Ising valley ordering, the conductivity exhibits a crossover rather than

a singularity at Tc. The orientational symmetry breaking of the Ising nematic phase is re-

flected in the anisotropic longitudinal conductivity of the QHIN where σxx/σyy 6= 1. Upon

adding disorder, the Ising transition is destroyed and at T = 0 the system is in the QHRFPM

phase. Above this at finite temperature (shaded region in Fig. 2.2) we once again find zero

longitudinal conductivity and quantized Hall conductivity (both with exponentially small

corrections), but the response is now isotropic: σxx/σyy = 1. With similar caveats as in the

clean case we will refer to the entire shaded region above the T = 0 line as the QHRFPM.

In contrast to the QHIN, there is no thermodynamic phase transition into the QHRFPM

at T > 0, only a crossover in the conductivity at a temperature scale T ∗ (dashed line in

Fig. 2.2.)

We emphasize that there is an important qualitative difference between the QHIN and

the QHRFPM, over and above the anisotropy in the former. Namely, the crossover into a

quantized Hall response in transport is governed by different physical mechanisms. In the

QHIN, this crossover occurs at a scale set by the exchange energy, effectively the single-

particle gap, ∆sp ∼ e2/ε`B in the QH ferromagnetic ground state. This also sets the scale

of the Ising Tc, upto a numerical factor that depends on the mass anisotropy. In contrast

the QHRFPM is, as we have noted, characterized by multiple domains of differing Ising

22

polarization. Here, the lowest-energy charged excitations are localized on one-dimensional

domain boundaries, [80] which in the strong-anisotropy limit can be understood in terms

of a pair of counterpropagating QH edge states of opposite pseudospin. The stability of

the QHE then rests on the gap to creating domain-wall excitations. As this is induced by

weak pseudospin symmetry-breaking terms in the Hamiltonian from both disorder and in-

teractions, it is expected to be small and the concomitant conductance quantization is thus

fragile. At weak disorder, the dominant source 1 of symmetry breaking is from intervalley

Coulomb scattering, Viv, which thus sets the domain-wall gap ∆dw and hence the crossover

scale T ∗. For sufficiently strong disorder above a critical strength Wc, the energy gap sta-

bilizing the QHRFPM collapses via the Fogler-Shklovskii scenario[38] originally devised

to describe the collapse of spin-splitting in quantum Hall ferromagnets in GaAs quantum

wells.

Whether a particular experimental sample will display the transport anisotropy charac-

teristic of the QHIN, or the isotropic domain-wall dominated transport of the QHRFPM

is a matter of quantitative detail, determined by the comparative energetics of the Ising

exchange energy and the disorder. Their competition sets a characteristic “Imry-Ma”[55]

domain size ξIM in the random-field phase. The question then turns on whether the system

consists of a single Ising domain or multiple domains, i.e. it depends on how the domain

size compares to the sample dimensions, LS (see Fig. 2.1). The exchange strength is de-

termined by the electron-electron interactions, while for the heterostructures of interest the

disorder is sensitive to the density of dopant impurities and their typical distance from the

plane of the 2DEG. The effective mass anisotropy is important to estimates of both these

quantities, for in the isotropic limit there is a full SU(2) pseudospin symmetry, and poten-

tial disorder does not exhibit a preference for any particular pseudospin orientation. Thus,

1While disorder can also lead to scattering between valleys, this is suppressed owing to the mismatchbetween the separation of the valleys in momentum space – roughly an inverse lattice spacing – and the scaleof the random potential fluctuations – typically several tens of nanometers. (We neglect short-range disorder.)Thus, interactions are the dominant source of intervalley scattering in this limit.

23

�xx,�yy ! 0

�xx/�yy ! 1

�xy = e2/h

Tc ⇠ �sp

T

QHIN

QHRFPM

PM

WWc

�xx = �yy ⇠ e��dw/2T

�xy 6= 0

�xx,�yy ⇠ e��sp/2T

�xx 6= �yy

�xy 6= 0

�xx,�yy ! 0

�xx/�yy ! ↵ 6= 1

�xy = e2/h

Ising

Fogler-Shklovskiicollapse

T ⇤ ⇠ �dw

Figure 2.2: Phase diagram as function of temperature (T ) and disorder strength(W ), showing behavior of conductivity. The phases and critical points are defined in theintroduction.

accurate estimates of these quantities picture are essential to make a quantitative connection

with experiments.

The introduction of an externally applied valley Zeeman field—experimentally

achieved via application of uniaxial strain to the 2DEG—provides a convenient probe of

the transport scales in the QHIN and QHRFPM. First, this field introduces a single-particle

splitting between valleys ∆v that stabilizes the Ising nematic against the effects of disorder.

Thus for sufficiently weak disorder and sufficently large ∆v, the anisotropic longitudinal

conductivity should be clearly established. Second, in the case when for ∆v = 0 the

disorder is sufficient that the sample is in the QHRFPM with multiple domains (for

instance, along the dotted line in the inset of Fig. 2.3), application of the valley Zeeman

field causes a crossover in the the longitudinal conductivity as a function of ∆v. A sketch

of this is provided in Fig. 2.3, and can be understood as follows. For a disorder strength

24

corresponding to the dotted line in the inset, the system crosses over from multiple domain

to single domain behavior. This is reflected in the activation gap for longitudinal transport:

in the multiple domain regime, the gap is dominated by the domain wall scale ∆dw. Deep

in the single-domain regime, the gap is essentially set by the single-particle gap, which

itself scales linearly with ∆v; the intercept of the asymptotic linear dependence can be

used to extract the characteristic single-particle energy scale at zero Zeeman splitting.

The sharp crossover between the two regimes can be understood qualitatively in terms of

tuning domain walls in a random-field Ising model away from percolation by applying a

constant symmetry-breaking field. The reader will note that the behavior of the energy gap

as a function of valley zeeman field here is very similar to that expected for the case where

valley skyrmions are the lowest energy charged excitations, but that is not the case for our

model.

2.3 Microscopic Theory

We will begin by developing a microscopic theory of the QHIN using the Hartree-Fock

(HF) approximation.[37] We will focus on the case relevant to AlAs, with two valleys

denoted by index κ = 1, 2 and centered at K1 = (K0, 0) and K2 = (0, K0) respectively,

with mass anisotropy λ2 = m1,x

m1,y= m2,y

m2,x(a schematic dispersion is sketched in Fig. 2.1.) In

each valley, the single-particle kinetic energy is

Tκ =∑i=x,y

(pi −Kκ,i + e

cAi)2

2mκ,i

(2.1)

Working in Landau gauge, A = (0,−Bx), we find that the lowest LL eigenfunctions are

ψκ,X(x, y) =eipyy√Ly`B

(uκπ

)1/4

e−uκ(x−X)2

2`2B (2.2)

25

�tr ⇠ �2T log �xx

�tr / �v

⇠ �dw �v

W

singledomain

multipledomain

�v

�sp

�⇤v

�⇤v

Figure 2.3: Valley symmetry-breaking field permits transport to probe the energyscales of the QHIN/QHRFPM. (Inset) Domain structure as function of disorder strengthand valley splitting; dashed line shows a representative path in ∆v leading to a transportsignature similar to that in the main figure. ∆∗v is the valley splitting for which the systemis single-domain dominated.

Here, u1 = 1/u2 = λ, and we have labeled states within a LL by their momentum py,

which translates into a guiding-center coordinate via X = py`2B. Henceforth, we shall ac-

count for the spatial structure of the LL eigenstates by the standard procedure of projecting

the density operators onto the lowest LL. [82]

2.3.1 Hartree-Fock Formalism

We consider a rectangular system of dimensions Lx, Ly. Since we are interested in a ν = 1

state, the total number of electrons in the system (which we take to be even for convenience)

is N = NΦ = LxLy2π`2B

. For periodic boundary conditions in the y-direction, the guiding

26

center coordinate along the x-direction is given by Xn =2π`2BLy

n, with n an integer between

−N2

+ 1 and N2

. For the sake of brevity, we shall continue to label states by X , but with

the understanding that it is now a discrete index. Unless otherwise mentioned, all sums and

products are over the full (finite) range of X .

In addition to the electron-electron interaction energy, the lowest Landau level Hamilto-

nian must include the energy of the electrons interacting with the potential of the positively

charged background, which depends on the form of the background charge density. We

shall take the positive charges to have orbitals of the same form as electronic states in the

two valleys, and corresponding occupation numbers n(b)κ :

ρb(r) =∑X,κ

n(b)κ ψ

∗X,κ(r)ψX,κ(r) (2.3)

with κ = 1, 2 as before and n(b)1 + n

(b)2 = 1. Note that for any choice of n(b)

κ satisfying the

latter constraint, ρb(r) is the same uniform constant. However, a judicious choice of the

background charges will allow us to cancel divergences of the Hartree contribution, as we

will see below.

In order to model boundaries between valley domains we also add a spatially varying

single-particle pseudospin splitting that increases linearly in X from negative to positive

across the system2 which models the external random valley Zeeman field from disorder.

This serves a twofold purpose: first, it pins the domain wall3 near n = 0, which is desirable

for a stable numerical solution even in the clean limit; second, it allows us to study how

the domain wall properties change as we vary the characteristic length scale and typical

strength of the random field that leads to domain formation.

With these preliminaries, the second-quantized Hamiltonian projected to the lowest

Landau level can now be written in the Landau basis:2Since the number of particles is even, this corresponds to vanishing in between the orbitals at n = 0 and

n = 1.3Without such pinning, for a finite system the energy optimization would force the domain wall to the

boundary where the loss of exchange energy is minimized.

27

H =1

2

∑κ,κ′

∑X,YX′,Y ′

V κY,κ′Y ′

κ′X′,κXc†κY c

†κ′Y ′cκ′X′cκX −

∑κ

∑X,Y

[n

(b)1 V κY,1X

1X,κY c†κY cκY + n

(b)2 V κY,2X

2X,κY c†κY cκY

]

+g∑X

(Ly

2π`2B

X − 1

2

)(c†1Xc1X − c†2Xc2X

)+ Eself

[ρ2b

](2.4)

The first term is the electron-electron interaction, the second is the interaction between

the electrons and the positive background, and the third term is the single-particle splitting.

As discussed in the next section, the characteristic energy scale of this is ∆SBd , and it varies

over a characteristic distance d corresponding to the correlation length of the random field;

rewriting this carefully, leads to the expression given, with g =∆SBd

2d

2π`2BLy

(which has units

of energy). Note that because d � `B, this term is fairly small even at the two ends

of the system, where it is maximal. The final term is the self-energy of the background

charge distribution, a positive constant that we omit forthwith. In writing (2.4), we have

ignored ‘umklapp’ terms that lead to a net transfer of electrons between valleys (as these

are exponentially suppressed in a/`B, as well as terms that exchange a pair of electrons in

the two valleys (suppressed by a factor of (a/`B)2). At the scales of interest, even the latter

term only contributes a small energy correction (. 1% of the terms kept), and we do not

expect their inclusion to significantly alter our conclusions.

The matrix elements of the Coulomb interaction are given by the usual second-

quantized form:

V κX,κ′X′

κ′Y ′,κY =

∫d2rd2r′ ψ∗κX(r)ψ∗κ′X′(r

′)V (r− r′)ψκ′Y ′(r′)ψκY (r) (2.5)

where the single-particle wave functions were defined in the previous section. Note that

momentum conservation requires that X +X ′ = Y + Y ′.

28

Energy scales

Throughout the remainder of this Chapter, we present our results in dimensionless units.

We measure energy in units of the Coulomb energy e2/ε`B where ε is the dielectric constant

appropriate to the heterostructure under consideration. For the AlAs devices which are our

primary focus, ε ≈ 10. The magnetic length, `B ≈ 8 nm for a magnetic field of 10 T, so

that e2/ε`B ≈ 200 K. The surface tension of Ising domain walls is measured in units of

e2/ε`2B, roughly 25 K/nm for this choice of parameters.

2.3.2 Estimates of Tc

Our first application of the microscopic theory will be to estimate the Ising ordering temper-

ature Tc for the clean system via finite-temperature Hartree-Fock theory. A standard mean-

field decoupling of the Hamiltonian (2.4) in the density channel, 〈c†κXcκ′Y 〉 = nκδκκ′δXY ,

where the occupation numbers are assumed independent of position, yields

HMF =1

2

∑κ,κ′

∑X,Y

(nκ′ − 2n

(b)κ′

)V κY,κ′Xκ′X,κY c

†κY cκY −

1

2

∑κ

∑X,Y

nκVκY,κXκY,κX c

†κY cκY (2.6)

We simplify the Hartree term by taking n(b)κ = nκ. For N → ∞, we may use translation

invariance of the potential to write HMF =∑

X,κ εκc†κXcκX , where

εκ = −1

2

∑κ′,Y

nκ′VκX,κ′Yκ′Y,κX −

1

2

∑Y

nκVκX,κYκY,κX (2.7)

is independent of X .

We seek a solution where the ground state spontaneously breaks valley symmetry; with-

out loss of generality we may assume it is polarized in valley 1, and take the energy splitting

to be ∆, whence

n1 =e∆/kBT

1 + e∆/kBT, n2 =

1

1 + e∆/kBT(2.8)

29

For this ansatz the self-consistency condition corresponds to ∆ = ε2 − ε1 = A1n1 −A2n2,

where after a tedious calculation we find (defining 1 = 2, 2 = 1)

Aκ =1

2

∑Y

[−V κX,κY

κY,κX + V κX,κYκX,κY + V κX,κY

κY,κX

]=

1

2

∑Y

V κX,κYκX,κY (2.9)

(This cancellation of the Hartree contributions from the two valleys is the reason for the

choice of background charge made previously.) Valley symmetry requires that A1 = A2,

which yields the self-consistency condition ∆ = Aκ tanh ∆2kBT

. By the standard compar-

ison of the slope of both sides of this equation at ∆ = 0, we find for the (mean-field)

transition temperature

kBTMFc =

1

2Aκ =

1

4

∑Y

V κX,κYκX,κY

=1

16π2

e2

ε`B

∫ ∞−∞

dx

∫ ∞−∞

dye− 1

2

(x2

λ+λy2

)√x2 + y2

=1

2(2π)3/2

e2

ε`B

K (1− 1/λ2)√λ

(2.10)

whereK is the complete elliptic integral of the first kind.4 Note that this mean-field expres-

sion for Tc has some unphysical aspects—most notably it is nonzero even in the Heisenberg

limit (λ→ 1), and decreases with increasing mass anisotropy. This will be corrected in an

RPA spin-wave calculation of quadratic fluctuations about the mean-field ground state. In

particular, the fluctuations drive TMFc to zero in the isotropic Heisenberg limit. Furthermore,

as spin-wave gap scales roughly with the Ising anisotropy, the debilitating effect of spin

waves on TMFc is suppressed at strong anisotropy, offsetting the decrease in the energy scale

predicted by the mean-field theory. As the spin-wave calculation is technically involved

and not too informative, we provide instead an alternative estimate of Tc for comparison:

T σc ∼ 4πρs log−1[ρs/α`2B], obtained from the NLσM with stiffness ρs ≈ 0.025 e2

16√

2πε`B

4Note that the λ → λ−1 symmetry while not manifest in the final expression for TMFc is nevertheless

obtained from an identity satisfied by the elliptic integral K.

30

2 4 6 8 100.0

0.1

0.2

0.3

0.4

0.5

0.6

kBTMFc

kBT �c

�2

kB

Tc

(in

e2/"` B

)

Figure 2.4: Mean-field and NLσM estimates of Tc. Dashed line shows the anisotropy(λ2 ≈ 5.5) appropriate to AlAs.

and Ising anisotropy α ≈ 0.01 e2

ε`3B(λ− 1)2, whose leading dependence of λ was computed

in a gradient expansion in [1]. We plot both estimates in Fig. 2.4.

2.3.3 Properties of Sharp Domain Walls

We turn now to an analysis of ‘sharp’ domain walls. These are solutions to the HF equa-

tions where the valley pseudospin abruptly changes orientation from one Landau gauge

orbital to the next. We will determine the properties of the sharp domain wall as a function

of the anisotropy. While analytically tractable, this approximation is expected to be a good

description of the domain wall only at strong anisotropy, but nevertheless provides a valu-

able complementary perspective of its properties in a regime where the NLσM is no longer

valid. If we take as the ground state a fully pseudospin polarized Slater determinant with

all the electrons in valley 1:

|ΨG〉 =∏X

c†1X |0〉 (2.11)

31

then a domain wall is captured by a Slater determinant of the form

|ΨDW〉 =∏X

(uXc

†1X + vXc

†2X

)|0〉. (2.12)

The sharp wall corresponds to the case uX = 1, vX = 0 for X ≤ 0 and uX = 0, vx = 1 for

X >. We once again consider the Hamiltonian (2.4) with g = 0 and assume a background

charge distribution polarized in valley 1, i.e. n(b)1 = 1. Two properties of the domain wall

will be of especial interest to us: its dipole moment and its surface tension.

Surface Tension

The first quantity of interest is the domain wall surface tension—the energy per unit length

of the wall. This provides a measure of the Ising exchange energy appropriate to the strong-

anisotropy limit. Note that within the NLσM the domain wall surface tension depends both

on the stiffness and the Ising anisotropy. In the microscopic theory, we find the surface

tension (energy per unit length along the wall) of a sharp domain wall to be the sum of

three contributions:

σ(λ) = limLx→∞

1

Ly(〈H〉DW − E0) = EI + EII + EIII. (2.13)

Here, E0 and (〈H〉DW − E0) are the energies of the ground state and the sharp domain wall.

The three contributions are individually convergent, and can be written as follows. The first

term,

EI =1

2

∞∑X,X′=1

(V 1X,1X′

1X′,1X + V 2X,2X′

2X′,2X − 2V 1X,2X′

2X′,1X

)(2.14)

32

measures the Hartree cost, and can be simplified as

EI =1

32π2

e2

ε`2B

∫ ∞−∞

dxdx′∫ Ly

`B

0

dyfλ(x)fλ(x

′)√(x− x′)2 + y2

(2.15)

where

fλ(x) = erfc(−xλ−1/2

)− erfc

(−xλ1/2

)(2.16)

with erfc the complementary error function. The second term,

EII =1

2

∞∑X,X′=1

(V 1X,1X′

1X,1X′ − V 2X,2X′

2X,2X′

)(2.17)

is the difference in the ‘bulk’ exchange energy between ground state and domain wall

state from orbitals near the center or the edge.The final contribution measures the loss of

exchange energy since the two valleys have vanishing exchange matrix elements:

EIII =0∑

X=−∞

∞∑X′=1

V 1X,1X′

1X,1X′ (2.18)

We find σ(λ) by numerically computing the convergent sums EI, EII and EIII, in each of

which we can truly take the upper bounds on X to infinity. Note that σ(λ) depends log-

arithmically on Ly from the upper bound in the integral in (2.15). Ignoring this weak

dependence, we can take Ly →∞ in integrals over qy and numerically integrate each term

to obtain the surface tension as a function of anisotropy, plotted in Fig. 2.5 (a).

Dipole Moment

Consider for a moment a long-wavelength description of an Ising nematic in terms of a

single-component Ising order parameter field ϕ, and consider a domain wall parallel to the

33

1 2 3 4 5 6 7 8 9 100

0.02

0.04

0.06

0.08

0.1

0.12

h2

Dip

ole

mom

ent p

er u

nit l

engt

h(p

/Ly, i

n un

its o

f e)

1 2 3 4 5 6 7 8 9 100.007

0.008

0.009

0.01

0.011

0.012

0.013

0.014

h2

Surfa

ce te

nsio

n (in

uni

ts o

f e2 /¡l

B)D

ipol

em

omen

tper

unit

lengt

h(p

/Ly,in

e)

�2

�2

Surf

ace

tensi

on(i

ne2

/"`2 B

)

(a.)

