studies on time-reversal invariant topological insulatorsfv691mt5830/maciejko_phd_the… · studies...

STUDIES ON TIME-REVERSAL INVARIANT TOPOLOGICAL

INSULATORS

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF PHYSICS

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Joseph Maciejko

August 2011

http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/fv691mt5830

© 2011 by Joseph Maciejko. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.

ii



http://purl.stanford.edu/fv691mt5830

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Shoucheng Zhang, Primary Adviser


David Goldhaber-Gordon


Xiaoliang Qi

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

Abstract

This dissertation brings together a number of topics in the theory of time-reversal (TR) in-

variant topological insulators. The first four chapters aredevoted to the transport properties

of the 2D quantum spin Hall (QSH) state. We explain nonlocal transport measurements in

HgTe quantum wells in terms of a Landauer-Buttiker theory of helical edge transport and

confirm the discovery of the QSH state in this material. We findthat decoherence can lead

to backscattering without breaking microscopic TR symmetry. As an example of incoherent

scattering, we study a Kondo impurity in an interacting helical edge liquid. A renormaliza-

tion group analysis shows the existence of an impurity quantum phase transition governed

by the Luttinger parameter of the edge liquid between a localhelical Fermi liquid withT 6

scaling of the low-temperature conductance, and an insulating strongly correlated phase

with fractionally charged emergent excitations. In the presence of a TR symmetry break-

ing magnetic field, it is known that even coherent scatteringcan lead to backscattering.

Through exact numerical diagonalization we find that nonmagnetic quenched disorder has

a strong localizing effect on the edge transport if the disorder strength is comparable to the

bulk gap. The predicted magnetoconductance agrees qualitatively with experiment.

The last two chapters are devoted to 3D topological insulators. We propose a com-

bined magnetooptical Kerr and Faraday rotation experimentas a universal measure of the

Z2 invariant. Finally, we propose a fractional generalization of 3D topological insulators

in strongly correlated systems, characterized by ground state degeneracy on topologically

nontrivial spatial 3-manifolds, a quantized fractional bulk magnetoelectric polarizability

without TR symmetry breaking, and a halved fractional quantum Hall effect on the sur-

face.

iv

Acknowledgments

First and foremost, I wish to thank my advisor Shou-Cheng Zhang. His remarkable

ability to distill beautiful and profound ideas from complex condensed matter problems,

his unwavering commitment to Dirac’s principle of simplicity and elegance in theoretical

physics, and his no-nonsense approach to the sometimes frustrating intricacies of day-to-

day problem-solving and overall project management alike have been a great source of

inspiration to me. I am most grateful for his constant reminder to trust in my own abilities

and judgment, as well as for his renewed encouragement to risk thinking outside the box

and choose challenging yet important research problems.

A special thanks goes to my other primary mentor and collaborator, Xiao-Liang Qi.

His generosity in sharing his amazing talent and enthusiasmfor physics with me as well as

his patient guidance have made working with him a truly formative experience, and a real

pleasure.

I would like to thank the faculty in the condensed matter group at Stanford, in particular

Steve Kivelson, Tom Devereaux, David Goldhaber-Gordon, Ian Fisher, and Hari Manoha-

ran, for many interesting discussions and for their help in various academic matters.

Wisdom is not found in books alone, and it is a pleasure to acknowledge my col-

laborators who did not skimp in sharing theirs with me: Laurens Molenkamp, Hartmut

Buhmann, Ewelina Hankiewicz, Eun-Ah Kim, Yuval Oreg, Andreas Karch, Congjun Wu,

Dennis Drew, Tadashi Takayanagi, Tom Devereaux, Rajiv Singh, Chao-Xing Liu, Cheng-

Chien Chen, Andreas Roth, Christoph Brune, Suk Bum Chung, Adam Sorini, and Brian

Moritz. I would also like to acknowledge financial support I have received over the past

five years from the Stanford Graduate Fellowship (SGF) Program, the National Science

and Engineering Research Council of Canada (NSERC), and theFonds Quebecois de la

v

Recherche sur la Nature et les Technologies (FQRNT), without which the work reported

in this dissertation would not have been possible. I also wish to thank Tom Devereaux for

kindly providing me access to the computing facilities of the Shared Hierarchical Academic

Research Computing Network (SHARCNET) and the Stanford Institute for Materials and

Energy Science (SIMES) at the SLAC National Accelerator Laboratory. Likewise, I thank

Laurens Molenkamp and Ewelina Hankiewicz for kindly providing me access to the com-

puting facilities of the Leibniz Rechenzentrum Munich.

A special thanks goes to Cheng-Chien Chen, my friend and fellow condensed matter

theorist at the physics department and CMITP. It has been an honor and a privilege to

share the highs and lows of graduate school with him. I also wish to thank my other

colleagues at the CMITP, Jiun-Haw Chu, Stathis Ilonidis, and George Karakonstantakis,

for their friendship and support.

I wish to thank my research group colleague Taylor Hughes forhis encouraging men-

torship. Benefitting from the guidance of such a fine physicist and friendly senior col-

league helped me through each of the new steps of my Ph.D. I also wish to thank my other

group colleagues, Rundong Li, Sri Raghu, Suk Bum Chung, Zhong Wang, Qin Liu, An-

drei Bernevig, Lukas Muchler, and Maissam Barkeshli for sharing their knowledge and

perspective on various physics topics.

I have benefitted a lot from various professional and personal interactions with my

fellow physics students and friends at Stanford, Hong Yao, Erez Berg, Andrei Garcia, John

Robertson, Wei-Feng Tsai, Katherine Luna, Reza Jamei, Ileana Rau, Alex Fried, Weejee

Cho, Yeming Shi, Sho Yaida, Li Liu, Francois Amet, Nachum Plonka, Daniel Harlow, and

Tzen Ong. You have all contributed to making the last five years truly enjoyable — thank

you!

I would like to express sincere thanks to Maria Frank, Roberta Edwards, Sybille Katz,

Corrina Peng, Ellie Lwin, and Catherine Meng for their invaluable help in dealing with

various administrative matters. Thank you for always finding time to help despite your

busy schedules.

I distinctly remember flipping through my dad’s Ph.D. dissertation on gauge theories of

particle physics as a kid, and being fascinated by the myriadof strange symbols and funny

little diagrams with wiggly lines that covered its pages. Little did I, or dad, know that this

vi

was the beginning of my career in theoretical physics, and that I would be writing my own

dissertation twenty or so years later, with its own collection of mysterious equations. Thank

you dad for introducing me to the amazing world of physics.

I am especially grateful to my parents Romain and Helene, my sister Eva, and all of

my family. Through your own example and your sense of humor, you have shown me that

belief in self and perseverance go a long way. Your unconditional love and support have

kept me plodding through during difficult times — thank you!

Finally, it is a joy to thank my wonderful fiancee, Michelle.Being engaged to a woman

who is both an incredibly smart Stanford Ph.D. student and anamazingly loving and sup-

portive person is a true blessing. I thank her for taking a genuine interest in all my endeav-

ors, academic and otherwise; for being the most balanced, optimistic, and joyful person

I know, especially during big research headaches; and most of all, for believing in me at

every step of the way. This dissertation is dedicated to her.

vii

Contents

Abstract iv

Acknowledgments v

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3 Dissertation overview . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 12

2 Nonlocal transport in the QSH state 14

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Ohm’s law versus nonlocal transport . . . . . . . . . . . . . . . . .. . . . 15

2.3 Device structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 Transport on the edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5 Dissipationless transport . . . . . . . . . . . . . . . . . . . . . . . .. . . 18

2.6 Helical versus chiral . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19

2.7 Dephasing and nonlocal transport . . . . . . . . . . . . . . . . . . .. . . 23

2.7.1 Quantum kinetic formalism and Landauer-Buttiker equations . . . . 24

2.7.2 Dephasing in the QSH regime: numerical study . . . . . . . .. . . 27

2.7.3 Dephasing in the QH regime: numerical study . . . . . . . . .. . . 32

2.7.4 Self-energy and microscopic time-reversal symmetry. . . . . . . . 34

2.7.5 Nonlocal resistance in H-bar structure: numerical study . . . . . . . 38

2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

viii

3 Kondo effect in the QSH state 41

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2 Theoretical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.3 Weak coupling regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.4 Strong coupling regime . . . . . . . . . . . . . . . . . . . . . . . . . . . .47

3.5 Experimental realization . . . . . . . . . . . . . . . . . . . . . . . . .. . 50

4 Magnetoconductance of the QSH state 52

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2 Theoretical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5 Spin AB effect and topological spin transistor 62

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.2 Phenomenological scattering matrix analysis . . . . . . . .. . . . . . . . . 65

5.3 Minimal model description . . . . . . . . . . . . . . . . . . . . . . . . .. 66

5.4 Experimental realization in HgTe QW . . . . . . . . . . . . . . . . .. . . 70

5.5 Conclusion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . .74

6 Topological quantization in units ofα 75

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.2 Magnetooptical Kerr and Faraday rotation . . . . . . . . . . . .. . . . . . 77

6.3 Reflectivity minima and total surface Hall conductance .. . . . . . . . . . 79

6.4 Reflectivity maxima and topological magnetoelectric effect . . . . . . . . . 80

6.5 Kerr-only measurements . . . . . . . . . . . . . . . . . . . . . . . . . . .82

6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

7 Fractional topological insulators in 3D 85

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

7.2 Projective construction and trial wave function . . . . . .. . . . . . . . . . 87

7.3 Effective gauge theory . . . . . . . . . . . . . . . . . . . . . . . . . . . .88

ix

7.4 Time-reversal symmetry and quantization of the axion angle . . . . . . . . 91

7.5 Bulk topology and surface “half” fractional quantum Hall effect . . . . . . 92

7.6 Three classes of parton models . . . . . . . . . . . . . . . . . . . . . .. . 93

7.6.1 Deconfined models . . . . . . . . . . . . . . . . . . . . . . . . . . 98

7.6.2 Higgs models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

7.7 Fractional quantized magnetoelectric effect . . . . . . . .. . . . . . . . . 107

7.7.1 Chiral anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.7.2 Topological quantization of the axion angle . . . . . . . .. . . . . 110

7.8 Ground state degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . .115

7.8.1 Ground state degeneracy on the 3-torus . . . . . . . . . . . . .. . 115

7.8.2 Ground state degeneracy on 3-manifolds with boundaries . . . . . . 123

7.9 Gapless surface states . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 126

7.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

8 Conclusion 130

A Theoretical methods 133

A.1 Weak-coupling renormalization group equations . . . . . .. . . . . . . . . 133

A.1.1 Operator algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

A.1.2 Scaling dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 138

A.1.3 Operator product expansions . . . . . . . . . . . . . . . . . . . . .140

A.2 Weak-coupling Kubo formula calculation of the edge conductance . . . . . 144

A.2.1 Analytic continuation in the time domain . . . . . . . . . . .. . . 158

A.2.2 Bosonic correlators . . . . . . . . . . . . . . . . . . . . . . . . . . 163

A.3 Coulomb gas description of the strong coupling regime . .. . . . . . . . . 165

A.3.1 Euclidean action . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

A.3.2 Semiclassical instanton calculation . . . . . . . . . . . . .. . . . . 168

A.3.3 Dilute instanton gas approximation . . . . . . . . . . . . . . .. . 171

A.3.4 Tree-level renormalization . . . . . . . . . . . . . . . . . . . . .. 174

A.3.5 Mapping to dual boundary sine-Gordon model . . . . . . . . .. . 176

A.3.6 Current operator in strong coupling regime . . . . . . . . .. . . . 179

A.4 Keldysh calculation of the shot noise . . . . . . . . . . . . . . . .. . . . . 180

x

A.4.1 Basics of noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

A.4.2 Schwinger-Keldysh action . . . . . . . . . . . . . . . . . . . . . . 185

A.4.3 Nonequilibrium current . . . . . . . . . . . . . . . . . . . . . . . . 186

A.4.4 Nonequilibrium noise . . . . . . . . . . . . . . . . . . . . . . . . 187

A.4.5 Schottky relation and fractional Fano factor . . . . . . .. . . . . . 188

A.5 Mean-field description of half-charge tunneling . . . . . .. . . . . . . . . 189

A.6 Phase space derivation of theT 6 behavior in the noninteracting case . . . . 192

A.7 Transport in a spin Aharonov-Bohm ring . . . . . . . . . . . . . . .. . . . 195

A.7.1 S-matrix analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

A.7.2 Scattering at the junction . . . . . . . . . . . . . . . . . . . . . . .197

A.7.3 Tight-binding model . . . . . . . . . . . . . . . . . . . . . . . . . 198

A.8 Magnetooptical Kerr/Faraday rotation in thick film geometry . . . . . . . . 199

A.9 Surface Chern-Simons theory for an Abelian fractional 3D TI . . . . . . . . 202

Bibliography 207

xi

List of Tables

7.1 Gauge charge assignments of parton fields: Abelian model. . . . . . . . . 98

7.2 Gauge charge assignments of parton fields: Higgs model . .. . . . . . . . 105

xii

List of Figures

1.1 Chiral versus helical: QH versus QSH effects . . . . . . . . . .. . . . . . 5

2.1 2- and 4-terminal resistances . . . . . . . . . . . . . . . . . . . . . .. . . 16

2.2 Devices D3 and D4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 2- and 4-terminal resistances for device D3 . . . . . . . . . . .. . . . . . 21

2.4 Nonlocal resistances for H-bar device D4 . . . . . . . . . . . . .. . . . . 22

2.5 2-terminal geometry in QSH regime with dephasing . . . . . .. . . . . . . 29

2.6 2-terminal conductance in QSH regime from TB model . . . . .. . . . . . 30

2.7 2- and 4-terminal conductances in QSH regime with dephasing . . . . . . . 31

2.8 Cartoon picture of edge transport in QSH regime with dephasing . . . . . . 32

2.9 2-terminal geometry in QH regime with dephasing . . . . . . .. . . . . . 33

2.10 2-terminal conductance in QH regime,Bz = 1 T . . . . . . . . . . . . . . 34

2.11 2-terminal conductance in QH regime,Bz = −1 T . . . . . . . . . . . . . 35

2.12 Cartoon picture of edge transport in QH regime with dephasing . . . . . . . 36

2.13 Numerical calculation of 4-terminal resistance in H-bar geometry . . . . . . 38

3.1 Edge conductance of the QSH state at finite temperature . .. . . . . . . . 42

3.2 Edge conductance in the strong coupling regime . . . . . . . .. . . . . . . 47

4.1 MagnetoconductanceG(B) of a QSH edge . . . . . . . . . . . . . . . . . 56

4.2 MagnetoconductanceG(B) for various disorder strengths . . . . . . . . . . 58

4.3 Effect onG(B) of BIA term,B-field orientation, and Dirac mass . . . . . . 60

5.1 Schematic picture of the spin AB effect . . . . . . . . . . . . . . .. . . . 64

5.2 Illustration of the minimal model describing a FM/QSH junction . . . . . . 67

xiii

5.3 Phenomenological analysis of the two-terminal conductance . . . . . . . . 69

5.4 Numerical study of the spin AB effect in HgTe QW . . . . . . . . .. . . . 71

6.1 Kerr and Faraday angles for a topological insulator thick film . . . . . . . . 77

6.2 Reflectivity and universal functionf(θ) . . . . . . . . . . . . . . . . . . . 79

6.3 Kerr-only measurement setup and universal functionfK(θ) . . . . . . . . . 82

7.1 Cartoon picture of fractional 3D topological insulator. . . . . . . . . . . . 89

7.2 Global center symmetry ofSU(Nc) effective gauge theory . . . . . . . . . 119

A.1 Integration contour for evaluating Matsubara sum . . . . .. . . . . . . . . 159

A.2 Analytic structure of the time-ordered boson propagator . . . . . . . . . . . 162

A.3 Incident, reflected, and transmitted light for Kerr/Faraday rotation . . . . . 200

xiv

Chapter 1

Introduction

Symmetry appears to be a profound organizational principleof nature. Beautifully sym-

metric natural patterns are ubiquitous, ranging from spiral galaxies and nearly spherical

planets to the five-fold symmetry of starfish and the helical symmetry of DNA. Symmetry

also turns out to be a remarkably useful theoretical principle by which we can structure our

understanding of complex physical systems. It is perhaps inthe field of condensed matter

physics — where one studies matter arranged in an endless variety of forms — that the

symmetry principle finds its most spectacular application.Paradoxically, one of the most

instructive ways to study stable phases of matter is to classify them according to the sym-

metries theybreak. Most classical and quantum gases and liquids do not break any sym-

metry at all; they enjoy the full translational and rotational symmetries of free space. Most

classical and quantum solids break these symmetries down toa discrete subgroup of trans-

lations and rotations (the space group). Magnetically ordered quantum spin systems such

as ferromagnets and antiferromagnets break spin rotation symmetry. Thisbroken symme-

try principle is at the heart of the phenomenological Ginzburg-Landau theory [1] of phase

transitions which, combined with microscopic many-body theories of condensed matter

systems, constitutes the cornerstone of pre-1980s condensed matter physics. In Ginzburg-

Landau theory, a stable phase of matter is characterized by alocal order parameter, which

is nonzero in an ordered phase but vanishes in a disordered phase. Phases with nonzero

order parameters are further distinguished by the way the order parameter transforms un-

der symmetry operations, i.e. by the representation of the symmetry group of the system

1

CHAPTER 1. INTRODUCTION 2

Hamiltonian to which it belongs.

The 1980s were marked by the discovery of the integer [2] and fractional [3] quantum

Hall (QH) effects. The QH effect occurs when a two-dimensional electron gas (2DEG), for

instance formed by electrons trapped in the inversion layerof a metal-oxide-semiconductor

structure or electrons in a semiconductor quantum well (QW), is subjected to a large mag-

netic field perpendicular to the plane of the 2DEG. A perpendicular magnetic field causes

the electrons to travel along circular cyclotron orbits, the radius of which becomes smaller

with increasing magnetic field. For large enough magnetic fields, electrons in the bulk of

the material form small, closed cyclotron orbits. On the other hand, electrons near the edge

of the sample can trace extended, open orbits that skip alongthe edge. At low tempera-

tures, quantum effects become important and two things happen. First, the area of closed

orbits in the bulk becomes quantized, bulk electrons becomelocalized (since they only

trace small, closed orbits) and the bulk turns into an insulator. Second, the skipping edge

orbits form extended one-dimensional channels with a quantized conductance ofe2/h per

channel. Furthermore, the transverse (Hall) conductanceσxy is quantized in integer (integer

QH) or rational (fractional QH) multiples ofe2/h.

It was soon realized that the bulk of a QH state is a featureless insulating state which

does not break any symmetries other than time-reversal (TR)symmetry, and thus cannot be

characterized by a local order parameter. Nevertheless, QHstates with different values of

the Hall conductance are truly distinct phases of matter, and correspond to quantum ground

states which cannot be adiabatically connected to each other without closing a spectral gap,

i.e. without going through a QH plateau transition. Even more surprising is the fact that

the quantization of the Hall conductance is found to be extremely accurate even in disor-

dered samples, where one would expect the randomizing effect of disorder to destroy any

quantization phenomenon. Indeed, if conduction proceeds only through one-dimensional

channels, one would naively expect these to be strongly affected by disorder due to Ander-

son localization [4].

The lack of a bulk local order parameter description a la Ginzburg-Landau and the

existence of boundary states robust to disorder can both be understood as defining charac-

teristics of a topological state of quantum matter. A usefulconcept in this context is that of

bulk-edge correspondence[5], of which the integer QH state provides a clear illustration.


A topological state of matter is insulating in the bulk, but supports gapless boundary states

which are perturbatively robust to disorder. Rather than being characterized by a local or-

der parameter, the bulk is characterized by atopological invariantwhich, in the case of the

integer QH state, is an integer denoted as the TKNN number [6]or Chern number [7]. The

bulk topological invariant is in turn related to the number of stable gapless boundary states.

In the integer QH state, the Chern number is simply equal to the number of stable gapless

edge states, and is also the value of the quantized Hall conductance in units ofe2/h. In that

sense, one says that the edge states areprotected by the bulk topology. But more concretely,

what is the mechanism for this “topological protection”? The answer is the following: the

bulk topology is responsible for some kind of fractionalization, in some sense, on the edge.

More precisely, the usual degrees of freedom of the electronare spatially separatedon

opposite edges. The usual degrees of freedom of an electron in a one-dimensional chan-

nel are twofold: right-moving and left-moving. However, ina QH sample, the top edge

has only right-moving electrons and the bottom edge has onlyleft-moving electrons (or

vice-versa, depending on the sign of the magnetic field). Backscattering on a given edge

is thus suppressed due to the inability of an electron to reverse its direction of motion, and

the QH edge channels completely evade Anderson localization. Since a single direction of

propagation is present on a given edge, the QH edge channels are termedchiral.

