Quantum Hall Ferromagnets

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QUANTUM HALL FERROMAGNETS

AKSHAY KUMAR

ADISSERTATION

PRESENTEDTOTHE FACULTY

OF PRINCETON UNIVERSITY

IN CANDIDACYFORTHE DEGREE

OF DOCTOROF PHILOSOPHY

RECOMMENDEDFOR ACCEPTANCE

BYTHE DEPARTMENT OF

PHYSICS

ADVISER:SHIVAJI L.SONDHI

APRIL 2016 c Copyright by Akshay Kumar, 2016.

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 occurs 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 number 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 curvature 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 student 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 multilayer graphene. Phys. Rev. B 87, 241108(R) (2013)

2. Akshay Kumar, S.A.Parameswaran and S. L. Sondhi. Microscopic theory of a quantum 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 ferromagnetism 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 doping 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 h i 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

3 global phase diagram, showing how the G = [SU(2)] 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 ∗ ...... 50 ∼ E-S 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 particular 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 0 with m = 1.8. (b) Low energy many- o − body spectrum for 8 fermions on a 4 4 lattice for the case of the single- × particle part of Hamiltonian chosen as H 0 with m = 1.8 and V = 3U. o − (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 K0 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 include 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 order 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 perturbations, 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 presence 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 topological 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 mass m 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~ = Bzˆ is turned on ? The Hamiltonian for this system is given by H = (~p + eA/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,

1 1 2 2 2 ikyy 2 −(x+kylB ) /2lB ψn,ky = n 1/2 1/2 1/2 e Hn(x + kylB)e (1.1) (2 n!π lB) Ly

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

~c 1/2 1 ( eB ) is the magnetic length. The corresponding eigenvalues are En = ~ωc(n + 2 ) eB where ωc = mc 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, energy 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 observations. 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

VH B arguments [9] the Hall resistance is given as RH = I = ne , 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 approximately 0 within the plateaus and peaks at the steps between the plateaus. The quantized

e2 values of Hall conductance are m h where 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 quantum 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 microscopic 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 localized 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 = 0 as the starting point of our discussion. In the plot of density of states versus en- 6 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 using a semi-classical model of electron dynamics in a smooth random potential [40]. This

e2 explains the existence of plateaus of Hall conductance at integral multiples of h . When the magnetic field is varied and the Fermi energy crosses the LL centers, the Hall conductivity 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

1 1 ν = 3 and 5 [70]. Jain’s construction- which involves building trial wavefunctions from p ﬁlled pseudo-Landau levels of composite fermions [57]- explains plateaus at ν = 2pk+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 number. 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 potential. 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 theory [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 eigenstate is defined only up to a global ~k dependent phase. Hence all the information about the eigenstates is encoded in U~ which belongs to the set U(2)/U(1) U(1). Moreover all k × information about the band structure is encoded in a mapping: ~k U~ . Thus, information → k about the state’s topology is present in the homotopy classes of this mapping. If the mapping 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 | i a closed loop in the Brillouin zone it acquires a phase given by the line integral of the Berry potential A~ = i ~k ~ ~ ~k . Using Stokes’ theorem, it can be rewritten as a surface integral h |∇k| i of Berry flux F = ~ A~. The Chern invariant is the total Berry flux through the Brillouin ∇k × zone. 1 Z C = d2~k F (~k) (1.2) 2π

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

hc another in general [39], but they commute if a unit cell with eB 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].

2 Z 2 ie 2 1~k pˆx 2~k 2~k pˆy 1~k 2~k pˆx 1~k 1~k pˆy 2~k e σxy = 2 d k h | | ih | | i − h 2| | ih | | i = C (1.3) hLxLym (E1 E2) h −

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

e2 Chern number close to the filling fraction (in units of 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 fractional 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 levels, 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 ferromagnet? 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 freedom. 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 electrons 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-

1 ized to other ﬁllings, for instance ν = 3 [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 symmetries 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 symmetry. 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 dispersion 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 supports 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 simultaneously on the internal index and on the spatial degrees of freedom. This occurs naturally in a multi-valley system where different valleys are related by a discrete rotation so that valley pseudospin and rotational symmetries are intertwined. 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 degeneracy [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 symmetries 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 order 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 theory. 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 symmetry 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 particularly 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 number. 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 distance (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 valleys (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 freedom. 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 transport 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 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 different orientations of the nematic order parameter, and is extremely sensitive to the application 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 k y IM

1 1 kx LS

L IM S

Figure 2.1: (a) Model band structure used in this chapter, appropriate to describing AlAs wide quantum wells. (b) Different phases as determined by comparing 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. ∼ 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 symmetries 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 properties 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 orientation 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 properties 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 order 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 ordering 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 temperature, 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 = 1. Upon 6 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-

2 particle gap, ∆ e /`B in the QH ferromagnetic ground state. This also sets the scale sp ∼ 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 interactions, 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 characteristic 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 determined 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 potential 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 mismatch between the separation of the valleys in momentum space – roughly an inverse lattice spacing – and the scale of 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 T Ising /2T , e sp xx yy ⇠ xx = yy PM 6 Tc sp =0 ⇠ xy 6 Fogler-Shklovskii T ⇤ dw ⇠ collapse /2T QHIN = e dw xx yy ⇠ =0 xy 6 W QHRFPM Wc

, 0 xx, yy 0 xx yy ! ! / ↵ =1 xx/yy 1 xx yy ! 6 ! 2 2 xy = e /h xy = e /h

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 the introduction.