(b.)

Figure 2.5: (a) Surface tension and (b) dipole moment of a sharp DW as a function ofthe effective mass anisotropy. Dashed line shows the anisotropy (λ2 ≈ 5.5) appropriate toAlAs.

34

y-axis at x = 0, between regions with opposite Ising polarization (i.e., ϕ → ±1 as x →

∓∞.) The remaining rotational symmetry is a rotation Rπ that takes x → −x, y → −y.

For the given configuration, we have Rπϕ(x, y) = ϕ(−x,−y) = −ϕ(x, y). Observe that

under this symmetry, ∂xϕ(x, y) is left invariant. From this it is not too difficult to show

that ∇ϕ(r) transforms as a vector under a rotation by π, and thus has the same symmetry

as that of a dipole moment normal to the domain boundary and oriented in the direction

of decreasing Ising polarization. It is quite straightforward to find a microscopic origin

for the dipole moment. Recall that the spatial extent of the Landau gauge orbitals in the

X-direction is different in the two valleys. At a domain wall, the charge distribution from

valley 1 decays with a smaller Gaussian envelope than the growth of charge from valley 2.

Assuming a uniform positive background, this leads to a dipole moment associated with the

interface between the two valleys and oriented as above. Therefore the theory of an Ising

nematic should properly include long-range interactions between dipoles tied to gradients

in the Ising order parameter. However, these appear only at higher orders in the gradient

expansion than those used to obtain the leading terms in the long-wavelength theory and

represent a small perturbation in the weak-anisotropy limit. It is easier to compute the

dipole moment at a domain wall within the microscopic theory: for our choice of back-

ground charge, it is straightforward to show that the charge distribution associated with a

sharp domain wall is

ρtot(r) = ρe(r) + ρbg(r)

= −N/2∑

X=−N/2+1

{(uK − 1)ψ∗1X(r)ψ1X(r) + vKψ∗2X(r)ψ2X(r)}

= −N/2∑X=1

{ψ∗1X(r)ψ1X(r)− ψ∗2X(r)ψ2X(r)} .

(2.19)

35

Performing the summations and using the explicit form of the single-particle wavefunc-

tions, we can verify that ρtot(r) corresponds to a pair of dipolar charge distributions, one

located at the domain wall (X = 0) and the other at the right edge of the system (since the

background falls off with a different exponential than the electronic density.) Some care

must be taken to separate the contribution of just the dipole at the center so that we have a

controlled Lx →∞ limit; after some work, we find the dipole moment per unit length of a

domain wall is given by

1

Lyp(λ) = xe

(λ− λ−1

)× 1

4π3/2

∫ ∞−∞

t2e−t2

dt =e

8π

(λ− λ−1

)x (2.20)

Note that the dipole moment changes sign under λ → 1/λ, reflecting the fact its exis-

tence is directly tied to the mass anisotropy; we plot this in Fig. 2.5 (b). We reiterate that

the dipole moment associated with the DW is a generic feature of an Ising model in which

the two phases are distinguished by an orientational symmetry-breaking order parameter;

however it is not captured by the NLσM description of [1] at leading order in the limit of

weak anisotropy.

2.3.4 Does the Dipole Moment Matter?

A central result of our microscopic study is that there is indeed a nonzero dipole moment

at the domain wall as suggested by the symmetries of the system. However, as we have

emphasized this physics is invisible in the weak-anisotropy NLσM treatment on the basis

of which we sketched the phase diagram of the system with temperature and disorder and

discussed qualitative features of these phases. As a consequence of this dipole moment,

there are long-range interactions between different portions of a domain wall and between

different domain walls. Do these perturbations to the original long-wavelength theory af-

fect the physics? We will address two separate questions: the role they play at the Ising

transition in the absence of disorder, as well as the interplay of the long-range couplings

36

with the formation of domains in the Ising phase. As both questions should have universal

answers independent of the microscopic model, it will suffice to consider the role of the

dipole-dipole interactions in the long-wavelength theory. Therefore we consider the free

energy of the 2D Ising model,

F ∼∫d2r[(∇ϕ)2 + rϕ2 + uϕ4

](2.21)

and add to it a perturbation appropriate to a long-range interaction between dipoles:

δF ∼ v

∫d2r

∫d2r′

pr · pr′ − (pr · r)(pr′ · r′)|r− r′|3 (2.22)

and determine its effect on the critical theory and domain formation with disorder.

(i) Irrelevance at Tc. Using the fact that pr ∼ ∇ϕ(r), we have for dimensional purposes

δF ∼ v

∫d2r

∫d2r′

ϕ(r)ϕ(r′)

|r− r′|5 (2.23)

where we have ignored angular factors as we are really only interested in power-

counting. Recall[20] that a long-ranged spin-spin interaction scaling as 1/xd+σ is

irrelevant at the short-ranged Ising critical point if σ > 2 − ηSR where ηSR is the

anomalous dimension of the Ising field in the short-ranged theory. For dipolar inter-

actions in the d = 2 Ising model we have ηSR = 1/4 and σ = 3, and thus (2.22)

represents an irrelevant perturbation at the finite-temperature Ising critical point Tc.

(ii) Imry-Ma domain formation at T = 0. Recall that the standard Harris criterion[48]/Imry-

Ma[55, 5] argument in the 2D Ising ordered phase proceeds as follows: we flip spins

to orient with the random field to gain an energy ∝ L, at the cost of a introducing

a smooth domain wall whose energy also scales as L; thus, for a sufficiently weak

random field there is no advantage to introducing domains. However, a more sophis-

ticated argument[15] notes that domain wall roughening can increase the energy gain37

from the random field so that it scales as L logL. Thus, disorder always destroys the

Ising ordered phase in d = 2. We have verified that long-range dipolar interactions do

not affect the qualitative features of this argument, so that disorder remains a relevant

perturbation that destroys Ising order at zero temperature.

Although the universal physics and the critical points are unaffected, one physical mani-

festation of the dipolar interactions is to increase the numerical value of the surface tension

and thus renormalize the Ising stiffness upwards. As a consequence, the characteristic size

of an Ising domain in the nematic phase is enhanced—note that owing to the exponential

dependence of the domain size on the stiffness this can be a quite significant effect.

2.3.5 Domain Wall Texturing

Thus far we have focused on a sharp domain wall. Within the NLσM, we find that do-

main walls are always textured: there is a length scale, set by the competition between the

Ising anisotropy (that breaks the SU(2) symmetry down to Z2) and the stiffness. Does the

texturing persist even when the NLσM is no longer valid? We answer this partially via a

self-consistent numerical solution of a domain wall, which reveals that some texturing does

indeed persist into the strong anisotropy regime; we also study the texturing as a function

of the random field gradient at the wall, as it provides additional information about how the

domain wall structure is altered in the presence of disorder.

We take (2.11) as the ground state as before, and the domain wall solution is given by

(2.12) subject now to the constraint |uX |2 + |vX |2 = 1, and with the boundary condition

that uX and vX approach 1 for X = (−N/2 + 1)2π`2B/Ly and X = N/2 × 2π`2

B/Ly

respectively. This corresponds to a domain wall where the pseudospin rotates from valley

1 to valley 2 as we move from left to right. Note that, unlike in the sharp case, the wall is

allowed to ‘texture’, i.e. cross over from one valley to the other over a finite length scale.

In our simulations, we will take Lx = 10π`B, Ly = 30`B corresponding to N = 150, and

once again take the background to be fully polarized in valley 1.38

Using Wick’s theorem and the HF trial wavefunction in (2.4), we find

〈H〉DW =∑X

(u∗X v∗X

)A

uX

vX

A =

UH1 (X) + U ex

1 (X) + g(LyX

2π`2B− 1

2

)U c(X)

U c∗(X) UH2 (X) + U ex

2 (X)− g(LyX

2π`2B− 1

2

)

where the Hartree-Fock potentials are

UH1 (X) =

∑X′

[V 1X′,1X

1X,1X′

(|uX′|2 − 1

)+ V 2X′,1X

1X,2X′ |vX′ |2]

UH2 (X) =

∑Y

[V 1X′,2X

2X,1X′

(|uX′|2 − 1

)+ V 2X′,2X

2X,2X′ |vX′ |2]

U ex1 (X) = −

∑X′

V 1X′,1X1X′,1X |uX′ |2

U ex2 (X) = −

∑X′

V 2X′,2X2X′,2X |vX′|2

U c(X) = −∑X′

V 2X′1X1X′,2Xv

∗X′uX′

In the above expressions we have subtracted off the energy of the ground state, so that

we may consistently compare domain wall energies for different values of the anisotropy.

The optimization procedure proceeds iteratively, as follows. We begin with a trial wave-

function satisfying the boundary conditions, and in each iteration find the values of um, vm

which optimize the HF energy, which are then used to generate the HF potentials for the

next iteration. Eventually, the procedure converges to a self-consistent solution.

We estimate the degree of texturing by computing the magnitude of the x-component

of the pseudospin in the domain wall configuration, since this is nonzero near the wall and

vanishes far from it. In Fig. 2.6, we plot contours of constant 〈Sx〉 in the anisotropy-field

gradient plane, as well as the degree of texturing as a function of field gradient at λ2 ≈ 5.5,

the anisotropy appropriate to AlAs.

39

�2

g

0.005 0.01 0.015 0.02 0.025 0.0310−3

10−2

10−1

g

S x/Ly (i

n un

its o

f 1/l B)

g

textu

ring

Domain Wall Texturing, hSxi`B/Ly

Figure 2.6: Domain-wall texturing from Hartee-Fock Theory. (Top) Contour plot of theaverage in-plane valley pseudospin 〈Sx〉 per unit magnetic length along the domain wall, asa function of the mass anisotropy λ2 and the valley Zeeman field gradient g, with the latteron a logarithmic scale. The dashed line marks the anisotropy λ2 ≈ 5.5 relevant to AlAs;note that there is still some texturing in this limit. (Bottom) Cut along dashed line, with gon a linear scale.

40

2.4 Disorder in the Microscopic Theory

As discussed previously, disorder plays a central role in destabilizing the QHIN towards

the QHRFPM. There are two primary sources of disorder: (i) random strains in the system

can lead to a position-dependent shift of the energies in the two valleys– while the average

strain (pseudomagnetic field) can be externally controlled, fluctuations of the strain are

inevitable; and (ii) random fluctuations of the smooth electric potential, Ud that arises from

the screening of the potential due randomly placed donor impurities by electrons in the

2DEG also give rise to a random valley field. The random valley Zeeman field from the

strain is difficult to quantify precisely, but is related to the anisotropy of the displacement

field u(r) of the crystal from its equilibrium position: ∆strv (r) ∝ (∂x − ∂y)u(r). The

random electric field mechanism can be understood via a straightforward application of

perturbation theory and its value estimated from the sample mobility, as we now describe.

2.4.1 Random Fields from Impurity Potential Scattering

We briefly summarize the argument that leads to a coupling between a local anisotropy

in the disorder potential and the Ising order parameter. Since the form factors of the two

valleys are different, we expect that the portion of the disorder potential that is antisym-

metric in valley indices will lead to a spatially dependent single-particle splitting between

valleys; in the limit when the cyclotron gap diverges, i.e. when the lowest Landau level

approximation is exact, this is the only contribution, and we can argue from symmetry that

the corresponding random field should take the form (∂2x − ∂2

y)Ud (at least in the small-

anisotropy limit). Note, however, that this term is a total derivative, and contributes signif-

icantly only at the boundary of a domain. To go beyond this, we must relax the ωc → ∞

limit, and allow for the effects of Landau-level mixing to first order in Ud; since this al-

lows for terms of order U2d/~ωc, the random field now receives contributions of the form

((∂xUd)2 − (∂yUd)

2)/~ωc, which is not simply a boundary term.

41

To derive the higher-order contribution to the single-particle valley splitting from the

Landau level mixing terms, we make a simplifying assumption: namely, we ignore interac-

tions while computing the effect of mixing. While the interactions may combine with the

effects of disorder to modify details of the calculation, we expect that their neglect does not

change the qualitative features of our results. The Hamiltonian for noninteracting electrons

in AlAs is, in the Landau basis

Hni =∑n,X

(n~ωc − µ) c†n,κ,Xcn,κ,X +∑

q,X,κ,n,m

Umnd (−q)eiqxXc†m,κ,X+

cn,κ,X−(2.24)

where we have defined X± = X ± qy`2B2

. Here, we have expanded the notation of Section I

to include Landau level indices n, m. In this basis, we have defined the matrix elements of

the disorder potential Ud via Umnd (−q) ≡ Ud(−q)Fmn

κκ (q), which naturally introduces the

form factors

F nmκκ (q) =

m!

n!

(iqx√2uκ− qy

√uκ2

)n−mLn−mm

(q2x

2uκ+q2yuκ

2

)e−

q2x4uκ−q2yuκ

4 (2.25)

for n ≥ m, with F nmκκ (q) = Fmn

κκ (−q)∗, where Lαn is the generalized Laguerre polynomial.

Next, we compute a renormalized effective potential[47, 140] within the lowest Landau

level (where m = n = 0) by including Landau level mixing in perturbation theory. We find

ULLL =∑q,X,κ

Uκ,00d,eff (−q)eiqxXc†0,κ,X+

c0,κ,X− (2.26)

where, to first order in Landau level mixing,

Uκ,00d,eff (−q) = Ud(−q)F 00

κκ(q) +∑

q′,n 6=0

[Ud(−q′)Ud(q′ − q)

n~ωcei

q×q′`2B2

×F 0nκκ (q− q′)F n0

κκ (q′)

]+O

(|Ud|3/~ωc

)(2.27)

42

We are primarily interested in the valley symmetry-breaking contribution from this

term, so we consider only the portion antisymmetric in κ. Assuming that the disorder

potential is smooth on the scale of `B, we may expand in gradients of Ud; to quadratic

order in qx, qy, only the n = 1 term in the sum contributes, and we find

USBd (−q) = U1

d,eff(−q)− U2d,eff(−q)

= −1

4

(λ− λ−1

) [(∂2x − ∂2

y

)Ud]−q +

1

2~ωc(λ− λ−1

) [(∂xUd)

2 − (∂yUd)2]−q

(2.28)

The leading piece vanishes except on domain boundaries, as discussed; thus, the dominant

valley splitting arising from impurities is due to the second term.

We focus our attention on a domain boundary, and assume that the distance between

the centers of two domains is roughly the correlation length d of Ud. In this case, we

simply assume that the single-particle energy splitting changes sign linearly over a distance

d, corresponding to the final term in (2.4), with the overall energy scale ∆SBd set by the

characteristic scale of the spatially varying random potential USBd (r).

2.4.2 Estimating Disorder Strength from Sample Mobility

We may estimate the strength Ud of the smooth random potential from the measured sample

mobility µ and the distance d of the dopant atoms from the plane of the 2DEG, and using

the results of the previous section, deduce the parameters of the random Zeeman field h.

Taking the dopants to be Poisson-distributed, and assuming that the potential fluctuations

are screened by electrons in the 2DEG, we can estimate the fluctuations of the potential[32]

in the plane of the 2DEG to be

〈|Ud(q)|2〉 = (U0d)2e−2qd (2.29)

43

where U0 is determined by the screening length and should be proportional to the impurity

density. The scattering rate due to this potential is

Wp,p′ =2π

~|Ud(p− p′)|2δ (Ep − Ep′) (2.30)

A straightforward Boltzmann transport calculation of the transport relaxation time, as-

suming that it is dominated by the Fermi surface yields

1

τtr=

m

2π~2

∫ π

−π

dθ

2π

(2π

~

)(1− cos θ)U2

0d2e−2kF sin θ

2×2d (2.31)

where the (1 − cos θ) factor suppresses the contribution of small-angle scattering, which

does not contribute to charge relaxation. For kFd� 1, we have

1

τtr=

m

π~2(U0d)2

√π~2

8(mvFd)3. (2.32)

Using the fact that 1/τtr = e/(mµ) where µ is the mobility,

U0 ≈(

8√πe~3k3

Fd

µm2

)1/2

(2.33)

where we take m =√mxmy.

Finally, we note that the characteristic length scale of the disorder potential is roughly

the distance of the dopant plane from the 2DEG, allowing us to estimate that |∇Ud| ∼

U0/d. Using the results of the previous section, the characteristic value of the symmetry

breaking term ∆SBd is given by

∆SBd ∼

1

2π~2(mx −my)

`4B

d2U2

0 . (2.34)

44

This corresponds to a random field h ∼ ∆SBd /`

2B in the NLσM. Since h is correlated roughly

over a distance d, we find that the characteristic width of the random field distribution is

W ∼ (hd)2; this is the parameter that quantifies the strength of disorder in our model.

2.5 Experiments

As promised, we now turn to a discussion of probes of valley-nematic ordering via trans-

port measurements. We will assume the ability to apply a valley-symmetry-breaking strain.

Furthermore, we shall also assume that the maximal valley splitting that can be thus pro-

duced is sufficient to fully polarize the system in one of the valleys. We note that this is

already feasible for the samples studied experimentally thus far. We will also assume that

the sample is engineered in a Hall bar geometry with principal axes parallel to the sample

boundaries, so that we may assume that the nematic anisotropy is oriented along the x- or

y- direction of the sample. This removes ambiguity in the definition of components of the

conductivity, but more importantly ensures that the anisotropies are observable in the Hall

bar geometry.5

The cleanest probe of the valley ordering is to examine the longitudinal conductivity

for anisotropy. A proxy for the orientational symmetry-breaking order parameter is the

quantity ζ ≡ σxx/σyy − 1. Note that it is important that both σxx, σyy are measured in

simultaneously, which can be conveniently accomplished in a four-terminal geometry.. The

behavior of ζ will exhibit quite distinct behavior as a function of temperature and disorder

strength, and will be affected by the application of a strain field. The principal distinction

due to disorder is between ‘clean’ samples dominated by the properties of a single Imry-Ma

domain, and ‘dirty’ ones which contain several domains. We identify four different cases:

5This would not be the case, for instance, if one of the principal axes of the Hall bar was oriented alongthe [110] direction of the quantum well, since the projection of the anisotropic valleys along [100] and [010]onto the [110] direction are identical and thus transport anisotropy no longer reflects valley polarization.

45

(i) Clean Sample, Zero Strain. Here, we expect that at high temperatures, the system is

in the Ising thermal paramagnet phase, with no anisotropy, so ζ = 0; furthermore,

ζ remains flat as the filling is tuned across the Hall plateau. As the temperature is

lowered below the Ising Tc, the sample should enter the valley-ordered phase. Here, ζ

remains pinned to zero exactly at ν = 1 i.e., the center of the Hall plateau. However,

upon tuning the filling about ν = 1, ζ will change sign. This follows from the fact that

the longitudinal conductivity goes from being dominated by hopping between hole-

like levels of one valley to that between electron-like states of the opposite valley

as the doping level crosses the center of the Hall plateau. The resulting longitudinal

conductivities inherit the local anisotropy of Landau orbitals of the two valleys.[1, 33]

The maximum value attained for T � Tc can be estimated as ζmax ≈ |√mx/my −

1| = λ− 1.

(ii) Clean Sample, Under Strain. Application of strain to a clean sample should have little

effect on the transport below Tc for one orientation of the strain, but should suppress

the anisotropy for the opposite orientation. In the paramagnetic phase, a strong valley

polarization should result in transport signatures similar to that of the Ising ordered

phase.

