geometric distortions and quantum criticality in the ......the second part of the thesis...
TRANSCRIPT
Geometric Distortions and Quantum
Criticality in the Lowest Landau Level
Matteo Ippoliti
A Dissertation
Presented to the Faculty
of Princeton University
in Candidacy for the Degree
of Doctor of Philosophy
Recommended for Acceptance
by the Department of
Physics
Adviser: Ravindra N. Bhatt
September 2019
c© Copyright by Matteo Ippoliti, 2019.
All rights reserved.
Abstract
The quantum Hall effect encompasses a large variety of phases and phenomena aris-
ing from the combination of topological bands (the Landau levels), strong electron-
electron interactions, and disorder. This thesis consists of an in-depth numerical
study of some of these fascinating phenomena.
In the first part, we study the effect of geometric distortions of the underlying
electron band on fractional quantum Hall (FQH) and composite Fermi liquid (CFL)
states in the lowest Landau level (LLL). Through extensive density matrix renor-
malization group numerical simulations, we map the shape of correlations in both
types of states in the presence of band mass anisotropy. We find that the geometry
of FQH states depends on the LLL filling fraction, in agreement with a microscopic
model of flux attachment. At half filling, the system forms a gapless CFL state with
an emergent Fermi contour. We quantify its anisotropy and compare it to that of
the zero-field carriers, finding an approximate square-root relationship between the
two which is in excellent agreement with concurrent experiments on strained GaAs
quantum wells. In contrast, we find the CFL Fermi contour is very weakly affected
by other types of band distortions.
The second part of the thesis investigates quantum phase transitions in the LLL.
We focus first on the integer quantum Hall plateau transition for disordered, non-
interacting electrons. We study this transition in the presence of point impurities
which remove a fraction of the states from the Landau band without altering its
topological character. We then characterize the quasi-one dimensional limit of the
transition, which reveals a surprising interplay between topology and disorder. We
conclude with a study of quantum criticality in graphene Landau levels for clean,
interacting electrons, focusing on a critical point between an antiferromagnet and a
valence bond solid which is conjectured to exhibit ‘deconfined’ quantum criticality.
iii
Acknowledgements
First of all I want to thank my advisor, Ravin Bhatt. He has been incredibly sup-
portive throughout the years of work that led to this thesis, and I am deeply grateful
for all the time and energy he has invested into my growth as a researcher. His group
has been a great learning environment, too; I was lucky to share my time there with
postdocs Scott Geraedts and Kartiek Agarwal and fellow students Akshay Krishna
and Fan Chen. I thank Scott in particular for suggesting the project that would
subsequently lead to the entire first part of this thesis, and for his collaboration on
the material presented in Chapters 4, 5 and 6.
Many of the results in this thesis were made possible by infinite DMRG libraries1
for the quantum Hall problem developed by Michael Zaletel, Roger Mong and Frank
Pollmann. I am especially grateful to Mike for helping me along the code’s rather
steep learning curve during his time at Princeton, and for continued collaboration
afterwards. I would also like to thank Mike, Roger and Fakher Assaad for their
collaboration on the project presented in Chapter 8.
During my time at Princeton I have had the privilege of interacting with several
faculty members in the departments of Physics and Electrical Engineering. I would
like to thank Duncan Haldane for illuminating discussions and for his contribution to
the results presented in Chapter 3; Mansour Shayegan for sharing the experimental
data that motivated our initial investigation of anisotropy in the CFL and for many
subsequent discussions; Shivaji Sondhi for his collaboration on various projects that
are not part of this thesis and for his helpful comments as a second reader; Silviu
Pufu for helpful discussions; Thomas Gregor for serving as my experimental project
advisor; David Huse and Waseem Bakr for serving on my thesis committee.
I want to take this opportunity to express my gratitude to some of the people
and institutions that, in one way or another, have made this whole adventure pos-
1The underlying implementation of infinite DMRG in Python, maintained by the TenPy collab-oration, is freely available at github.com/tenpy/tenpy.
iv
sible: Vittorio Giovannetti, Rosario Fazio and Leonardo Mazza at Scuola Normale
Superiore (Pisa); Andrea Basile, Pietro Dal Borgo and Stefano De Toffol at Liceo
Scientifico Galilei (Belluno); and Francesco Lin, who was at both places and then
also at Princeton!
Finally, I want to thank my family in Italy (Massimo, Rita and Maria Antonietta),
my Princeton friends, and especially Jessica Frick, Ravioli the lizard and Gnocchi the
birb. I could not have done this without your support.
The research presented in this thesis was funded by the Department of Energy
BES grant DE-SC0002140 and by a Harold W. Dodds fellowship from the Graduate
School of Princeton University.
v
Publications Associated with this Thesis
Matteo Ippoliti, Scott D. Geraedts and R. N. Bhatt
Physical Review B 95, 201104 (Rapid Communications), May 2017
Physical Review B 96, 045145, July 2017
Physical Review B 96, 115151, September 2017
Physical Review B 97, 014205, January 2018
Matteo Ippoliti, R. N. Bhatt and F. D. M. Haldane
Physical Review B 98, 085101, August 2018
Matteo Ippoliti, Roger S. K. Mong, Fakher F. Assaad and Michael P. Zaletel
Physical Review B 98, 235108, December 2018
Matteo Ippoliti and R. N. Bhatt
arXiv:1905.13171, May 2019
vi
Presentations Associated with this Thesis
Gordon Research Conference on Correlated Electron Systems, Mount Holyoke
College (South Hadley, MA), 26 June 2016. Poster on Chapter 6.
APS March Meeting 2017 (New Orleans, LA), 16 March 2017. Presentation on
Chapter 4.
Electronic Properties of Two-Dimensional Systems (EP2DS), Penn State Uni-
versity (State College, PA), 31 July 2017. Presentation on Chapters 4 and 5.
APS March Meeting 2018 (Los Angeles, CA), 5 March 2018. Presentation on
Chapters 4 and 5.
Summer School on Emergent Properties in Quantum Materials, Cornell Uni-
versity (Ithaca, NY), 11 June 2018. Poster on Chapters 4 and 5.
Condensed Matter Theory Seminar, MIT (Boston, MA), 26 October 2018. Pre-
sentation on Chapters 3, 4 and 5.
APS March Meeting 2019 (Boston, MA), 6 and 7 March 2019. Presentations
on Chapters 3 and 8.
Quantum Fluids and Solids 2019 (Edmonton, AB), 10 August 2019. Presenta-
tion on Chapters 4 and 5 given by R. N. Bhatt.
vii
To Ravioli the lizard
and Gnocchi the birb
viii
Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Publications and Presentations Associated with this Thesis . . . . . . . . . vi
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
1 Introduction 1
1.1 Geometry of quantum Hall states . . . . . . . . . . . . . . . . . . . . 2
1.2 Disorder, localization and quantum criticality . . . . . . . . . . . . . 4
1.3 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Quantum Hall effect and Density Matrix Renormalization Group 8
2.1 Electrons in a magnetic field . . . . . . . . . . . . . . . . . . . . . . 8
2.1.1 Landau quantization . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.2 GaAs, AlAs and graphene . . . . . . . . . . . . . . . . . . . . 11
2.2 The integer quantum Hall effect . . . . . . . . . . . . . . . . . . . . 14
2.2.1 Role of topology . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.2 Role of disorder . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 The fractional quantum Hall effect . . . . . . . . . . . . . . . . . . . 18
2.3.1 The Laughlin wavefunction . . . . . . . . . . . . . . . . . . . 18
2.3.2 Other states: hierarchy and composite fermions . . . . . . . . 20
ix
2.3.3 Structure factor . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 The density matrix renormalization group . . . . . . . . . . . . . . . 24
2.4.1 Matrix product states . . . . . . . . . . . . . . . . . . . . . . 25
2.4.2 Variational optimization . . . . . . . . . . . . . . . . . . . . . 28
2.5 Infinite DMRG for the quantum Hall problem . . . . . . . . . . . . . 30
I Geometric distortions of quantum Hall states 33
3 Geometry of flux attachment in anisotropic fractional quantum Hall
states 34
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3 Guiding center structure factor of anisotropic FQH states . . . . . . . 39
3.4 Numerical method and symmetry considerations . . . . . . . . . . . 42
3.5 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5.1 First Jain sequence: ν = 1/3, 2/5, 4/9 . . . . . . . . . . . . . 47
3.5.2 Second Jain sequence: ν = 1/5 . . . . . . . . . . . . . . . . . 50
3.5.3 Absence of non-elliptical distortions . . . . . . . . . . . . . . . 51
3.6 Microscopic model of flux attachment . . . . . . . . . . . . . . . . . 53
3.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Appendices 61
3.A Effective interaction potential in real space . . . . . . . . . . . . . . 61
3.B Numerical data on ν = 2/5, 4/9 . . . . . . . . . . . . . . . . . . . . . 63
4 Band mass anisotropy in the composite Fermi liquid at ν = 1/2 66
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2 Mapping the Fermi contour . . . . . . . . . . . . . . . . . . . . . . . 70
4.3 Benchmarking the method . . . . . . . . . . . . . . . . . . . . . . . . 75
x
4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5 Distortions of the composite Fermi liquid beyond band mass
anisotropy 81
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.2 Exact results for radial distortions . . . . . . . . . . . . . . . . . . . . 83
5.3 Systems with multiple Fermi pockets . . . . . . . . . . . . . . . . . . 88
5.4 Systems with N -fold rotational symmetry . . . . . . . . . . . . . . . 96
5.4.1 Model CN -symmetric bands . . . . . . . . . . . . . . . . . . . 98
5.4.2 Anisotropic Pseudopotentials . . . . . . . . . . . . . . . . . . 100
5.4.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Appendices 109
5.A Electron and CF Fermi seas with different topologies . . . . . . . . . 109
II Quantum criticality in the lowest Landau level 111
6 Integer quantum Hall transition in a fraction of a Landau level 112
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.2 Delta-function potentials in the LLL . . . . . . . . . . . . . . . . . . 115
6.3 Hall conductance of LLL with δ-impurities . . . . . . . . . . . . . . . 117
6.3.1 Slater determinant wavefunction . . . . . . . . . . . . . . . . . 117
6.3.2 Direct computation of the Chern number . . . . . . . . . . . . 119
6.4 δ-function lattice potentials . . . . . . . . . . . . . . . . . . . . . . . 121
6.5 Numerical study of the plateau transition . . . . . . . . . . . . . . . . 125
6.5.1 Impurity-free problem . . . . . . . . . . . . . . . . . . . . . . 126
6.5.2 Square lattice of δ-impurities . . . . . . . . . . . . . . . . . . 128
xi
6.5.3 Different spatial distributions of δ functions . . . . . . . . . . 133
6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Appendices 138
6.A Kubo formula for the Hall conductance with δ-impurities . . . . . . . 138
7 Dimensional crossover of the integer quantum Hall plateau transi-
tion and disordered topological pumping 141
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7.2 Density of Chern states . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.3 Thin-torus limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.4 Dimensional crossover . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.5 Thouless conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Appendices 157
7.A Details of numerical calculation . . . . . . . . . . . . . . . . . . . . . 157
7.B Additional data on density of Chern states . . . . . . . . . . . . . . . 160
8 Deconfined quantum criticality in graphene Landau levels 163
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
8.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
8.3 Infinite DMRG simulations . . . . . . . . . . . . . . . . . . . . . . . . 171
8.3.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
8.3.2 Cylinder diagnostics of the 2D phases . . . . . . . . . . . . . . 172
8.3.3 Continuous transition . . . . . . . . . . . . . . . . . . . . . . . 173
8.3.4 Scaling dimensions . . . . . . . . . . . . . . . . . . . . . . . . 175
8.4 Sign-free Determinantal Quantum Monte Carlo . . . . . . . . . . . . 178
8.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
xii
Appendices 184
8.A Equivalent parametrizations of SU(4) anisotropies . . . . . . . . . . . 184
9 Concluding remarks 187
9.1 Geometry of quantum Hall states . . . . . . . . . . . . . . . . . . . . 187
9.2 Quantum criticality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
Bibliography 192
xiii
List of Tables
6.1 Number of disorder realizations used for each size in the impurity-free
problem, Sec. 6.5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.2 Number of disorder realizations used for each size in the problem with
a square lattice of δ-impurities, Sec. 6.5.2. . . . . . . . . . . . . . . . 129
xiv
List of Figures
1.1 Transport measurements on fractional quantum Hall systems. . . . . 3
2.1 Schematics of the Fermi sea and dispersion of GaAs, AlAs and graphene. 13
3.1 Structure factor of the ν = 1/3 state computed with iDMRG. . . . . 43
3.2 Possible distortions of the shape of equal-value contours of S(q) near
q = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3 Anisotropy of FQH states: comparison between numerical method and
exact result for Gaussian electron-electron interaction [237]. . . . . . 46
3.4 Numerical results for the anisotropy of the ν = 1/3 state. . . . . . . . 48
3.5 Coefficients for polynomial fit of data in Fig. 3.4 as a function of L for
the ν = 1/3 state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.6 Same plots as Fig. 3.4, but for the ν = 1/5 state. . . . . . . . . . . . 50
3.7 Coefficients for polynomial fit of data in Fig. 3.6 as a function of L for
the ν = 1/5 state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.8 Probing the shape of the small-q structure factor by tilting the band
mass tensor relative to the cylinder axis. . . . . . . . . . . . . . . . . 53
3.9 Orbitals for the relative motion of two electrons in the LLL with
anisotropic interaction. . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.10 Profiles of ρ3(x, y) (total probability density of “excluded orbitals” at
filling ν = 1/3) along the lines x = 0 and y = 0. . . . . . . . . . . . . 57
xv
3.11 Comparison between the anisotropy estimate σq=0.2 and the best fit to
the iDMRG data of Fig. 3.4 and 3.6. . . . . . . . . . . . . . . . . . . 58
3.A.1Real-space potential between two electrons in the LLL with anisotropic
mass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.B.1Numerical data for the states at filling ν = 2/5 and ν = 4/9 as a
function of electron mass anisotropy γ and cylinder circumference L. 64
3.B.2Coefficients of the polynomial fit of data in Fig. 3.B.1 for the states at
filling ν = 2/5 and ν = 4/9. . . . . . . . . . . . . . . . . . . . . . . . 65
4.1 Mapping the CFL Fermi contour from the guiding center structure
factor S(q). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2 Location of the Fermi contour extracted from data similar to that in
Fig. 4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3 Benchmarking the method to map the CFL Fermi contour against the
exact result for a Gaussian interaction. . . . . . . . . . . . . . . . . . 76
4.4 Composite fermion anisotropy αCF as a function of bare electron
anisotropy αF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.1 Landau levels and interacting phase diagram of a system with annular
zero-field Fermi sea. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.2 Examples of zero-field Fermi seas obtained from the kinetic energy
Eq. (5.10). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.3 Zero-field Fermi sea split into well-separated pockets leads to simply
connected CF Fermi contour. . . . . . . . . . . . . . . . . . . . . . . 89
5.4 The two lowest energies of the Hamiltonian in Eq. (5.10) in a high
magnetic field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.5 Valley polarization vs anisotropy for the ground state of the Hamilto-
nian Eq. (5.10) with Coulomb interaction at half filling. . . . . . . . . 94
xvi
5.6 Scaling of critical anisotropy with the magnetic field B. . . . . . . . . 95
5.7 Landau level mixing coefficients for the ground state of the 2-, 4- and
6-fold symmetric dispersion in a high magnetic field. . . . . . . . . . . 100
5.8 Anisotropic pseudopotentials for the effective interactions in Eq. (5.24)
and Eq. (5.27). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.9 Fermi wavevector kCF as a function of angle θ for N -fold symmetric
band anisotropy, N = 4 and 6. . . . . . . . . . . . . . . . . . . . . . . 104
5.10 CF Fermi contour anisotropy αCF for 4- and 6-fold symmetric dispersions.105
5.11 Summary of Fermi contours for the zero-field electrons and the ν = 1/2
CFs, for 2, 4 and 6-fold symmetric band distortions. . . . . . . . . . . 106
5.A.1Dispersion of Hamiltonian in Eq. (5.28) with zero-field Fermi sea made
of arbitrarily many pieces and connected CF Fermi sea. . . . . . . . . 110
6.1 Hofstadter butterfly spectrum of the δ-function square lattice potential
in the LLL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.2 Density of states of a sample with a square lattice of repulsive δ-
impurities and weak white-noise disorder in the LLL. . . . . . . . . . 126
6.3 Disorder-averaged Hall and Thouless conductances for an impurity-free
Landau level. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.4 Same plot as Fig. 6.3, but for a fraction of the LLL definied by Nφ flux
quanta and Nδ = 12Nφ δ-impurities or Nδ = 4
5Nφ. . . . . . . . . . . . . 130
6.5 Comparison between Landau levels with different concentration of δ-
impurities and fixed Nφ −Nδ = 4096. . . . . . . . . . . . . . . . . . . 132
6.6 The three distributions of δ-impurities considered in Sec. 6.5. . . . . . 134
6.7 Comparison between plateau transitions in the E = 0 subspace of the
three δ-impurity potentials represented in Fig. 6.6. . . . . . . . . . . . 135
xvii
7.1 Density of trivial and topological states in a disordered LLL under
quasi-1D scaling of system size. . . . . . . . . . . . . . . . . . . . . . 145
7.2 Sketch of a thin torus system and partitioning of the boundary angle
torus used to calculate the Chern number. . . . . . . . . . . . . . . . 148
7.3 Example of a Thouless pump cycle showing the identity between Chern
number and winding number. . . . . . . . . . . . . . . . . . . . . . . 150
7.4 Scaling of the density of Chern states with the torus aspect ratio. . . 151
7.5 Data on the Thouless conductance in a disordered LLL for 1D scaling
of the system size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.B.1Density of Chern states for rectangular tori with Lx = 10`B and vari-
able Nφ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
7.B.2Same as Fig. 7.B.1, but for Lx = 20`B instead of 10`B. . . . . . . . . 161
7.B.3Disorder-averaged Hall conductance under one-dimensional scaling . . 162
8.1 Kekule valence-bond-solid pattern on the honeycomb lattice. . . . . . 167
8.2 Schematics of two possible phase diagrams of the model in Eq. (8.1):
a DQCP and a Landau-allowed scenario. . . . . . . . . . . . . . . . . 168
8.3 MPS correlation length of the model in Eq. (8.1) obtained from nu-
merical iDMRG simulations. . . . . . . . . . . . . . . . . . . . . . . . 173
8.4 Two-parameter scaling collapse of the squared magnetization M2i on
the SO(5) line of the model in Eq. (8.1). . . . . . . . . . . . . . . . . 176
8.5 Measured scaling dimension ∆V along two cuts in parameter space for
the model in Eq. (8.1). . . . . . . . . . . . . . . . . . . . . . . . . . . 177
8.6 DQMC result for the embedding of the transverse field Ising model
into the half-filled Landau level. . . . . . . . . . . . . . . . . . . . . . 181
xviii
Chapter 1
Introduction
Almost forty years ago, transport measurements on two-dimensional electron gases
at low temperature and in a strong perpendicular magnetic field revealed a surpris-
ing quantization of the transverse (or Hall) conductance, at either integer or simple
rational multiples of a fundamental unit. This discovery, known as the quantum Hall
effect, would have profound effects on condensed matter physics in the decades to
follow.
The integer quantum Hall effect [111], discovered in 1980, was the first observation
of a topological quantization in a solid state system. Topological invariants now play
a central role in our understanding of bands in solids, and have guided theorists to
predict remarkable new phenomena – the quantum ‘anomalous’ [62] and ‘spin’ [104]
Hall effects, topological insulators and superconductors [176], etc.
The discovery of the fractional quantum Hall effect [213], in 1982, proved even
more transformative. As the first example of a strongly-correlated topological phase
of matter, it led to the introduction of ground-breaking new concepts such as any-
onic statistics and topological order – a type of quantum order encoded in long-range
entanglement, which eludes Landau’s classification based on symmetry breaking and
1
local order parameters. This redefined of our very understanding of phases of mat-
ter [228].
With the passage of time, the quantum Hall phase diagram got richer and richer,
with more and more phases being discovered in transport measurements. Fig. 1.1
shows two examples of such measurements: one from high-mobility electrons in gal-
lium arsenide (GaAs) quantum wells, and a recent one from graphene. Both reveal
a large set of integer and fractional plateaus, with a rich structure of symmetries
and self-similarities. Each plateau represents a distinct phase, with its own unique
topological signature.
However, both the integer and the fractional effects display a very rich phe-
nomenology beyond their topological nature. The focus of this thesis lies within
this type of non-topological features of quantum Hall physics, as we will now explain
in more detail.
1.1 Geometry of quantum Hall states
Much of the theory of the fractional quantum Hall effect has relied on the assumption
of rotational symmetry, or isotropy. Insofar as topological aspects are concerned, this
is assumption is innocuous: robustly quantized properties of the state cannot change
under a weak breaking of rotational symmetry (barring nematic instabilities).
However, it was recognized by Haldane [64] that imposing isotropy on the problem
hides a collective geometric degree of freedom of quantum Hall fluids. This degree of
freedom is the fluid’s ability to smoothly deform its correlations in order to minimize
its energy. In an isotropic system, the minimization is trivial due to symmetry. That
is however not the case in the absence of rotational symmetry, and in particular
when distinct geometries are at play in the problem. As an example, this may occur
2
Figure 1.1: Transport measurements on fractional quantum Hall systems. Top: lon-gitudinal and Hall resistance of a two-dimensional electron gas in a GaAs quantumwell as a function of perpendicular magnetic field, from Ref. [204] (1999). Plateausof the Hall resistance accompanied by vanishing longitudinal resistance correspondto incompressible quantum Hall phases, labeled by their filling fraction ν. Bottom:conductance of a graphene sheet in a Corbino disk geometry as a function of fill-ing fraction ν, from Ref. [124] (2019). Vanishing conductance indicates incompress-ible quantum Hall phases. Among the many features, one can see integer plateaus(ν = 1, 2, 3, 4 in GaAs, ν = 0, 1 in graphene), as well as many fractional ones, theclearest being ν = 1
3. Sequences of plateaus at ν = 1
3, 2
5, 3
7. . . and ν = 2
3, 3
5, 4
7. . .
accumulate towards ν = 12, where the system exhibits a gapless state (as evidenced by
a nonzero diagonal conductance or resistance) known as the composite Fermi liquid.
3
whenever the band mass and dielectric tensor are not proportional to each other,
which is not uncommon in semiconductors.
The statics and dynamics of this collective mode of quantum Hall fluids have
been studied intensely over the past decade, with the development of a variety of
theoretical tools. Among these are quantum field theories that relate this degree of
freedom to the order parameter of a nematic phase [137], with interesting connections
to gravity [25, 58].
Remarkably, a particular state – the composite Fermi liquid at half filling of the
lowest Landau level, visible in both panels of Fig. 1.1 as the limit point of two sym-
metric sequences of plateaus – has allowed direct experimental investigation, as well.
The first part of the thesis studies the effects of various types of geometric distortions
on fractional quantum Hall and composite Fermi liquid states.
1.2 Disorder, localization and quantum criticality
In the second part of the Thesis we study three different problems whose common
thread is quantum criticality : the framework for describing continuous phase transi-
tions based on the emergence of scale invariance and universality.
Electron localization [5] can be described within this framework. A scaling the-
ory [3] uses the conductance of a sample as the scaling variable and the sample size as a
renormalization group scale. Surprisingly, this approach predicts that non-interacting
two-dimensional electron systems are localized for arbitrarily weak disorder.
In the quantum Hall setting, however, the topologically quantized Hall conduc-
tance is incompatible with a fully localized spectrum. This introduces a critical en-
ergy where the electron localization length diverges. This corresponds to the plateau
transition: a peculiar “metal-insulator transition” where the phases on either side are
insulating, the critical point is multifractal (critically delocalized), and no true metal-
4
lic phase is present. The divergent localization length enables a scaling theory similar
to those of conventional quantum phase transitions characterized by the closing of a
gap. A review of scaling ideas in the quantum Hall regime is given in Ref. [77].
Clean (i.e., non-disordered) quantum Hall systems can exhibit many types of quan-
tum order and phase transitions between them, if the underlying two-dimensional elec-
tron gas comes with internal degeneracies. The typical example is quantum Hall ferro-
magnetism: the spin polarization of the state at unit filling that occurs for arbitrarily
weak Zeeman coupling. This is an interaction-driven spontaneous symmetry breaking
phenomenon. Similar phenomena can take place with various ‘pseudospins’ in lieu of
the physical spin, e.g. the valley index in multi-valley semiconductors. [193, 42]
Because of the absence of kinetic energy (i.e., the flatness of the Landau band),
interactions are always dominant in clean quantum Hall systems. This makes them
an ideal playground for the study of interaction-driven quantum criticality.
1.3 Structure of the thesis
In Chapter 2 we complete this introduction by giving a brief review of the integer
and fractional quantum Hall effects, matrix product states, and the density matrix
renormalization group algorithm.
The rest of the thesis is divided in two parts. Chapters 3, 4 and 5 comprise the
first part, which is devoted to geometric distortions of quantum Hall states.
In Chapter 3 we start with a study of the archetypical fractional quantum Hall
state – the Laughlin state at filling ν = 1/3 in the lowest Landau level. Using the
density matrix renormalization group method, we are able to accurately characterize
the shape of correlations in the quantum Hall fluid with anisotropic band mass and
isotropic electron-electron interactions. In a similar fashion we study incompressible
quantum Hall states at various other filling fractions, and find a nontrivial dependence
5
of the geometric response on filling that is well captured by a simple microscopic
model.
In Chapter 4 we investigate the same problem in the composite Fermi liquid state
at filling ν = 1/2. Unique among quantum Hall states, composite Fermi liquids offer
the possibility of a direct, parameter-free comparison to experimental measurements.
This is due to their emergent Fermi contour, which can be detected in transport
measurements and mapped in our numerical simulations. For the case of band mass
anisotropy, we find an approximate square-root relationship between the Fermi con-
tour anisotropy of the non-interacting zero-field electrons and that of the composite
Fermions. This is in very good agreement with concurrent experimental measure-
ments, without adjustable parameters, and is one of the main results of the thesis.
We conclude the first part by studying the response of the composite Fermi liquid
to more general band distortions in Chapter 5. We find that the composite Fermi
liquid is quite resilient to distortions beyond the elliptical ones represented by band
mass anisotropy. In particular, its Fermi contour remains connected even when the
zero-field carriers are split into distinct, well-separated Fermi pockets, and it responds
extremely weakly to distortions with more than two-fold rotational symmetry.
The second part of the thesis, comprised of Chapters 6, 7 and 8, deals with
quantum criticality in the quantum Hall regime, including (i) the disorder-driven
integer quantum Hall plateau transition for non-interacting electrons, and (ii) an
interaction-driven transition between magnetically ordered phases in a degenerate
Landau level in the absence of disorder.
In Chapter 6, we examine the plateau transition for non-interacting electrons in the
presence of an arbitrary number of point-like impurities, which pin down individual
electrons and remove them from the Landau band. We characterize the plateau
transition within the leftover band – a “fraction” of a Landau level – and find the
behavior of the Hall and diagonal conductances to be quantitatively identical to that
6
in the entire Landau level. This suggests an enhanced notion of universality for
plateau transitions in bands with the same Hall conductance, besides the universality
of the exponents.
Chapter 7 analyzes the same transition in quasi-one dimensional geometry, where
only one dimension is scaled to infinity. Doing so reveals surprising and counterintu-
itive behavior: while the system maps onto a disordered one-dimensional chain, where
Anderson localization is well established, computing the topological invariant associ-
ated to the Hall conductance (the Chern number) reveals a proliferation of topological
states. We reconcile these seemingly contradictory results by carefully analyzing the
one-dimensional limit as a disordered ‘Thouless pump’.
Finally, in Chapter 8 we consider an interaction-driven phase transition taking
place in the ‘zeroth Landau level’ of graphene, where the spin and valley degrees of
freedom of electrons give rise to several possible types of magnetic order. We focus in
particular on a transition between an antiferromagnet and a continuum version of a
valence bond solid. This transition has been conjectured as an example of ‘deconfined’
quantum criticality – loosely speaking a Landau-forbidden continuous transition be-
tween different ordered phases that are dual to each other. We find indications that
the transition is either continuous or very weakly discontinuous, in line with existing
results on lattice models. While our results are not conclusive, this serves as a proof
of principle for how Landau levels with internal degeneracy can be used to regularize
critical points in two-plus-one dimensions.
We conclude by summarizing our results and outlining directions for future re-
search in Chapter 9.
7
Chapter 2
Quantum Hall effect and Density
Matrix Renormalization Group
This Chapter provides a brief review of concepts that are necessary to understand
the rest of this thesis. We begin by reviewing the problem of quantum-mechanical
electrons in an external magnetic field, and in the process we fix various notations
and conventions.
2.1 Electrons in a magnetic field
2.1.1 Landau quantization
The free-electron Hamiltonian in the presence of a vector potential A is
H0 =1
2m(p− eA)2 , (2.1)
where e and m are the electron charge and mass respectively. Planck’s constant ~ is
set to 1. The vector potential A is such that ∇×A = B, where B is the magnetic
field. In this thesis we will focus on two-dimensional systems in the presence of
8
a perpendicular magnetic field. We therefore fix B = Bz, and consider canonical
position and momentum operators in two dimensions: r = (x, y) and p = (px, py).
Cross products, where used, are understood to yield a number (instead of the usual
vector): a × b ≡ εijaibj, where ε is the Levi-Civita symbol in two dimensions, ε01 =
−ε10 = 1, ε00 = ε11 = 0.
The Hamiltonian in Eq. (2.1) is expressed in terms of the kinetic momentum,
π ≡ p− eA. The components of π have a nontrivial commutator:
[πa, πb] = ie(∂aAb − ∂bAa) = ieBεab ≡ i`−2B εab , (2.2)
where we have introduced the magnetic length `B ≡ 1/√eB. This canonical com-
mutation relation leads to an uncertainty relation between components of the kinetic
momentum: the simultaneous measurement of πx and πy cannot be arbitrarily pre-
cise, making momentum space “fuzzy”. The same is true of real space: one can define
guiding center coordinates
Ra ≡ ra + `2Bε
abπb , (2.3)
which obey
[Ra, πb] = 0 , [Ra, Rb] = −i`2Bε
ab . (2.4)
It is also convenient to define the cyclotron coordinates ξa ≡ −`2Bε
abπb, so that the
position operator decomposes into ra = Ra + ξa. This has a clear intuitive interpre-
tation: much like in the classical problem, electrons move in cyclotron orbits with
a center and a radius; the position of the center is irrelevant (in the absence of a
real-space potential), hence Ra does not appear in the Hamiltonian in Eq. (2.1); the
radius on the other hand is directly tied to the velocity and thus to the kinetic en-
ergy. In contrast to the classical problem, though, quantum mechanics makes both
the guiding center and cyclotron coordinates fuzzy on a scale of `B.
9
The Hamiltonian in Eq. (2.1) can be solved by the substitution
ξx 7→ `Ba− a†
i√
2, ξy 7→ `B
a+ a†√2
, Rx 7→ `Bb+ b†√
2, Ry 7→ `B
b− b†
i√
2, (2.5)
where a, a† and b, b† are canonical bosonic variables, [a, a†] = [b, b†] = 1, [a, b] =
[a, b†] = 0. The Hamiltonian becomes that of a harmonic oscillator,
H0 = ωc
(a†a+
1
2
), (2.6)
where ωc ≡ 1/m`2B = eB/m is the cyclotron frequency. The solution to the eigen-
problem is
EN,k = ωc(N + 1/2) , |N, k〉 ≡ (a†)N(b†)k√N !k!
|0〉 . (2.7)
The energy EN,k only depends on N ; thus, each energy level is macroscopically degen-
erate. The degeneracy, parametrized by k, arises from all the possible positions of the
guiding center. As a consequence of the commutation relations, each state occupies
an irreducible area O(`2B) in the guiding center space, which limits the degeneracy on
finite systems. On a manifold with finite area A, the degeneracy is
g =A
2π`2B
=AB2π/e
=Φ
Φ0
≡ Nφ . (2.8)
We have introduced the magnetic flux piercing the system, Φ = BA, and the quantum
of magnetic flux Φ0 ≡ 2π/e. The degeneracy is therefore equal to the number of
magnetic flux quanta through the system, Nφ. This must be an integer for a compact
manifold due to Dirac’s monopole quantization condition [35].
The Nφ-fold degenerate eigenspace at E = ωc(N+1/2) is known as the N th Landau
level (LL). While the discussion up to this point has been gauge-independent, it is
convenient to set the system on a cylinder, y ≡ y + L, and use the Landau gauge
A = Bxy. This allows one to write explicit forms for the orbitals representing
10
eigenstates |N, k〉 above in terms of Hermite polynomials. We will in particular focus
on N = 0, i.e. the lowest Landau level (LLL):
ψ0,n(x, y) =1√π1/2L
eikny exp
(− 1
2`2B
(x− kn`2B)2
), kn =
2π
Ln . (2.9)
2.1.2 GaAs, AlAs and graphene
The above discussion has assumed a quadratically dispersing, isotropic Hamiltonian,
which is appropriate for the case for 2D electron gases in gallium arsenide (GaAs)
quantum wells. Here we consider two more cases of interest: aluminum arsenide
(AlAs), which (in the two-dimensional situation relevant for the QH regime) has two
anisotropic, quadratically dispersing valleys, and graphene, which has two isotropic
but linearly dispersing valleys. The Fermi contours and dispersions of these materials
is summarized in Fig. 2.1.
We start with AlAs. Its two valleys can be described by the kinetic energy
H0,valley i =1
2m
(αip
2x + α−1
i p2y
), (2.10)
where i = ± labels the valleys and αi parametrizes each valley’s anisotropy. Sym-
metry dictates that α+ = α−1− . AlAs has α+ ≈ 2.3 (i.e. the mass anisotropy is
myy,+/mxx,+ = α2+ ≈ 5.5) [193]. In a high magnetic field, one could apply Eq. (2.5),
obtain the Hamiltonian in the basis of GaAs (isotropic) Landau levels, and solve
the eigenproblem. A simpler and more insightful solution is to instead define the
“squeezed” operators
aα ≡α1/2πx + iα−1/2πy√
2`B, a†α ≡
α1/2πx − iα−1/2πy√2`B
, (2.11)
11
which satisfy the bosonic commutation relation for every α > 0. Substituting into
Eq. (2.10) with the value of α appropriate for each valley yields
H0,valley i = ωc
(a†αiaαi +
1
2
), (2.12)
which has the same spectrum as the isotropic case, EN = ωc(N + 1/2). The eigen-
states in valleys i = ± are related to the ones in GaAs by the linear transformation
(x, y) 7→ (α1/2i x, α
−1/2i y). Notice that the cyclotron frequency ωc here is defined via
the geometric mean of the masses, ωc = eB(mxxmyy)−1/2, which is the same for
both valleys. Each Landau energy EN is thus two-fold degenerate due to the ‘valley
pseudospin’ i = ±. Symmetry-breaking perturbations, such as strain, act as ‘valley
Zeeman’ fields and lift the degeneracy. We will study multi-valley models similar to
AlAs in Chapter 5.
In graphene, the honeycomb lattice structure gives rise to two bands which touch
at Dirac points – isolated points in the Brillouin zone where the low-energy dispersion
is linear, resembling the massless dispersion of a relativistic particle: E ∼ ±v|δk|,
where v is the Fermi velocity. The Dirac points are situated at the six vertices of the
hexagonal Brillouin zone; two of them, which we call K and K ′, are not equivalent
under reciprocal lattice translations (see Fig. 2.1). For now let us focus on one such
Dirac point, say K. At small momenta, one can linearize the dispersion to get
H0,valley K = vp · σ = v
0 px − ipy
px + ipy 0
, (2.13)
where the internal index represents a sublattice pseudospin σ and v is the Fermi
velocity. Switching on the perpendicular magnetic field and using Eq. (2.5) one
12
GaAs AlAs graphene
k y
kx
E
kx
Figure 2.1: Schematics of the Fermi sea (top) and dispersion (bottom) of GaAs,AlAs and graphene. Fermi pockets of the same color are related by reciprocal latticetranslations; the “Fermi pockets” of graphene are points (enlarged for visual clarity).The Landau level energies for each dispersion are represented by gray horizontalsegments. Landau levels of AlAs and graphene have a two-fold valley degeneracy.Graphene’s LL spectrum is particle-hole symmetric and features a zero-energy LL.
obtains
H0,valley K =√
2v
`B(aσ+ + a†σ−) , (2.14)
whose eigenvalues are easily found e.g. by observing that the square of H is diagonal.
One gets a particle-hole symmetric spectrum,
En =v
`B
√2|n|sign(n) , (2.15)
with eigenvalues stretching all the way to positive and negative infinity, with exact
particle-hole symmetry (the large negative energies are an artifact of linearizing the
13
dispersion). The spectrum features a non-degenerate zeroth Landau level (ZLL) at
E0 = 0 which shares many features with the LLL in GaAs. The eigenfunction for
this level is |ZLL〉 = (0, |0〉)T , which is none other than the LLL eigenfunction |0〉
completely polarized onto one sub-lattice pseudospin component.
In order to account for both valleys K and K ′, we must introduce an additional
pseudospin τ . The dispersion is then described by [251]
H0 = v(pxσxτ 0 + pyσ
yτ z) , (2.16)
which accounts for the opposite chirality of the two valleys. As a result the ZLL
is two-fold degenerate, with (0, |0〉, 0, 0)T and (0, 0, |0〉, 0)T both in the kernel of H0.
Notice that the ZLLs of valleys K and K ′ are polarized into distinct sub-lattices. In
other words, after projecting onto the ZLL, the valley and sub-lattice pseudospins
merge into one. Inter-valley scattering, induced e.g. by disorder, would introduce
off-diagonal terms ∝ τx, τ y and lift the degeneracy.
Including the physical electron spin enhances the multiplicity of the ZLL to N = 4.