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 = m2,y (a schematic dispersion is sketched in Fig. 2.1.) In m1,y m2,x each valley, the single-particle kinetic energy is

e 2 X pi Kκ,i + Ai T = − c (2.1) κ 2m i=x,y κ,i

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

2 ipyy 1/4 − uκ(x−X) e uκ 2`2 ψκ,X (x, y) = p e B (2.2) Ly`B π

25 2T log tr ⇠ xx tr / v

sp W

multiple single domain domain

v⇤ ⇠ dw v

v v⇤

Figure 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.

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

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

LxLy is N = NΦ = 2 . For periodic boundary conditions in the y-direction, the guiding 2π`B

26 2π`2 center coordinate along the x-direction is given by X = B n, with n an integer between n Ly N + 1 and N . For the sake of brevity, we shall continue to label states by X, but with − 2 2 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

(b) two valleys, and corresponding occupation numbers nκ :

X (b) ∗ ρb(r) = nκ ψX,κ(r)ψX,κ(r) (2.3) X,κ

(b) (b) (b) with κ = 1, 2 as before and n1 + n2 = 1. Note that for any choice of nκ 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 1 X X κY,κ0Y 0 † † X X h (b) κY,1X † (b) κY,2X † i H = V 0 0 c c 0 0 cκ0X0 cκX n V c cκY + n V c cκY 2 κ X ,κX κY κ Y − 1 1X,κY κY 2 2X,κY κY κ,κ0 X,Y κ X,Y X0,Y 0 X Ly 1 † † 2 +g X c c1X c c2X + E ρ (2.4) 2π`2 − 2 1X − 2X self b X B

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.

SB As discussed in the next section, the characteristic energy scale of this is ∆d , and it varies over a characteristic distance d corresponding to the correlation length of the random ﬁeld;

∆SB 2π`2 rewriting this carefully, leads to the expression given, with g = d B (which has units 2d Ly

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

2 the two valleys (suppressed by a factor of (a/`B) ). 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:

Z κX,κ0X0 2 2 0 ∗ ∗ 0 0 0 V 0 0 = d rd r ψ (r)ψ 0 0 (r )V (r r )ψκ0Y 0 (r )ψκY (r) (2.5) κ Y ,κY κX κ X −

where the single-particle wave functions were deﬁned in the previous section. Note that momentum conservation requires that X + X0 = Y + Y 0.

28 Energy scales

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

2 We measure energy in units of the Coulomb energy e /`B where is the dielectric constant appropriate to the heterostructure under consideration. For the AlAs devices which are our

primary focus, 10. The magnetic length, `B 8 nm for a magnetic ﬁeld of 10 T, so ≈ ≈ 2 that e /`B 200 K. The surface tension of Ising domain walls is measured in units of ≈ 2 2 e /`B, 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 temperature 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 cκ0Y = nκδκκ0 δXY , h κX i where the occupation numbers are assumed independent of position, yields

1 X X (b) κY,κ0X † 1 X X κY,κX † H = nκ0 2n 0 V 0 c cκY nκV c cκY (2.6) MF 2 − κ κ X,κY κY − 2 κY,κX κY κ,κ0 X,Y κ X,Y

(b) We simplify the Hartree term by taking nκ = nκ. For N , we may use translation → ∞ P † invariance of the potential to write HMF = X,κ κcκX cκX , where

1 X κX,κ0Y 1 X κX,κY κ = nκ0 V 0 nκV (2.7) −2 κ Y,κX − 2 κY,κX κ0,Y Y is independent of X. We seek a solution where the ground state spontaneously breaks valley symmetry; without loss of generality we may assume it is polarized in valley 1, and take the energy splitting to be ∆, whence

e∆/kB T 1 n1 = , n2 = (2.8) 1 + e∆/kB T 1 + e∆/kB T 29 For this ansatz the self-consistency condition corresponds to ∆ = 2 1 = A1n1 A2n2, − − where after a tedious calculation we ﬁnd (deﬁning 1¯ = 2, 2¯ = 1)

1 X h κX,κY¯ κX,κY κX,κY i 1 X κX,κY Aκ = V + V + V = V (2.9) 2 − κY,κX¯ κX,κY κY,κX 2 κX,κY Y Y

(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 . By the standard compar- 2kB T ison of the slope of both sides of this equation at ∆ = 0, we ﬁnd for the (mean-ﬁeld) transition temperature

1 1 X k T MF = A = V κX,κY B c 2 κ 4 κX,κY Y 2 − 1 x +λy2 1 e2 Z ∞ Z ∞ e 2 λ = dx dy 2 p 2 2 16π `B −∞ −∞ x + y 1 e2 K (1 1/λ2) = 3/2 − (2.10) 2(2π) `B √λ

where K 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

MF particular, the ﬂuctuations drive Tc to zero in the isotropic Heisenberg limit. Furthermore, as spin-wave gap scales roughly with the Ising anisotropy, the debilitating effect of spin

MF waves on Tc 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:

2 σ −1 2 √e Tc 4πρs log [ρs/α`B], obtained from the NLσM with stiffness ρs 0.025 ∼ ≈ 16 2π`B 4 1 MF Note that the λ λ− symmetry while not manifest in the ﬁnal expression for Tc is nevertheless obtained from an identity→ satisﬁed by the elliptic integral K.