(iii) Dirty Sample, Zero Strain. For dirty samples, the anisotropy from the different do-

mains cancel and we have ζ = 0 for zero strain, at all temperatures.

(iv) Dirty Sample, Under Strain. Once again, application of strain to a dirty sample should

polarize the system, and lead to transport signatures similar to the clean limit at zero

strain, below Tc. As discussed previously, the activation gap measured via longitudi-

nal transport will be highly sensitive to the application of strain, and increase dramati-

cally as the sample crosses over from multiple-domain to single-domain behavior and

thus from domain-wall dominated to single-particle longitudinal transport.

46

As noted in the preceding section, the Imry-Ma domain size is exponentially sensitive to

changes in microscopic parameters and thus estimating the domain size is a challenge.

This can be circumvented to some degree by studying transport in samples of different

sizes and/or doping levels. For a given doping level, smaller samples are more likely to be

in the clean limit as defined above, while lowering the doping level for samples of a fixed

size should weaken disorder to some extent. Also, the identification of clean and dirty

samples is somewhat loose; samples of intermediate size may show significant anisotropy

even though there is no net Ising ordering, since the anisotropies of different domains may

not fully cancel.

Note that while the four-terminal probes are particularly unambiguous and striking,

there is also useful information that can be gleaned from two-terminal transport measure-

ments which only have access to a single longitudinal transport coefficient. Here, the ne-

matic symmetry breaking is encoded in the behavior of ρxx as a function of the doping

level. This will be minimal in the center of the Hall plateau, and grow as the filling is

detuned from ν = 1 in either direction. The mismatch in ρxx for ν < 1 and ν > 1 will

exhibit behavior similar to that described for ζ in the different cases above.

Finally, we note that random field Ising order is typically accompanied by a host of

hysteretic effects[21] that might also be observable in experiments, particularly with an

applied valley Zeeman field.

2.6 Concluding Remarks

We have spent the majority of this chapter focussing on a specific instance of valley order-

ing relevant to experiments: the Ising-nematic order in AlAs quantum wells. As the reader

no doubt appreciates by now, much of the richness of the phenomena discussed above

stems from the inequivalence of the low-energy electronic dispersion in the two valleys.

More specifically, the key observation underpinning our analysis is that the inequivalence

47

between valleys is encoded by the fact that rotating between them necessarily requires a

simultaneous interchange of spatial axes; this has three striking consequences. First, in

the presence of interactions the naıve SU(2) symmetry associated with a generic ‘internal’

index is reduced to an Ising symmetry. Second, the intertwining of pseudospin and spatial

rotations results in the transmutation of quenched spatial disorder into a random field act-

ing on the Ising order, driving the transition into the paramagnetic QH phase. Finally, the

same coupling permits strain to act as a valley Zeeman field, and anisotropy to serve as a

probe of transport—both important to experimental studies of nematic ordering.

48

Chapter 3

Order by Disorder and by Doping in

Quantum Hall Valley Ferromagnets

3.1 Introduction

In this chapter, we revisit the interplay between broken symmetry and topological order

in the context of the QH states of multi-valley semiconductors, specifically those recently

observed in two-dimensional electron gases (2DEGs) confined in Si(111) quantum wells [8,

34, 65]. We find several striking new phenomena embedded in surprisingly intricate phase

diagrams—even while considering only integer quantum Hall states in the lowest Landau

level (LLL). This system exhibits six-fold valley degeneracy in the electronic dispersions,

as shown in Figure 3.1. Consequently, it exhibits large symmetry group—[SU(2)]3oD3—

in the standard limit where the magnetic length `B is much longer than the lattice constant a.

Here,Dn is the dihedral group of symmetries of a regular n-gon, and the semidirect product

structure (denoted by o) reflects the fact that these discrete symmetries act upon the SU(2)

axes. The rich phase structure derives from the various possibilities for breaking these

symmetries, and how these manifest at different ν. For our primary example—the (111)

system (Sec. 3.2)—we find a finite-temperature Z3 transition into a nematic phase where

49

kx

ky

A

B

C

A

B

C

⌫

T

1 2 3H0 =U(1)2⇥SU(2)

Z3 nematic

G = [SU(2)]3 o D3

H0 =U(1)⇥SU(2)2

HT = [SU(2)2 o D2] ⇥ SU(2)

T ⇤E-S,2 T ⇤

E-S,3

Tc

first-order

second-order

H0 = U(1)3

HT = G

O = S2⇥S2⇥S2O = S2 O = S2 ⇥ S2

O = Z3

⌫12c ⌫23

c

Figure 3.1: Valley ordering in Si(111) QH states. (Inset) Model Fermi surface. Ellipsesdenote constant-energy lines in k-space. (Main figure) Schematic global phase diagram,showing how the G = [SU(2)]3 o D3 symmetry is broken to H0, HT at zero and finitetemperature. The order parameter spaces are O = G/HT for T > 0, and O = HT/H0

at T = 0. For ν = 1, 2, D3 symmetry breaks continuously at Tc, but this becomes first-order around ν = 3. Near ν = 2, 3 order by doping yields to thermal order-by-disorder atT ∼ T ∗E-S.

the discrete factor is broken, and zero-temperature phases where the continuous [SU(2)]3

symmetry is broken down to various subgroups. We give a detailed group-theoretic analysis

of symmetry breaking in Sec. 3.5. We sketch the phase diagram resulting from fitting

together these possibilities in Fig. 3.1.

The mechanisms of symmetry breaking are also unusual. While the nematicity is driven

by Hartree-Fock exchange interactions as is standard in QH ferromagnetism [66], the T →

0 ordering involves entropic selection—order by disorder [130, 81, 22, 26]—and selection

via the energetics of Skryme lattices that form in the vicinity of integer ν, a new mechanism

that we term order by doping in tribute to its entropic cousin.

50

We also discuss experiments in Sec. 3.3 and the very similar case of Si(110) in Sec.

3.4. Finally we discuss valley skyrmion crystals in Sec. 3.6 and then close with a few final

remarks in Sec. 3.7.

3.2 Silicon(111)

We begin our discussion by listing some salient features of Si(111) quantum wells relevant

to understanding the QHE in these systems. The valley degeneracy of the Si 2DEG de-

pends on the orientation of the interface, as this choice can break the crystal symmetries

responsible for the exact valley degeneracy in bulk Si. (We ignore spin.) In case of the

(111) interface, effective mass theory predicts a six-fold degeneracy [121] (Fig. 3.1, inset).

This degeneracy is quite robust—for instance, it cannot be lifted by changing the width of

the confining well or by an interface potential. For the bulk of this Chapter, we take this

degeneracy to be exact, surely an idealization; we comment on corrections to this scenario

—arising due to wafer miscut and lattice mismatch— in Sec. 3.7.

We label valleys as shown in Fig. 3.1 (inset). Valley κ is centered at ~Kκ, where

~KA = (√

3K0

2, K0

2), ~KB = (0, K0) and ~KC = (−

√3K0

2, K0

2), with ~Kκ = − ~Kκ. Here

K0 ≈ 1/a where a is the lattice constant [K0 =√

2/3∆m, where ∆m is the distance in

K-space within the Brillouin zone from the Γ point to the minimum-energy point in the

conduction band]. Note that in each valley the effective mass tensor is anisotropic; this

is most evident in a coordinate system in which the mass tensor is diagonal. The single-

particle Hamiltonian in valley κ (where κ = A,B,C) is Hκ =∑

i=1,2((~p+e ~A/c− ~Kκ)·~ηκi)2

2mi,

where ~ηA/C1 = 12(∓1,

√3), ~ηA/C2 = 1

2(±√

3, 1), ~ηB1 = (1, 0) and ~ηB2 = (0, 1); Hκ is

obtained by taking K0 → −K0 in these expressions.

In Landau gauge ~A = (0, Bx), the LLL eigenfunctions labeled by momentum ky are

given by

φκ,ky =(fκ)

1/4

(π1/2`BLy)1/2ei

~Kκ.~reikyye−(fκ+igκ)

(x+ky`2B)2

2`2B , (3.1)

51

where (f, g)A,A = (4√λ

λ+3,√

3(1−λ)λ+3

), (f, g)B,B = ( 1√λ, 0), (f, g)C,C = (4

√λ

λ+3, −√

3(1−λ)λ+3

),

λ = (m2/m1) ≈ 3.55 [64] and the magnetic length `B =√

~ceB

. We focus on filling

fractions ν < 6 and ignore mixing between different Landau levels (LLs). As is usual in

QH ferromagnets, even if we restrict to (near-)integer filling, the exact degeneracy between

the valley degrees of freedom at single-particle level is lifted by interactions, which select

a ground state at each integer filling ν < 6, and in doing so break one or more symmetries

spontaneously. The question of precisely how this happens is our focus in the remainder.

3.2.1 Effective Hamiltonian

Since we are working in a degenerate manifold of the electron kinetic energy—quenched by

the magnetic field—the effective Hamiltonian is comprised solely of interaction terms, that

inherit the kinetic anisotropies through their dependence on the single-particle LL eigen-

functions. In the limit K0`B � 1, the electron-electron interaction term is

H =1

2S

∑~q,κ,κ′

V (~q)ρκκ(~q)ρκ′κ′ (−~q) (3.2)

where S = LxLy is the total area, ρκκ is the density operator within valley κ projected

to the LLL and V (~q) = 2πe2

qis the matrix element of the Coulomb interaction. [A static

background is omitted from (3.2) for clarity.]

The Hamiltonian (3.2) has an approximate G = [SU(2)]3 o D3 symmetry. To see this,

note that H is invariant under SU(2) rotations between the two valleys (κκ) in the pair,

explaining the [SU(2)]3, as well as under a D6 discrete point-group symmetry. However,

any element of D6 that only interchanges the two valleys (κ, κ) in a pair is equivalent to

an SU(2) π-rotation; the D6 elements not of this type form a D3 subgroup that acts on the

3 SU(2) indices, leading to the semidirect product structure. (See Sec. 3.5 for details.)

Recent work on wide (001) AlAs quantum wells studied a symmetry similar to the discrete

rotation above [1].

52

We now consider the cases of various integer filling fractions using the vanishing mass

anisotropy limit as the starting point. (In the theoretically convenient case of λ = 1, H is

SU(6) symmetric; qualitative pictures obtained for |λ−1| � 1 remain valid even when the

anisotropy is no longer small.) Note that, owing to particle-hole symmetry about ν = 3 the

problems at ν = 1; 5 are equivalent, as are those at ν = 2; 4. This leaves us three distinct

fillings to consider.

3.2.2 ν = 1

At the SU(6) symmetric point λ = 1, the degenerate ground states with one filled LL are

|ψ〉 =∏

kyd†ky |0〉 where d†ky =

∑κ∈{A,B,C}(ακc

†κky

+ακc†κky

), with∑

κ(|ακ|2 + |ακ|2) = 1

The anisotropy splits this degeneracy. At |λ−1| � 1, first-order perturbation theory yields

〈ψ|H|ψ〉 =∑κ,κ,′σ∈{A,B,C}

δV σκκ′(|ακ|2 + |ακ|2)(|ακ′ |2 + |ακ′ |2) (3.3)

where δV σκκ′ = 1

2|εσκκ′|(V σσ

σσ − V κ′κκκ′ ), and

V κ′κκκ′ =

∑k,k′

∫~r,~r′φ∗κk(~r)φ

∗κ′k′(~r

′)V~r−~r′φκ′k(~r′)φκk′(~r). (3.4)

The δV σκκ′ are all positive and proportional to N , the number of electrons. Hence

the approximate new ground states in the thermodynamic limit are of the form |κ〉 =∏ky

(ακc†κ,ky

+ ακc†κ,ky

)|0〉; these break the [SU(2)]3 o D3 symmetry down to H0 =

U(1)× [SU(2)2oD2] (where the second factor refers to rotations of the unoccupied pairs),

leading to a single Goldstone mode (as G/H0 = S2). Working in the vicinity of ν = 1 at

T = 0, from standard energetic arguments [120] we conclude that for ν & 1, skyrmions are

created within the occupied-valley subspace (κ, κ); similarly, for ν . 1, anti-skyrmions

are created [120]. At any T 6= 0, statistical averaging over Goldstone modes restores the

broken SU(2) symmetry, so that the invariance group is HT = SU(2) × [SU(2)2 o D2].

53

⌫ = 1 ⌫ = 2 ⌫ = 3

Class I Class II Class I Class II

Figure 3.2: Possible valley-ordered states at ν = 1, 2, 3, including representatives ofClass I and II states for ν = 2, 3. Unfilled and fully-filled valleys are shown as empty andfilled ellipses; valleys partially-filled due to a particular choice of SU(2) vector within thetwo-valley subspace are shaded with different colors.

The order parameter of the resulting phase lies in G/HT = Z3. (See Sec. 3.5 for details.)

We conclude that valley ferromagnetic order onsets via a finite temperature Z3 transition

into a nematic phase with broken orientational symmetry (Fig. 3.1).

3.2.3 ν = 2

We next consider the case when two LLs are filled. Again, we begin at λ = 1

where the degenerate ground states are given by |ψ〉 =∏

i=1,2

∏kyd†i,ky |0〉 where

d†i,ky =∑

κ∈{A,B,C}(ακic†κky

+ ακic†κky

), and∑

κ(|ακi|2 + |ακi|2) = 1. Moving away

from the SU(6) point, but keeping |λ − 1| � 1, the ground state manifold has two

kinds of states that remain degenerate even upon inclusion of the anisotropic terms

(Fig. 3.2). “Class I” ground states are obtained by filling both valleys in a pair, and

take the form |κκ〉 =∏

kyc†κ,kyc

†κ,ky|0〉. “Class II” ground states on the other hand

are constructed by picking two pairs and setting each to have ν = 1 by spontaneously

breaking the residual SU(2) symmetry of rotations within the pair. These are of the form

|κκ′〉 =∏

ky(ακc

†κ,ky

+ ακc†κ,ky

)(ακ′c†κ′,ky

+ ακ′c†κ′,ky

)|0〉. While Class I states break the

[SU(2)]3oD3 symmetry down to [SU(2)]3, Class II states break it down to U(1)2×SU(2);

the order parameter spaces are Z3 and S2 × S2 for class I and II states, respectively, which

54

therefore host no and two Goldstone modes. This disparity leads to selection of the latter

by thermal fluctuations as we discuss below.

First, however, we demonstrate that selection occurs due to charge doping – incommen-

suration – at T = 0. To see why this is so, observe that doping Class II states to a filling

ν & 2 (ν . 2) proceeds by creating skyrmions (anti-skyrmions) in the two-dimensional

subspaces of the occupied valley pairs. For Class I states on the other hand, the charge

added or subtracted is accomodated in a conventional quasielectron (quasihole) Wigner

crystal. As skyrmion (anti-skyrmion) lattices have lower energy [120], we argue that dop-

ing selects Class II states.

Turning now to T > 0, we observe that the combination of a high ground state de-

generacy and a disconnected ground state manifold—there is no continuous path in the set

of ground states that connects a state in Class I to a state in Class II— is ideal for seeing

“order by disorder”. This phenomenon, in which entropic considerations select a ground

state, occurs often in frustrated spin systems [130, 81, 22, 26]. Since there are gapless

excitations about Class II states, they are selected by thermal fluctuations as the free en-

ergy of fluctations about the degenerate ground state manifold is peaked about states with

a large number of soft modes. However this mechanism comes into play above a crossover

temperature scale T ∗ (Fig. 3.1). Below T ∗, energetic, rather than entropic, considerations

favor Class II states—the order by doping mechanism. The crossover between selection by

energetic considerations and selection by entropy—that we dub the “E-S crossover”—takes

place at T ∗E-S ∼ e2

`B.

We observe that while Goldstone modes are responsible for order by disorder, the

SU(2) symmetries remain unbroken at any T 6= 0 so that the invariance group is HT =

[SU(2)2 oD2]×SU(2), (the D2 reappears as we may once again interchange between the

filled pairs when SU(2) is restored). As G/HT = Z3, we conclude that the system has a

transition at Tc > 0 described by a Z3 nematic order parameter, in which the D3 symmetry

is broken (Fig. 3.1). Since in the experimental systems of interest the filling is tuned with

55

field rather than by gating, we anticipate that T ν=2c < T ν=1

c , as the relevant energy scale

is the surface tension of Z3 domain walls, that depends in turn on the Coulomb energy

e2/`B ∝√B. (There is also a factor of 1/2 from the fact that at fixed electron number only

half of the electrons enter the domain wall energetics at ν = 2.)

3.2.4 ν = 3

We now examine the situation with three filled LLs. At the isotropic point λ = 1, the de-

generate ground states are |ψ〉 =∏

i=1,2,3

∏kyd†i,ky |0〉where d†i,ky =

∑κ∈{A,B,C}(ακic

†κky

+

ακic†κky

). As before, the symmetry is reduced for λ 6= 1, and the new ground state

manifold again has two kinds of states (Fig. 3.2). Class I states are obtained by filling

both valleys in a pair and then spontaneously breaking SU(2) in another pair, and are of

the form |κκκ′〉 =∏

kyc†κ,kyc

†κ,ky

(αc†κ′,ky + α′c†κ′,ky)|0〉. Class II ground states are ob-

tained by breaking each of the three SU(2)s by forming a ν = 1 state in each pair, so

|ABC〉 =∏

ky

∏κ=A,B,C(ακc

†κ,ky

+ ακc†κ,ky

)|0〉 with |ακ|2 + |ακ|2 = 1. Class I states are

invariant under H0 = SU(2)2 × U(1), and Class II states under U(1)3. By explicit con-

struction we find the order parameter spaces S2 and S2×S2×S2, leading to one and three

Goldstone modes respectively. (See Sec. 3.5 for details.)

Consider charge doping at T = 0. For ν & 3 (ν . 3), skyrmions (anti-skyrmions) are

created about both Class I and II states. However, the structure of the resulting triangular

lattices is quite different. Discussing skyrmions for specificity, for Class I states doping

proceeds by making a lattice of skyrmions that live in only one two-dimensional subspace,

whereas for Class II states the skyrmion lattice has a tripled unit cell, as it results from sym-

metrically combining three sublattices each built from skyrmions in one of the three differ-

ent two-dimensional subspaces. As there is no valley Zeeman energy, the skyrmion size is

set entirely by their density. In order to estimate the energies of the competing skyrmion

crystals, we utilize the fact that the relevant nonlinear sigma models optimize their gradient

energy for analytic (two-component) spinor solutions—non-analytic configurations gener-

56

ically have higher energy. The simplest such analytic solution that is (quasi-)periodic with

finite topological charge Q per unit cell, has Q = 2, as all Q = 1 configurations with these

desiderata are non-analytic and hence have higher energy. [See Sec. 3.6 for details.] Such

a quasi-periodicQ = 2 spinor solution [68] gives Class I and II states identical gradient en-

ergies, so the issue turns on the Coulomb energy which is lower for Class II. We therefore

conclude that doping selects Class II states.

For T > 0, the Goldstone mode fluctuations about Class II states restore the full G =

[SU(2)]3 o D3 symmetry and hence there is no sharp finite-temperature transition owing

to the lack of any broken symmetries. We nevertheless expect thermal selection of Class II

states owing to the excess of Goldstone modes compared to Class I states. As in the ν = 2

case this occurs above a scale T ∗E-S, below which order by doping dominates.

In combining the results for ν = 1, 2, 3 we note that their distinct symmetries and

doping energetics at T = 0 point to first-order transitions at ν12c and ν23

c , where 1 < ν12c <

2 < ν23c < 3, that we expect survive to T > 0. Thus, we arrive at the global phase diagram

of Fig. 3.1.