The TKNN integer relates the physical response of the Hall conductance to a topo-

logical invariant in momentum space. Whereas this work gives the first insight into the

topological nature of the QH state, it is limited to noninteracting systems. A more funda-

mental description of the QH effect is given by the topological field theory based on the

Chern-Simons term in2 + 1 dimensions [8, 9]. In this approach, the problem of electrons

in a 2DEG subject to an external perpendicular magnetic fieldB is exactly mapped to that

of bosons coupled to both the external magnetic field and an internal, emergent statistical

magnetic fieldb. This statistical magnetic field, the dynamics of which is described by

the Chern-Simons term, is responsible for the transmutation of the fermionic electrons into

bosons. At the magic filling fractionsν = 1/m (with m an odd integer) at which the QH

effect occurs, the external and statistical magnetic fieldsprecisely cancel each other, and

the bosons condense into a superfluid state. The effective field theory of a boson superfluid

is the(2 + 1)-dimensional Maxwell electrodynamics [10, 11]. In the long-wavelength and


low-energy limit, the Chern-Simons term dominates over theMaxwell term, and the ef-

fective theory of the QH state is just the topological Chern-Simons term. This topological

field theory is generally valid in the presence of disorder and interactions.

Until very recently, QH states were the only topological states the existence of which

had been firmly established by experimental observation. Compared to the rich variety

of “traditional” broken-symmetry states, one is led to the obvious question: should there

not be other topological states remaining to be discovered?In particular, the QH insulator

requires a large magnetic field for its existence. A natural question to ask is whether a

magnetic field, which breaks TR symmetry, is a necessary condition to obtain a topological

state. The first answer to this question was provided by the independent theoretical pre-

diction by Kane and Mele [12, 13] and Bernevig and Zhang [14] of a new state of matter,

the 2D time-reversal invariant topological insulator or quantum spin Hall (QSH) insulator.

This state displays robust quantized properties but does not require a TR symmetry break-

ing magnetic field for its observation. Roughly speaking, the QSH state can be viewed as

two copies of the QH state with opposite Hall conductances. The proposal of Kane and

Mele is based on the spin-orbit interaction of graphene, andis mathematically motivated

by the earlier work of Haldane [15] on the so-called quantum anomalous Hall (QAH) ef-

fect. The proposal of Bernevig and Zhang is based on the spin-orbit interaction induced

by strain in semiconductors. Neither proposals have yet been realized in actual condensed

matter systems, mostly because of the small spin-orbit interaction in the proposed systems.

However, they provided an important conceptual framework in which the stability of the

QSH state can be investigated.

In what sense is the QSH state a topological state of matter? This question is most

clearly answered by looking at whether this state supports stable gapless boundary modes,

robust to disorder. Let us proceed by comparison with the QH edge modes discussed pre-

viously (Fig. 1.1). As mentioned previously, the edge states of the QH state are such that

electrons can only propagate in a single direction on a givenedge. Compared with a one-

dimensional system of spinless electrons (Fig. 1.1, top left), the top edge of a QH system

contains only half the degrees of freedom (Fig. 1.1, bottom left). The QH system can thus

be compared to a “freeway” where electrons traveling in opposite directions have to be

“driving in different lanes”. This spatial separation resulting in chiral edge channels can be


Spinless 1D chain Spinful 1D chain

2 = 1 + 1 4 = 2 +22 1 + 1 4 2 +2

QH QSH

Figure 1.1: Chiral vs helical: spatial separation is at the heart of both the QH and QSH ef-fects. A spinless one-dimensional system (top left) has both right-moving and left-movingdegrees of freedom. Those two basic degrees of freedom are spatially separated in a QHsystem (bottom left), as illustrated by the symbolic equation “2 = 1 + 1”. The upper edgehas only a right-mover and the lower edge a left-mover. Thesechiral edge states are ro-bust to disorder: they can go around an impurity (green dot) without backscattering. Onthe other hand, a spinful one-dimensional system (top right) has twice as many degrees offreedom as the spinless system due to the twofold spin degeneracy. Those four degrees offreedom are separated in a TR invariant way in a QSH system (bottom right). The top edgehas a right-mover with spin up (denoted by a red dot) and a left-mover with spin down(denoted by a blue cross), and conversely for the lower edge.That separation is illustratedby the symbolic equation “4 = 2+ 2”. Thesehelicaledge states are robust to nonmagneticdisorder, i.e. impurities which preserve the TR symmetry ofthe QSH state.

illustrated by the symbolic equation “2 = 1+ 1” where each “1” corresponds to a different

chirality. This “chiral traffic rule” is particularly effective in suppressing electron scatter-

ing: since electrons travel always in the same direction, they are forced to avoid impurities

(Fig. 1.1, bottom left, green dot), and thus cannot backscatter.

On the other hand, as mentioned earlier, the QSH state can be roughly understood as

two copies of the QH state, with one copy for each spin. The edge state structure of the QSH

state (Fig. 1.1, bottom right) can thus be described pictorially by superposing two copies of

QH edge states (Fig. 1.1, bottom left), with opposite chirality for each spin. Compared to a

spinful one-dimensional system (Fig. 1.1, top right), the top edge of a QSH system contains


only half the degrees of freedom. The resulting edge states are termedhelical, because spin

is correlated with the direction of propagation. This new pattern of spatial separation can

be illustrated by the symbolic equation “4 = 2+2” where each2 corresponds to a different

helicity. Although electrons are now allowed to travel bothforward and backwards on

the same edge, there is a new “traffic rule” which suppresses backscattering: in order to

backscatter, an electron needs to flip its spin, which requires the breaking of TR symmetry.

If TR symmetry is preserved, as is the case for nonmagnetic impurities, no backscattering

is allowed.

What is the mechanism which allows this spatial separation?In the case of the QH

effect, the separation is achieved by an external magnetic field or, in the case of the QAH

effect, by some internal field which breaks TR symmetry. Thisinternal field takes the form

of a relativistic mass term for emergent Dirac fermions in2 + 1 dimensions, with the sign

of the internal field (and hence the chirality of the QAH edge states) dictated by the sign of

the mass. In the case of the QSH effect, the separation is achieved through the TR invariant

spin-orbit coupling.

Since the QSH state is characterized by a bulk insulating gapand gapless boundary

states robust to disorder (in the presence of TR symmetry), the QSH state is indeed a new

topological state of matter. However, because the Hall conductance of the QSH state van-

ishes, it is clear that the TKNN or Chern number discussed above, which corresponds to

the the value of the Hall conductance in units ofe2/h, cannot provide a useful classifica-

tion of the QSH state. This question has been addressed both within the topological band

theory [12] and the topological field theory [16], and it turns out that the proper topological

invariant is valued in theZ2 group containing only two elements,0 or 1, with 1 correspond-

ing to the topologically nontrivial QSH insulator and0 corresponding to a topologically

trivial insulator with no robust gapless edge states. Physically, thisZ2 invariant simply

counts the number of stable gapless edge states modulo2.

As mentioned previously, Kane and Mele [13] proposed graphene — a monolayer of

carbon atoms — as a possible candidate for the QSH effect. Unfortunately, this proposal

turned out to be unrealistic because the spin-orbit gap in graphene is extremely small [17,

18]. The QSH effect was also independently proposed in semiconductors in the presence of

strain gradients [14], but this proposal also turned out to be hard to realize experimentally.


Soon afterwards, Bernevig, Hughes, and Zhang [19] initiated the search for the QSH state in

semiconductors with an “inverted” bandstructure, and predicted a quantum phase transition

in type-III HgTe/CdTe QW between a trivial insulator phase and a QSH phase governed

by the thicknessd of the QW. The QSH state was observed experimentally soon after by

Konig et al. [20].

The QSH insulator in HgTe/CdTe QW is characterized by insulating gap in the bulk and

a single pair of helical edge states at each edge. A topological phase transition occurs due to

the band inversion at theΓ point driven by the spin-orbit interaction. The helical edge state

forms a single one-dimensional massless Dirac fermion withcounter-propagating states

forming a Kramers doublet under TR symmetry. In a strip geometry with two edges, each

edge supports one propagating mode of a given chirality withconductancee2/h, hence

the predicted longitudinal conductance in such a geometry is2e2/h [19]. Furthermore, the

helical state consisting of a single massless Dirac fermionis “holographic”, in the sense

that it cannot exist in a purely one-dimensional system, butcan only exist as the boundary

of a two-dimensional system [21].

The model Hamiltonian for the two-dimensional topologicalinsulator in HgTe/CdTe

QW [19] also gives a basic template for a generalization to three dimensions, leading to

a simple model Hamiltonian for a class of materials: Bi2Se3, Bi2Te3 and Sb2Te3 [22, 23].

Similar to their two-dimensional counterpart the HgTe/CdTe QW, these materials can be

described by a simple but realistic model, where the spin-orbit interaction drives a band

inversion transition at theΓ point. In the topologically nontrivial phase, the bulk states

are fully gapped, but there is a topologically protected surface state consisting of a single

two-dimensional massless Dirac fermion. This two-dimensional massless Dirac fermion is

“helical”, where the spin of the electron points perpendicularly to its momentum, forming

a left-handed helical texture in momentum space. Similarlyto the one-dimensional heli-

cal edge state, a single two-dimensional massless Dirac surface state is “holographic”, in

the sense that it cannot occur in a purely two-dimensional system with TR symmetry, but

can exist as the boundary of a three-dimensional insulator.A TR invariant single-particle

perturbation can not introduce a gap for the surface state. Agap can open up for the sur-

face state when a TR breaking perturbation is introduced on the surface. In this case, the


system becomes a full insulator, both in the bulk and on the surface. The topological prop-

erties of the fully gapped insulator are characterized by a novel topological magnetoelectric

effect [16].

Soon after the theoretical prediction of the three-dimensional topological insulator in

the Bi2Se3, Bi2Te3, and Sb2Te3 class of materials [22, 24], angle-resolved photoemission

(ARPES) experiments observed the surface state with a single Dirac cone [24, 25, 26].

Spin-resolved ARPES experiments subsequently observed the predicted left-handed helical

spin texture of the massless Dirac fermion [26]. These pioneering works inspired much of

the subsequent developments both in theory and experiment.

The general theory of the three-dimensional topological insulator has been developed

along two different routes. The topological band theory gives a general description of the

topological invariant in the single-particle momentum space [27, 28, 29]. In particular, a

method due to Fu and Kane [30] gives a simple algorithm to determine the topological

properties of any complex electronic structure with inversion symmetry. This method pre-

dicts that the semiconducting alloy BixSb1−x is a topological insulator for a certain range

of compositionx. ARPES experiments [31] have indeed observed topologically nontrivial

surface states in this system. However, the surface states in BixSb1−x are rather compli-

cated, and cannot be described by a simple model Hamiltonian.

The topological band theory is only valid for noninteracting systems in the absence of

disorder. The topological field theory is a more general theory which describes the electro-

magnetic response of the topological insulator [16]. In Ref. [16], Qi, Hughes, and Zhang

found that the electromagnetic response of three-dimensional topological insulators is de-

scribed by the Maxwell equations with an added topological term proportional toE·B. This

exact modification [16] had been proposed earlier in the context of high-energy physics [32]

as a modification to conventional electrodynamics due to thepresence of the Peccei-Quinn

axion field [33]. In this approach [16], theZ2 topological invariant from topological band

theory corresponds to a quantized emergent axion angleθ which is constrained by TR in-

variance to take only two values,0 (the trivial insulator) orπ (the topological insulator).

The equivalence between the two definitions has been proven recently [34]. Several unique

experiments based on axion electrodynamics in three-dimensional topological insulators

have been proposed: a topological Kerr and Faraday effect [16, 35, 36, 37], a topological


magnetoelectric effect [16], an image magnetic monopole effect [38, 39], and emergent dy-

namical axion particles [40, 41]. Efforts towards the discovery of these exotic phenomena,

as well as intensive searches for new three-dimensional topological insulator materials, are

ongoing.

1.1 Motivation

The work described in this dissertation can be divided in twoparts which roughly follow

the chronological evolution of the field of topological insulators from the years 2007 to

2010.

The first part of this dissertation is concerned with the transport properties of 2D QSH

insulators. Our work was mainly motivated by the experimental observation of a quantized

longitudinal conductance of2e2/h in HgTe QW [20]. Although the measurements reported

in Ref. [20] were strong evidence for the existence of robustedge channels, a number of

questions awaited explanation:

• Could the observed quantized conductance simply result from the usual subband

quantization in quasi-1D mesoscopic systems rather than from the existence of edge

channels?

• Why does the conductance deviate from2e2/h in long enough samples?

• Why does the conductance deviation increase rather than decrease with decreasing

temperature?

• Is there a way to directly measure theZ2 invariant in this material?

• Why does the conductance decrease so steeply with an appliedperpendicular mag-

netic field, and throughout the bulk gap, even if the edge gap opened by the magnetic

field is extremely small?

• Can we use the helical property of the edge channels to designa novel type of elec-

tronic device?


The first four chapters of this dissertation deal with answers to these questions.

The second part of this dissertation is concerned with 3D topological insulators. Here

the motivation was both experimental and theoretical. Our work was started soon after ex-

perimental observation of gapless surface states by ARPES in the Bi2Se3, Bi2Te3, Sb2Te3

class of materials [24, 25, 26]. Although this provided strong evidence for the existence of

the 3D topological insulator state in these materials, it was still unclear how to observe the

topological magnetoelectric effect predicted in Ref. [16], which is the true topologically

quantized physical observable in a 3D topological insulator (i.e. the analog of measuring

the quantized Hall conductance in a 2D QH insulator). Finally, the motivation for the last

part of our work came from the existence of two “competing” definitions of topological

order in condensed matter systems. Topological order as described earlier in this chapter

is fully characterized by band-theoretic invariants and arises in noninteracting or weakly

interacting systems. It can be called topological order of Thouless type, of which the in-

teger QH is a prime example. On the other hand, another notionof topological order was

pioneered by Wen [42] and is characterized by ground state degeneracies, fractionalization,

and emergent gauge theories, and arises in strongly correlated systems. FQH systems are

the canonical example of topologically ordered systems in the sense of Wen. Our work on

fractional 3D topological insulators was an effort to generalize the Thouless-type topolog-

ical order in ordinary 3D topological insulators to Wen-type topological order.

1.2 Contributions

This dissertation is built on the six publications listed and summarized below:

(1) Andreas Roth, Christoph Brune, Hartmut Buhmann, Laurens W. Molenkamp, Joseph

Maciejko, Xiao-Liang Qi, and Shou-Cheng Zhang,Nonlocal transport in the quan-

tum spin Hall state, Science325, 294 (2009).

This work confirms the existence of the QSH state in HgTe QW through an extensive

set of multi-terminal transport measurements. It is the result of a collaboration be-

tween the experimental group of L. W. Molenkamp at WurzburgUniversity and the

Stanford theory group. The latter developed a Landauer-Buttiker theory of transport


in the QSH state which produced excellent agreement with theexperimental data.

The theory also explains the different role played by voltage probes in the QH and

QSH states.

(2) Joseph Maciejko, Chao-Xing Liu, Yuval Oreg, Xiao-LiangQi, Congjun Wu, and

Shou-Cheng Zhang,Kondo effect in the helical edge liquid of the quantum spin Hall

state, Physical Review Letters102, 256803 (2009).

Using linear response and renormalization group methods, we calculate the low-

temperature edge conductanceG of a QSH insulator in the presence of a magnetic

impurity. We find a quantum phase transition between a weaklycoupled metallic

“local helical liquid” with unusual power lawsG ∼ T x, and a strongly coupled

insulating phase where transport proceeds by weak tunneling of quasiparticles with

half an electron charge.

(3) Joseph Maciejko, Xiao-Liang Qi, and Shou-Cheng Zhang,Magnetoconductance of

the quantum spin Hall state, Physical Review B82, 155310 (2010).

We study numerically the edge magnetoconductance of the QSHstate in the presence

of quenched nonmagnetic disorder. For a disorder strength on the order of the bulk

gap, the conductance decreases roughly linearly with the magnetic field, in qualitative

agreement with experiments on HgTe QW. We conjecture that for disorder small

compared to the gap the edge liquid suffers 1D Anderson localization, while for

disorder larger than the gap the electrons can hop to the 2D bulk and undergo 2D

diffusive motion and 2D antilocalization.

(4) Joseph Maciejko, Eun-Ah Kim, and Xiao-Liang Qi,Spin Aharonov-Bohm effect and

topological spin transistor, Physical Review B82, 195409 (2010).

Ever since its discovery, the electron spin has only been measured or manipulated

through the application of an electromagnetic force actingon the associated magnetic

moment. In this work, we use the helical property of the QSH edge states to propose

a spin Aharonov-Bohm effect in which the electron spin is controlled by a magnetic

flux while no electromagnetic field is acting on the electron.


(5) Joseph Maciejko, Xiao-Liang Qi, H. Dennis Drew, and Shou-Cheng Zhang,Topo-

logical quantization in units of the fine structure constant, Physical Review Letters

105, 166803 (2010).

Fundamental topological phenomena in condensed matter physics are associated

with a quantized electromagnetic response in units of fundamental constants. The

3D topological insulator is predicted to exhibit a topological magnetoelectric effect

quantized in units of the fine structure constantα = e2/~c. In this work, we propose

an optical experiment to directly measure this topologicalquantization phenomenon,

independent of material details.

(6) Joseph Maciejko, Xiao-Liang Qi, Andreas Karch, and Shou-Cheng Zhang,Frac-

tional topological insulators in three dimensions, Physical Review Letters105, 246809

(2010).

Topological insulators can be generally defined by a topological field theory with

an axion angleθ of 0 or π. In this work, we introduce the concept of fractional

topological insulator defined by a fractional axion angle and show that it can be

consistent with time-reversal invariance. The fractionalaxion angle can be measured

experimentally by a “halved” FQH effect on the surface with Hall conductance of the

form σH = pqe2

2hwith p, q odd integers.

1.3 Dissertation overview

This dissertation is structured as follows. Chapters 2-5 pertain to the transport properties

of the 2D QSH insulator. Chapter 2 covers publication (1) andincludes experimental work

by the group of L. W. Molenkamp at Wurzburg University, as well as supporting theoret-

ical material (Sec. 2.7). The latter section describes the effect of coherent and incoherent

scattering on the transport properties of the QSH state, andemphasizes the similarities

and differences with the QH state. Analytical and numericalcalculations allow us to re-

solve a paradox between the theoretically predicted absence of backscattering in the helical

edge state and the experimentally observed nonzero resistance induced by a voltage probe.

Chapter 3 covers publication (2). A key result of this chapter is the prediction of a fractional


Fano factor of1/2 at low temperatures, which is a direct measure of theZ2 invariant of the

QSH state. Details of the calculations involved in this chapter are given in Sections A.1-

A.6 of Appendix A. Chapter 4 covers publication (3). Chapter5 covers publication (4),

and details of theS-matrix transport calculations are given in Sec. A.7. Chapters 6 and 7

are concerned with 3D topological insulators and generalizations thereof. Chapter 6 covers

publication (5), and details of the Kerr/Faraday calculation are given in Sec. A.8. Sec-

tions 7.1-7.5 of Chapter 7 cover publication (6), while Sections 7.6-7.10 cover unpublished

material. An explicit derivation of the boundary Chern-Simons theory for an Abelian frac-

tional 3D topological insulator is given in Sec. A.9. Finally, we conclude in Chapter 8.

Chapter 2

Nonlocal transport in the quantum spin

Hall state

2.1 Introduction

The search for topological states of quantum matter has become an important goal in con-

densed matter physics. Inside a topological insulator, theconventional laws of electrody-

namics are dramatically altered [43], which may have applications in constructing novel

devices for processing of (quantum) information. The QSH state [13, 14] is a topologi-

cally nontrivial state of matter which exists in the absenceof any external magnetic field.

It has a bulk energy gap but gapless helical edge states protected by TR symmetry. In

the QSH regime, opposite spin states forming a Kramers doublet counter-propagate at the

edge [21, 44]. Recently, the QSH state has been theoretically predicted in HgTe QW [19].

There is a topological quantum phase transition at a critical thicknessdc of the QW, sepa-

rating the trivial insulator state ford < dc from the QSH insulator state ford > dc. Soon

after the theoretical prediction, evidence for the QSH state has been observed in transport

measurements [20]. In the QSH regime, experiments measure aconductanceG close to

twice the quantum unit of conductanceG = 2e2/h, which is consistent with quantum

transport due to helical edge states. However, such a conductance quantization in small

Hall-bar geometries does not allow us to distinguish experimentally between ballistic and

edge channel transport in a convincing manner. Thus it is important to be able to prove

14

CHAPTER 2. NONLOCAL TRANSPORT IN THE QSH STATE 15

experimentally in an unambiguous manner the existence of edge channels in HgTe QW.