At charge neutrality, the system is at filling ν = 2 (out of 4) of the ZLL and can
exhibit various types of magnetic order. For example, the LLs (K, ↑) and (K ′, ↑)
give a ferromagnet; (K, ↑) and (K ′, ↓) give a Neel antiferromagnet (as the two sub-
lattices have staggered electron spin); (K, ↑) and (K, ↓) give a valley-polarized state,
equivalent to a charge density wave (as the electrons only occupy one sub-lattice
only). Other types of order are possible, as well. In Chapter 8 we study a phase
transition in this model.
2.2 The integer quantum Hall effect
Consider a 2D electron gas. Its response to an applied voltage is a longitudinal
current, E = ρJ, ρab ∝ δab. However, in the presence of a perpendicular magnetic
14
field B, one also gets a transverse current, due to the Lorentz force acting on charge
carriers: a simple Drude model gives ρxy ' Bnq
, where n is the density of carriers and
q is their charge. This is the classical Hall effect, discovered in 1879 [67] and still
widely used as a tool to characterize charge carriers in semiconductors.
It came as a huge surprise when, in 1980, the Hall resistance of a 2D electron
gas at low temperature T . 1.5K was found to be sharply quantized, rather than
varying linearly with B. [111]. The quantization is best expressed in terms of the Hall
conductance,
σxy =e2
hn, n ∈ N , (2.17)
and it is so astonishingly precise that as of 2019 it is being used to define the unit of
resistance in the international system of units [217]. This phenomenon is the integer
quantum Hall (IQH) effect. That such a precise quantization is found in a solid state
system, and independent of details such as the purity of the sample or the device
geometry, is extremely remarkable. This discovery has spurred enormous progress in
theoretical condensed matter physics in the decades to follow.
2.2.1 Role of topology
The first key ingredient to solve the puzzle came when Laughlin [118] put forward
an argument relying on topology and gauge invariance. The argument relies on a
finite cylinder geometry (equivalent to a Corbino disk geometry), which has a non-
contractible loop. By threading magnetic flux through the loop, the electron orbitals
in the system undergo spectral flow, moving from one edge of the cylinder to the
other (or from the inner to the outer edge of the Corbino disk). Upon threading
an exact quantum of magnetic flux, gauge invariance mandates that the bulk of the
system be back to its initial state; the spectral flow must then result in the transport
of an integer number of electrons from one edge to the other. The Hall conductance
15
σxy = ne2/h is then transparently interpreted as σxy = ne/Φ0, the number of electron
charges transported per quantum of flux Φ0 = h/e.
The role of topology in the quantization was further illuminated by Thouless, [211,
210] who explicitly connected the integer (h/e2)σxy to a topological invariant known
as the Chern class, or number, C:
C =1
2πi
∫λ∈T 2
d2λεab〈∂λaψ|∂λbψ〉 . (2.18)
C is an invariant associated to the mapping λ 7→ |ψ(λ)〉 from a two-dimensional torus
T 2, where the parameters λ = (λ1, λ2) live, into the Hilbert space. The parameters
λ may take on different meanings, including:
1. A 2D quasi-momentum k, in which case |ψ(k)〉 represents a band of Bloch
eigenstates, as in the original case studied in Ref. [211];
2. A pair of generalized boundary conditions θ = (θx, θy) for a system on a torus [7],
which we will use in Chapter 6;
3. A pair (k, t), where k is a 1D quasi-momentum and t represents time in a
periodic drive. This case gives the quantized transport in a one-dimensional
Thouless pump [210], and will be central in Chapter 7.
The connection between quantized Hall effect and topological invariants has marked
the beginning of a revolution in condensed matter physics which has had deep con-
sequences for our understanding of phases of matter.
2.2.2 Role of disorder
While the quantization in the IQH is ultimately due to the topological invariant dis-
cussed above, this does not fully explain the observation of sharp transitions between
16
plateaus. As the magnetic field is varied, the filling of the valence LL changes con-
tinuously; why then should σxy jump from a quantized value to another at a specific
value of B, rather than change smoothly?
Some useful intuition came from early work by Prange [169], who studied the effect
of a single point impurity on the Hall conductance. Surprisingly, while an individual
Landau orbital is bound at the impurity, this does not lower the overall conductance
of the LLL: the other electron states compensate for the missing conductance of the
bound state, and the quantization is maintained. This suggested that the disconti-
nuities between conductance plateaus could be explained by the presence of disorder.
Disorder broadens the LLs in energy; if all states away from the band center are
localized, and we assume they do not contribute to the Hall conductance, then all
the states contributing to σxy must lie in a narrow energy window around the band
center, giving sharp transitions between plateaus.
The intuition is indeed correct, and can be refined into a scaling theory of the
localization problem in the LL [77]. The key idea is to consider the localization
length ξ(E) of electrons in the presence of disorder (projected into the LL). This
would be everywhere finite in a regular Anderson problem, but the magnetic field
and the ensuing topological constraint on σxy cause ξ to diverge at a critical energy,
ξ(E) ∼ 1
|E − Ec|ν. (2.19)
Current-carrying states must have ξ(E) & L (L being the linear size of the system),
which gives an energy window of width δE ∼ L−1/ν . This vanishes in the thermody-
namic limit, explaining the observation of sharp plateau transitions.
The accurate determination of the critical exponent ν and of the effective theory at
the critical point remain open problems. Numerics based on single-particle models [79,
197, 252, 175] gives ν ∼ 2.3− 2.6. The lower end of this interval is in agreement with
17
experiment [226, 41, 125] (ν ' 2.4), suggesting interactions may be irrelevant at the
critical point. The higher end of the interval, especially supported by transfer matrix
calculations on long strips of Chalker-Coddington networks [27], is incompatible with
this conclusion; the issue is not yet settled. The IQH plateau transition is the topic
of Chapters 6 and 7.
2.3 The fractional quantum Hall effect
Soon after the observation of the IQH effect, a new surprise came out of similar
measurements on high-mobility 2D electrons in GaAs-AlGaAs heterojunctions [213].
Tsui, Stormer and Gossard found a fractional conductance plateau, σxy = 13e2/h,
at magnetic fields corresponding to partial filling of the LLL: the fractional quan-
tum Hall (FQH) effect. Contrary to the IQH plateaus, this was only visible in very
clean samples, and was immediately understood to be a product of strong electron-
electron correlations. However, identifying the exact mechanism proved much more
complicated.
Laughlin’s gauge invariance argument for the IQH effect [118] seemingly only
allows integer values for (h/e2)σxy. However, a key assumption in the derivation is
the existence of a nondegenerate ground state. The proof breaks, and allows fractional
values σxy = 1me2/h, if a multiplet of m degenerate ground states is present. But why
should such a degeneracy be present in the quantum Hall system, and why is a plateau
observed at m = 3 but not m = 2?
2.3.1 The Laughlin wavefunction
A key step in addressing these questions came a few years later again from Laugh-
lin [119], who proposed the following wavefunction as a guess for the ground state at
18
filling ν = 1/m:
Ψ1/m(zi, zi) =∏i<j
(zi − zj)m∏i
e−|zi|2/4`2B . (2.20)
This is expressed in the symmetric gauge, where Landau orbitals are ψn(z) =
zne−|z|2/4`2B , and z = x+ iy is the position on the plane. Indices i, j run over electrons
in the system. For Eq. (2.20) to describe fermions, the integer m must be odd: this
immediately singles out ν = 1/3 as the first fractional state of this kind. Laughlin’s
guess was successful in the sense that the proposed wavefunction has high overlap
with the actual ground state of the Coulomb Hamiltonian, for numerically accessible
sizes. Even stronger indication that Eq. (2.20) correctly describes the Coulomb
ground state came when Haldane [61] showed that the model wavefunction is in fact
the exact ground state of a pseudopotential interaction Pm =∑
i,j P(m)i,j for m = 1,
where P(m)i,j is a projector on states where electrons i and j have relative angular
momentum m (which must be odd for fermions). One can decompose the Coulomb
Hamiltonian into such pseudopotentials, HCoul. =∑
m VmPm, and show numerically
that the energy gap above the ground state does not close as one continuously
changes the Vm values to the model interaction Vm = δm,1. Therefore the Laughlin
ground state is in the same phase as the realistic, Coulomb-interacting state.
The state in Eq. (2.20) describes a droplet of the FQH fluid on a plane, which
implicitly assumes a confining potential. This is the unique, nondegenerate ground
state in such geometry. However, placing the system on a topologically nontrivial
manifold, such as a torus, makes the ground state degenerate. This degeneracy con-
stitutes a loophole in Laughlin’s gauge invariance argument for the quantization of
σxy, which allows it to be fractional rather than integer, and is a unique feature of
topological phases of matter.
19
2.3.2 Other states: hierarchy and composite fermions
Subsequent measurements [36, 204] revealed a rich variety of fractional plateaus,
most of them not of the ν = 1/m type, as shown in Fig. 1.1. One approach to
understanding the other fractions in terms of “fundamental”, better-understood ones
goes by the name of “hierarchy” [61]. This approach views filling fractions near 1/3
as, essentially, the Laughlin ν = 1/3 state with a certain density of quasiparticles (or
quasiholes) on top. Interactions between these can then drive the formation of FQH
states of the excitations. By this mechanism one can derive new plateaus; e.g., by
taking ν = 1/3 as the “parent” state one can construct “daughter” states at ν = 2/5
and ν = 2/7 by adding quasiparticles and quasiholes, respectively. The procedure
does not stop at this level. Adding excitations on top of derived states gives rise to
yet more states. Quantitatively this is described by a continued fraction expansion,
and predicts an infinite hierarchy of states (hence the name). Not all of these states
should be expected to be energetically favorable; this will depend on the details of
the interaction.
A different view, which is less rooted in the energetics of the states but has been
extremely successful in predicting filling fractions for FQH states, is that of composite
fermions (CFs) [92, 93]. This picture considers Eq. (2.20) and decomposes it as
Ψ1/m(zi, zi) = Ψ1(zi, zi)∏i<j
(zi − zj)m−1 . (2.21)
The above is an IQH wavefunction where additionally each electron i sees an an even
number (m−1) of “vortex” factors attached to every other electron j. This motivates
the notion of flux attachment : the idea that the FQH problem can be described in
terms of objects (the CFs) made by “gluing” an even number of magnetic flux tubes
to an electron. These objects then see a reduced magnetic field and can then form
IQH states, if the filling is right. Their emergent Landau levels are sometimes called
20
“Λ-levels”. As a function of the number 2k of fluxes attached to each electron and
the number p of filled Λ-levels, one predicts a FQH effect of electrons at a fraction
νp =p
2kp+ 1. (2.22)
The simplest case is that of k = 1, which gives a sequence of fractions νp = p/(2p+1)
that includes Laughlin’s ν = 1/3 state and correctly describes the accumulation of
fractions near ν = 1/2 observed in experiment [36], and clearly visible in Fig. 1.1.
Perhaps most strikingly, the CF picture predicts the possibility of gapless, Fermi
liquid-like states at even denominators ν = 12k
= limp→∞p
2kp+1. These fractions
are accumulation points for incompressible FQH states with progressively smaller
gaps, as shown in Fig. 1.1. Equivalently, once flux attachment is performed, no
magnetic flux is left, so there is nothing preventing the CFs from forming a Fermi
sea. This state was described with a field theoretic approach by Halperin, Lee and
Read (HLR) [69], and the emergent Fermi contour of CFs has been experimentally
measured in GaAs quantum wells at ν = 1/2 [103]. Recently this state has again
attracted much attention due to the proposal by Son [199] that the CFs may be Dirac
particles. Among other things, this theory predicts exact particle-hole symmetry
(which is not explicit in the HLR theory) and a cone singularity at the center of the
Fermi sea carrying a nontrivial Berry phase. The CFL state at filling ν = 1/2 will be
the subject of Chapters 4 and 5.
2.3.3 Structure factor
A quantity that will be used extensively throughout Chapters 3, 4 and 5 is the guiding
center structure factor,
S(q) =1
Ne
〈δρqδρ−q〉 , (2.23)
21
where Ne is the number of electrons and ρ is the guiding center density operator,
ρ(r) =∑i
δ(Ri − r) , ρq =∑i
eiq·Ri . (2.24)
In Eq. (2.23), δρ is the density fluctuation, which differs from the total density by a
singularity at q = 0:
δρq ≡ ρq − 〈ρq〉 = ρq − 2πνδ(q`B) . (2.25)
S(q) contains important information about the correlations in a FQH state. It fea-
tures prominently in the theory of the neutral collective excitations of incompressible
states by Girvin, MacDonald and Platzmann (GMP) [53, 54]. In incompressible FQH
states, long-range fluctuations of density are strongly suppressed. This is manifested
in the small-q behavior of S(q):
S(uq) ∼ u4 , u 1 . (2.26)
In the Laughlin wavefunction, the quartic coefficient can be determined exactly
through a mapping to the 2D one-component plasma [23, 53], which gives S(q) =
m−18q4 +O(q6) for the state at filling ν = 1/m.
The guiding center structure factor S(q) in Eq. (2.23) is essentially the LL projec-
tion of the electron structure factor s(q) ≡ 1Ne〈δρqδρ−q〉, where the operators ρ here
depend on the electron position operators ri, which include both the guiding center
and cyclotron coordinates. The state where the expectation is taken is understood as
a product between the FQH ground state (expressed purely in terms of the guiding
center variables Ri) and a projector onto a specified LL |n〉 (expressed purely in terms
of the cyclotron variables ξi). Therefore s(q) is related to the guiding center S(q)
22
defined in Eq. (2.23) as follows:
s(q) =1
Ne
〈δρqδρ−q〉 =1
Ne
∑i,j
⟨eiq·(Ri−Rj)eiq·(ξi−ξj)
⟩=
1
Ne
∑i,j
⟨eiq·(Ri−Rj)
⟩〈n|e−i`2Bq×πi |n〉〈n|ei`2Bq×πj |n〉
= |Fn,n(q)|2S(q) . (2.27)
Throughout the rest of this thesis, we are going to use the guiding center S(q) exclu-
sively. We will sometimes call it ‘structure factor’ for brevity.
The function F (q) in Eq. (2.27) is the form factor,
Fn1,n2(q) ≡ 〈n1|e−i`2Bq×π|n2〉 , (2.28)
where n1 and n2 are integers labeling the Landau levels. For the usual Landau levels
of a quadratic dispersion the form factors are given analytically by
Fn1,n2(q) =
√n2!
n1!
(qx − iqy`B√
2
)n1−n2
L(n1−n2)n2
(`2Bq
2/2)e−`2Bq
2/4 , (2.29)
where L(n1−n2)n2 is a generalized Laguerre polynomial. Eq. (2.29) assumes n1 > n2; for
n1 = n2 = n, one has Fn,n(q) = Ln(`2Bq
2/2)e−`2Bq
2/4, where Ln is an ordinary Laguerre
polynomial, whereas for n1 < n2 one can use the identity Fn1,n2(q) = Fn2,n1(−q)∗.
For a non-parabolic or anisotropic dispersion, the electrons will populate “gen-
eralized Landau levels” obtained by solving for the eigenstates of the dispersion in
a strong magnetic field. These are generally expressible as a superposition of the
parabolic, isotropic LLs: |m〉 ≡∑
n um,n|n〉, where |m〉 is a generalized LL, |n〉 is a
“traditional” one, and the um,n are LL mixing coefficients. The form factors for
23
the generalized LLs are then given as linear combinations of the ones in Eq. (2.29):
Fm1,m2(q) =∑n1,n2
um1,n1Fn1,n2(q)u∗m2,n2. (2.30)
Generalized LLs and form factors will be used in Chapter 5.
A derivation analogous to the one in Eq. (2.27) allows us to write the LL-projected
Hamiltonian as
PLLHPLL =∑q
VqPLLρqρ−qPLL =∑q
Vq|F (q)|2ρqρ−q , (2.31)
which defines the effective interaction Vq ≡ Vq|F (q)|2.
This concludes our introductory discussion of the quantum Hall effect. The rest of
this Chapter introduces a computational method which we use extensively in Chap-
ters 3, 4, 5 and 8.
2.4 The density matrix renormalization group
The density matrix renormalization group (DMRG) was introduced in 1992 by
White [230] as a computational tool capable of effectively reducing the Hilbert space
dimension of certain local one-dimensional quantum systems. It allowed the accurate
numerical study of unprecedented sizes with modest computational resources and
very high accuracy (unlike e.g. exact diagonalization, whose cost grows exponentially
in system size), and quickly became the method of choice for 1D lattice systems.
However, DMRG’s success did not straightforwardly extend beyond one dimension,
nor beyond ground state searches.
A deeper understanding of the method, capable of explaining its unusual strengths
and weaknesses, came a only a few years later, when it was realized [37] that DMRG
is equivalent to a variational optimization over matrix product states (MPSs). This
24
class of quantum states was in fact already known – they notably appear in the exact
expression for the Affleck-Kennedy-Lieb-Tasaki (AKLT) ground state [4] – but their
computational power was not fully realized until their connection to DMRG.
The name of the method is due to its original interpretation as a peculiar real-
space renormalization group (where the Hilbert space basis, rather than real-space
degrees of freedom, was iteratively decimated). However the modern understanding
of DMRG is fully embedded in the more general field of tensor networks [196]. This
has a firm theoretical foundation in quantum information theory, and encompasses
many related but different computational methods, notably including PEPS [215] for
2D systems and MERA [216] for critical states.
2.4.1 Matrix product states
A pure state of a 1D spin chain of length L can be written as
|ψ〉 =∑s1=0,1
· · ·∑sL=0,1
cs1,...sL|s1, . . . sL〉 , (2.32)
where c encodes 2L independent complex coefficients – an enormous amount of infor-
mation, even for modestly sized systems. This is the crucial reason why the quantum
many-body problem is hard. However, for some states, including the ground states
of short-ranged, gapped 1D Hamiltonians, this amount of information is hugely re-
dundant, as we will see shortly.
By performing a sequence of singular-value decompositions, one can always rewrite
Eq. (2.32) as
|ψ〉 =∑s
∑a
(A1)aL,a1s1(A2)a1,a2s2
· · · (AL−1)aL−2aL−1sL−1
(AL)aL−1,aLsL
|s〉 , (2.33)
25
where s = (s1, . . . sL) are physical indices (s = 0 for spin up, 1 for spin down),
a = (a1, . . . aL) are internal indices, and the single rank-L tensor c in Eq. (2.32) has
been replaced by the trace of a product of matrices: cs → Tr[(A1)s1 · · · (AL)sL ]. We
call the range of the internal index ai at a certain bond i the bond dimension, χi. The
bond dimension in Eq. (2.33) is χ0 ≡ χL = 1 at the edges, and grows exponentially
towards the center, where it reaches χL/2 = 2L/2.
Crucially, the bond dimension χ has a physical interpretation, related to quantum
entanglement. A product state (which has no entanglement) can trivially be written
as a χi ≡ 1 MPS. A randomly-generated pure state, which typically has extensive
entanglement, requires the full χ = 2L/2 as described above. But between these
two extremes lies an important class of states. Ground states of local, short-ranged,
gapped Hamiltonians in d dimensions have “area-law” entanglement, i.e. the entan-
glement entropy of a subsystem of linear size r in those states is proportional to rd−1,
rather than to rd as is the case for random states (‘volume law’). [40] In dimension
d = 1 this implies a constant, rather than extensive, entanglement entropy. There-
fore one can take Eq. (2.33) and truncate the bond dimension to some fixed χ, while
retaining as much entanglement as possible given the constraint (this corresponds to
keeping the largest Schmidt values, or lowest ‘entanglement energies’, at each bond).
The resulting expression encodes the state with high accuracy in fewer than 2Lχ2
parameters.
MPSs have exponentially decaying correlation functions (as is expected for gapped
ground states). This is best seen by introducing the transfer matrix,
T(a1,b1),(a2,b2) ≡∑s
(A)a1,a2s (A∗)b1,b2s , (2.34)
where the four indices have been grouped into pairs to view T as a matrix. Its
eigendecomposition is T =∑χ2−1
n=0 λn|λRn )(λLn |, where |λL) and |λR) denote left and
26
right eigenvectors (T is generally not Hermitian). We use |·) to denote vectors with
internal indices, to differentiate them from states with physical indices denoted by
|·〉. It can be shown that |λn| ≤ 1 and that normalization implies |λ0| = 1 in the
thermodynamic limit (we assume the |λn| are sorted from highest to lowest).
One can express any two-point correlator in terms of T and its eigendecompo-
sition. For simplicity, let us consider a single-site operator O and a translation-
ally invariant MPS on a periodic chain, specified by matrices A0 and A1: |ψ〉 =∑s Tr(As1 · · ·AsL)|s〉. The two-point function for operators at sites i and i+ r is
〈O(i)O(i+ r)〉 =∑s,s′
Tr(A∗s1 · · ·A
∗sL
)Tr(As′1 · · ·As′L
)Osi,s′i
Osi+r,s′i+r
∏j 6=i,i+r
δsj ,s′j
= Tr(TL−r−1TOTr−1TO)
=∑n,n′
λL−r−1n λr−1
n′ (λLn |TO|λRn′)(λLn′ |TO|λRn ) , (2.35)
where (TO)(a1,b1),(a2,b2) ≡∑
s,s′(A∗)a1,a2s Os,s′(A)b1,b2s′ and we have expanded the transfer
matrix in its eigenbasis in the last line. Taking the thermodynamic limit L → ∞ at
fixed r selects the n = 0 eigenvalue, |λ0| = 1, and leaves a sum of terms that are either
constant or exponentially decaying in r. The slowest decay is given by the n′ = 1
term in the sum:
〈O(i)O(i+ r)〉 ∼∣∣∣∣λ1
λ0
∣∣∣∣r = e−r log |λ0/λ1| , r 1 . (2.36)
This defines the MPS correlation length
ξMPS ≡1
log |λ0/λ1|. (2.37)
This quantity will be used in “finite-entanglement scaling” studies of critical states
in Chapters 4 and 8.
27
2.4.2 Variational optimization
In this Section we provide a simplified overview of DMRG. This is by necessity very
incomplete, and we refer the reader to specialized reviews for more details [186, 187,
33, 161].
DMRG is essentially a variational optimization of the ground state energy over
the class of MPSs of a given bond dimension χ. In order to set up the variational
energy functional, one needs to write the Hamiltonian in matrix product operator
(MPO) form, i.e.
H =∑s,s′
Hs,s′|s〉〈s′| , Hs,s′ ≡ Tr(Ws1,s′1· · ·WsL,s
′L) , (2.38)
where the trace is taken over internal indices with their own bond dimension D
(this describes the complexity of the Hamiltonian). An MPO by construction can
only encode finite-range or exponentially decaying interactions. Once the MPO is
constructed, one can write down the energy of an MPS as a contraction of the matrices
A, A∗ and W :
〈ψ|H|ψ〉 =∑s,s′
Tr((A∗1)s1 · · · (A∗L)sL)Tr(Ws1,s′1· · ·WsL,s
′L)Tr((A1)s′1 · · · (AL)s′L)
= Tr
[L∏n=1
(A∗nWAn)
]≡ F [A,A∗] , (2.39)
where A∗nWAn ≡∑
s,s′(A∗n)sWs,s′(An)s′ is a matrix with internal indices with bond
dimension χ2D. F in Eq. (2.39) is the cost functional to be minimized in order to
find the best approximation to the ground state, subject to the constraint 〈ψ|ψ〉 = 1.
The variational equation is
δ
δA∗nF [A,A∗] = E
δ
δA∗nN [A,A∗] , (2.40)
28
where N is the norm of the MPS (defined by taking Eq. (2.39) and replacing W by
the identity) and E is a Lagrange multiplier. Both F and N are quadratic forms,
therefore Eq. (2.40) yields a generalized linear eigenproblem. Choosing a suitable
gauge (i.e. basis in the internal indices) for the MPS, this can be turned into a
regular eigenproblem:
M(a,b),(a′,b′)s,s′ (An)a
′,b′
s′ = E(An)a,bs , (2.41)
M(a,b),(a′,b′)s,s′ = La,u,bn W u,v
s,s′Ra′,v,b′
n . (2.42)
Ln and Rn are the left and right “environments”: essentially the contraction of all
A∗WA matrices to the left and right of site n, respectively.
Starting from one end of the 1D chain, Eq. (2.41) can be solved to find the optimal
A1. The ansatz for the MPS can then be updated, and the eigensystem for A2 can
be set up. One can iterate this step and “sweep” from site 1 to L and back, lowering
the energy of the ansatz at each step. After a sufficient number of sweeps, the MPS
converges and the algorithm stops.
The dimension of the eigenproblem in Eq. (2.41) is 2χ2. Its complexity varies
depending on sparsity, but is safely bounded from above by O(χ6) (cost of full dense
diagonalization). A sweep takes time linear in L, and the number of sweeps needed for
convergence may depend strongly on the system under consideration. Convergence in
the bond dimension χ must be tested by repeating the algorithm at increasing values
of χ and ensuring results do not change significantly.
An important variant to the finite-system DMRG described above is infinite
DMRG (iDMRG) [139]. The idea is to find the ground state of a translationally-
invariant, infinite 1D chain in the form of an MPS specified by a single tensor A,
through a self-consistency condition. One can initialize A with a simple guess (e.g.
a product state), create left and right environments from the initial A∗WA, and set
29
up the eigenproblem Eq. (2.41). This can be solved for a new A, which is then used
to “grow” the environments by performing the update
La,u,b 7→ La′,u′,b′ = La,u,b(A∗)a,a
′
s W u,u′
s,s′ Ab,b′
s′ , (2.43)
and similarly for the right environment R. This can be iterated indefinitely: a new
eigenproblem is set up, the optimization is repeated, the environments on both sides
updated again, etc. The method continues until convergence is achieved, i.e. until
the environment update rule Eq. (2.43) is at a fixed point.
We conclude by noting that the single-site update outlined above has very poor
convergence properties (e.g., it can never build entanglement if initialized in a product
state), and was used purely for clarity. In practice it is common to use two sites.
Moreover, one can use a unit cell comprised of more sites and follow the same logic
to simulate systems with a nontrivial unit cell.
2.5 Infinite DMRG for the quantum Hall problem
In this thesis, we will make extensive use of an extension of iDMRG to the quantum
Hall problem developed by Zaletel, Mong and Pollmann [247, 248]. In this Section
we briefly review some of its key aspects.
First of all, in order to be amenable to DMRG, the quantum Hall problem must
be mapped onto a 1D fermionic chain (equivalent to a spin chain). This is done by
setting the system on an infinite cylinder and taking a basis of Landau orbitals as the
one given in Eq. (2.9). The x axis is taken to be along the infinite direction, while
the y axis wraps around the finite circumference (0 ≤ y < L). One can then use the
30
following basis for the many-body Hilbert space,
|s〉 ≡∏n
(c†n)sn|Ω〉 , (2.44)
where |Ω〉 is the vacuum and c†n creates an electron in Landau orbital n, localized
near x = 2πL`2Bn, in the desired LL.
The above is formally equivalent to a spin chain (up to fermionic minus signs).
Of course, this merely hides the actual two-dimensional nature of the problem, and
does not eliminate it. The spacing between sites on the chain (i.e. Landau orbitals) is
2π`2B/L, while the spatial extent of each orbital in the x direction is constant O(`B).
Therefore, even contact interactions become long-ranged as the planar limit L→∞
is approached.
The Hamiltonian is given in general by
H =∑n,m,k
Vm,kc†n+mc
†n+k−mcncn+k , (2.45)
for appropriate coefficients Vm,k dependent on the type of interaction and on the LL
projection. This is considerably more complicated than a typical spin chain Hamil-
tonian, and converting it to MPO form is highly nontrivial. The solution to this
problem is presented in Ref. [247] and involves a mapping to a finite-state machine
represented by a two-dimensional graph.
A second problem comes from the algebraic nature of the Coulomb interaction.
As we saw, MPOs can only capture exponentially decaying interactions. This issue
is avoided by considering a regularized interaction
V (r) =1
rexp
(− r2
2λ2
), (2.46)
31
where λ is a cutoff much larger than `B. In practice, even modest values λ ≈ 6`B
yield accurate results, as the crucial ingredient of FQH physics is the short-distance
repulsion taking place at inter-particle distances r . `B.
The iDMRG method allows the study of multicomponent problems (involving
e.g. several Landau levels, or internal degrees of freedom such as spin) by essentially
splitting the 1D chain into alternating sublattices: in a problem with Nl components,
or ‘layers’, mode c†µ,n (where n indicates the orbital and 0 ≤ µ < Nl the layer) is
mapped onto site nNl + µ in the MPS. This extends the unit cell and the range
of interactions and generally lowers the accuracy of the method at a fixed bond
dimension χ. Nonetheless, the complexity scales linearly in the number of layers Nl,
in contrast with the exponential scaling in exact diagonalization. Moreover, if the
multicomponent problem has any additional conserved charges, such as spin, these
can be used to make the tensors block-diagonal and considerably reduce complexity.
This makes the DMRG method especially well suited for multicomponent problems.
We consider a problem with two bands in Chapter 5 and one with internal degeneracy
in Chapter 8.
32
Part I
Geometric distortions of quantum
Hall states
33
Chapter 3
Geometry of flux attachment in
anisotropic fractional quantum
Hall states
3.1 Introduction
This Chapter and the following two investigate the response of fractional quantum
Hall and composite Fermi liquid states to geometric distortions. This topic has be-
come the focus of much research in the last decade, since Haldane’s seminal work on
the geometric description of FQH states [64]. It was well known even before then that
FQH physics does not necessarily rely on continuous rotational symmetry (isotropy):
anisotropic FQH states had received significant attention in the context of nematic,
smectic and crystalline phases [151, 44, 47, 1, 149, 233, 112], and the effects of band
mass anisotropy on certain states had also been studied before [11]. Nonetheless,
in Haldane’s framework geometry assumes a new, crucial role in explaining the in-
compressibility of quantum Hall fluids. In particular, it is associated with a genuine
variational parameter for Laughlin-type model wavefunctions in the form of an in-
34
trinsic metric – a spin 2 degree of freedom that the FQH state can use to minimize its
energy. This feature is obscured in the presence of rotational symmetry, where this
degree of freedom is fully constrained by the presence of a unique “privileged” metric
in the problem, while it becomes manifest when rotational symmetry is broken. Real-
istic effects such as band mass anisotropy, mechanical strain or in-plane components
of the magnetic field can then be used as probes of this collective degree of freedom.
On the theoretical side, a variety of tools have been developed to study this
physics. In particular, quantum field theories can be built by assuming proximity to
an isotropic-nematic critical point and considering long-distance fluctuations of the
nematic order parameter [137]. The field-theoretic point of view shows interesting
connections to gravity [24, 25, 21], and has led to the recent development of bi-metric
theory [58, 57]. Aside from field theories, other approaches include descriptions based
on anisotropic model wavefunctions [177, 12], a generalization of the Hamiltonian
theory of FQH states [150], and Haldane pseudopotentials [61] for general anisotropic
interactions [234, 235]. In parallel with these developments, important analytical and
numerical progress has been made on related problems, such as nematic instabilities
and anisotropy-driven transitions of FQH fluids [243, 244, 254, 253, 123] and the
out-of-equilibrium dynamics of geometric degrees of freedom [131].
On the experimental side, the CFL state at filling ν = 1/2 has been intensely
studied. Measurements on patterned GaAs quantum wells at filling close to ν = 1/2
can be used to map the shape of the Fermi contour [55, 100], allowing the mea-
surement of the CFL’s response to different kinds of anisotropy [97, 148, 101]; the
numerical simulation of these results is the topic of Chapters 4 and 5. However,
incompressible states, lacking a Fermi contour, do not offer this experimental route
to the intrinsic geometry. Transport measurements can sharply detect the onset of
anisotropic symmetry-broken phases [74], such as stripes [182], but they are not as
informative when it comes to anisotropic FQH states that are smoothly connected to
35
their isotropic counterpart. A comparison with theory in this case is complicated by
two facts: first, the gaps generally depend on anisotropy; secondly, and more impor-
tantly, the longitudinal conductivities involve the (non-universal) anisotropy of the
transport relaxation time in addition to that of the FQH state, while the transverse
(Hall) conductiance is a topological invariant, hence insensitive to geometry. Though
alternative ways to probe the intrinsic geometry of gapped states have been put for-
ward, e.g. by measuring acoustic wave absorption [238], at this stage our knowledge
comes almost exclusively from theory and numerics.
Most of the activity in this area has focused on the state at filling fraction ν =
1/3, where both numerical exact diagonalization and anisotropic model wavefunctions
allow significant progress. Examples include the effect of anisotropic interaction [220,
6] or band mass [236], in-plane magnetic fields [162], and spatial curvature in the form
of a position-dependent metric [98]. Being mostly limited to the ν = 1/3 state, this
body of work has left an important open question: how does a FQH state’s response
to anisotropy depend on the filling fraction ν and on the corresponding phase? In this
Chapter we address this question by performing extensive numerical simulations using
the infinite density matrix renormalization group (iDMRG) method on incompressible
states at various fillings, including different states in the first hierarchy/Jain sequence
νp = p2p+1
and the state at ν = 1/5, which belongs to the second sequence νp = p4p+1
.
The Chapter is organized as follows. In Sec. 3.2 we briefly review the model and
its Hamiltonian. Then, in Sec. 3.3, we present a detailed discussion of the guiding
center structure factor (a key quantity for our numerical method) in the absence of
rotational symmetry. Sec. 3.4 offers a review of the numerical method itself, while
results are presented in Sec. 3.5 and interpreted theoretically in terms of a minimal
microscopic model of anisotropic flux attachment in Sec. 3.6. Finally, in Sec. 3.7 we
summarize and discuss our conclusions. The results presented in this Chapter have
been published in Ref. [82].
36
3.2 Model
We consider the familiar setting for the fractional quantum Hall problem introduced
in Chapter 2: a two-dimensional electron gas in a high perpendicular magnetic field
B, in the presence of electron-electron interactions and in the absence of disorder.
The system is described by the following Hamiltonian:
H =∑i
1
2
(m−1
)abπi,aπi,b +
∑i<j
V (|ri − rj|) , (3.1)
where π = p − eA is the kinetic momentum, m−1 is the inverse mass tensor, and
V (r) is the Coulomb interaction V (r) = (εabrarb)−1/2, dependent on the dielectric
tensor εab. Indices i and j enumerate the electrons in the system (1, . . . N), while a
and b run over spatial components (x, y). We can choose our coordinates so that the
dielectric tensor is the identity (εab = δab) while the mass tensor is diagonal:
mab = m
1/α 0
0 α
. (3.2)
Here α ≡√myy/mxx is a measure of the mismatch between the metric tensors εab
and mab, which characterizes the anisotropy of the system (α = 1 corresponds to the
isotropic case). The electron kinetic energy can be rewritten as
H0 =1
2m
(απ2
x +1
απ2y
). (3.3)
We take the cyclotron gap ~ωc to be much larger than all other energy scales in the
problem, so that the dynamics can be projected into the lowest Landau level (LLL)
and mixing with higher Landau levels can be safely neglected. The Hamiltonian
37
projected in the LLL is
HLLL =∑i<j
∑q
V (q)eiq·(Ri−Rj) (3.4)
where Rj is the guiding center operator for the j-th electron and
V (q) ≡ V (q)|F0(q)|2 =2π
qexp
[−1
2`2B
(q2xα + q2
y/α)]
(3.5)
is the effective interaction, given by the Coulomb repulsion V (q) between two
anisotropic LLL orbitals whose shape is encoded in the form factor F0(q).
The goal of our investigation is to measure the intrinsic geometry of the FQH
ground state of the Hamiltonian in Eq. (3.4). Previous studies have considered
the overlap of numerically obtained ground states with anisotropic model wavefunc-
tions [236]. However, this method is subject to the assumption that anisotropic model
wavefunctions are a faithful description of the state, and becomes more complicated
for non-Laughlin states at filling ν 6= 1/m. In this work, we use a different approach
which is enabled by our numerical method. The infinite cylinder geometry allows us
to probe a continuum of wavevectors along the cylinder axis. This, in turn, gives
us a new method to access the intrinsic geometry of the state. The method, which
we thoroughly describe in Sec. 3.4, is based on the calculation of the static guiding
center structure factor. In the absence of isotropy, this quantity contains important
information about the geometry of the FQH state, which we describe in the next
Section.
38
3.3 Guiding center structure factor of anisotropic
FQH states
In Sec. 2.3.3 we introduced the static guiding center structure factor of quantum Hall
states,
S(q) =1
Ne
〈δρ(q)δρ(−q)〉 . (3.6)
A crucial property of S(q), related to the incompressibility of the state, is that its
small-q behavior is quartic [53, 54]: S(uq) ∼ u4 as u → 0. The long-wavelength
structure of S(q) encodes important information about the geometry of the state [63].
In particular, for an incompressible, translationally invariant ground state, as u→ 0
one has
S(uq) ∼ 1
4(u`B)4Γabcdqaqbqcqd , (3.7)
where the Γ tensor is
Γabcd =1
Ne
(⟨1
2Λab,Λcd
⟩−⟨Λab⟩ ⟨
Λcd⟩)
, (3.8)
Λab =1
2`2B
∑i
Rai , R
bi . (3.9)
Equations (3.7), (3.8) and (3.9) can be proved by starting from the following formula
for the structure factor, obtained by substituting the definition of δρ in Eq. (2.24):
S(q) =1
Ne
∑i,j
〈eiq·Rie−iq·Rj〉 − 〈eiq·Ri〉〈e−iq·Rj〉 . (3.10)
Expanding S(uq) in powers of u around u = 0 yields
S(uq) =∞∑k=0
uk∞∑
m,n=0
δm+n,kCm,n . (3.11)
39
The coefficients Cmn are
Cmn ≡im(−i)n
m!n!Ne
∑i,j
〈(q ·Ri)m(q ·Rj)
n〉 − 〈(q ·Ri)m〉〈(q ·Rj)
n〉 . (3.12)
All coefficients C0,m and Cm,0 vanish trivially. Coefficients C1,m and Cm,1 also vanish,
because∑
iRai = −`2
BεabPb, where P is the total momentum operator (generator of
center-of-mass translations), which annihilates the translationally invariant ground
state. Therefore the lowest-order contributions is O(u4), coming from m = n = 2:
C2,2 =`4B
4Ne
qaqbqcqd(〈ΛabΛcd〉 − 〈Λab〉〈Λcd〉) . (3.13)
This finally yields
S(uq) =(u`B)4
4Γabcdqaqbqcqd +O(u6) (3.14)
with Γabcd given by Eq. (3.8).