3.3 Experiments

We expect that nematic order leads to measurable anisotropies in longitudinal conductivi-

ties σxx, σyy, though the orientation of the valleys with respect to the symmetry axes may

present an added complication, even for samples oriented along crystallographic axes. For

ν = 1, 2, the anisotropy should show order-parameter onset behavior at Tc, typically a

few kelvin at B ≈ 10T for systems with comparable mass anisotropy and dielectric con-

stant [1]. Class II state selection at ν = 2 will be reflected by the extreme sensitivity of

the activation gap to strain-induced valley Zeeman splitting [1] absent in Class I states that

lack skyrmion excitations. The selection of Class II states at ν = 3 is challenging to detect

as they lack nematic order, and Class I states also host skyrmions. However, restoration of

57

orientational symmetry in going from ν = 2 to ν = 3 coupled with observation of the QHE

would bolster this scenario. Similar considerations apply, mutatits mutandis to the case of

Si(110) quantum wells.

3.4 Silicon(110)

We now briefly discuss the case of Si(110) where in the presence of a weak interface po-

tential, effective mass theory predicts a fourfold valley degeneracy [8] (Fig. 3.3). (See

Sec. 1.4 for the discussion of strong interface potential case.) The valleys are centered at

~KA = − ~KA = (K, 0) and ~KB = − ~KB = (0, K). In the Landau gauge ~A = (0, Bx), the

LLL eigenfunctions are given by Equation 4.4 with (f, g)A,A = (√λ, 0) and (f, g)B,B =

( 1√λ, 0). The interaction Hamiltonian has [SU(2)]2 oD2 symmetry where the SU(2)s are

independent rotations in the pair subspaces (A, A) and (B, B) and the D2 symmetry inter-

changes these pairs. (A group-theoretic analysis of this symmetry structure is provided in

Sec. 3.5.) This symmetry structure is clearly very different from that in the two valley case

discussed in Sec. 1.4. Thus order by disorder and by doping are expected to occur in this

case, in a scenario that closely parallels the six-valley case of Si(111) described earlier.

For |λ−1| � 1, the ground states at ν = 1 are given by |ψ〉 =∏

ky(ακc

†κ,ky

+ακc†κ,ky

)|0〉

and (|ακ|2 + |ακ|2) = 1 (Fig. 3.3). This case resembles ν = 1 for Si(111) and we expect

analogous results. The anisotropy in longitudinal conductivities should show up in the form

of order-parameter onset behavior at Tc.

At ν = 2 the ground state manifold supports two types of states (Fig. 3.3). Class I

states have the form |κκ〉 =∏

kyc†κ,kyc

†κ,ky|0〉. Class II states are of the form |AB〉 =∏

ky

∏κ=A,B(ακc

†κ,ky

+ ακc†κ,ky

)|0〉 where |ακ|2 + |ακ|2 = 1. Though the T = 0 mode

counting—zero versus a pair of Goldstone modes—is similar to ν = 2 for Si(111), at

T > 0 the full [SU(2)]2 oD2 symmetry is restored to Class II states by thermal averaging,

analogous to the Si(111) ν = 3 Class II state. Moreover unlike Class II states, the Class

58

kx

ky

A

B

A

B ⌫ = 1 ⌫ = 2

Class I Class II

Figure 3.3: Model Fermi surface and possible valley-ordered states for Si(110) quan-tum wells.

I states lack skyrmion excitations. Therefore, we expect selection of Class II by thermal

fluctuations and charge doping, but no finite-temperature transition about ν = 2.

3.5 Group-theoretic analysis of symmetry breaking

We discuss the simpler four-valley case first as a warm-up, before moving to the six-valley

example. In each case, we first discuss the high-temperature symmetry-group G, the finite-

temperature invariance subgroup of the broken-symmetry states HT , and finally its zero-

temperature counterpart H0. The nonlinear sigma models (NLσM) governing the T > 0

and T = 0 transitions have order-parameter spaces given by the group manifolds G/HT

and HT/H0, respectively. Valley indices are as described in the main text.

3.5.1 Four-Valley Case

We first show that high-temperature valley symmetry group is [SU(2)]2 o D2, where the

D2 interchanges the two SU(2) axes. To see this, we note that the valley Hamiltonian (after

including the anisotropy terms) has the following symmetries (refer to Fig. 3 of the main

text for valley labeling): two distinct SU(2) symmetries that each act within a valley pair

(A, A) and (B, B), and the dihedral group of symmetries of the square, that we denote

D4. The full symmetry group G is obtained by combining these symmetries. Clearly

59

N = [SU(2)]2 is a subgroup of G. Turning to D4, we recall that this is generated by two

operations, a permutation r = (ABAB) and a swap ρ = (AA), where we use conventional

notation to describe the action of finite groups: for instance (a1 . . . am)(b1 . . . bn) denotes

the cyclic permutations a1 → a2 → . . . → am → a1 and b1 → b2 → . . . → bn → b1 with

all other ‘letters’ left invariant. Now, it is clear that r4 = e, the identity; also, we note that

r2 simply interchanges the two valleys in a pair, and is therefore equivalent to a π rotation

within each valley pair, i.e. r2 ∈ N . Therefore it follows that the coset r2N = N = Nr2.

A similar argument reveals that ρN = N = Nρ, since ρ corresponds to a π-rotation

in the pair (AA) coupled with an identity operation in the other valley pair. Finally, we

observe that since r preserves the valley pair structure, transforming valley indices by r,

performing independent SU(2) rotations within each pair of valleys, and undoing the index

transformation, must be equivalent to a product of independent rotations within each pair,

so that r−1Nr = N . Since D4 = {e, r, r2, r3, ρ, rρ, r2ρ, r3ρ}, we see that full list of cosets

is {N, rN}. Thus, (i) N is a normal subgroup of G, N / G and (ii) G = {N, rN}, so

that G/N ∼= D2. We therefore conclude that G = N o D2 = [SU(2)]2 o D2, where in

identifying the D2 structure we used the fact r2 ∼ e in the coset space since r2N = N =

eN .

We now turn to the breaking of symmetries at different fillings. We will discuss the

symmetry breaking in two stages: first, we will determine the residual symmetry group

HT for T > 0, where the Mermin-Wagner theorem precludes the breaking of continuous

symmetries; the corresponding NLσM has target space G/HT . Then, we will discuss how

HT is further broken down toH0 at T = 0, described by a NLσM with target spaceHT/H0.

At ν = 1, and finite temperature we choose to fill a single valley pair while leaving the

other unfilled. The invariant subgroupHT of the resulting state is SU(2)2 (corresponding to

rotating in the filled and unfilled pairs – since for T > 0 Goldstone modes lead to averaging

over all possible superpositions of valleys within the filled pair), but the semidirect product

structure does not survive as we can distinguish the filled and unfilled pairs and therefore

60

their interchange is not a symmetry. The corresponding NLσM target space is G/HT =

([SU(2)]2 oD2)/[SU(2)2] = D2∼= Z2, consistent with our argument that the symmetry is

broken via a finite-temperature Ising transition. As T → 0 the Goldstone mode fluctuations

responsible for the finite-temperature restoration of SU(2) symmetry (of rotating between

valleys (κ, κ) in the filled pair) are suppressed, and this symmetry is broken by a specific

choice of SU(2) vector in the (κ, κ) subspace. This leaves a residual U(1) phase, but

the SU(2) symmetry between the unfilled valleys is still preserved, and therefore we have

the residual symmetry group H0 = U(1) × SU(2), giving the target space HT/H0 =

[SU(2)× SU(2)]/[U(1)× SU(2)] = SU(2)/U(1) = S2.

For Class II states at ν = 2, we fill both valley pairs to ν = 1, but as before the SU(2)

symmetry between the two valleys in each pair is restored at any finite temperature. As

a consequence, we can still interchange SU(2) axes in this case, so we have HT = G =

[SU(2)]2 oD2, and so there is no finite-temperature phase transition as G/HT is the trivial

group. We may parameterize any state in this base space ~z ∈ HT by ~z = g · ~z0, where

g =

g1 0

0 g2

, (3.5)

with g1, g2 ∈ SU(2), and ~z0 = (1, 0, 1, 0)T is a reference spinor; note that this implicitly

respects the semidirect product structure of HT . As T → 0, we break each of the two

SU(2) symmetries down to U(1). This remaining U(1) invariance within each valley pair

is generated by matrices h ∈ H0, where

h =

h1 0

0 h2

, (3.6)

with h1, h2 ∈ U(1). Using the equivalence relation on ~z, ~z′ given by ~z ∼ ~z′ ⇐⇒ ~z

′=

h · g′~z for h ∈ H0, we see that HT/H0∼= S2 × S2. For Class I states at ν = 2, we fill

61

both valleys in a pair, and therefore the residual symmetry group is HT = SU(2)× SU(2)

as this corresponds to rotating within the filled and unfilled pairs—we can not interchange

between the pairs. So, there is a putative finite-temperature transition possible for Class I

states, as G/HT = D2∼= Z2. However, as T → 0, we observe that there is no additional

structure that emerges as the full SU(2)×SU(2) symmetry is preserved. In contrast, Class

II states have Goldstone modes that will lead to their selection over Class II states as T → 0.

Therefore starting from T = 0 and restoring symmetry in stages we see that Class I states

do not emerge in the phase diagram.

3.5.2 Six-Valley Case

In the six-valley case, we begin by observing that the high-temperature symmetry group

is G = [SU(2)]3 o D3, where D3 is the dihedral group of symmetries of an equilat-

eral triangle (isomorphic to S3, the symmetric group) that acts on the three SU(2) axes.

To see this, we first observe that the symmetries of the valley Hamiltonian are (i) three

SU(2) symmetries that rotate between the two valleys in each pair (AA), (BB), (CC) and

(ii) the dihedral group D6 of symmetries of a regular hexagon. Clearly, N = SU(2)3

is a subgroup. Turning next to D6, we observe that it is generated by the sixfold rota-

tion r = (ABCABC) and the reflection ρ = (AA)(BC)(BC), so that we may write

D6 = {e, r, r2, r3, r4, r5, ρ, ρr, ρr2, ρr3, ρr4, ρr5}. [Note that, unlike the swap in the four-

valley case, here we can not write ρ as equivalent to a rotation; while it is indeed a rotation

in the indices (AA), it is not on the remaining indices.] In listing cosets, we first ob-

serve that left and right cosets must be equivalent, i.e. s−1Ns = N for any s ∈ D6,

following the same logic as in the four-valley case: performing a discrete transforma-

tion s on the valley indices, performing independent SU(2) rotations in each valley pair

and then transforming back to the original valley indices using s−1 should be simply

equivalent to three independent SU(2) rotations, as long as s preserves the pairing of

the valleys, and it is clear this is satisfied by r, ρ, and hence by any combination of

62

their powers. From this reasoning, we conclude that N is a normal subgroup. In ad-

dition, we note that ρ2 = e, and furthermore r3 = (AA)(BB)(CC), corresponding to

π-rotations in each valley pair, whence r3N = N . Putting these arguments together, we

find the list of cosets to be {N, rN, r2N, ρN, ρrN, ρr2N}. Now, the fact that left and right

cosets are equivalent means that the coset space has a group structure, obtained by sim-

ply writing g1Ng2N = g1g2N . We can verify that the operations r, ρ satisfy the identity

ρ−1r−1ρr = r2, so that under the coset multiplication rule (rρ)−1ρrN = r2N 6= N i.e. the

quotient group G/N is non-Abelian. Since the unique non-Abelian group of order 6 is D3,

we see thatG/N ∼= D3, from which the structureG = [SU(2)3]oD3 follows immediately.

We now discuss symmetry breaking using the same conventions as previously. At ν = 1

and T > 0, we fill a single valley, breaking the D3 structure, but thermal fluctuations of

Goldstone modes restore the SU(2) symmetry within the filled pair. We still have the

ability to perform SU(2) rotations within the unfilled pairs as well as swap their axes; we

argue that this leads to an SU(2)2 o D2 structure within the unfilled subspace, so that in

total we have HT∼= SU(2) × [SU(2)2 o D2]. From this, we see that the NLσM target

space is given by G/HT = [SU(2)3 o D3]/[SU(2) × [SU(2)2 o D2]] ∼= Z3. At T = 0,

we argue in analogy with the 4-valley case that SU(2) in the filled pair is broken down to

U(1) so that the residual symmetry H0 = U(1) × [SU(2)2 oD2], so that the target space

is HT/H0∼= S2.

Turning now to ν = 2 and T > 0, we first consider class II states where we fill a

pair of valleys breaking the S3 structure but preserving SU(2) via thermal restoration of

symmetry, so thatHT = [SU(2)2oD2]×SU(2) again (but the role of the filled and unfilled

valleys are interchanged), and once againG/HT∼= Z3. At T = 0, the situation is similar to

ν = 2 for the four-valley case: the residual symmetry is U(1)2×SU(2), yielding the target

space HT/H0∼= S2 × S2. For Class I states at ν = 2, we fill both valleys in a pair, leading

to residual symmetry group HT = SU(2)× [SU(2)2 oD2]; since G/HT∼= Z3, it appears

that there an alternative finite-temperature transition. However, proceeding to T = 0, we

63

see that there is no additional symmetry breaking, i.e. H0 = HT , leading to a lack of

Goldstone modes, and therefore near T = 0 Class II states are selected. Since the thermal

fluctuations of the Goldstone modes about class II states restores [SU(2)2 oD2]× SU(2)

symmetry, which returns to to the full symmetry G via a finite temperature transition, we

never access Class I states.

At ν = 3, for Class II states we fill each of the three valley pairs to ν = 1, and and

T > 0 we have HT = G = [SU(2)]3 o S3; once again G/HT is trivial, reflecting the fact

that there is no finite-temperature transition. To examine further symmetry breaking we

may parametrize this new base space HT similarly to the four-valley case: we write any

spinor in this space as ~z = g~z0 with

g =

g1 0 0

0 g2 0

0 0 g3

, (3.7)

with g1, g2, g3 ∈ SU(2), and ~z0 = (1, 0, 1, 0, 1, 0)T is a reference spinor. As T → 0,

each SU(2) breaks to U(1), so once again we consider states ~z, ~z′ that are connected by a

product of U(1) rotations in each valley to be equivalent, yielding target space HT/H0∼=

S2 × S2 × S2. For Class I states, we fill both members of one valley pair to ν = 1,

while filling one of the other pairs at ν = 1, and leaving the third pair unfilled. The

corresponding symmetry is simply HT = SU(2)3, and therefore G/HT∼= D3, suggesting

a potential finite-temperature transition. However, as T → 0, we break the SU(2) of the

filled pair down to U(1), so that H0 = U(1) × SU(2)2, and HT/H0∼= S2; since the

corresponding theory has one Goldstone mode compared to three about Class II states,

T = 0 state selection favors Class II states, and therefore we do not see Class I states as

thermal fluctuations for T 6= 0 restore the full symmetry G immediately.

64

3.6 Valley Skyrmion Crystals

In the case of Hamiltonians having SU(2) internal symmetry, Coulomb interactions lead

to the formation of skyrmion crystals at a finite (but small) density of skyrmions. Their

properties in the presence of a Zeeman term have been extensively studied using Hartree-

Fock theory [28, 126, 103, 19, 131]. Skyrme crystals have also been studied in nuclear

physics (three dimensions), but in the presence of an interaction term different from the

Coulomb interaction [61, 118, 52]. In this section, we will be interested in Si quantum Hall

systems near filling factor ν = 1 in the absence of a Zeeman term associated with internal

degrees of freedom. Here, density alone sets the size of skyrmions. What is the ground

state of such a two-dimensional system of repulsively interacting skyrmions? The main

objective of this section is to show that even though the lowest energy charged excitations

are charge one skyrmions, they in fact bind together to form a higher charge unit cell of the

lowest energy lattice.

3.6.1 Analytics

Consider a quantum Hall ferromagnet at ν ∼ 1 with internal degrees of freedom described

by the CP1 model. We collectively denote the 2 complex fields by v(~x) = (v1(~x), v2(~x)).

They are subject to the constraint |v1|2 + |v2|2 = 1. The energy functional of this system is

the sum of a gradient term Ho and a Coulomb interaction term.

Ho = 4∑µ=x,y

∫d2x [(∂µv)†(∂µv) + (v†(∂µv))2] (3.8)

where the integration is over the unit cell [82]. We consider the case of a very low density

of skyrmions and thus the interaction term can be treated perturbatively. First we find the

minimum energy configurations for Ho and then minimize the interaction term within the

degenerate subspace of those configurations.

65

All finite energy configurations can be classified into various topological sectors in-

dexed by their topological charge Q, which can only take integer values,

Q =i

2π

∫d2x εµν∂µ(v†(∂νv)) (3.9)

where the integration is over the unit cell. It is known that Ho ≥ 8πQ [102, 36] and

this bound can only be satisfied by fields that satisfy the following equation: Dµv =

±iεµνDνv. As suggested by Ref. [102], this equation can be converted into a Cauchy-

Riemann condition: ∂µw = ±iεµν∂νw, where the w field is defined patchwise. (In regions

with v1 6= 0, we define w = v2/v1 and similarly in regions with v2 6= 0, we define

w = v1/v2. The different definitons can be analytically continued from one to another.)

The formula for charge takes the following form:

Q =1

4π

∫d2x ∇2ln(1 + |w|2) (3.10)

For our case of a lattice, we are interested in writing down a complex analytic function w

that satisfies the following quasi-periodicity conditions:

w(z + 1) = w(z) ; w(z + τ) = w(z)eiφ (3.11)

where φ is a constant real number. But according to Liouville’s theorem, if a complex

valued function is bounded and analytic for all coordinates, then that function is a constant.

Hence an entire quasi-periodic complex function is constant and any v made out of constant

fields belongs to the trivial Q = 0 sector. So we must allow for w to have poles and thus

solve the above equation for w locally. (Here topological charge is equal to the number of

poles counted with multiplicities.)

Before we look at quasi-periodic meromorphic functions, we rephrase our lattice re-

quirement as follows: We can map our CP1 model to the O(3) model and work with a new

66

real vector of fields: ~φ = v†~σv where ~σ is the vector of three Pauli matrices. Actually we

want this vector field to be doubly periodic. We make use of this in our next argument.

Let us now study quasi-periodic meromorphic functions. We show that such a function

with φ 6= 0 does not exist. Now suppose a meromorphic function is constructed as a

ratio f(z) = h1(z)h2(z)

, where h1(z) and h2(z) are entire functions which satisfy the following

properties: hα(z + pτ + q) = hα(z)eλα(p,q,x,y)+iλα(p,q,x,y), where p and q are integers,

and α = 1, 2. Now we construct v1 = h1/√h2

1 + h22 and v2 = h2/

√h2

1 + h22, and then

construct ~φ. We observe that ~φ is doubly periodic only if λ1(p, q, x, y) = λ2(p, q, x, y) and

λ1(p, q, x, y) = λ2(p, q, x, y). But this implies that f(z) is doubly-periodic. Hence it is not

possible to have φ 6= 0.