30 0.6 MF ) 0.5 kBTc B "`

/ 0.4 2 e k T 0.3 B c (in c

T 0.2 B k 0.1 0.0 2 4 6 8 10 2

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

e2 2 and Ising anisotropy α 0.01 `3 (λ 1) , whose leading dependence of λ was computed ≈ B − 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 equations 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:

Y † ΨG = c 0 (2.11) | i 1X | i X 31 then a domain wall is captured by a Slater determinant of the form

Y † † Ψ = uX c + vX c 0 . (2.12) | DWi 1X 2X | i X

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

(b) charge distribution polarized in valley 1, i.e. n1 = 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:

1 σ(λ) = lim ( H DW E0) = EI + EII + EIII. (2.13) Lx→∞ Ly h i −

Here, E0 and ( H E0) are the energies of the ground state and the sharp domain wall. h iDW − The three contributions are individually convergent, and can be written as follows. The ﬁrst term,

∞ 1 X 1X,1X0 2X,2X0 1X,2X0 E = V 0 + V 0 2V 0 (2.14) I 2 1X ,1X 2X ,2X − 2X ,1X X,X0=1

32 measures the Hartree cost, and can be simpliﬁed as

Ly 2 Z ∞ Z ` 0 1 e 0 B fλ(x)fλ(x ) EI = dxdx dy 2 2 p 0 2 2 32π `B −∞ 0 (x x ) + y − (2.15)

where

−1/2 1/2 fλ(x) = erfc xλ erfc xλ (2.16) − − −

with erfc the complementary error function. The second term,

∞ 1 X 1X,1X0 2X,2X0 E = V 0 V 0 (2.17) II 2 1X,1X − 2X,2X X,X0=1 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:

0 ∞ X X 1X,1X0 EIII = V1X,1X0 (2.18) X=−∞ X0=1

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 (a.) 0.014

0.013 ) B l / ) 2

2 B 0.012 "` / 2

e 0.011

(in 0.01 Surface tension

0.009 Surface tension (in units of e 0.008

0.007 1 2 3 4 5 2 6 7 8 9 10 (b.) 0.12

0.1

0.08 ) e

,in 0.06 y /L , in units of e) p y

( 0.04 (p/L 0.02 Dipole moment per unit length Dipole moment per unit length 0 1 2 3 4 5 22 6 7 8 9 10 Figure 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 y-axis at x = 0, between regions with opposite Ising polarization (i.e., ϕ 1 as x → ± → ˆ .) The remaining rotational symmetry is a rotation Rπ that takes x x, y y. ∓∞ → − → − ˆ 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 background 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 ∗ ∗ = (uK 1)ψ (r)ψ1X (r) + vK ψ (r)ψ2X (r) − { − 1X 2X } X=−N/2+1 N/2 X ∗ ∗ = ψ (r)ψ1X (r) ψ (r)ψ2X (r) . − { 1X − 2X } X=1 (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 1 Z 2 e p ˆ −1 2 −t −1 ˆ (λ) = xe λ λ 3/2 t e dt = λ λ x (2.20) Ly − × 4π −∞ 8π −

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 affect 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,

Z d2r ( ϕ)2 + rϕ2 + uϕ4 (2.21) F ∼ ∇

and add to it a perturbation appropriate to a long-range interaction between dipoles:

Z Z 0 p p 0 (p ˆr)(p 0 ˆr ) δ v d2r d2r0 r · r − r · r · (2.22) F ∼ r r0 3 | − | and determine its effect on the critical theory and domain formation with disorder.

(i) Irrelevance at Tc. Using the fact that p ϕ(r), we have for dimensional purposes r ∼ ∇

Z Z ϕ(r)ϕ(r0) δ v d2r d2r0 (2.23) F ∼ r r0 5 | − |

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 η where η is the − SR SR 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 gain 37 from the random field so that it scales as L log L. 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 domain 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 2 (2.12) subject now to the constraint uX + vX = 1, and with the boundary condition | | | | 2 2 that uX and vX approach 1 for X = ( N/2 + 1)2π` /Ly and X = N/2 2π` /Ly − B × B 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

  uX X ∗ ∗ H DW = u v A   h i X X   X vX   H ex LyX 1 c U1 (X) + U1 (X) + g 2π`2 2 U (X) A =  B −   c∗ H ex LyX 1  U (X) U2 (X) + U2 (X) g 2π`2 2 − B − where the Hartree-Fock potentials are

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 h i 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 Domain Wall Texturing, S ` /L h xi B y

−1 10 ) B

−2 10 (in units of 1/l y /L x texturing S

−3 10 0.005 0.01 0.015 0.02 0.025 0.03 gg

Figure 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 λ2h andi 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 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

str ﬁeld u(r) of the crystal from its equilibrium position: ∆ (r) (∂x ∂y)u(r). The v ∝ − 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 antisymmetric in valley indices will lead to a spatially dependent single-particle splitting between valleys; in the limit when the cyclotron gap diverges, i.e. when the lowest Landau level approximation is exact, this is the only contribution, and we can argue from symmetry that

2 2 the corresponding random ﬁeld should take the form (∂ ∂ )Ud (at least in the small- x − y 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-

2 lows for terms of order Ud /~ωc, the random ﬁeld now receives contributions of the form

2 2 ((∂xUd) (∂yUd) )/~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 interactions 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

X † X mn iqxX † Hni = (n~ωc µ) cn,κ,X cn,κ,X + Ud ( q)e cm,κ,X cn,κ,X−(2.24) − − + n,X q,X,κ,n,m

2 qy`B where we have deﬁned X± = X . Here, we have expanded the notation of Section I ± 2 to include Landau level indices n, m. In this basis, we have deﬁned the matrix elements of

mn mn the disorder potential Ud via U ( q) Ud( q)F (q), which naturally introduces the d − ≡ − κκ form factors

n−m r 2 2 2 q2u m! iqx uκ q qyuκ qx y κ nm n−m x − 4u − 4 Fκκ (q) = qy Lm + e κ (2.25) n! √2uκ − 2 2uκ 2 for n m, with F nm(q) = F mn( q)∗, where Lα is the generalized Laguerre polynomial. ≥ κκ κκ − n 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