The symmetric matrix Λab consists of three independent Hermitian operators (Λxx,
Λxy, Λyy) which generate area-preserving diffeomorphisms of the plane: depending on
the matrix of coefficients ηab, the unitary
U(η) ≡ eiηabΛab
can be a squeezing transformation (det η < 0), a shear transformation (det η = 0) or
a rotation (det η > 0). The latter case allows for a general, coordinate-independent
notion of rotational symmetry: if there exists a metric g such that
[H, gabΛab] = 0 , (3.15)
40
then the rotation group generated by the angular momentum operator
Lz(g) = gabΛab
is a continuous symmetry of the system. The system is thus isotropic (though that
may not be manifest in a coordinate system where gab 6= δab).
In the most general case, the tensor Γ is defined by two unimodular metrics, g1
and g2, and two real coefficients κ, ξ as follows:
Γabcd = κ(gac1 gbd1 + gad1 g
bc1 ) + ξgab2 g
cd2 . (3.16)
If there exists a metric g that satisfies Eq. (3.15), then symmetry requires g1 = g2 = g.
Furthermore, since the ground state is an eigenstate of Lz(g), one has
gabΓabcd =
1
Nφ
(⟨Lz(g),Λcd/2
⟩− 〈Lz(g)〉
⟨Λcd⟩)
= 0
which forces ξ = −κ, giving
Γabcd = κ(gacgbd + gadgbc − gabgcd) . (3.17)
Under this assumption of isotropy, the quartic part of the structure factor becomes
the perfect square of a quadratic:
S(uq) ∼ κ
4(u`B)4
(|q|2g)2
,
where |q|2g ≡ gabqaqb. In general, the metric tensors g1 and g2 need not be the same,
and the long-wevelength structure factor can be an arbitrary quartic. Additional
symmetries (e.g. reflection or discrete rotations) can put restrictions on it, as will be
the case in our numerical setup.
41
3.4 Numerical method and symmetry considera-
tions
We use the infinite density matrix renormalization group (iDMRG) algorithm for
quantum Hall states [247, 248], briefly reviewed in Sec. 2.5. We parametrize the
electron mass tensor as
mab = m
e−γ 0
0 eγ
, γ ∈ R . (3.18)
This parametrization is related to Eq. (3.2) by γ = lnα, and has the advantage1 that
a π/2 rotation acts simply as γ 7→ −γ.
At any of the fillings discussed below, we compute the matrix product state (MPS)
approximation to the many-body ground state for a range of values of the cylinder
circumference L and the anisotropy parameter γ. For each ground state, we compute
the guiding center structure factor S(qx, 0), as discussed in Chapter 2. An example
for the ν = 1/3 state is shown in Fig. 3.1, where the quartic behavior S(qx, 0) ∼ q4x is
clearly visible near the origin.
Based on the discussion in the previous Section, the long-wavelength limit of the
structure factor must be a homogeneous quartic polynomial. The reflection sym-
metries qx ↔ −qx, qy ↔ −qy force the coefficients of odd-power terms to vanish,
leaving
S(uq) ∼ u4(Aq4x +Bq2
xq2y + Cq4
y) (3.19)
1There are infinitely many such parametrizations. They can all be obtained by acting on γ withan odd diffeomorphism between R and some interval around the origin.
42
6 4 2 0 2 4 6qx
0.0
0.5
1.0
1.5
S(q x
,0)
1.50 1.25 1.00 0.75 0.50 0.25 0.00 0.25 0.50log10 qx
5
4
3
2
1
0
1
log 1
0S(
q x,0
)
Figure 3.1: Top: structure factor S(qx, 0) for the ν = 1/3 state, computed numericallywith iDMRG on a cylinder with circumference L = 18`B at bond dimension χ =1024. Bottom: same quantity on a bilogarithmic scale (dots) compared to the quarticmonomial q4
x (dashed line, slope 4).
for u → 0. A more useful re-parametrization of the above form is in terms of the
variables
D ≡ 1
4lnAC , σ ≡ 1
4lnA
C, β ≡ B
2√AC− 1 , (3.20)
so that
A = e2(D+σ), B = 2e2D(1 + β), C = e2(D−σ) . (3.21)
This results in
S(uq) ∼ u4e2D((eσq2
x + e−σq2y)
2 + 2βq2xq
2y
). (3.22)
43
D combined
Figure 3.2: Effect of the terms in Eq. (3.22) on the shape of equal-value contours of theguiding center structure factor S(q) near q = 0. From left to right: D parametrizesan isotropic rescaling; σ a unimodular metric (uniaxial stretching); β a C4-symmetricdistortion. An example where all three are present is shown on the right. The dashedcircle corresponds to D = σ = β = 0, for comparison.
A π/2 rotation (C4) acts by mapping γ 7→ −γ while exchanging q2x and q2
y. It follows
that the parameters transform under C4 as
D(−γ) = D(γ), σ(−γ) = −σ(γ), β(−γ) = β(γ) . (3.23)
The physical meaning of these parameters is apparent from Eq. (3.22): D describes
a change of the overall magnitude of S(q), in an isotropic fashion; σ parametrizes
an internal unimodular metric of the FQH state that describes a uniaxial squeezing;
and β represents a possible C4-symmetric anisotropy that remains once a coordinate
transformation qx 7→ e−σ/2qx, qy 7→ eσ/2qy removes the uniaxial squeezing. Each term
is illustrated graphically in Fig. 3.2.
If we assume the absence of the C4-symmetric distortion parametrized by β, the
long-wavelength limit of the structure factor takes the particularly simple form of a
perfect square,
S(uq) ∼ u4(gabqaqb)2, gab = diag(eD+σ, eD−σ) . (3.24)
44
This assumption will be tested numerically in Sec. 3.5.3, where we will show that it
applies to the ν = 1/3 state by tilting the band mass tensor relative to the cylin-
der axis. A consequence of Eq. (3.24) is that all the information about the long-
wavelength limit of S(q), contained in the parameters D and σ, can be extracted by
measuring S along the qx direction only! This is important because our numerical
resolution in momentum space, while continuous in qx, is discretized in increments of
2π/L in qy (L is the finite cylinder circumference). We have
S(qx, 0) ' λ(γ)q4x ≡ e2(D(γ)+σ(γ))q4
x , (3.25)
and by exploiting the different parity of D(γ) and σ(γ) we obtain
σ(γ) =1
4log
λ(γ)
λ(−γ), D(γ) =
1
4log λ(γ)λ(−γ) . (3.26)
On a cylinder with finite circumference, the values of γ and −γ need not be related by
an exact duality, as the boundary conditions explicitly break C4 rotation symmetry.
In that case, Eq. (3.24) does not rigorously hold and there can be more information
in the geometry of the state which is not captured by Eq. (3.26). Therefore, care
must be taken to analyze finite-size effects appropriately.
In Fig. 3.3 we benchmark this method against the known exact result [237] for
a Gaussian electron-electron interaction with characteristic length scale s, V (r) =
e−12
(r/s)2 . This problem is actually isotropic: it is invariant under the generator
Lz(g) = gabΛab for the metric gab = diag(eσ, e−σ), with
σ =1
2log
eγ + s2/`2B
e−γ + s2/`2B
. (3.27)
We numerically compute the interacting ground states at ν = 1/3 and ν = 2/5 in a
cylinder with circumference L = 18`B for electron mass anisotropy parameter eγ = 3
45
0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8log10(s/ B)
0.0
0.2
0.4
0.6
0.8
1.0
(s)
exactnumerical, = 1/3numerical, = 2/5
Figure 3.3: Comparison between numerical method and exact result for Gaussianelectron-electron interaction. The numerical data is obtained with iDMRG on acylinder with circumference L = 18`B for filling fractions ν = 1/3 and 2/5. The bonddimension is χ = 1024. We find the ground states for γ = log 3 and use Eq. (3.26)to estimate σ. For the ν = 1/3 state the agreement is nearly perfect in the entirerange of s, whereas for the ν = 2/5 state there are small but visible deviations at thelargest values of σ (small s).
and a wide range of interaction length scales s. We find very good agreement between
our estimate of σ and the exact result, showing how in this simple case the L → ∞
formulas appear to be valid for all practical purposes.
For the rest of this Chapter, we will study genuinely anisotropic FQH states that
in some cases exhibit significant finite-circumference effects. In order to take into
account the effects of finite L, it is helpful to assume smoothness around the isotropic
point γ = 0, and Taylor expand
log(√λ) = D + σ =
∑n
cnγn . (3.28)
By construction, the even-n terms contribute to D and the odd-n ones contribute
to σ. By fitting log√λ to a polynomial at each size, it is possible to analyze the
finite-size drift of the contributions to D and σ and extrapolate their L → ∞ limit.
In all cases we present in the following, keeping only terms with n ≤ 3 is sufficient
46
to get a good fit to the data. Further, we find c2 to become consistent with 0 as
L is increased, indicating that the overall scale parameter D is compatible with a
constant. Coefficients c1 and c3 characterize the dependence of the internal metric
parameter σ on the electron anisotropy γ and are found to approach finite values as
L→∞.
3.5 Numerical results
3.5.1 First Jain sequence: ν = 1/3, 2/5, 4/9
We start from the FQH state at filling ν = 1/3, which has the largest gap and exhibits
minimal finite-size effects. We gather data for 50 equally spaced values of γ in the
range −1.2 . γ . 1.2 (corresponding to 0.3 . α . 3.3) and cylinder circumferences
14`B ≤ L ≤ 22`B. We extract λ(γ) from a fit of S(qx, 0) to λq4x, and decompose it
into the even and odd parameters D(γ), σ(γ) as explained in Eq. (3.26). These are
shown in Fig. 3.4 as a function of γ for all the values of L. As can be seen, finite-size
effects are rather small.
Following the scheme described in the previous Section, and in particular
Eq. (3.28), we fit log√λ(γ) to a cubic polynomial in γ:
log√λ(γ) = c0 + c1γ + c2γ
2 + c3γ3 . (3.29)
In the planar limit (L → ∞), coefficients c0 and c2 represent contributions to D(γ),
while c1 and c3 contribute to σ(γ). Though this is only expected to hold at infinite
L, by probing a range of sizes and doing the fit in Eq. (3.29) at each size we can
extrapolate the L → ∞ limit. We show the results in Fig. 3.5. As L increases,
c2 → 0, while c1 and c3 approach finite values. c3, in particular, seems to have a
47
1.0 0.5 0.0 0.5 1.00.6
0.4
0.2
0.0
0.2
0.4
0.6(a)
1.0 0.5 0.0 0.5 1.01.2
1.0
0.8
0.6
0.4
0.2
0.0
D
(b) L/ B141516
171819
202122
Figure 3.4: Numerical data for the ν = 1/3 state as a function of electron anisotropy γand cylinder circumference L. The data is obtained with iDMRG at bond dimensionχ = 1024, by fitting S(qx, 0) to λq4
x near the origin. (a) σ(γ), defined as the odd partof log
√λ. (b) D(γ), defined as the even part of log
√λ. σ is approximately linear,
while D approaches a constant as L increases. Finite-size effects are rather limited.
small, non-zero limit. In summary, our extrapolation to the planar L→∞ limit is
σ ' 0.43γ + 0.01γ3 , D ' −0.66 (3.30)
(notice the absence of the γ2 term in D). Even for the smallest sizes, D is constant
within ∼ 4%, while σ is approximately linear in γ. We also show the results of a
linear fit, σ = c1γ with c3 constrained to zero. This would correspond to a power-
law relationship between the band mass metric and the internal FQH state metric,
gFQH ∝ (gm)c1 . The result of this fit is c1 ' 0.445.
We repeat the same analysis on the states at filling ν = 2/5 and ν = 4/9. In
the hierarchy scheme [61], these states are derived from the ν = 1/3 state by the
addition of suitable numbers of quasielectrons. In the composite fermion picture [92,
93] they are obtained by filling to p = 2 and p = 4 “Λ levels”, respectively, with
CFs made from one electron and two magnetic flux quanta. The resulting Jain
sequence νp = p2p+1
starts from ν = 1/3 and approaches the ν = 1/2 CFL state
48
14 16 18 20 22L/ B
0.1
0.0
0.1
0.2
0.3
0.4
0.5
fit c
oeffi
cien
tsc1, linear fitc1c2c3
0.42
0.43
0.44
0.45
0.46
15 18 21L/ B
0.02
0.00
0.02
0.04
Figure 3.5: Coefficients for polynomial fit of log(√λ(γ)) as a function of γ for the
ν = 1/3 state (data in Fig. 3.4). Coefficients from a linear fit (c1) as well as the cubicfit (c1, c2, c3) are shown together. Insets: c1 and c2,3 coefficients shown separatelyfor clarity. c2 seems to approach 0 (black line), while c3 appears to oscillate arounda small but finite limit.
for p→∞. Based on these ideas, we expect the response of these states to be similar
to that of ν = 1/3. We indeed obtain similar numerical results, which are shown in
detail in Appendix 3.B. These states have smaller energy gaps, which imply longer
correlation lengths, resulting in stronger finite-size effects. Especially for the ν = 4/9
state, the oscillations of the c2 and c3 coefficients as a function of size still have a
significant amplitude at the largest size we consider, L = 22`B. This increases the
uncertainty on the estimate of c1, as well. Nonetheless, the results for both states
appear compatible with the values obtained for the ν = 1/3 state. This supports the
view that the geometric response of all these states is ultimately determined by their
common “building block”.
49
1.5 1.0 0.5 0.0 0.5 1.0 1.50.6
0.4
0.2
0.0
0.2
0.4
0.6(a)
1.5 1.0 0.5 0.0 0.5 1.0 1.50.6
0.4
0.2
0.0
0.2
0.4
0.6
D
(b) L/ B161718
192021
222324
Figure 3.6: Same plots as Fig. 3.4, but for the ν = 1/5 state. Despite strongerfinite-size effects, this state exhibits similar qualitative behavior as the ν = 1/3 state.
3.5.2 Second Jain sequence: ν = 1/5
Having found the same result for all the states derived from ν = 1/3, a natural
question to ask is whether the response to applied anisotropy changes for states that
are not derived from ν = 1/3. We consider the simplest such state, ν = 1/5. This
belongs to a different Jain sequence, where flux attachment combines each electron
to four flux quanta, rather than two: νp = p4p+1
. In the hierarchy picture, this is the
parent state of a different tree of daughter states.
In Fig. 3.6 we show data collected for the state at ν = 1/5 for cylinder circumfer-
ences L = 16`B to 24`B. Here, finite-size effects are more severe than for the ν = 1/3
state, requiring somewhat larger values of the cylinder circumference L and a wider
range of electron anisotropies γ. Again, we define D + σ = log√λ and perform
a polynomial fit of this quantity as a function of γ. The fit coefficients, shown in
Fig. 3.7, exhibit wide fluctuations with system size, which have not yet decayed at
circumference L = 24`B. Nonetheless, the value of the c1 coefficients, obtained from
either a cubic or a linear fit of the data in Fig. 3.6, is consistently below 0.4 and
appears to approach an asymptotic value between 0.25 and 0.30. We can therefore
50
16 17 18 19 20 21 22 23 24L/ B
0.1
0.0
0.1
0.2
0.3
0.4
0.5
fit c
oeffi
cien
ts
c1, linear fitc1, cubic fit
c2c3
Figure 3.7: Coefficients for polynomial fit of log(√λ(γ)) as a function of γ for the
ν = 1/5 state (data in Fig. 3.6). The black line highlights c = 0. The behavior isqualitatively the same as that of the states in the first Jain sequence, albeit withfluctuations persisting to larger sizes. Quantitatively, both estimates of c1 are consis-tently below 0.4, and appear to stabilize between 0.25 and 0.3, a much smaller valuethan the ν = 1/3 state.
conclude that, for a fixed value of the electron mass anisotropy γ, the ν = 1/5 state
is much less anisotropic than states in the first Jain/hierarchy sequence.
3.5.3 Absence of non-elliptical distortions
We conclude this Section by presenting numerical evidence that the long-wavelength
limit of S(q) is consistent with the square of a quadratic function, S(q) ∼ (gabqaqb)2,
which we have assumed in the calculations of Sec. 3.5.1 and 3.5.2. Physically, this
implies the shape of the state is entirely characterized by an emergent metric gab, and
the shape of long-wavelength correlations is elliptical. In the notation of Eq. (3.22),
this means that the coefficient β, parametrizing non-elliptical distortions, is consistent
with zero within the numerical and finite-size accuracy of our method.
51
To probe the distortions parametrized by β, we need to evaluate S(q) along lines
where neither qx nor qy vanish. This is in principle possible within the setup considered
in the main text, but the discretization of qy into multiples of 2π/L due to the finite
circumference makes the procedure rather inaccurate. To circumvent this issue, we
consider the situation in which the band mass tensor is tilted relative to cylinder axis
and circumference directions:
mab = Rθ
eγ 0
0 e−γ
R−θ (3.31)
where Rθ is a rotation by an angle θ. For θ = π/4, the general form in Eq. (3.22)
becomes
S(qx, 0) = e2D[(coshσ)2 + β/2]q4x ≡ λq4
x , (3.32)
which allows us to directly probe β, given that D and σ are known from the θ = 0
calculations.
We perform iDMRG simulations of the ν = 1/3 state on a cylinder with circumfer-
ence L = 18`B over a range of values of the anisotropy parameter γ, for mass tensors
tilted by θ = π/4 relative to the cylinder axis. The values of λ so obtained can be
compared to the prediction of Eq. (3.32) for β = 0,
√λ(γ) = eD(γ) coshσ(γ) , (3.33)
to confirm the absence of non-elliptical distortions. The data for D and σ at tilt
angle θ = 0 for the same filling and size is in Fig. 3.4; we also show log√λ = D + σ
in Fig. 3.8(a) for convenience. Fig. 3.8(b) shows iDMRG data for the λ coefficient
at tilt angle θ = π/4, along with the prediction in Eq. (3.32), where we set D(γ) to
its isotropic value D(0), neglecting small variations with γ which have been shown
52
1.5
1.0
0.5
0.0
0.5
ln
(a)
1.0 0.5 0.0 0.5 1.00.7
0.6
0.5
0.4
ln
(b) = 0 predictioniDMRG data
= 0
= /4
Figure 3.8: Probing the shape of the small-q structure factor by tilting the bandmass tensor relative to the cylinder axis. (a) Numerical data for the coefficient λ inS(qx, 0) ∼ λq4
x for the ν = 1/3 state as a function of the anisotropy parameter γ,when the electron mass tensor is aligned with the cylinder axis (ellipses shown on theright, the dashed line is the cylinder axis direction). (b) Same data for band masstensors tilted by π/4 relative to the cylinder axis (ellipses shown on the right). Thedata agrees with a form S(q) ∼ (gabqaqb)
2, obtained when the C4-symmetric distortionparameter β is set to 0. The cylinder circumference is L = 18`B and the iDMRGbond dimension is χ = 1024.
to be a finite-size effect (Sec. 3.5.1). As can be seen, the comparison reveals very
good agreement. We conclude that the distortion of S(q) is consistent with a purely
uniaxial stretching, parametrized by a unimodular metric whose anisotropy parameter
is σ.
3.6 Microscopic model of flux attachment
In order to understand the observed behavior of states at different filling fractions,
we introduce a minimal microscopic model of flux attachment. We consider two
53
electrons with anisotropic band mass mab in the lowest Landau level interacting with
each other via the Coulomb interaction V (r) = 1/r. The system is governed by an
effective interaction V (r) whose Fourier transform is
V (q) =2π
qexp
[−1
2`2B
(eγq2
x + e−γq2y
)]. (3.34)
The two-body problem can be reduced to a single-particle problem in the relative
coordinate, which describes an electron orbiting around a fixed potential V (r) gen-
erated by an anisotropic cloud of charge pinned to the origin. At filling ν = 1/m,
the “excluded region” around each electron in the incompressible many-body state is
approximated by the m highest-energy orbitals of this potential. This is in analogy
with the isotropic Laughlin wavefunction, where each factor of (zi−zj)m can be inter-
preted as arising from the exclusion of orbitals z0, · · · zm−1 in the relative coordinate
problem.
This simple model makes some unambiguous predictions. The effective interaction
V is isotropic at large distance (small q) and becomes more anisotropic at shorter
distance (large q). Therefore, the innermost orbitals in the excluded region are the
most anisotropic, and as one moves outwards the orbitals approach circular symmetry
more and more closely. Overall, the anisotropy of the composite boson is expected to
decrease with m. This is explored in more detail in Appendix 3.A, and is in qualitative
agreement with our numerical results from Sec. 3.5, where ν = 1/5 was found to be
less anisotropic than ν = 1/3.
To be more quantitative, we start from the LLL Hamiltonian in Eq. (3.4) for two
electrons,
H2−body =∑q
V (q)eiq·(R1−R2) . (3.35)
54
We define the center-of-mass and relative guiding center coordinates as
RCM ≡R1 +R2√
2, δR ≡ R1 −R2√
2, (3.36)
respectively. Notice the normalization, which is necessary to maintain the correct
canonical algebra, i.e. [δRx, δRy] = −i`2B. The center of mass coordinate has trivial
dynamics, leaving a single-particle Hamiltonian for the relative motion:
Hrel. =∑q
V (q)eiq·δR√
2 ≡ V(√
2δR). (3.37)
The factor of√
2 is important because V is not a homogeneous function. It is expected
to reduce anisotropy, as λV (q/λ) approaches the isotropic Coulomb interaction as
λ→∞.
A first approximation to the geometry of the excluded region can be obtained
by looking at equipotential contours of V (δR). Contours that enclose an area of
2π`2B(n + 1/2) should approximate the semiclassical electron trajectories. We follow
this approach in Appendix 3.A, and find that it gives the correct qualitative depen-
dence on m and α; however, the quantitative agreement is not satisfactory. This
is not surprising, as the semiclassical approach works best for large n, while we are
interested in the first few orbitals. To get a more accurate quantitative description,
one has to find the shape of the actual quantum eigenstates of Hrel..
We study this single-particle problem numerically on a torus with sides large
enough that finite-size effects are negligible (a torus with side L ' 40`B is enough
for this purpose) and numerically obtain eigenvalues Ek and eigenvectors |ψk〉 .
We then calculate the total real-space probability density of the m highest-energy
wavefunctions,
ρm(x, y) =1
m
m∑k=1
|ψk(x, y)|2 . (3.38)
55
3 0 3x/ B
6
3
0
3
6
y/B
(a) k = 1
3 0 3x/ B
k = 2
3 0 3x/ B
k = 3
0.000.010.020.030.040.050.06
|k|2
6 4 2 0 2 4 6x/ B
4
2
0
2
4
y/B
(b)
0.000.020.040.060.080.100.120.14
3(x,
y)
Figure 3.9: (a) Probability density of the three highest-energy orbitals ψk of therelative Hamiltonian in Eq. (3.37) for γ = log 3. The orbitals have been calculatednumerically on a torus with side L =
√512π`B ' 40`B, where finite-size effects are
completely negligible. (b) Probability density ρm defined in Eq. (3.38), for m = 3.The data are obtained by summing the three orbital probability densities above.
The result for α = 3 is shown in Fig. 3.9.
If V (q) is obtained by stretching a function of q2, i.e. if the problem is actually
isotropic with respect to some metric gab 6= δab (as in the case of the Gaussian electron-
electron interaction used in Fig. 3.3), then ρm(x, y) is also isotropic with respect to
the same gab. Therefore its profiles along the y = 0 and x = 0 lines, ρm(r, 0) and
ρm(0, r), should have the same shape up to a rescaling factor a:
ρm(r/a, 0) = ρm(0, ra) ∀ r ∈ R .
56
8 6 4 2 0 2 4 6 8r/ B
0.0
0.2
0.4
0.6
0.8
1.0
q = 0.2
q = 0.5
q = 0.83(r, 0)3(0, r)
Figure 3.10: Profiles of ρ3(x, y) along the lines x = 0 and y = 0, normalized so themaximum height is 1. The two shapes are clearly different and no single rescalingcan collapse them. In particular, the ratio of widths at a fraction q of the maximumheight, defined as σq in Eq. (3.39), depends significantly on q (we highlight q = 0.2,0.5 and 0.8 as examples).
In that case the parameter σ is unambiguously determined by eσ = a2. For Coulomb
interaction, though, the problem is genuinely anisotropic. We find that the two
profiles have significantly different shapes, as demonstrated in Fig. 3.10, and no such
rescaling exists. In particular, one can estimate σ from the ratio of the widths of the
probability density profiles at various fractions q of their height:
σq ≡ lnyqxq, xq, yq : ρm(xq, 0) = ρm(0, yq) = qρm(0, 0) . (3.39)
Each value 0 < q < 1 defines an estimate σq of σ. The σq estimates are found to
depend significantly on q, which might seem to make the model not predictive. This,
however, turns out not to be the case, as we now discuss.
In general, once q is picked so as to best approximate the iDMRG result at certain
values of anisotropy γ and inverse filling m, there is no guarantee that the same esti-
mate σq will work well at other values of γ and m. However, we find that the estimate
57
1.5 1.0 0.5 0.0 0.5 1.0 1.50.8
0.6
0.4
0.2
0.0
0.2
0.4
0.6
0.8
m = 3 m = 5q = 0.2
iDMRGq = 0.2
iDMRG
Figure 3.11: Comparison between the anisotropy estimate σq=0.2, defined in Eq. (3.39),and the best fit to the iDMRG data of Fig. 3.4 and 3.6 in the form σ = c1γ + c3γ
3.For m = 3 we set c1 = 0.43, c3 = 0.01. For m = 5 we set c1 = 0.28 and c3 = 0.03.The estimate σ0.2 shows good agreement with the data for both values of m, in theentire range − log 4 ≤ γ ≤ log 4.
σq=0.2, i.e. the anisotropy of the equal-probability contour of ρm from Eq. (3.38) at
20% of its maximum, reproduces very accurately the numerical results of Sec. 3.5 for
the ν = 1/3 state (m = 3) in a wide range of values of γ, as shown in Fig. 3.11. The
same quantity can then be evaluated for the ν = 1/5 state (m = 5) and compared to
the iDMRG data, and again shows fairly accurate agreement (also in Fig. 3.11). So,
with the ad hoc tuning of just one parameter, this remarkably simple model quan-
titatively reproduces the many-body data with quite good accuracy. Physically, the
optimization over q corresponds to the choice of which contour of the composite boson
informs the shape of long-distance correlations in the FQH state. This is ultimately
decided by the energetics of the many-body problem, and goes beyond the scope of
this minimal two-body model.
58
3.7 Discussion
We have numerically investigated the response of incompressible fractional quantum
Hall states to band mass anisotropy. We did so by applying a new technique based
on the long-wavelength limit of the guiding center structure factor, encoded in its
quartic coefficient. This is made possible by infinite DMRG, which enables access to
a continuum of momentum values in one direction and makes the determination of
the quartic coefficient very accurate.
We discussed the general response of a FQH state, and in particular of its long-
wavelength structure factor, to applied anisotropy. With the symmetries of our
infinite-cylinder geometry, this reduces to three parameters: D, describing an isotropic
change in the magnitude of S(q); σ, corresponding to a uniaxial squeezing (or uni-
modular metric); and β, which describes any residual C4-symmetric distortion when
the uniaxial squeezing is eliminated through a coordinate transformation. We find,
within numerical and finite-size accuracy, that β vanishes and D is a constant, leav-
ing only the unimodular metric (parametrized by σ) to characterize the geometric
response of the FQH state to band mass anisotropy.
We found that states in the first Jain/hierarchy sequence appear to share the
same response to band mass anisotropy, consistent with σ ' 0.43γ to first order near
the isotropic point. The response of the second Jain sequence, on the other hand, is
found to be radically different. The only state we could study in this case is ν = 1/5,
due to stronger finite-size effects. We find σ ≈ 0.28γ near the isotropic point, a value
which is clearly different from that of the first Jain sequence, even when the larger
uncertainty is taken into account.
Our numerical findings are consistent with a minimal microscopic model of flux
attachment, where the geometry of the quantum Hall state is fixed by the shape of
the composite boson formed by attaching m “excluded orbitals” to each electron. We
compute these orbitals in the relative motion problem of two electrons in the lowest
59
Landau level and find that the shape of the composite boson correctly reproduces our
numerical results as a function of electron anisotropy and filling (which is represented
by the number of “excluded orbitals” surrounding each electron).
This microscopic model, as formulated here, only applies to Laughlin-like “prim-
itive states” at filling ν = 1/m, and does not directly explain the observed behavior
of descendants such as ν = 2/5. Our numerical results on the sequence of states at
filling ν = p2p+1
find compatible geometric response. This would arise naturally in
a composite fermion picture: if all states in the sequence are interpreted as integer
quantum Hall states of the same anisotropic composite fermion, then clearly their
geometry should be the same. Our data on states at fillings ν = 1/3, 2/5 and 4/9 do
not show evidence of a drift of the exponent; however, a slow drift cannot be ruled
out, since the numerical uncertainties grow as one moves along the sequence. This
raises an interesting question about the ν = 1/2 CFL state. As the limiting point
of the first Jain sequence, one may reasonably expect it to show the same response
as the ν = 1/3 state. At the same time, it is a qualitatively different state – it is
compressible, it exhibits a Fermi contour, etc. For these reasons, and in light of its
unique connection to experiment, its response to band mass anisotropy is especially
interesting, and will be investigated in the next Chapter.
60
Appendices
3.A Effective interaction potential in real space
In this Appendix we derive a formula for the real-space effective interaction between
two electrons in the LLL with anisotropic mass and use it to obtain a first, semi-
classical approximation to the problem of anisotropic flux attachment. This informs
the discussion in Sec. 3.6.
We start from the effective potential for the relative coordinate problem as com-
puted in Eq. (3.37),
V (q/√
2) =23/2π
qexp
[−1
4`2B(eγq2
x + e−γq2y)
]. (3.40)
Its Fourier transform is
V (x, y) =
∫dqxdqy
1√2πq
e−14`2B(eγq2x+e−γq2y)eiqxx+iqyy . (3.41)
61
We do the integral in polar coordinates q, θ and introduce shorthand notations c ≡
cos θ, s ≡ sin θ:
V (x, y) =√
2
∫ 2π
0
dθ
2π
∫ ∞0
dq e−`2Bq
2
4(eγc2+e−γs2)−iq(cx+sy)
=√
2
∫ π
0
dθ
2π
∫ +∞
−∞dq e−
`2Bq2
4(eγc2+e−γs2)−iq(cx+sy)
=
∫ π
0
dθ
π
√2π`−2
B
eγc2 + e−γs2exp
(−`−2
B
(cx+ sy)2
eγc2 + e−γs2
). (3.42)
Simple manipulations and a change of integration variable θ 7→ 2θ lead to the expres-
sion
V (x, y) =1
`B
∫ 2π
0
dθ
2π
√2π
cosh γ + cos θ sinh γ
× exp
(− 1
2`2B
(x2 + y2) + (x2 − y2) cos θ + 2xy sin θ
cosh γ + cos θ sinh γ
). (3.43)
The integral can be done analytically in the isotropic case γ = 0, but must be evalu-
ated numerically for γ 6= 0.
We show a plot of V (x, y) for γ = log 2 in Fig. 3.A.1, alongside equipotential
contours that enclose areas 2π`2B(n + 1/2), which represent an approximation of the
semiclassical trajectories. More accurately, the semiclassical orbitals should be con-
tained between the contours enclosing areas 2π`2Bn and 2π`2
B(n+1), with n = 0 giving
a disk and n > 0 giving annuli [66]. The n-th annulus can be though of as a fattened
version of the orbit enclosing area 2π`2B(n+ 1/2), which is the one we consider.
A first approximation of the anisotropy of the quantum Hall state σ is then given
by taking the ratio of the semiaxes of those contours. The result underestimates |σ|,
i.e. suggests a less anisotropic state, relative to the numerical results of Sec. 3.5.
Nonetheless, it shows qualitatively correct behavior, with the shape of the composite
boson becoming less anisotropic as m (the number of attached fluxes) increases.
62
4 2 0 2 4x/ B
4
2
0
2
4
y/B
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
V(x,
y)
Figure 3.A.1: Color plot of the real-space potential in Eq. (3.43) for γ = log 2. Thedashed lines are equipotential contours enclosing areas 2π`2
B(n+ 1/2), for n = 0 to 4.
A more accurate approximation is given by solving the quantum mechanical prob-
lem of a LLL electron moving the potential V (r) and describing the shape of the actual
excluded orbitals, rather than their semiclassical approximation. This approach in
described in Sec. 3.6.
3.B Numerical data on ν = 2/5, 4/9
Here we show additional numerical data on other states in the first Jain sequence
beyond ν = 1/3, namely ν = 2/5 and ν = 4/9. As the filling fraction approaches
ν = 1/2 along the sequence ν = p2p+1
, the gap of the associated incompressible state
becomes smaller and its correlation length become longer. This implies stronger
finite-size effects compared to the ν = 1/3 state.
63
0.3
0.2
0.1
0.0
0.1
0.2
0.3(a)
0.6
0.5
0.4
0.3
0.2
0.1
0.0
D
(b)
0.75 0.50 0.25 0.00 0.25 0.50 0.750.3
0.2
0.1
0.0
0.1
0.2
0.3(c)
0.75 0.50 0.25 0.00 0.25 0.50 0.750.6
0.5
0.4
0.3
0.2
0.1
0.0
D
(d) L/ B15161718
192021
Figure 3.B.1: Numerical data for the states at filling ν = 2/5 (panels a, b) andν = 4/9 (panels c, d), as a function of electron mass anisotropy γ and cylindercircumference L. The data is obtained with iDMRG at bond dimension χ = 1024.These states display the same qualitative behavior as the ν = 1/3 state (Fig. 3.4),with progressively stronger finite-size effects.
The data we obtain from the quartic coefficient λ in S(qx, 0) ≈ λq4x for filling
fractions ν = 2/5 and ν = 4/9 is shown in Fig. 3.B.1. The coefficients c1,2,3 for
polynomial fits of D + σ are shown, for both fillings, in Fig. 3.B.2. They display
wider fluctuations with the circumference L compared to the ν = 1/3 state (Fig. 3.5),
but appear to be consistent with the same asymptotic values, c1 ' 0.42, c2 ' 0, c3 '
0.02. These data support the idea that all incompressible states in the sequence ν =
p2p+1
respond quantitatively in the same way to applied anisotropy. Any drift in the
response with filling fraction is smaller than the numerical and finite-size uncertainties
of the method.
64
14 16 18 20L/ B
0.3
0.2
0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
fit c
oeffi
cien
ts
c1, linear fitc1, cubic fit
c2c3
16 18 20L/ B
Figure 3.B.2: Coefficients of the polynomial fit of log√λ to γ (data from Fig. 3.B.1)
for the states at filling ν = 2/5 (left) and ν = 4/9 (right). Despite the increasingfinite-size effects, the coefficients c1, c2, c3 are consistent with those of the ν = 1/3state, shown in Fig. 3.5.
65
Chapter 4
Band mass anisotropy in the
composite Fermi liquid at ν = 1/2
4.1 Introduction
In Chapter 3 we reviewed the geometric degree of freedom of FQH states and numer-
ically analyzed its response to band mass anisotropy. In this Chapter we will extend
the study to a very different state: the gapless phase at half filling of the LLL, known
as the composite Fermi liquid (CFL) [69, 179]. This state can be understood as the
limit point of the first Jain sequence [93]. The CF picture predicts an incompressible
FQH state at filling νp = p2p+1
for every finite p ∈ Z, and interprets it as an IQH state
with p completely filled ‘Λ-levels’ of CFs. However, as p → ∞, by filling infinitely
many Λ-levels one recovers a continuous, Fermi liquid-like spectrum. Equivalently, as
ν → 1/2, no magnetic flux is left after performing flux attachment, so the CFs are
free to form a Fermi sea. The resulting state in many ways resembles a Fermi liquid
– chief among them, it has well defined Fermi contour. However, strictly speaking,
it is a non-Fermi liquid, due to the strong coupling between the fermions and the
emergent Chern-Simons gauge field that describes flux attachment [69, 14].
66
The existence of this state is a crucial success of the composite fermion picture. It
uniquely highlights that CFs are real in some concrete physical sense, and not merely
a convenient theoretical tool to explain FQH fractions. Perhaps most significantly,
at magnetic fields slightly removed from ν = 1/2, it has been possible to observe sig-
natures of the semiclassical cyclotron orbits of CFs: by patterning a semiconductor
quantum well periodically in space, one observes features (commensurability oscilla-
tions) in transport whenever the cyclotron orbit radius is resonant with the spatial
periodicity of the pattern. [102, 100, 99, 148, 147]. This experimental avenue allows
the mapping of the CF Fermi contour, which is ultimately what motivates our study
of geometric distortions in this state.
Recently there has been a revival of theoretical interest in this state, as well.
This is mainly in regards to the nature of the effective field theory describing CFs
and the role of particle-hole (PH) symmetry in the half-filled LLL. The Halperin-
Lee-Read (HLR) theory [69] describes CFs as massive particles without explicit PH
symmetry. A more recent proposal by Son [199] suggests the CFs may be Dirac
fermions described by a relativistic theory, in which PH symmetry plays a crucial
role. The two scenarios have proven hard to distinguish, as most of their computable
predictions turn out to be identical. A difference is thought to be encoded in the Berry
phase accumulated by taking a CF adiabatically around the Fermi contour: the Dirac
cone is thought to host a singular π Berry phase at its center, which should instead
be absent in the massive dispersion of HLR composite fermions. This issue has been
investigated numerically with iDMRG [52], quantum Monte Carlo techniques [222]
and exact diagonalization [51], which support the existence of such singular term in
the Berry curvature distribution. This conclusion has been extended to the CFL at
filling ν = 1/4 as well, suggesting the Dirac fermion theory applies more broadly and
with a generalized notion of PH symmetry [221].