2.2 Ohm’s law versus nonlocal transport

In conventional diffusive electronics, bulk transport satisfies Ohm’s law. The resistance is

proportional to the length and inversely proportional to the cross-sectional area, implying

the existence of a local resistivity or conductivity tensor. However, the existence of edge

states necessarily leads to nonlocal transport which invalidates the concept of local resis-

tivity. Such nonlocal transport has been experimentally observed in the QH regime in the

presence of a large magnetic field [45], and the nonlocal transport is well described by a

quantum transport theory based on the Landauer-Buttiker formalism [46]. These measure-

ments constitute definitive experimental evidence for the existence of edge states in the QH

regime.

In this chapter, we first describe nonlocal transport measurements in HgTe QW per-

formed by the group of Laurens W. Molenkamp at the Universityof Wurzburg which

demonstrate the existence of the predicted extended edge channels. Device structures that

are more complicated compared to a standard Hall bar allow a detailed investigation of

the transport mechanism. After describing the measurements, we present the theory of

quantum transport in the QSH regime, and uncover the effectsof macroscopic time irre-

versibility on the helical edge states.

2.3 Device structure

We present experimental results on four different devices.The behavior in these struc-

tures is exemplary for the around 50 devices we studied. The devices are fabricated from

HgTe/(Hg,Cd)Te QW structures with well thicknesses ofd = 7.5 nm (samples D1, D2 and

D3) and 9.0 nm (sample D4). Note that all wells have a thickness d > dc ≃ 6.3 nm, and

thus exhibit the topologically non-trivial inverted band structure. At zero gate voltage, the

samples aren-type and have a carrier density of aboutns = 3× 1011 cm−2 and a mobility

of 1.5× 105 cm2/(V·s), with small variations between the different wafers. Thedevices are

lithographically patterned using electron-beam lithography and subsequent Ar ion-beam


I1

2 3

4

56

V

I1

2 3

4

56

V

-0.5 0.0 0.5 1.0 1.5 2.00

5

10

15

20

25

30

35

40

R (

kΩ)

V* (V)

(1 x 0.5) µm2

(2 x 1) µm2

R14,14=3/2 h/e2

R14,23=1/2 h/e2

Figure 2.1: Two-terminal (R14,14) and four- terminal (R14,23) resistance versus (normalized)gate voltage for the Hall bar devices D1 and D2 with dimensions as shown in the insets.The dotted blue lines indicate the resistance values expected from the Landauer-Buttikerapproach.

etching. Devices D1 and D2 are micrometer-scale Hall bars with exact dimensions as indi-

cated in the insets of Fig. 2.1. D3 and D4 are dedicated structures for identifying non-local

transport, with schematic structure given in Fig. 2.2. All devices are fitted with a 110-nm-

thick Si3N4/SiO2 multilayer gate insulator and a 5/50 nm Ti/Au gate electrodestack. By

applying a voltageVg to the top gate the electron carrier density of the QW can be ad-

justed, going from ann-type behavior at positive gate voltages through the bulk insulator

state into ap-type regime at negative gate voltages. For reasons of comparison, the exper-

imental data in Fig. 2.1, 2.3, and 2.4 are plotted as a function of a normalized gate voltage

V ∗ = Vg−Vthr (Vthr is defined as the voltage for which the resistance is largest). Measure-

ments are performed at a lattice temperature of 10 mK for samples D1, D2 and D3 and at

1.8 K for sample D4 using low-frequency (13 Hz) lock-in techniques under voltage bias.

The four terminal resistance (Fig. 2.1) shows a maximum at about h/2e2, in agreement


with the results of Ref. [20]. The contact resistance shouldbe insensitive to the gate volt-

age, and can be measured from the resistance deep in the metallic region. By subtracting

the contact resistance we find that the two terminal resistance has its maximum of about

3h/2e2 (Fig. 2.1). This value is exactly what is expected from the theory of QSH edge

transport obtained from the Landauer-Buttiker formula.

2.4 Transport on the edge

Within the general Landauer-Buttiker formalism [47], thecurrent-voltage relationship is

expressed as

Ii =e2

h

∑

j

(TjiVi − TijVj), (2.1)

whereIi is the current flowing out of thei-th electrode into the sample region,Vi is the

voltage on thei-th electrode, andTji is the transmission probability from thei-th to thej-

th electrode. The total current is conserved in the sense that∑

i Ii = 0. A voltage leadj is

defined by the condition that it draws no net current,i.e. Ij = 0. The physical currents are

left invariant if the voltages on all electrodes are shiftedby a constant amountµ, implying

that∑

i Tij =∑

i Tji. In a TR invariant system, the transmission coefficients satisfy the

conditionTij = Tji.

For a general two-dimensional sample, the number of transmission channels scales

with the width of the sample, so that the transmission matrixTij is complicated and non-

universal. However, a tremendous simplification arises if the quantum transport is entirely

dominated by the edge states. In the QH regime, chiral edge states are responsible for the

transport. For a standard Hall bar withN current and voltage leads attached (see the insets

of Fig. 2.1 withN = 6), the transmission matrix elements for theν = 1 QH state are given

by T (QH)i+1,i = 1, for i = 1, . . . , N , and all other matrix elements vanish identically.

Here we periodically identify thei = N + 1 electrode withi = 1. Chiral edge states are

protected from backscattering, therefore, thei-th electrode transmits perfectly to the neigh-

boring (i+1)th electrode on one side only. In the example of current leads on the electrodes

1 and4, and voltage leads on the electrodes2, 3, 5 and6, one finds thatI1 = −I4 ≡ I14,


V2 − V3 = 0 andV1 − V4 = (h/e2)I14, giving a four-terminal resistance ofR14,23 = 0 and

a two-terminal resistance ofR14,14 = h/e2.

In the case of helical edge states in the QSH regime, oppositespin states form a Kramers

pair, counter-propagating on the same edge. The helical edge states are protected from

backscattering due to TR symmetry, and the transmission from one electrode to the next

is perfect. From this point of view, the helical edge states can be viewed as two copies of

chiral edge states related by TR symmetry. Therefore, the transmission matrix is given by

T (QSH) = T (QH) + T †(QH), implying that the only non-vanishing matrix elements are

given by

T (QSH)i+1,i = T (QSH)i,i+1 = 1. (2.2)

Considering again the example of current leads on the electrodes1 and4, and voltage leads

on the electrodes2, 3, 5 and6, one finds thatI1 = −I4 ≡ I14, V2 − V3 = (h/2e2)I14

andV1 − V4 = (3h/e2)I14, giving a four-terminal resistance ofR14,23 = h/2e2 and a two-

terminal resistance ofR14,14 = 3h/2e2. The experimental data in Fig. 2.1 confirm this

picture. For both micro Hall-bar structures D1 and D2 that differ only in the dimensions of

the area between the voltage contacts 3 and 4, we observe exactly the expected resistance

values forR14,23 = h/2e2 andR14,14 = 3h/2e2 for gate voltages where the samples are in

the QSH regime.

2.5 Dissipationless transport

Conceptually, one might sense a paradox between the dissipationless nature of the QSH

edge states and the finite four-terminal longitudinal resistanceR14,23, which vanishes for the

QH state. We can generally assume that the microscopic Hamiltonian governing the voltage

leads is invariant under TR symmetry, therefore, one would naturally ask how such leads

could cause the dissipation of the helical edge states, which are protected by TR symmetry?

In nature, the TR symmetry can be broken in two ways, either atthe level of the microscopic

Hamiltonian, or at the level of the macroscopic irreversibility in systems whose microscopic

Hamiltonian respects the TR symmetry. When the helical edgestates propagate without

dissipation inside the QSH insulator between the electrodes, neither forms of TR symmetry


breaking are present. As a result, the two counter-propagating channels can be maintained

at two different quasi chemical potentials, leading to a netcurrent flow. However, once

they enter the voltage leads, they interact with a reservoircontaining infinitely many low-

energy degrees of freedom, and the TR symmetry is effectively broken by the macroscopic

irreversibility. As a result, the two counter-propagatingchannels equilibrate at the same

chemical potential, determined by the voltage of the lead. Dissipation occurs with the

equilibration process. The transport equation (2.1) breaks the macroscopic TR symmetry,

even though the microscopic TR symmetry is ensured by the relationshipTij = Tji. In

contrast to the case of QH state, the absence of dissipation of the QSH helical edge states

is protected by Kramers’ theorem, which relies on the quantum phase coherence of wave

functions. Thus dissipation can occur once the phase coherence is destroyed in the metallic

leads. On the contrary, the robustness of QH chiral edge states does not require phase

coherence. A more rigorous and microscopic analysis on the different role played by a

metallic lead in QH and QSH states is provided in Sec. 2.7, theresult of which agrees

with the simple transport equations (2.1) and (2.2). These two equations correctly describe

the dissipationless quantum transport inside the QSH insulator, and the dissipation inside

the electrodes. One can subject these two equations to more stringent experimental tests

than the two- and four-terminal experiments of Fig. 2.1 by considering devices D3 and D4

(Fig. 2.2).

2.6 Helical versus chiral

A further difference between helical and chiral edge channels is evident from our experi-

ments on the six-terminal device D3 (Fig. 2.3). When the longitudinal resistance of device

D3 is measured by passing a current through contacts 1 and 4 and by detecting the voltage

between contacts 2 and 3 (R14,23), we find, similarly to the results of Fig. 2.1, the resistance

value ofh/2e2 when the bulk of the device is gated into the insulating regime (Fig. 2.3A).

However, the longitudinal resistance is significantly different in a slightly modified config-

uration, where the current is passed through contacts 1 and 3and the voltage is measured

between contacts 4 and 5 (R13,45) (Fig. 2.3B). We now findR13,45 ≈ 8.6 kΩ, which is

markedly different from what one would expect for either theQH transport, or the purely


1 μm 2 μm

1 2 3

6 5 4

1 μm

1 μm

5 μm

1 2

34

A B

Figure 2.2: Schematic layout of devices D3 (A) and D4 (B). Thegrey areas are the mesa’s,the yellow areas the gates, with dimensions as indicated in the figure. The numbers indicatethe coding of the leads.

diffusive transport, where this configuration would be equivalent to the previous. The ap-

plication of equations (2.1) and (2.2) actually predicts indeed that the observed behavior is

what one expects for helical edge channels. One finds that this resistance value can again be

expressed as an integer fraction of the inverse conductancequantae2/h: R13,45 = h/3e2.

This result shows that the current through the device is influenced by the number of ohmic

contacts in the current path. These ohmic contacts lead to the equilibration of the chemical

potentials between the two counter-propagating helical edge channels inside the contact.

There are also some devices for which the maximal resistancedoes not match the theo-

retical value obtained from Eq. (2.1) and (2.2), but still remains an integer fraction of the

quantumh/e2. This result can be naturally understood as due to inhomogeneities in the

gate action, e.g. due to interface trap states, inducing some metallic droplets close to the

edge channels while the bulk of the sample is insulating. A metallic droplet can cause

dephasing of the electronic wave function, leading to fluctuations in the device resistance.

For full dephasing, the droplet plays the role of an additional Ohmic contact, just as for

the chiral edge channels in the QH regime [45]. More details on the effects of additional

Ohmic contacts in the QSH state are given in Sec. 2.7.

Another measurement that directly confirms the nonlocal character of the helical edge


A B

-1 0 1 2 30

5

10

15

20

25

30

35

40

R(k

Ω)

V* (V)

I: 1-4

V: 2-3

R14,23=1/2 h/e2

R14,14=3/2 h/e2

I: 1-3

V: 5-6

R13,13=4/3 h/e2

R13,56=1/3 h/e2

-1 0 1 2 3 4

V* (V)

Figure 2.3: Four- and two-terminal resistance measured on device D3: (A)R14,23 (red line)andR14,14 (green line) and (B)R13,56 (red line) andR13,13 (green line). The dotted bluelines indicate the expected resistance value from a Landauer-Buttiker calculation.

channel transport in the QSH regime is shown in Fig. 2.4, which displays data obtained

from device D4, in the shape of the letter “H”. In this 4-terminal device the current is passed

through contacts 1 and 4 and the voltage is measured between contacts 2 and 3. In the

metallicn-type regime (low gate voltage) the voltage signal tends to zero. In the insulating

regime, however, the nonlocal resistance signal increasesto approximately6.5 kΩ, which

again fits perfectly to the result of Laudauer-Buttiker considerations:R14,23 = h/4e2 ≈6.45 kΩ. Classically, one would expect only a minimal signal in thisconfiguration (from

Poisson’s equation, assuming diffusive transport, one estimates a signal of about 40Ω), and

certainly not one that increases so strongly when the bulk ofthe sample is depleted. This

signal measured here is fully nonlocal, and can be taken (as was done twenty years ago for

the QH regime) as definite evidence of the existence of edge channel transport in the QSH

regime. A similar nonlocal voltage has been studied in a metallic spin Hall system with

the same H-bar geometry [48], in which case the nonlocal voltage can be understood as

a combination of the spin Hall effect and the inverse spin Hall effect [49]. The quantized

nonlocal resistanceh/4e2 we find here is the quantum counterpart of the metallic case.


Fig. 4

0.0 0.5 1.0 1.5 2.00

5

10

15

20

25

R (

kΩ)

V* (V)

I: 1-4

V: 2-3

1

3

2

4

R14,23=1/4 h/e2

R14,14=3/4 h/e2

Figure 2.4: Nonlocal four-terminal resistance and two-terminal resistance measured on theH-bar device D4:R14,23 (red line) andR14,14 (green line). Again, the dotted blue linerepresents the theoretically expected resistance value.

Assuming for example that the chemical potential in contact1 is higher than that in contact

4 (see the layout of D4 in Fig. 2.2B), more electrons will be injected into the upper edge

state in the horizontal segment of the H-bar than into the lower edge state. Because on

opposite edges, the right-propagating edge states have opposite spin, this implies that a

spin-polarized current is generated by an applied biasV1 − V4, comparable to a spin Hall

effect. When this spin-polarized current is injected into the right leg of the device, the

inverse effect occurs. Electrons in the upper edge flow to contact 2 while those in the lower

edge will flow to contact 3, establishing a voltage difference between those two contacts

due to the charge imbalance between the edges. The right leg of the device thus acts as a

detector for the injected spin-polarized current, which corresponds to the inverse spin Hall

effect.


2.7 Dephasing and nonlocal transport

In this section, we clarify the role played by contacts in edge transport in QSH systems by

answering the following questions:

• In the absence of microscopic TR symmetry breaking, why doesa contact contribute

an additional longitudinal resistance, if edge state backscattering is forbidden by TR

symmetry?

• How can we understand the different role played by contacts in QH and QSH sys-

tems?

• Besides a voltage probe, can other processes cause an additional resistance?

The answers are as follows:

• A contact isnot a TR symmetry breaking single-particle potential with matrix ele-

ments connecting counter-propagating edge channels with opposite spins. An contact

is, ultimately, a reservoir of electrons which populates both channels incoherently.

Furthermore, anideal contact populates both channelswith equal weight, i.e. it in-

jects spin up and spin down electrons with equal probability. A right-moving electron

entering a contact will disappear into the reservoir. Sincethe contact is assumed not

to draw any current, the reservoir has to inject a new electron into the device. This

new electron can be of either spin, and bears no phase relationship with the previous

electron. For an ideal contact, this electron has1/2 probability of being injected as a

left-mover, which is the origin of the resistance contributed by the contact. Note that

this incoherent momentum-relaxing event is different fromusual coherent momen-

tum relaxation caused by potential scattering in an ordinary (non-helical) 1D system.

• In the QH case, the channels on each edge are chiral such that the longitudinal two-

terminal conductance is always quantized tone2/h with n the number of channels

on each edge, regardless of the presence of additional lateral contacts. Quantization

of the conductance is maintained independently of whether transport is coherent or

incoherent.


• Based on analytical and numerical results, we show that a finite-sized region with

dephasing processes can cause additional longitudinal resistance in the same way as

a contact does.

The structure of this section is as follows. Starting from the quantum kinetic formalism

embodied in the Meir-Wingreen formula (Sec. 2.7.1), we firstshow how arbitrary dephas-

ing processes contributing a self-energy to the one-particle Green function can be formally

thought of as a contact. In particular, we show that the equilibrium two-terminal conduc-

tance calculated from the quantum kinetic formalism agreeswith the conductance calcu-

lated from thethree-terminal Landauer-Buttiker formula if the third terminal is identified

with the “dephasing bath”. Then, using a four-band tight-binding model on a square lattice

for HgTe/CdTe QW, we perform numerical calculations of the conductance in the QSH

regime (Sec. 2.7.2). We show how dephasing processes contribute the expected additional

resistance, which constitutes an explicit justification ofLandauer-Buttiker phenomenology.

By adding an orbital magnetic field to the model, we also perform conductance calculations

in the QH regime (Sec. 2.7.3) and compare to the previous QSH results. We then proceed

with a short discussion of the transformation properties ofthe self-energy under time re-

versal (Sec. 2.7.4) to emphasize that resistance in a QSH system is caused by macroscopic,

and not microscopic, TR symmetry breaking. Finally, we perform an explicit numerical

calculation of the nonlocal resistance in a H-bar structure(Sec. 2.7.5) similar to that of

sample D4 (Fig. 2.4) using the four-band tight-binding model. We find good agreement

with both the prediction of the Landauer-Buttiker formalism and the experimental results,

thus validating the picture of transport by edge states.

2.7.1 Quantum kinetic formalism and Landauer-Buttiker equations

The Meir-Wingreen formula [50] for the current through a terminal connected to a meso-

scopic region is a natural starting point since it is an exactresult for noninteracting leads

derived from the quantum kinetic formalism, valid out of equilibrium. It encompasses the

Kubo formula and the Landauer-Buttiker formula as specialcases. It expresses the current


flowing into the mesoscopic region through terminali as

Ii =ie

htr

∫

dE [Γi(E)G<(E)− Σ<

i (E)A(E)], (2.3)

whereA = i(G − G†) is the nonequilibrium spectral function,G,G< are nonequilibrium

Green functions, andΣ<i ,Γi = i(Σi − Σ†

i) are the lesser self-energy and broadening func-

tions of the semi-infinite leads, respectively.

Dephasing region and Landauer-Buttiker equations

The Green functionG contains the total self-energyΣ which comprises not only the self-

energies of the leadsΣj , but also the self-energyΣd of any processes occurring in the

mesoscopic region that cannot be described by a single-particle Hermitian Hamiltonian,

Σ = Σd +∑

j

Σj .

We will refer to the processes described byΣd asdephasingprocesses.G< andA are given

by [51, 52]

G< = G

(

Σ<d +

∑

j

Σ<j

)

G†, A = G

(

Γd +∑

j

Γj

)

G†.

One then makes the following ansatze,

Σ<j = ifjΓj, Σ<

d = ifdΓd, (2.4)

that is, the self-energies should obey the fluctuation-dissipation theorem [53]. Due to the

self-energies, the eigenstates of the isolated mesoscopicregion acquire a finite lifetime.

Equations (2.4) mean that electrons “leaking out” of the mesoscopic region end up in a

“reservoir” with distribution functionfj,d according to which they are equilibrated. It is

this precise requirement which enables us to derive the Landauer-Buttiker result. That is,

an arbitrary dephasing region and a contact can be treated onthe same footing if they both

obey the fluctuation-dissipation theorem.


Using the preceding equations, and restricting ourselves to linear response

fi,d ≃ f +

(

− df

dE

)

(µi,d − µ), (2.5)

wheref(E) = (eβ(E−µ) + 1)−1, we find easily

Ii =e

h

∫

dE

(

− df

dE

)

[

∑

j

Tij(µi − µj) + Tid(µi − µd)

]

,

where

Tij(E) ≡ tr ΓiGΓjG†, Tid(E) ≡ tr ΓiGΓdG

†. (2.6)

By virtue of the second equality in Eq. (2.4) it makes sense toconsider the current drawn

by the dephasing reservoir characterized by the distribution functionfd. After linearization

[Eq. (2.5)] we obtain

Id =e

h

∫

dE

(

− df

dE

)

∑

j

Tdj(µd − µj),

whereTdj(E) ≡ tr ΓdGΓjG†. We now require that the dephasing reservoir does not draw

any current,

Id = 0.

In other words, it acts as a voltage probe. DenotingGij ≡ e2

h

∫

dE(

− dfdE

)

Tij(E), we have

the set of equations

Ii =∑

j

Gij(Vi − Vj) +Gid(Vi − Vd),

Id =∑

j

Gdj(Vd − Vj) = 0,

which are just the multiterminal Landauer-Buttiker equations with a voltage probe.