ˆ X κ,00 iqxX † ULLL = U ( q)e c c0,κ,X (2.26) d,eff − 0,κ,X+ − q,X,κ¯ where, to ﬁrst order in Landau level mixing,

0 0 0 2 κ,00 X Ud( q )Ud(q q) q×q `B 00 i 2 Ud,eff ( q) = Ud( q)Fκκ(q) + − − e − − n ωc q0,n6=0 ~ 0n 0 n0 0 3 Fκκ (q q )Fκκ (q ) + Ud /~ωc × − O | | (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

U SB( q) = U 1 ( q) U 2 ( q) d − d,eff − − d,eff − 1 1 = λ λ−1 ∂2 ∂2 U + λ λ−1 (∂ U )2 (∂ U )2 x y d −q x d y d −q −4 − − 2~ωc − − (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

SB d, corresponding to the ﬁnal term in (2.4), with the overall energy scale ∆d set by the

SB characteristic scale of the spatially varying random potential Ud (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

2 2 −2qd Ud(q) = (U0d) e (2.29) h| | i

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

2π 0 2 Wp,p0 = Ud(p p ) δ (Ep Ep0 ) (2.30) ~ | − | −

A straightforward Boltzmann transport calculation of the transport relaxation time, assuming that it is dominated by the Fermi surface yields

π 1 m Z dθ 2π θ 2 2 −2kF sin 2 ×2d = 2 (1 cos θ)U0 d e (2.31) τtr 2π~ −π 2π ~ −

where the (1 cos θ) factor suppresses the contribution of small-angle scattering, which − does not contribute to charge relaxation. For kF d 1, we have

2 1 m 2 √π~ = 2 (U0d) 3 . (2.32) τtr π~ 8(mvF d)

Using the fact that 1/τtr = e/(mµ) where µ is the mobility,

3 3 1/2 8√πe~ kF d U0 (2.33) ≈ µm2

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

SB breaking term ∆d is given by

4 SB 1 `B 2 ∆d (mx my) U0 . (2.34) ∼ 2π~2 − d2

44 This corresponds to a random field h ∆SB/`2 in the NLσM. Since h is correlated roughly ∼ d B 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 transport 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 produced 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 along the [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] p The maximum value attained for T Tc can be estimated as ζ mx/my max ≈ | − 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 domains 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 longitudinal 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 measurements 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 ordering 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 na¨ıve 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 acting 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,

3 as shown in Figure 3.1. Consequently, it exhibits large symmetry group—[SU(2)] oD3— 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 ﬁrst-order T ky second-order B C A 3 G =[SU(2)] o D3 kx

A¯ C¯ Z3 nematic 2 Tc B¯ HT =[SU(2) o D2] SU(2) ⇥ O = Z3

HT = G T ⇤ E-S,2 TE-S⇤ ,3

12 23 ⌫ 1 ⌫c 2 ⌫c 3 2 2 3 H0 =U(1) SU(2) H =U(1) SU(2) H0 = U(1) ⇥ 0 ⇥ O = S2 O = S2 S2 O = S2 S2 S2 ⇥ ⇥ ⇥ Figure 3.1: Valley ordering in Si(111) QH states. (Inset) Model Fermi surface. Ellipses denote constant-energy lines in k-space. (Main figure) Schematic global phase diagram, 3 showing how the G = [SU(2)] 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 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 depends 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 √ √ ~ 3K0 K0 ~ ~ 3K0 K0 ~ ~ KA = ( , ), KB = (0,K0) and KC = ( , ), with Kκ¯ = Kκ. Here 2 2 − 2 2 − p 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-

~ ~ 2 P ((~p+eA/c−Kκ)·~ηκi) particle Hamiltonian in valley κ (where κ = A, B, C) is Hκ = , i=1,2 2mi 1 1 where ~ηA/C1 = ( 1, √3), ~ηA/C2 = ( √3, 1), ~ηB1 = (1, 0) and ~ηB2 = (0, 1); Hκ¯ is 2 ∓ 2 ± obtained by taking K0 K0 in these expressions. → − ~ In Landau gauge A = (0, Bx), the LLL eigenfunctions labeled by momentum ky are given by 2 2 1/4 (x+ky`B ) −(fκ+igκ) (fκ) iK~ .~r ik y 2`2 κ y B φκ,ky = 1/2 1/2 e e e , (3.1) (π `BLy)

51 √ √ √ √ 4 λ 3(1−λ) 1 4 λ − 3(1−λ) where (f, g) ¯ = ( , ), (f, g) ¯ = ( √ , 0), (f, g) ¯ = ( , ), A,A λ+3 λ+3 B,B λ C,C λ+3 λ+3 q ~c λ = (m2/m1) 3.55 [64] and the magnetic length `B = . We focus on filling ≈ eB 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 eigenfunctions. In the limit K0`B 1, the electron-electron interaction term is

1 X H = V (~q)ρκκ(~q)ρ 0 0 ( ~q) (3.2) 2S κ κ − ~q,κ,κ0

where S = LxLy is the total area, ρκκ is the density operator within valley κ projected

2πe2 to the LLL and V (~q) = q is 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,

3 explaining the [SU(2)] , 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

Q † † P † † P 2 2 ψ = d 0 where d = (ακc + ακ¯c ), with ( ακ + ακ¯ ) = 1 | i ky ky | i ky κ∈{A,B,C} κky κk¯ y κ | | | | The anisotropy splits this degeneracy. At λ 1 1, ﬁrst-order perturbation theory yields | − |

X σ 2 2 2 2 ψ H ψ = δV 0 ( ακ + ακ¯ )( ακ0 + ακ¯0 ) (3.3) h | | i κκ | | | | | | | | κ,κ,0σ ∈{A,B,C}