67
In this Chapter we are going to study the effect of band mass anisotropy on this
state. This is generally present in many-valley semiconductors [193, 55], but can
also be artificially induced by means of in-plane magnetic fields [127, 103, 129] or
mechanical strain [97]. We model this with the same Hamiltonian as in the previous
Chapter: electron kinetic energy
H0 =π2x
2mxx
+π2y
2myy
=1
2m
(αFπ
2x +
1
αFπ2y
), (4.1)
where π = p − eA is the kinetic momentum, and isotropic Coulomb interaction
V (r) = 1/r. The anisotropy parameter αF directly determines the shape of the
(non-interacting, zero-field) Fermi contour: the ratio of the Fermi wavevectors in
perpendicular directions x and y is
kF,ykF,x
=
√myy
mxx
= αF . (4.2)
The zero-field Fermi contour observed in the gallium arsenide (GaAs) quantum well in
Ref. [97] is more complicated than the elliptical one represented by the above Hamil-
tonian; nevertheless, we might expect the above model to describe the substantial
x− y anisotropy seen in experiment reasonably well. In fact, the model turns out to
be very accurate. We elucidate the reason for this high accuracy in Chapter 5, where
we deal with more general types of rotational symmetry breaking.
While the discussion on the intrinsic metric from Chapter 3 strictly applies only
to incompressible FQH fluids, there is still a meaningful notion of anisotropy in the
incompressible CFL, which is sharply defined from the geometric shape of the Fermi
contour of CFs. From this we can define, in the same way as Eq. (4.2), the CFL
anisotropy parameter as
αCF =kCF,ykCF,x
. (4.3)
68
In the notation of Chapter 3, these parameters would correspond to αF ≡ eγ and
αCF ≡ eσ. However, given the different context, and to facilitate contact with exper-
imental results, we are going to adopt the α notation.
The work presented in this Chapter, published in Ref. [85], was done concurrently
with experiments from the Shayegan group, also at Princeton [97]. There, the Fermi
contour anisotropy for CFs in a GaAs quantum well subjected to tunable strain
was measured and found to be smaller than that of the zero-field carriers (holes),
approximately following the relationship αCF '√αF .
A number of analytical results for the state at ν = 1/2 exist in the literature.
Ref. [11] used Chern-Simons theory to argue that αCF = αF (as we shall see, this
is only true for contact interactions). Ref. [12] used this as an assumption to derive
exact results on anisotropic model wavefunctions. Ref. [237] replaced the realistic
Coulomb interaction with a Gaussian interaction, where the system becomes isotropic
with respect to a different metric (as discussed in Chapter 3); this result is filling-
independent and thus in particular allows an analytical determination of αCF in terms
of αF .
On the other hand, most pre-existing numerical work on anisotropy in the quan-
tum Hall regime deals with incompressible states such as ν = 1/3, where tech-
niques ranging from exact diagonalization [220] to comparisons with model wavefunc-
tions [177, 236] are viable due to the energy gap and the relatively simple structure
of the Laughlin state. The CFL at ν = 1/2 does not lend itself to this type of anal-
ysis for a number of reasons. The variational wavefunction for a CFL has additional
variational parameters representing the shape of the Fermi surface [179]. Within the
finite-size systems accessible numerically, these variational parameters take discrete
values, and cannot capture small changes in the anisotropy. Since the CFL is gapless,
its energy spectrum is strongly size-dependent, which makes it more difficult to in-
terpret. The iDMRG method presented in Chapter 2 enables a wholly new numerical
69
route for the study of this problem. This technique has been successful in the study
of the isotropic CFL, where a circular Fermi surface was detected [52].
4.2 Mapping the Fermi contour
The DMRG method, which we described in detail in Chapter 2, is a variational
technique within the space of matrix product states (MPS). As such it comes with a
cutoff, the bond dimension χ, which limits the amount of entanglement in the MPS.
This is in principle not an issue for gapped ground states: their area-law entanglement
entropy can be fully captured by a large enough χ (in practice this still poses a
nontrivial limitation as the required χ diverges exponentially with L). However for
the critical CFL state, entanglement scales with a log-volume law, which on an infinite
cylinder implies infinite entanglement entropy. This can never be achieved with
a finite χ. A complementary but equivalent way of stating this limitation is that
a critical ground state (without symmetry breaking or long-range order) features
algebraic correlations, 〈X(0)X(r)〉 ∼ r−∆X for some local operator X and critical
exponent ∆X . But this cannot be reproduced by any MPS. By construction, an
MPS, as discussed in Sec. 2.4, has exponentially decaying correlations for large r,
〈X(0)X(r)〉 ∼ e−r/ξX where the finite correlation length ξX is a function of the MPS
transfer matrix spectrum. For these reasons, the DMRG method may seem hopelessly
ill-suited to this particular state.
However, there is a way around this limitation: finite-entanglement scaling [168].
By repeatedly optimizing the MPS ansatz at increasing values of χ, the MPS en-
tanglement entropy S and correlation length ξ (defined in Sec. 2.4) must obey a
well-defined scaling relation,
S = const. +c
6log ξ , (4.4)
70
where c is the central charge of the conformal field theory [34, 247] (loosely speak-
ing, the number of independent gapless modes in the theory). Physically, Eq. (4.4)
captures the critical log-volume entanglement scaling: while the system is infinite,
the entanglement cutoff χ introduces a lengthscale ξ that can be used as a finite-size
scaling cutoff. This method was introduced and used to map an isotropic CFL Fermi
contour in Ref. [52].
We use iDMRG to optimize the MPS ansatz for the ground state wavefunction at
increasingly large values of χ, from χ ∼ 100 up to χ = 6000. At each step we compute
S and ξ, then we check that Eq. (4.4) is obeyed. This allows us to verify that the
iDRMG has found an approximation to the correct ground state and is refining it at
each step.
After obtaining the approximate ground states with iDMRG, we compute the
guiding center structure factor, S(q) ≡ 〈:ρ(q)ρ(−q):〉, whose properties we discussed
in detail in Chapter 3 for incompressible FQH states. In the CFL state, which is
compressible, S(q) is not expected to exhibit quartic behavior at small q. On the
other hand, it displays singular features that allow us to map the Fermi contour of
the underlying composite fermions. The finite cylinder circumference L discretizes qy
in steps of κ ≡ 2π/L, while qx can in principle take on continuous values. Therefore
at any finite L the CF Fermi sea consists of a sequence of one-dimensional segments,
q : |qx| < Qn/2, qy = κ(n + δ), defined by the momenta Qn and the parameter
δ = 0, 12
depending on whether the emergent CF boundary conditions are peri-
odic or antiperiodic in y.1 The Fermi contour then consists of the isolated points
q = (±Qn/2, κ(n+ δ)), resembling a collection of one-dimensional Fermi liquids.
Observables in 1D Fermi liquid states generally show singularities at 2kF owing to the
non-analyticity in the occupation number at q = ±kF . For this reason, S(q) is ex-
pected to be singular whenever q connects two different points belonging to the Fermi
1This in turn is determined by whether the flux attachment gauge field has 0 or π flux through thecylinder, and in practice depends on the momentum of the product state used to seed the iDMRG.
71
0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5qx
S(q x
,qy)
(a) (b)
qy = 0
qy =
qy = 2
qy = 3
Figure 4.1: Mapping the CFL Fermi contour from the guiding center structure factorS(q). (a) Fermi sea for the composite fermions at L = 17`B and αF = 0.445. Theshaded area represents the 2D Fermi sea in the planar (L→∞) limit. The gray linescontain allowed values of q on the cylinder with finite L. Colored arrows show allpossible CF scattering processes between points on the Fermi contour. (b) Numericaldata for S(q) at the lowest four allowed values of qy, with bond dimension χ = 3000.The location of the singularities can be used to determine the coordinates of Fermicontour points, and thus the CF anisotropy (αCF = 0.667 in this case).
contour. Locating these singularities in S(q) allows us to extract the Qn momenta
and effectively map the Fermi contour.
Since the states obtained by DMRG are approximations of the true ground state,
these singularities will not be reproduced perfectly. In particular, as discussed above,
finite bond dimension χ introduces a length cutoff ξ whose effect is to broaden the
singular features in S(q) over a momentum scale δq ∼ ξ−1. In practice, for the sizes
we study, δq can be made small enough with χ ≤ 6000 that the gapless nature of the
CFL is not the dominant source of uncertainty of the method.
Fig. 4.1 illustrates an example of our method of mapping the Fermi contour from
analyzing S(q). The ellipse in Fig. 4.1(a) represents the anisotropic Fermi sea whose
shape we want to determine. The horizontal gray lines represent the allowed values
of momentum q in the infinite cylinder geometry. The colored arrows represent
72
vectors ∆q connecting points on the Fermi contour. If we fix ∆qy = 0 (horizontal
arrows), we expect singularities in S(∆qx, 0) at ∆qx = Qm. Such singularities are
indeed visible in the data of Fig. 4.1(b), and are in principle all we need to map
the Fermi contour. However, as a consistency check, we can also find singularities
at other values of δqy, corresponding to vectors with a non-zero vertical component:
∆q =(Qm±Qn
2, κ(m− n)
), for m 6= n. A given estimate of the singularities Qm
at ∆qy = 0 naturally comes with a prediction for the whole set of singularities at
|∆qy| > 0, which provides a robust test of the existence of the Fermi contour.
We consider values of the Fermi contour anisotropy αF ranging from 0.16 to 6.25.
The dynamic range in αF is limited by convergence of the iDMRG algorithm: very
small values αF 1 elongate the correlation length along the circumference of the
cylinder, giving rise to increasingly strong finite-size effects. Conversely, very large
values αF 1 elongate correlations along the axis of the cylinder, requiring larger
values of the bond dimension χ for convergence and causing a rapid increase in com-
putational complexity. Despite these numerical limitations on the dynamic range of
αF , we emphasize that the range covered is significantly larger than that covered in
experiment.
At each value of αF , we map the Fermi contour as described in the previous section,
using several values of L (3 to 5 distinct values in the range of 13 to 27`B, depending
on αF ). Examples representative of the cases αF < 1, αF = 1 and αF > 1 are shown
in Fig. 4.2. We extract αCF from the optimal fit of the Fermi contour points to an
ellipse. Though in the thermodynamic limit we expect an elliptical Fermi contour,
our finite-size data necessarily deviate from it due to Luttinger’s theorem, which in
our system fixes the total length of segments making up the finite-size Fermi sea to
∑n
Qn = νL`−2B . (4.5)
73
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4qx
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
q y
F = 0.326, CF = 0.582F = 1.000, CF = 1.002F = 3.062, CF = 1.713
Figure 4.2: Location of the Fermi contour for three selected values of αF , extractedfrom data similar to that in Fig. 4.1. The dashed lines are the best fits of the data toellipses of fixed area π, with the value of αCF shown in the legend. Different symbolsof the same color (circle, square, triangle and diamond) correspond to different systemsizes for the same αF .
This leads to an error in the anisotropy of our fitted ellipse because of the finite
values of L. We can estimate this error by considering the isotropic point αF = 1,
where αCF = 1 by symmetry. Our result based on the total data from three system
sizes is αCF = 1.002. However, randomly removing one system size from the dataset
causes the result to fluctuate between 0.98 and 1.01. On this basis, we estimate the
uncertainty on αCF due to Luttinger’s theorem and to the finite number of system
sizes considered to be of order 1− 2%.
We fit the discrete set of (qx, qy) coordinates obtained through the process de-
scribed above to an ellipse of area π (which is fixed by the carrier density at half
filling). From the optimal ellipse (k/kCF,x)2 + (k/kCF,y)
2 = 1, we obtain αCF =
kCF,y/kCF,x. As we discussed in Chapter 3, in the planar limit L → ∞ a π/2 ro-
74
tation exchanges the major and minor axes of the elliptical Fermi surface, mapping
α to 1/α. This implies that log(αCF ) is an odd function of log(αF ), so its Taylor
expansion around the isotropic point αF = αCF = 1 contains only odd powers:
log(αCF ) =∞∑n=0
c2n+1(logαF )2n+1 = c1 logαF +O((logαF )3) . (4.6)
As we will see, the first term in this expansion, corresponding to a power law αCF =
αc1F , already provides a very good approximation to the data for Coulomb interaction.
4.3 Benchmarking the method
So far we have assessed the accuracy of the method only on the isotropic CFL,
where the result αCF = 1 is trivial. We now benchmark the numerical accuracy
of the method at measuring a nontrivial αCF . To do so, we use the only known ex-
act formula, Yang’s prediction [237] that for a Gaussian electron-electron interaction
V (r) = V0e−r2/2s2 one has
αCF =
√αF + (s/`B)2
1/αF + (s/`B)2. (4.7)
For this purpose, we pick two nearly reciprocal values of the electron anisotropy,
αF = 2.25 and αF = 0.445, and compute αCF with our method at different values
of s of the order of a magnetic length `B. The results, displayed in Fig. 4.3, show
a good agreement with the prediction Eq. (4.7). Our method appears to slightly
underestimate αCF by 1 to 2%. This small bias however, being present for both
αF > 1 and αF < 1, should not significantly affect our estimates of the coefficient c1,
as that only takes into account the odd part of the dependence on log(αF ). Either
way, an uncertainty of . 2% is consistent with what we estimated from finite-size
fluctuations in the isotropic CFL.
75
0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4s/ B
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
CF
F = 0.445 predictionF = 2.250 prediction
datadata
Figure 4.3: Benchmarking our method against the exact result Eq. (4.7) for a Gaussianinteraction V (r) ∝ e−r
2/2s2 [237]. We consider αF = 2.25 and αF = 0.445, two nearlyreciprocal values to test the method in opposite regimes for varying s in the vicinityof a magnetic length. This test reveals good agreement between the numerics andthe analytical prediction, despite a possible systematic underestimation of αCF by∼ 1− 2%.
4.4 Results
Having established the reliability of the method, we next consider electrons interacting
via the realistic Coulomb potential, V (r) = 1/r. In analogy to our results on the
ν = 1/3 FQH state, we find that the first term in the expansion Eq. (4.6), a power-
law αCF = αc1F , provides an excellent approximation to the data in the anisotropy
range we explore. This is shown in Fig. 4.4. A linear fit yields c1 = 0.493 ± 0.008,
close to a square-root dependence αCF =√αF . Remarkably, this is in very good
agreement with experiments on holes in GaAs quantum wells under application of
in-plane strain [97].
Allowing a constant term c0 in our fit, i.e. fitting to αCF = ec0αFc1 , yields c1 =
0.498 ± 0.010 and c0 = −0.02 ± 0.02. The constant term is symmetric, rather than
antisymmetric, under π/2 rotation, and is thus forbidden in the planar limit. The fact
that we find a value for c0 compatible with zero, despite the significant x-y asymmetry
76
0.1 0.2 0.5 1.0 2.0 5.0 10.0F
0.2
0.5
1.0
2.0
5.0
CF
CF = 0.49F
CF = 0.79F
Coulomb interactionDipolar interaction
Figure 4.4: Composite fermion anisotropy αCF , obtained by mapping the Fermicontour of the CFL state obtained numerically with iDMRG, as a function of bareelectron anisotropy αF for Coulomb interaction V (r) = 1/r and dipolar interactionV (r) = 1/r3. The solid lines correspond to power-law fits αCF = (αF )c1 as shown inthe legend.
inherent to our method, is an encouraging sign that finite-size effects in our result are
limited.
In order to test the universality of our result, we replace the Coulomb interaction
with a dipolar interaction V (r) ∝ r−3, also shown in Fig. 4.4). This interaction could
be realized in systems of cold atoms [231, 30, 31, 242, 250]. We find again that a single
power law (i.e. a truncation of Eq. (4.6) to the first term) fits the data well, but we
measure a significantly larger exponent c1 = 0.795 ± 0.005. This is unambiguously
different from a square root dependence. In particular it implies that αCF is much
closer to αF (i.e. the CFL is much more anisotropic) than for the case of Coulomb
interaction.
Since the CFL anisotropy arises from the competition between the kinetic energy
term (which has anisotropy αF ) and the interaction (which is isotropic), this result
means that the dipolar interaction is less effective than the Coulomb one at countering
the effect of band mass anisotropy. This is intuitively reasonable: as r−3 decays more
77
quickly than r−1, its effect on the shape of long-distance correlations will be weaker.
A comparison with the Gaussian interaction discussed in Sec. 4.3 is useful: there, as
the interaction length scale s goes to zero (approaching a contact interaction), the
CFL state becomes more anisotropic, eventually saturating the maximal anisotropy
αCF = αF . While V (r) = r−3 is still a long-ranged interaction, it is intuitively closer
to a contact interaction than r−1 is. This intuition can be made rigorous [114] by
considering a pseudopotential decompostion of LLL-projected power law interactions
V (r) = r−n, which reveals that for ν = 1/3 the interaction becomes effectively point-
like at n ≥ 4. In the same work, the CFL with V (r) = r−2 is studied and the value of
the exponent c1 is found to be c1 ' 0.61. These values are consistent with a saturation
c1(n) ≡ 1 for n ≥ 4 and overall indicate that the geometric response of the CFL is
similar to that of the ν = 1/3 FQH state, though possibly slightly stronger.
4.5 Discussion
In summary, we have numerically mapped the composite Fermi contour of a half-
filled LLL for a two-dimensional electron gas with varying mass anisotropy, with
three different forms of the electron-electron interaction: (i) the realistic Coulomb
interaction V = 1/r, (i) a Gaussian interaction, used as a theoretical benchmark, and
(iii) the dipolar interaction V = r−3, used as a test of universality. We have found that
the CFL Fermi contour is less anisotropic than that of the zero-field, non-interacting
electrons. When the electrons in the system interact via the Coulomb interaction, our
data indicates a dependence consistent with αCF =√αF . This result is in agreement
with recent experimental measurements [97], but not with some earlier theoretical
work [11, 12]. The relationship between αCF and αF does however depend on the
form of the electron-electron interaction. For example, we find a larger CF anisotropy
78
for the dipolar 1/r3 interaction, and for a Gaussian interaction we recover the known
results from Ref. [237].
The relation between our results on the CFL and the behavior of FQH states
examined in Chapter 3 is interesting and not entirely clear. The geometric response
of the CFL is broadly similar to that of states in the first Jain sequence, νp = p2p+1
.
This follows naturally from composite fermion theory: if all these states are indeed
constructed from a common building block, then it stands to reason that they should
exhibit similar (or identical) geometric response. However, quantitatively, we find
the response is slightly (but significantly) stronger in the CFL (c1 ' 0.49) than in
the ν = 1/3 FQH state (c1 ' 0.43). It is possible that the value of the response
coefficient c1 slowly drifts with p along the Jain sequence νp, though we were not
able to detect any such drift up to p = 4 in Chapter 3. Another possibility is a much
slower finite-size drift in the CFL; while we do not see evidence of it, it cannot be ruled
out, especially given the gapless nature of the state. Finally, the different definitions
of anisotropy for the two cases (the long-wavelength limit of S(q) versus the shape
of the Fermi contour) may account for the observed difference. In conclusion, this
small numerical discrepancy hides an interesting fundamental question that deserves
further investigation.
Another interesting consequence of the peculiar square root relationship between
the electron and composite fermion anisotropies is explored in Ref. [120]. The authors
argue, via a combination of analytical and numerical methods applied to a variety
of models, that our finding αCF '√αF may in fact be a special case of a universal
Fermi contour renormalization exhibited by two-dimensional fermions interacting via
the Coulomb potential. This would connect the geometric response of FQH states to
Fermi liquid physics. While interesting, it is not clear that this result should apply
to the CFL. Composite fermions are inherently “dressed” by interactions; as such
they are not expected to interact via the Coulomb potential as the bare electrons do.
79
Moreover the CFL is a non-Fermi liquid, owing to the presence of the Chern-Simons
flux attachment gauge field. This is likely to affect the Fermi contour renormalization
significantly. Either way, the idea of connecting the geometric response of CFL and
FQH states to Fermi liquid physics is a fascinating research direction.
Though our results match the experimental data in Ref. [97] very well, there
are some important differences between the experimental system and our theoretical
model. Our simulations use an anisotropic band mass, leading to a purely elliptical
zero-field Fermi contour. The Fermi contour of holes in the GaAs quantum well used
in experiment, on the other hand, has a more complicated shape. Simulating models
that more closely resemble such realistic Fermi contours is a natural extension of this
work. From the theoretical standpoint this is especially interesting, as relatively little
is known about the response of FQH fluids and CFLs to geometric distortions more
general than the purely quadrupolar one represented by band mass anisotropy. This
will be the focus of the next Chapter.
80
Chapter 5
Distortions of the composite Fermi
liquid beyond band mass
anisotropy
5.1 Introduction
In the previous Chapter we have studied the effect of band mass anisotropy on the
CFL, finding results in excellent agreement with experiments on gallium arsenide
(GaAs) quantum wells subjected to tunable strain [97]. However, as we have noted,
the band structure of the zero-field carriers in the GaAs quantum well used in exper-
iment is not isotropic even in the absence of strain: besides the strain-induced x-y
asymmetry, it possesses distortions with four-fold rotational symmetry (C4). This
is to be expected in realistic systems. Near the bottom of a band, the electron dis-
persion is well approximated by a quadratic (defined by the band mass mab), which
constrains the Fermi contour to be elliptical (or circular if mab ∝ δab). But away from
the bottom, at larger Fermi energy, much more general shapes are possible (within
81
lattice symmetry constraints). A natural question then is the following: what, if any,
is the effect of these distortions on the CFL Fermi contour?
The answer to this question, as we will see, is subtle. Due to crystallographic
constraints, experiments using commensurability oscillations can only measure the
Fermi wavevectors in two perpendicular directions, so making definitive statements
on the shape of the CF Fermi sea is difficult. Nonetheless, in Ref. [97] the assumption
that the measured dimensions represent the major and minor axes of an ellipse gives
the correct Fermi sea area (which is fixed by the carrier density). It is safe to say,
then, that the result of that experiment is consistent with the C4-symmetric distortion
not being transferred to the CFL. Interestingly though, Ref. [148], which studies a
hole system in a similar quantum well with C4-symmetric warping of the zero field
Fermi contour, does find evidence of this warping in the CF Fermi contour once an
in-plane magnetic field is turned on.
These issues motivate us to consider a broader question: to what extent can
the zero-field electronic band structure be used to engineer different states in the
high-field quantum Hall regime? Could such an approach parallel the enormously
successful zero-field band-structure engineering in traditional semiconductor physics?
In this Chapter we show that this expectation is too optimistic, and provide
examples that demonstrate how the connection between the zero-field and high-field
regimes is subtle, even tenuous in some instances. Perhaps the most striking example
is provided by rotationally symmetric band distortions, which we study in Sec. 5.2.
We exhibit a number of cases that display a wide variety of zero-field Fermi seas, but
nevertheless share the same Fermi sea for CFs at high field. In Sec. 5.3 we consider
anisotropic cases where the zero-field Fermi sea has two disconnected Fermi pockets,
and demonstrate that the CF Fermi sea, while anisotropic, remains fully connected.
This suggests that only coarse features of the zero-field Fermi contour are retained
in the high field limit, while the finer details are wiped out. This expectation is
82
largely confirmed by the study of distortions with specific angular number N , which
we examine in Sec. 5.4: finer features, represented by higher N , are very strongly
suppressed in the CFL, while N = 2 (corresponding to band mass anisotropy) is
by far the dominant distortion. Finally, in Sec. 5.5 we summarize our results and
conclude this part of the dissertation. This Chapter is based on results published in
Ref. [83] and [84].
5.2 Exact results for radial distortions
We start by considering the LLL problem for a single, non-degenerate electron band
with a circularly symmetric, but otherwise arbitrary, electron dispersion. We note
in passing that non-parabolic dispersions could occur in any strongly interacting
fermionic system with continuous translational and full rotational symmetry (i.e.
Galilean invariance). In fact, significant deviations have been predicted and seen
experimentally in Galilean invariant three-dimensional dilute mixtures of 3He in
4He [214, 18] which depend on the effective interaction pseudopotentials [75].
We show that the problem in the high field limit reduces to the usual Landau
problem with the same extensively degenerate Landau bands (and in particular the
same eigenfunctions), but with eigenvalues that generally differ from the canonical
harmonic oscillator values. As a result, in the high magnetic field limit, the problem
can be projected onto a single Landau level; but which one? This depends on which
how the nontrivial dispersion rearranges the original Landau level energies.
Two possibilities arise here: the N = 0 LL may retain the lowest energy, or there
may be a nontrivial rearrangement of the ground levels whereby a N ≥ 1 LL has the
lowest energy. For now we will assume the former scenario. Given that the single-
particle wavefunctions are exactly the same as those for the traditional parabolic
dispersion, the interacting problem at half filling is unchanged. It follows that the
83
CF Fermi sea is completely invariant under a whole range of distortions of the electron
dispersion, where the B = 0 Fermi sea may change from a circle, to an annulus, to a
set of disconnected annular pieces. In short, while the zero-field Fermi sea undergoes
Lifshitz transitions, the high field problem remains thoroughly unaffected.
To get more quantitative, we consider a zero-field Hamiltonian with an arbitrary,
but isotropic kinetic energy term in two dimensions:
H0 = E0f
(|p|2
2mE0
), (5.1)
wherem is the free electron mass, p is the canonical momentum, and E0 is an arbitrary
energy scale that makes the argument of f dimensionless. For non-relativistic free
fermions, one has f(x) = x (and the energy scale E0 cancels out). In order for
the problem to be well defined, we require that f(x) be analytic and bounded from
below, but set no other constraints on it. In the presence of a magnetic field B in the
z-direction, we apply the minimal coupling substitution p→ π = p− eA.
In the usual Landau problem with a quadratically dispersing Hamiltonian,
Hquadratic0 = |π|2/2m, the eigenstates |φN,m〉 have eigenvalues
εquadraticN = (N + 1/2)~ωc = (N + 1/2)βE0. (5.2)
Here N is the Landau level index, m labels the intra-LL degeneracy, ωc = eB/m is the
cyclotron frequency, and β is the dimensionless ratio ~ωc/E0. Since the Hamiltonian
H0 in Eq. (5.1) is itself a function of Hquadratic0 , the |φN,m〉 are eigenstates for any
f(x):
H0|φN〉 = εN |φN〉, εN = E0f ((N + 1/2)β) . (5.3)
Before proceeding further, we establish conventions that we follow in the rest of
the Chapter. We use as independent variables the Landau level filling fraction ν, and
84
either the zero-field Fermi energy εF or the electron density ne. For a given dispersion,
fixing ν and εF (or ν and ne) fixes the magnetic field B and thus the cyclotron energy
~ωc. We define A(εF ) as the area in k-space occupied by the Fermi sea at given
chemical potential εF (with dimensions of inverse length squared). These quantities
are connected by various identities, including:
ne =ν
2π`2B
, (5.4)
ne =
∫Fermi sea
d2k
(2π)2=A(εF )
4π2, (5.5)
β ≡ ~ωcE0
=A(εF )
2πνk20
. (5.6)
In Eq. (5.6), k0 ≡√E0m/~ is the wavevector associated to the energy scale E0.
Having established the notation, we now discuss the simplest non-trivial example
for f(x):
f(x) = −Cx+ 4x2 . (5.7)
which gives
H0(C) = −C |π|2
2m+
1
E0
(|π|2
m
)2
. (5.8)
This kinetic energy yields a Fermi sea which is an annulus if C > 0 and εF < 0, and
a circle otherwise. The annular case can be seen as a simplified model for a realistic
energy dispersion that has recently been observed [96] for holes in GaAs quantum
wells.
In order to demonstrate the lack of correspondence between the zero-field and
composite fermion Fermi contours at magnetic field corresponding to a half-filled
lowest Landau level (ν = 12), it suffices to exhibit a pair of values (C, εF ) where (i)
the zero-field Fermi sea is annular and (ii) the N = 0 Landau level has the lowest
energy. Such a combination of parameters is in fact quite generic. Fig. 5.1(a) shows
an example of the band structure of a system with C = 2.5 and εF = −0.34E0. The
85
0.0 0.5 1.0 1.5k/k0
0.5
0.0
0.5
1.0
1.5
2.0
2.5
E/E
0
(a) (b)ε(k)
εF
εN
0.0 1.0 2.0 3.0 4.0C
0.00
0.02
0.04
0.06
0.08
0.10
ne CFL
MRStripes / Bubbles
Figure 5.1: (a) Dispersion of Hamiltonian Eq. (5.8) with C = 2.5 (solid line), Fermienergy εF = −0.34E0 (dashed line) and energies of the lowest three Landau levels(dots). The B = 0 Fermi sea (consisting of k-states below the dashed line) is anannulus. At these values of C and εF , the N = 0 Landau level has the lowest energy,so the ground state is a CFL with a circular Fermi sea. (b) Phase diagram of thesystem described by Eq. (5.8), as a function of C and electron density ne (expressed inunits of k2
0), in a magnetic field tuned to half-filling. Depending on which N minimizesεN , the system can be in a CFL phase (N = 0), a Moore-Read phase (N = 1), or astripe or bubble phase (N ≥ 2). The dashed line in the CFL phase corresponds toεF = 0, where the B = 0 electron Fermi sea goes from circular to annular. The stardenotes the parameters used in (a).
system clearly has an annular Fermi sea and the N = 0 Landau level is the one with
lowest energy. Fig. 5.1(b) shows a phase diagram in C and the electron density
ne =k2
0
2π
√(C/4)2 + εF/E0 , (5.9)
displaying a large region of parameter space where these conditions are met. At
smaller C or larger density, the zero-field Fermi sea becomes a circle (though the
CFL is completely insensitive to this transition).
Fig. 5.1(b) also contains other phases. For a general function f(x), there are a
number of different possibilities depending on the values of N ≡ argminN ′(εN ′):
86
i. If N = 0, the ground state of the interacting system will be a CFL with a
circular Fermi contour, despite the fact that the bare electron Fermi contour
may have a more complicated (albeit circularly symmetric) shape.
ii. If N = 1, the interacting system will be in the Moore-Read phase [145], with
ground state described either by the Pfaffian or anti-Pfaffian model wavefunc-
tions (which one is chosen depends generically on the effects of Landau level
mixing, which is affected by the particular choice of f(x)).
iii. If N ≥ 2, we generically expect the ground state at every filling 0 < ν < 1 (not
just the value ν = 12
used in Fig. 5.1) to be in a symmetry-broken phase, such
as a stripe or bubble phase [141]. This opens exciting possibilities for studies
of such phases in the extreme high-field limit, a regime opposite to the one in
which they are normally observed.
We can see that the latter case is realized even in the simple example of Eq. (5.8)
for large enough C and low enough density. For example, we find that these phases
are predicted to occur at a realistic carrier density ∼ 1011cm−2 for band parameters
C ∼ 3 and E0 ∼ 3meV. The recently observed annular Fermi sea for holes in GaAs
quantum wells [96] would most likely not give rise to these phases, due to the fact that
the annular Fermi pocket is included in a larger Fermi sea (i.e. there is no gap between
the band of interest and other bands). On the other hand, band structures that are
reasonably well approximated by Eq. (5.8) are expected to occur quite generically in
band-inverted semiconductors, such as HgTe, when a gap is opened e.g. by application
of strain [143].
The above analysis is sufficient to demonstrate that there is not necessarily a
relationship between zero-field and ν = 12
composite fermion Fermi contours. This
analysis could be easily repeated for more complicated forms of f(x), such that the
electron Fermi sea may consist of multiple disconnected pieces. In order to show that
87
our conclusion is completely generic, in Appendix 5.A we consider an example of
f(x) which allows us to generate a zero-field Fermi sea consisting of arbitrarily many
disconnected components, while at the same time having argminN(εN) = 0 when the
magnetic field is tuned to half filling.
5.3 Systems with multiple Fermi pockets
The previous Section shows that one can have an electron Fermi sea made of multiple
disconnected pieces at zero field, and yet only a single CF Fermi sea at high field. We
can observe a similar phenomenon for Fermi seas consisting of disconnected “pockets”
without rotational symmetry, such as those shown in Fig. 5.2. These Fermi contours
are generated from the following zero-field Hamiltonian:
H0(α) = −(αp2
x
2m+
p2y
2mα
)+
1
E0
(p2x + p2
y
m
)2
. (5.10)
This is the same as Eq. (5.8) with C = 1, except that we have included an anisotropy
parameter α in the quadratic part which explicitly breaks rotational symmetry.
Fermi contours with shapes similar to those in Fig. 5.2 have been observed in GaAs
systems in a parallel field [101], as well as in the surface states of Sn1−xPbxSe [39] and
bismuth [160, 42] (though in the latter case the valleys are elongated radially, rather
than tangentially). In such systems it is tempting to assume that multiple zero-field
Fermi pockets imply that the system can be treated as having a single Fermi pocket
and an additional ‘valley pseudospin’ at high field. However, we will show that this
assumption is not always correct.
To prove this point, for the dispersion Eq. (5.10), we choose the Fermi energy εF
such that there are two disconnected zero-field pockets (as in Fig. 5.3). Since we work
at ν = 12, the Fermi energy also sets the cyclotron energy. We solve the single-particle
Schrodinger equation using Eq. (5.10) in the strong magnetic field determined by the
88
0.40.20.00.20.4
ky/k
0
−0.068E0
α= 1.05−0.095E0
α= 1.25
0.40.20.00.20.4
ky/k
0−0.060E0 −0.075E0
1.0 0.5 0.0 0.5 1.0kx/k0
0.40.20.00.20.4
ky/k
0
−0.055E0
1.0 0.5 0.0 0.5 1.0kx/k0
−0.050E0
Figure 5.2: Examples of zero-field Fermi seas obtained from the kinetic energyEq. (5.10) at small values of the anisotropy parameter, α = 1.05 (left) and α = 1.25(right), at various Fermi energies shown in the upper right corner of each panel.Tuning these parameters allows continuous interpolation between the annular shapediscussed in Sec. 5.2 and the Fermi pockets discussed in Sec. 5.3.
2 1 0 1 2kx B
1.0
0.5
0.0
0.5
1.0
k yB
L/ B121314
Figure 5.3: Zero-field Fermi sea for the Hamitonian in Eq. (5.10) (filled shapes) andν = 1/2 CF Fermi contour (dashed line), obtained numerically from singularitiesin S(q) at the sizes indicated in the legend. The parameters are α = 4 and εF =−0.49E0. The zero-field Fermi pockets are separated by ≈ 2`−1
B .
89
εF and ν. This is done by replacing
πx 7→a+ a†
`B√
2, πy 7→
a− a†
i`B√
2, (5.11)
where a, a† are canonical bosonic creation-annihilation operators. By applying the
above substitution we get
H0
~ωc= − cosh γ
(a†a+
1
2
)− sinh γ
a2 + (a†)2
2+ 4β
(a†a+
1
2
)2
, (5.12)
where γ ≡ logα. It is convenient to express Eq. (5.12) in the basis of isotropic LLs
|n〉 ≡ (a†)n√
n!|0〉 : n = 0, 1, . . .
, (5.13)
where it is a tight-binding problem on a semi-infinite line with quadratically divergent
on-site potential and range-2 hopping. While the problem infinite-dimensional, for the
purpose of finding low-lying state it can be safely truncated to a finite n ≤ nmax given
the confining effect of the on-site potential. The inversion symmetry of Eq. (5.10) is
manifested in the fact that only even powers a2, (a†)2 appear, decoupling the even-n
and odd-n sectors. As a consequence, each generalized Landau level (eigenvector of
Eq. (5.12)) is either even or odd under inversion, but in either case has equal weight
in both valleys.
So when can one treat the valley as an internal, pseudospin-like degree of freedom?
For that description to apply, the N = 0 and N = 1 LLs must be very close in energy,
compared to the interaction strength. Then interactions can hybridize the states and
lead to valley polarization, where the electron orbits reside purely in one valley or
the other [198]. It would be a mistake, however, to assume that this hybridization
always happens when the zero-field Fermi surface has multiple pockets. If the splitting
between the two lowest-lying single-particle solutions is much larger than the strength
90
1 2 3 4 5 6 7 8 9 1020
15
10
5
0
5
E/E c
groundfirst excited
= 4
100 101 10210 6
10 3
100
|gap
|
Figure 5.4: The two lowest energies (in units if the cyclotron energy ~ωc = βE0) ofthe Hamiltonian in Eq. (5.10) placed in a magnetic field such that the electron densityof Fig. 5.3 corresponds to filling ν = 1
2. As the α parameter changes, the chemical
potential is changed so that electron density and magnetic field stay constant. Inset:difference between the two energies. The dashed lines indicate α = 4, the value usedin Fig. 5.3. Even with a Fermi sea made of two well-separated pockets, there is stilla substantial energy gap between the lowest-lying Landau levels (roughly half of thatat the isotropic point).
of interactions, ∼ e2/`Bε, then LL mixing effects are modest, and the quantum Hall
problem takes place in one generalized LL that has equal weight in both valleys.
Fig. 5.4 shows the lowest two energy levels of the single-particle problem as a
function of α. The value α = 4, which was used to generate the Fermi contours in
Fig. 5.3, is indicated by the dashed line. We see that, as one turns α up from the
isotropic point (α = 1), a large splitting between the two energies persists long after
the problem has developed two disconnected Fermi contours. This energy splitting
should be compared to the interaction energy to determine whether the system can
exhibit valley polarization. In quantum Hall problems we often assume that the
interaction energy is much smaller than the cyclotron energy: κ 1, where κ is the
ratio of interaction energy to cyclotron energy, known as the Landau level mixing
parameter. We can see that at large α, even a small κ will be enough to hybridize
91
the levels and give well-defined valleys. But up to α ∼ 4, the energy gap is still
substantial, and a value of κ ≈ 1 would be required to effectively hybridize the LLs.