Two-terminal case with dephasing

We study as a particular case the two-terminal geometry which we will use for the numer-

ical analysis. We consider two physical contacts1, 2 with I = I1 = −I2. Solving the

previous equations we obtain the two-terminal conductance,

G2t ≡I

V1 − V2= G12 +

G1dGd2

Gd1 +Gd2,

which is just the three-terminal Landauer-Buttiker result if we considerd as a terminal. We

write

G2t = Gcoh +Gincoh,

with Gcoh = G12 the coherent part of the conductance andGincoh = G1dGd2/(Gd1 + Gd2)

the incoherent part. This result is valid in the absence of TRsymmetry (hence holds for the

QH case to be considered later). If TR symmetry is preserved,thenG1d = Gd1 and we can

write

G2t = G12 +

(

1

G1d+

1

Gd2

)−1

,

which corresponds to two resistors in series, combined in parallel with a third.

2.7.2 Dephasing in the QSH regime: numerical study

To understand the effect of a dephasing region in the quantumspin Hall system, we now

perform an explicit numerical calculation of the two-terminal conductanceG2t in the HgTe

system. The geometry (Fig. 2.5) consists of a finite QSH region connected tosemi-infinite

terminals 1 and 2, and to afinitedephasing region 3. Corresponding to the continuumk ·pHamiltonian of Ref. [19], one can derive the following tight-binding (TB) model on a 2D

square lattice,

H =∑

i

c†iVici +∑

i

(

c†iTxci+x + c†iTyci+y + h.c.)

, (2.7)


where theci are four-component spin-3/2 spinors, and the4 × 4 matricesVi, Tx, Ty are

defined as

Tx =1

2(2D14×4 + 2B12×2 ⊗ σz + 2βσy ⊗ σy)

+1

2i

(

vσz ⊗ σx +∆e +∆h

2σx ⊗ 12×2 +

∆e −∆h

2σx ⊗ σz

)

,

Ty =1

2(2D14×4 + 2B12×2 ⊗ σz + 2βσy ⊗ σy)

+1

2i

(

−v12×2 ⊗ σy −∆e +∆h

2σy ⊗ σz −

∆e −∆h

2σy ⊗ 12×2

)

,

Vi = (C − 4D − µ+ Eg(i))14×4 + (M − 4B)12×2 ⊗ σz + (∆z − 4β)σy ⊗ σy,

whereµ is the chemical potential andEg is the gate potential used to tune the Fermi level

inside the bulk energy gap in the device region. The parameters v, B, C, D, M , β, ∆e,h,z

are obtained fromk · p theory [54]. The self-energies of the semi-infinite leadsΣj are

calculated by a transfer matrix method [55]. The dephasing region 3 is a metallic region

of finite sizeNy × L and differs from the other regions in that an imaginary term−iη is

added to the energy of each site, i.e. we have a local self-energy (Σd)iα,jβ = −iηδijδαβfor sitesi, j with spin indicesα, β in region 3. The Green functionG of the QSH region

in the presence of all three self-energies is calculated by arecursive method [56], and the

transmission functionsTij(µ) andTid(µ) at the Fermi level are subsequently obtained from

Eq. (2.6). We then study the dependence of the zero-temperature conductancesGcoh,Gincoh

andG2t onη andL.

First of all, one can understand the two limit cases intuitively. For a fixed small value of

η, the region 3 is a true terminal in the limit ofL → ∞, and the two-terminal conductance

G2t agrees with the prediction of the Landauer-Buttiker formalism. From Fig. 2.5 we can

just read offT12 = 1, T13 = 1 andT32 = 1 such that we expectG2t =32e2

h. On the other

hand, in the limit ofL→ 0 the top edge is unperturbed and we expectG2t = 2e2/h. To see

how a crossover between the two limits occurs, we study the two-terminal conductance for

genericL numerically.

Results of the numerical calculation are plotted in Fig. 2.6. For each fixed value ofη,

we see the crossover between the two limit values2e2/h (L → 0) and3e2/2h (L → ∞)


Figure 2.5: Two-terminal geometry in the QSH regime with dephasing. Regions 1 and 2are semi-infinite leads and region 3 has finite sizeNy×L and contains dephasing processesmodeled by a local self-energyΣiα,jβ = −iηδijδαβ . In the QSH region the Fermi level liesin the bulk gap, and it lies in the conduction band in regions1, 2, 3.

as expected from the Landauer-Buttiker analysis. The off-resonance conductanceG2t (see

Fig. 2.7 and the discussion of resonances in the next paragraph) scales withηL (Fig. 2.6B).

Upon increasingL, the convergence ofG2t to the asymptotic value of3e2/2h occurs at a

length scaleLc ∝ 1/η. Indeed, forη = 0, region 3 is just a single-particle TR invariant

Hamiltonian which cannot induce any backscattering. Transport is then entirely coherent

(Fig. 2.8A). Asη is increased from zero, there is a growing incoherent contributionGincoh

which increases at the expense of the coherent contributionGcoh in order to preserve cur-

rent conservation. The electron flux incoming from the left is partially decohered with

probability per unit time∼ η/~. The remaining flux contributes to the remaining coher-

ent contributionGcoh (Fig. 2.8B). AsηL increases, the coherent conductance eventually

levels off toe2/h which is just the contribution from the unperturbed edge. Transport on

the top edge then becomes totally incoherent (Fig. 2.8C) with Gcoh = 0.5 e2/h. This is

the signature of an ideal contact which populates the two counter-propagating edge states

equally. In this regime, the finite dephasing region completely mimicks a true semi-infinite


2040

6080

100

0

0.005

0.010

0.5

1

1.5

2

L (a)η (eV)

G (

e2 /h)

G2t

Gcoh

Gincoh

A

0 20 40 60 80 50 60 70 801.4

1.5

1.6

1.7

1.8

1.9

2

ηL (meV⋅a)

G2t

(e2 /h

)

0.0010.0680.130.200.270.330.400.470.672.0

Bη (meV)

Figure 2.6: (A) Two-terminal conductance in the QSH regime.Top curve (red): totalconductance, middle curve (green): coherent partGcoh, bottom curve (blue): incoherentpartGincoh. We useNy = 40, Nx = 120, and a TB lattice constanta = 60 A for allcalculations. (B) Scaling plot for the conductanceG2t ∼ G2t(ηL) for values ofL awayfrom transmission resonances (see Fig. 2.7).

lead (Fig. 2.8D).

There are however sharp resonant dips in the conductance forsmall but nonzero values

of η (Fig. 2.7). These resonances appear because forη smaller than the level spacing in the

dephasing region∆E ∼ 1/L, the self-energy contributed by the dephasing region to the

QSH region has sharp features. AsL is varied, these features cross the Fermi energy and

lead to a sharp decrease in the coherent conductance. From Fig. 2.7 we see that even for

small ηL, these resonances can be strong enough as to completely block coherent trans-

port on the top edge (G2t → 1.5e2/h). Consequently, even a small dephasing region may

have the same effect as a Buttiker probe. Experimentally, in some of the devices D3 we

have observed deviation of the four-terminal conductance from the value predicted by the

Landauer-Buttiker formalism. For example, in the data shown in Fig. 2.7B, the six-terminal

Landauer-Buttiker formula predicts a four-terminal resistance ofR45,12 = h/6e2, but the

experimental value of the maximal resistance is smaller andclose toh/7e2 instead. Such a

result is consistent with the existence of an additional dephasing region, as shown schemat-

ically by the red solid circle in the inset of Fig. 2.7B. Such dephasing regions can exist due

to the inhomogeneity of the sample. Experimentally, we havenoticed that changing the

gate voltage also influences the homogeneity of the sample bycharging and discharging


0 20 40 60 80 1001.5

1.6

1.7

1.8

1.9

2

Ny = 40, N

b = 40, E

g = 0.015 eV, η (meV) is varied

L (a)

G (

e2 /h)

0.0010.010.050.10.250.5

A

-1,5 -1,0 -0,5 0,0 0,5 1,00

1

2

3

4

5

6

R (

kΩ

)

V* (V)

B

I: 4-5

V: 1-2

R45,12=1/6 h/e2

R45,12=1/7 h/e2

Figure 2.7: (A) Total two-terminal conductanceG2t in the QSH regime for small valuesof dephasing strengthη. On resonance, coherent transport on the top edge can be entirelyblocked withG2t dropping to the completely incoherent limit 1.5e2/h. (B) Four-terminalresistanceR45,12 in a device D3. The blue dashed line shows the theoretical valueh/6e2

from the Landauer-Buttiker formalism. The green dashed line shows the theoretical valueh/7e2 when considering the effect of an additional probe, as shownschematically by thered solid circle in the inset (see text).

trap states at the semiconductor interface. This tends to aninhomogeneous potential profile

in the gated area so that metallic islands still exist when most of the gated area is insulating.

As shown by the theoretical analysis here, a metallic islandcan lead to a similar effect as

an additional probe.

The characteristic length for dephasingLc, which can be defined operationally as the

lengthL required for the conductance to reach the completely incoherent limit G2t = 1.5

e2/h, is thus seen to depend on the dephasing strength asLc ∼ 1/η whereτ ∼ ~/η

would correspond to the decoherence time associated with some physical dephasing pro-

cess, e.g. electron-phonon or electron-electron interactions. Note that the dephasing self-

energyΣ ∝ η does not have to break TR symmetry (see Sec. 2.7.4) to decrease the con-

ductance. Decoherence comes from the presence of the dephasing bath with a distribution

function included inΣ<, not fromΣ breaking TR symmetry.


Figure 2.8: Cartoon picture of edge transport in the QSH regime in the presence of de-phasing. Only right-moving states are shown. Bottom edge isunaffected by dephasing andcontributes a fixede2/h toG2t. Coherent transport is denoted by solid lines and incoherenttransport by dotted lines. (A) Forη = 0, top edge state propagates coherently without re-sistance. (B) Forη 6= 0, top edge state is phase-randomized (yellow cloud) with probabilityper unit timeη/~ and contributes to incoherent transport, or is not phase-randomized andcontributes to coherent transport. For increasingηL, incoherent transport is enhanced atthe expense of coherent transport. (C) ForηL large enough, transport is completely inco-herent. This case is equivalent to (D) which represents the true semi-infinite lead attachedto an electron reservoir.

2.7.3 Dephasing in the QH regime: numerical study

It is instructive to investigate the effect of dephasing in the QH regime and compare it to

the QSH case just studied. To reach the QH regime we add to the HgTe/CdTe TB model

an orbital magnetic fieldB = Bz perpendicular to the 2D electron gas. When the Fermi

level lies inside the Landau level gap, transport is carriedby chiral edge states (Fig. 2.9).

We chooseB andEF such that there is a single chiral state on each edge.

From the Landauer-Buttiker analysis we expect the two-terminal conductance to be

G2t = e2/h since there is only one right-moving channel, on the top or bottom edge de-

pending on the sign of the magnetic field. The numerical results for the conductance are


Figure 2.9: Two-terminal geometry in the QH regime with dephasing. The Fermi level liesbetween Landau levels and transport is carried by chiral edge states.

given in Fig. 2.10 forBz = 1 T and Fig. 2.11 forBz = −1 T.

In Fig. 2.10 withBz = 1 T, the right-mover is on the bottom edge and is therefore

unperturbed by the dephasing region (Fig. 2.12D). Transport is therefore entirely coherent,

with G2t = Gcoh = e2/h andGincoh = 0 (numerically, one obtains a very small but finite

Gincoh due to interedge tunneling across the finite device widthNy. Upon increasingNy,

Gincoh gets vanishingly small).

On the other hand, if we reverse the sign of the magnetic field,in Fig. 2.11 forBz = −1

T the right-mover is on the top edge and is therefore affectedby the dephasing region. For

weak enoughη, as in the QSH case transport is mostly coherent (Fig. 2.12A), except for

resonances. AsηL increases, the incoherent partGincoh increases (Fig. 2.12B). For large

enoughηL, transport is completely incoherent (Fig. 2.12C). However, the sum ofGcoh and

Gincoh (not shown in Fig. 2.11) remains quantized toe2/h. This is because of the chiral

nature of the edge states: unlike the QSH case, in the QH case decoherence does not lead

to momentum relaxation.


2040

6080

100

0

5

100

0.2

0.4

0.6

0.8

1

L (a)η (meV)

G (

e2 /h)

Gcoh

Gincoh

Figure 2.10: Two-terminal conductance in the QH regime withBz = 1 T, correspondingto Fig. 2.12D. Top curve (red): coherent partGcoh, bottom curve (blue): incoherent partGincoh.

2.7.4 Self-energy and microscopic time-reversal symmetry

In this section, we describe briefly the transformation properties of the self-energy under

TR and show that the dephasing self-energy used in the numerical calculation does not

break TR symmetry.

Consider a Hermitian single-particle Hamiltonian operator H = H† which is invariant

under the action of the TR operatorT , T H = HT . This impliesΘHΘ−1 = H∗ for the

single-particle Hamiltonian matrixH in a given basis, withT = ΘK the representation of

T in that basis,K is the complex conjugation operation which acts only onc-numbers and

Θ2 = −1. In the single-particle case, the retarded Green function in frequency space is


0

20

40

60

80

100

0

5

100

0.5

1

L (a)η (meV)

G (

e2 /h)

Gcoh

Gincoh

Figure 2.11: Two-terminal conductance in the QH regime withBz = −1 T, correspondingto Fig. 2.12A,B,C. Top curve (red): incoherent partGincoh, bottom curve (blue): coherentpartGcoh.

given by the matrix inverseG = (ω + iδ −H)−1. We then have

G∗ = (ω − iδ −H∗)−1

= (ω − iδ −ΘHΘ−1)−1

= Θ(ω − iδ −H)−1Θ−1

= ΘG†Θ−1,

or

ΘGΘ−1 = GT , (2.8)

usingΘ† = Θ−1.

We now show that Eq. (2.8) also holds in the case thatH is a general many-body

Hamiltonian as long as it is TR invariant. In other words, we want to show that Eq. (2.8)

holds even if the Green function contains a self-energy term. Consider the zero temperature

Green function〈0|ciα(t)c†jβ(t′)|0〉 whereiα denotes the single-particle basis,i being a


Figure 2.12: Cartoon picture of edge transport in the QH regime in the presence of de-phasing. Only right-moving states are shown. Bottom edge isunaffected by dephasing.Coherent transport is denoted by solid lines and incoherenttransport by dotted lines. (A)Forη = 0, top edge state propagates coherently without resistance,in exactly the same wayas in Fig. 2.8A. (B) Forη 6= 0, partial phase-randomization occurs as in Fig. 2.8B, exceptthat decohered electrons can only be injected as right-movers since there is no left-movingchannel on the top edge. (C) For large enoughηL, transport is totally incoherent. (D) Ifthe right-mover is on the bottom edge, transport is undisturbed by the dephasing region andthus completely coherent.

site index andα a spin index. (At finite temperature the result is the same since the density

matrix is TR invariant.) Since the ground state is TR invariant T |0〉 = |0〉, we have

〈0|ciα(t)c†jβ(t′)|0〉 =

(

c†iα(t)|0〉, c†jβ(t′)|0〉)

=

(

c†iα(t)T |0〉, c†jβ(t′)T |0〉)

=

(

T T−1c†iα(t)T |0〉, T T−1c†jβ(t′)T |0〉

)

.


Now the TR transformation of the creation operators is

T−1c†iα(t)T = θαα′c†iα′(−t),

with a sum over repeated indices, andt → −t since the complex conjugation operationK

acts on the factors ofi in the exponential of the Heisenberg operatorsc(t) = eiHtce−iHt (but

not on the Hamiltonian operatorH itself for which complex conjugation is not defined).

For fermions we haveθ2 = −1. We obtain

〈0|ciα(t)c†jβ(t′)|0〉 = θαα′θββ′

(

T c†iα′(−t)|0〉, T c†jβ′(−t′)|0〉)

.

Using the antiunitary property(

T |φ〉, T |χ〉)

= 〈χ|φ〉 of T , we obtain

〈0|ciα(t)c†jβ(t′)|0〉 = θαα′θββ′

(

c†jβ′(−t′)|0〉, c†iα′(−t)|0〉)

= θαα′θββ′〈0|cjβ′(−t′)c†iα′(−t)|0〉.

In equilibrium, the correlation functions are time-translationally invariantG(t, t′) = G(t−t′), hence

〈0|ciα(t)c†jβ(t′)|0〉 = θββ′〈0|cjβ′(t)c†iα′(t′)|0〉(θT )α′α,

or

ΘGΘ−1 = GT ,

whereΘ = θ⊕· · ·⊕θ, the transpose is with respect to the single-particle basisand we used

ΘT = Θ−1. We have carried out the derivation forG> but this property holds regardless of

the time ordering and is thus valid for the various Green functionsG<,>,R,A,T,T . It is also

valid in frequency space. The retarded self-energy is defined byΣ = G−10 − G−1, where

G0 andG are the unperturbed and perturbed retarded Green functions, respectively. From

Eq. (2.8) which is valid for an arbitrary interacting Hamiltonian, it immediately follows

that

ΘΣΘ−1 = ΣT .


Then a local self-energy, diagonal in spin space,

Σiα,jβ(ω) = −iηδijδαβ,

will satisfy this requirement. Hence this means that our additional self-energy does not

break TR symmetry.

2.7.5 Nonlocal resistance in H-bar structure: numerical study

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Eg (eV)

R12

,34 (

h/e2 )

B

Figure 2.13: (A) Geometry of the H-bar structure. Semi-infinite metallic leads in then-typeregime are attached to contacts 1-4. Contacts 1 and 2 are current probes and contacts 3 and 4are voltage probes that draw no current. The calculation is performed forW = 0.24µm andL = 1.2µm, corresponding to the same aspect ratio as that of device D4(see Fig. 2.2). (B)Solid line: disordered-averaged nonlocal resistanceR12,34 as a function of gate potentialEg, dotted line: prediction of the Landauer-Buttiker formalism in the edge state picture,R12,34 = h/4e2.

Using the TB model [Eq. (2.7)], one can directly calculate the nonlocal resistanceR12,34

for the H-bar geometry illustrated in Fig. 2.13A. In the linear response regime, the nonlocal

resistance is simply obtained by solving the Landauer-Buttiker equations [57],

R12,34 ≡V3 − V4

I=

h

e2T31T42 − T32T41

D,


whereI ≡ I1 = −I2 and

D ≡ det

T12 + T13 + T14 −T12 −T13−T31 −T32 T31 + T32 + T34

−T41 −T42 −T43

,

where we consider for simplicity the case of zero temperature withTij ≡ Tij(µ), and the

transmission coefficientsTij are obtained from the Green function of the whole device as

in Eq. (2.6). For ad = 89.9 A device, the mass gap parameterM obtained fromk · pcalculations is|M | = 24.5 meV and the velocity is~v = 3.34 eV·A, such that the effective

mass of the carriers at the bottom of the conduction band is approximatelym∗ ≃ M/v2 =

0.0167m0 wherem0 is the bare electron mass. Considering the experimental mobility of

µ = 1.5 × 105 cm2/(V·s), we can estimate the inverse lifetime due to disorder as~/τ =e~

m∗µ= 0.46 meV.

The results of the calculation are plotted in Fig. 2.13B. We have averaged the resis-

tance over 50 configurations of Gaussian on-site disorder. The lattice constant of the tight-

binding model isa = 30 A. The resistance plateau in the nominally insulating regime

agrees very well with the predictionR12,34 = h4e2

of the Landauer-Buttiker formalism for

pure edge state transport. These numerical results thus confirm the edge state picture of

transport in the quantum spin Hall insulator regime.

2.8 Conclusion

The multi-terminal and nonlocal transport experiments on HgTe microstructures in the

QSH regime demonstrate that charge transport occurs through extended helical edge chan-

nels. We have extended the Landauer-Buttiker model for multi-terminal transport in the

QH regime to the case of helical QSH edge channels, and have shown that this model con-

vincingly explains the observations. Currently, logic devices based on the CMOS design

generate significant heating due to the ohmic dissipation within the channel. Our work

on the conductance quantization demonstrates that electrons can be transported coherently


within the edge channel without ohmic dissipation. Such an effect can be used to construct

a new generation of logic devices with improved performance.

Chapter 3

Kondo effect in the helical edge liquid of

the quantum spin Hall state

3.1 Introduction

The QSH insulator is a topologically nontrivial state of matter [54] that has recently been

observed in transport experiments carried out in HgTe QW [20] following its theoretical

prediction [19]. The two-dimensional QSH insulator has a charge excitation gap in the

bulk but supports one-dimensional gapless edge states forming a so-called “helical liquid”:

on each edge there exists a Kramers’ pair of counter-propagating states with opposite spin

polarization. The QSH insulator is robust against weak single-particle perturbations which

preserve TR symmetry such as weak potential scattering [13,21, 44].

This theoretical picture is consistent with experimental observations: the longitudinal

conductanceG in a Hall bar measurement is approximately quantized toG0 = 2e2/h,

independent of temperature, for samples of about a micron length [20, 58]. However,

larger samples exhibitG < G0 andG decreases with decreasing temperature [58]. De-

viations from the expected quantized value have been attributed to the presence of local

doped regions due to potential inhomogeneities within the sample arising from impurities

or roughness of the well/barrier interface [58]. Although pure potential scattering can-

not backscatter the edge states, the role of these potentialinhomogeneities is to trap bulk

electrons which may then interact with the edge electrons. These localized regions act as

41

CHAPTER 3. KONDO EFFECT IN THE QSH STATE 42

dephasing centers for the edge channels due to interaction effects and may cause backscat-

tering.