So we must focus on the doubly periodic case. According to Abel’s theorem, there is a

meromorphic doubly-periodic function with zeros ai of order ni and poles bj of order mj

if and only if∑

i ni =∑

jmj and∑

i niai =∑

jmjbj . (It is assumed that the position of

zeros and poles is specified within the same unit cell.) Moreover such a function g(z) is

unique up to a constant factor. It is made by using modified theta functions (see appendix

A).

g(z) =h(z)

t(z)=

∏i θ( 1

2+ τ

2)(z − ai)ni∏

j θ( 12

+ τ2

)(z − bj)mj

h(z + 1)

h(z)=t(z + 1)

t(z)= 1

h(z + τ)

h(z)=t(z + τ)

t(z)= e−2πi(z+ 1

2+ τ

2)∑i nie2πi

∑i niai (3.12)

So now we have a recipe for making doubly periodic meromorphic functions. The zeros

and poles would be the degrees of freedom. (Kovrizhin et al [68] have used a slightly

different approach to study the case of valley textures with Q > 1.)

Notice that in the case of one zero of order one and one pole of order one, g(z) simplifies

to a trivial constant function. Hence as far as doubly periodic meromorphic functions are

67

Figure 3.4: Unit cell Γ of a skyrmion lattice with L = 1/√

sin γ.

concerned, we must look into Q > 1 sectors for field configurations that satisfy the lower

bound of energy. So we have analytically shown that in the absence of any interactions,

the lowest energy valley textured lattice with charge Q = 1 per unit cell has higher energy

than the lowest energy lattice with Q > 1. Kovrizhin et al [68] have explicitly calculated

the minimum energy valley skyrmion crystal picked out by Coulomb interactions out of the

various degenerate possibilities in the higher charge sectors.

Next, we will explicitly verify our analytical result by finding the minimum energy

configuration within the Q = 1 sector using a stochastic algorithm. Since the variational

approach of Ref. [68] can not be extended to Q = 1 sector, we must use numerical mini-

mization methods. As a check, we first carry out the optimization process for the topologi-

cal charge two sector.

3.6.2 Numerical Minimization

For the purpose of this section, we will work with the O(3) model. Now Ho is just the

gradient term:

Ho =

∫Γ

d2x ((∂x~φ)2 + (∂y~φ)2) (3.13)

68

where Γ is the unit cell as shown in Fig. 3.4. So ~φ(~x+Ln1) = ~φ(~x+Ln2) = ~φ(~x). Once

again, the set of all possible ~φ can be classified by their topological chargeQ =∫

Γd2xQ(~x)

where Q(~x) = 14π~φ · (∂x~φ× ∂y~φ).

We discretize the unit cell along the n1 and n2 directions. For numerically calculating

derivatives along n1 and n2, the backward finite difference method is used. They are related

to the partial derivatives along x and y by the following equations:

∂~φ

∂x=

∂~φ

∂n1

,∂~φ

∂y= − cot γ

∂~φ

∂n1

+1

sin γ

∂~φ

∂n2

(3.14)

where γ is the angle in Fig. 3.4.

The energy of the system is minimized by repeating the following steps: a point on the

grid is chosen at random. The field at that grid point is tilted by an angle ε chosen randomly

in the interval (0, δ) and then rotated about the old vector by an angle φ chosen randomly

in the interval (0, 2π) to form a new field vector. Then the change in energy is calculated.

If the energy decreases and the change in the field vector doesn’t take the total charge of

the system out of the range (1.99, 2.01), the modification of the field is accepted; otherwise

it is discarded. Hence the system’s energy decreases throughout the procedure. (We will

later observe that there is no need to use advanced methods like simulated annealing.) The

procedure is repeated with the value of δ being gradually decreased. This is done until no

further decrease in energy is observed.

We ran the numerical minimization procedure with 2 different initial guesses within

the Q = 2 sector: (i) ~φo1 made from the eigenvector corresponding to the lowest

eigenvalue of a model Hamiltonian H = [(1 − cos 2πn1/L − cos 2πn2/L)σz +

sin 2πn1/Lσx + sin 2πn2/Lσy] where σx, σy, σz are the usual Pauli matrices and (ii)

~φo2 = (sin(f)cos(2θ), sin(f)sin(2θ), cos(f)) where f(r) = πe−r/λ and r, θ are the usual

polar coordinates. We expect the system to end up in a final configuration having a charge

distribution that is pretty uniformly distributed over the unit cell. But we find that the

69

system ends up in a configuration having most of it’s charge concentrated at the center.

This is most likely a result of the attractive energy terms which have been artificially

introduced because of the discretization. In fact, we find that the size of the concentrated

charge distribution isn’t independent of the number of grid points. It decreases with

increase in the number of grid points.

In order to stabilize the exploratory process built into our algorithm, we have to include

a Coulombic interaction term in the energy function. So now we numerically minimize the

following energy functional:

H = Ho + g

∫Γ

d2xd2yQ(~x)Q(~y)

|~x− ~y| (3.15)

where g is inversely proportional to the skyrme lattice constant. We are interested in the

limit g → 0, which is equivalent to a large lattice constant, i.e. an extremely dilute density

of skyrmions. We find the following results: (i) the final energy of the system is propor-

tional to g for all γ, (ii) the gradient part of the final energies is 8π for all γ and (iii) the

final charge distributions are spread out over the unit cell and (iv) the coulombic energy

of the minimum configuration is the same as that obtained by Kovrizhin et al [68]. These

results make us confident of the correctness of our numerical procedure and we do not need

to resort to methods like simulated annealing. (We also observed that the total charge of

the final configuration is always 1.99.)

Now let us look at Q = 1 case. In the absence of the interaction term, we face the same

problem as seen earlier in Q = 2 case. The system ends up in a configuration having most

of it’s charge concentrated at one place in the unit cell. These results aren’t trustworthy.

The case with Coulombic interaction term is left for future research.

70

3.7 Concluding Remarks

In closing we comment briefly on complications ignored in our discussion. Foremost

among these is the neglect of various terms in (3.2) that while suppressed by O((a/`B)2)

relative to the dominant Coulomb energy scale could compete with the selection mecha-

nisms discussed above. For B ≈ 10T, we find that these terms split energy levels by a few

millikelvin, so that there is a large window of temperature where their neglect is justified.

We note that competition between quantum selection by high-order effects and thermal

order-by-disorder was studied in quantum magnets [25, 56]; similar situations may arise

here once the neglected terms become significant.

Secondly, in a more realistic situation, the six-fold valley degeneracy can be lifted due

to wafer miscut and strain arising from lattice mismatch. While valley splitting due to the

former mechanism is negligible compared to the cyclotron gap [78], the latter can be more

significant [110, 128]. Although this problem has been largely solved by working with

2DEGs on a H-terminated Si(111) surface [34, 65], both mechanisms can still change the

E-S crossover temperature T ∗E-S and possibly even the Class of states that are selected below

T ∗.

Third, we have ignored Landau level mixing [16, 95, 116], likely to be significant in

any realistic situation. Our confidence rests on the fact that estimates based on LLL approx-

imations have met with noteworthy success to date—particularly in the present context of

QHFM—and as such represent a standard approximation in the field. As an example, a

study by Sinova et al [117] that includes disorder effects within a Hartree-Fock treatment

similar to that used in our analysis, and ignores LL mixing, gives results in good qualita-

tive agreement with experiment. A second reason is that once again, we may fall back on

general symmetry arguments: we do not anticipate that LL mixing will split the symmetry

directly, though it could potentially lift the degeneracy of the different ground state classes,

this is an effect that we are unable to discuss in detail at present and therefore prefer to

leave out. We note that a full-fledged calculation of LL mixing with interactions is ex-

71

tremely challenging (including numerically), and as such lies well beyond the scope of this

Chapter. (a good example of the complexity of the relevant many-body problem is in Ref.

[16].)

Fourth, let us add spin to the problem. In the absence of Zeeman splitting, there are two

classes of degenerate ground states at filling = 2: fill 2 out of set A (e.g.) = A↑, A↓, A↑, A↓

or fill 1 out of set A and 1 out of set B (e.g.). On turning on the Zeeman field, we are back

to results mentioned earlier. A similar story holds for filling ν = 3. So, our results for

fillings ν = 1, 2 and 3 still hold. Now, the results for ν = 4, 5, 6 depend on the relative

size of the spin Zeeman energy ∆Z = gµBB and the valley splitting ∆v. For ∆v � ∆Z ,

we expect that the relevant selection mechanisms involve splitting the near-perfect valley

splitting, and we can safely ignore the spin degeneracy; therefore, the results for filling

4, 5, 6 are obtained by effectively ‘particle hole conjugating’ the ν = 1, 2, 3 results. In this

limit, we see that we need not consider the spin physics in the QHFM problem.

Finally, spatial disorder induces a random field acting on the nematic order parameter,

which is a relevant perturbation. An infinitesimal field destroys the nematic order in the

thermodynamic limit [15, 55] at ν = 1, 2 by proliferating domain walls but the QHE sur-

vives as long as disorder is sufficiently weak [1]. The interplay of disorder with the novel

selection mechanisms discussed above is likely intricate and worthy of further study.

72

Chapter 4

Generalizing Quantum Hall

Ferromagnetism to Fractional Chern

Bands

4.1 Introduction

In this chapter we study analogs of quantum Hall ferromagnets in fractionally filled Chern

bands, specifically in Chern bands with Chern number C > 1 (Table 4.1). These are

states that exhibit topological order at T = 0 and discrete symmetry breaking for 0 ≤

T ≤ Tc, specifically they break discrete sublattice symmetries. As such they generalize

recent theoretical work in the quantum Hall effect wherein a discrete global symmetry acts

simultaneously on an internal and a spatial degree of freedom [1]. This situation occurs in

multi-valley systems where different valleys are related by a discrete rotation and gives rise

Table 4.1: Analogies between gas and lattice systems.Integer quantum Hall effect Fully filled Chern bandsFractional quantum Hall effect Partially filled flat Chern bandQuantum Hall Ferromagnet ?

73

to interesting phenomena. For example in case of AlAs heterostructure, the Hamiltonian

has Z2 symmetry which involves the operation of π/2 rotation in real space combined with

interchange of the 2 valley indices. There is an interesting interplay between the Ising order

and topological order. In a clean system, ferromagnetism onsets via a finite temperature

Ising transition and exists without topological order at T > 0. By contrast disorder induces

a random field acting on the Ising order parameter destroying the Ising order but leaving

the topological order intact; the resulting phase called the QH random field paramagnet

(QHRFP).

In the following we will see that these features do arise in fractionally filled Chern

bands as well, specifically in multiple Chern number bands. As has been noted previously

[11, 97] and we will sharpen below, multiple Chern number bands resemble Landau levels

with multiple components. A central part of our analysis will be working in a limiting

case where this analogy is sharp. Specifically, we will focus on a family of square lattice

models with flat C = 2 bands and a wide band gap at 1/2 filling. We show that near-

est neighbor density-density repulsive interactions pick QH Ising ferromagnets as ground

states (Sec. 4.2, 4.3). We also study properties of domain walls germane to the QHRFP

phase on lattice (Sec. 4.2, 4.3) and discuss the alternative interpretation of the states con-

sidered in this Chapter as topological Mott insulators. Our ideas can be generalized to flat

C = n > 2 bands at 1/n filling (Sec. 4.4) and also to fillings hosting quantum Hall states

with fractional Hall conductance (Sec. 4.5). We close with a summary in Sec. 4.6.

Before proceeding we would like to draw the reader’s attention to two related pieces of

work in single Chern (C = ±1) band systems. Neupert et al [89, 88] have studied a model

for Z2 insulators with an additional global Ising symmetry—but now at fractional fillings

and with a Hubbard interaction. They showed that the system exhibits Ising ferromagnetic

order along with quantum Hall ordering at fillings 1/2 and 2/3. Kourtis and Daghofer [67]

have presented numerical results indicating the coexistence of charge density wave order

and quantum Hall order at filling 2/5 of a C = ±1 band system.

74

4.2 A Special flat C=2 band at 1/2 filling

We consider C = 2 band Hamiltonians on a square lattice. The analogy between C = 2

bands and multi-layer/flavor quantum Hall systems is especially transparent at a special set

of points in the space of C = 2 band Hamiltonians. At these points the Hamiltonian can be

written as the tensor sum of Hamiltonians associated with the two inter-penetrating square

sub-lattices A and B (say) with no hopping between the sites of A and B. The Hamiltonians

associated with each sublattice are identical due to translation symmetry and if they each

have Chern number, C = 1, the lower band of the tensor sum Hamiltonian has C = 2. An

example of such a Hamiltonian with C = 2 with just second nearest neighbor hoppings is

given by

Ho =∑~k

(c†~k↑ c†~k↓

)H11(~k) H12(~k)

H∗12(~k) −H11(~k)

c~k↑c~k↓

(4.1)

where H11(~k) = m+ cos(kx +ky) + cos(kx−ky), H12(~k) = sin(kx +ky)− i sin(kx−ky),

↑ and ↓ represent two kinds of orbitals at every lattice site and m is a tunable parameter

which leads to a C = 2 lower band in the range (−2, 0). (We take the lattice constant

a = 1.)

Ho can be rewritten as ∆(1 − t + tE+,~k)|γ+,~k〉〈γ+,~k| + (−1 + t + tE−,~k)|γ−,~k〉〈γ−,~k|

where t = 1, ∆ = 1, |γ−,~k〉 and |γ+,~k〉 are the eigenfunctions for the lower and upper

band, respectively and E−,~k, E+,~k are their corresponding eigenvalues. The eigenstates are

related to the old set of states by |cη,~k〉 =∑

κ=−,+ uηκ(~k)|γκ,~k〉. If we tune t → 0 and

∆ → ∞ (which can always be done by suitably added further nearest neighbor hoppings

which preserve the sub-lattice tensor structure) to end up with a flat C = 2 lower band

separated from the upper band by an infinitely large band gap.

We then add interactions:

Hint. = V∑~i

n~i↑n~i↓ + U∑〈~i~j〉

∑κ,κ′=↑,↓

n~iκn~iκ′ (4.2)

75

(a)

(b)

0 2 4 6 8 10 12 14 163

3.5

4

4.5

5

5.5

6

Kx+L

xK

y

E /

U

Figure 4.1: (a) Lower band Chern flux distribution over the Brilliouin zone for the single-particle Hamiltonian Ho with m = −1.8. (b) Low energy many-body spectrum for 8fermions on a 4× 4 lattice for the case of the single-particle part of Hamiltonian chosen asHo with m = −1.8 and V = 3U . (Energies are resolved using total many-body momenta(Kx, Ky) which are in units of 1/a.)

where V is the on-site,intra-sublattice, interaction strength and U is the nearest neighbor

(NN), inter-sublattice, interaction strength. Since we are interested only in the low energy

part of the many-body spectrum, we will restrict our analysis to Hint. projected onto the

76

lower flat band. Hproj. can be obtained by making the change cη,~k → uη−(~k)γ−,~k in Hint..

We will work at half filling.

Intuitively we expect two regimes for the Hamiltonian (4.2). At V � U the particles

can efficiently avoid the NN repulsion by segregating on one sublattice while for V � U

they will minimize the on-site repulsion by inhabiting both sublattices. In the first regime,

the many-body spectrum has 2 degenerate ground states (Fig. 4.1(b)) corresponding to fully

filling the C = 1 bands that live on sublattice A/B. The reader can check that these states

are always eigenstates of the projected Hint as they are tensor products of the empty state

on A/B with the fully filled state on B/A. By explicit computation for the case of m =

−1.8 which we will use for illustration purposes in this Chapter, we find that they are the

ground states for V < 4.3U which covers the physically interesting regime where V is not

smaller than U . Both ground states break the Ising symmetry between the two sublattices

or alternatively are (π, π) charge density waves, one having all its weight on the A lattice

and the other on B lattice (Fig. 4.2). These results can be interpreted using approximate

Slater determinant ground states constructed from single particle wavefunctions which are

formed from linear combinations of the Bloch states at a given k and k + (π, π) such that

the wavefunctions have support only on one of the lattices. Clearly both states exhibit a

Figure 4.2: Ising ordered ground state for Hproj. at half filling.

77

Hall conductance σH = e2

h. For V > 4.3U the many-body spectrum does not have two

degenerate ground states and hence a phase transition has ocurred.

The excitations about these symmetry broken ground states are two-fold. First there are

particle hole excitations in which a hole is created on sublattice A (say) and a particle is

created on sublattice B. While the quasiparticle band is flat, the quasihole band has finite

dispersion. The minimum particle-hole pair creation energy asymptotes to a value which

depends on the choice of C = 1 Hamiltonian. For the case of m = −1.8 it asymptotes to

4U − 0.6V . Note that this gap does not close for V < 4.3U so the transition that we report

in the ground state is first order.

The second set of excitations consists of domain walls across which sub-lattice occu-

pation changes. These are the topological defects of the Ising order ferromagnet discussed

above. Two of the many possible orientations of these domain walls are depicted in Fig. 4.3.

The energy of the domain walls is set by the interactions. For example in the case of do-

main wall oriented as shown in Fig. 4.3(b), the energy per unit length is approximately

0.2V + 2.4U for m = −1.8. The electronic structure of these domain walls is of interest

as it reflects the interaction between the topological order and symmetry breaking. In the

current limit a pair of counter-propagating gapless, chiral modes of opposite sub-lattice in-

dex exist at a domain wall. In the absence of inter-sublattice hopping both sides “see” the

domain wall as the boundary to a topologically trivial vacuum.

With these excitations identified we can describe the response of the state to tempera-

ture, doping and disorder. At half filling in the clean system, topological order is lost at

any T > 0 by the proliferation of particles and holes which, following the standard lore

in the QHE, one can think of these as vortices in the topological gauge field. [Admittedly

the topological order is somewhat trivial here being that of the Integer Hall effect. We

will present a fractional case later.] Domain walls are bounded in size at small T but will

proliferate above a critical temperature Tc leading to a finite temperature Ising phase tran-

sition. In our present limit domain walls conduct whence we expect the AC longitudinal

78

conductivity to be sizeable on the scale of the domains and to increase sharply in the DC

limit around Tc. Weak doping around half filling will be accommodated by the inclusion

of a density of particles/holes in the ground state. Absent disorder and for our short ranged

model, doping will destroy QH order while continuing to preserve the sublattice order—

whence the finite T physics described above will survive. Finally with disorder two new

effects will enter. First, the physics of the random field Ising model [55, 15] will enter

and restore sublattice symmetry at and near half filling. Second, disorder will localize the

particles/holes leading to a quantized Hall effect for a finite range of doping.

We have described the parent state at half filling as a quantum Hall ferromagnet in the

above. Here we would like to note that it is also a species of Mott insulator by which

we mean a state with the constituent particles localized on account of strong interactions.

Indeed, for |m| > 2, the lower band is topologically trivial (C = 0) and if we again

flatten the bands as described above and add the same interactions, then the resulting state

is reasonably described as a Mott insulator with additional sublattice symmetry breaking.

In this case one can also construct localized Wannier states with support entirely on one

of the sub-lattices, A or B and the filled band corresponds to a occupation of all of the

Wannier states on one or the other sub-lattice. For our C = 2 system, Wannier states

exponentially localized in both dimensions can no longer be defined. Nevertheless, the

resulting ground state is insulating in the bulk and may be regarded as a form of Mott

Chern insulator with sublattice symmetry breaking. Topological Mott phases have been

previously been proposed in some very different settings [99, 100, 53].

4.3 Other flat C=2 bands at 1/2 filling

When we move away from the fixed points where the band Hamiltonians sum structure by

adding inter-sublattice hopping, we expect the many body gap to generically remain stable.

In Sec. 4.2, the single-particle Hamiltonian is such that the Berry curvature corresponding

79

(a)

(b)

Figure 4.3: 2 species of domain wall considered in the main text.

to the lowest band is symmetric under translations by (π, π) in the Brillouin zone and is

concentrated in two pockets in the Brillouin zone (Fig. 4.1(a)). Our earlier statements

regarding the ground states should hold irrespective of the kind of Chern flux distribution.