To make this analysis more concrete, we perform iDMRG calculations on the
system described by Eq. (5.10) placed on an infinite cylinder with circumference
L = 13`B. We retain the lowest two generalized LLs, |0〉 and |1〉, whose single-
particle energies are shown in Fig. 5.4. The two generalized LLs have different form
factors which are determined by the corresponding orbitals. Decomposing them in
the basis of isotropic LLs as
|0〉 =∑n
un|n〉, |1〉 =∑n
vn|n〉 , (5.14)
where the coefficeints u and v are found numerically as eigenvectors of Eq. (5.12), we
can express their form factors as
F00(q) =∑n1,n2
u∗n1un2Fn1,n2(q) , (5.15)
and similarly for F01, F10 and F11. The isotropic form factors Fij(q) are expressed
exactly in terms of Laguerre polynomials [170] and given in Sec. 2.3.3. We run the
iDMRG with the interaction determined by projecting the Coulomb V (q) = 2π/q
potential with the form factors in Eq. (5.15), and the single-particle energy offset
between the two generalized LLs obtained from the numerical solution of Eq. (5.12),
at total filling ν = 1/2. We obtain the many-body ground state and evaluate the
difference in single-particle orbital occupations, δn ≡ 〈n0〉 − 〈n1〉. When the energy
separation is large we expect all electrons to form a ν = 12
state in the lowest-energy
Landau level, as the interactions should not be able to significantly hybridize the
single-particle orbitals:
|Ψunpolarized〉 = F [c†m,0]|Ω〉 , (5.16)
92
where |Ω〉 is the vacuum, cm,N is the mth electron mode in the N th generalized LL, and
F is the CFL ground state wavefunction. In this case, 〈n0〉 = 12
and 〈n1〉 = 0, or δn =
12. As the energy separation gets smaller, the system could valley polarize, i.e. pick a
coherent, equal-amplitude superposition of the Landau levels, cm,± = 1√2(cm,0± cm,1).
Given the symmetry of the problem, the exact ground state should be a “cat state”
superposition of the two states in which all electrons are in the same valley. We can
schematically write such a state as
|Ψcat〉 =1√2
(F [cm,+] + F [cm,−]) |Ω〉 . (5.17)
However, a “cat state” like Eq. (5.17) has larger entanglement than an unpolarized one
like Eq. (5.16). The entanglement cutoff built into the DMRG (the bond dimension
χ) favors lower entanglement, so the symmetry gets spontaneously broken [248] and
the numerics yields a (randomly chosen) valley-polarized state of the type
|Ψvalley〉 = F [c†m,+]|Ω〉 . (5.18)
Such a state has 〈n0〉 = 〈n1〉 = 14
by symmetry, hence δn = 0.
The results of our numerical calculation of δn can be seen in Fig. 5.5. They indicate
that the system does not valley-polarize until the interaction strength e2/`B is several
times larger than the inter-Landau-level gap δE, namely κ & 5. Nearly all electrons
populate the inversion-even N = 0 Landau level up to α ≈ 6. After the Landau
levels cross at α ' 6.45, we see that a small energy difference (≈ 0.4~ωc) in favor of
the inversion-odd N = 1 Landau level is enough to again prevent valley polarization,
making δn significantly negative. Valley polarization is thus only possible in the
immediate vicinity of the crossing (α ' 6.45) and beyond α ≈ 12. We remark that
our choice of Landau level mixing parameter κ at the isotropic point, κ ≈ 0.5, is
conservative; the desirable scenario often assumed in studies of quantum Hall physics
93
1 2 4 8 16α
0.4
0.2
0.0
0.2
0.4
0.6
δn
δn
δE/Ec
α= 4
4
2
0
2
4
6
δE/E
c
Figure 5.5: Difference in orbital densities (δn ≡ 〈n0〉−〈n1〉) as a function of α for theground state of the Hamiltonian Eq. (5.10) with Coulomb interaction at filling ν = 1
2
(dots). The values are computed via infinite DMRG on a cylinder of circumference13`B, with bond dimension χ ∼ 3000. Magnetic field and electron density are thesame as in Fig. 5.4. The strength of e-e interactions is taken to be e2/`B = E0 ∼2.4~ωc. Also shown is the inter-Landau-level gap δE in units of cyclotron energy(solid line).
is that of negligible Landau level mixing, κ 1. Such a choice of parameters would
further hinder valley polarization and reinforce our conclusion.
By studying the structure factor S(q) of states with δn ' 1/2, in particular for
α < 6, we are able to map the Fermi contour of CFs with the method discussed in
Chaper 4. This shows that the composite Fermi contour goes from being a circle at
the isotropic point (in agreement with the analysis of Sec. 5.2) to an ellipse which
becomes more and more elongated as the parameter α is increased. The ratio of
longest to shortest Fermi wavevector varies between 1 and ' 2 in the available range
of α. Thus, we again find a lack of correspondence between the Fermi contours of zero-
field electrons and CFs: while the electron Fermi sea is made of two well-separated
pockets like in Fig. 5.3, the CFL has a Fermi sea consisting of a single connected
component, generally elliptical, but smoothly connected to the circular one at α = 1.
In the same situation, but with total filling ν = 1 (i.e. 1/2 per Fermi pocket),
the system simply forms a ν = 1 IQH state with the generalized LLL orbitals. If we
94
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.71
3
5
7
9
11
crit.
Figure 5.6: Dependence of αcrit. (value of α at which the N = 0 and N = 1 LLsfirst cross) on the magnetic field B, expressed through the dimensionless ratio β =~ωc/E0 = B(~e/mE0). For each value of β, we fix the density so filling is ν = 1
2
and increase α starting from the isotropic point until we encounter the first crossing.Fig. 5.4 correspond to β = 0.41 and has the first crossing at α ' 6.45.
take α to be very large, to the point that the inter-LL gap becomes much smaller
than the interaction strength, then the system is described by a bilayer with linearly-
independent form factors. At total filling ν = 1, such a system is expected to spon-
taneously valley-polarize [198], again forming an IQH state. At total filling ν = 12,
the system is expected to have a rich phase diagram [165] including potentially a
Halperin 331 state [68], a single-component Moore-Read state [145], two ν = 1/4 CF
Fermi contours (one in each valley), or a valley-polarized ν = 1/2 CFL. Thus even
when the two Fermi pockets are extremely well separated and it is possible to model
the system as a quantum Hall bilayer, it is far from obvious that the two zero-field
electron pockets translate into two Fermi pockets for the CFs. Confirmation of this
lack of correspondence also comes from experiment. Measurements on electrons in
GaAs quantum wells with in-plane magnetic field [101] find that the CF Fermi con-
tour remains connected up to the highest achievable values of the parallel fields, even
as the bare electron Fermi sea is disintegrated into two well-separated pockets.
Finally, we comment on the dependence of our results on magnetic field. In
general, we expect two Fermi pockets to lead to well-defined quantum Hall valleys if
95
the separation between them is large compared to `−1B , the typical spread of a Landau
orbital in momentum space. At the same time, following Eq. (5.6), we know that the
inverse magnetic length is
`−1B =
√A(εF )
2πν. (5.19)
Therefore the valley approximation holds well if the separation between the Fermi
pockets is large enough relative to their typical linear size, defined as the square root
of their area. Moreover, Eq. (5.19) implies that the same set of Fermi pockets may
or may not lead to quantum Hall valleys depending on filling, with larger ν being
more likely to have well-defined valleys. For Fermi energy εF near the bottom of the
band, we have small, widely separated pockets at zero field, so that at high field there
are well-defined quantum Hall valleys. Increasing εF causes the Fermi pockets to get
closer and eventually merge, by which point the notion of valleys has clearly lost
meaning. When exactly this transition occurs depends on the value of the LL mixing
parameter κ, but a reasonable, κ-independent estimator is the value αcrit. at which the
two LLs cross for the first time. We compute αcrit. and find an approximately linear
dependence on B, shown in Fig. 5.6. This confirms that larger electron densities,
which require stronger magnetic fields to get into the LLL regime, tend to oppose the
onset of a “valley pseudospin” regime and instead favor the population of a single
non-degenerate, valley-symmetric LLL.
5.4 Systems with N-fold rotational symmetry
In this Section we consider the effects of band distortions with discrete, N -fold rota-
tional symmetry (CN), which generalize the case of band mass anisotropy (N = 2)
studied in Chapter 4.
There are generally two ways to introduce anisotropy in a quantum Hall system:
either via the one-electron dispersion, or via the electron-electron interaction. In
96
the case of elliptical (N = 2) anisotropy, the two approaches are equivalent, since
a linear coordinate transformation can move the anisotropy between the band mass
tensor mab and the dielectric tensor εab. This is not the case for generic anisotropy,
and thus band anisotropy and interaction anisotropy are not equivalent. Here we
concentrate on the former. We consider a simple set of dispersions, one for each CN
symmetry group with even N . Completely general distortions such as those studied
in experiments can then be thought of as arising from a sum of terms with different
CN symmetries, so the response of the CFL can be approximately broken down into
its response to each individual “harmonic”.
We quantify the anisotropy by αF , defined as the ratio of the longest and shortest
Fermi wavevectors for the system at zero magnetic field, directly generalizing the
definition for band mass anisotropy. The corresponding CF quantity, αCF , is defined
analogously. We note that for N > 2 we can no longer use the language of metric
tensors to describe anisotropy. In Ref. [114] we have used the anisotropy of equal-value
contours of S(q) near q = 0 to characterize the effect of C4-symmetric distortions on
incompressible FQH states, effectively probing the Γabcd rank-4 tensor from Eq. (3.8).
However this approach does not work beyond N = 4. While higher-rank tensor
generalizations can presumably be defined for every N , CFL states have the desirable
property of allowing a definition of anisotropy (αCF as the ratio of Fermi wavevectors)
that applies straightforwardly regardless of N .
In Sec. 5.4.1 we formulate dispersion relations with N -fold anisotropy and deter-
mine the corresponding form factors. We express the effective interaction between
one-electron orbitals with these form factors in terms of generalized pseudopotentials.
We find that for fixed αF the strength of the anisotropic pseudopotentials decreases
rapidly with N . In Sec. 5.4.3 we present the results of a numerical density matrix
renormalization group (DMRG) study with anisotropic dispersion, in particular we
show αCF as a function of αF for N = 2, 4, and 6. In agreement with the result for
97
anisotropic pseudopotentials, we find that the response of αCF to αF drops sharply
as N increases. Our results suggest that distortions with angular number N > 2 have
a very small effect on the CFL for reasonable values of αF such as those encountered
in experiment.
5.4.1 Model CN-symmetric bands
We study the following Hamiltonian:
H =∑i
E(πi) +Hint, (5.20)
where E(π) is the electron dispersion andHint is the e-e interaction, which we normally
take to be the Coulomb potential. We want a dispersion E(k) that satisfies the
following requirements:
i. It has CN symmetry.
ii. It is homogeneous: E(k) = kaE(k) (this guarantees the shape of the Fermi
contour is independent of εF ).
iii. It is a polynomial in the components kx, ky (this guarantees it is well-defined
as an operator once the magnetic field is turned on, k 7→ π).
iv. It is bounded from below (this guarantees it has a finite-energy ground state).
Such a function can easily be constructed by taking the homogeneous exponent a to
be N :
Eλ(k) ≡ E0(`0k)N(1 + λ cos(Nθ)) , (5.21)
where (k, θ) are polar coordinates in two-dimensional momentum space and E0, `0
are arbitrary units of energy and length. This clearly has the desired CN symmetry.
Moreover, by observing that kN cos(Nθ) = Re(keiθ)N = Re(kx + iky)N , we see that
98
this is a homogeneous polynomial of degree N in kx, ky. To ensure it is bounded from
below we must choose −1 < λ < 1. The zero field electron Fermi contour for the
dispersion in Eq. (5.21) is given by
kF (θ) =A
`0
(1
1 + λ cos(Nθ)
)1/N
(5.22)
where the dimensionless prefactor is A = (εF/E0)1/N . The dispersions E−λ and Eλ are
related by a π/N rotation, so we can assume λ ≥ 0 without loss of generality. The
anisotropy parameter αF can be expressed in terms of λ:
αF =kF (π/N)
kF (0)=
(1 + λ
1− λ
)1/N
. (5.23)
While the choice of dispersion Eq. (5.21) is guided by simplicity, we expect it to
capture the main features of realistic distortions with the given symmetry. We thus
contend that our main result, which is the rapid decrease with N of the effect of
N -fold anisotropy of the zero-field Fermi contour on the CFL, is generic and not
particular to this special class of anisotropic dispersions.
As in the situation discussed in Sec. 5.3, the eigenfunctions of Eq. (5.21), which
we denote by |0〉, |1〉, . . . , can be expressed in terms of the isotropic Landau levels
|0〉, |1〉, . . . . Once we project to the generalized LLL 0, electrons interact via the
effective potential
V (q) = |F00(q)|2V (q), (5.24)
where V (q) = 2π/q is the Coulomb potential and F00(q) is the form factor of the
|0〉 anisotropic LL, defined in Eq. (5.15). Examples of the LL mixing coefficients un
in |0〉 =∑
n un|n〉, as a function of the anisotropy parameter αF and the rotational
symmetry N , are given in Fig. 5.7. A few properties of the mixing are apparent
from the figure. First of all, a CN -symmetric dispersion only mixes Landau levels
99
1.0
0.01|un|
N=2 αF = 2.00
αF = 4.00
αF = 8.00
1.0
0.01|un|
N=4
0 20 40 60 80 100n
1.0
0.01|un|
N=6
Figure 5.7: Landau level mixing coefficients un for the ground state of the two-, four-and six-fold symmetric dispersion in a high magnetic field, at several values of theFermi contour anisotropy αF . Larger anisotropy leads to a stronger mixing withhigher Landau levels. Only Landau levels with n multiple of N are involved.
differing by an integer multiple of N ; in particular, the ground state |0〉 is a mixture
of |0〉, |N〉, |2N〉, . . . . Secondly, the amplitudes un decay exponentially at large n,
and the decay gets slower for increasing αF and N . Finally, the leading anisotropic
contribution (n = N) gets weaker for increasing N .
5.4.2 Anisotropic Pseudopotentials
It is instructive to express the effective potential in Eq. (5.24) in terms of the gener-
alized pseudopotentials of Ref. [234]. Such pseudopotentials can be written as V σm,n,
with m,n non-negative integers and σ = ±. The generalized pseudopotentials with
n = 0, σ = + are the traditional Haldane pseudopotentials, and are the only terms
that appear for isotropic interactions. For N -fold anisotropy, terms with n ≥ N
100
appear in addition to n = 0. As in the case of traditional pseudopotentials, only
terms with odd m contribute for fermions. In this approach, the true anisotropy of
the system in a high magnetic field is best estimated not from the dispersion relation
in Eq. (5.21) but from the effective potential in Eq. (5.24), whose anisotropy can be
quantified by means of the generalized pseudopotentials. Since these form a complete
basis, we can expand Eq. (5.24) in terms of these pseudopotentials and compare the
size of the contributions at n = 0 (the isotropic part) with those at n > 0.
Some technical comments are in order. The effective interaction Eq. (5.24) is not
normalizable, i.e.∫d2q(V (q))2 = ∞ due to the q → 0 singularity in the Coulomb
potential. While this is in principle cured by charge screening, we note that this only
appears in the m = n = 0 pseudopotential, which does not affect fermions. The only
terms which are relevant to the physics of the system under consideration are those
with odd m, with n multiple of N , and with σ = + (σ = − would appear if we
rotated our coordinates). We thus measure anisotropy as follows: we compute the
coefficients
C+m,n ≡
∫d2qV (q)V +
m,n(q), (5.25)
then we compute the total norm of terms at a given n as
χn ≡√∑
m odd
(C+m,n)2 , (5.26)
and finally we normalize it by the norm of the isotropic part, χ0, computed in the same
way. The sum over m should range over all odd positive integers, but it converges at
finite m (typically m . 100) for the anisotropies studied here.
Our results for the leading anisotropic contribution, χN/χ0, are shown in Fig. 5.8
and reveal that the effect of a given kinetic energy anisotropy becomes rapidly smaller
for increasing N , i.e. the effective Hamiltonian Eq. (5.24) is more robust to higher an-
gular momentum distortions. Importantly, this statement is true at the Hamiltonian
101
1.0 2.0 3.0 4.0 5.0αF
0.00
0.05
0.10
0.15
0.20
χN/χ
0
(a)N = 2N = 4N = 6
1.0 2.0 3.0 4.0 5.0αF
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40(b)
Figure 5.8: (a) Relative weight of leading anisotropic pseudopotentials in the effectiveinteraction Eq. (5.24), as a function of anisotropy αF , for N = 2, 4 and 6. Thequantity χn is defined in Eq. (5.26). (b) Same quantity plotted for the anisotropicCoulomb interaction defined in Eq. (5.27). In both cases the growth with αF is slow(logarithmic or sub-logarithmic) and the effect decreases quickly for increasing N .
level, and thus in principle applies to states at all fillings. A practical consequence
is that, for gapped states, anisotropy-driven transitions to states with discrete CN
rotational symmetry (N > 2) are expected to occur only at very large values of the
symmetry-breaking field 1, if at all. For the gapless CFL state at filling ν = 12, the
expectation is that the Fermi contour should be significantly stiffer than it is for the
elliptical (N = 2) case [85], where αCF ∼ α0.5F .
Before testing this expectation numerically, we can also ask what happens when
the interaction, rather than the dispersion relation, is anisotropic. We note that this
scenario is not as well-motivated experimentally. The most natural way to engineer
an anisotropic interaction with CN -symmetry in a real system would be to use a
cold atomic system composed of “molecules” of positively and negatively charged
atoms with N -fold symmetry. Note that the molecules cannot be neutral, since in
that case they could orient themselves to have attractive interactions, which in a LL
generically leads to bound states instead of a quantum Hall phase. For molecules
1This is under the assumption that said transitions occur when the effective interaction hasanisotropic pseudopotential components of order O(1).
102
with a net charge, the long-range interactions would be dominated by the isotropic
Coulomb repulsion. This falls off as r−1, while the anisotropic CN -symmetric part,
which comes from higher multipoles of the molecule, decays as r−(2N+1) (e.g. r−3
for two dipoles, N = 1). Therefore such molecules will have only weakly anisotropic
interactions. We attempted to make an analogue of Fig. 5.8(a) for such molecular
objects, and found that increasing N by 2 decreased χN/χ0 by approximately four
orders of magnitude.
Though it is less physically motivated, we can also study an anistropric interaction
by making the following change to the Coulomb interaction:
1
r→ 1
r[1 + λ cos(Nθ)]1/N. (5.27)
This yields equipotential contours that have the same shape as the Fermi contours
described by Eq. (5.22). χN/χ0 for such potentials are shown in Fig. 5.8(b). We
see qualitatively similar behavior as for anisotropic dispersion relations, with χN/χ0
growing sublinearly with αF and shrinking as N is increased. Thus we see that
the interaction anisotropy, for N > 2, is extremely small for physical models, or
qualitatively similar to anisotropic dispersion for ad hoc models. Consequently, in
the next Section, we consider only anisotropy arising from the dispersion.
5.4.3 Numerical results
We numerically calculate αCF as a function of αF for N = 4 and 6. We obtain the
ground state of a system of electrons at half filling of the generalized LLL of the
dispersion in Eq. (5.21) and isotropic Coulomb interactions on an infinite cylinder
using iDMRG. We then map out the CFL Fermi contour by locating singularities in
the guiding center structure factor, as described in Chapter 4.
103
0.96
0.98
1.00
1.02
1.04
kCF
N=4
αF = 4
N=6
0 π/8 π/4 3π/8 π/2
θ
0.96
0.98
1.00
1.02
1.04
kCF
0 π/8 π/4 3π/8 π/2
θ
αF = 14
L = 13L = 18L = 19L = 20L = 21
Figure 5.9: Fermi wavevector kCF as a function of angle θ, for N = 4 (left) andN = 6 (right) and band anisotropy αF = 4 (top) and αF = 1/4 (bottom). ChangingαF 7→ 1/αF rotates the dispersion by π/N and thus these values of αF providea consistency check for our method. The solid line represents a fit to kCF (θ) =A/(1 + λCF cos(Nθ))1/N , Eq. (5.22). From λCF one can estimate αCF via Eq. (5.23).The N = 4 data shows anisotropy, whereas the N = 6 Fermi contour appears circularwithin numerical accuracy (i.e. the effect of band anisotropy on the CFs, if present,is too small to be detected).
Examples of the data that we gather with this method are shown in Fig. 5.9. In
the figure we show the locations of the observed singularities in polar coordinates,
i.e. the Fermi wavevector kF as a function of the angle around the Fermi contour
θ. The resulting data should satisfy Eq. (5.22), but with λ replaced by λCF , from
which αCF can be derived using Eq. (5.23). Each plot aggregates singularities from
different system sizes (three to five distinct values of the cylinder circumference rang-
ing between 13`B and 21`B) at the same N and αF . The prefactor A in Eq. (5.22)
for the CFs is set by Luttinger’s theorem. In a purely two-dimensional system, the
Fermi contour would need to enclose an area of π`−2B in momentum space, and this
would set A = 1. At the finite sizes we can simulate, our requirement is instead that
the sum of the length of all the cuts across the Fermi surface totals L`−2B /2 – see
Eq. (4.5). A is modified from 1 to satisfy this constraint. In practice, we find this
104
1 2 4 8αF
1.00
1.02
1.04
1.06
1.08
1.10
αCF
N=2N = 4N = 6
Figure 5.10: CF Fermi contour anisotropy αCF , extracted from fits such as thoseshown in Fig. 5.9, for αF in the range of 1 to 4
√2, for the four- and six-fold sym-
metric one-electron dispersions defined in Eq. (5.21). Each datapoint is obtainedby combining the estimates obtained at αF and 1/αF , which are related by a π/Nrotation (as in the top and bottom panels of Fig. 5.9).
deviation is typically of order 1 to 2%, which can be comparable in magnitude to the
modulation that we seek to measure. Therefore we rescale each kF by the average of
all the kF values measured at that size. This allows a more accurate comparison of
different system sizes.
By fitting the data points at each αCF to Eq. (5.22), we can extract the CF Fermi
contour anisotropy parameter αCF as a function of αF . The results are shown in
Fig. 5.10. We see that the dependence of αCF on αF gets much weaker as N is
increased. While the results for N = 2 suggested αCF ≈ α0.5F (see Chapter 4, in this
case we consistently find αCF to be very close to 1 for N = 4 and 6. In particular,
the data for N = 4 deviates from a power law, possibly saturating at αCF ' 1.05 (for
dipolar 1/r3 interaction, we find αCF goes up to ∼ 1.1, but also seems to saturate).
For N = 6 the data is entirely consistent with αCF = 1 within numerical uncertainty.
The situation is summarized in Fig. 5.11. The CF Fermi contour (thin red line) for
N = 6 is not distinguishable from a circle (green dots), whereas for N = 4 there is a
small but systematic difference, visible upon close examination. This illustrates the
105
N=2
kF
kCF N=4 N=6
Figure 5.11: Fermi contours for the electrons (thick blue) and the CFs (thin red), fordifferent values ofN , for the same value of zero-field Fermi contour anisotropy, αF = 4.αCF is obtained from Fig. 5.10. The green dots correspond to a circular (isotropic)Fermi contour for reference. The gray lines indicate the accessible momenta on acylinder of circumference Ly = 20`B. The guiding center structure factor S(qx, 0) issingular at values of qx for which the gray lines connect two points on the CF Fermicontour. We use this feature to determine αCF .
drastic reduction in the magnitude of the effect when going from N = 2 to N = 4
and then N = 6.
5.5 Discussion
We have investigated the connection between the Fermi contours of zero-field carriers
(electrons or holes) and their high-field CF counterparts. We divided our analysis
into three cases of interest: isotropic zero-field dispersions, potentially giving rise to
a Fermi sea consisting of one or more annuli; anisotropic dispersions giving rise to
multiple disconnected pockets; and anisotropic dispersions with higher-order discrete
rotational symmetry (CN with N > 2).
In all cases, we find the effects on the CFL to be quite subtle. An exact analytical
argument allows us to prove that the CFL is completely insensitive to the rotationally-
symmetric distortion, so that its Fermi contour remains a circle, regardless of whether
the system at zero field develops a disconnected Fermi sea. A similar conclusion
extends to the many-valley case, where the CFL either has a single connected Fermi
106
sea, or transitions into different phases, depending on the separation of the pockets
and the magnetic field. In the third case, while we do observe transference of the
C4-symmetric anisotropy to the CFL, the magnitude of the effect is very small. For
N ≥ 6 the effect is smaller than our numerical accuracy, and thus most likely not
measurable in experiment.
The model bands we have used to study the effects of CN symmetry were designed
ad hoc and as such may be unrealistic. In particular, the absence of k2 terms for N > 2
is non-generic. However, restoring such isotropic terms would make the problem less
anisotropic overall. Thus our model bands capture the essential features of distortions
with the given symmetry, and a bound on their effect is a fortiori a bound on the
effect of more realistic dispersions. Our results show that the quadrupolar (N = 2)
distortion, corresponding to band mass anisotropy, has by far the strongest effect,
with the effect of higher angular numbers decaying rapidly. This indicates that finer
structures in the zero-field problem are more efficiently washed out at high field.
Finally we remark on how our results apply to quantum Hall states beyond the
CFL. The discussion in Sec. 5.4.2 focuses on the Hamiltonian itself, so its conclu-
sions should apply regardless of filling fraction, and in particular to compressible and
incompressible states alike. It suggests that the effect of anisotropy with CN symme-
try, as measured from the anisotropic pseudopotential decomposition, drops rapidly
with N . It would be interesting to test this prediction numerically for N > 2. One
barrier to doing this is that the unlike the N = 2 case, for N > 2 it is not obvious
how to define the anisotropy parameter for a gapped FQH state. In Ref. [114] we
do so by using the equal-value contours of S(q) near q = 0, generalizing the method
introduced in Chapter 3. For C4-symmetric anisotropy, we find that the ν = 1/3
and 1/5 FQH states behave similarly to the ν = 1/2 CFL, though with a somewhat
stronger response. As S(q) is quartic, this method enables the study of CN -symmetric
107
distortions for N ≤ 4 only; for higher N , the anisotropic effect becomes sub-leading,
and thus hard to detect.
Another issue in going form band mass to higher-order anisotropy is that, while in
the former case the anisotropy parameter is related to the intrinsic metric, and can be
thought of as a coordinate rescaling that makes the state approximately isotropic, no
such interpretation is currently available for N > 2. Model wavefunctions with higher-
order anisotropy can be obtained by displacing the zeros of the Laughlin wavefunction
in patterns of suitable rotational symmetry [151, 29]. This is subject to a filling
constraint: the ν = 1/m Laughlin state allows CN -symmetric deformations with
N ≤ m− 1 (one zero is fixed by antisymmetry). The finding that ν = 1/5 responds
more strongly than ν = 1/3 to C4-symmetric distortions [114] may indicate that this
type of model wavefunctions do indeed play a role. However it would be desirable to
extend the coordinate rescaling available for N = 2 to a more general transformation
that captures the essential effects of N > 2 anisotropy. Whether this is possible, and
how it would connect to a higher-rank tensor generalization of the intrinsic metric, is
a challenging but fascinating direction for future research.
108
Appendices
5.A Electron and CF Fermi seas with different
topologies
In this Appendix we show an example of circularly symmetric dispersion where the
electron Fermi sea can be made to have any number of disconnected components by
tuning the Fermi energy, while the CFs always have the same circular Fermi sea.
This complements and generalized the result of Sec. 5.2 about the annular Fermi sea
obtained from a quartic dispersion.
Consider the function f(x) = 1 − sin(2πx)/2πx, i.e. the one-electron isotropic
dispersion
E(k) = E0
(1− sin(πk2/k2
0)
πk2/k20
). (5.28)
This dispersion asymptotically approaches E0 as k →∞, with infinitely many minima
that get progressively shallower (we neglect effects of the finite Brillouin zone size
here). Thus, as the Fermi energy approaches E0 from below, one gets a Fermi sea
consisting of arbitrarily many rings. Specifically, A(εF ) (the area of the Fermi sea in
k-space at chemical potential εF ) is a continuous, monotonically increasing function
that goes from A = 0 at εF ≤ 0 to +∞ at εF ≥ E0 (where the Fermi sea would include
all but a finite area of momentum space). It diverges like |εF − E0|−1 as εF → E−0 .
Since A(εF ) spans all positive real numbers as εF → E−0 , a value of εF can be chosen
109
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0k/k0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
E/E
0
ε(k) εF εN
Figure 5.A.1: Dispersion of Hamiltonian in Eq. (5.28) (solid line) with Fermi energyεF = 0.953E0 (dashed line) and energy eigenvalues for the lowest 3 Landau levels(dots). Inset: shape of the corresponding zero-field Fermi sea, with a circle and 3rings.
such that the lowest Landau level energy ε0 (in the notation of Eq. (5.3)) falls in a
local minimum of f(x), which occurs near each integer value of x. Given one such
minimum xn ' n, f is such that f(x) > f(xn) for all values x > xn. Therefore, all
higher Landau levels N > 0 are guaranteed to have εN > ε0, so that argminN(εN) = 0
and the ground state is a ν = 12
CFL made from the familiar N = 0 Landau orbitals;
hence its Fermi sea is circular.
The (arbitrary) choice of minimum n also determines the number of rings in the
corresponding zero-field Fermi sea. It is straightforward to prove that this number
scales as O(n) and that therefore it can be made arbitrarily large. An example is
illustrated in Fig. 5.A.1, where n = 1 gives rise to a circle and 3 annuli.
110
Part II
Quantum criticality in the lowest
Landau level
111
Chapter 6
Integer quantum Hall transition in
a fraction of a Landau level
6.1 Introduction
This Chapter presents a study of the phase transition between two conductance
plateaus in the integer quantum Hall (IQH) regime, in which the Hall conductance
σxy jumps from zero on one side to the topologically quantized value σxy = e2/h on
the other. This is known as an IQH plateau transition, and is briefly reviewed in
Sec. 2.2. Such a transition results from the interplay of a topological invariant (the
Landau level’s Chern number) and electron localization due to disorder. We leave
out electron-electron interactions for this study.
Focussing on the LLL for concreteness, one has a two-dimensional disordered
non-interacting problem, whose fate is generically Anderson localization; at the same
time, a topologically quantized Hall conductance σxy = C e2
h(C = 1 is the Chern
number) must be carried by some single-particle states. As a result, the localization
length ξ, which characterizes the decay of localized wavefunctions at long distance
as ψ(r) ∼ e−|r|/ξ, cannot be finite across the entire spectrum. The resolution to this
112
seeming paradox is that the localization length must diverge at a critical energy Ec:
ξ(E) ∼ 1
|E − Ec|ν, (6.1)
where ν is a critical exponent. Single-particle eigenstates in the vicinity of Ec can
carry nontrivial Chern numbers (both positive and negative, adding up to +1), while
away from Ec states are exponentially localized and topologically trivial. The presence
of a divergent length scale makes it possible to formulate scaling theories for this
problem [77, 19, 20].
The critical exponent ν is a quantity of special interest: due to its universality,
it offers the possibility of a direct comparison between single-particle theory and
experimental measurements [226, 41, 125]. Such a comparison may tell us whether
interactions are relevant or irrelevant (in the renormalization-group sense) at the
critical point, which would be a crucial step towards a more complete understanding
of the quantum Hall phase diagram [110]. For this reason, many numerical studies
have tried to determine the value of ν with a variety of methods: directly calculating
ξ(E) from electron transmission calculations, either for the original continuum LLL
problem [78, 76] or the Chalker-Coddington network model [27, 121, 197, 159]; using
the Fermi energy dependence of the Hall [20, 86], Thouless [171, 86], or longitudinal
conductance [224]; or studying the Chern number of individual eigenstates in the
disordered problem [79, 7, 239, 240].
To calculate ν from a finite-size scaling analysis, it is crucial to achieve as large
a system size as possible. In this Chapter, we introduce an idea that has the poten-
tial of dramatically increasing the reach of single-particle numerical simulations, by
removing from the problem orbitals that do not participate in the critical behavior
and thus reducing the dimensionality of the computational problem for a fixed phys-
ical system size. The approach is motivated by Prange’s realization [169] that the
113
Hall conductance is robust to the presence of impurities, which came soon after the
experimental discovery of the IQH effect [111], and before Laughlin [118] and Thou-
less [211] discovered its connection to topology. Prange’s work showed that if a single
impurity (represented by a δ-function potential) is placed in a quantum Hall system,
the Hall conductance does not change: while one electron’s cyclotron orbit becomes
bound to the impurity, the other ones avoid the impurity and exactly compensate
the loss of conductance due to the bound state. These impurity-bound orbitals have
been recently observed[42] in spectacular STM images of IQH states on the surface
of bismuth.
In this Chapter, we consider a generalization of the above problem to an arbi-
trary number Nδ of impurities represented by δ-function potentials (which we call
“δ-impurities” in the following). We find that the same conclusion as Prange’s single-
impurity case applies: there is a manifold of states which are not bound by the
δ-impurities, and which carry the quantized Hall current corresponding to the entire
Landau level. In other words, we show that the subspace of electron states which
avoid all δ-impurities has a total Chern number [71] C = +1. We thus have a sub-
space of degenerate electron states with C = 1, analogous to a Landau level of reduced
dimension N ′φ ≡ Nφ − Nδ. As such, this “fraction” of a Landau level should exhibit
an integer quantum Hall transition. Projecting the dynamics onto this lower dimen-
sional space, which can be made arbitrarily smaller than the original Landau level
subspace, would seem to offer access to much larger sizes than otherwise possible in
the full LLL problem. Based on this intuition, we carry out a study of Hall as well
Thouless (longitudinal) conductance for varying degrees of “dilution” of the Landau
level Hilbert space and various geometric distributions of δ-impurities.
With a lattice of δ-impurities with identical or periodically varying strength, in the
regime Nδ < Nφ, one has, in addition to the flat C = 1 band, one or more dispersive
bands with varying Chern character, depending on the nature of the lattice. This
114
allows one to create Chern insulator models of different kinds, which have been of
increasing interest in tight binding models; here they arise out of a single Landau
level. Thus this model may also offer a rich variety of phases upon addition of
electron-electron interactions, such as fractional Chern insulators[178, 130, 164, 201].
It also offers the possibility of many-body localization[116, 115], which appears to
not be possible in a single Landau level subject to a random (e.g. white-noise)
potential [157, 52].
This Chapter is structured as follows. In Sec. 6.2 we present general facts about δ-
function potentials in the lowest Landau level, including the existence of a degenerate
subspace at the original Landau level energy. In Sec. 6.3 we discuss the total Hall
conductance of the above subspace and show that its total Chern number is +1.
Sec. 6.4 discusses δ-function potentials with lattice symmetry and the structure of
the subbands they give rise to. Sec. 6.5 we present numerical calculations of the
Hall and Thouless conductance in the presence of disorder, both with and without
δ-impurities, and show that the quantum Hall transition seems to only depend on the
number of states left at the Landau level energy, while being completely unaffected
by the states localized by the δ-impurities. Finally, we summarize our results and
discuss avenues for future research in Sec. 6.6. The results presented in this Chapter
have been published in Ref. [86].
6.2 Delta-function potentials in the LLL
We consider a two-dimensional electron system in a strong magnetic field, so that the
cyclotron gap is infinite for practical purposes. We add a set of δ-impurities so that
the Hamiltonian within the lowest Landau level is H = 12~ωc + V (r), with
V (r) =
Nδ∑i=1
λiδ(r− ri) . (6.2)
115
Here ri : i = 1, . . . Nδ are the positions of the impurities and λi denote the
strength of each impurity. In the following we discard the constant energy shift 12~ωc
and refer to the Landau level energy as “zero energy”, or E = 0, for simplicity. Let
|ψn〉 : n = 1, . . . Nφ be an orthonormal basis of states in the LLL. The matrix
element of V between any two such basis states is
Vmn = 〈ψm|V (r)|ψn〉 =∑i
λiψ∗m(ri)ψn(ri) = (v†Λv)mn , (6.3)
where we defined the Nδ ×Nφ matrix vin ≡ ψn(ri) and the Nδ ×Nδ diagonal matrix
Λij ≡ λiδij. Eq. (6.3) implies that the kernel of V contains the kernel of v, which by
definition consists of wavefunctions that vanish on all impurities: a vector α in the
kernel of v defines a wavefunction ψ(r) such that
ψ(ri) =∑n
αnψn(ri) =∑n
vinαn = 0 ∀ i . (6.4)
There are Nφ−Nδ independent states with this property (provided Nφ ≥ Nδ). Gener-
ically these are the only zero-energy states present, i.e., for random values of the
positions ri and strengths λi of the δ-impurities, all other eigenvalues are non-
zero with probability 1. An important special case which we consider later is that of
δ-impurity lattices; in that case, it is possible to have zero-energy states even when
Nδ > Nφ, but only if the potential is not of a definite sign. If V is, for example,
positive-definite (i.e. if all δ-impurities are repulsive), then an eigenstate ψn that
does not vanish on some impurity i satisfies
En = 〈ψn|V |ψn〉 ≥ λi|ψn(ri)|2 > 0 . (6.5)
116
Thus if the δ-function potential is positive-definite, it has exactly (Nφ − Nδ) zero-
energy states. These have vanishing amplitude on all impurities. The remaining Nδ
states have E > 0 and have non-zero amplitude on at least one impurity.
Throughout the rest of the paper, we use the notation L(Nφ, Nδ) to denote the ker-
nel of a positive-definite potential with Nδ δ-functions in the lowest Landau level of
a system with Nφ flux quanta. In particular, L(Nφ, 0) denotes the whole Hilbert
space of a LLL with no impurities. Based on the previous discussion, we have
dimL(Nφ, Nδ) = Nφ −Nδ.
6.3 Hall conductance of LLL with δ-impurities
In this Section we compute the Chern number [71] of the completely filled L(Nφ, Nδ)
subspace and show that it always equals +1, regardless of the strength and position
of the δ-impurities, provided Nφ > Nδ. For simplicity we consider a rectangular torus
with sides Lx, Ly, but this assumption is not essential.