σ 1 σσ κ0κ where δV 0 = σκκ0 (V V 0 ), and κκ 2 | | σσ − κκ Z κ0κ X ∗ ∗ 0 0 Vκκ0 = φκk(~r)φκ0k0 (~r )V~r−~r0 φκ0k(~r )φκk0 (~r). (3.4) 0 k,k0 ~r,~r

σ The δVκκ0 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 = | i Q † † 3 (ακcκ,k + ακ¯cκ,k¯ ) 0 ; these break the [SU(2)] o D3 symmetry down to H0 = ky y y | i 2 U(1) [SU(2) oD2] (where the second factor refers to rotations of the unoccupied pairs), × 2 leading to a single Goldstone mode (as G/H0 = S ). 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 = 0, statistical averaging over Goldstone modes restores the 6 2 broken SU(2) symmetry, so that the invariance group is HT = SU(2) [SU(2) o D2]. × 53 Class I Class II Class I Class II ⌫ =1 ⌫ =2 ⌫ =3

Figure 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 particular choice of SU(2) vector within the two-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 = Q Q d† 0 where | i i=1,2 ky i,ky | i † P † † P 2 2 d = (ακic + ακi¯ c ), and ( ακi + ακi¯ ) = 1. Moving away i,ky κ∈{A,B,C} κky κk¯ y κ | | | | 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¯ = Q c† c† 0 . “Class II” ground states on the other hand | i ky κ,ky κ,k¯ y | i 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

0 Q † † † † κκ = (ακc + ακ¯c )(ακ0 c 0 + ακ¯0 c 0 ) 0 . While Class I states break the | i ky κ,ky κ,k¯ y κ ,ky κ¯ ,ky | i 3 3 2 [SU(2)] oD3 symmetry down to [SU(2)] , Class II states break it down to U(1) SU(2); × 2 2 the order parameter spaces are Z3 and S S 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 doping selects Class II states. Turning now to T > 0, we observe that the combination of a high ground state degeneracy 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 energy 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 ∗ e2 . E-S ∼ `B We observe that while Goldstone modes are responsible for order by disorder, the

SU(2) symmetries remain unbroken at any T = 0 so that the invariance group is HT = 6 2 [SU(2) o D2] 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 ν=2 ν=1 ﬁeld rather than by gating, we anticipate that Tc < Tc , as the relevant energy scale is the surface tension of Z3 domain walls, that depends in turn on the Coulomb energy

2 e /`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- Q Q † † P † generate ground states are ψ = d 0 where d = (ακic + | i i=1,2,3 ky i,ky | i i,ky κ∈{A,B,C} κky † ακi¯ c ). As before, the symmetry is reduced for λ = 1, and the new ground state κk¯ y 6 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

0 Q † † † 0 † the form κκκ¯ = c c (αc 0 + α c 0 ) 0 . Class II ground states are ob- | i ky κ,ky κ,k¯ y κ ,ky κ¯ ,ky | i tained by breaking each of the three SU(2)s by forming a ν = 1 state in each pair, so

Q Q † † 2 2 ABC = (ακc + ακ¯c ) 0 with ακ + ακ¯ = 1. Class I states are | i ky κ=A,B,C κ,ky κ,k¯ y | i | | | | 2 3 invariant under H0 = SU(2) U(1), and Class II states under U(1) . 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 different 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-periodic Q = 2 spinor solution [68] gives Class I and II states identical gradient energies, 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 =

3 [SU(2)] 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 TE-S, below which order by doping dominates. In combining the results for ν = 1, 2, 3 we note that their distinct symmetries and

12 23 12 doping energetics at T = 0 point to ﬁrst-order transitions at νc and νc , where 1 < νc <

23 2 < νc < 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 conductivities 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 potential, 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 = K ¯ = (K, 0) and KB = K ¯ = (0,K). In the Landau gauge A = (0, Bx), the − A − B

LLL eigenfunctions are given by Equation 4.4 with (f, g)A,A¯ = (√λ, 0) and (f, g)B,B¯ =

( √1 , 0). The interaction Hamiltonian has [SU(2)]2 D symmetry where the SU(2)s are λ o 2 ¯ ¯ independent rotations in the pair subspaces (A, A) and (B, B) and the D2 symmetry interchanges 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. Q † † For λ 1 1, the ground states at ν = 1 are given by ψ = (ακc +ακ¯c ) 0 | − | | i ky κ,ky κ,k¯ y | i 2 2 and ( ακ + ακ¯ ) = 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¯ = Q c† c† 0 . Class II states are of the form AB = | i ky κ,ky κ,k¯ y | i | i Q Q † † 2 2 (ακc + ακ¯c ) 0 where ακ + ακ¯ = 1. Though the T = 0 mode ky κ=A,B κ,ky κ,k¯ y | i | | | | counting—zero versus a pair of Goldstone modes—is similar to ν = 2 for Si(111), at

2 T > 0 the full [SU(2)] o D2 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 ky B

A¯ kx A Class I Class II

B¯ ⌫ =1 ⌫ =2

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

2 We ﬁrst show that high-temperature valley symmetry group is [SU(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 2 N = [SU(2)] is a subgroup of G. Turning to D4, we recall that this is generated by two operations, a permutation r = (ABA¯B¯) 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,

−1 2 3 2 3 so that r Nr = N. Since D4 = e, r, r , r , ρ, rρ, r ρ, r ρ , 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 { } { } 2 that G/N ∼= D2. We therefore conclude that G = N o D2 = [SU(2)] o D2, where in 2 2 identifying the D2 structure we used the fact r e in the coset space since r N = 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 to H0 at T = 0, described by a NLσM with target space HT /H0. At ν = 1, and ﬁnite temperature we choose to ﬁll a single valley pair while leaving the