Figure 3.1: Temperature dependence of the conductance: thebehavior is logarithmic athigh temperatureT ≫ T ∗ and power-law at low temperatureT ≪ T ∗. At T = 0, ametal-insulator quantum phase transition is driven by Coulomb interactions in the helicalliquid: the system is a “Kondo metal” forK > 1/4 and a “Luttinger liquid insulator” forK < 1/4. The Fano factore∗ is defined as the ratio between shot noise and current, andreflects the charge of the current-carrying excitations.

In this work, we study theoretically the temperature dependence of the edge conduc-

tance of a QSH insulator. We consider the case where a local doped region in the vicinity

of the edge contains an odd number of electrons and acts as a magnetic impurity coupled

to the helical edge liquid. Our main results are as follows (Fig. 3.1):

1. At high temperatures, the conductance is logarithmic,−∆G ≡ −(G − G0) = η +

γ ln(D/T ) whereη, γ are interaction-dependent parameters andD is an energy scale

of order the bulk gap.

2. For weak Coulomb interactionsK > 1/4 whereK is the Luttinger parameter of the

edge liquid, the conductance is restored to unitarity atT = 0 due to the formation of


a Kondo singlet. This is in stark contrast with the Kondo problem in a usual spinful

1D liquid where the conductance vanishes atT = 0 for all Kρ < 1 whereKρ is the

Luttinger parameter in the charge sector [59]. At low but finite T the conductance

decreases as an unusual power-law∆G ∝ −T 2(4K−1) due to correlated two-particle

backscattering. The edge liquid being helical, the decrease in conductance is a direct

measure of the spin-flip rate [60].

3. For strong Coulomb interactionsK < 1/4, two-particle backscattering processes

are relevant and the system becomes insulating atT = 0. At low but finite T , the

conductance is restored by tunneling of excitations with fractional chargee/2 and we

obtainG(T ) ∝ T 2(1/4K−1).

3.2 Theoretical model

We model the impurity by aS = 12

local spin coupled by exchange interaction to the 1D

helical liquid with Coulomb interactions. The helical liquid having the same number of

degrees of freedom as a spinless fermion, a single nonchiralbosonφ is sufficient for its

description in the bosonized language [21]. The system is described by the Hamiltonian

H = H0 +HK +H2 whereH0 is the usual Tomonaga-Luttinger Hamiltonian [61],

H0 =v

2

∫

dx

[

KΠ2 +1

K(∂xφ)

2

]

,

with K the Luttinger parameter andv the edge state velocity. The Kondo HamiltonianHK

has the form

HK =J‖a

2πξ

(

S− : e−i2

√πφ(0) : + H.c.

)

− Jza√πSzΠ(0), (3.1)

whereS± = Sx ± iSy andSz are the spin operators for the impurity localized atx = 0.

a is the lattice constant of the underlying 2D lattice and corresponds to the size of the

impurity (we assume that the impurity occupies a single lattice site). ξ, the penetration

length of the helical edge states into the bulk, acts as a short-distance cutoff for the 1D

continuum theory in the same way that the magnetic length, the penetration length of the


chiral edge states, acts as a short-distance cutoff for the chiral Luttinger liquid theory of the

QH edge [62]. In addition to Kondo scattering, two-particlebackscattering is allowed by

TR symmetry [21, 44]. In HgTe QW the wavevector at which the edge dispersion enters the

bulk is usually much smaller thanπ/2a such that the uniform two-particle backscattering

(umklapp) term requiring4kF = 2π/a can be ignored. The impurity potential can however

provide a4kF momentum transfer and we must generally also consider a local impurity-

induced two-particle backscattering term [63]

H2 =λ2a

2

2π2ξ2: cos 4

√πφ(0) : ,

whereλ2 is the two-particle backscattering amplitude.

3.3 Weak coupling regime

We first consider the weak coupling regime whereJ‖, Jz andλ2 are small parameters.

The calculation of the conductance proceeds in two steps. Wefirst perform an explicit

perturbative calculation of the conductance to quadratic order in the bare couplingsJ‖ and

λ2 using the Kubo formula. This result is then extended to include all leading logarithmic

terms in the perturbation expansion by means of a weak coupling renormalization group

(RG) analysis of the scale-dependent couplingsJ‖(T ) andλ2(T ) where the scale is set by

the temperatureT .

The forward scattering termJz can be removed from the Hamiltonian by a unitary

transformation [64] of the formU = eiλSzφ(0) with λ = −Jza/vK√π, at the expense

of modifying the scaling dimension of the vertex operator: ei2√πφ : . The transformed

Hamiltonian reads

UHU † = H0 +H2 +J‖a

2πξ

(

S− : e−i2√πχφ(0) : + H.c.

)

,

whereχ = 1 − νJz/2K with ν = a/πv the density of states of the helical liquid. The

scaling dimension of: ei2√πχφ : is K ≡ Kχ2.

Using the transformed Hamiltonian we obtain the correctionto the conductance∆G(T ) =


∆GK(T ) + ∆G2(T ) to quadratic order in the couplingsJ‖ andλ2 using the Kubo formula

(see Sec. A.2), where∆GK is the correction due to spin-flip Kondo scattering1,

∆GK

e2/h= −Γ(1

2)Γ(K)

Γ(12+ K)

π2

3S(S + 1)(νJ‖(T ))

2, (3.2)

whereνJ‖(T ) = νJ‖(T/D)K−1 to O(J‖) andD = ~v/πξ is a high-energy cutoff of order

the bulk gap.∆G2 is the correction due to two-particle backscattering (see Sec. A.2),

∆G2

e2/h= −Γ(1

2)Γ(4K)

Γ(12+ 4K)

a4

2π4ξ4

(

λ2(T )

D

)2

, (3.3)

whereλ2(T ) = λ2(T/D)4K−1. There are no crossed terms of the formO(J‖λ2) orO(Jzλ2)

for a S = 12

impurity since the two-particle backscattering operator flips two spins but

a S = 12

spin can be flipped only once. We however expect that such terms would be

generated for impurities with higher spin.

These results can be complemented by a RG analysis. The RG equation forλ2 follows

by dimensional analysisdλ2

dℓ= (1−4K)λ2 with ℓ = ln(D/T ), so thatλ2 is relevant forK <

1/4 and irrelevant forK > 1/4. The renormalized coupling isλ2(T ) = λ2(T/D)4K−1, and

second order renormalized perturbation theory∆G2(T ) ∝ −λ2(T )2 simply reproduces the

Kubo formula result (3.3). Perturbation theory fails forT . T ∗2 whereT ∗

2 ∝ (λ02)1/(1−4K)

is a scale for the crossover from weak to strong two-particlebackscattering withλ02 the bare

two-particle backscattering amplitude.

The one-loop RG equations [66, 21] for the Kondo couplingsJ‖, Jz read (see Sec. A.1)

dJ‖dℓ

= (1−K)J‖ + νJ‖Jz,dJzdℓ

= νJ2‖ . (3.4)

The family of RG trajectories is indexed by a single scaling invariantc = (νJ‖)2 − (νJz)

2

whereνJz ≡ νJz + 1 −K, which is fixed by the couplings at energy scaleD. In contrast

to the spinful case [59], the absence of spin-flip forward scattering in the helical liquid

preserves the stability of the ferromagnetic fixed line, as is the case in the usual Kondo

1This result is valid in the high-temperature regime~v/L ≪ T < D for Fermi liquid leads whereL isthe length of the QSH region (see Ref. [65]).


problem. The renormalized spin-flip amplitudeJ‖(T ) is given in terms of the bare param-

etersJ0‖ , J

0z by

νJ‖(T ) =ανJ0

‖

sinh(

ανJ0‖ ln(T/T

∗K)) , (3.5)

such that∆GK is obtained to all orders in perturbation theory in the leading-log approxi-

mation by substituting Eq. (3.5) in Eq. (3.2). In Eq. (3.5),

α =

(

J0z

J0‖

)2

− 1

1/2

is an anisotropy parameter2 andT ∗K is the Kondo temperature,

T ∗K = D exp

(

− 1

νJ0‖

arcsinhα

α

)

.

In the isotropic caseα = 0, one recovers the usual formT ∗K = De−1/νJ0

‖ . In the limit 1 −K ≫ νJ0

‖ , νJ0z , Coulomb interactions dominate over Kondo physics and we obtainT ∗

K ≃

D

(

νJ0‖

1−K

)1/(1−K)

, a power-law dependence similar to results previously obtained [67] for

Kondo impurities in spinful Luttinger liquids. From the scaling exponent we see thatT ∗K

corresponds to the scale of the mass gap opened in a spinless Luttinger liquid by a point

potential scatterer of strengthνJ0‖ and the corresponding crossover is that of weak to strong

single-particle backscattering.

In the high temperature limitmaxT ∗2 , T

∗K ≪ T . D, both the Kondo and two-

particle scattering processes contribute logarithmically to the suppression of the conduc-

tance,−∆G(T ) ∼ η+γ ln(D/T ) whereη, γ are functions of the bare couplingsK, J0‖ , J

0z

andλ02.

2One can show that the Kondo model derived from the Anderson model for a single level coupled to the

helical liquid is isotropic due to time-reversal symmetry with J0‖ = J0

z = (|t|2 + |u|2)(

1ǫF−ǫd

+ 1ǫd+U−ǫF

)

whereǫF is the Fermi energy,ǫd is the impurity level with on-site Coulomb repulsionU , andu andt are thespin-flip and non-spin-flip hopping amplitudes, respectively. Coulomb interactions (K 6= 1) may howeverinduce an effective anisotropy (α 6= 0) even if the original Kondo model is isotropic.


Figure 3.2: Strong coupling regime: the Kondo singlet effectively removes one site fromthe system. a) Luttinger liquid withKρ < 1: a punctured 1D lattice is disconnected, b)QSH edge liquid withK > 1/4: the edge liquid follows the boundary of the deformed 2Dlattice. c) Half-charge tunneling forK < 1/4 by flips of the Ising order parameterm.

3.4 Strong coupling regime

We now investigate the low temperature regime below the crossover temperaturesT ≪minT ∗

2 , T∗K. The topological nature of the QSH edge state as a “holographic liquid”

living on the boundary of a 2D system [21] results in a drasticchange of the low-energy

effective theory in the vicinity of the strong coupling fixedpoint as compared to that of a

usual 1D quantum wire. As suggested by the perturbative RG analysis, the nature of the

T = 0 fixed point depends on whetherK is greater or lesser than1/4.

We first consider the case whereK > 1/4. In this case, two-particle backscattering is

irrelevant and∆G2 flows to zero. On the other hand, for antiferromagneticJz the Kondo

strong coupling fixed pointJ‖, Jz → +∞ is reached atT = 0, with formation of a local

Kramers singlet and complete screening of the impurity spinby the edge electrons. As a

result, the formation of the Kondo singlet effectively removes the impurity site from the

underlying 2D lattice (Fig. 3.2b). In a strictly 1D spinful liquid, this has the effect of

cutting the system into two disconnected semi-infinite 1D liquids (Fig. 3.2a) and transport


is blocked atT = 0 for all Kρ < 1 [59]. In contrast, due to its topological nature, the QSH

edge state simply follows the new shape of the edge and we expect the unitarity limitG =

2e2/h to be restored atT = 0. For finiteT ≪ T ∗K, the effective low-energy Hamiltonian

contains the leading irrelevant operators in the vicinity of the fixed point. In the case of

spinful conduction electrons, the lowest-dimensional operator causing a reduction of the

conductance is single-particle backscattering. However,the helical property of the QSH

edge states forbids such a term and it is natural to conjecture that the leading irrelevant

operator must be the two-particle backscattering operatorwith scaling dimension4K. We

thus expect a correction to the conductance at low temperaturesT ≪ T ∗K for K > 1/4 of

the form∆G ∝ − (T/T ∗K)

2(4K−1).

In particular, in the noninteracting caseK = 1 we predict aT 6 dependence in marked

contrast to both the usual Fermi liquid [68] and spinful 1D liquid [59] behaviors. This

dependence characteristic of a “local helical Fermi liquid” can be understood from a sim-

ple phase space argument (see Sec. A.6). The Pauli principlerequires the two-particle

backscattering operator to be defined through a point-splitting procedure [21] with the

short-distance cutoffξ, which translates into a derivative coupling in the limit ofsmall

ξ,

ψ†R(0)ψ

†R(ξ)ψL(0)ψL(ξ) → ξ2ψ†

R∂xψ†RψL∂xψL.

In the absence of derivatives, the four fermion term contributesT 2 to the inverse lifetime

τ−1k . The derivatives correspond to four powers of momenta closeto the Fermi points in

the scattering rateΓk,k′→p,p′ ∝ (k− k′)2(p− p′)2, which translates into an additional factor

of T 4. Furthermore, since at temperaturesT ≪ T ∗K suppression of the conductance is

entirely due to two-electron scattering, we expect the effective chargee∗ ≡ S/(2|〈IB〉|)obtained from a measurement of the shot noiseS in the backscattering current3 〈IB〉 to be

e∗ = 2e [63].

ForK < 1/4, the|λ2| → ∞ fixed point is reached atT = 0 and the system becomes

insulating. The fieldφ(x = 0, τ) is pinned at the minima of the cosine potentialH2 located

at±(2n + 1)√π/4 for λ2 > 0 and±2n

√π/4 for λ2 < 0, with n ∈ Z. The conductance

3In the weak coupling or high temperature regimeT ≫ T ∗2 , T

∗K, both the Kondo (e∗ = e) and 2P backscat-

tering (e∗ = 2e) contributions to the effective carrier charge are present, such that we expect a non-universalvalue for the Fano factor.


is restored at finite temperatures by instanton processes corresponding to the tunneling

between nearby minima separated by∆φ =√π/2. From the relationje = e∂tφ/

√π

between electric currentje andφ field, the charge pumped by a single instanton (Fig. 3.2c)

is obtained as

∆Qinst =e√π∆φ =

e

2.

This fractionalized tunneling current can be understood asthe Goldstone-Wilczek cur-

rent [69, 70, 16] for 1D Dirac fermions with a mass termδL = gΨ(m1 + iγ5m2)Ψ where

g ∼ λ2, and the mass order parametersm1 = cos 2√πφ, m2 = sin 2

√πφ change sign

during an instanton process with∆φ =√π/2 (see Sec. A.5). The order is Ising-like be-

cause the two-particle backscattering term explicitly breaks the spinU(1) symmetry of the

helical liquid

H0 =πv

2

∫

dx

(

Kσ2z +

1

Kρ2)

down toZ2, whereσz = ρ+ − ρ− andρ = ρ+ + ρ− are the spin and charge densities, and

ρ± are the chiral densities for the two members of the Kramers pair.

Fractionalization of the tunneling current is confirmed by asaddle-point evaluation of

the path integral for largeλ2 in the dilute instanton gas approximation (see Sec. A.3), which

yields a Coulomb gas representation of the partition function that can be mapped exactly

to the boundary sine-Gordon theory

S[θ] =K

β

∑

iωn

|ωn||θ(iωn)|2 + t

∫ β

0

dτ cos√πθ(τ),

wheret is the instanton fugacity. The RG equation fort follows as

dt

dℓ=

(

1− 1

4K

)

t,

and the conductanceG(T ) ∝ t(T )2 is a power-lawG ∝ (T/T ∗2 )

2(1/4K−1) for T ≪T ∗2 , K < 1/4. In contrast to the strong coupling regime in a usual Luttinger liquid where

t corresponds to a single-particle hopping amplitude [71, 72], the unusual scaling dimen-

sion of the tunneling operator in the present case corresponds to half-charge tunneling. In


particular, we calculate the shot noise in the strong coupling regime (see Sec. A.4) using

the Keldysh approach [73] and findS = 2e∗|〈I〉| where〈I〉 is the tunneling current and

e∗ = e/2.

3.5 Experimental realization

We find that the experimental results of Ref. [58] are consistent with our theoretical expres-

sions for the weak coupling regime with a weak Luttinger parameterK ≃ 1, but the small

number of available data points does not allow for a reliabledetermination of the model

parameters. The temperature dependence of the conductancebeing exponentially sensitive

to K, our predictions can be best verified in QW with stronger interaction effects. Due

to reduced screening of the Coulomb interaction4, we expect to see a steeper decrease of

conductance with decreasing temperature in HgTe samples with only a backgate.

Because of lower Fermi velocitiesvF , we expect even stronger interaction effects to

occur in InAs/GaSb/AlSb type-II QW [74] which have been recently predicted to exhibit

the QSH effect [75]. Interestingly, some recent experimental results [76, 77] suggest that

the QSH state is indeed realized in these materials. For QW widthswInAs = wGaSb = 10

nm in the inverted regime [75], and considering only screening from the front gate closest

to the QW layer, from ak ·p calculation of material parameters we obtainK ≃ 0.2 < 1/4,

making the insulating phase observable at low temperatures. Although the backgate will

cause additional screening,vF can be further decreased by adding a thin AlSb barrier layer

between the InAs and GaSb QW layers. The Fermi velocity is controlled by the overlap

between electron and hole subband wave functions [19] whichare localized in different

QW layers in the type-II configuration [75], and an additional barrier layer will decrease

this overlap. A lowervF also translates into higher Kondo temperatures sinceνJ ∝ 1/v2F ,

where one power ofvF comes from the matrix element of the localized impurity potential

between edge states, and one power comes from the density of statesν. SinceT ∗K depends

4K can be estimated [71] byK = [1 + α ln(d/ℓ)]−1/2 whereα = 2π2

e2/ǫ~vF

andǫ is the bulk dielectricconstant. The distanced from the QW layer to a nearby metallic gate acts as a screeninglength for theCoulomb potential, andℓ is a microscopic length scaleℓ = maxξ, w which acts as a short-distance cutofffor the Coulomb potential, wherew is the thickness of the QW layer.


on νJ exponentially, we expect experimentally accessible Kondotemperatures in type-II

QW.

Chapter 4

Magnetoconductance of the quantum

spin Hall state

4.1 Introduction

A great deal of interest has been generated recently by the theoretical prediction [19] and

experimental observation [20, 78, 79] of the QSH insulator state [13, 14, 54]. The QSH

state is a novel topological state of quantum matter which does not break TR symmetry, but

has a bulk insulating gap and gapless edge states with a distinct helical liquid property [21].

The gaplessness of the edge states is protected against weakTR symmetry preserving per-

turbations by Kramers degeneracy [21, 44]. As a result, the QSH state exhibits robust

dissipationless edge transport [20, 78, 79] in the presenceof nonmagnetic disorder.

However, in the presence of an external magnetic field which explicitly breaks TR sym-

metry, the gaplessness of the edge states is not protected. This can be simply understood

by considering the generic form of the effective 1D HamiltonianH for the QSH edge [70]

to first order in the magnetic fieldB,H = H0 +H1(B), whereH0 = ~vkσ3 is the Hamil-

tonian of the unperturbed edge, andH1(B) =∑

a=1,2,3(ta · B)σa is the perturbation due

to the field. k is a 1D wave vector along the edge,v is the edge state velocity,σ1,2,3 are

the three Pauli spin matrices, andt1,2,3 are model-dependent coefficient vectors [70]. IfB

points along a special direction in spacet∗ ≡ t1 × t2, thenH1(B) ∝ σ3 commutes with

H0, the wave vectork is simply shifted, and the edge remains gapless, unless mesoscopic

52

CHAPTER 4. MAGNETOCONDUCTANCE OF THE QSH STATE 53

quantum confinement effects become important [80]. IfB ∦ t∗, then[H0, H1(B)] 6= 0 and

a gapEgap ∝ |B| opens in the edge state dispersion.

Experimentally [20, 58], one observes that the conductanceG(B) of a QSH device ex-

hibits a sharp cusp-like peak atB = 0, andG decreases for increasing|B|. Although the

explanation of a thermally activated behaviorG(B) ∝ e−Egap(|B|)/kBT with T the tempera-

ture accounts qualitatively for the observed cusp, it does so only if the chemical potential

µ lies inside the edge gap which, according to theoretical estimates [54], is rather small

(Egap ∼ 1 meV). Experimentally, a sharp peak is observed [58] throughout the bulk gap

(Eg ∼ 40 meV). Furthermore, this explanation ignores the effects ofdisorder. In the ab-

sence of TR symmetry, the QSH edge liquid is topologically equivalent to a spinless 1D

quantum wire, and is expected to be strongly affected by disorder due to Anderson localiza-

tion. Although the effect of disorder on transport in the QSHstate has been the subject of

several recent studies [21, 44, 81, 82, 83, 84, 85], except for studies addressing the effect of

magnetic impurities [21, 86] there have been no theoreticalinvestigations of the combined

effect of disorder and TR symmetry breaking on edge transport in the QSH state.