6.3.1 Slater determinant wavefunction
Consider the fully-occupied lowest Landau level L(Nφ, 0). The corresponding many-
electron wavefunction is obtained as a Slater determinant from single-electron wave-
functions and can be written in the Landau gauge as [65]
Ψ(zi) = e−12
∑i y
2iFcm(Z)
∏i<j
f(zi − zj) , (6.6)
where f and Fcm are holomorphic functions of the complex argument z = x + iy,
f is odd and Z =∑
i zi is the center-of-mass coordinate. Quasiperiodicity can be
used to constrain F and f , and eventually this wavefunction can be used to prove the
quantization of the Hall conductance for the ν = 1 integer quantum Hall effect.
117
In a similar way, we introduce the wavefunction for the completely filled L(Nφ, Nδ)
subspace. This contains Nφ−Nδ electrons which are constrained to avoid the impurity
sites, ηi. We require the ηi’s to be non-degenerate but make no other assumptions
about their spatial distribution. The most general wavefunction for N ′φ ≡ Nφ − Nδ
electrons in the lowest Landau level which vanishes on all impurities is given by
Ψ0(zi) = exp
−1
2
N ′φ∑i=1
y2i
Fcm(Z)
N ′φ∏i<j=1
f(zi − zj)N ′φ∏i=1
Nδ∏j=1
f(zi − ηj) . (6.7)
Following Ref. [65], we have that f is expressed in terms of Euler’s Theta function
ϑ1(z|τ) ≡ −∑n∈Z
eiπ(n+1/2)+iπτ(n+1/2)2+2πi(n+1/2)z (6.8)
as f(z) = ϑ1(z/Lx|iLy/Lx). Similarly, for the center-of-mass wavefunction we have
Fcm(Z) = eiKZϑ1
(Z − Z0
Lx
∣∣∣∣iLyLx)
(6.9)
so that the only remaining degrees of freedom in the ansatz are the quasi-momentum
K and the center-of-mass node Z0. By imposing generalized periodic boundary con-
ditions with twist angles θx, θy, these are found to be
K =θxLx− π
Lx(2b+Nφ) , (6.10)
Z0 =Lx2
(Nφ +
θyπ
+ 2a
)+ iNφK −
∑i
ηi , (6.11)
where a, b are integers chosen so that Z0 is in the torus unit cell −Lx2≤ x < Lx
2,
−Ly2≤ y < Ly
2. It can be seen from Eq. (6.11) that the node of the center-of-mass
wavefunction, Z0, can be moved to an arbitrary position on the torus by suitably
adjusting the boundary twist angles θx, θy. This is known to be a signature of
118
non-zero Chern number [7]. However, we also prove that the C = +1 by a direct
computation of the Berry curvature integrated over the torus of boundary angles θi.
6.3.2 Direct computation of the Chern number
The Chern number is defined as the suitably normalized integral of the Berry curva-
ture over the Brillouin zone:
C = − 1
2π
∫d2θ~∇θ × ~A(θ) (6.12)
where ~A = 〈ψ(θ)|i~∇θ|ψ(θ)〉 is the Berry connection. Reducing Eq. (6.12) to a bound-
ary integral gives
C =1
2πi
∮dθi〈Ψ0(θ)| ∂
∂θi|Ψ0(θ)〉 , (6.13)
which can be directly computed for the wavefunction in Eq. (6.7). The key observation
is that the θ dependence is only in the center-of-mass part of the wavefunction,
Fcm(Z; ~θ). This allows us to rewrite the integral as
C =1
2πi
∮dθi
∫dN′φx dN
′φy |Ψ0(zi; ~θ)|2
∂
∂θilogFcm(Z; ~θ) . (6.14)
Next, we split the contour integral into the four sides of the rectangle. Since a
2π change in the boundary angles can cause at most a phase shift, the weight
|Ψ0(zi; ~θ)|2 is the same for corresponding points on opposite sides; so contributions
from the four sides can be grouped as follows:
C =1
2πi
∫ 2π
0
dθ
∫d2N ′φzi
(|Ψ0(zi; θ, 0)|2 ∂
∂θlog
Fcm(Z; θ, 0)
Fcm(Z; θ, 2π)
+|Ψ0(zi; 2π, θ)|2 ∂∂θ
logFcm(Z; 2π, θ)
Fcm(Z; 0, θ)
). (6.15)
119
Figure 6.1: Spectrum of the δ-function square lattice potential as a function of mag-netic field. The ratio p/q equals the magnetic flux per unit cell, in units of Φ0 = h/e
(quantum of magnetic flux). The spatial density of δ-impurities is nδ = q/p
2π`2B. At small
p/q, nδ diverges; the electrons see a nearly uniform potential of strength nδ ∼ q/pand all energy levels diverge. For p/q > 1, a zero-energy flat subband appears. Forp/q 1, the δ-impurities are so dilute that hopping is exponentially suppressed, thusthe bandwitdth of all E > 0 bands decays exponentially with
√p/q.
Exploiting the quasiperiodicity of ϑ1 we finally get
C =1
2πi
∫ 2π
0
dθ
∫dN′φx dN
′φy |Ψ0(zi; 0, θ)|2 ∂
∂θ
(π
(i+
LyLx
)− 2πi
Z
Lx+ iθ
)=
∫ 2π
0
dθ
2π
∫dN′φx dN
′φy |Ψ0(zi; 0, θ)|2
=
∫ 2π
0
dθ
2π〈Ψ0(0, θ)|Ψ0(0, θ)〉 = +1 . (6.16)
While a much more straightforward proof applies to the case of lattices of δ-impurities,
this shows that the result applies independently of the position, strength or number
of the impurities, provided Nδ < Nφ.
120
6.4 δ-function lattice potentials
In this Section we discuss δ-function potentials with discrete translational symmetry,
where the δ-impurities are arranged on a lattice. We assume a Bravais lattice gener-
ated by vectors a1, a2 (though it would be easy to generalize this to a lattice with a
basis). We further assume the torus has sides L1 = N1a1, L2 = N2a2, so that there
is a total of Nδ = N1N2 δ-impurities. The potential is then given in real space by
Vδ(r) = λ
N1∑n1=1
N2∑n2=1
δ(r− n1a1 − n2a2) . (6.17)
Furthermore, we assume that the magnetic flux through each unit cell is Φ = pqΦ0,
where p and q are co-prime integers with p > q. In other words, we require |a1×a2| =
2π`2Bp/q.
The lattice symmetry allows us to pick a basis of eigenstates of Vδ which are
also eigenstates of the magnetic translations [246] τ(qa1), τ(a2) (the translations
commute only if they enclose an area containing an integer number of flux quanta).
The eigenvalues of magnetic translations define a quasi-momentum k, and the orbitals
can be written in a quasi-Bloch form as
ψk,n(r) = eik·ruk,n(r) , (6.18)
where n is a band index and the pseudo-Bloch wavefunction u has the quasi-
periodicity
uk,n(r + qa1) = e−2πi(qa1)×ruk,n(r) ,
uk,n(r + a2) = uk,n(r) .
(6.19)
121
The matrix Vδ is then block-diagonalized into quasi-momentum sectors, with each
block given by
V abδ (k) = 〈ψk,a|Vδ|ψk,b〉 = λ
N1∑n1=1
N2∑n2=1
ψ∗k,a(n1a1 + n2a2)ψk,b(n1a1 + n2a2) . (6.20)
We can replace the ψ functions with the pseudo-Bloch u functions, since the plane-
wave phase factors cancel out. Furthermore, by using Eq. (6.19), we obtain that all
terms related by a magnetic-unit-cell translations are identical, and thus the sum
reduces to
V abδ (k) = λ
Nδ
q
q∑n=1
u∗k,a(na1)uk,b(na1) . (6.21)
By defining the q × p matrix vn,a(k) ≡ uk,a(na1), we get the form V (k) ∝ v(k)†v(k).
The resulting bands are found by diagonalizing Vδ(k), which is a p × p matrix
of rank q. As such, it has q non-zero eigenvalues and a kernel of dimension p − q.
Practically, the projector P on the kernel of Vδ, which we will use in Sec. 6.5.2, is
obtained by diagonalizing Vδ(k) in the basis of quasi-Bloch wavefunctions as discussed
earlier, and then transforming to the basis of usual Landau orbitals on a torus,
φn(x, y) =1√Lπ1/2
∑p∈Z
e2πiy(n+Nφp)/Le− 1
2(x+ n
NφL+pL)2
.
The whole process takes computational time O(Nφp2), as opposed to the numerical
diagonalization of a generic δ-function potential without lattice symmetry which takes
O(N3φ). Furthermore, the resulting projector is sparse due to the quasi-momentum
quantum number, which speeds up the calculation of PVnP .
From the structure of the Hamiltonian block in Eq. (6.21), Vδ(k) ∝ v†(k)v(k), we
know that each Hamiltonian block is a p × p matrix of rank at most q, so (p − q)
eigenvalues are guaranteed to be zero. Thus we find that there are p subbands; of
122
these, (p− q) have E ≡ 0 identically. As an example, the spectrum of a square lattice
of δ-impurities as a function of the magnetic flux per unit cell p/q is shown in Fig. 6.1.
It displays the peculiar Hofstadter butterfly fractal pattern [73]. Moreover, the flat
zero-energy bands are clearly visible for p/q > 1.
The existence of this flat, zero-energy, C = 1 band is consistent with previous
studies of δ-function lattices in the presence of a magnetic field [50, 91, 89, 90] and
with the discussion in Sec. 6.2 about general δ-function potentials. There, we showed
that in the presence of Nδ impurities one has Nφ−Nδ exact zero-energy states. In the
lattice case, we have p Hofstadter subbands, hence Nφ/p states per subband; having
p− q flat E = 0 bands gives a total ofNφp
(p− q) = Nφ −Nδ zero-energy states. The
result also fits with the general theory of periodic potentials in the lowest Landau
level [211], which states that the Chern number C when the Fermi level sits in the
rth gap must solve the Diophantine equation
pC + qS = r , (6.22)
where S is another integer. One can equivalently express this constraint as C ≡
rp−1(mod q), where p−1 denotes the multiplicative inverse of p in Zq (which is well
defined as p and q are co-prime). As the flat band consists of p − q degenerate
Hofstadter subbands, the gap immediately above E = 0 corresponds to r = p − q,
which gives
C ≡ p−1(p− q) ≡ 1 (mod q) , (6.23)
which is always satisfied by C = 1 regardless of p, q.
While we have shown that C = 1 directly and for more general distributions
of δ-impurities in Sec. 6.3.2, in this case we can prove the same result much more
straightforwardly. The Chern number associated to a certain gap in the Hofstadter
band structure equals the derivative of the total number of states below the gap with
123
respect to Nφ, with the lattice potential held constant [205]. In our case this yields
C =∂(Nφ −Nδ)
∂Nφ
∣∣∣∣Nδ
= +1 . (6.24)
We conclude this Section by discussing some interesting features of the subband
structures introduced above. First, we see from Eq. (6.22) that, by tuning p and
q, one can engineer subbands with arbitrarily large Chern numbers. In the pres-
ence of interactions, these high-|C| subbands may host interesting strongly-correlated
phases, such as fractional Chern insulator states [178, 130, 164] (recently observed in
graphene-boron nitride heterostructures [201]) or analogues of multilayer FQH states
with topological defects known as genons [15, 94, 88].
Secondly, since the zero-energy subbands take up the entire Chern character of
the Landau level, the remaining q subbands taken together must have C = 0. The
simplest instance of this occurs for q = 1, when there is only one dispersive C = 0
subband with bandwidth decreasing exponentially in p. This provides a setting to
study localization in quantum Hall systems without the critical states that are nor-
mally present due to the topological character. This removes one crucial obstruction
to many-body localization in the system; other possible obstructions (the dimension-
ality, the continuum nature of the problem) remain. This platform therefore allows
us to isolate the role of topology in delocalizing the many-body spectrum. Results
of numerical exact diagonalization of the disordered many-body problem reveal an
ergodic-to-MBL crossover at finite disorder strength [116, 115]. While finite-size lim-
itations make it impossible to rule out a slow drift of the crossover with system size,
the behavior of trivial bands differs dramatically from that of topological bands. This
shows that the many-body delocalization of the LLL [52] is overwhelmingly due to
critical single-particle states.
124
6.5 Numerical study of the plateau transition
Having established the existence and basic properties of the zero-energy, C = 1 sub-
space in a LLL with Nφ flux quanta and Nδ δ-impurities, which we call L(Nφ, Nδ), we
can move forward with the study the integer quantum Hall transition by broadening
the subspace with the addition of disorder. Specifically, we consider the following
Hamiltonian,
H =1
2m
(p− e
cA)2
+ Vn(r) + λVδ(r) , (6.25)
where Vn(r) is a disordered potential (e.g. Gaussian white noise [77, 79]), Vδ(r) is
a sum of repulsive δ function potentials, and λ is a coefficient that determines the
relative strength of the δ-impurities compared to the disorder. Once the Hamiltonian
Eq. (6.25) is projected into the lowest Landau level, the first term is reduced to the
constant cyclotron energy offset 12~ωc. Denoting the LLL projector as PLLL, this
leaves
H =1
2~ωc + PLLL(Vn + λVδ)PLLL . (6.26)
Taking λ to be much larger than the disorder strength, but still much smaller than the
Landau level gap ~ωc so as to avoid Landau level mixing, the density of states looks
like the one depicted in Fig. 6.2: the δ-function potential singularity at E = 12~ωc gets
broadened by the white noise to a narrow peak; the rest of the states, with vanishing
total Chern number, lie above a large gap O(λ).
It is not obvious a priori how having some fraction of the wavefunction nodes
pinned down at specified locations should affect various quantities, such as the Hall
conductance σxy(E), the diagonal conductance σxx(E) or the Thouless conductance
g(E), as a function of the Fermi energy E. These quantities are expected to display
finite-size scaling collapse with a localization length critical exponent ν, since the
system undergoes a plateau transition. It is therefore interesting to check if the
exponent ν is the same as that of the “original” IQH transition. Further, if it is
125
0.50 0.25 0.00 0.25 0.50 0.75 1.00 1.25 1.50E/
(E)
Figure 6.2: Density of states ρ(E) of a typical sample with Nφ = 384 magnetic fluxesand Nδ = 256 repulsive δ-impurities arranged in a 16× 16 square lattice, in the LLLwith the addition of weak white-noise disorder. As Nφ/Nδ = 3/2 there are threesubbands with Chern numbers C = +1,−1,+1; the first, which has E ≡ 0 in theclean problem, displays a quantum Hall transition at a critical energy Ec ≈ 0. Theother states sit above a large gap O(λ) and have total Chern number C = 0.
indeed the same, finite-size scaling may be ruled by the Hilbert space dimension of
L(Nφ, Nδ), which is Nφ −Nδ, or by the physical size of the system L, in turn set by
Nφ. The latter scenario is the original motivation of this work: attaining effectively
larger system sizes for a given Hilbert space dimension, and thus pushing finite-size
scaling to sizes beyond current limits.
6.5.1 Impurity-free problem
First, we consider the problem without δ-impurities in the presence of white noise. We
assume a square geometry with periodic boundary conditions. We consider systems
with Nφ = 4n, n = 3, 4, 5, 6. These values correspond to tori of linear size L =
2n√h/eB. For each system size, we exactly diagonalize a large number of disorder
realizations shown in Table 6.1 to obtain the ensemble-averaged quantities. Each
disorder realization is represented by a Gaussian white noise potential Vn(r) of unit
strength, projected into the lowest Landau level [77]. For each realization, we compute
126
Nφ No. of realizations
64 106
256 106
1024 105
4096 104
Table 6.1: Number of disorder realizations used for each size in the impurity-freeproblem, Sec. 6.5.1.
the Hall conductance via the Kubo formula
σxy(E) =e2
hν(E) + ∆σxy(E) , (6.27)
where ν(E) is the filling fraction when all states up to energy E are occupied, ν(E) =
1Nφ
∫ E−∞ dερ(ε), and
∆σxy(E)
e2/h= − 2
Nφ
Im∑a,b
Ea<EF<Eb
〈a|∂xVn|b〉〈b|∂yVn|a〉(Ea − Eb)2
. (6.28)
We also compute the Thouless number [212]:
g(E) =〈|δE|〉E〈∆E〉E
, (6.29)
where ∆E is the level spacing and δE is the change in energy of an eigenstate under a
change of boundary conditions (periodic to anti-periodic) along one direction. 〈· · · 〉E
denotes averaging over eigenstates near energy E. The function g(E) is related to
the diagonal conductance, σxx, and is a measure of how localized a state is: high
sensitivity to boundary condition twists is a feature of extended or critical states,
whereas exponentially localized states are insensitive to such global operations. The
quantity g(E)e2/h is also known as the Thouless conductance.
127
0.4 0.2 0.0 0.2 0.4E
0.00.20.40.60.81.0
g(E
)/g(
0)
3 2 1 0 1 2 3E ·L 1/2.4
0.00.20.40.60.81.0
σxy(E
)Nφ = 64
Nφ = 256
Nφ = 1024
Nφ = 4096
Figure 6.3: Left: disorder-averaged conductances for an impurity-free Landau levelwith Nφ flux quanta, L(Nφ, 0). From top to bottom: Hall conductance σxy andThouless number g, defined in Eq. (6.29), normalized to 1. Right: the same quantitiesplotted as a function of E ·L1/ν . σxy and g show scaling collapse for ν ≈ 2.4, consistentwith the existing literature on this problem [77, 79].
The disorder-averaged curves for σxy and g, plotted in Fig. 6.3, show scaling
collapse when plotted as a function of E ≡ L1/νE. This collapse gives a localization
length critical exponent ν = 2.4± 0.1, consistent with known results for these types
of calculations [77, 79]. More recent calculations [197, 159] with transfer matrix
techniques on Chalker-Coddington network models suggest a larger exponent ν ≈ 2.6.
However, such large exponents are not seen in the continuum Landau level problem
even with larger sizes [252], and recent work on topologically disordered Chalker-
Coddington network models [59] suggests a possible reason for the discrepancy.
6.5.2 Square lattice of δ-impurities
We then add a square lattice of δ function potentials. We consider systems with 2
magnetic flux quanta per δ-function, i.e. (p, q) = (2, 1), and 5 magnetic flux quanta
for every 4 δ-functions, i.e. (p, q) = (5, 4). We fix Nφ−Nδ = 4n, so that the dimension
of L(Nφ, Nδ) is N ′φ ≡ Nφ −Nδ = 4n, allowing for a straightforward comparison with
128
Nφ Nδ N ′φ No. of realizations
128 64 64 106
320 256 64 106
512 256 256 106
1280 1024 256 105
2048 1024 1024 105
5120 4096 1024 104
8192 4096 4096 104
20480 16384 4096 2× 103
Table 6.2: Number of disorder realizations used for each size in the problem with asquare lattice of δ-impurities, Sec. 6.5.2. We take N ′φ = Nφ − Nδ = 4n and considertwo different concentrations of δ-impurities: Nδ/Nφ = 1/2, 4/5.
the system sizes studied in the previous case, without δ-impurities. Lattice symmetry
allows for an efficient diagonalization of this potential, as discussed in Sec. 6.4. The
resulting energy bands include one flat, zero-energy band and q dispersive bands lying
above a gap.
For each size, we diagonalize Vδ and obtain the projector P onto its kernel
L(Nφ, Nδ), then generate a large number of disorder samples, given in Table 6.2.
For each sample, we project the white noise potential Vn and obtain the effective
Hamiltonian Htrunc = PVnP . We diagonalize Htrunc and compute σxy and g as in the
previous case. While the computation of the Thouless number translates straightfor-
wardly, in the Hall conductance one cannot simply take Eq. (6.28) and replace Vn by
the projected PVnP . Contributions coming from virtual hopping of electrons to the
high-energy band of Vδ must be taken into account as well. Indeed, absent any extra
terms, the Hall conductance of the zero-energy band, based on Eq. (6.27), would
take the value σxy = e2
h(1 − q/p), inconsistent with the Chern number. A careful
perturbative analysis (see Appendix 6.A) shows that Vδ induces terms in the Kubo
129
0.4 0.2 0.0 0.2 0.4E
0.00.20.40.60.81.0
g(E
)/g(
0)
3 2 1 0 1 2 3E ·L 1/2.4
0.00.20.40.60.81.0
σxy(E
)
(a)
Nφ = 128, Nδ = 64
Nφ = 512, Nδ = 256
Nφ = 2048, Nδ = 1024
Nφ = 8192, Nδ = 4096
0.4 0.2 0.0 0.2 0.4E
0.00.20.40.60.81.0
g(E
)/g(
0)
3 2 1 0 1 2 3E ·L 1/2.4
0.00.20.40.60.81.0
σxy(E
)
(b)
Nφ = 320, Nδ = 256
Nφ = 1280, Nδ = 1024
Nφ = 5120, Nδ = 4096
Nφ = 20480, Nδ = 16384
Figure 6.4: (a) Same plot as Fig. 6.3, but for a fraction of the lowest Landau leveldefinied by Nφ flux quanta and Nδ = 1
2Nφ δ-impurities, L(Nφ, Nφ/2). The choice of
sizes is such that, for each curve, the truncated Hilbert space dimension (Nφ − Nδ)coincides with that of a curve in Fig. 6.3. (b) Same plot, but with a larger truncationNδ = 4
5Nφ. In both cases, the curves show scaling collapse with the same localization
length critical exponent ν = 2.4± 0.1.
formula which stay finite even in the λ → ∞ limit of strong impurities, as the large
energy denominators are compensated by equally large matrix elements.
The main benefit of using a lattice configuration of δ-functions is computational
speed: lattice symmetry ensures P has a sparse (block-diagonal) form, so that the
additional manipulations required to compute σxy are substantially faster than they
would be for a generic distribution of δ functions (for which P is generally dense).
130
Numerical results are shown in Fig. 6.4. The data again exhibits scaling collapse
with a localization length critical exponent ν = 2.4 ± 0.1. Moreover, the functional
forms of the Hall and Thouless conductances look the same as in the Landau level
problem without δ-impurities, Fig. 6.3. On the other hand, if for each pair (Nφ, Nδ)
we compare the data for L(Nφ, Nδ) and L(Nφ − Nδ, 0) from the two figures, we see
that in the former the slope of σxy(E) is steeper and the peak in g(E) is narrower.
This suggests the possibility that the observed behavior of L(Nφ, Nδ) might capture
some information about the plateau transition in a larger Landau level Hilbert space
L(N effφ , 0), with Nφ ≥ N eff
φ ≥ Nφ − Nδ. Following this hypothesis, by matching the
width of the scaling function features, we find N effφ ≈ 1.7Nφ at p/q = 2 and N eff
φ ≈
6.1Nφ at p/q = 5/4. This is manifestly unphysical, as the size of the entire Landau
level, before projection, is only Nφ: no information about the plateau transition in a
Landau level of larger size is present at any stage of the scheme.
There is another explanation of the observed behavior, which arises more naturally
by looking at the density of states. In Fig. 6.5 we compare the scaling functions and
density of states for L(Nφ, Nδ), at the two ratios we considered, and L(Nφ − Nδ, 0).
For clarity, we only show the largest size we studied, Nφ−Nδ = 4096, as all other sizes
yield analogous results. We find that the presence of the δ-impurities has a significant
effect: it reduces the effective noise strength, and thus the width of the density of
states. Upon rescaling the energy axis to account for the change in effective disorder
strength, we observe that the scaling functions σxy and g of L(Nφ, Nδ) overlap with
those of the whole Landau level without δ-impurities, L(Nφ −Nδ, 0). This holds for
all the sizes we studied.
The renormalization of the noise strength can be understood heuristically as fol-
lows. Localized orbitals are sensitive to the random potential averaged over a surface
area ∼ `2B,
Vn(`B) = `−2B
∫r<`B
d2rVn(r) . (6.30)
131
0.00.20.40.60.81.0
σxy(E
)
0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4E
0.00.20.40.60.81.0
g(E
)/g(
0)
0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4E/f
0.00.20.40.60.81.0
ρ(E
)
Nφ = 4096, Nδ = 0, f= 1.00
Nφ = 8192, Nδ = 4096, f= 0.77
Nφ = 20480, Nδ = 16384, f= 0.49
Figure 6.5: Comparison between Landau levels with different concentration of δ-impurities and fixed Nφ − Nδ = 4096. Left: data for density of states ρ, Hall con-ductance σxy and Thouless number g as a function of energy E. Right: same dataon a rescaled energy axis E/f that matches the density of states in all three cases.The rescaling factor is f = 1 for L(4096, 0), f = 0.77 for L(8192, 4096), and f = 0.49for L(20480, 16384). Upon rescaling the energy axis, σxy and g also coincide withinnumerical accuracy.
Assuming uncorrelated Gaussian white noise, such that 〈Vn(r1)Vn(r2)〉 = W 2δ(r1−r2)
(W has units of length times energy), the variance of the locally averaged potential
scales like `−2B :
〈Vn(`B)2〉 =1
(π`2B)2
∫r1,r2<`B
d2r1d2r2〈Vn(r1)Vn(r2)〉
=1
(π`2B)2
∫r1<`B
d2r1W2 =
W 2
π`2B
. (6.31)
The pinned nodes of the wavefunctions act as magnetic flux tubes which let a fraction
q/p of the total magnetic field B through the sample. The remaining field B(1− q/p)
sets an effective magnetic length `?B = (1−q/p)−1/2`B. Thus localized orbitals average
132
the white noise over a larger area and, based on Eq. (6.31), experience a reduced
disorder strength Weff = W (1 − q/p)1/2. This simple derivation is in qualitative
agreement with what we observed for a variety of p/q ratios; for the cases discussed in
this Section (p/q = 2 and p/q = 5/4) it gives f = 1/√
2 ' 0.71 and f = 1/√
5 ' 0.45
respectively, reasonably close to the observed values of 0.77 and 0.49.
Remarkably, up to the aforementioned rescaling of the disorder strength (or, equiv-
alently, of the magnetic length), the plateau transition in a fraction of the lowest
Landau level looks exactly the same as in the full lowest Landau level of correspond-
ing dimension. This result suggests a notion of universality: besides the localization
length critical exponent, all quantitative details of the plateau transition in a flat,
C = 1 band depend solely on the dimensionality of the Hilbert space; they do not
seem to depend on other details of the states that make up such band, such as the
pinned nodes of the wavefunctions in the model we considered.
6.5.3 Different spatial distributions of δ functions
In order to test the universality of our result beyond the square lattice distribution
examined in the previous Section, we will repeat the analysis for different geometric
distributions of δ functions – a triangular lattice and a random distribution, shown
in Fig. 6.6.
We start with a triangular lattice. Given the irrational aspect ratio of the unit cell
(√
3/2), it is impossible to accommodate an integer number of δ functions on a square
torus in a perfectly triangular lattice. Either the torus or the lattice configuration
must be made slightly anisotropic. We use an anisotropic triangular lattice with
unit cell aspect ratio of 13/15, which approximates√
3/2 to one part in 103. We
fit 13 × 15 = 195 δ-impurities on a square torus this way, set the magnetic flux per
unit cell to p/q = 2, and study the plateau transition in the subspace L(390, 195) by
diagonalizing 106 disorder realizations.
133
Figure 6.6: The three distributions of δ-impurities we consider in Sec. 6.5. Fromleft to right: a 14 × 14 square lattice; a 13 × 15 triangular lattice; 195 δ-impuritiesdistributed randomly as discussed in the text.
Secondly, we consider a random distribution of δ functions, again in a system with
Nφ = 390 and Nδ = 195. We sample the positions from a uniform distribution on the
square torus, but require any two δ-impurities to be at leas 0.5`B away from each other
(for comparison, the spacing in the lattices we considered is ∼ 3`B) to ensure that the
projection on L(Nφ, Nδ) is numerically well-behaved. This is necessary because, if two
δ-impurities get very close to each other, Vδ can have very small, non-zero eigenvalues
arising from wavefunctions that have a node on one δ-impurity but not on the other.
This can close the gap above the E = 0 and make the projection ill-defined. A cut-off
of 0.5`B is sufficient to avoid this. We generate 102 such configurations and, for each
one, we diagonalize 104 disorder realizations.
These two test cases can be compared to the transition in the square lattice
problem L(392, 196), where the δ-impurities are arranged on a 14× 14 square lattice,
which we know is equivalent to L(195, 0) from the analysis presented in the previous
section (up to an O(10−3) error due to the small discrepancy in electron number).
For this case, too, we average 106 disorder realizations. The results are shown in
Fig. 6.7. For the two lattices, the density of states and scaling functions overlap
within uncertainty at all energies. For the random distribution, the density of states
134
0.00.20.40.60.81.0
ρ(E
)
0.00.20.40.60.81.0
σxy(E
)
0.6 0.4 0.2 0.0 0.2 0.4 0.6E
0.00.20.40.60.81.0
g(E
)/g(
0)
Nφ = 392, Nδ = 196, squareNφ = 390, Nδ = 195, triangularNφ = 390, Nδ = 195, random
Figure 6.7: Comparison between plateau transitions in the E = 0 subspace of thethree δ-impurity potentials represented in Fig. 6.6: a square lattice, a triangularlattice, and a random distribution. 106 disorder realizations are averaged for eachcase (for the random case, 102 distributions of δ-impurities are considered and foreach of them 104 realizations of white-noise disorder are averaged).
has larger tails and the scaling functions also show small deviations away from E = 0.
These tails arise from the spatial non-uniformity of the random distribution, as can be
seen in Fig. 6.6: states localized in regions without δ-impurities, or with an abnormally
low concentration of them, experience a stronger effective disorder. Nonetheless, in
the critical region, the forms of ρ, g and σxy coincide for all distributions. This
quantitative match further supports a notion of universality that goes beyond the
critical exponent: the diagonal and Hall conductivities match quantitatively for any
C = 1 flat band of a given dimensionality – be it a whole Landau level or a fraction
of a larger one.
135
6.6 Discussion
We have presented a method to isolate a “fraction” of the lowest Landau level (LLL)
by using δ-function potentials. This subspace shares many features with the original
LLL, including its total topological Chern character and its vanishing bandwidth. It
therefore undergoes a quantum Hall plateau transition, but its lower dimensionality
makes numerical study easier. Physically, each δ-impurity effectively binds, or local-
izes, one electronic state, so that Nδ impurities in an Nφ-dimensional LLL give rise to
a subspace L(Nφ, Nδ) of dimension N ′φ ≡ Nφ − Nδ consisting of electron states that
avoid all the δ-impurities.
We studied the integer quantum Hall plateau transition in this subspace by numer-
ical diagonalization of large numbers of disorder realizations. Our numerical results
indicate that the transition in the “fraction” of the LLL L(Nφ, Nδ) quantitatively
matches the one occurring in the whole LLL of a smaller system with (Nφ − Nδ)
magnetic flux quanta and no δ-impurities. The only effects of the δ-impurities are
(i) an effective reduction of the magnetic field through the system, or equivalently
an effective increase in the magnetic length; and, as a consequence of that, (ii) an
effective reduction in the strength of the disorder.
From this we conjecture that the plateau transition, and in particular the localiza-
tion length critical exponent, is the same for all flat C = 1 bands. Our results suggest
that a computational speedup for finite-size scaling studies of the plateau transition
cannot be achieved by retaining only a fraction of the LLL. It remains to be seen
how this conclusion changes in the presence of interactions. A particularly interest-
ing application of this setup in the interacting case may enable the study of FQH
states that would otherwise require exceedingly high magnetic fields to be observed.
Attractive δ-impurities may be used to artificially reduce the density of carriers in
quantum Hall systems by binding a fraction 0 < f < 1 of the carriers into inert,
topologically trivial states at large negative energies. Then one could obtain effective
136
filling fractions ν? of the flat C = 1 band with a larger overall LLL filling fraction,
ν = f + ν?(1 − f). This could allow, for example, to observe a ν? = 1/3 FQH state
at a physical filling ν much closer to 1, by trapping a fraction f = ν−ν?1−ν? of the single-
particle orbitals. This can lower the required magnetic field by a factor of up to 3,
in this example. The advantage over other ways of splitting the LLL into Hofstadter
subbands is the exact flatness of the resulting C = 1 subband. While generic periodic
potentials give rise to dispersive subbands, which can be partially filled to give rise to
strongly-correlated fractional Chern insulator, here one may actually replicate FQH
states. Recent progress towards the realization of artificial super-lattices in various
two-dimensional electron systems [45, 72] appears very promising in this regard.
137
Appendices
6.A Kubo formula for the Hall conductance with
δ-impurities
In this Appendix we do a perturbative analysis of the Kubo formula for the Hall
conductance σxy of states in the kernel of a δ-function potential in the LLL, in the
limit of very strong δ-functions.
Starting from the Hamiltonian Eq. (6.26) for the full LLL, we assume λ 1 and
work in perturbation theory in the small parameter η ≡ 1/λ 1. For convenience,
we drop the constant 12~ωc (which does not affect the Hall conductance), rescale H
by λ and define V ≡ Vδ + ηVn. The Kubo formula for the exact eigenstate |ψa〉 of V
reads
σxy(a) =e2
Nφh+ ∆σxy(a) , (6.32)
with
∆σxy(a)
e2/h= −2Im
∑b 6=a
〈ψa|∂xV |ψb〉〈ψb|∂yV |ψa〉(Ea − Eb)2
. (6.33)
States a that belong to the broadened E = 0 band have a perturbative expansion
|ψa〉 = |ψ(0)a 〉+ η|ψ(1)
a 〉+ . . . (6.34)
138
with Vδ|ψ(0)a 〉 = 0. Sorting the states based on increasing energy, the kernel of Vδ
corresponds to a ≤ Nφ −Nδ.
We expand the Kubo formula Eq. (6.33) using perturbation theory in η 1
and retain all contributions of order 1. We split the summation over b in Eq. (6.33)
between b > Nφ −Nδ and b ≤ Nφ −Nδ.
Terms with b > Nφ − Nδ have a O(1) denominator, so only O(1) terms in the
numerator matter. The only such term is 〈ψ(0)a |∂xVδ|ψ(0)
b 〉〈ψ(0)b |∂yVδ|ψ
(0)a 〉. So the sum
over b > Nφ −Nδ, in the limit η → 0, is
〈ψ(0)a |∂xVδ
∑b>Nφ−Nδ
|ψ(0)b 〉〈ψ
(0)b |
E2b
∂yVδ|ψ(0)a 〉 = 〈ψ(0)
a |∂xVδQV −2δ Q∂yVδ|ψ(0)
a 〉 , (6.35)
where Q is the orthogonal complement to P , the projector on the kernel of Vδ.
Terms with b ≤ Nφ −Nδ have an O(η2) denominator. For the limit η → 0 to be
finite, all terms O(1) or O(η) must vanish in the numerator. This is indeed the case,
since
〈ψ(0)a |∂iVδ|ψ
(0)b 〉 =
Nδ∑j=1
∂i(ψ(0)∗a ψ
(0)b )∣∣∣r=Rj
= 0 (6.36)
as the product ψ(0)∗a (r)ψ
(0)b (r) has a double zero at all impurity locations. This leaves,
as the next leading terms, products of pairs of the following terms, all O(η):
〈ψ(1)a |∂iVδ|ψ
(0)b 〉 = −〈ψ(0)
a |Vn∆i|ψ(0)b 〉 , (6.37)
〈ψ(0)a |∂iVδ|ψ
(1)b 〉 = −〈ψ(0)
a |∆†iVn|ψ
(0)b 〉 , (6.38)
〈ψ(0)a |∂xVn|ψ
(0)b 〉 , (6.39)
where we introduced the operators ∆i ≡ QV −1δ Q∂iVδ for i = x, y.
139
Putting the two contributions together, the overall result in the limit η → 0 is
∆σxy(a)
e2/h= − 2
Nφ
Im〈ψ(0)a |
∆†x∆y +∑
b<Nφ−Nδb 6=a
DxVn|ψ(0)b 〉〈ψ
(0)b |DyVn
(E(0)a − E(0)
b )2
|ψ(0)a 〉 , (6.40)
DiVn = ∂iVn − Vn∆i −∆†iVn, i = x, y . (6.41)
All the disorder-dependent data in Eq. (6.40) comes from the projected problem: the
|ψ(0)a 〉 and E
(0)a are respectively eigenvectors and eigenvalues of the projected potential
PVnP . The ∆i operators are solely functions of Vδ. As they do not depend on the
disorder realization Vn, they can be computed only once. This guarantees that the
computation of σxy projected to the subspace L(Nφ, Nδ) is almost as efficient as that
of σxy in L(Nφ −Nδ, 0).
140
Chapter 7
Dimensional crossover of the
integer quantum Hall plateau
transition and disordered
topological pumping
7.1 Introduction
In the previous Chapter we studied the IQH plateau transition in a square geome-
try: we placed the electron gas on a square torus and scaled both sides concurrently,
Lx = Ly ∼ N1/2φ (Nφ is the number of magnetic flux quanta through the system). As
the system size increases, the number of states with nonzero Chern number (here-
after simply called Chern states) diverges subextensively: only states with localization
length ξ & L, which appear delocalized in the finite-size system, can carry the Hall
current; but since the localization length diverges as ξ ∼ |E −Ec|−ν , these states oc-
cupy an energy window δE ∼ L−1/ν , hence a fraction ∼ N− 1
2νφ of the spectrum. Based
on this scaling argument, the localization length critical exponent ν can be extracted
141
from the distribution of Chern states [79]. The idea of using the Chern numbers of
individual eigenstates to characterize a transition has been applied successfully to a
variety of settings [239, 240, 195, 194, 218, 252].
The success of these methods in square geometry motivates their application to
rectangular geometries Lx > Ly with varying aspect ratio a = Lx/Ly, and particu-
larly in the quasi-one-dimensional limit a→∞ at fixed thickness. This is especially
interesting because the defining feature of the 2D problem (the presence of a topo-
logically robust Hall conductance, encoded in the Chern number C) does not have an
obvious one-dimensional counterpart. While the mathematical definition of C holds
regardless of system size or aspect ratio, on physical grounds the system in the quasi-
1D limit must be described by a local, disordered free-fermion chain – essentially the
Anderson model. The question then arises of what is the fate of Chern states in this
limit, and how the topological character of the LLL is manifested once the system is
mapped onto a 1D Anderson insulator.