2 other unfilled. The invariant subgroup HT of the resulting state is SU(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 =

2 2 ([SU(2)] o D2)/[SU(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 =

2 [SU(2)] o D2, 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 0  1  g =   , (3.5) 0 g2

T with g1, g2 SU(2), and ~z0 = (1, 0, 1, 0) 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 0  1  h =   , (3.6) 0 h2

0 0 0 with h1, h2 U(1). Using the equivalence relation on ~z, ~z given by ~z ~z ~z = ∈ ∼ ⇐⇒ 0 2 2 h g ~z for h H0, we see that HT /H0 = S S . 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

3 is G = [SU(2)] 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

3 (ii) the dihedral group D6 of symmetries of a regular hexagon. Clearly, N = SU(2) is a subgroup. Turning next to D6, we observe that it is generated by the sixfold rotation r = (ABCA¯B¯C¯) and the reﬂection ρ = (AA¯)(BC)(B¯C¯), so that we may write

2 3 4 5 2 3 4 5 D6 = e, r, r , r , r , r , ρ, ρr, ρr , ρr , ρr , ρr . [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-

−1 serve that left and right cosets must be equivalent, i.e. s Ns = N for any s D6, ∈ following the same logic as in the four-valley case: performing a discrete transformation 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 addition, 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 = N i.e. the 6 quotient group G/N is non-Abelian. Since the unique non-Abelian group of order 6 is D3,

3 we see that G/N ∼= D3, from which the structure G = [SU(2) ]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

2 argue that this leads to an SU(2) o D2 structure within the unﬁlled subspace, so that in

2 total we have HT = SU(2) [SU(2) o D2]. From this, we see that the NLσM target ∼ × 3 2 space is given by G/HT = [SU(2) o D3]/[SU(2) [SU(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

2 U(1) so that the residual symmetry H0 = U(1) [SU(2) o D2], so that the target space × 2 is HT /H0 ∼= S . 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

2 symmetry, so that HT = [SU(2) oD2] SU(2) again (but the role of the filled and unfilled × valleys are interchanged), and once again G/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 × 2 2 space HT /H0 = S S . For Class I states at ν = 2, we fill both valleys in a pair, leading ∼ × 2 to residual symmetry group HT = SU(2) [SU(2) o D2]; 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

2 fluctuations of the Goldstone modes about class II states restores [SU(2) o D2] 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

3 3 T > 0 we have HT = G = [SU(2)] o S ; 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

  g1 0 0     g =  0 g 0  , (3.7)  2    0 0 g3

T with g1, g2, g3 SU(2), and ~z0 = (1, 0, 1, 0, 1, 0) is a reference spinor. As T 0, ∈ → each SU(2) breaks to U(1), so once again we consider states ~z, ~z0 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

3 corresponding symmetry is simply HT = SU(2) , and therefore G/HT ∼= D3, suggesting a potential finite-temperature transition. However, as T 0, we break the SU(2) of the → 2 2 filled pair down to U(1), so that H0 = U(1) SU(2) , and HT /H0 = S ; 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 = 0 restore the full symmetry G immediately. 6

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)).

2 2 They are subject to the constraint v1 + v2 = 1. The energy functional of this system is | | | | the sum of a gradient term Ho and a Coulomb interaction term.

Z X 2 † † 2 Ho = 4 d x [(∂µv) (∂µv) + (v (∂µv)) ] (3.8) µ=x,y

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 indexed by their topological charge Q, which can only take integer values,

i Z Q = d2x ∂ (v†(∂ v)) (3.9) 2π µν µ ν

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 =

iµνDνv. As suggested by Ref. [102], this equation can be converted into a Cauchy- ± Riemann condition: ∂µw = iµν∂νw, where the w field is defined patchwise. (In regions ± with v1 = 0, we define w = v2/v1 and similarly in regions with v2 = 0, we define 6 6 w = v1/v2. The different definitons can be analytically continued from one to another.) The formula for charge takes the following form:

1 Z Q = d2x 2ln(1 + w 2) (3.10) 4π ∇ | |

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 requirement 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 = 0 does not exist. Now suppose a meromorphic function is constructed as a 6 h1(z) ratio f(z) = , where h1(z) and h2(z) are entire functions which satisfy the following h2(z)

λα(p,q,x,y)+iλˆα(p,q,x,y) properties: hα(z + pτ + q) = hα(z)e , where p and q are integers,

p 2 2 p 2 2 and α = 1, 2. Now we construct v1 = h1/ h1 + h2 and v2 = h2/ h1 + h2, and then ~ ~ construct φ. We observe that φ is doubly periodic only if λ1(p, q, x, y) = λ2(p, q, x, y) and ˆ ˆ 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 = 0. 6 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 P P P P if and only if i ni = j mj and i niai = j mjbj. (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).