In this work, we study numerically the edge magnetoconductanceG of a QSH insu-

lator in the presence of quenched nonmagnetic disorder. Ourmain findings are: (1) For

a finite magnetic fieldB and disorder strengthW on the order of the bulk energy gap

Eg, G deviates from its quantized valueG(0) = 2e2/h at zero field [20] by an amount

∆G(B) ≡ G(B)−G(0) which seems roughly linear in|B| at smallB, at least in the range

of fields we study. We observe this behavior forµ across the bulk gap (Fig. 4.1c), which

agrees qualitatively with the cusp-like features reportedin Ref. [20]. (2) The slope∂G/∂B

of G(B) at smallB steepens rapidly whenW > Eg (Fig. 4.2b), which suggests that bulk

states play an important role in the backscattering of the edge states. (3)G is unaffected by

an orbital magnetic field in the absence of inversion symmetry breaking terms (Fig. 4.3a).

In the absence of such terms,t1 andt2 are entirely in thexy plane of the device [54], hence

t∗ ∝ z is out-of-plane and a perpendicular fieldB ‖ t∗ cannot lead to backscattering, as

discussed earlier. In the presence of inversion symmetry breaking terms, the effective edge

Hamiltonian becomesH ′ = ~vkσ′3+∑

a=1,2,3(t′a ·B)σ′

a, whereσ′3 has nonzero components

along the1 and2 directions. Thent′∗ = t′1×t′2 is not alongz anymore, and a perpendicular

field B = Bz can lead to backscattering.


4.2 Theoretical model

We start from a simple 4-band continuum model Hamiltonian [19, 54] used to describe the

physics of the QSH state in HgTe QW,

H(k) =

(

H(k) ∆(k)

∆†(k) H∗(−k)

)

, (4.1)

written in the(E1+, H1+, E1−, H1−) basis whereE1, H1 are the relevant QW subbands

close to the Fermi energy and± denotes time-reversed partners. The diagonal blocks

H(k), H∗(−k) with H(k) = ǫ(k) + vk · σ +M(k)σz are related by TR symmetry and

correspond to decoupled 2D Dirac-like Hamiltonians, wherek = (kx, ky), σ = (σx, σy) is

a vector of Pauli matrices, and the velocityv is obtained fromk ·p theory. We also define a

quadratic kinetic energy termǫ(k) = C−Dk2 and the Dirac mass termM(k) =M−Bk2,

whereC,D,M,B arek · p parameters. The off-diagonal block∆(k) is given by [87]

∆(k) =

(

∆ek+ −∆z

∆z ∆hk−

)

, (4.2)

where∆e,∆h,∆z arek · p parameters andk± = kx ± iky. It originates from the bulk

inversion asymmetry (BIA) of the underlying microscopic zincblende structure of HgTe

and CdTe [88]. A nearest-neighbor TB model on the square lattice can be derived from

Eq. (4.1),

H =∑

i

c†iV ci +∑

i

(

c†iTxci+x + c†iTyci+y + h.c.)

, (4.3)

where the4× 4 matricesV, Tx, Ty depend solely on thek ·p parameters introduced above.

Equations (4.1) and (4.3) correspond to a translationally invariant system in the absence

of magnetic field or disorder. In the presence of disorder andan external magnetic field

B = (Bx, By, Bz), we perform the substitutions

V −→ V +HZ‖ +HZ⊥ +Wi,

Tx −→ Tx exp

(

2πi

φ0

∫ i+x

i

dℓ ·A)

= Txe−2πinzy/a,


whereWi is a Gaussian random on-site potential with standard deviation W mimicking

quenched disorder,A = (−Bzy, 0) is the in-plane electromagnetic vector potential in the

Landau gauge,φ0 = h/e is the flux quantum, andnz = Bza2/φ0 is the number of flux

quanta per plaquette witha the lattice constant. We usea = 30 A which is a good approx-

imation to the continuum limit. The in-plane Zeeman termHZ‖ is given by [54]

HZ‖ = g‖µB

0 0 B− 0

0 0 0 0

B+ 0 0 0

0 0 0 0

, (4.4)

whereB± = Bx ± iBy, µB is the Bohr magneton, and the in-planeg-factorg‖ is obtained

from k · p calculations [87]. The out-of-plane Zeeman termHZ⊥ is given by [54]

HZ⊥ = µBBz diag(

gE⊥, gH⊥,−gE⊥,−gH⊥)

, (4.5)

and the out-of-planeg-factorsgE⊥, gH⊥ are also obtained fromk ·p calculations [87]. The

k ·p parameters used in the present work correspond to a HgTe QW thickness ofd = 80 A.

We calculate numerically theT = 0 disordered-averaged two-terminal conductanceG

and conductance fluctuationsδG of a finite QSH strip (Fig. 4.1a) using the standard TB

Green function approach [89, 90, 91]. We find thatNdis ∼ 100 disorder configurations are

enough to achieve good convergence forG andδG. For a strip of widthLy comparable

to the edge state penetration depthλ, interedge tunneling [92] backscatters the edge states

even atB = 0 and the system is analogous to a topologically trivial quasi-1D quantum

wire. To ensure that we are studying effects intrinsic to thetopologically nontrivial QSH

helical edge liquid, we first need to suppress interedge tunneling. The naive way to achieve

this is to use a very largeLy; however, this can be computationally rather costly. We usea

geometry (Fig. 4.1a) which allows us to effectively circumvent this problem while keeping

Ly reasonable. By adding a local Dirac mass term [54]δM < 0 on the first horizontal chain

of our TB model (Fig. 4.1a, red dots), the penetration depthλ2 at the top edge becomes

much smaller than that at the bottom edgeλ1 ≫ λ2. We then add disorder only to the last

Ldis/a chains of the central region withLdis ≫ λ1 andLy − Ldis ≫ λ2. The resulting top


a) M

Ly

!2

1

Ldis

-1

-0.5

0

B [

T]

0 1 2 3G [e

2/h]c)

Lx

1 4

1.6

1.8

2

e2/h

]

0.12

0.18

0.24

Ly [ m]b) 0.5

1

B

-1

-0.5

]

Ee

Ed

Ec

Eb

Ead)

-1 -0.5 0 0.5 1

1

1.2

1.4

B [T]

G [

e

0

0.5

1-0.05 0 0.05

kx [ /a

]!, E(k

x) [eV]

Figure 4.1: MagnetoconductanceG of a QSH edge: a) TB model with asymmetric edgestatesλ2 ≪ λ1 to study a single disordered edge; b) dependence ofG on sample widthLy

for disorder strengthW = 55 meV larger than the bulk gap, lengthLx = 2.4µm, fixedclean widthLy − Ldis = 0.03µm, and local mass termδM = −70 meV, with error bars(plotted forLy = 0.12µm andB > 0 only) corresponding to conductance fluctuationsδG;c) dependence ofG on chemical potentialµ; d) quasi-1D spectrum of the device illustratedin a) for zeroW,B, showing bulk states (blue), top edge states (green) and bottom edgestates (red).

edge states are very narrow, contribute an uninteresting background quantized conductance

independent ofB andW , and are essentially decoupled from the bottom edge states (whose

magnetoconductance we wish to study) that are effectively propagating in a semi-infinite

disordered medium.

4.3 Numerical results

For µ inside the bulk gap, we expect edge transport to dominate thephysics. The typical

behavior of the magnetoconductanceG(B) for B = Bz and disorder strengthW larger

than the bulk gapEg ≃ 40meV is shown in Fig. 4.1b. The cusp-like feature atB = 0 agrees


qualitatively with the results of Ref. [20].G(B) is independent ofLy, which suggests that

transport is indeed carried by the edge states.G(B = 0) is quantized toG0 ≡ 2e2/h

independent ofW up toW = 71 meV with extremely small conductance fluctuations

δG(B = 0)/G0 ∼ 10−5, which confirms that interedge tunneling is negligible evenfor

strong disorder. Furthermore,G tends toG0/2 for large |B| ∼ 1 T, which indicates that

the disordered bottom edge is completely localized for large W and |B|, and only the

unperturbed top edge conducts. ForW < Eg, G is approximately quadratic inB (not

shown), and|G(B) − G0|/G0 ≪ 1 even for large|B| ∼ 1 T. ForB 6= 0, we observe that

the amplitude of the fluctuationsδG does not decrease upon increasingNdis, and is roughly

independent ofW with δG/G0 ∼ O(10−1) for large enough disorderW & Eg. Since in

the absence of TR symmetry the QSH system is a trivial insulator and the edge becomes

analogous to an ordinary spinless 1D quantum wire with no topological protection, we

conclude thatδG corresponds to the well-known universal conductance fluctuations [89].

The dependence ofG(B) on µ is plotted in Fig. 4.1c. We considerW = 55 meV

slightly larger thanEg (Fig. 4.1d). This is not unreasonable as the bulk mobilityµ∗ of

the HgTe QW in Ref. [20] is estimated asµ∗ ≃ 105 cm2/(V·s), which corresponds to

a momentum relaxation timeτ = µ∗m∗/e ≃ 0.57 ps. The bulk carriers at the bottom

of the conduction subband have an effective massm∗ ≃ 0.01me whereme is the bare

electron mass.τ is given by~/τ ≃ 2πν(Wa)2, with ν the bulk continuum density of

states at the Fermi energy given byν ≃ m∗/π~2. This yieldsW ≃ 22 meV. However,

this estimate considers only bulk disorder and we expect edge roughness to yield a higher

effectiveW on the edge. Furthermore, this estimate is perturbative inW and neglects

interband effects which are expected to occur forW ∼ Eg. For the chosen value ofW

we observe that the bulk states (Fig. 4.1d, blue lines) are strongly localized withG ≪ G0

for µ < Ea andµ > Ee in the bulk bands, while the cusp-like feature atB = 0 with

G(B = 0) = G0 remains prominent forEb < µ < Ed in the bulk gap and even at the

bottom of the conduction bandEd < µ < Ee where the top edge states (Fig. 4.1d, red

lines) coexist with the bulk states. The sudden dip inG(B 6= 0) for µ ∼ Ec ≃ 15 meV

corresponds to the opening of the small edge gap discussed earlier. Finally, G ≃ G0 is

almost independent ofB for Ea < µ < Eb, where the disordered bottom edge and bulk

states are mostly localized while the clean top edge supports another channel (Fig. 4.1d,


0 0.025 0.05

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

|B| [T]

G [e

2 /h]

0 20 40 600

5

10

15

20

25

30

W [meV]

−dG

/dB

[(e2 /h

).T

−1 ]

293337414549535761656971

a)

W [meV]

b)

Figure 4.2: a) Magnetoconductance for various disorder strengthsW ; b) small-B slope ofthe magnetoconductance (obtained by linear regression for0 < B < 15 mT). Device sizeis (Lx × Ly) = (2.4× 0.12)µm2.

dashed green line), with a total top edge conductance ofG = G0.

The magnetoconductance forB = Bz and various values ofW is plotted in Fig. 4.2.

Although not evident from the figure,G(B) is approximately quadratic inB for W < Eg,

and approximately linear in|B| at smallB for W > Eg (Fig. 4.2a). The slope ofG(B)

at small fields (obtained by linear regression for0 < B < 15 mT where the dependence

is approximately linear) is plotted in Fig. 4.2b, and is seento increase rapidly forW &

Eg ≃ 40 meV. ForB = 0, we have essentiallyG = G0 independent ofW (Fig. 4.2a).

This contrasts with the results of Ref. [81, 84] where deviations fromG = G0 atB = 0

occur forW larger than some critical valueWc > Eg. The reason for this difference is

that in Ref. [81, 84], disorder-induced collapse of the bulkgap is accompanied by the edge

states penetrating deeper into the bulk and eventually reaching the opposite edge, such

that interedge tunneling takes place and causes backscattering. Here, due to our special

geometry (Fig. 4.1a) the top edge state is unperturbed and always remains localized near

the edge, out of reach of the bottom edge state, even as the latter penetrates deeper into the

disordered bulk for increasingW .


The BIA term∆k has an important effect onG for B = Bz (Fig. 4.3a). For simplicity,

we set∆e = ∆h = 0 and consider only the effect of∆z. For∆z = 0, the perturbation

H′ = ej · A due to an orbital field, withe the electron charge andj the current operator,

has no matrix element between the spin states of a counterpropagating Kramers pair on

a given edge [54], andG is unaffected. For an in-plane field,HZ‖ does have a nonzero

matrix element between these states, and there is a nontrivial magnetoconductance even in

the absence of BIA.

The dependence ofG(B) on the orientation ofB is plotted in Fig. 4.3b. Theg-

factors [87] used in the Zeeman terms are such that the Zeemanenergies for in-plane and

out-of-plane fields are of the same order [54]. The in-plane vs out-of-plane anisotropy

(Fig. 4.3b,x, y vs z) arises from the orbital effect of the out-of-plane fieldB = Bz, which

is absent for an in-plane field. In our model, the in-plane anisotropy is very weak (some-

what visible on Fig. 4.3b for|B| ∼ 1 T), and is due to the inequivalence between the

transportx and confinementy directions. Finally, theB = 0 peak inG is more pronounced

for a smaller mass termM [54] in the Dirac HamiltoniansHk, H∗−k

(Fig. 4.3c). Since

Eg ∝ |M | approximately, a smaller|M | results in a larger dimensionless disorder strength

W/Eg, which is equivalent to an increase inW (see Fig. 4.2b).

Although the mechanism behind the observed negative magnetoconductance∆G ∝−|B| (Fig. 4.1,4.2) for an orbital fieldB = Bz cannot be unambiguously inferred from our

numerical results, a dependence linear in|B| for smallB and the requirement of “strong”

disorderW & Eg for its observation seem to indicate that the effect has a nonperturbative

character. A treatment which is perturbative inW andB yields at most, to leading order, the

result−∆G ∝ ℓ−1 ∝ W 2eff(B) ∝ B2, whereℓ is the mean free path [93] andWeff(B) ≡

W |B|/B0 is some effective disorder strength, withB−10 ∝ ∆z if only the effect of∆z

is considered for simplicity. For “weak” disorderW < Eg, the 1D edge states which

enclose a negligible amount of flux are the only low-energy degrees of freedom, and the

magnetic field only has a perturbative effect on them. Indeed, if we choose the gaugeA =(

Bz(Ly − y), 0)

, for sufficiently smallBz we have thatA is small forLy − λ1 . y < Ly

with λ1 ≪ Ly where the bottom edge state wavefunction has finite support (Fig. 4.1a), and

the effect of an orbital fieldBz on a single edge can be treated perturbatively. In this case,

the amplitude∝ Weff(B) in perturbation theory for a leading order backscattering process


1

1.2

1.4

1.6

1.8

2

G [

e2/h

]

-0.5 0 0.5

1.7

1.75

1.8

1.85

1.9

1.95

2

B [T]

G [

e2/h

]

x

y

z

-1 -0.5 0 0.5 1

1.4

1.5

1.6

1.7

1.8

1.9

2

B [T]

G [

e2/h

]

15

10

5

2

1

z = 0

z = -0.5

z = -1.5

z = -1

a) b)

c)

-M [meV]

Figure 4.3: Dependence of the magnetoconductanceG on a) strength of thek-independentBIA term∆z with ∆e = ∆h = 0; b) magnetic field orientation; c) Dirac mass termM < 0.Sample size is(Lx × Ly) = (2.4 × 0.12)µm2, disorder strength isW = 55 meV for a),b)andW = 30 meV for c).

on a single edge involves one power of∆z and one power ofBz to couple the spin states of

the counterpropagating Kramers partners [54] (with no momentum transfer as our choice of

gauge preserves translational symmetry in thex direction), and one power ofW to provide

the necessary momentum transfer for backscattering. Our observation that∆G ∝ −B2

for W < Eg corroborates this physical picture. On the other hand, the cusp-like feature

atB = 0 (Fig. 4.1b) occurs for “strong” disorderW & Eg, which seems to indicate that

the bulk states play an important role. This leads us to a different physical picture. For

W & Eg, the edge electrons easily undergo virtual transitions to the bulk. In other words,

the emergent low-energy excitations forW & Eg extend deeper into the bulk than the

“bare” edge electrons. The electrons spend a significant amount of time diffusing randomly

in the bulk away from the edge, with their trajectories enclosing finite amounts of flux

before returning to the edge, which endows the orbital field with a nonperturbative effect. In


this way the conventional picture of 2D antilocalization [94] can apply, at least qualitatively,

to a single disordered QSH edge. We are thus led to the interesting picture, peculiar to the

QSH state, of a dimensional crossover between 1D antilocalization [95, 96, 97] in the weak

disorder regimeW < Eg with the orbital field having a perturbative effect, to an effect

analogous to 2D antilocalization in the strong disorder regimeW > Eg with the orbital

field having a nonperturbative effect.

4.4 Conclusion

We have shown that “strong” disorder effectsW/Eg ∼ 1 in a QSH insulator in the presence

of a magnetic fieldB and inversion symmetry breaking terms can give rise to a cusp-like

feature in the two-terminal edge magnetoconductance with an approximate linear depen-

dence∆G(B) ∝ −|B| for smallB. These results are in good qualitative agreement with

experiments. A possible physical intepretation of our results consists of a dimensional

crossover scenario where a weakly disordered, effectivelyspinless 1D edge liquid crosses

over, for strong enough disorder, to a state where disorder enables frequent excursions of

the edge electrons into the disordered flux-threaded 2D bulk, resulting in a behavior remi-

niscent of 2D antilocalization.

Chapter 5

Spin Aharonov-Bohm effect and

topological spin transistor

5.1 Introduction

The spin of the electron is one of the most fundamental quantum mechanical degrees of

freedom in Nature. Historically, the discovery of the electron spin helped to lay the foun-

dation of relativistic quantum mechanics. In recent years,the electron spin has been pro-

posed as a possible alternate state variable for the next generation of computers, which led

to extensive efforts towards achieving control and manipulation of the electron spin, a field

known as spintronics [98]. Despite the great variety of currently used or theoretically pro-

posed means of manipulating the electron spin, a feature common to all of them is that they

all make use of theclassicalelectromagnetic force or torque actinglocally on the magnetic

moment associated with the spin.

On the other hand, it is known that due to the Aharonov-Bohm (AB) effect [99], elec-

trons in a ring can be affected in a purelyquantum mechanicalandnonlocalway by the

flux enclosed by the ring even though no magnetic field — hence no classical force —

is acting on them. This effect could be termed “charge AB effect”, as it relies only on

the electron carrying an electric charge. In systems with spin-orbit coupling or magnetic

fields, a spin-dependent phase factor can be obtained and leads to modifications to the AB

62

CHAPTER 5. SPIN AB EFFECT AND TOPOLOGICAL SPIN TRANSISTOR 63

effect [100, 101, 102, 103]. However, these effects usuallyinvolve classical forces act-

ing on the spin such as electromagnetic fields, and the pure gauge potential leading to the

charge AB effect does not directly couple to spin. This observation leads naturally to the

question of whether it is possible to observe a “spin AB effect” which would enable one

to manipulate the electron spin in a purely nonlocal and quantum mechanical way, without

any classical force or torque acting locally on the spin magnetic moment.

In this work, we show that the spin AB effect is indeed possible by making use of the

edge states of the recently discovered QSH insulators. In recent years, the QSH insulator

state has been proposed in several different materials [13,14, 104, 19, 75, 105]. In particu-

lar, this topologically nontrivial state of matter has beenrecently predicted [19] and realized

experimentally [20, 78, 79] in HgTe QW. The QSH insulator is invariant under TR, has a

charge excitation gap in the bulk, but has topologically protected gapless edge states that

lie inside the bulk insulating gap. These edge states have a distinct helical property: two

states with opposite spin polarization counterpropagate at a given edge [13, 21, 44]. The

edge states come in Kramers doublets, and TR symmetry ensures the crossing of their en-

ergy levels at TR invariant points in the Brillouin zone. Because of this level crossing, the

spectrum of a QSH insulator cannot be adiabatically deformed into that of a topologically

trivial insulator without closing the bulk gap. The helicity of the QSH edge states is the

decisive property which allows the spin AB effect to exist: the perfect correlation between

spin orientation and direction of propagation allows the transmutation of a usual charge AB

effect into a spin AB effect, as will be explained in detail below.

The mechanism we propose to realize the spin AB effect is illustrated in Fig. 5.1. Con-

sider a two-terminal device consisting of a bounded QSH insulator region pierced by a

hole which is threaded by a magnetic fluxφ. If the edge electrons propagating clock-

wise have their spin pointing out-of-plane alongz (spin up |↑〉 , red trajectory), due to

TR symmetry the electrons propagating counterclockwise must have opposite spin along

−z (spin down|↓〉 , blue trajectory). If we inject electrons spin-polarized along thex di-

rection |→〉 = 1√2(|↑〉 + |↓〉) from a ferromagnetic (FM) lead on the left, the electron

beam will be split coherently upon entering the QSH region atthe left junction into a|↑〉beam propagating along the top edge and a|↓〉 beam propagating along the bottom edge.