As a check, we consider a single-particle HamiltonianH ′o for which the lower band’s Chern

flux is concentrated in only one pocket (Fig. 5.2(a)). (There is always some contribution

80

(a)

(b)

0 2 4 6 8 10 12 14 163

3.5

4

4.5

5

5.5

6

Kx+L

xK

y

E /

U

Figure 4.4: (a) Lower band Chern flux distribution over the Brilliouin zone for the single-particle Hamiltonian H

′o with m = −1.8. (b) Low energy many-body spectrum for 8

fermions on a 4× 4 lattice for the case of the single-particle part of Hamiltonian chosen asH′o with m = −1.8 and V = 3U . (Energies are resolved using total many-body momenta

(Kx, Ky) which are in units of 1/a.)

coming from the other regions of the Brillouin zone.) This can be made possible by turning

on hopping between A and B lattice sites. One such instance can be given in the form of

Equation 4.1 where H ′11(~k) = m+ cos(kx) + cos(ky), H ′12(~k) = 12(cos(2ky)− cos(2kx)) +

i(cos(kx − ky)− cos(kx + ky)) and −2 < m < 0.

81

The new projected Hamiltonian H ′proj. is diagonalized for 8 particles on a 4 × 4 lat-

tice (Fig. 5.2(b)). A 2-fold degenerate ground state and a gap of order U are again ob-

served. We also check for broken sublattice symmetry by computing the Fourier transform

of 〈ρ′(~x)ρ′(0)〉 where ρ′ is the density operator projected onto the lowest flat band. Peaks

are observed at (0, 0) and (π, π) indicating the presence of sublattice symmetry breaking

(CDW order).

The response of the state to T , doping and disorder is qualitatively the same as before.

The one significant difference is in the electronic structure of the domain walls. Upon inclu-

sion of inter-sublattice hopping we may wonder whether they gap generically or are remain

protected by some symmetry of the problem. This examination is partly motivated by anal-

ogous objects in the Ising quantum Hall FM in the AlAs problem [1] where the domain

walls are gapless to an excellent approximation. To this end we have examined domain

walls in the (1,0) and and (1,1) orientations produced by the added one-body potential

H =∑~i

V (~i)(ρ↑(~i) + ρ↓(~i))projected (4.3)

where the projection is on to the lowest band of a generic Hamiltonian of the form given

by Equation 4.1. For purposes of explicit computation below we will take H11(~k) = m +

cos(kx+ky)+cos(kx−ky), H12(~k) = sin(kx+ky)− i sin(kx−ky)−t(sin(qx)+ i sin(qy))

where t is the parameter for hopping between the two sublattices. The V (~i) corresponding

to two orientations of domain wall are:

V1(~i) =

(−1)ix+iy , ix < N/4

(−1)ix+iy+1, N/4 < ix < 3N/4

(−1)ix+iy , 3N/4 < ix

(4.4)

82

V2(~i) =

(1 + (−1)ix−iy)/2, (ix − iy) < N/4

(1 + (−1)ix−iy+1)/2, N/4 < (ix − iy) < 3N/4

(1 + (−1)ix−iy)/2, 3N/4 < (ix − iy)

(4.5)

whereN is the number of sites along x/y direction and we are assuming periodic boundary

conditions along both x and y directions. In Equation 4.4, domain walls run parallel to y

direction and in Equation 4.5 they run parallel to the diagonal. In both cases, we find a gap

in the spectrum (Fig. 4.5) and this should hold for generic orientations of the walls.

4.4 Generalization to higher Chern bands

Our results can be generalized to flat C = n > 2 bands at filling factor 1/n . An obvious

way to begin is to make a flat C = n > 1 lower band by putting n decoupled lattices

together, each independently having a flat C = 1 lower band. Then one can arrange for

repulsive interactions to pick the n fully occupied single sublattice bands as ground states.

For example in the C = 4 case, on-site, nearest neighbor, next-nearest neighbor and next-

next-neighbor repulsive interactions pick 4 degenerate ground states having both many-

body Chern number C = 1 and broken sublattice/translational symmetry. Such a state is

the Chern band analog of a QHFM in a system with a Zn global symmetry, examples with

n = 3 are the Si (111) QH system at filling factors 1 and 5. In addition to domain walls the

system will now exhibit proto-vortices where n distinct domains come together at a point.

For n > 4 the system will also exhibit a T > 0 Kosterlitz-Thouless phase [58].

4.5 Fractional states

Let us return to the C = 2 band but now turn to filling 1/6 with the Hamiltonian as the one

in Equation 4.2 plus a set of further intra-lattice and inter-lattice further neighbor repulsive

interaction terms. In the decoupled limit, we can tune the intra-lattice interactions to be83

those that stabilize a state that forms a fractional Chern insulator state at 1/3 of a single

C = 1 band [107]. Adding a sufficiently large U and a number of further neighbor

inter-lattice repulsive terms of adequate magnitude will ensure that domain walls between

regions which reside on one sub-lattice and the other are energetically unfavorable and

thus favor a state that breaks sublattice symmetry and exhibits the topological order of the

ν = 1/3 Laughlin state and a quantized Hall conductance σH = e2/(3h). Evidently this

construction can be repeated for other known QH states in a C = 1 Chern band.

4.6 Concluding Remarks

To summarize, we found analogs of QH ferromagnets in lattice systems. The case of a flat

C = 2 band at 1/2 filling is analogous to AlAs QH system at filling factor 1. The two

ground states are the broken sublattice symmetry states having topological order. Unlike in

the AlAs system the domain walls come naturally with gapped electronic excitations.

In future work it would be interesting to locate the phases that we have discussed in a

larger phase diagram in which we restore dispersion to the band and also allow its Chern

number to go through a transition. This will introduce the trivial Mott insulator/FM and the

trivial and topological half filled metal into the phase diagram and it would be interesting

to see exactly how the various phases fit together and the nature of the transitions between

them.

84

(a)

−4 −3 −2 −1 0 1 2 3 4Momentum along domain wall

−0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Energ

y

(b)

−4 −3 −2 −1 0 1 2 3 4Momentum along domain wall

−0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Energ

y

Figure 4.5: Spectra of the Hamiltonian in Equation 4.3 with m = −1 and choice oforientation of domain wall as shown in Fig. 4.3(b). (a) t = 0. (b) t = 0.3. (Momentumalong domain wall is in units of 1/

√2a.) Similar results are obtained for the other choice

of V (~i).

85

Chapter 5

Flat bands with local Berry curvature in

multilayer graphene

5.1 Introduction

The behavior of strongly interacting electrons and the effect of quantum geometry are two

of the most exciting fields of research in modern condensed matter physics. Strong interac-

tions can give rise to effects such as fractionalization [101, 71, 77], where the elementary

excitations carry some fraction of the ‘fundamental’ electronic quantum numbers. One

popular way to access strong correlation physics is to consider band structures that are flat

(non-dispersive), since in such band structures interactions naturally dominate over kinetic

energy. Meanwhile, the two ingredients of quantum geometry are integrated Berry cur-

vature (a.k.a. Chern number for two dimensional systems) and Fubini-Study metric, the

metric measuring the quantum distance. While non-zero integrated Berry curvature can

lead to the existence of states that are protected against disorder [18, 49, 46] and to effects

like quantized Hall conductance of a band insulator [125], non-zero Fubini Study met-

ric can lead to phenomena such as pseudospin conservation laws in single layer graphene

and unusual features in the current noise spectrum of a band insulator [86]. Recently,

86

a strong interest has emerged in flat band systems as playgrounds for investigating the

interplay of strong correlation effects and non-trivial quantum geometry. However, at-

tention has been mostly focused on systems with non-zero integrated Berry curvature,

[87, 123, 93, 83, 138, 72, 122, 76, 127], while the effect of the Fubini-Study metric has

largely been ignored.

The influence of quantum geometry on the strong correlation physics stems from the

fact that the interactions couple to electron density, and the electron density operators pro-

jected onto the flat band obey a non-trivial commutation relation if the band has a non-trivial

quantum geometry [93, 108]. The general commutation relation for projected densities in

a band with non-uniform Berry curvature is [108]:

[ρ ~q1, ρ ~q2] ≈ i~q1 ∧ ~q2

∑~k,b

[B(~k)u∗b(~k+)ub(~k−)γ†~k+

γ~k− ] (5.1)

where ~q1 ∧ ~q2 = z.(~q1 × ~q2), ~k± = ~k ± ~q1+~q22

, B(~k) is the local Berry curvature for the

band of interest for the single-particle hamiltonian H =∑

~k,a,b c†~k,ahab(~k)c~k,b and the cor-

responding eigenstate is given by∣∣∣~k⟩ = γ†~k |0〉 =

∑a ua(

~k)c†~k,a |0〉. Thus far, attention has

been focused on systems where the Berry curvature is nearly uniform [87, 123, 93, 83, 138,

72, 122, 76, 127]. However, it is apparent from (5.1) that non-trivial quantum geometry

effects do not require a uniform Berry curvature. The influence of the quantum distance

metric may be most clearly revealed in a flat band system with local Berry curvature, but

zero integrated Berry curvature, since here the non-trivial quantum geometry effects (en-

coded e.g. in the projected density commutator) arise purely due to the metric. Thus far,

theoretical studies have largely ignored this exciting direction of research, in part because

of the lack of experimental realizations.

In this Chaper we show that flat bands with local Berry curvature (but vanishing inte-

grated Berry curvature) arise naturally in chiral multilayer graphene. Our proposal exploits

the fact that ABC stacked multilayer graphene in the presence of a perpendicular elec-

87

tric field has a bandstructure with flat pockets that possess Berry curvature. Placing the

graphene on a hexagonal Boron Nitride (BN) substrate then produces a superlattice po-

tential such that the reduced Brillouin zone lies entirely within the flat pocket. Umklapp

scattering opens a bandgap at the reduced zone edge. For a N layer system with N > 5

layers, the lowest band is nearly flat, with a bandwidth∼ 5meV . We have verified that this

nearly flat band has a non-vanishing local Berry curvature. Chiral multilayer graphene thus

provides an ideal platform for investigating the interplay of strong correlations and quan-

tum geometry, with the quantum geometry effects coming from the (hitherto neglected)

channel of the Fubuni-Study metric, rather than the more conventional channel of non-zero

Chern number.

5.2 ABC stacked graphene

A single graphene sheet consists of a honeycomb lattice of carbon atoms. The honey-

comb lattice consists of two sublattices, A and B. Chiral multilayer graphene consists of

graphene sheets with an ABC stacking order (each succeeding sheet is rotated by 2π/3

relative to the preceding sheet). Every lattice site in the bulk is either directly above or

directly below another lattice site. In the (ψ1A~kψ1B~kψ2A~kψ2B~k · · · · · ·ψNA~kψNB~k) basis, the

nearest neighbor tight binding Hamiltonian for an N layer system takes the form [145]

H0 =

0 t~p 0 0 0 0 · · ·

t∗~p 0 γ 0 0 0 · · ·

0 γ 0 t~p 0 0 · · ·

0 0 t∗~p 0 γ 0 · · ·

0 0 0 γ 0 t~p · · ·

0 0 0 0 t∗~p 0 · · ·...

......

......

... . . .

(5.2)

88

Here, t~p = t0(

exp(ikxa)+2 exp(−ikxa2

) cos(√

3kya

2)), represents nearest neighbor hop-

ping within each graphene layer (t0 ≈ 3eV ), and γ ≈ 300meV represents interlayer hop-

ping between two sites that lie on top of each other. Here a is the lattice constant of

graphene.

In the absence of interlayer hopping, γ = 0, the bandstructure consists of N copies of

the graphene bandstructure, E = ±|t~p| [132]. Near the two inequivalent corners of the

Brillouin zone, conventionally labelled K and K ′, the function t~p vanishes as t~p ≈ v~p+ and

t~p ≈ −v~p− respectively, where ~p± = px ± ipy and v = 3ato/2 is the Fermi velocity for

graphene.

We now consider interlayer hopping γ 6= 0. This interlayer hopping causes all the bulk

sites to dimerize, opening up a bulk gap of size γ at the Dirac points. On the top and bottom

surfaces, there are undimerized lattice sites, which give rise to gapless surface states. The

low energy single particle Hamiltonian for the surface states of an N layer graphene takes

the form [63, 51, 50, 84]

HK(~p) =vN

(−γ)N−1

0 pN+

pN− 0

; p± = px ± ipy (5.3)

Here HK is the Hamiltonian in the K valley, and the Hamiltonian in the K ′ valley

is given by HK′(~p) = H∗K(−~p). The basis is such that (1, 0) is a Bloch state in the A

sublattice of the top layer, and (0, 1) is a Bloch state in the B sublattice of the bottom layer.

Only the lowest order terms have been written in each matrix element. This Hamiltonian is

valid up to an energy scale γ. Note that this energy scale is independent of N , the number

of layers of graphene. The above effective Hamiltonian consists of a single bandcrossing

with non-trivial Berry phase Nπ, and low energy dispersion E = ±vNpN/γN−1. Thus, the

conduction band has a very flat pocket of a size p ∼ γ/v around the K and K ′ points, and

this pocket becomes perfectly flat in the limit N → ∞, corresponding to ‘rhombohedral

graphite’. The emergence of this flat pocket has previously been discussed in [51, 50].

89

−0.06 −0.04 −0.02 0 0.02 0.04 0.06−1

−0.5

0

0.5

1

ka

E /

γ

N=3N=5N=10

Figure 5.1: Low energy band structure for N layer ABC stacked graphene in thepresence of a vertical electric field. The band structure is plotted in the vicinity of the~K point, assuming that the potential difference between the top and bottom layers ∆ =0.167t0 ≈ 50meV .

Now consider the application of a vertical electric field leading to potential difference

V between any 2 consecutive layers. We assume that γ � e(N − 1)V . This ensures that

the surface bands are much below the bulk bands. The low energy effective Hamiltonian

then takes the form

HK(~p) =

∆2

vN

(−γ)N−1pN+

vN

(−γ)N−1pN− −∆

2

(5.4)

The resulting band structure is plotted in Fig.5.1. We can see that a gap of magnitude

∆ = (N − 1)eV opens in the surface state spectrum. We can also see that the band near

the K point is extremely flat, asymptoting to perfect flatness in the limit N →∞.

5.3 Effect of BN substrate

BN is a popular substrate for graphene. It has the same hexagonal structure, but with a

slight lattice-mismatch (lattice constant∼ 1.02a) and is a large band-gap insulator (Egap ∼

5eV ). Placing multilayer graphene on BN substrate introduces a superlattice potential with

90

a Z × Z supercell, where Z = 56. The primitive lattice vectors corresponding to the

reciprocal lattice are ~B1

~B2

=2π

3Za

1√

3

1 −√

3

~x~y

(5.5)

The strong superlattice potential is seen primarily by states on the lower surface of the mul-

tilayer graphene, close to the substrate. These states are either conduction band or valence

band states, depending on the sign of the vertical electric field. We assume the electric field

is chosen such that it is the conduction band that mainly sees the superlattice potential. If

the multilayer graphene were sandwiched between BN sheets, then both conduction and

valence bands would see a strong superlattice potential, irrespective of the sign of electric

field.

The reduced ‘Brillouin zone’ for the superlattice is hexagonal, but scaled by a factor

1/Z with respect to the original Brillouin zone. If Z is chosen such that | ~K|/Z < γ/v, i.e.

Z > Zc, where

Zc = 20π/√

3 = 36.3 (5.6)

then the ‘flat pockets’ atK andK ′ extend over the entire reduced Brillouin zone. Moreover,

umklapp scattering at the boundaries of the reduced Brillouin zone opens a gap between

the flat pocket and the rest of the band. Since for BN substrate, Z = 56 > Zc, it follows

that placing chiral multilayer graphene on BN leads to flat bands that extend over the entire

reduced Brillouin zone, and which are separated from the rest of the bandstructure by the

umklapp energy scale λ.

We now explicitly calculate the bandstructure in the reduced zone. It may be readily

determined that for the above superlattice potential, the K and K ′ points in the original

zone get folded to inequivalent corners K and K ′ of the reduced zone. Meanwhile, the

reciprocal lattice vectors of the superlattice satisfy | ~B1,2| < γ/v < 2| ~B1,2|, where γ/v is

the width of the flat pocket. Since Bragg scattering is only effective between states that

are near degenerate, we restrict our attention to Bragg scattering by a single reciprocal

91

4 5 6 7 8 9 100

5

10

15

20

25

30

N

Λ (

meV

)

Figure 5.2: Bandwidth Λ of the lowest conduction band for N layer chiral graphene.For N > 5, the bandwidth comes mainly from umklapp scattering at the zone boundary.

lattice vector, and neglect higher order Bragg scattering events. The matrix elements of the

superlattice potential 〈~k|V | ~k +B1,2〉 were determined by modelling the BN superlattice as

a positive δ-function potential at each B site and a negative δ-function potential at each N

site. (See Sec. 5.4 for details.) Hence we obtained the bandstructure in the reduced zone.

Although the bandstructure contains Z2 = 562 = 3136 conduction bands per spin and

valley, only the lowest conduction band is flat, and is separated from the higher bands by

an energy scale of order λ, where the gap may be estimated from the DFT calculations in

[109], and is of order 10meV . Meanwhile, the bandgap ∆ between conduction and valence

bands can be externally controlled using gates, and may be made as large as desired. We

should take ∆� 10meV to ensure that we do not mix conduction and valence bands. For

specificity, we suggest using ∆ = 50meV , which is easily achievable by gating [147].

The superlattice potential from the BN substrate also introduces intervalley tunneling

of magnitude λ/Z (See Sec. 5.4 for details.), which turns the bandcrossings of the two

nearly flat bands coming from K and K ′ valleys into avoided crossings. The resulting

bandstructure contains two strictly non-degenerate flat conduction bands, separated by a

minigap λ/Z ≈ 0.2meV . The lowest energy band comes mostly from the K valley in

92

10

15

20

25

Γ

E (

meV

)

K M K

Figure 5.3: Dispersions of the three lowest conduction bands forN = 7 along the highsymmetry directions.

regions closest to the K corner of the reduced zone, and comes mostly from the K ′ valley

in regions closest to the K ′ corner. From the bandstructure calculations, we can extract the

bandwidth of the two low-lying conduction bands. In Fig.5.2, we plot the bandwidth of

the lowest conduction band as a function of N . We see that for N > 5 the two low-lying

conduction bands are nearly flat, with a small residual bandwidth around 5 meV which

comes mainly from umklapp scattering at the zone boundary. The N = 7 layer system

actually has minimum bandwidth of ∼ 3.6 meV, due to a cancellation between ‘intrinsic’

curvature of the band and the effect of umklapp scattering.

We also note that there is an indirect band gap separating the two flat bands from the

higher energy non-flat bands (Fig.5.3). It vanishes for N < 6 and then decreases with

increasing N . Transitions across the indirect band-gap must be phonon assisted and should

be weak, but nevertheless it is essential that we work with N > 5 to have a truly isolated

flat band. Fortunately, N = 7 is optimal not only through having the flattest band (Fig.5.2),

but also because its indirect band gap is of order the direct band gap. For N = 7 the direct

band gap is ∼ 6meV .