Q ni h(z) i θ( 1 + τ )(z ai) g(z) = = 2 2 − Q mj t(z) j θ( 1 + τ )(z bj) 2 2 − h(z + 1) t(z + 1) = = 1 h(z) t(z)

h(z + τ) t(z + τ) −2πi(z+ 1 + τ ) P n 2πi P n a = = e 2 2 i i e i i i (3.12) h(z) t(z)

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 minimization methods. As a check, we first carry out the optimization process for the topological 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:

Z 2 ~ 2 ~ 2 Ho = d x ((∂xφ) + (∂yφ) ) (3.13) Γ

68 ~ ~ ~ where Γ is the unit cell as shown in Fig. 3.4. So φ(~x + Lnˆ1) = φ(~x + Lnˆ2) = φ(~x). Once ~ R 2 again, the set of all possible φ can be classiﬁed by their topological charge Q = Γ d xQ(~x) 1 ~ ~ ~ where Q(~x) = φ (∂xφ ∂yφ). 4π · × We discretize the unit cell along the nˆ1 and nˆ2 directions. For numerically calculating derivatives along nˆ1 and nˆ2, the backward ﬁnite difference method is used. They are related to the partial derivatives along xˆ and yˆ by the following equations:

∂φ~ ∂φ~ ∂φ~ ∂φ~ 1 ∂φ~ = , = cot γ + (3.14) ∂x ∂n1 ∂y − ∂n1 sin γ ∂n2 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 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) ~ −r/λ φo2 = (sin(f)cos(2θ), sin(f)sin(2θ), cos(f)) where f(r) = πe 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:

Z Q(~x)Q(~y) H = H + g d2xd2y (3.15) o ~x ~y Γ | − | 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 proportional 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

2 among these is the neglect of various terms in (3.2) that while suppressed by O((a/`B) ) relative to the dominant Coulomb energy scale could compete with the selection mechanisms 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 TE-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 qualitative 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 bands Fractional quantum Hall effect Partially filled flat Chern band Quantum 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 nearest 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 considered 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     ~ ~ X H11(k) H12(k) c~k↑ H = † †     (4.1) o c~ c~ k↑ k↓  ∗ ~ ~    ~ H (k) H11(k) c~ k 12 − k↓ ~ ~ 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 ~ ) γ ~ γ ~ + ( 1 + t + tE ~ ) γ ~ γ ~ − +,k | +,kih +,k| − −,k | −,kih −,k| where t = 1, ∆ = 1, γ ~ and γ ~ are the eigenfunctions for the lower and upper | −,ki | +,ki band, respectively and E−,~k, E+,~k are their corresponding eigenvalues. The eigenstates are P ~ related to the old set of states by c ~ = uηκ(k) γ ~ . If we tune t 0 and | η,ki κ=−,+ | κ,ki → ∆ (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:

X X X Hint. = V n~i↑n~i↓ + U n~iκn~iκ0 (4.2) ~i h~i~ji κ,κ0=↑,↓

75 (a)

(b)

5.5

4.5 E / U 4

3.5

3 0 2 4 6 8 10 12 14 16 K +L K x x y

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 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.)

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 c ~ uη−(k)γ ~ in Hint.. η,k → −,k 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 e2 Hall conductance σH = 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 occupation 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 domain 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 index 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 temperature, 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 transition. 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.

0 As a check, we consider a single-particle Hamiltonian Ho 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)

5.5

4.5 E / U

3.5

3 0 2 4 6 8 10 12 14 16 K +L K x x y

Figure 4.4: (a) Lower band Chern ﬂux distribution over the Brilliouin zone for the single- 0 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 H 0 with m = ×1.8 and V = 3U. (Energies are resolved using total many-body momenta o − (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 0 ~ 0 ~ 1 Equation 4.1 where H (k) = m + cos(kx) + cos(ky), H (k) = (cos(2ky) cos(2kx)) + 11 12 2 − i(cos(kx ky) cos(kx + ky)) and 2 < m < 0. − − −

81 The new projected Hamiltonian H 0 is diagonalized for 8 particles on a 4 4 lat- proj. × tice (Fig. 5.2(b)). A 2-fold degenerate ground state and a gap of order U are again observed. We also check for broken sublattice symmetry by computing the Fourier transform of ρ0(~x)ρ0(0) where ρ0 is the density operator projected onto the lowest ﬂat band. Peaks h i 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 inclusion 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 analogous 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

X H = V (~i)(ρ↑(~i) + ρ↓(~i))projected (4.3) ~i 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:

  ix+iy ( 1) , ix < N/4  −  ~ ix+iy+1 V1(i) = ( 1) ,N/4 < ix < 3N/4 (4.4)  −   ix+iy ( 1) , 3N/4 < ix −

82   ix−iy (1 + ( 1) )/2, (ix iy) < N/4  − −  ~ ix−iy+1 V2(i) = (1 + ( 1) )/2,N/4 < (ix iy) < 3N/4 (4.5)  − −   ix−iy (1 + ( 1) )/2, 3N/4 < (ix iy) − − where N 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 be 83 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

2 ν = 1/3 Laughlin state and a quantized Hall conductance σH = e /(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 1.2

1.0

0.8

0.6

Energy 0.4

0.2

0.0

−0.2 −4 −3 −2 −1 0 1 2 3 4 Momentum along domain wall (a)

1.2

1.0

0.8

0.6

Energy 0.4

0.2

0.0

−0.2 −4 −3 −2 −1 0 1 2 3 4 Momentum along domain wall (b)

Figure 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 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 interactions 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 curvature (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 metric 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, attention 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 projected 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]:

X ~ ∗ ~ ~ † [¯ρq~1, ρ¯q~2] i~q1 ~q2 [B(k)ub (k+)ub(k−)γ~ γ~k ] (5.1) ≈ ∧ k+ − ~k,b

~ ~ ~q1+ ~q2 ~ where q~1 q~2 =z. ˆ (~q1 ~q2), k± = k , B(k) is the local Berry curvature for the ∧ × ± 2 P † band of interest for the single-particle hamiltonian H = ~ c h (~k)c~ and the cor- k,a,b ~k,a ab k,b

~E † P ~ † responding eigenstate is given by k = γ 0 = ua(k)c 0 . Thus far, attention has ~k | i a ~k,a | i 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 (encoded 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 integrated 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 potential 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 quantum 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 honeycomb 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 (ψ ~ ψ ~ ψ ~ ψ ~ ψ ~ ψ ~ ) basis, the 1Ak 1Bk 2Ak 2Bk ······ NAk NBk nearest neighbor tight binding Hamiltonian for an N layer system takes the form [145]

  0 t~p 0 0 0 0  ···     t∗ 0 γ 0 0 0   ~p ···     0 γ 0 t 0 0   ~p   ···  H =  ∗  (5.2) 0  0 0 t~p 0 γ 0   ···     0 0 0 γ 0 t~p   ···     0 0 0 0 t∗ 0   ~p ···   ......  ......