When the electron beams are recombined on the right side of the ring, the electrons along


Figure 5.1: Schematic picture of the spin AB effect. A ring ofQSH insulator threaded by amagnetic fluxφ is connected to two magnetic leads. Spin polarized electrons injected fromthe left lead enter the QSH region as a superposition of spin up and down states. The spinup (down) state can only propagate along the top (bottom) edge of the QSH ring, and thetwo spin states thus acquire an AB phase difference proportional toφ. Consequently, uponexiting the QSH region the two edge states recombine into a state with spin rotated withrespect to the injected direction. The magnetization direction of the right lead generallydiffers from that of the left lead by an angleθ. The two-terminal conductanceG = G(φ, θ)of the device depends on the relative angle between the spin polarization of the outgoingstate and that of the right lead.

top and bottom edges will acquire a phase difference ofϕ = 2πφ/φ0 due to the AB ef-

fect, whereφ0 = hc/e is the flux quantum. Consequently, the output state is given by1√2

(

|↑〉+e−iϕ |↓〉)

, such that the electron spin is rotated by an angleϕ in thexy plane. The

magnetic flux being confined to the hole in the device (Fig. 5.1), the electromagnetic fields

are zero in the region where the electrons propagate, and thespin is rotated by a purely

quantum mechanical Berry phase effect. In particular, for collinear FM leads (θ = 0 in

Fig. 5.1), one expects the conductance to be maximal forφ = 0 (modφ0) and minimal for

φ = 12φ0 (modφ0), thus realizing a “topological” spin transistor [Fig. 5.3(c)]. This effect is

topological in the sense that the spin is always rotated by one cycle for each period of flux

φ0, regardless of the details of the device, such as the size of the system or the shape of the

ring.


5.2 Phenomenological scattering matrix analysis

Before considering any microscopic model of transport in a QSH system, generic features

of two-terminal transport in the device of Fig. 5.1 that depend only on symmetry consid-

erations can be extracted from a simple phenomenological scattering matrix orS-matrix

analysis [106]. The left and right junctions are each described by a scattering matrixSL

andSR, respectively (e.g. Fig. 5.2(a) for the left junction). Considering the left junc-

tion first, SL consists of four submatricestL, t′L, rL, r′L which correspond respectively to

transmission from left to right, transmission from right toleft, reflection from the left, and

reflection from the right. One can define similar submatricesfor SR. We wish to obtain an

effectiveS-matrixS [see Eq. (A.63)] for the whole device, by combining theS-matrices of

the junctions together with theS-matrix for the central QSH region. Inside the QSH region,

the AB effect is described by the matrixΦ ≡ e−iϕσz/2 whereσx, σy, σz are the three Pauli

matrices. In addition to the geometric phaseϕ, the edge electrons also acquire a dynamical

phaseλ = 2kF ℓ identical for both spin polarizations, whereℓ is the distance travelled by

the edge electrons from left to right junction andkF is the edge state Fermi wave vector.

Details of the analysis are presented in Sec. A.7; here we discuss only the main results. We

obtain the effective2× 2 device scattering matrixS,

S(φ, θ) =(

1− eiλΦr′L(0)ΦrR(θ))−1

eiλ/2Φ, (5.1)

where the junction reflection matricesr′L(θL) andrR(θR) depend on the anglesθL, θR of

the magnetizationML,R in the left and right leads. For simplicity we considerθL = 0 and

defineθ ≡ θR (Fig. 5.1).

The two-terminal conductanceG of the device can be written as

G =e2

htr ρRSρLS

†, (5.2)

using Eq. (A.63) of Sec. A.7. HereρL, ρR are2 × 2 effective spin density matrices for the

FM leads, and have the form

ρα(θα) =12Tα(θα)

(

1 +Pα(θα) · σ)

, (5.3)


with α = L,R, whereTα = tr ρα is the transmission coefficient of the junction andPα

is a polarization vector. For simplicity, we can assume the device to have aπ-rotation

symmetry, which together with TR symmetry restricts the generic form of the reflection

matricesr′L andrR in Eq. (5.1) to be

r′L(θ) =

(

αθ βθ

γθ αθ+π

)

, rR(θ) =

(

αθ+π βθ

γθ αθ

)

. (5.4)

Physically,αθ is a non-spin-flip reflection amplitude whereasβθ, γθ are spin-flip reflection

amplitudes, withβθ corresponding to a|↓〉 → |↑〉 reflection andγθ to a |↑〉 → |↓〉 reflec-

tion. These amplitudes are generally different due to the breaking of TR symmetry at the

junctions by the nearby FM leads.

5.3 Minimal model description

These expressions being so far very general, to make furtherprogress it is useful to consider

a simple continuum Hamiltonian model for the FM/QSH junctions in which the reflection

matricesr′L andrR can be calculated explicitly. This model satisfies the symmetries in-

voked earlier and will be seen to be a good description of the realistic HgTe system in

spite of its simplicity. We model the FM leads as 1D spin-12

fermions with a term which

explicitly breaks theSU(2) spin rotation symmetry [107],

HFM =

∫

dxΨ†(

− 1

2m

∂2

∂x2−M(θ) · σ

)

Ψ,

whereM(θ) =M n, with n = x cos θ+ y sin θ, is an in-plane magnetization vector andΨ

is a two-component spinorΨ ≡(

ψ↑ ψ↓

)T

. In the absence of AB flux, the QSH edge

liquid consists of 1D massless helical fermions [21, 44]. When the spins of the edge states

are polarized along thez direction, the Hamiltonian is given by

HQSH = −iv∑

α=t,b

ηα

∫

dx(

ψ†α↑∂xψα↑ − ψ†

α↓∂xψα↓)

,


Figure 5.2: Illustration of the minimal model describing a FM/QSH junction. (a) Schematicpicture of the junction between the left FM lead and the QSH insulator. Incoming channelsal,1, . . . , al,pL from the left lead scatter at the junction into transmitted QSH edge channelsbl′,↑, bl′,↓ and reflected lead channelsbl,1, . . . , b1,pL. This scattering process is described bya scattering matrixSL. (b) Minimal model description of the junction. The FM lead isdescribed by 1D parabolic bands with a spin splitting2M , while the QSH edge states arelinearly dispersing and TR invariant, with opposite spin states counter-propagating.

wherev is the edge state velocity andα = t, b refers to the top and bottom edge, respec-

tively, with ηt = 1 andηb = −1.

In this simple model, the junction is described as a sharp interface between the FM

region and the QSH region, from which the reflection matrixr′L in Eq. (5.1) and the spin

density matrixρL in Eq. (5.2) can be obtained. The calculation yields the reflection matrices

precisely in the form of Eq. (5.4) withαθ = a andβθ = γ∗θ = be−iθ. In the limit of small

spin splittingM/εF ≪ 1 whereεF is the Fermi energy in the leads [Fig. 5.2(b)], the


reflection amplitudesa andb are given by

a ≃ v − vFv + vF

, b ≃ M

2εF

v3Fv(v + vF )2

, (5.5)

wherevF =√

2εF/m is the Fermi velocity in the FM leads. The off-diagonal spin-flip

reflection amplitudeb is proportional to the magnetizationM and along with its accompa-

nying scattering phase shifte±iθ is an explicit signature of TR symmetry breaking at the

junction. The diagonal non-spin-flip reflection amplitudea does not break TR symmetry

and is the same as would be obtained in the scattering from a nonmagnetic metal with

M = 0. The lead spin density matricesρL, ρR can also be calculated explicitly and are

found to follow the form of Eq. (5.3) as expected from the general S-matrix analysis. In

the limitM/εF ≪ 1, we obtainTL = TR = 8vvF/(v + vF )2 and

PL(θ) = PR(θ) ≡ P(θ) = −M(θ)

4εF

v2Fv(v + vF )

, (5.6)

i.e. the spin polarization vector is directly proportionalto the magnetizationM.

From the results obtained above, we can readily evaluate theconductanceG, which has

the following expression in the limitM/εF , P ≡ |P(θ)| ≪ 1 andλ = 0,

G(ϕ, θ;λ = 0) =e2

h

TLTR/2

1− 2a2 cosϕ+ a4

×[

1 +cos(θ − ϕ) + (1− t2)2 cos(θ + ϕ) + C(ϕ, θ)

1− 2a2 cosϕ+ a4P 2 +O(P 4)

]

,

(5.7)

wheret = 1 − a andC(ϕ, θ) ≡ γ cosϕ + δ cos θ with γ, δ some constants depending

only ona. The effect of a finiteλ will be addressed in the next section, where we study

numerically a more realistic model of the QSH state in HgTe QW. Physically,a and t

can be interpreted as reflection and transmission coefficients for theSz spin current. The

generic behavior of Eq. (5.7) is illustrated in Fig. 5.3. ThetermC(ϕ, θ) is an uninteresting

background term which manifests no correlation between AB phaseϕ and rotation angle

of the electron spinθ. The term∝ cos(θ−ϕ) corresponds to a rotation of the electron spin


Figure 5.3: Phenomenological analysis of the two-terminalconductance (top view ofFig. 5.1). (a) The two leading contributions to the spin AB rotation. The purple pathstands for the process with no spin flips, which leads to a spinrotation ofϕ ≡ 2πφ/φ0.The orange path stands for the process with spin-dependent reflections, which leads to aspin rotation of−ϕ. (b) Schematic intensity map of the two-terminal conductanceG(ϕ, θ).The conductance reaches its maximum along the linesϕ = θ (purple) andϕ = −θ (or-ange), which are contributed by the purple and orange paths in panel (a), respectively. (c)The on and off states of the topological spin transistor are defined forθ = 0 by ϕ = 0 andϕ = π, respectively, as also indicated in panel (c).

byϕ, and the term∝ cos(θ+ϕ) corresponds to a rotation by−ϕ. The conductance is thus

maximal forϕmax = ±θ [Fig. 5.3(b)], manifesting the desired flux-induced spin rotation

effect. Physically, theϕmax = θ term corresponds to a process in which electrons traverse

the device without undergoing spin flips (Fig. 5.3(a), purple trajectory) while theϕmax =

−θ term corresponds to a process involving at least one TR breaking spin-flip reflection

(Fig. 5.3(a), orange trajectory). As can be seen from Eq. (5.7), the relative intensity of the

two contributions to the conductance isI−θ/Iθ = (1 − t2)2 which can be close to unity

for strongly reflecting junctionst ≪ 1. As both contributions are minimal forϕ = π at

θ = 0, one can considerϕ = π, θ = 0 as the “off” state of a spin transistor (Fig. 5.3(c),

right) where the rotation of the spin is provided by a purely quantum mechanical Berry

phase effect. This is in contrast with the famous Datta-Das spin transistor [108] where


the rotation of the spin is achieved through the classical spin-orbit force. The “on” state

corresponds to the absence of spin rotation forϕ = 0 (Fig. 5.3(c), left).

5.4 Experimental realization in HgTe QW

We now show that this proposal can in principle be realized experimentally in HgTe QW.

We model the device of Fig. 5.1 as a rectangular QSH region threaded by a magnetic AB

flux through a single plaquette in the center, and connected to semi-infinite metallic leads

on both sides by rectangular QSH constrictions modeling quantum point contacts (QPC)

[Fig. 5.4(a)]. The QSH region is described by an effective4 × 4 tight-binding Hamil-

tonian [19, 54] with the chemical potential in the bulk gap, while the metallic leads are

described by the same model with the chemical potential in the conduction band. The de-

tailed form of the model is given in Sec. A.7.3. The injectionof spin-polarized carriers

by the FM layers of Fig. 5.1 is mimicked by the inclusion of an effective Zeeman term

in the Hamiltonian of the semi-infinite leads. We calculate numerically the two-terminal

conductance through the device of Fig. 5.4(a) for a QW thicknessd = 80 A. We use the

standard lattice Green function Landauer-Buttiker approach [89] in which the conductance

is obtained from the Green function of the whole device, the latter being calculated recur-

sively [90].

The results of the numerical calculation are plotted in Fig.5.4(b), (c), (d). In the ab-

sence of phase-breaking scattering processes, one distinguishes two temperatures regimes

T ≪ Tℓ andT ≫ Tℓ separated by a crossover temperatureTℓ = π~v/kBℓ with v the edge

state velocity, defined as the temperature for which a thermal spread∆µ ∼ kBT in the

energy distribution of injected electrons corresponds to aspread in the distribution of dy-

namical phasesλ = 2kF ℓ of ∆λ ∼ 2π. In the low temperature regimeT ≪ Tℓ, ∆λ ≪ 2π

and the dynamical phase is essentially fixed such thatG(T ≪ Tℓ) ≃ G(T = 0). In this

regime,G(T = 0, µ) is approximately periodic inµ for µ within the bulk gap, with period

∆µ ∼ kBTℓ. A crossing pattern (Fig. 5.4(b), top) occurs periodicallyand can be obtained

by tuning the chemical potential. It corresponds to the flux-induced spin rotation effect

(Fig. 5.3). In the high temperature regimeT ≫ Tℓ, one could expect that the crossing

pattern, and thus the spin rotation effect, would be washed out by thermal self-averaging of


Figure 5.4: Numerical study of the spin AB effect in HgTe QW. (a) Device geometry usedfor the numerical two-terminal conductance calculation:a = 30 A is the lattice constant ofthe tight-binding model,L = 18 nm, Lx = ℓ = 240 nm, Ly = 120 nm, φ is the AB flux,andW is the QPC width. (b) Intensity map of the conductanceG(φ, θ) for fixed chemicalpotentialµ = 0.06 eV (top panel) and averaged chemical potential over energy range∆µ =5meV corresponding to an average over∼ 2π dynamical phase (bottom panel). These twosituations correspond to low and high temperature, respectively (see text). (c) Logarithmicplot of on/off ratioGon/Goff of topological spin transistor as a function of spin polarizationP of injected carriers for fixed chemical potentialµ = 0.06 eV and different values of theQPC widthW . (d) Plot of on/off ratio as a function of QPC widthW for fixed chemicalpotentialµ = 0.06 eV and different values of the spin splitting∆s in the bulk leads.


the dynamical phase. Surprisingly, the pattern remains (Fig. 5.4(b), bottom), and actually

acquires a more symmetric structure through the self-averaging procedure. In both temper-

ature regimes, the conductance pattern agrees qualitatively with the result of the simple 1D

Hamiltonian model [Fig. 5.3(b)].

So far, our discussion has ignored the existence of phase-breaking processes. Such

processes introduce an additional characteristic temperatureTϕ, defined as the temperature

above which the phase coherence lengthℓϕ(T ) becomes smaller than the system sizeℓ,

that isℓϕ(Tϕ) = ℓ andℓϕ(T > Tϕ) < ℓ. As explained in the Introduction, the stability of

the QSH state is protected by Kramers’ theorem. However, Kramers’ theorem requires the

quantum phase coherence of electronic wave functions, hence for T > Tϕ the QSH state

can be destroyed [20, 78, 79]. Thus, the observation of the spin AB effect requiresT <

Tϕ, with the precise value ofTϕ depending on the particular nature of the phase-breaking

mechanisms. With this first requirement satisfied, two scenarios are possible depending on

the relative value of the two characteristic temperaturesTℓ andTϕ. If Tℓ < Tϕ, the scenario

described in the previous paragraph applies, with the existence of a low temperature regime

T ≪ Tℓ < Tϕ with well-defined dynamical phase and a high temperature regime Tℓ ≪T < Tϕ with completely randomized dynamical phase. On the other hand, if Tℓ > Tϕ, then

since we requireT < Tϕ for the observation of the spin AB effect the high temperature

regimeT ≫ Tℓ can be never be achieved. This could correspond for instanceto a very

small, fully phase-coherent deviceℓ ≪ ℓϕ, with no noticeable thermal fluctuation effects.

In transport measurements on HgTe QW [20, 78, 58] a robust QSHstate has been observed

in devices of sizeℓ ≃ 1 µm up to temperatures of4.2 K. This gives us a lower bound

estimate of a few Kelvins forTϕ, for a device of such size. For a typical edge state velocity

~v ∼ 3.5 eV·A one obtains a crossover temperatureTℓ ∼ 13 K & Tϕ, which indicates that

one would probably be in the low temperature regime with weakthermal fluctuation effects

and good tunability of the crossing pattern with chemical potentialµ. The other scenario

requiringTℓ < Tϕ can be realized if the lower bound estimate of4.2 K turns out to be too

conservative and we actually haveTϕ > 13 K, or if the edge state velocity is significantly

smaller than the value of3.5 eV·A used above. The latter possibility can occur in type-

II QW [75] where the edge state velocity is about one order of magnitude smaller, hence

Tℓ ∼ 1 K and the conditionTℓ < Tϕ would in principle be satisfied.


In our calculations, for simplicity we have assumed that electrons on both the top and

bottom edges acquire the same dynamical phaseλ. In a real system, the two arms of

the ring are not perfectly symmetric and the electrons propagating on different arms can

certainly acquire different dynamical phasesλbottom 6= λtop. However, the dynamical phase

differenceδ ≡ λbottom − λtop only leads to an additional flux-independent rotation of the

spin of the outgoing electrons, which leads to a shift of the conductance pattern in the angle

θ by an amountδ (see Eq. (A.64) of Sec. A.7). Thus the transistor remains effective if one

usesθ = δ instead ofθ = 0 in the right FM lead. If one prefers to useθ = 0, one can cancel

out the phase asymmetry by patterning an electrostatic gateon top of one given arm. By

tuning the potential of this gate, one can adjust the Fermi wave vector locally and introduce

a dynamical phase offset which cancels out the phase asymmetry δ.

In Fig. 5.4(c), (d) we plot the on/off ratioGon/Goff of the topological spin transistor,

which can be taken as the figure of merit of the device. We defineGon ≡ G(φ = 0, θ = 0)

andGoff ≡ G(φ = 12φ0, θ = 0) [Fig. 5.3(c)]. We use two parameters, the junction spin

polarizationP and the bulk spin splitting∆s to quantify the degree of spin polarization

of the injected carriers. An actual experimental implementation of the transistor concept

described here will require optimization of these or similar parameters. The junction spin

polarizationP is obtained for a given junction geometry, i.e. a given choice of QPC width

W and lengthL [Fig. 5.4(a)], by calculating the transfer matrix [55] of the junction di-

rectly from the TB model and using equation (5.3) withP ≡ |P|. The spin splitting∆s

is obtained from the continuumk · p HgTe QW Hamiltonian mentioned earlier, and is de-

fined as the energy difference between “spin up” (E1+) and “spin down” (E1−) energy

levels [54] at theΓ point. The on/off ratio increases rapidly for a polarization P of order

unity [Fig. 5.4(c)]. It is reasonable to expect that optimized junction designs, better that

the simplistic proof-of-concept geometry used here, wouldyield even higher on/off ratios.

There is also an optimal widthWopt ≃ 0.29Ly for the junction QPC [Fig. 5.4(d)]. For

W < Wopt, interedge tunneling [92] strongly backscatters the incoming electrons and re-

ducesGon, which suppresses the on/off ratio. ForW > Wopt, the edge states on opposite

edges are too far apart to recombine coherently and to produce the desired spin rotation

effect, which increasesGoff and also suppresses the on/off ratio. In our calculation, we

did not take into account the possible structural inversionasymmetry (SIA) which induces


Rashba spin-orbit coupling in the QW [88]. However, it should be noticed that the usual

contributions of SIA to the AB effect, such as the Aharonov-Casher (AC) effect [109], are

absent because there are no 2D bulk carriers in the QSH state.Since the only conducting

channels in the QSH state are the 1D edge states, the only effect of SIA is some global

rotation of the edge state spin direction. The topological spin rotation induced by half of

a flux quantum is simply a consequence of the spatial separation of opposite spins on op-

posite edges, which is determined by the topological properties of the QSH state and thus

remains robust.

5.5 Conclusion and outlook

In this work, we have shown the possibility of using a topologically nontrivial state of

matter, the QSH insulator state, to manipulate the spin of the electron by purely nonlocal,

quantum mechanical means, without recourse to local interactions with classical electro-

magnetic fields. This spin AB effect, which is a spin analog ofthe usual charge AB effect,

relies on the helical and topological nature of the QSH edge states which is peculiar to that

state of matter, combined with a Berry phase effect. In addition, we have shown that the

spin AB effect can be used to design a new kind of spin transistor which is fundamentally

different from the previous proposals, in that there is no classical force or torque acting on

the spin of the electron. Furthermore, edge transport in theQSH regime being dissipation-

less [20, 78, 79], the proposed topological spin transistorwould have the advantage of a

lower power consumption in comparison to previous proposals for spin transistors. More

generally, such a quantum manipulation of the electron spin, if observed, could open new

directions in spintronics research and applications, and would at the same time demonstrate

the practical usefulness of topological states of quantum matter.