93

We have now verified that the lowest conduction band can be made flat. Now we show

that this band has non-vanishing Berry curvature. The momentum-space Berry curvature

for the nth band is given by [125]

Bn(~k) = −∑n′ 6=n

2Im⟨n~k|vx|n′~k

⟩⟨n′~k|vy|n~k

⟩(En′ − En)2

(5.7)

where vx(y) is the velocity operator and En is the eigenenergy corresponding to the∣∣∣n~k⟩

eigenstate. It may be readily determined that the K and K ′ bands have opposite signs

of local Berry curvature. The conduction and valence bands also have opposite signs of

local Berry curvature at every point in momentum space. As a result, when the chemical

potential is placed between conduction and valence bands, the system displays quantum

valley Hall effect [85]. However, flat band physics will manifest itself when the chemical

potential is placed inside either the conduction or the valence bands. In this case we can

focus on the band that contains the chemical potential.

We consider the conduction band for specificity. Due to avoided crossings between

bands coming from the two valleys, the lowest conduction band contains regions with pos-

itive and negative local Berry curvature respectively while the integrated Berry curvature

(Chern number) is zero (Fig.5.4). Meanwhile, there is a second nearly flat conduction band

which is separated by a mini gap equal to the inter valley scattering amplitude.

Thus, we conclude that placing N = 7 layer ABC graphene on BN substrate allows us

to realize two nearly flat bands with local Berry curvature but zero Chern number which

are separated from the non-flat bands by a band gap of 6meV and are separated from each

other by a minigap of 0.2meV , equal to the inter valley scattering amplitude. Chiral mul-

tilayer graphene thus offers a promising playground for investigating the effect of strong

interactions in the presence of a non-trivial quantum distance metric.

94

px (3Za/4/π)

p y (3Z

a/4/

π)

−0.5 0 0.5−0.5

0

0.5

−10

0

10

Figure 5.4: Contour plot of Berry curvature in the lowest flat conduction band forN = 7. The red/yellow regions come mainly from theK valley and have positive curvature,while the blue regions come mainly from the K ′ valley and have negative curvature. TheBerry curvature integrated over the band is zero.

5.4 Details of Bandstructure Calculation

Now we calculate the matrix elements of the external potential V coming from the BN be-

tween Bloch states, 〈~k|V |k+~q〉. This is just equivalent to calculating the Fourier transform

of the BN potential. We model the BN potential as λδ(~ri) − λδ(~rj), where ~ri are the po-

sitions of boron atoms and ~rj are the positions of nitrogen atoms. This is a natural model,

since the atomic numbers of B and N are one less than and one more than carbon respec-

tively. Taking the delta functions to have slightly different weights will not qualitatively

alter our results. The boron atoms sit on the A sublattice of the hexagonal superlattice, and

the nitrogen atoms sit on the B sublattice.

The Fourier transform of the above potential takes the form

〈~k|V |~k + ~q〉 = V (~q) = δ~k, ~Qf1(~q) (5.8)

where ~Q denotes a reciprocal lattice vector of the BN lattice (which is equal to ~B1,2 in

Eq.5.5 of the main text modulo reciprocal lattice vectors of the graphene lattice) and f1(~q)

95

is a form factor coming from the two site nature of the BN unit cell. For the model potential

under consideration, f1(~q) ∝ (ei~q.~ri − ei~q. ~rj). Hence, we obtain 〈~k|V |~k + ~B1〉 = iλ,

〈~k|V |~k + ~B2〉 = iλ, 〈~k|V |~k + ~B1 + ~B2〉 = iλ, and f1(−~k) = f ∗1 (~k).

Thus far we have assumed that the BN superlattice potential can be modelled as a delta

function array. In fact, the B and N atoms carry + and − charge respectively. The BN

superlattice potential may thus be better modelled as a delta function array convolved with

a 1/r envelope. The 1/r envelope simply reflects the Coulomb potential arising from a

local charge imbalance. This Fourier transforms to a delta function array multiplied by a

1/k envelope. Thus intervalley scattering by Z reciprocal lattice vectors is weaker than

intravalley scattering by one reciprocal lattice vector by a factor of 1/Z. However as we

will see below, this weak intervalley scattering is still important along lines in the reduced

zone where the K and K ′ bands are degenerate.

Now we calculate the band structure. First we consider the low energy hamiltonian of

multilayer graphene without BN substrate in Eq.5.4. It is written in a basis such that (1, 0) is

a Bloch state in the A sublattice of the top layer and (0, 1) is a Bloch state in the B sublattice

of the bottom layer. Now we consider the case of multilayer graphene sandwiched between

BN sheets. Without considering the folding of the bands at the reduced zone edges, the 4

low energy bands corresponding to a particular hexagonal unit cell with center ~Q are found

from the eigenvalues of

m(~k, ~Q) =

∆/2 α λZ

0

α∗ −∆/2 0 λZ

λZ

0 ∆/2 β

0 λZ

β∗ −∆/2

(5.9)

where α =vN ((kx−Q′x)−i((ky−Q′y))N

(−γ)(N−1), β =

vN (−(kx−Q′′x)−i((ky−Q′′y ))N

(−γ)(N−1)and ~Q′ points to the K

point of the hexagon, for which the distance between ~k and ~Q′ is the minimum. A similar

definition holds for ~Q′′ and K ′ point. The matrix is written in a basis where (1, 0, 0, 0)

96

denotes a state on sub lattice A and with valley K, (0,1,0,0) denotes a state on sub lattice

B and valley K, (0, 0, 1, 0) denotes a state on sub lattice A and valley K ′ and (0, 0, 0, 1)

denotes a state on sub lattice B and valley K ′. In the off diagonal blocks we have included

a weak inter-valley scattering coming from the superlattice potential.

The band folding at the zone edges can be taken into account by considering the four

low energy bands arising from a central hexagonal unit cell in the reciprocal space and

those arising from it’s 6 nearest neighbouring cells. The corresponding band structure is

easily found from the eigenvalues of the following 7× 7 matrix:

m(~k, ~K0) n∗ n n n n∗ n∗

n m(~k, ~K1) n 0 0 0 n∗

n∗ n∗ m(~k, ~K2) n 0 0 0

n∗ 0 n∗ m(~k, ~K3) n∗ 0 0

n∗ 0 0 n m(~k, ~K4) n∗ 0

n 0 0 0 n m(~k, ~K5) n∗

n n 0 0 0 n m(~k, ~K6)

(5.10)

where ~K0 points to the center of the central hexagon, ~K1, ~K2, ~K3, ~K4, ~K5 and ~K6

point to the centers of the 6 adjoining hexagons. The matrix is written in a basis where

(1, 0, 0, 0, 0, 0, 0) is a state near K0, (0, 1, 0, 0, 0, 0, 0) is a state near K1, (0, 0, 1, 0, 0, 0, 0)

is a state near K2 etc. and the matrix corresponding to the intra-valley scattering between

the valleys of neighbouring hexagonal unit cells is

n =

iλ 0 0 0

0 iλ 0 0

0 0 iλ 0

0 0 0 iλ

(5.11)

97

This 4× 4 matrix is written in a basis where (1, 0, 0, 0) denotes a state on sub lattice A and

with valley K, (0,1,0,0) denotes a state on sub lattice B and valley K, (0, 0, 1, 0) denotes

a state on sub lattice A and valley K ′ and (0, 0, 0, 1) denotes a state on sub lattice B and

valley K ′.

5.5 Chern number from adatoms

The key feature of chiral multilayer graphene is that it allows access to a system with a non-

trivial quantum geometry but without Chern number. However, a non-zero Chern number

may also be obtained by making use of adatom deposition on the outermost graphene layers

to open up a gap between conduction and valence bands, rather than using vertical electric

field. Adatom deposition also introduces a superlattice potential. For the appropriate choice

of (time reversal symmetry breaking) adatoms, the two valleys acquire the same sign of

Berry curvature [98, 146, 30]. The rest of the analysis proceeds exactly as before, only

now the Berry curvature has the same sign everywhere in the flat band, and thus does not

cancel.

5.6 Concluding Remarks

We have shown that ABC stacked multilayer graphene placed on BN substrate has a band-

structure containing flat bands. The seven layer system is ideal for this purpose. The flat

bands have nonzero local Berry curvature but zero Chern number. Thus, chiral multilayer

graphene represents an exciting new frontier in the study of interaction effects in systems

with non-trivial quantum geometry, allowing access to an interaction dominated system

with a non-trivial quantum distance metric but without the complication of a non-zero

Chern number.

98

Chapter 6

Conclusion

In this dissertation, we have mainly studied a number of systems that host quantum Hall

ferromagnets. First, we looked at two multi-valley systems —AlAs and Si— where a global

symmetry acts simultaneously on the internal valley index and on the spatial degrees of

freedom. In Chapter 2, we provided a microscopic analysis of domain walls found in AlAs

quantum Hall system in the presence of random-field disorder. We also discussed the effect

of dipole moment on critical behavior and domain wall energetics. Next, in Chapter 3 on

Si quantum Hall systems, we gave examples of two unusual selection mechanisms —order

by disorder and order by doping— operating near integer fillings. To our knowledge, this

is the first time either selection mechanism has been shown to operate in the QHE setting.

Study of ground states of multicomponent quantum Hall systems like Si at fractional filling

is a challenging open question.

Then in Chapter 4, we exploited the analogy between multicomponent quantum hall

systems and lattice systems with higher Chern number bands to discover analogs of quan-

tum hall ferromagnets in the zoo of fractional Chern insulator phases. We also pointed

out the similarities and differences between such phases in gas (AlAs and Si(111) at filling

factor one) and lattice systems. Finally in Chapter 5, we showed that multilayer graphene

on BN provides a platform for investigating the effect of interactions in a system with

99

Fubini-Study metric, without the complication of nonzero Chern numbers. This offers a

new frontier in the study of systems with an interplay of strong correlations and quantum

geometry.

100

Appendix A

Theta Functions

A.1 Basic Theta Function

θ(z) =∑n∈Z

eiπn2τe2πinz

θ(z + 1) = θ(z); θ(z + τ) = e−iπτ−2πizθ(z)

θ(z) = 0 if and only if z = τ2

+ 12

+ pτ + q, where p and q are integers. All zeros are

simple zeros.

A.2 Modified Theta Function

For ξ = aτ + b and a, b ∈ R,

θξ(z) =∑n∈Z

eiπ(n+a)2τe2πi(n+a)(z+b)

θξ(z + 1) = e2πiaθξ(z); θξ(z + τ) = e−2πibe−πiτe−2πizθξ(z)

θξ(z) = 0 if and only if z = τ2

+ 12− ξ + pτ + q, where p and q are integers. All zeros

are simple zeros. Lastly for a = 0 and b = 0, we get back the basic theta function.

101

Bibliography

[1] D. A. Abanin, S. A. Parameswaran, S. A. Kivelson, and S. L. Sondhi. Nematic valleyordering in quantum hall systems. Phys. Rev. B, 82(3):035428, Jul 2010.

[2] D. A. Abanin, S. A. Parameswaran, and S. L. Sondhi. Charge 2e skyrmions in bilayergraphene. Phys. Rev. Lett., 103:076802, Aug 2009.

[3] Dmitry A. Abanin, Benjamin E. Feldman, Amir Yacoby, and Bertrand I. Halperin.Fractional and integer quantum hall effects in the zeroth landau level in graphene.Phys. Rev. B, 88:115407, Sep 2013.

[4] Dmitry A. Abanin, Patrick A. Lee, and Leonid S. Levitov. Randomness-induced xyordering in a graphene quantum hall ferromagnet. Phys. Rev. Lett., 98:156801, Apr2007.

[5] Michael Aizenman and Jan Wehr. Rounding of first-order phase transitions in sys-tems with quenched disorder. Phys. Rev. Lett., 62:2503–2506, May 1989.

[6] A. Altland and B. Simon. Condensed Matter Field Theory. Cambridge UniversityPress, Cambridge, 2006.

[7] J. Anandan and Y. Aharonov. Geometry of quantum evolution. Phys. Rev. Lett.,65:1697–1700, Oct 1990.

[8] Tsuneya Ando, Alan B. Fowler, and Frank Stern. Electronic properties of two-dimensional systems. Rev. Mod. Phys., 54:437–672, Apr 1982.

[9] N.W. Ashcroft and N.D. Mermin. Solid state physics. Holt, Rinehart and Winston,New York, 1976.

[10] Aritra Banerjee, BenoˆFauque, Koichi Izawa, Atsushi Miyake, Ilya Sheikin, JacquesFlouquet, Bertrand Lenoir, and Kamran Behnia. Transport anomalies across thequantum limit in semimetallic bi0.96sb0.04. Phys. Rev. B, 78:161103, Oct 2008.

[11] E.J. Bergholtz and L. Zhao. Topological flat band models and fractional chern insu-lators. Int. Mod. Phys. B, 27:1330017, 2013.

[12] G. Bergmann. Weak localization in thin films. Phys. Rep., 107:1–58, 1984.

[13] B.A. Bernevig and T.L. Hughes. Topological Insulators and Topological Supercon-ductors. Princeton University Press, Princeton, 2013.

102

[14] M.V. Berry. Quantal phase factors accompanying adiabatic changes. Proc. R. Soc.London Ser. A, 392:45, 1984.

[15] K. Binder. Random-field induced interface widths in ising systems. Zeitschrift furPhysik B Condensed Matter, 50:343–352, 1983.

[16] Waheb Bishara and Chetan Nayak. Effect of landau level mixing on the effectiveinteraction between electrons in the fractional quantum hall regime. Phys. Rev. B,80:121302, Sep 2009.

[17] D. J. Bishop, R. C. Dynes, B. J. Lin, and D. C. Tsui. Anisotropy in weakly localizedelectronic transport: A parameter-free test of the scaling theory of localization. Phys.Rev. B, 30:3539–3541, Sep 1984.

[18] B. Blok and X. Wen. Many-body systems with non-abelian statistics. Nucl. Phys. B,374:615, 1992.

[19] L. Brey, H. A. Fertig, R. Cote, and A. H. MacDonald. Skyrme crystal in a two-dimensional electron gas. Phys. Rev. Lett., 75:2562–2565, Sep 1995.

[20] J. Cardy. Scaling and Renormalization in Statistical Physics. Cambridge LectureNotes in Physics. Cambridge University Press, 1996.

[21] E. W. Carlson, K. A. Dahmen, E. Fradkin, and S. A. Kivelson. Hysteresis and noisefrom electronic nematicity in high-temperature superconductors. Phys. Rev. Lett.,96(9):097003, Mar 2006.

[22] J. T. Chalker, P. C. W. Holdsworth, and E. F. Shender. Hidden order in a frus-trated system: Properties of the heisenberg kagome antiferromagnet. Phys. Rev.Lett., 68:855–858, Feb 1992.

[23] A. R. Champagne, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West. Evidence fora finite-temperature phase transition in a bilayer quantum hall system. Phys. Rev.Lett., 100:096801, Mar 2008.

[24] Xie Chen, Zheng-Cheng Gu, Zheng-Xin Liu, and Xiao-Gang Wen. Symmetry pro-tected topological orders and the group cohomology of their symmetry group. Phys.Rev. B, 87:155114, Apr 2013.

[25] A. L. Chernyshev and M. E. Zhitomirsky. Quantum selection of order in an xxzantiferromagnet on a kagome lattice. Phys. Rev. Lett., 113:237202, Dec 2014.

[26] Andrey Chubukov. Order from disorder in a kagome antiferromagnet. Phys. Rev.Lett., 69:832–835, Aug 1992.

[27] N. R. Cooper. Optical flux lattices for ultracold atomic gases. Phys. Rev. Lett.,106:175301, Apr 2011.

103

[28] R. Cote, A. H. MacDonald, Luis Brey, H. A. Fertig, S. M. Girvin, and H. T. C.Stoof. Collective excitations, nmr, and phase transitions in skyrme crystals. Phys.Rev. Lett., 78:4825–4828, Jun 1997.

[29] C.R. Dean, A.F. Young, P. Cadden-Zimansky, L. Wang, H. Ren, K. Watanabe,T. Taniguchi, P. Kim, J. Hone, and K.L. Shepard. Multicomponent fractional quan-tum hall effect in graphene. Nature Physics, 7:693–696, 2012.

[30] Jun Ding, Zhenhua Qiao, Wanxiang Feng, Yugui Yao, and Qian Niu. Engineeringquantum anomalous/valley hall states in graphene via metal-atom adsorption: Anab-initio study. Phys. Rev. B, 84:195444, Nov 2011.

[31] R. L. Doretto and C. Morais Smith. Quantum hall ferromagnetism in graphene:Su(4) bosonization approach. Phys. Rev. B, 76:195431, Nov 2007.

[32] A. L. Efros, F. G. Pikus, and V. G. Burnett. Density of states of a two-dimensionalelectron gas in a long-range random potential. Phys. Rev. B, 47(4):2233–2243, Jan1993.

[33] G. T. Einevoll and C. A. Lutken. Anisotropic quantum hall effect. Phys. Rev. B,48:11492–11495, Oct 1993.

[34] K. Eng, R. N. McFarland, and B. E. Kane. Integer quantum hall effect on a six-valleyhydrogen-passivated silicon (111) surface. Phys. Rev. Lett., 99:016801, Jul 2007.

[35] Andrew M. Essin and Michael Hermele. Classifying fractionalization: Symmetryclassification of gapped z2 spin liquids in two dimensions. Phys. Rev. B, 87:104406,Mar 2013.

[36] Z. Ezawa. Quantum Hall Effects: Field Theoretical Approach and Related Topics.World Scientific, Singapore, 2008.

[37] H. A. Fertig, Luis Brey, R. Cote, A. H. MacDonald, A. Karlhede, and S. L. Sondhi.Hartree-fock theory of skyrmions in quantum hall ferromagnets. Phys. Rev. B,55(16):10671–10680, Apr 1997.

[38] M. M. Fogler and B. I. Shklovskii. Collapse of spin splitting in the quantum halleffect. Phys. Rev. B, 52:17366–17378, Dec 1995.

[39] E. Fradkin. Field theories of Condensed Matter systems. Addison Wesley publishinghouse, 1991.

[40] S. Girvin. The quantum Hall effect: Novel excitations and broken symmetries, Topo-logical Aspects of Low Dimensional Systems, Proceedings of the LXIX Les HouchesSummer School. A. Comtet, T. Jolicoeur, S. Ouvry, and T. David, eds., 1999.

[41] T. Gokmen, Medini Padmanabhan, and M. Shayegan. Transference of transportanisotropy to composite fermions. Nature Physics, 6:621–624, 2010.

104

[42] T. Gokmen and M. Shayegan. Density and strain dependence of ν = 1 energy gapin a valley-degenerate alas quantum well. Phys. Rev. B, 81(11):115336, Mar 2010.

[43] G. P. Guo, X. J. Hao, T. Tu, Y. J. Zhao, Z. R. Lin, G. Cao, H. O. Li, C. Zhou, G. C.Guo, and H. W. Jiang. Probing a quantum hall pseudospin ferromagnet by resistivelydetected nuclear magnetic resonance. Phys. Rev. B, 81:041306, Jan 2010.

[44] G. P. Guo, Y. J. Zhao, T. Tu, X. J. Hao, X. C. Zhang, G. C. Guo, and H. W. Jiang.Observation of an in-plane magnetic-field-driven phase transition in a quantum hallsystem with su(4) symmetry. Phys. Rev. B, 78:233305, Dec 2008.

[45] F. D. M. Haldane. Model for a quantum hall effect without landau levels:Condensed-matter realization of the ”parity anomaly”. Phys. Rev. Lett., 61:2015–2018, Oct 1988.

[46] F. D. M. Haldane. Geometrical description of the fractional quantum hall effect.Phys. Rev. Lett., 107:116801, Sep 2011.

[47] F. D. M. Haldane and K. Yang. Landau level mixing and levitation of extended statesin two dimensions. Phys. Rev. Lett., 78(2):298, Jan 1997.