88 √ kxa 3kya Here, t~p = t0 exp(ikxa)+2 exp( i ) cos( ) , represents nearest neighbor hop- − 2 2 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 ±| | 0 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 = 0. This interlayer hopping causes all the bulk 6 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]

  vN 0 pN  +  HK (~p) = N−1 ; p± = px ipy (5.3) ( γ)  N  ± − p− 0

0 Here HK is the Hamiltonian in the K valley, and the Hamiltonian in the K valley

∗ is given by HK0 (~p) = H ( ~p). The basis is such that (1, 0) is a Bloch state in the A K − 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 = vN pN /γN−1. Thus, the ± conduction band has a very flat pocket of a size p γ/v around the K and K0 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 1 N=3 N=5 0.5 N=10 γ 0 E /

−0.5

−1 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 ka

Figure 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 . ≈

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   ∆ vN N 2 (−γ)N−1 p+ HK (~p) =   (5.4)  N  v pN ∆ (−γ)N−1 − − 2 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 2π 1 √3 xˆ   =     (5.5)  ~  3Za   ~ B2 1 √3 yˆ − The strong superlattice potential is seen primarily by states on the lower surface of the multilayer 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’ at K and K0 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 K0 points in the original zone get folded to inequivalent corners K˜ and K˜ 0 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 30 25 20 15 (meV) Λ 10 5 0 4 5 6 7 8 9 10 N

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 ~ ˆ ~ superlattice potential k V k + B1,2 were determined by modelling the BN superlattice as h | | i 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 K0 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 25

20 E (meV) 15

10 K Γ M K

Figure 5.3: Dispersions of the three lowest conduction bands for N = 7 along the high symmetry directions. regions closest to the K˜ corner of the reduced zone, and comes mostly from the K0 valley in regions closest to the K˜ 0 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]

D ED E ~ 0~ 0~ ~ 2Im nk vx n k n k vy nk ~ X | | | | Bn(k) = 2 (5.7) − (En0 En) n06=n −

E ~ where vx(y) is the velocity operator and En is the eigenenergy corresponding to the nk eigenstate. It may be readily determined that the K and K0 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 positive 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 multilayer graphene thus offers a promising playground for investigating the effect of strong interactions in the presence of a non-trivial quantum distance metric.

0.5 10 ) π

0 0 (3Za/4/ y p −10 −0.5

−0.5 0 0.5 p (3Za/4/π) x

Figure 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 K0 valley and have negative curvature. The Berry curvature integrated over the band is zero.

5.4 Details of Bandstructure Calculation

Now we calculate the matrix elements of the external potential Vˆ coming from the BN between Bloch states, ~k Vˆ k +~q . This is just equivalent to calculating the Fourier transform h | | i 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 respectively. 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

~ ˆ ~ ˆ k V k + ~q = V (~q) = δ~ ~ f1(~q) (5.8) h | | i k,Q

~ ~ 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

i~q.~ri i~q. ~rj ~ ˆ ~ ~ under consideration, f1(~q) (e e ). Hence, we obtain k V k + B1 = iλ, ∝ − h | | i ~ ˆ ~ ~ ~ ˆ ~ ~ ~ ~ ∗ ~ k V k + B2 = iλ, k V k + B1 + B2 = iλ, and f1( k) = f (k). h | | i h | | i − 1 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 K0 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

  ∆/2 α λ 0  Z     α∗ ∆/2 0 λ  ~ ~  − Z  m(k, Q) =   (5.9)  λ 0 ∆/2 β   Z    0 λ β∗ ∆/2 Z −

N 0 0 N N 00 00 N v ((kx−Qx)−i((ky−Qy)) v (−(kx−Qx)−i((ky−Qy )) ~ 0 where α = (−γ)(N−1) , β = (−γ)(N−1) and Q points to the K point of the hexagon, for which the distance between ~k and Q~ 0 is the minimum. A similar deﬁnition holds for Q~00 and K0 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 K0 and (0, 0, 0, 1) denotes a state on sub lattice B and valley K0. 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, K~ ) n 0 0 0   2     ∗ ∗ ~ ~ ∗   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, K~ ) n∗   5    ~ ~ 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

  iλ 0 0 0      0 iλ 0 0    n =   (5.11)  0 0 iλ 0      0 0 0 iλ

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 K0 and (0, 0, 0, 1) denotes a state on sub lattice B and valley K0.

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 bandstructure 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 quantum 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

X 2 θ(z) = eiπn τ e2πinz n∈Z θ(z + 1) = θ(z); θ(z + τ) = e−iπτ−2πizθ(z)

τ 1 θ(z) = 0 if and only if z = 2 + 2 + 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, ∈

X iπ(n+a)2τ 2πi(n+a)(z+b) θξ(z) = e e n∈Z

2πia −2πib −πiτ −2πiz θξ(z + 1) = e θξ(z); θξ(z + τ) = e e e θξ(z)

τ 1 θξ(z) = 0 if and only if z = + ξ + pτ + q, where p and q are integers. All zeros 2 2 − are simple zeros. Lastly for a = 0 and b = 0, we get back the basic theta function.

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