One may reasonably expect, given the stronger tendency towards localization in
one-dimensional systems, that quasi-1D scaling will lead to a faster decay of the
fraction of Chern states relative to the 2D case, where the fraction falls off as N− 1
2νφ ;
the fastest possible decay, achieved if all states but one have C = 0, is N−1φ . In fact,
we find quite the opposite: Chern states do not vanish under 1D scaling. On the
contrary, they represent a finite fraction of all states and, as we argue later, they
asymptotically take over the entire spectrum!
As a byproduct of our Chern number calculations, we also obtain the (longitudinal)
Thouless conductance g [212]. Both the typical and average g decay exponentially
with Lx, as is expected for localized one-dimensional systems. Interestingly though,
we find that the average g retains a memory of the 2D critical scaling.
Existing studies of one-dimensional scaling of the integer quantum Hall prob-
lem [106, 206] focus on open boundary conditions, where the crossover is seen through
142
mixing of topological edge states on opposite edges of the strip. Our edge-free torus
geometry offers a different perspective on the problem and reveals fascinating and
unexpected behavior. Guided by our surprising numerical findings, we develop a
theoretical understanding based on a mapping to a disordered Thouless pump [210],
and uncover the invariant corresponding to the 1D limit of the Chern number. A
quantitative description of the proliferation of Chern states follows naturally from
this perspective. The results presented in this Chapter have been submitted for pub-
lication [81].
7.2 Density of Chern states
We consider a continuum model of two-dimensional (2D) electrons in a high perpen-
dicular magnetic field such that the dynamics can be projected onto the LLL. The
model is set on a rectangular torus with sides Lx, Ly such that LxLy = 2πNφ`2B,
where `B =√eB/~ is the magnetic length, which we set to 1 henceforth. We de-
fine the aspect ratio a = Lx/Ly and take a ≥ 1. Disorder in the system is modeled
by a Gaussian white noise potential. The torus has generalized periodic boundary
conditions with angles θx,y. These also represent magnetic fluxes through the two
nontrivial loops in the torus and are needed to define and compute Chern numbers
of individual eigenstates in the disordered problem.
For each disorder realization, we compute and diagonalize the single-particle
Hamiltonian on a lattice of boundary angles θ and store the eigenvalues En(θ) and
eigenvectors |ψn(θ)〉. The energies are used to calculate the Thouless conductance
gn ≡ Eθy [σθxEn(θ)] (E denotes averaging, σ denotes standard deviation), a measure
of sensitivity to boundary conditions in the long direction; the wavefunctions are
used to compute each eigenstate’s Chern number Cn via a standard numerical tech-
143
nique [49]. Further details on the model and the numerical method are provided in
Appendix 7.A.
With the method outlined above, we calculate the density of states with Chern
number C, ρC(E) : C ∈ Z. These obey
∑C∈Z
ρC(E) = ρ(E) ,
∫ E
−∞dε∑C∈Z
CρC(ε) = σxy(E) , (7.1)
where ρ is the total density of states and σxy is the Hall conductance. Past studies [79,
252] have characterized the 2D critical behavior by looking at the density of “current-
carrying states”, ρtop.(E) ≡ ρ(E)−ρ0(E). The width of ρtop. scales as N−1/2ν2Dφ in the
2D thermodynamic limit [henceforth we use ν2D (≈ 2.4) to denote the localization
length critical exponent in two dimensions].
In the present context, we observe completely different behavior. Namely, the
width of ρtop. does not vanish as Lx is increased. It stays roughly constant for a & 1,
and eventually starts increasing for a 1, as shown in Fig. 7.1. This increase is due
both to the broadening of ρ±1(E) (i.e. more pairs of Chern ±1 states appearing away
from the band center), and to an increase in higher-|C| states. Despite these effects,
the Hall conductance remains unchanged, and is determined by the shortest side of
the torus (see Appendix 7.B).
This extensive number of topological “current-carrying states” seems to be in-
compatible with the localized nature of the spectrum (which we verify independently
by means of the Thouless conductance). Reconciling these facts requires a careful
analysis of the fate of Chern numbers as the dimensionality is tuned from d = 2 to
d = 1 by increasing the aspect ratio a.
144
0.00
0.01
0.02
0.03
0(E)
(a)
1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00E
0.00
0.01
0.02
top(
E)
(b) N163264
128196256
Figure 7.1: Density of (a) C = 0 and (b) C 6= 0 states for fixed Ly = 10 and increasingNφ. The density of Chern C 6= 0 states ρtop.(E) grows and broadens at the expenseof ρ0(E).
7.3 Thin-torus limit
The above question is best addressed in the thin-torus limit Ly 1, though (as we
shall clarify later) the answer we find also applies to finite Ly, provided a is sufficiently
large. The LLL Hamiltonian in the thin-torus limit is approximated by the following
fermionic chain,
H1D =∑n
vnc†ncn + (tnc
†n+1cn + h.c.) , (7.2)
with
vn = V0(xn) , tn = eiθx/NφV1(xn) , xn =2πn+ θy
Ly. (7.3)
The Vm are Fourier transforms of the LLL-projected real-space disordered potential,
V (x, y), given by
Vm(x) ≡∫ Ly
0
dy
Lye2πimy/Ly V (x, y) . (7.4)
LLL-projection suppresses non-zero wave vectors due to the form factor e−q2/4, giving
t/v ∼ e−π2/L2
y 1. Further-neighbor hopping terms in Eq. (7.2) are exponentially
smaller than t and hence can be neglected. In the following, we take tn ≡ teiθx/Nφ
for simplicity, as the precise values are unimportant. The angles θ assume very
145
different roles in this limit: θx is the magnetic flux through the ring, while θy is
the parameter of a Thouless pump [210] which smoothly moves the Landau orbits
relative to the background potential. At any fixed θ, the Hamiltonian of Eq. (7.2)
is Anderson localized. As the pump parameter θy is adiabatically taken through a
cycle, the random on-site potentials vn(θy) change smoothly and the system undergoes
spectral flow: at the end of the cycle, vn(2π) = vn+1(0), so the initial and final spectra
coincide up to a n 7→ n + 1 translation. However, following each eigenstate through
the adiabatic cycle reveals an interesting picture.
Adiabatically changing a local chemical potential in an Anderson insulator leads
to non-local charge transport [109] due to avoided resonances between the site being
manipulated and arbitrarily distant ones (the distance is practically limited by log τ ,
where τ is the time scale of the adiabatic manipulation; for the purpose of calculating
C, we can take τ →∞). In the present setting, varying θy adiabatically manipulates
all random fields at once, giving rise to a complicated network of resonances and thus
more intricate patterns of charge transfer across the system. However, as a conse-
quence of adiabaticity, energy levels can never cross, so an electron that starts the
cycle in orbital n ends in orbital n−1 (i.e., at the same point in real space). Whenever
an avoided crossing between orbitals localized at sites n1 and n2 takes place, charge
is transported through the shortest path. That is because the perturbative hopping
amplitude for a path of length d is ∼ (t/v)d, so the shortest path is exponentially
favored. Randomness in vn does not change this conclusion, unless the two paths
have almost equal lengths, which is rare in the thermodynamic limit.
To show this, consider the amplitudes A1,2 for the two paths P1,2:
|Ai| ∼ t
∣∣∣∣∣∏n∈Pi
t
δvn
∣∣∣∣∣ = t exp
[−∑n∈Pi
ln(|δvn/t|)
]≡ te−w(Pi) , (7.5)
146
where δvn ≡ vn+1 − vn is an energy denominator arising from perturbation theory in
the nearest-neighbor hopping t. The exponent w(Pi) in Eq. (7.5) is a random variable
with mean and variance given by
E[w(Pi)] = LiE[ln |δv/t|] ≡ Liµ (7.6)
σ2[w(Pi)] = LiE[(ln |δv/t| − µ)2
]≡ LiS
2 , (7.7)
where Li is the length of path Pi (and L1 + L2 = Nφ, total length of the chain).
For a well-behaved distribution of on-site chemical potentials, such as uncorrelated
Gaussian or box distributions, the quantities µ and S are finite. By the central
limit theorem, w(P1)−w(P2) is then approximated by a normal distribution of mean
µ(L1−L2) = µ(2L1−Nφ) and standard deviation S√L1 + L2 ≡ S
√Nφ. We conclude
that the ratio of the two amplitudes is
∣∣∣∣A1
A2
∣∣∣∣ = ew(P2)−w(P1) ∼ exp
[Nφ
(µ(1− 2L1/Nφ) +
s√Nφ
)], (7.8)
where s is a zero-mean, unit-variance normal variable. This is exponentially large (or
small) in system size unless L1 ' 12Nφ (i.e. the two sites are diametrically opposite on
the ring) within an error O(N1/2φ ), which occurs with probability O(N
−1/2φ ). Thus in
the (1D) thermodynamic limit it is safe to assume that charge always moves through
the shortest path even with randomness in the problem.
Having established that charge moves through the shortest path, one may expect
each electron to take a local random walk in the vicinity of its initial site ni and end
the cycle at site ni − 1. However, this cannot be true for every electron: at least
one must wind around the entire ring during the cycle. Indeed, if Wn is the winding
number of the electron initially at site n, then∑
nWn = 1, as we show next.
As avoided crossings (absent fine-tuning) happen at isolated points in the θx in-
terval and involve only a pair of states each, proving the above identity is equivalent
147
x
y
(a)
x
y
0 2
2
1
2
3
4
(b)
Figure 7.2: (a) Sketch of a thin torus system. The two boundary angles correspondto fluxes through the long (θx) and short (θy) non-trivial loops of the torus, respec-tively. (b) The partitioning of the θ torus used to calculate the Chern number. Allslices contribute the same amount. The sides of the highlighted slice are numberedaccording to the discussion in the text.
to this simple combinatorics problem: showing that any decomposition of the cyclic
permutation C : (1, 2, . . . N) 7→ (2, 3, · · ·N, 1) (representing the spectral flow) as a
product of pairwise swaps Sij (representing the avoided crossings) gives a total wind-
ing number of 1. This can be shown as follows. Let one such decomposition be
C =∏
c Sn1(c),n2(c). Define a matrix of integers dc,i as the distance element i traverses
during move c, with d < 0 for clockwise and d > 0 for counterclockwise moves. This
is 0 for all elements except the two being swapped, i = n1,2(c), which have opposite
values: dc,n1(c) + dc,n2(c) = 0. It follows trivially that∑
i dc,i = 0. The sum of each
column i is the total distance traversed by element i, which must be −1 mod N , as
C moves every element one step to the left. We set∑
c dc,i = NWi − 1, defining the
winding number Wi. Putting the two together, we get
∑c,i
dc,i = N
(−1 +
∑i
Wi
)= 0 , (7.9)
hence∑
iWi = 1.
148
This bears intriguing similarity to the total Chern number of the Landau level,∑nCn = 1. In fact, such an identification is correct: the Chern number Cn reduces to
the winding number Wn in the thin-torus limit. This can be seen by considering the
phase acquired during a loop around the “Brillouin zone” defined by θ. To do so, we
split the torus θ ∈ [0, 2π)2 in many vertical strips, θy ∈ [0, 2π), θx ∈ [mε, (m + 1)ε),
for ε = 2π/M 1, m,M ∈ Z, as sketched in Fig. 7.2. We calculate the phase
for transporting the wavefunction around each side of each thin rectangle, then add
all the contributions (the numerical calculation for the general case follows a similar
technique). The cycle goes as follows:
1. Increase θx from mε to (m+ 1)ε at fixed θy. This twist of boundary conditions
has no effect on a perfectly Anderson-localized orbital.
2. Increase θy from 0 to 2π at fixed θx. During this step, the electron winds
Wi times around the circle and acquires an Aharonov-Bohm phase eiθyWi =
ei(m+1)εWi , where θy = (m + 1)ε is the flux through the circle. An example is
sketched in Fig. 7.3.
3. Decrease θx from (m+ 1)ε to mε. Like step 1, this has no effect.
4. Decrease θy from 2π to 0. The electron winds −Wi times around the circle (as
the evolution is perfectly reversible) and acquires a phase e−iWimε, completing
the cycle.
The net phase acquired by the electron wavefunction is Um = eiWiε. Since ε 1, one
can take the logarithm unambiguously and obtain the Chern number:
Ci =1
2πi
M∑m=1
lnUm = Wi . (7.10)
This derivation relies on a few assumptions, namely that (i) the electron orbital is
Anderson-localized at every θx = mε, and (ii) Wi does not depend on θx. The
149
0 /2 3 /2 2y
E
0 6
Ay
By
1 0
Ay
By
2 1
Ay
By
3 2
Ay
By
4 3
Ay
By5 4
Ay
By
6 5Ay
By
(a) y = 0(b) y = Ay(c) y = B
y(d) y = 2(e)
012
34
5 6
(f)
012
34 5
6
(g)01
23
4 56
(h)01
23
4 56
(i)
Figure 7.3: (a) Energy levels (thick colored lines) and on-site chemical potentials(thin black lines) as a function of Thouless pump parameter θy for a simple examplewith N = 7 orbitals. All states have C = 0 except the one starting at orbital 5. (b-e)Evolution of the chemical potentials at four moments during the cycle, θy = 0, θAx ,θBy , 2π. White circles denote empty orbitals, red circles occupied orbitals. At θA,By
the occupied orbital would be at resonance with an empty orbital, a narrow avoidedcrossing opens, and the electron hops. (f-i) View of the same cycle in real space,showing how the electron’s path winds once around the system during each cycle.
first assumption is satisfied in the thin-torus limit. The electron typically resonates
between orbitals at a distance ∼ O(Nφ), so the gap at the avoided crossing is ∼
(t/v)Nφ . Therefore the fraction of the interval θx ∈ [0, 2π) spent in a non-local
superposition is exponentially small in system size and asymptotically has measure
0. That Wi doesn’t depend on θx in this limit follows from similar considerations.
The identification in Eq. (7.10) is the key to explaining the observed proliferation
of Chern states under 1D scaling. In essence, during a Thouless pump cycle, every
electron hops randomly and non-locally across the chain many times, generically
acquiring a large winding number, which translates to a large Chern number. We
conclude this Section by noting that our result also holds for higher-|C| bands. In a
Chern-k band, one would have to replace the cyclic permutation C used in the proof
with Ck, as the spectral flow cycles the orbitals k times in a Chern-k band. Under
this modification, Eq. (7.9) correctly yields∑
iWi = k.
150
0.6 0.4 0.2 0.0 0.2 0.4 0.6E
10 3
10 2
10 1
100
1(E)
/1(
0)
(a)
N = 64, a = 4.02N = 128, a = 8.04N = 196, a = 12.32N = 256, a = 16.08
0.50 0.25 0.00 0.25 0.50E/ ln(a)
10 3
10 2
10 1
100
1(E)
/1(
0)
(c)
0.75 1.00 1.25 1.50 1.75ln(a)
0.3
0.4
0.5
0.6
FWHM
(b)
Figure 7.4: (a) Density of C = 1 states ρ1(E) for Ly = 10 and different aspect ratios a.(b) Full width at half maximum of ρ1 indicates broadening consistent with Eq. (7.14).(c) Rescaling the energy by
√ln(a) collapses the curves for different sizes.
7.4 Dimensional crossover
Even though the thin-torus limit Ly 1 is a helpful simplification, the physics
described above remains valid for Ly > 1, as long as a 1. Hopping matrix el-
ements are not negligible up to a real-space distance O(1), i.e. a number of sites
O(Ly). These matrix elements are responsible for local level repulsion and strongly
suppress energy fluctuations during the Thouless pump cycle. To get a quantitative
measure of this, we start from a square torus. There, we know from numerics that
the average dimensionless Thouless conductance obeys an approximately Gaussian
form g(E,L) ' G(EL1/ν2D), where G is a scaling function well approximated by
G(x) ∼ g0e−x2/2σ2
(g0 and σ are O(1) constants). Inverting the definition of g yields
an estimate of the energy fluctuations of a typical state during the pump cycle:
δE ∼ 2πvg0
L2exp
(− E
2
2σ2L2/ν2D
). (7.11)
151
Here v is the bandwidth and 2πv/L2 is the typical level spacing. As δE is determined
by the range of local hopping matrix elements, Eq. (7.11) remains true if we consider
a rectangular torus and replace L with the short circumference Ly. The expected
number of avoided resonances encountered during a pump cycle, Nr, is simply the
number of states in the spectrum with energies within the range of fluctuations δE.
Approximating
ρ(E) ' LxLy2πv
e−12
(E/σ′)2 (7.12)
(the exact expression [225] deviates slightly from a Gaussian) yields
Nr ∼ ρ(E)δE ∼ g0ae− 1
2(E/E0)2 , (7.13)
where E0 is an a-independent energy scale. Thus, even away from the band center,
and even for Ly > 1, increasing a eventually leads to Nr & 1. At that point the
crossover between 2D and 1D behavior takes place, with typical states acquiring
nontrivial winding and thus Chern number. This crossover happens unevenly in
energy: it starts at the band center (where one already has Chern states even in the
2D thermodynamic limit) and spreads towards the band edges. The contour defining
the dimensional crossover is approximately fixed by setting Nr = 1:
E ∼√
ln(a) . (7.14)
This prediction is borne out by numerical data on the density of Chern states, ρC(E).
Fig. 7.4 shows that the broadening of ρtop., already visible in Fig. 7.1, is explained
fairly accurately as a scaling collapse of ρC(E/√
ln(a)), for large enough a (the ar-
gument is formulated for a 1, and the proposed rescaling is singular at a = 1,
hence the poor collapse of the a ∼ O(1) curves). While numerical instability prevents
152
0 50 100 150Lx
10 6
10 5
10 4
10 3
10 2
10 1
g(0)
(a)
Ly = 10, av.Ly = 10, typ.Ly = 20, av.Ly = 20, typ.
10 8 6 4 2 0log10g(0)
10 5
10 4
10 3
10 2
10 1
P
(b) N163264128196256
1.0 0.5 0.0 0.5 1.0E
10 7
10 5
10 3
10 1
g av(E
)
(c)
2 1 0 1 2EL1/ 2D
x
0.00
0.25
0.50
0.75
1.00
g av(E
)/gav
(0) (d) N
3264128256
Figure 7.5: (a) Average and typical values of the Thouless conductance at the centerof the band, g(0), for fixed Ly, as a function of Lx. (b) Distribution of log10 g(0)for Ly = 10, increasing Nφ. (c) Average Thouless conductance gav(E) for Ly = 14decays with increasing Nφ. (d) The normalized quantity gav(E)/gav(0) shows scaling
collapse under E 7→ EL1/ν2Dx with critical exponent ν2D ' 2.4.
us from following the crossover to a 1, where all states are predicted to become
topological, results are consistent with the above picture.
7.5 Thouless conductance
As mentioned earlier, while 1D scaling causes the proliferation of Chern states across
the spectrum, it also removes the critical energy characteristic of the 2D problem and
makes the entire spectrum Anderson-localized. We verify this numerically by calcu-
lating the disorder- and eigenstate-averaged Thouless conductance gav(E), Fig. 7.5.
Unlike the 2D case, where at the center of the band gav(0) ∼ O(1) as L → ∞,
153
here it decays exponentially in Lx, as expected for a 1D problem: gav(0) ∼ e−Lx/ξ1
with ξ1 ' 1.7Ly. However, surprisingly, the normalized quantity gav(E)/gav(0) dis-
plays scaling collapse with the same critical exponent as the two-dimensional case,
ν2D ∼ 2.4 (this collapse is not seen for the typical g). These results seem contra-
dictory: on the one hand, a finite ξ1 suggests localization across the spectrum with
no critical energy; nevertheless, we observe signatures of a divergent ξ2 ∼ E−ν2D ,
reproducing the 2D critical behavior, even as the scaling is purely one-dimensional.
The variation of g across samples and eigenstates sheds light on this issue. At
the center of the band, the distribution of ln(g) broadens as Lx is increased, so that
the typical value gtyp ≡ e〈ln g〉 decays exponentially faster than the average value gav.
In the thermodynamic limit, the peak in gav(E) near E = 0 is thus dominated by
rare states that are abnormally extended in the long direction (x). We attribute the
appearance of ν2D to these atypical states: states percolating across the sample in Lx
but not in Ly are by construction unaware of the aspect ratio, and thus display the 2D
critical behavior as Lx is increased. As Lx grows, these states become exponentially
rare, which explains the vanishing amplitude of the signal and its presence in gav but
not gtyp.
7.6 Discussion
We have investigated the fate of the quantum Hall plateau transition when finite-size
scaling is performed for one of the two dimensions only. Through numerical diago-
nalization, we have uncovered surprising and counter-intuitive behavior: Anderson
localization across the spectrum, accompanied by the proliferation of Chern states.
This led us investigate the fate of the Chern number, a two-dimensional topological
invariant, in the quasi-one-dimensional limit defined by a = Lx/Ly 1. In the thin-
torus limit Ly 1, the system maps onto a 1D fermionic chain and the flux through
154
the short loop of the torus becomes a Thouless pump parameter that smoothly shifts
the random chemical potentials of the fermion chain. We have shown that in this
limit the Chern number C of a wavefunction maps onto the winding number W of
the electron around the system (now a one-dimensional ring) over the course of a
pump cycle, which is determined by a random network of avoided crossings in the
spectrum of the system. This identification consistently gives the total Chern num-
ber of the band and leads to some striking predictions, e.g. that generic states in
this limit have large, random Chern number, even though they may percolate in one
direction only.
We have further shown that this picture is valid away from the thin-torus limit, i.e.
for Ly > 1, as long as the torus aspect ratio a is large enough. The crossover between
2D and 1D behavior as a is increased starts at the band center and spreads towards
the band edges. The broadening is predicted to be extremely slow, ∼√
ln(a), but it
is nonetheless visible in our numerics at Ly ∼ O(10), quite far from the thin-torus
limit.
On a theoretical level, our findings provide a new example of subtle interplay
between topology and disorder [172, 80, 17, 10, 163]. The idea of topological pumping,
which goes back to Thouless [210], is a subject of rising theoretical interest, especially
in connection to Floquet physics [138, 227, 113, 48] and synthetic dimensions [166].
Here it is applied in a new, disordered context, and provides the key to interpret the
quasi-1D limit of the quantum Hall plateau transition.
We conclude with some remarks related to experiment. As the non-local avoided
crossings that underpin the picture presented here are generally very narrow (expo-
nentially in system size), the adiabatic time scales required to observe this behavior
in macroscopic systems are unphysically long. However, for microscopic systems, the
manipulations required may still be performed adiabatically. The ingredients required
are (i) adiabatically tunable, pseudo-random on-site chemical potentials, (ii) nearest-
155
neighbor hopping, and (iii) sufficiently long coherence times (relative to the required
adiabatic time scale). Clean Thouless pumps have been successfully engineered using
ultracold bosonic [132, 133] or fermionic [154] atoms in optical superlattices, single
spins in diamond [136], Bose-Einstein condensates [135] and quantum dots [207, 22];
adding disorder could be an interesting new direction for these and other experimen-
tal platforms. Finally, while periodic boundary conditions are required to observe the
winding, the striking coexistence of Anderson localization and non-local charge trans-
port across the length of the system would be observable even with open boundary
conditions.
156
Appendices
7.A Details of numerical calculation
We consider a continuum model of electrons in a high magnetic field projected into
the LLL. The system is set on a rectangular torus with sides Lx, Ly and generalized
periodic boundary conditions parametrized by angles θ ≡ (θx, θy) ∈ [0, 2π)2. The
torus is pierced by Nφ quanta of magnetic flux, i.e. LxLy = 2πNφ. The wavefunctions
in Landau gauge are given by
ψn(x, y) =1√
Lyπ1/2
∑p∈Z
eipθxeikn,p(θy)y− 12
(x−kn,p(θy))2 (7.15)
with kn,p(θy) = 2π(n+pNφ+θy/2π)/Ly. Disorder is represented by a Gaussian white
noise potential Vq obeying
〈Vq1Vq2〉 = V 20 δ
2(q1 + q2) . (7.16)
A straightforward calculation yields the Hamiltonian matrix elements in the basis of
Eq. (7.15):
Hn1,n2 =∑
p,mx,my∈Z
eipθxδ(n2 − n1 + pNφ = my) Vq e−2πi(n1+my/2+θy/2π)mx/Nφ (7.17)
where Vq ≡ Vqe− 1
4q2 is the LLL-projected potential, and q = 2π(mx/Lx,my/Ly).
157
In order to calculate the Chern number of each eigenstate in the spectrum of H
from Eq. (7.17), we follow the standard method described in Ref. [49] and split the
torus of boundary angles θ into a lattice, θi,j = 2π(i/Nx, j/Ny). Appropriate values of
the mesh size were discussed in Ref. [252], which takes Nx = Ny =√
4πNφ/3. Unlike
the present case, Ref. [252] only deals with square systems where Nx = Ny is clearly
optimal. We fix the product to the same value, NxNy & 4πNφ/3, but we find that a
rectangular mesh with Nx/Ny ' Ly/Lx is optimal. This is physically reasonable as
the angle in the short direction affects the long boundary, and vice versa.
For each site on this lattice, we diagonalize Eq. (7.17) numerically and obtain the
spectrum |θij, n〉, n = 0, . . . Nφ − 1. We then assign a U(1) gauge variable to each
bond in the lattice:
Axn(i, j) ≡ 〈θij, n|θi+1,j, n〉|〈θij, n|θi+1,j, n〉|
, Ayn(i, j) ≡ 〈θij, n|θi,j+1, n〉|〈θij, n|θi,j+1, n〉|
(7.18)
for horizontal and vertical bonds, respectively. The curvature of this gauge field is
given by the “integral” around a plaquette:
Un(i, j) ≡ Axn(i, j)Ayn(i+ 1, j)Axn(i, j + 1)∗Ayn(i, j)∗ .
If the mesh is fine enough (i.e. Nx, Ny are large enough), |Un(i, j)− 1| 1 and one
can define γn(i, j) ≡ lnUn(i, j) without ambiguity in the choice of branch cut for the
logarithm. The Chern number is then
Cn ≡1
2πi
∑i,j
γn(i, j) . (7.19)
This method also gives us access to all the energies En(i, j), which we use to compute
the Thouless conductance.
158
A subtlety implicit in formulae Eq. (7.18) is that the inner products involve wave-
functions with different boundary conditions. The matrices implementing such inner
products are not equal to the identity and must be calculated. As we sample θ from
a rectangular lattice and only need inner products between neighboring points in that
lattice, we are interested in
Nmn(θ;θ + µj) ≡ 〈ψm(θ)|ψn(θ + µj)〉 , j = x, y . (7.20)
For j = y, we have
Nmn(θ,θ + µy) =1
Ly√π
∑P,p∈Z
eipθx∫dy eikn−m,p(µ)y
∫ (P+1)Lx
PLx
dx
exp
[−(x− kn+m
2, p2(θy + µ/2)
)2
− 1
4k2n−m,p(µ)
]=∑p∈Z
eipθxeikn−m,p(µ)Ly − 1
ikn−m,p(µ)Lye−
14k2n−m,p(µ) , (7.21)
where kn,p(µ) = 2πLy
(n+Nφp+µ/2π). One can verify that this reduces to the identity
for µ→ 0. Moreover the entries only depend on the difference n−m. For j = x, we
have instead
Nmn(θ,θ + µx) =1
Ly√π
∑P,p∈Z
eipθx+i(p+P )µ
∫dy eikn−m,p(0)y
∫ Lx
0
dx
exp
[−(x− kn+m
2,P+ p
2(θy)
)2
− 1
4k2n−m,p(0)
]= δmn
1√π
∑P∈Z
eiPµ∫ (P+1)Lx
PLx
dx e−(x−kn,0(θy))2 , (7.22)
where we used the fact that the y integral gives a δ function to arrive to the result.
This matrix is diagonal and reduces to the identity for µ→ 0 as well. Moreover, we
verify numerically that for Lx = Ly the two matrices are conjugate to each other.
159
10 5
10 4
10 3
10 2
10 1
C(E
)
N = 16, a = 1.01 N = 32, a = 2.01 N = 64, a = 4.02 C01-12-23-3
1.0 0.5 0.0 0.5 1.0E
10 5
10 4
10 3
10 2
10 1
C(E
)
N = 128, a = 8.04
1.0 0.5 0.0 0.5 1.0E
N = 196, a = 12.32
1.0 0.5 0.0 0.5 1.0E
N = 256, a = 16.08
Figure 7.B.1: Density of states with Chern number C, ρC(E), for rectangular toriwith Lx = 10`B and variable Nφ indicated on each panel. As Nφ increases thesystem gradually approaches the 1D regime: more and more states become topological(C 6= 0), due to the broadening of ρ±1(E) and to states with higher |C| becomingmore common at the band center.
This reflects the fact that the two must map onto each other under a π/2 rotation
followed by a gauge transformation.
7.B Additional data on density of Chern states
Here we report additional data on the density of states ρC(E) parsed by Chern number
C beyond what is shown in Fig. 7.1 in the main text. We consider rectangular tori
with Lx = 10`B (Fig. 7.B.1) and Lx = 20`B (Fig. 7.B.2). Increasing Ly causes both
a broadening of ρ±1 and an increase in the density of higher-|C| states. Comparing
the two Lx values shows how this behavior is due to the 2D-1D crossover controlled
by the aspect ratio a = Ly/Lx, as opposed to 2D behavior controlled solely by the
system’s area Nφ.
160
10 5
10 4
10 3
10 2
10 1
C(E
)
N = 96, a = 1.51 N = 144, a = 2.26 C01-12-2
1.0 0.5 0.0 0.5 1.0E
10 5
10 4
10 3
10 2
10 1
C(E
)
N = 216, a = 3.39
1.0 0.5 0.0 0.5 1.0E
N = 324, a = 5.09
Figure 7.B.2: Same as Fig. 7.B.1, but for Lx = 20`B instead of 10`B. Despite thelarger system size Nφ, the aspect ratio we can achieve is smaller and the systemis further away from the 1D limit: C = ±3 states are still uncommon (below thedisplayed range) and the broadening of ρ±1 is not as clear. Nonetheless the growthof C = ±2 states is visible.
While the density of Chern-C states broadens for each C, the linear combination∑C CρC remains constant, as seen by looking at the disorder-averaged Hall conduc-
tance,
σxy(E) =
∫ E
−∞dε∑C∈Z
CρC(ε) , (7.23)
shown in Fig. 7.B.3. Under 2D scaling, σxy(E) is known to collapse onto F (EL1/ν2D),
with F a scaling function in the shape of a sigmoid. We conclude that the width of
the crossover interval between the two conductance plateaus is fixed by the smaller
side of the torus.
161
0.4 0.2 0.0 0.2 0.4E
0.0
0.2
0.4
0.6
0.8
1.0
xy
Ly = 10.0 B
N163264128196
0.4 0.2 0.0 0.2 0.4E
Ly = 14.0 B
N3264128256
0.4 0.2 0.0 0.2 0.4E
Ly = 20.0 B
N6496144216324
Figure 7.B.3: Disorder-averaged Hall conductance σxy calculated from the densitiesρC as in Eq. (7.23).
162
Chapter 8
Deconfined quantum criticality in
graphene Landau levels
8.1 Introduction
After devoting Chapters 6 and 7 to the integer quantum Hall transition, we switch
our focus to a very different example of quantum criticality in the quantum Hall
regime – this time in a clean system with interactions. We will describe how internal
degeneracies of a Landau level can be used to engineer interesting quantum critical
points. From a theoretical standpoint this is appealing as it allows a continuum
regularization of phase transitions otherwise only seen in lattice models, where spatial
symmetries are explicitly broken and only recovered in the infrared (IR), i.e. at long
distances and low energies. At the same time, the model we introduce is motivated
by the four-fold spin and valley degeneracy of Landau levels in graphene, which has
intriguing experimental implications.
Our understanding of quantum phase transitions, based on notions such as scale
invariance, renormalization and universality, is ultimately supported by conformal
field theories (CFT) [34]. These are quantum field theories where the usual spacetime
163
Lorentz symmetries are supplemented with the additional requirement of scale invari-
ance.1 One-plus-one-dimensional (1+1D) CFTs are remarkably well understood, so
much so that the most accurate determination of the Ising critical exponents currently
available is provided by the non-perturbative conformal bootstrap method [167]. How-
ever, understanding the space of two-plus-one dimensional (2+1D) CFTs remains a
central challenge in strongly-interacting physics. In contrast to the 1+1D case, com-
paratively little is known about the space of possible fixed points beyond large-N ,
supersymmetric, or perturbative approaches. Where available, our knowledge relies
heavily on numerical Monte-Carlo simulations. More recently, the conformal boot-
strap has made it possible to compare numerical estimates of scaling exponents with
rigorous analytical bounds.
A class of 2+1D CFTs of particular interest is given by the ‘deconfined quantum
critical points’ (DQCPs) [190, 188, 146]. These are of interest both to condensed
matter, where they arise as Landau-forbidden phase transitions between magnetic
orders with differing order parameters [142, 46], and to high-energy theory, where they
are thought to provide realizations of the non-compact CP1 nonlinear sigma model and
QED3 [219]. Numerical studies of lattice models where such critical points are thought
to arise support the theoretical prediction of an emergent SO(4) or SO(5) symmetry
larger than the microscopic one [183, 153, 152]. However, it has proven difficult
to obtain converged scaling exponents, or even conclusively determine whether the
transition point is a CFT [117, 95, 153, 191]. More perplexing, numerical estimates
of the vector operator’s scaling dimension appear to contradict bounds from the
conformal bootstrap [156, 167].
Previous numerical studies of the DCQP considered lattice models of spins [183,
140, 184, 13, 105, 16, 28, 192, 223], 3D loop models [152, 153, 191], hard-core
1Strictly speaking, CFTs possess conformal invariance which is a stronger requirement. Scaleinvariance implies conformal invariance in d = 2; whether this holds in general is an open ques-tion [155].
164
bosons [249] or fermions [8, 185]. In these models many of the symmetries, both
internal and spatiotemporal, emerge only in the IR. Here, we instead introduce
a continuum regularization of the DQCP and other 2+1D CFTs which preserves
these symmetries exactly in the ultraviolet (UV): rather than discretizing space, the
Hilbert space is made finite by Landau level quantization. The idea is to embed
the critical fluctuations into an N -component “flavor” degree of freedom carried by
itinerant fermions in the continuum. The motion of the fermions is then quenched by
a strong magnetic field. This introduces a soft cut-off in the form a finite magnetic
length, which smoothly suppresses short-wavelength fluctuations within a LL. When
the fermions fill N/2 of the N -fold degenerate LLs, fluctuations in the flavor space
give rise to a nonlinear sigma model (NLSM). This is the famous problem of quantum
Hall ferromagnetism [200] realized experimentally both in GaAs (spin degeneracy,
N = 2) and graphene (spin and valley degeneracies, N = 4). In the latter case,
the resulting SO(5) NLSM has the Wess-Zumino-Witten term thought to stabilize a
DQCP [2, 209, 189, 122]. We demonstrate that this model can be studied with both
DMRG and sign-free determinantal quantum Monte-Carlo (DQMC).
Models with exact UV symmetries have several potential numerical advantages.
First, the continuum formulation allows the model to be defined on any manifold,
such as a sphere, without introducing lattice defects. Second, this realization of the
DQCP has an exact SO(4) or SO(5) symmetry, whereas on a lattice such a symmetry
putatively emerges only in the IR at the critical point. Because the model is essentially
an explicit regularization of an SO(5) NLSM, it is straightforward to identify the
microscopic operators corresponding to the stiffness, vector, and symmetric-tensor
perturbations of the NLSM. As such the DQCP should exist as a phase, i.e., without
tuning, which greatly simplifies scaling collapses. The chief question is whether the
model actually flows to a CFT.
165
This Chapter is structured as follows. In Sec. 8.2 we review the model of electrons
in graphene with N = 4 flavors, its Neel and valence bond solid (VBS) ordered phases,
and the SU(N) anisotropies that drive the transition between them. Sec. 8.3 contains
the results of iDMRG simulations, which are consistent with a direct, continuous
transition between a Neel and VBS phase up to the largest system sizes. However,
our estimate of the SO(5) vector operator’s scaling dimension ranges from ∆V ∼ 0.55
to 0.7, depending on model parameters (essentially the stiffness of the NLSM). Due
to the limited DMRG system size (cylinder circumference L . 12`B), it is unclear
whether this is a finite-size artifact or a signature of a weakly-first order transition. In
Sec. 8.4 we show that the model can be solved with sign-free determinantal quantum
Monte Carlo, allowing for simulations with polynomial complexity in system size,
for which we present a numerical benchmark and discuss the prospects for large-scale
simulations. We conclude by summarizing our results and discussing future directions
in Sec. 8.5. This Chapter is based on results published in Ref. [87].
8.2 Model
The model is motivated by the physics of graphene in a magnetic field, where N = 4
flavors of a two-component Dirac fermion Ψa, a = 1, 2, 3, 4, arise from the combination
of valley and spin degeneracy [108, 107, 232, 122]. To a first approximation, they are
related by an U(4) flavor symmetry; letting Pauli matrices τµ act on valley and σµ
on spin (µ = 0 indicates the identity), the generators are the 1 + 15 bilinears τµσν .
In reality the SU(4) part is broken down to spin SO(3) (generated by σµ) and a near-
exact SO(2) valley conservation (generated by τ z). Microscopically, the two strongest
instabilities [108, 107, 232, 245, 255] which may spontaneously break the SO(3) ×
SO(2) symmetry are antiferromagnetism, with three-component Neel vector N = τ zσ,
and valence-bond-solid (VBS) order in the Kekule pattern on the honeycomb lattice,
166
Figure 8.1: Kekule valence-bond-solid pattern on the honeycomb lattice. Each seg-ment represents a spin singlet, (| ↑↓〉−| ↓↑〉)/
√2. The pattern has a unit cell (dashed
blue line) three times the size of that of the honeycomb lattice.
shown in Fig. 8.1. The latter has order parameter eiφK = τx+iτ y: because the valleys
are at different momenta, inter-valley coherence produces a VBS distortion. Together
these form a maximal set of anti-commuting terms Γi = τ zσx, τ zσy, τ zσz, τx, τ y,
the Clifford algebra for SO(5).