[48] A B Harris. Effect of random defects on the critical behaviour of ising models.Journal of Physics C: Solid State Physics, 7(9):1671, 1974.

[49] M. Z. Hasan and C. L. Kane. Colloquium : Topological insulators. Rev. Mod. Phys.,82:3045–3067, Nov 2010.

[50] T.T. Heikkila, N.B. Kopnin, and G.E. Volovik. Flat bands in topological media.JETP Lett., 94:233–239, 2011.

[51] T.T. Heikkila and G.E. Volovik. Dimensional crossover in topological matter: Evo-lution of the multiple dirac point in the layered system to the flat band on the surface.JETP Lett., 93:59–65, 2011.

[52] Itay Hen and Marek Karliner. Hexagonal structure of baby skyrmion lattices. Phys.Rev. D, 77:054009, Mar 2008.

[53] Chen-Hsuan Hsu, S. Raghu, and Sudip Chakravarty. Topological density wave statesof nonzero angular momentum. Phys. Rev. B, 84:155111, Oct 2011.

[54] Ling-Yan Hung and Xiao-Gang Wen. Quantized topological terms in weak-couplinggauge theories with a global symmetry and their connection to symmetry-enrichedtopological phases. Phys. Rev. B, 87:165107, Apr 2013.

[55] Yoseph Imry and Shang-keng Ma. Random-field instability of the ordered state ofcontinuous symmetry. Phys. Rev. Lett., 35:1399–1401, Nov 1975.

[56] S. V. Isakov and R. Moessner. Interplay of quantum and thermal fluctuations in afrustrated magnet. Phys. Rev. B, 68:104409, Sep 2003.

105

[57] J. K. Jain. Composite-fermion approach for the fractional quantum hall effect. Phys.Rev. Lett., 63:199–202, Jul 1989.

[58] Jorge V. Jose, Leo P. Kadanoff, Scott Kirkpatrick, and David R. Nelson. Renor-malization, vortices, and symmetry-breaking perturbations in the two-dimensionalplanar model. Phys. Rev. B, 16:1217–1241, Aug 1977.

[59] T. Jungwirth and A. H. MacDonald. Pseudospin anisotropy classification of quantumhall ferromagnets. Phys. Rev. B, 63:035305, Dec 2000.

[60] A. Karlhede, S.A. Kivelson, and S.L. Sondhi. Quantum Hall Effect: The Arti-cle, Correlated electron systems: Proceedings of the 9th Jerusalem Winter School,Jerusalem, Israel, 30 Dec. 91-8 Jan. 92. V.J. Emery, ed., World Scientific, 1993.

[61] I. Klebanov. Nucl. Phys. B, 262:133, 1985.

[62] K. v. Klitzing, G. Dorda, and M. Pepper. New method for high-accuracy determi-nation of the fine-structure constant based on quantized hall resistance. Phys. Rev.Lett., 45:494–497, Aug 1980.

[63] Mikito Koshino and Edward McCann. Trigonal warping and berry’s phase nπ inabc-stacked multilayer graphene. Phys. Rev. B, 80:165409, Oct 2009.

[64] T.M. Kott. Measurements of Correlated 2D Electrons in the Lowest Landau Levelon Silicon-(111). PhD thesis, University of Maryland, 2012.

[65] Tomasz M. Kott, Binhui Hu, S. H. Brown, and B. E. Kane. Valley-degenerate two-dimensional electrons in the lowest landau level. Phys. Rev. B, 89:041107, Jan 2014.

[66] A. A. Koulakov, M. M. Fogler, and B. I. Shklovskii. Charge density wave in two-dimensional electron liquid in weak magnetic field. Phys. Rev. Lett., 76:499, 1996.

[67] Stefanos Kourtis and Maria Daghofer. Combined topological and landau order fromstrong correlations in chern bands. Phys. Rev. Lett., 113:216404, Nov 2014.

[68] D. L. Kovrizhin, Benoıt Doucot, and R. Moessner. Multicomponent skyrmion lat-tices and their excitations. Phys. Rev. Lett., 110:186802, Apr 2013.

[69] J. Lambert and R. Cote. Quantum hall ferromagnetic phases in the landau leveln = 0 of a graphene bilayer. Phys. Rev. B, 87:115415, Mar 2013.

[70] R. B. Laughlin. Anomalous quantum hall effect: An incompressible quantum fluidwith fractionally charged excitations. Phys. Rev. Lett., 50:1395–1398, May 1983.

[71] R. B. Laughlin. Nobel lecture: Fractional quantization. Rev. Mod. Phys., 71:863–874, Jul 1999.

[72] Ching Hua Lee, Ronny Thomale, and Xiao-Liang Qi. Pseudopotential formalismfor fractional chern insulators. Phys. Rev. B, 88:035101, Jul 2013.

106

[73] Y. Lee, J. Velasco, D. Tran, F. Zhang, W. Bao, L. Jing, K. Myhro, D. Smirnov, andC.N. Lau. Broken Symmetry Quantum Hall States in Dual Gated ABA TrilayerGraphene. ArXiv e-prints, 2012.

[74] Lu Li, J. G. Checkelsky, Y. S. Hor, C. Uher, A. F. Hebard, J. Cava, and N. P. Ong.Phase transitions of dirac electrons in bismuth. Science, 321:547, Jul 2008.

[75] Zhao Liu and Emil J. Bergholtz. From fractional chern insulators to abelian and non-abelian fractional quantum hall states: Adiabatic continuity and orbital entanglementspectrum. Phys. Rev. B, 87:035306, Jan 2013.

[76] Yuan-Ming Lu and Ying Ran. Symmetry-protected fractional chern insulators andfractional topological insulators. Phys. Rev. B, 85:165134, Apr 2012.

[77] J. et al Martin. Science, 305:(5686):9803, 2004.

[78] R.N. McFarland. Multi-Valley Physics of Two-Dimensional Electron Systems onHydrogen-Terminated Silicon (111) Surfaces. PhD thesis, University of Maryland,2010.

[79] Andrej Mesaros and Ying Ran. Classification of symmetry enriched topologicalphases with exactly solvable models. Phys. Rev. B, 87:155115, Apr 2013.

[80] Aditi Mitra and S. M. Girvin. Electron/nuclear spin domain walls in quantum hallsystems. Phys. Rev. B, 67(24):245311, Jun 2003.

[81] R. Moessner and J. T. Chalker. Low-temperature properties of classical geometri-cally frustrated antiferromagnets. Phys. Rev. B, 58:12049–12062, Nov 1998.

[82] K. Moon, H. Mori, Kun Yang, S. M. Girvin, A. H. MacDonald, L. Zheng, D. Yosh-ioka, and Shou-Cheng Zhang. Spontaneous interlayer coherence in double-layerquantum hall systems: Charged vortices and kosterlitz-thouless phase transitions.Phys. Rev. B, 51:5138–5170, Feb 1995.

[83] G. Murthy and R. Shankar. Composite fermions for fractionally filled chern bands.arXiv:1108.5501, 2011.

[84] Masaaki Nakamura and Lila Hirasawa. Electric transport and magnetic properties inmultilayer graphene. Phys. Rev. B, 77:045429, Jan 2008.

[85] Rahul Nandkishore and Leonid Levitov. Quantum anomalous hall state in bilayergraphene. Phys. Rev. B, 82:115124, Sep 2010.

[86] Titus Neupert, Claudio Chamon, and Christopher Mudry. Measuring the quantumgeometry of bloch bands with current noise. Phys. Rev. B, 87:245103, Jun 2013.

[87] Titus Neupert, Luiz Santos, Claudio Chamon, and Christopher Mudry. Fractionalquantum hall states at zero magnetic field. Phys. Rev. Lett., 106:236804, Jun 2011.

107

[88] Titus Neupert, Luiz Santos, Shinsei Ryu, Claudio Chamon, and Christopher Mudry.Fractional topological liquids with time-reversal symmetry and their lattice realiza-tion. Phys. Rev. B, 84:165107, Oct 2011.

[89] Titus Neupert, Luiz Santos, Shinsei Ryu, Claudio Chamon, and Christopher Mudry.Topological hubbard model and its high-temperature quantum hall effect. Phys. Rev.Lett., 108:046806, Jan 2012.

[90] Qian Niu, D. J. Thouless, and Yong-Shi Wu. Quantized hall conductance as a topo-logical invariant. Phys. Rev. B, 31:3372–3377, Mar 1985.

[91] Kentaro Nomura and Allan H. MacDonald. Quantum hall ferromagnetism ingraphene. Phys. Rev. Lett., 96:256602, Jun 2006.

[92] Medini Padmanabhan, T. Gokmen, and M. Shayegan. Ferromagnetic fractionalquantum hall states in a valley-degenerate two-dimensional electron system. Phys.Rev. Lett., 104(1):016805, Jan 2010.

[93] S. A. Parameswaran, R. Roy, and S. L. Sondhi. Fractional chern insulators and theW∞ algebra. Phys. Rev. B, 85:241308, Jun 2012.

[94] Siddharth A. Parameswaran and Vadim Oganesyan. Valley nematics: Unfinishedbismuth. Nat Phys, 8(1):7–8, 01 2012.

[95] Michael R. Peterson and Chetan Nayak. More realistic hamiltonians for the frac-tional quantum hall regime in gaas and graphene. Phys. Rev. B, 87:245129, Jun2013.

[96] S. Prabhu-Gaunkar, S. Birner, S. Dasgupta, C. Knaak, and M. Grayson. Valleydegeneracy in biaxially strained aluminum arsenide quantum wells. Phys. Rev. B,84:125319, Sep 2011.

[97] Xiao-Liang Qi. Generic wave-function description of fractional quantum anomaloushall states and fractional topological insulators. Phys. Rev. Lett., 107:126803, Sep2011.

[98] Zhenhua Qiao, Shengyuan A. Yang, Wanxiang Feng, Wang-Kong Tse, Jun Ding,Yugui Yao, Jian Wang, and Qian Niu. Quantum anomalous hall effect in graphenefrom rashba and exchange effects. Phys. Rev. B, 82:161414, Oct 2010.

[99] Stephan Rachel and Karyn Le Hur. Topological insulators and mott physics fromthe hubbard interaction. Phys. Rev. B, 82:075106, Aug 2010.

[100] S. Raghu, Xiao-Liang Qi, C. Honerkamp, and Shou-Cheng Zhang. Topological mottinsulators. Phys. Rev. Lett., 100:156401, Apr 2008.

[101] R. Rajaraman. Fractional charge. arXiv:cond-mat/0103366.

[102] R. Rajaraman. Solitons and Instantons. North Holland, Amsterdam, 1982.

108

[103] Madan Rao, Surajit Sengupta, and R. Shankar. Shape-deformation-driven structuraltransitions in quantum hall skyrmions. Phys. Rev. Lett., 79:3998–4001, Nov 1997.

[104] M. Rasolt. Continuous symmetries and broken symmetries in multivalley semicon-ductors and semimetals. Solid State Physics, 43:93–228.

[105] Mark Rasolt, B. I. Halperin, and David Vanderbilt. Dissipation due to a “valleywave” channel in the quantum hall effect of a multivalley semiconductor. Phys. Rev.Lett., 57(1):126–129, Jul 1986.

[106] Mark Rasolt, F. Perrot, and A. H. MacDonald. New gapless modes in the fractionalquantum hall effect of multicomponent fermions. Phys. Rev. Lett., 55:433–436, Jul1985.

[107] N. Regnault and B. Andrei Bernevig. Fractional chern insulator. Phys. Rev. X,1:021014, Dec 2011.

[108] Rahul Roy. Band geometry of fractional topological insulators. Phys. Rev. B,90:165139, Oct 2014.

[109] B. Sachs, T. O. Wehling, M. I. Katsnelson, and A. I. Lichtenstein. Adhesion andelectronic structure of graphene on hexagonal boron nitride substrates. Phys. Rev. B,84:195414, Nov 2011.

[110] A. A. Shashkin, A. A. Kapustin, E. V. Deviatov, V. T. Dolgopolov, and Z. D.Kvon. Strongly enhanced effective mass in dilute two-dimensional electron systems:System-independent origin. Phys. Rev. B, 76:241302, Dec 2007.

[111] M. Shayegan, E. P. De Poortere, O. Gunawan, Y. P. Shkolnikov, E. Tutuc, andK. Vakili. Quantum hall effect in a multi-valley two-dimensional electron system.International Journal of Modern Physics B, 21(08n09):1388–1397, 2007.

[112] M. Shayegan, E.P. De Poortere, O. Gunawan, Y.P. Shkolnikov, E. Tutuc, and K. Vak-ili. Int. J. Mod. Phys. B, 21:1388, 2007.

[113] D.N. Sheng, Z.C. Gu, K. Sun, and L. Sheng. Fractional quantum hall effect in theabsence of landau levels. Nat. Commun., 2:389, 2011.

[114] Naokazu Shibata and Kentaro Nomura. Coupled charge and valley excitations ingraphene quantum hall ferromagnets. Phys. Rev. B, 77:235426, Jun 2008.

[115] Y. P. Shkolnikov, S. Misra, N. C. Bishop, E. P. De Poortere, and M. Shayegan.Observation of quantum hall “valley skyrmions”. Phys. Rev. Lett., 95(6):066809,Aug 2005.

[116] Steven H. Simon and Edward H. Rezayi. Landau level mixing in the perturbativelimit. Phys. Rev. B, 87:155426, Apr 2013.

[117] Jairo Sinova, A. H. MacDonald, and S. M. Girvin. Disorder and interactions inquantum hall ferromagnets near ν = 1. Phys. Rev. B, 62:13579–13587, Nov 2000.

109

[118] T.H.R. Skyrme. A non-linear field theory. Proc. R. Soc. B, 260:127, 1961.

[119] I. Sodemann and A. H. MacDonald. Broken su(4) symmetry and the fractionalquantum hall effect in graphene. Phys. Rev. Lett., 112:126804, Mar 2014.

[120] S. L. Sondhi, A. Karlhede, S. A. Kivelson, and E. H. Rezayi. Skyrmions and thecrossover from the integer to fractional quantum hall effect at small zeeman energies.Phys. Rev. B, 47:16419–16426, Jun 1993.

[121] Frank Stern and W. E. Howard. Properties of semiconductor surface inversion layersin the electric quantum limit. Phys. Rev., 163:816–835, Nov 1967.

[122] Kai Sun, Zhengcheng Gu, Hosho Katsura, and S. Das Sarma. Nearly flatbands withnontrivial topology. Phys. Rev. Lett., 106:236803, Jun 2011.

[123] Evelyn Tang, Jia-Wei Mei, and Xiao-Gang Wen. High-temperature fractional quan-tum hall states. Phys. Rev. Lett., 106:236802, Jun 2011.

[124] Zlatko Tesanovic and B. I. Halperin. Multivalley electron gas in a strong magneticfield. Phys. Rev. B, 36:4888–4901, Sep 1987.

[125] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs. Quantized hallconductance in a two-dimensional periodic potential. Phys. Rev. Lett., 49:405–408,Aug 1982.

[126] Carsten Timm, S. M. Girvin, and H. A. Fertig. Skyrmion lattice melting in thequantum hall system. Phys. Rev. B, 58:10634–10647, Oct 1998.

[127] Maximilian Trescher and Emil J. Bergholtz. Flat bands with higher chern number inpyrochlore slabs. Phys. Rev. B, 86:241111, Dec 2012.

[128] D. C. Tsui and G. Kaminsky. Observation of sixfold valley degeneracy in electroninversion layers on si(111). Phys. Rev. Lett., 42:595–597, Feb 1979.

[129] D. C. Tsui, H. L. Stormer, and A. C. Gossard. Two-dimensional magnetotransportin the extreme quantum limit. Phys. Rev. Lett., 48:1559–1562, May 1982.

[130] J. Villain, R. Bidaux, J.-P. Carton, and R. Conte. Order as an effect of disorder. J.Phys. France, 41(11):1263–1272, 1980.

[131] N.R. Walet and T. Weidig. Full 2d numerical study of the quantum hall skyrmecrystal. arXiv:cond-mat/0106157, 2011.

[132] P. R. Wallace. The band theory of graphite. Phys. Rev., 71:622–634, May 1947.

[133] Xin Wan and R. N. Bhatt. Search for multiple-step integer quantum hall transitions.Phys. Rev. B, 64:201313, Nov 2001.

[134] Fa Wang and Ying Ran. Nearly flat band with chern number c = 2 on the dicelattice. Phys. Rev. B, 84:241103, Dec 2011.

110

[135] Yi-Fei Wang, Hong Yao, Chang-De Gong, and D. N. Sheng. Fractional quantum halleffect in topological flat bands with chern number two. Phys. Rev. B, 86:201101, Nov2012.

[136] X. G. Wen and Q. Niu. Ground-state degeneracy of the fractional quantum hallstates in the presence of a random potential and on high-genus riemann surfaces.Phys. Rev. B, 41:9377–9396, May 1990.

[137] P. Wolfle and R. N. Bhatt. Electron localization in anisotropic systems. Phys. Rev.B, 30:3542–3544, Sep 1984.

[138] Yang-Le Wu, B. Andrei Bernevig, and N. Regnault. Zoology of fractional cherninsulators. Phys. Rev. B, 85:075116, Feb 2012.

[139] Ying-Hai Wu, J. K. Jain, and Kai Sun. Fractional topological phases in generalizedhofstadter bands with arbitrary chern numbers. Phys. Rev. B, 91:041119, Jan 2015.

[140] Kun Yang and R. N. Bhatt. Floating of extended states and localization transition ina weak magnetic field. Phys. Rev. Lett., 76:1316–1319, Feb 1996.

[141] Kun Yang, S. Das Sarma, and A. H. MacDonald. Collective modes and skyrmionexcitations in graphene su(4) quantum hall ferromagnets. Phys. Rev. B, 74:075423,Aug 2006.

[142] Shuo Yang, Zheng-Cheng Gu, Kai Sun, and S. Das Sarma. Topological flat bandmodels with arbitrary chern numbers. Phys. Rev. B, 86:241112, Dec 2012.

[143] N. Y. Yao, A. V. Gorshkov, C. R. Laumann, A. M. Lauchli, J. Ye, and M. D.Lukin. Realizing fractional chern insulators in dipolar spin systems. Phys. Rev.Lett., 110:185302, Apr 2013.

[144] A. F. Young, C. R. Dean, L. Wang, H. Ren, P. Cadden-Zimansky, K. Watanabe,T. Taniguchi, J. Hone, K. L. Shepard, and P. Kim. Spin and valley quantum hallferromagnetism in graphene. Nat Phys, 8(7):550–556, 07 2012.

[145] Fan Zhang, Jeil Jung, Gregory A. Fiete, Qian Niu, and Allan H. MacDonald. Spon-taneous quantum hall states in chirally stacked few-layer graphene systems. Phys.Rev. Lett., 106:156801, Apr 2011.

[146] Hongbin Zhang, Cesar Lazo, Stefan Blugel, Stefan Heinze, and Yuriy Mokrousov.Electrically tunable quantum anomalous hall effect in graphene decorated by 5dtransition-metal adatoms. Phys. Rev. Lett., 108:056802, Feb 2012.

[147] Z. Zhang, T. Tang, C. Girit, Z. Hao, M. Martin, A. Zettl, M. Crommie, Y. Shen,and F. Wang. Direct observation of a widely tunable bandgap in bilayer graphene.Nature, 459:820–823, 2009.

[148] Z. Zhu, A. Collaudin, B. Fauque, W. Kang, and K. Behnia. Field-induced polariza-tion of dirac valleys in bismuth. Nat Phys, 8(1):89–94, 01 2012.

111