For numerical purposes the Dirac fermions must be regularized, but rather than
falling back to the honeycomb lattice of graphene, we instead stick to the continuum
and introduce a uniform background magnetic field B orthogonal to the manifold.
The single particle spectrum collapses into N = 4-fold degenerate Landau levels, with
energy spectrum εn = ~v`B
sign(n)√
2|n|, where v is the Dirac velocity, as discussed in
Sec. 2.1.2. At zero density, the fermions should fill two of the four n = 0 LLs, i.e.,
the zeroth LL (ZLL) is half filled.
When the interactions are weak compared with to the cyclotron splitting ~v/`B,
we can project them into the ZLL. A phenomenological model capturing the result-
ing Neel and Kekule instabilities is an SU(4)-symmetric contact interaction U and
anisotropies ui,
HZLL =U
2
[ 4∑a=1
ψ†a(x)ψa(x)
]2
−5∑i=1
ui2
[ 4∑a,b=1
ψ†a(x)Γiabψb(x)
]2
. (8.1)
167
uK/U
u N/U
1st o
rder
Kekule
(VBS)
Neel (A
F)
O(5) C
FT(a)
uK/U1s
t orde
r
Kekule
(VBS)
Neel (A
F)
gapp
ed P
M(b)
Figure 8.2: Schematics of two possible phase diagrams of the model in Eq. (8.1).(a) DQCP scenario. The AF and VBS orders are separated by a line (uN = uK)with manifest O(5) symmetry. For ui/U < w the symmetry is spontaneously broken,giving a first-order transition (solid line); for ui/U > w there is a continuous transitioncharacterized by an O(5)-symmetric CFT (dotted line). This line realizes the DQCP.Alternatively, it may be that a true CFT does not exist and the transition is weaklyfirst-order for all U . (b) A Landau-allowed scenario. For ui/U > w the AF and VBSphases are separated by two independent continuous transitions (dotted lines) to agapped paramagnetic phase, with a multicritical point at uN = uK = wU .
Here ψa(x) is the field-operator of the ZLL, which can be decomposed as ψa(x) =∑Nφm=1 φm(x)ca,m for ZLL orbitals φm on a system pierced by Nφ flux quanta.2 Because
each LL has one state per magnetic flux the Hilbert space is now completely finite,
with NNφ single particle states on a surface pierced by Nφ flux.
The anisotropies ui favor either Neel (u1 = u2 = u3 = uN > 0) or Kekule (u4 =
u5 = uK > 0) order. A transition between the two orders is driven by the difference
uN − uK , and for uN = uK there is an exact O(5) symmetry (the inversion element
arises from the anti-unitary particle-hole symmetry ψ → ψ†). Alternatively, taking
u3 < u1 = u2, we have an “easy-plane” model with at most SO(4) symmetry.
The magnetic field quenches the kinetic energy, driving quantum Hall (anti)-
ferromagnetism, ni =⟨ψ†Γiψ
⟩6= 0.[200, 60] The order parameter n encodes which
two of the four LLs are filled. However, in contrast to the SU(4) symmetric case
2On manifolds withe genus not equal to one, there is technically a ‘shift’ between Nφ and thenumber of orbitals.
168
(ui = 0), where the order parameter commutes with H and hence does not fluctuate,
the anisotropies u lead to fluctuations. Extending the standard N = 2 theory of
quantum Hall ferromagnetism, [200] on the O(5) line these fluctuations are captured
by an SO(5) NLSM (Euclidean) action, [232, 122]
S =1
2γ
∫d3r(∂n)2 + SWZW[n] + . . . , (8.2)
SWZW[n] =2πi
vol(S4)
∫dt d3r εabcdena∂sn
b∂xnc∂yn
d∂tne. (8.3)
SWZW is the SO(5) Wess-Zumino-Witten term, whose presence we explain shortly. [122]
Note that with a magnetic field, the particle-hole symmetry CT still ensures the
symmetry n→ −n.
The stiffness 1/γ of the NLSM is controlled by the repulsion U ; the exchange en-
ergy from large U leads to a stiff (small γ) NLSM. [200] Perturbing away from the O(5)
line, uN 6= uK , will generate the “symmetric-tensor” anisotropies L 3 −(∑
i uini)2.
Hence there is a direct correspondence between the microscopic parameters U , ui and
the stiffness and symmetric-tensor perturbations of the NLSM respectively.
The SO(5)-NLSM with topological term has been argued to flow to the DCQP -
unless it is too stiff, in which case SO(5) symmetry may break spontaneously. [209,
189] So we conjecture a two-parameter phase diagram in uN/U and uK/U shown
in Fig. 8.2(a). Away from the O(5)-line, the anisotropies reduce the fluctuations
and render the WZW term inoperative, so we expect Neel or Kekule order. The
O(5)-line is a direct transition between the two, which may either be first or second
order. If the NLSM is stiff (ui/U < w for some threshold w) the O(5) symmetry
will be spontaneously broken, [232] which corresponds to a first-order transition. If
the NLSM is floppy (ui/U > w), the putative existence of DQCP could lead to
a continuous transition which will manifest as a critical line uN/U = uK/U > w
described by an O(5)-symmetric CFT on which the scaling dimensions are constant.
169
In contrast, the conventional Landau-Ginzburg-Wilson theory of phase transitions
requires either a first-order transition, or two independent continuous transitions. The
two transitions will generically be separated either by a region of phase coexistence
with both Neel and Kekule order, or by a gapped (possibly topologically ordered)
symmetric paramagnet, a possibility illustrated in Fig. 8.2(b). The transitions can
only coincide when fine-tuned to a multi-critical point. As we will see, the numerics
are in fact consistent with a direct transition along the whole O(5) line, though (at
present) we cannot precisely determine whether the transition at high ui is truly
continuous or just weakly first-order.
The presence of the SWZW term can be inferred by extending the theory of N = 2
ferromagnetic skyrmions [200, 144] to N = 4. [122] When half-filling N = 2 flavors, it
is well known that skyrmions in the ferromagnetic O(3) order parameter n = ψ†σψ
carry electrical charge.[200] This response is captured by the topological term Ltopo =
A(n) · ∂tn + εµνρ
8πAµn · ∂νn× ∂ρn, where A is the vector potential of a monopole and
A is a probe U(1) gauge-field. Moving on to N = 4, consider a skyrmion in the
anti -ferromagnetic order N = ψ†τ zσψ. The anti-ferromagnet has filling ν = 1 in
each valley, but with opposite spin. So the N = 4 skyrmion is equivalent to an N = 2
skyrmion in each valley independently, but with opposite handedness (due to τ z).
Thus in contrast to a ferromagnetic skyrmion, the total charge is zero, but there is
valley-polarization 1 − (−1) = 2 under τ z, the generator of the symmetry relating
τx/y. More generally, we invoke SO(5) to conclude a skyrmion in any 3 of the 5
components induces charge under the remaining two, and a vortex (meron) under 2
of the 5 components carries spin-1/2 under the remaining three. This is the physics of
SWZW. A second consequence of anti-ferromagnetism is the cancellation of the A·∂tn
term to leading order, with fluctuations generating (∂tn)2. [32]
170
8.3 Infinite DMRG simulations
In this Section we study the model on an infinitely long cylinder of circumference
L in order to use iDMRG, as described in Sec. 2.5. The complexity of the DMRG
diverges exponentially with the circumference, which restricts us to smaller system
sizes (L ∼ 12`B) than previous lattice Monte-Carlo simulations. Nevertheless our
results appear consistent with the conjectured phase diagram of Fig. 8.2(a).
8.3.1 Method
After projecting the Hamiltonian in Eq. (8.1) into the ZLL, the contact interactions
become familiar Haldane V0 pseudopotentials. We then solve for the ground state on
an infinitely-long cylinder of circumference L using the iDMRG algorithm as described
in Chapter 2. Our numerics exactly conserve the quantum numbers of charge, spin,
and valley, while the rest of the O(N) symmetry emerges as the numerics converge.
As we discussed in Chapter 4 (where we applied the iDMRG method to the gapless
CFL state at ν = 1/2 in the LLL), the MPS ansatz used in DMRG by construction
can only capture the algebraic correlations of critical states, 〈O(r)O(0)〉 ∼ r−∆O ,
out to a finite length ξ introduced by the finite bond dimension χ. Therefore, to
study 1+1D critical points, we are going to invoke the idea of ‘finite-entanglement
scaling’ [208, 168]: near a critical point, the length scale ξ can be treated as an
IR cutoff and used in finite-size scaling collapses. In the present case, the putative
2+1D critical point on the cylinder does not dimensionally reduce to a 1+1D critical
point, as explained below. This means that the MPS correlation length ξ does not
scale to infinity with χ, but rather converges to a finite “true” value (at finite L):
ξ∞(L) ≡ limχ→∞ ξ(χ, L). Nevertheless, at large enough L, we find that ξ is well
below ξ∞ at the achievable values of χ, so we will extract the critical properties from
two-parameter scaling collapses in the length scales L and ξ.
171
8.3.2 Cylinder diagnostics of the 2D phases
The 2+1D phases we wish to distinguish are:
i. An ordered phase in which SO(2) is spontaneously broken (e.g. VBS, XY, or
Kekule order);
ii. An ordered phase in which an SO(N) symmetry with N = 3, 4 or 5 is broken;
iii. A gapped paramagnetic phase;
iv. An SO(N)-symmetric CFT.
The subtlety, however, is that for fixed cylinder circumference L, each of ii, iii and
iv dimensionally reduces to a 1+1D gapped, symmetric paramagnet, so we must
elucidate how to distinguish them within our numerics.
To do so we place the O(N)-NLSM of Eq. (8.2) on a cylinder with x running along
the axis and y around the circumference. If the symmetry is spontaneously broken
in 2+1D (cases i and ii), then we can take ∂yn ∼ 0 and obtain
Scyl =L
2γ
∫dxdt(∂n(x, t))2 + · · · (8.4)
Here ∂ is the derivative in 1+1D, and the WZW term vanishes because ∂yn = 0
(the skyrmions are gapped on the cylinder). This is a 1+1D O(N) NLSM with
stiffness L/γ and without a topological term. For N = 2 (case i) this model has
algebraic order, unless L/γ is small enough to drive a Berezinskii-Kosterlitz-Thouless
transition into a disordered phase. For N > 2 (case ii) the system is gapped, with a
finite correlation length ξ1D ∼ ae2π NN−2
Lγ . [181] Hence cases i and ii would manifest
in a ξ1D which diverges exponentially with L, or which may be infinite. In contrast,
in a 2+1D gapped paramagnet (case iii), the ξ1D will saturate with L to the true
correlation length ξ of the 2+1D phase (which is finite). Finally, if the system is
172
0
2
4
6
8(a)
1603005008001200180027004000600090001200016000200002500032000
0
5
10
15
20(b)
0 2 4 6 8 10 12
L0
15
30
45
60
V
(c)
0 3 6 9 12
L
0.51.01.52.0
log 1
0V
Figure 8.3: Correlation length obtained from numerical iDMRG simulations, as afunction of cylinder circumference L for different values of the bond dimension χ. (a)On the O(5) line, with small stiffness: U = 2, u = u3 = 1, m = 0. As χ is increased,ξ approaches a linear dependence on size, ξ ∼ αL (dashed line), consistent with aCFT on the cylinder. (b) On the O(5) line, with large stiffness: U = 10, u = u3 = 1,m = 0. ξ(L) curves upwards (dashed line is an exponential fit to the first 4 pointsat largest χ), consistent with a weakly-first-order transition. (c) On the VBS side:U = 2, u = 1, u3 = 0, m = −0.1. The correlation length in the valley channel ξVdiverges exponentially with L (inset shows a semilogarithmic plot), clearly indicatinga symmetry-broken state.
a 2+1D CFT (case iv) we cannot approximate ∂yn = 0. Scale invariance instead
dictates that ξ1D ∝ L, and the behavior of other observables can be determined by
conformal finite-size scaling in L.
8.3.3 Continuous transition
To assess the plausibility of the scenario shown in Fig. 8.2(a) we first measure the
scaling of ξ1D with L. We set u1 = u2 = u + m and u4 = u5 = u − m, so that
173
the AF-VBS transition is driven by m (m = 0 defines the critical point), while
u3 ≤ u can be used to introduce easy-plane anisotropy. The repulsion U sets the
overall spin stiffness. In Fig. 8.3 we show the correlation length ξ, defined by the
dominant eigenvalue of the MPS transfer matrix as discussed in Sec. 2.4, for several
representative points. For m = 0 and small U (i.e., on the putative critical line), the
scaling of ξ with L appears to be perfectly linear. In contrast, for m 6= 0 (i.e., in an
ordered phase), or for m = 0 and large U (i.e., on the putative first-order transition
line), ξ grows super-linearly and is well fit by an exponential dependence in both
cases. For m 6= 0 the the exponential form is clear over more than a decade, while
for m = 0, large U , we can really only detect a positive second derivative (upwards
curvature).
The linear-in-L behavior for small U is consistent with scenario Fig. 8.2(a), though
we cannot rule out a gapped paramagnet, Fig. 8.2(b), with a correlation length ξ2D &
12`B larger than the circumference we can access. Likewise, while the super-linear
behavior for large U indicates a region of first-order behavior, we cannot rule out a
transition which is weakly first-order along the whole m = 0 line. As U varies along
the m = 0 line, the curvature in ξ(L) appears smoothly, and becomes clear in our
numerics for U & 5u.
The values of bond dimension used to obtain the data in Fig. 8.3, χ ≤ 32000, are
much larger than those used elsewhere in this thesis – e.g. in Chapter 4, where we also
study a critical state (the CFL), we are limited to χ . 8000. The reason for this is the
presence of additional symmetries in this problem, which give rise to the conservation
of spin and valley quantum numbers in addition to particle number and momentum.
This make the MPS tensors block-diagonal, thus lowering the computational cost of
DMRG at a fixed χ. Conversely, this also lowers the accuracy of DMRG at a fixed χ.
This means that even a nominally a huge value like χ = 32000 is not enough to fully
174
capture the correlations of the critical state for a modest size L = 12`B, as shown by
the ongoing drift of ξ with χ in Fig. 8.3(a).
8.3.4 Scaling dimensions
To investigate the intriguing possibility of a CFT in the small-U regime, we attempt
to measure the scaling dimension ∆V of the vector operator ni(r) = ψ†(r)Γiψ(r).
Assuming conformal invariance, on the plane the two-point function is
Cij(r) = 〈ni(r) nj(0)〉 ∝ δijr−2∆V (8.5)
where ∆V is the scaling dimension. Since the SO(5) symmetry is exact in our numer-
ics, we can restrict to a single Cii (for SO(4), i 6= 3). On the cylinder, we measure
∆V via the total squared “magnetization”, M ≡∫d2rn(r):
M2(L) =1
Vol〈MiMi〉 =
1
Vol
∫d2r1 d
2r2〈ni(r1)ni(r2)〉
=
∫d2r〈ni(r)ni(0)〉 ≡
∫Rdx
∫ L
0
dy Cii(x, y) . (8.6)
The dependence on the cylinder circumference L is easily isolated by rescaling the
integration variables and exploiting Eq. (8.5):
M2(L) = L2−2∆VM2(1). (8.7)
Thus in principle ∆V can be extracted from M2(L) using a one-parameter finite-size
scaling collapse.
This picture is complicated by the finite bond dimension χ in our iDMRG numer-
ics, which as discussed earlier introduces a second length cutoff in the problem in the
form of a finite correlation length ξ. Consequently, we calculate M2(L) for a range
of values of L and χ, with the latter parameterized via the MPS correlation length ξ,
175
0.35 0.40 0.45 0.50 0.55
/L0.70
0.72
0.74
0.76
0.78
0.80
0.82
0.84
M2 (
L,)/L
22
V V = 0.57
L/ B6789101112
0.6 0.8 1.0 1.2 1.4 1.6 1.8
/L
0.25
0.30
0.35
0.40
0.45
0.50
0.55
M2 (
L,)/L
22
V
V = 0.24
Figure 8.4: Two-parameter scaling collapse of M2i on the SO(5) line, ui ≡ u, with
U = 0.5u. The data is obtained from iDMRG simulations with bond dimensions χranging from 2000 to 32000 (leftmost to rightmost points at each size). The solidline shows a polynomial fit to data points with L ≥ 8`B and represents the scalingfunction f(ξ/L) in Eq. (8.8); ∆V is chosen so as to minimize the error of the fit. Inset:data for U = 10u (large stiffness) shows poor collapse and returns an estimate of ∆V
in severe violation of the unitarity bound, ∆V > 0.5.
and collapse the data using the scaling form
M2(L, ξ) = L2−2∆V f(ξ/L) . (8.8)
For a large enough circumference (L & 8`B), we find that there exists a value of
∆V , typically determined to within ±0.01, such that the data for different L up to
12`B collapse onto the scaling form of Eq. (8.8). An example is shown in Fig. 8.4.
Similar behavior is found across much of the parameter space (we sit at the critical
point, m = 0, and assume u > 0, leaving the two independent parameters U/u and
u3/u).
In Fig. 8.5 we show the variation of the estimated ∆V along two cuts in parameter
space, one on the O(5)-line and one in the SO(4)-region. The value of the scaling
176
3 2 1 0 1 2U
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
V4 2 0 2
U
0.85
0.90
0.95
1.00
1.05
u 3 SO(4) critical
O(5) criticalEasy-axis Neel
phas
e se
para
tion
u3 = 1 [O(5)]u3 = 0.9 [SO(4)]
Figure 8.5: Measured scaling dimension ∆V along two cuts in parameter space atm = 0 and u = 1: u3 = 1 (O(5) symmetry) and u3 = 0.9 (SO(4) symmetry). Notethat while we refer to U/ui as the stiffness, making the region U < 0 seem unphysical,the ui themselves lead to a repulsive interaction which prevents phase-separation forU & −2.6. For u3 > 1, the system polarizes into an easy-axis Neel state.
dimension ∆V (as well as the accuracy of the collapse) drifts with U and u3; large
U generally shows lower ∆V and worse collapse. The conjectured O(N)-symmetric
CFT should yield one well defined value for ∆V for N = 4, 5 respectively, so the
dependence we observe is either a finite-size effect or evidence that a weakly-first
order transition persists to higher ui/U than can be detected from the super-linear
scaling of ξ. Indeed, at large U , ∆V violates the unitarity bound ∆V ≥ d2− 1 = 1
2,
while lowering U takes ∆V up to ∼ 0.7 (U can be reduced down to U ' −2.6u, at
which point the attractive interaction leads to phase separation). Due to the limited
system size, it is difficult to determine where (if at all) the weakly first-order line
becomes a CFT.
The easy-plane anisotropy (u3 < u) breaks O(5) down to SO(4) and makes the
model stiffer. A moderate value like u3 = 0.9u (used in Fig. 8.5) lowers ∆V slightly,
while a large anisotropy like u3 = 0 makes the transition strongly first-order.
177
In conclusion, while iDMRG simulations do not provide a definitive numerical
prediction for the scaling dimension ∆V , they are consistent with a continuous tran-
sition characterized by an exponent ∆V somewhat larger than the unitarity bound,
in agreement with earlier calculations on the cubic dimer model [202, 203], the JQ
model [174, 173, 70, 134], loop models [152] or a large-N expansion of the CPN−1 field
theory [38], all of which place the vector dimension ∆V in the range 0.57 to 0.68.
8.4 Sign-free Determinantal Quantum Monte
Carlo
We now show that the model is amenable to sign-free determinantal quantum Monte
Carlo, due to a combination of particle-hole and flavor symmetry, leading to an algo-
rithm with polynomial complexity in system size.
We consider a quantum Hall Hamiltonian of the general form
H =1
2
∑i
∫d2r ni(r)U i(r− r′)ni(r′) =
1
2V
∑i
∫d2q ni−qU
i(q)niq (8.9)
in real and Fourier space respectively (V is the system’s “volume” 2πNφ`2B). On the
sphere, the Fourier transformation can be replaced by a spherical harmonic decom-
position. Here ni(r) = ψ†(r)Oiψ(r), where O acts on the flavor index. Without loss
of generality we take O = O†, so that the ni are Hermitian. After LL-projection
on a cylinder with a circumference L in the Landau gauge, the single particle orbitals
are labelled by their momenta around the cylinder, k = 2πLm for m ∈ Z, and flavor
index a. The density operators are expanded in annihilation operators ck,a as [170]
niq = e−q2`2B/2
∑a,b,k
e−ikqx`2B c†k+qy/2,a
Oiabck−qy/2,b. (8.10)
178
On the torus the same form carries through after identifying k ∼ k + L/`2B, up to
exponentially small terms in L/`B.
In the auxiliary field method the interactions are decoupled using bosonic
Hubbard-Stratonovich fields φ. There are a variety of possible channels for this
decomposition, including the Cooper channel, but as a proof-of-principle we present
here the choice niq − ni−q. We introduce Hermitian Hubbard-Stratonovich fields
φqi = φ−qi for each operator type, so that a small imaginary time step dτ can be
decomposed as
e−dτH ∼∏i,q
edτni−q(−U i(q)/V )niq ∼
∏i,q
∫dφqi e
−dτ |φqi|2+dτ√−U i(q)/V (niqφqi+h.c.) (8.11)
up to normalization and Trotter errors O(dτ 2). Note that because of LL-projection,
[ni(r), ni(r′)] 6= 0. Multiplying over imaginary time-steps and integrating out the
fermions, we obtain an auxiliary field path integral of the general form
Z = Tr(e−βH
)=
∫D[φ]e−S[φ]M [φ] (8.12)
S[φ] =∑i,q
∫dτ |φqi(τ)|2 (8.13)
where M [φ] is the fermion determinant for auxiliary field space-time configuration
φq(τ).
The problem is sign free if M [φ] ≥ 0 for all φ. A sufficient criterion for a sign-
free determinant is the existence of two anti-unitary symmetries T1, T2 such that
T 21 = T 2
2 = −1 and T1T2 = −T2T1. The symmetry must exist for any auxiliary field
configuration. [229, 9, 126] Time-reversal is broken by the magnetic field, but at half-
filling there is an anti-unitary particle-hole operation PH which exchanges empty and
179
occupied states of the LL:
PH αψa(r) PH−1 = αψ†a(r) (8.14)
PHckaPH−1 = c†ka (8.15)
Since the ni are Hermitian, PH acts as PHni(r) PH−1 = −ni(r), or PHniq PH−1 =
−ni−q (we assume Tr(Oi) = 0). PH can be combined with a unitary transformation X
acting on the flavor index. We take Tg = XgPH, g = 1, 2, and look for Xg satisfying
the symmetry conditions and the sign-free conditions.
The symmetry condition is
Oi = sign(U i(q)
)XgO
iX†g (8.16)
For a repulsive channel (U i > 0), Oi must be even under Xg, while for an attractive
one (U i < 0), Oi must be odd. The sign-free conditions are
T 2g = XgXg = −1 T1T2 = X1X2 = −X2X1 = −T2T1 . (8.17)
For the problem at hand we have Oi = Γi, i = 1, · · · , 5 with U i(q) = −ui < 0, plus
the density channel O0 = 1. But with this decomposition it is impossible to find the
two required Xg, because the Γ are by definition a maximally anti-commuting set.
Fortunately, for contact interactions we may use a Fierz identity (see appendix 8.A)
and consider instead [232]
H =g
2(ψ†ψ)2 +
1
2
∑µ=x,y,z
gµ(ψ†τµψ)2 (8.18)
where g = U + uN , gx = −uN − u4, gy = −uN − u5, and gz = 2uN . The region of
interest is g, gz > 0 and gx, gy < 0. Decomposing in the density channels associated
180
0.0 0.5 1.0 1.5 2.0 2.5 3.0J
2
4
6
8
10
M2 L
22
N12345678
Figure 8.6: DQMC result for the embedding of the transverse field Ising model intothe half-filled Landau level. We plot the squared Ising magnetization M2 at transversefield h = 0.1, as a function of the Ising coupling J . Data is scaled by L2∆−2, where∆ ' 0.259 is the known scaling dimension of the Ising magnetization. As expected,the data shows a crossing around J ≈ 1.
to these g, it is now easy to verify that X1 = iτ zσx and X2 = iτ zσy satisfy the sign
free condition. To handle the SO(4) case, we can reduce |u4| from its SO(5) value.
The sign-free condition can be seen more explicitly from Eq. (8.18) because the
determinant M factors by spin, M [φ] = M↑[φ]M↓[φ]. This is because the Oi do not
depend on the σµ spin Pauli matrices, so all the densities decompose as ni = ni↑+ni↓,
[ni↑, nj↓] = 0. The spin-exchanging anti-unitaries Tg ensure M↑ = M∗
↓ , so the partition
function can be evaluated by restricting to the ↑ orbitals,
Z =
∫D[φ]e−S[φ]|M↑[φ]|2 , (8.19)
which is manifestly sign-free and reduces the dimension of the linear algebra routines
from 4Nφ to 2Nφ.
As a simple test of the proposal we consider the 2+1D transverse field Ising model,
which can be embedded into the N = 4 model by choosing U = 0, u1 = u2 =
u3 = u4 = 0, u5 = J , and introducing an additional transverse field hψ†τxψ (fields
181
along τx/y preserve the sign-free condition). The ratio h/J should tune an ordered-
disordered transition with Ising order parameter M = 〈n5〉.
We present the results of a small projector DQMC simulation in Fig. 8.6. Rather
than extrapolating to zero-temperature, we evolve a transverse-polarized state | →〉
to finite β ∝√Nφ, i.e. calculate |β〉 = e−βH | →〉. In units where `B = e2
4πε0= 1,
we take β = 5√Nφ and ∆τ = 0.25 with a second-order Trotter decomposition and
measure the total squared magnetization M2 = 〈(n5q=0
)2〉Nφ . After scaling with the
known Ising critical exponents, the data shows the crossing predicted by an Ising
transition.
8.5 Discussion
In this Chapter we have discussed how several 2+1D quantum phase transitions, in-
cluding deconfined quantum critical points, can be realized in half-filled continuum
Landau levels which exactly preserve internal and spatial symmetries which would
otherwise be realized only in the IR. The approach can be understood as a con-
tinuum regularization of an O(N) nonlinear sigma model – albeit one with a soft
UV cut-off provided by the finite magnetic length. These models can be studied us-
ing DMRG, and despite the broken time-reversal symmetry, sign-free determinantal
quantum Monte Carlo.
To our knowledge, our DMRG results are the first study of a putative DCQP be-
yond QMC. While the DMRG system size (L = 12`B) is much smaller than previous
QMC results, we do see behavior in rough agreement with the existing literature.
Specifically, in QMC the exponents drift slowly with system size (∆V flows down-
ward) indicating that the transition is either weakly first-order or has unconventional
corrections to scaling. [152] While we cannot detect such a finite-size drift given our
small L, we do observe a complementary phenomenon. Our model allows us to tune a
182
parameter, the stiffness U , which (if the DCQP exists) should be irrelevant. Instead
we find that the estimate of the scaling dimension ∆V changes with U . For large
U , ∆V is reduced and eventually violates the unitarity bound before the transition
becomes clearly first-order. The largest value we observe is ∆V ∼ 0.7. This estimate
still slightly violates the best bounds from conformal bootstrap when assuming SO(5)
symmetry. [167, 156]
Going forward, the crucial question is whether sign-free DQMC simulations will
be able to reach the system sizes required to shed new light on this issue. If so,
the continuum realization may have significant advantages because it allows direct
identification of the NLSM stiffness, SO(5) vector operator, and symmetric-tensor
perturbations without tuning.
A second question is which other CFTs might be realized in this fashion. For
half-filled N = 4 LLs, we have a sign-free realization of the O(M) Wilson-Fisher
fixed point for M = 1, 2, 3 and O(M) DCQPs for M = 4, 5. It will be interesting to
investigate what other models are sign-free when using N > 4 LLs, or even attacking
5-dimensional CFTs using the quantum Hall effect in 4 + 1 dimensions.
Finally, we remark that the model and parameters we used to engineer the DQCP
are very far from those of realistic monolayer graphene. The tuning of large inter-
action anisotropies in the valley degree of freedom is especially problematic. How-
ever, multilayer graphene structures, which have attracted enormous attention re-
cently [26, 180, 241], can have much greater tunability; the possibility of replacing
the valley index examined here with a layer index appears especially promising.
183
Appendices
8.A Equivalent parametrizations of SU(4) anisotropies
Here we review the Fierz identities [43, 158] used to relate the two parametrizations
of SU(4) anisotropies used in this Chapter, i.e., that Eq. (8.1) is equivalent to
H = H0 +1
2
∑µ=x,y,z
gµ(ψ†τµψ)2 ,
with gx = gy ≡ g⊥. This parametrization allows a more direct conversion to ex-
perimental parameters [245, 255], and is crucial in the implementation of sign-free
determinant quantum Monte Carlo in Sec. 8.4.
The equivalence can be proven by making use of a version of the Fierz identities,
which we derive in the following. We start by considering the set of matrices Oi =
σaτ b, with a, b ∈ 0, 1, 2, 3. These form a basis of 4 × 4 matrices. Therefore the
tensor products Oi⊗Oj form a basis of 16× 16 matrices, and one can perform the
following decomposition:
OiαβO
iγδ =
∑j,k
bijkOjαδO
kγβ , (8.20)
184
where the Greek indices run over electron flavors and b is a matrix of coefficients. We
insert OmδαO
nβγ on both sides of Eq. (8.20) and contract all flavor indices, obtaining
Tr(OiOnOiOm) =∑j,k
bijkTr(OjOm)Tr(OkOn) .
The Oi are trace-orthogonal, with Tr(OiOj) = 4δij. Moreover, any two O operators
either commute or anti-commute, and each O squared yields the identity. Using these
facts we obtain
bimn =1
16Tr(OiOmOiOn) = ± 1
16Tr(OmOn) = ±1
4δmn ,
with the ± sign decided by whether Oi and Om commute or anti-commute. We can
finally rewrite Eq. (8.20) as
(ψ†(x)Oiψ(x))2 = −∑j
bij(ψ†(x)Ojψ(x))2 , (8.21)
bij =
+ 1/4 if OiOj = OjOi ,
− 1/4 if OjOi = −OiOj .
(8.22)
The extra sign comes from the Fermi statistics of the ψ operators.
Direct application of Eq. (8.21) shows that
(ψ†τ zψ)2 −∑a=4,5
(ψ†Γaψ)2 + (ψ†ψ)2 = −(ψ†τ zψ)2 −∑
a=1,2,3
(ψ†Γaψ)2 ,
which implies
(ψ†τ zψ)2 = −1
2(ψ†ψ)2 − 1
2
∑a=1,2,3
(ψ†Γaψ)2 +1
2
∑a=4,5
(ψ†Γaψ)2 (8.23)
185
This identity allows us to map the two parametrizations:
H =V
2(ψ†ψ)2 +
g⊥2
∑µ=x,y
(ψ†τµψ)2 +gz2
(ψ†τ zψ)2
=U
2(ψ†ψ)2 − uN
2
∑a=1,2,3
(ψ†Γaψ)2 − uK2
∑a=4,5
(ψ†Γaψ)2 (8.24)
with
U = V − 1
2gz, uN =
1
2gz, uK = −g⊥ −
1
2gz (8.25)
and the Γa matrices given in Sec. 8.2.
186
Chapter 9
Concluding remarks
Nearly four decades after its discovery, the quantum Hall effect continues to be a
source of interesting and exciting physics. The vast landscape of phases and phenom-
ena exhibited by a single physical system – a 2D electron gas in a magnetic field – is
truly remarkable. In this thesis we have explored some particular regions within this
landscape: the response of quantum Hall fluids to geometric distortions, and quan-
tum critical points (arising from disorder or interactions) in a Landau level. In this
concluding Chapter we discuss some open questions and directions for future research
for each of these topics.
9.1 Geometry of quantum Hall states
The geometry of fractional quantum Hall states has attracted much interest in the
past decade, resulting in a fairly detailed understanding of many of its aspects. The
work presented in this thesis gives some of the most detailed predictions available to
date for static distortions in both compressible and incompressible states. However,
many questions remain.
A question we have raised and not fully answered is that of the relation between
the CFL at ν = 1/2 and the states in the first Jain sequence νp = p2p+1
. Our re-
187
sults indicate a small but significant difference between the former and the latter in
terms of their response to band mass anisotropy. Is the discrepancy real, or is it a
numerical artifact? This may seem like a narrow technical point, but it carries some
implications for a fundamental question: how real are the composite fermions? The
incompressible states in the first Jain sequence are thought to arise as IQH states
of the same composite fermions making up the CFL. If the composite fermions are
objects with a well-defined shape, we may expect all these phases (whether com-
pressible or incompressible) to have the same geometry. Numerical evidence to the
contrary may cast doubt on the composite fermion as the fundamental building block
of these phases, and raise the question – is the more fundamental object a composite
boson instead? On the other hand, a small filling-dependent variation of the geomet-
ric response might also result form weak residual interactions between the composite
fermions, which may renormalize the geometry of these states in different ways. More
detailed studies are necessary to shed light on this issue.
Another goal for future research is a more complete theory of higher-order
anisotropy, i.e. distortions with N -fold rotational symmetry, N > 2. The original
theory of the geometry of FQH states [64] singles out the spin-2 ‘intrinsic metric’
degree of freedom as special; so does bi-metric theory [58]. Comparatively little is
known about distortions with other rotational symmetries. Our study in Chapter 5
is one of the first investigations in this direction. Recent field-theoretic work on
the CFL outlines a theory based on a “tower” of higher-spin modes representing
distortions of the Fermi contour [56]. More work is needed to achieve a better
understanding of such modes in incompressible states, and of how they may relate
to other descriptions (e.g., model wavefunctions).
On the experimental side, a major outstanding problem is the measurement of
the ‘intrinsic metric’ of incompressible states. Yang [238] has proposed using the
absorption of acoustic waves, which unlike electromagnetic probes would couple to
188
the spin-2 “graviton” mode. Alternatively, one could image quasiholes pinned by a
point impurity. This has been done successfully with scanning tunneling microscopy
for integer states [42], where band mass is the sole metric in the problem. For
fractional states the picture would be somewhat complicated: the density profile
of the quasihole would not necessarily have the same anisotropy at all distances; the
“intrinsic metric” would be encoded in the asymptotic shape of equal-density contours
at long distance (which could be extrapolated from a few maxima and minima).
Whether this approach is viable for fractional states remains to be seen.
Finally, in this thesis we have restricted our study to the statics of the geometric
degree of freedom: we introduced a fixed, constant distortion, and measured its effect
on the zero-temperature many-body ground state. The dynamics of this degree of
freedom is equally interesting, and its exploration has just begun [131, 128]. Out-
of-equilibrium and Floquet dynamics of the “quantum Hall graviton”, as well as
higher-spin modes, are promising directions for future research.
9.2 Quantum criticality
The integer quantum Hall plateau transition remains a subject of controversy due
to the unsolved issue of the agreement between experiment and single-particle nu-
merics. The nature of the fixed-point theory remains unclear, as well. While in this
thesis we have not directly attacked these long-standing issues, we have found new
and unexpected behavior of the transition in unusual setups.
In Chapter 6 we discussed a method to isolate a flat, Chern 1 subband of the LLL
by using point impurities, and showed that the plateau transition in this band looks
exactly the same as in the whole LLL, and that the scale in the renormalization-
group flow is set by the number of states in the band, rather than by the physical
system size. This indicates a possible universality for plateau transitions in general
189
Chern 1 bands. As a byproduct, one gets the possibility of engineering Chern 0
bands by picking trivial states from the LLL. This technique has enabled the study
of many-body localization in the continuum LLL setting [116, 115].
In Chapter 7 we considered the one-dimensional limit of the plateau transition,
and found an unexpected proliferation of Chern states coexisting with Anderson local-
ization across the spectrum. This behavior was understood by mapping the system
onto a disordered Thouless pump, where the Chern number equals the real-space
winding number of each electron during a pump cycle. While we argued this behav-
ior is not visible in macroscopic systems (as it is associated to exponentially long time
scales), it may be observed in microscopic quantum simulators, such as cold atoms
in optical lattices. This idea may also be generalized to higher dimension. The four-
dimensional quantum Hall effect can be dimensionally reduced to a two-dimensional
Thouless pump, which has already been realized experimentally in a clean case [133];
adding disorder to this problem would be interesting both theoretically and experi-
mentally.
Finally, Chapter 8 presents the possibility of using internal degeneracies of LLs to
realize various types of magnetic order, and thus engineer interesting critical points.
From the numerical standpoint this provides a regularization of the desired critical
point that does not rely on a lattice and therefore preserves all the continuum sym-
metries of the theory. This may have a number of applications. The example we
discussed – the conjectured ‘deconfined’ critical point between a Neel antiferromag-
net and a Kekule valence bond solid in graphene – has been the subject of a huge
amount of work in the past decade, both numerical and field-theoretic. The finite-size
limitations of our DMRG simulations may be improved upon by large-scale quantum
Monte Carlo, taking advantage of the sign-free condition we discussed. This may
clarify whether the critical point is described by a symmetric CFT, or by a theory
with weakly broken symmetry. Other 2+1D CFTs may be studied in the same way.
190
Beyond the numerical application, this raises the interesting possibility of using
graphene Landau levels as laboratories for quantum criticality. The goal would be
to engineer a system that can host different types of quantum Hall ‘ferromagnetism’
depending on couplings that are easily and accurately tunable by the experimentalist.
This is still far from a reality. However, multilayer graphene structures, which have
attracted explosive interest lately, may provide an avenue towards the realization of
this idea.
191
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