A recent paper by Usaj [110] discusses a similar effect in thespin-polarized edge states

of graphene ribbons. We expect our effect to be more robust toexternal perturbations due

to the topological protection of the QSH edge states. Indeed, the helical edge liquid of the

QSH state is a novel state of matter which is topologically distinct [21, 44] from the edge

states of graphene.

Chapter 6

Topological quantization in units of the

fine structure constant

6.1 Introduction

Topological phenomena in condensed matter physics are typically characterized by the

exact quantization of the electromagnetic response in units of fundamental constants. In

a superconductor, the magnetic flux is quantized in units of the flux quantumφ0 ≡ h2e

;

in the QH effect, the Hall conductance is quantized in units of the conductance quantum

G0 ≡ e2

h. Not only are these fundamental physical phenomena, they also provide the most

precise metrological definition of basic physical constants. For instance, the Josephson ef-

fect in superconductivity allows the most precise measurement of the flux quantum which,

combined with the measurement of the quantized Hall conductance, provides the most ac-

curate determination of Planck’s constanth to date [111]. The remarkable observation of

such precise quantization phenomena in these imprecise, macroscopic condensed matter

systems can be understood from the fact that they are described in the low-energy limit

by topological field theories with quantized coefficients. For instance, the QH effect is

described by the topological Chern-Simons theory [9] in2 + 1 dimensions, with coeffi-

cient given by the quantized Hall conductance. Superconductivity can be described by the

topologicalBF theory [112] with coefficient corresponding to the flux quantum.

More recently, a new topological state in condensed matter physics, the TR invariant

75

CHAPTER 6. TOPOLOGICAL QUANTIZATION IN UNITS OFα 76

topological insulator (TI), has been investigated extensively [113, 114, 115, 116]. The

concept of TI can be defined most generally in terms of the topological field theory [16]

with effective Lagrangian

L =1

8π

(

εE2 − 1

µB2

)

+θ

2π

α

2πE ·B, (6.1)

whereE andB are the electromagnetic fields,ε andµ are the dielectric constant and mag-

netic permeability, respectively, andθ is an angular variable known in particle physics as

the axion angle [32]. Under periodic boundary conditions, the partition function and all

physical quantities are invariant under shifts ofθ by any multiple of2π. SinceE ·B is odd

under TR symmetry, the only values ofθ allowed by TR symmetry are0 or π (modulo2π).

The second term of Eq. (6.1) thus defines a topological field theory with coefficient quan-

tized in units of the fine structure constantα ≡ e2

~c. The topological field theory is generally

valid for interacting systems, and describes a quantized magnetoelectric response denoted

topological magnetoelectric effect [16]. The quantization of the axion angleθ depends

only on the TR symmetry and the bulk topology; it is thereforeuniversal and independent

of any material details. More recently, it has been shown [34] that the topological field

theory [16] reduces to the topological band theory [12, 13, 27, 28] in the noninteracting

limit. Interestingly, the topological magnetoelectric effect is the first topological quantiza-

tion phenomenon in units ofα. It can therefore be combined with the two other known

topological phenomena in condensed matter, the QH effect and superconductivity, to pro-

vide a metrological definition of the three basic physical constants,e, h, andc.

The topological magnetoelectric effect has not yet been observed experimentally. An

insight into why this is so can be gained by comparing the3 + 1 dimensional topological

field theory (6.1) of TI to the2 + 1 dimensional Chern-Simons topological field theory of

the QH effect [9]. In2 + 1 dimensions, the topological Chern-Simons term is the only

term which dominates the long-wavelength behavior of the system, which leads to the

universal quantization of the Hall conductance. On the other hand, in3 + 1 dimensions

the topologicalθ-term in Eq. (6.1) and the Maxwell term are equally importantin the

long wavelength limit. Therefore, one has to be careful whendesigning an experiment

to observe the topological quantization of the topologicalmagnetoelectric effect, in which


Figure 6.1: Measurement of Kerr and Faraday angles for a TI thick film of thicknessℓ andoptical constantsε2, µ2 on a topologically trivial insulating substrate with optical constantsε3, µ3, in a perpendicular magnetic fieldB. (We consider normal incidence in the actualproposal but draw light rays with a finite incidence angle in the figure for clarity.) Theexternal magnetic field can be replaced by a thin magnetic coating on both TI surfaces, assuggested in Ref. [16].

the dependence on the non-topological materials constantsε andµ are removed.

6.2 Magnetooptical Kerr and Faraday rotation

In this chapter, we propose an optical experiment to observethe topological quantization

of the topological magnetoelectric effect in units ofα, independent of material properties

of the TIsuch asε andµ. This experiment could be performed on any of the available TI

materials, such as the Bi2Se3, Bi2Te3, Sb2Te3 family or the recently discovered thallium-

based compounds [116]. Consider a TI thick film of thicknessℓ with optical constants

ε2, µ2 and axion angleθ deposited on a topologically trivial insulating substratewith optical

constantsε3, µ3 (Fig. 6.1). The vacuum outside the TI hasε = µ = 1 and trivial axion angle


θvac = 0. The substrate being also topologically trivial, both interfaces atz = 0 andz = ℓ

support a domain wall ofθ giving rise to a surface QH effect with half-quantized surface

Hall conductanceσsH = (n + 1

2) e

2

hwith n ∈ Z [16]. The factor of1

2is a topological

property of the bulk and is protected by the TR symmetry. On the other hand, the value of

n depends on the details of the interface and may thus be different for the two interfaces. To

account for this general case we assignθsubs = 2pπ with p ∈ Z to the topologically trivial

substrate, corresponding toσs,0H = θ

2πe2

hon thez = 0 interface andσs,ℓ

H = (p − θ2π) e

2

hon

thez = ℓ interface. The experiment consists in shining normally incident monochromatic

light with frequencyω on the TI film, and measuring the Kerr angleθK of the reflected

light and Faraday angleθF of the transmitted light. However, the effective theory (6.1)

applies only in the regimeω ≪ Eg/~ whereEg is the surface gap [16]. Such a surface

gap can be opened by a thin magnetic coating on both surfaces of the TI, as first suggested

in Ref. [16], or by an applied perpendicular magnetic fieldB = Bz (Fig. 6.1) through the

surface Zeeman effect as well as the exchange coupling between surface electrons and the

paramagnetic bulk. We discuss the experimentally simpler case of the external magnetic

field. For incident light linearly polarized in thex directionEin = Einx, the Kerr and

Faraday angles are defined bytan θK = Eyr /E

xr andtan θF = Ey

t /Ext , respectively, with

Er = Exr (−x) + Ey

r y andEt = Ext x + Ey

t y the reflected and transmitted electric fields,

respectively (Fig. 6.1). Furthermore,θK andθF are to be measured as a function ofB. The

angles that we discuss in the following are defined as the linear extrapolation ofθK(B) and

θF (B) asB → 0+, in which limit the non-topological bulk contribution to optical rotation

is removed [16].

The problem of optical rotation at a TI/trivial insulator interface has been studied be-

fore [16, 35, 117]. In general,θK andθF depend on the optical constantsε2, µ2 of the TI. In

the thick film geometry considered here, they will also depend in a complicated manner on

the optical constantsε3, µ3 of the substrate, the film thicknessℓ, and the photon frequency

ω, due to multiple reflection effects at the two interfaces (see Sec. A.8). It seems therefore

dubious that one could extract the exact quantization of thetopological magnetoelectric

effect from such a measurement. However, we find that these multiple reflection effects

can be used for a universal measurement of the topological magnetoelectric effect, with no

explicit dependence onε2, µ2, ε3, µ3, ℓ, andω.


Figure 6.2: (a) ReflectivityR as a function of photon frequencyω in units of the character-istic frequencyωℓ for a topological insulator Bi2Se3 thick film on a Si substrate; universalfunctionf(θ) for different values of (b) the substrate dielectric constant ε3, (c) p, the totalsurface Hall conductance in units ofe2

h, and (d) the TI dielectric constantε2. The position of

the zero crossing is universal and provides an experimentaldemonstration of the quantizedtopological magnetoelectric effect.

6.3 Reflectivity minima and total surface Hall conductance

In Fig. 6.2(a) we plot the reflectivityR ≡ |Er|2/|Ein|2 as a function of photon frequencyω

in units of a characteristic frequencyωℓ ≡ c√ε2µ2

πℓ, for ε2 = 100, ε3 = 13, andµ2 = µ3 = 1,

appropriate for a topological Bi2Se3 [22] thin film on a Si substrate [118, 119, 120]. We

observe that minima inR occur whenω/ωℓ is an integer, corresponding toℓ being an

integer multiple ofλ2

2with λ2 = 2πc

ω√ε2µ2

the photon wavelength inside the TI. For radiation

in the terahertz range this corresponds toℓ ∼ 100 µm. Whenω is tuned to any of these


minima, we find

tan θ′K =4αp

Y 23 − 1 + 4α2p2

, tan θ′F =2αp

Y3 + 1, (6.2)

whereYi ≡√

εi/µi is the admittance of regioni, and the prime indicates rotation angles

measured at a reflectivity minimum, i.e. forω/ωℓ ∈ Z. We see thatθ′K andθ′F are indepen-

dent of the TI optical constantsε2, µ2. Equation (6.2) corresponds simply to the results of

Ref. [16, 35] for auniqueinterface with axion domain wall∆θ = 2pπ. Moreover, the two

angles can be combined [121] to obtain a universal result independent of both TIε2, µ2 and

substrateε3, µ3 properties,

cot θ′F + cot θ′K1 + cot2 θ′F

= αp, p ∈ Z. (6.3)

Since the rotation angles are measured at a reflectivity minimum, Eq. (6.3) has no explicit

dependence onℓ or ω either. Equation (6.3) clearly expresses the topological quantiza-

tion in units ofα solely in terms of experimentally measurable quantities, and is the first

important result of this work.

6.4 Reflectivity maxima and topological magnetoelectric

effect

However, neither Eq. (6.2) nor Eq. (6.3) depend explicitly on the TI axion angleθ, and

one may ask whether Eq. (6.3) is at all an indication of nontrivial bulk topology. In fact,

Eq. (6.3) describes the topological quantization of thetotal Hall conductance of both sur-

facesσs,totH = σs,0

H + σs,ℓH = p e2

h, which holds independently of possibleT breaking in the

bulk. In the special case that the two surfaces have the same surface Hall conductance, we

havep = 2σs,0H = θ

πand Eq. (6.3) is sufficient to determine the bulk axion angleθ. How-

ever, for a TI film on a substrate the two surfaces are generically different and can have

different Hall conductance. To obtain the axion angleθ in the more general case of dif-

ferent surfaces, we propose another optical measurement performed at reflectivitymaxima

ω = (n + 12)ωl, n ∈ Z [Fig. 6.2(a)]. We denote byθ′′K andθ′′F the Kerr and Faraday angles


measured at an arbitrary reflectivity maximum. In contrast to θ′K andθ′F [Eq. (6.2)], these

depend onε2, µ2 as well as onε3, µ3,

tan θ′′K =4α[

Y 22

(

p− θ2π

)

− Y 23

θ2π

]

Y 23 − Y 4

2 + 4α2[

2Y 22

θ2π

(

p− θ2π

)

− Y 23

(

θ2π

)2] ,

tan θ′′F =2α(

p− θ2π

+ Y3θ2π

)

Y3 + Y 22 − 4α2 θ

2π

(

p− θ2π

) , (6.4)

where we defineY 23 = Y 2

3 +4α2(

p− θ2π

)2. More importantly,θ′′K andθ′′F depend explicitly

on the TI axion angleθ. It is readily checked that Eq. (6.4) reduces to Eq. (6.2) in the single-

interface limitθ = 2pπ, Y2 = Y3 or θ = 0, Y2 = 1. In general however, from the knowledge

of p [Eq. (6.3)] and eitherθ′K or θ′F we can extractY3 by using Eq. (6.2) without performing

any separate measurement. Moreover,θ′′K andθ′′F can be combined to cancel the explicit

dependence on the TI propertiesε2, µ2. We solve forY 22 in Eq. (6.4) in terms ofθ′′F , say,

and substitute the resulting expressionY 22 = Y 2

2 (θ) into the equation forθ′′K in Eq. (6.4).

The result can be expressed in the formf(θ′K , θ′F , θ

′′K , θ

′′F ; p, θ) = 0 wheref is “universal”

in the sense that it does not depend explicitly on any material parameterεi, µi. Substituting

the experimental values ofθ′K , θ′F , θ

′′K , θ

′′F andp into this expression, we obtain a function

of a single variablef(θ). If we plot f as a function ofθ, the zero crossingf(θ) = 0 gives

the value of the bulk axion angleθ with no2π ambiguity. Plots of the universal functionf

are given in Fig. 6.2(b), (c), and (d) for different values ofthe material parametersε2, ε3, p

(settingµ2 = µ3 = 1 without loss of generality) and for a bulk axion angleθ = π. The

zero crossing point is independent of material parameters and, together with Eq. (6.3),

can provide a universal experimental demonstration of the quantization of the topological

magnetoelectric effect in the TI bulk. In athin film geometryℓ ≪ λ2

2corresponding to

ω ≪ ωℓ, the optical response is always given by the sum of the Hall conductivities of the

two surfaces. Therefore, thick filmsℓ ≥ λ2

4to allow destructive interference and reflectivity

maxima are essential to the measurement of the bulk topological magnetoelectric effect.


Figure 6.3: (a) Kerr-only measurement setup, with materialparameters the same as indi-cated in Fig. 6.1; (b), (c) and (d): universal functionfK(θ) for different material parameters[same as in Fig. 6.2(b), (c), (d)]. As in Fig. 6.2, the position of the zero crossing is universaland provides an experimental demonstration of the quantized topological magnetoelectriceffect.

6.5 Kerr-only measurements

Our proposal so far necessitates the measurement of both Kerr and Faraday angles. We now

show that it is possible to extractp andθ from Kerr measurements alone, if the Kerr angle

is measured in both directions [Fig. 6.3(a)]. Indeed, whilethe Faraday angle is generally

independent of the direction of propagation [122], the Kerrangle depends on it. Here we

exploit this asymmetry of the Kerr angle to extractp andθ. We denote byθ′13K andθ′′13K the

Kerr angles defined previously in Eq. (6.2) and (6.4), respectively. Conversely, we denote

by θ′31K andθ′′31K the Kerr angles for light traveling in the opposite direction, i.e. incident

from the substrate [Fig. 6.3(a)]. As before, the prime and double prime correspond to


angles measured at reflectivity minima and maxima, respectively. We find

tan θ′31K = − 4αpY3Y 23 − 1 + 4α2p2

, (6.5)

tan θ′′31K =4αY3

[

Y 22

θ2π

− γ(

p− θ2π

)]

γY 23 + 4γα2

[

p2 −(

θ2π

)2]

− Y 42 − 8α2Y 2

2

(

θ2π

)2,

where we defineγ ≡ 1 + 4α2(

θ2π

)2. As previously,θ′13K and θ′31K can be combined to

eliminateY3 and provide a universal measure ofp ∈ Z,

cot θ′13K − sgn p√

1 + cot2 θ′13K (1− tan2 θ′31K ) = 2αp, (6.6)

providedY 23 ≡ ε3/µ3 > 1 + 4α2p2, which is satisfied in practice for lowp sinceα2 ∼

10−4. Furthermore, comparing Eq. (6.5) forθ′31K to Eq. (6.2) forθ′13K we see thatY3 is

easily obtained asY3 = − cot θ′13K tan θ′31K . Finally, to extract the bulk axion angleθ, we

need to solve forY 22 in Eq. (6.4) in terms ofθ′′13K , and substitute the resulting expression

Y 22 = Y 2

2 (θ) into the equation forθ′′31K in Eq. (6.5). The result of this analysis can once

again be expressed in the formfK(θ′13K , θ′31K , θ′′13K , θ′′31K ; p, θ) = 0, wherefK is a “universal”

function which only depends on the measured Kerr angles. As before, we substitute intofK

the experimental values ofθ′13K , θ′31K , θ′′13K , θ′′31K andp [obtained from Eq. (6.6)] and obtain

a function of a single variablefK(θ) which crosses zero at the value of the bulk axion

angle with no2π ambiguity. In Fig. 6.3(b), (c) and (d) we plot the universal functionfK

for different values of the material parametersε2, ε3, p and for a bulk axion angleθ = π.

The zero crossing point is independent of material parameters and, together with Eq. (6.6),

provides another means to demonstrate experimentally the universal quantization of the

topological magnetoelectric effect in the bulk of a TI.

6.6 Discussion

Recent work [36] has addressed the similar problem of optical rotation on a TI film, and

found interesting and novel results for the rotation angles. However, these results hold only

in certain limits which are less general than the ones discussed in this work. First, Ref. [36]


considers a free-standing TI film in vacuum. Most films are grown on a substrate which can

affect the physics qualitatively. For instance, the giant Kerr rotationθK = tan−1(1/α) ≃π/2 found in Ref. [36] is a special case of our Eq. (6.2) withp = 1 andε3/µ3 = 1. It is

dramatically suppressed whenε3/µ3 − 1 is greater thanα2 ∼ 10−4, which is typically the

case in practice. Second, in Ref. [36] a correction proportional to∆/ǫc was introduced to

the surface Hall conductance, where∆ is the TR symmetry-breaking Dirac mass andǫc is

a non-universal high-energy cutoff. According to the general bulk topological field theory

of the TI [16], the surface Hall conductance is always quantized as long as the surface is

gapped and the bulk is TR invariant (in theB → 0 limit). Thus we conclude that such a

non-universal correction is absent and the requirement∆ ≪ ǫc is not necessary within the

topological field theory approach [16]. This difference clearly demonstrates the power of

the topological field theory approach [16] in predicting universally quantized topological

effects in condensed matter physics.

Chapter 7

Fractional topological insulators in three

dimensions

7.1 Introduction

Most states of quantum matter are classified by the symmetries they break. However, topo-

logical states of quantum matter [113] evade traditional symmetry-breaking classification

schemes, and are rather described by topological field theories in the low-energy limit. For

the quantum Hall effect, the topological field theory is the2+1 dimensional Chern-Simons

theory [9] with coefficient given by the quantized Hall conductance. In the noninteracting

limit, the integer QH [2] conductance in units ofe2

his given by the TKNN invariant [6] or

first Chern number. In the presence of strong correlations, one can also observe the frac-

tional QH effect [3, 123], where the Hall conductance is quantized in rational multiples ofe2

h. In both cases however, these topological states can exist only in a strong magnetic field

which breaks TR symmetry.

More recently, TR invariant TI have been studied extensively [113, 114, 115]. The

TI state was first predicted theoretically in HgTe QW, and observed experimentally [19,

20, 12, 14] soon after. The theory of TI has been developed along two independent routes.

Topological band theory identifiedZ2 topological invariants for noninteracting band insula-

tors [12, 27, 28]. The topological field theory of TR invariant insulators was first developed

in 4 + 1 dimensions, where the Chern-Simons term is naturally TR invariant [124, 125].

85

CHAPTER 7. FRACTIONAL TOPOLOGICAL INSULATORS IN 3D 86

Dimensional reduction then gives the topological field theory for TI in 3 + 1 and2 + 1

dimensions [16]. The topological field theory is generally valid for interacting systems,

and describes the experimentally measurable quantized magnetoelectric response. The co-

efficient of the topological term, the axion angleθ, is constrained to be either0 or π by TR

invariance. The topological field theory has been further developed in Ref. [126, 35]. More

recently, it has been shown that it reduces to the topological band theory in the noninteract-

ing limit [34].

By analogy with the relation between the integer QH effect and fractional QH effect,

one is naturally led to the question whether there can exist a“fractional TI”. In 2+1 dimen-

sions, an explicit wave function for the fractional QSH state was first proposed in Ref. [14],

and the edge theory was investigated in Ref. [127]. The TR invariant fractional topological

state has also been constructed explicitly in4 + 1 dimensions [124]. Since TR invariant

TI form a dimensional ladder in4, 3 and2 dimensions [16, 128, 129], it is natural to in-

vestigate the TR invariant TI in3 + 1 dimensions. Fractional states generally arise from

strong interactions. Since topological band theory cannotdescribe such interactions, we

formulate the general theory in terms of the topological field theory. The TI is generally

described by the effective actionSθ = θ2π

e2

2π

∫

d3x dtE · B whereE andB are the elec-

tromagnetic fields [16]. Under periodic boundary conditions, the partition function and all

physical quantities are invariant under shifts ofθ by multiples of2π. SinceE · B is odd

under TR, it appears that the only values ofθ allowed by TR symmetry are0 or π mod2π.

In this paper, we show that there exist TR invariant insulating states in3+1 dimensions

with P3 ≡ θ2π

quantized in non-integer, rational multiples of12

of the formP3 = 12pq

with

p, q odd integers. The magnetoelectric polarizationP3 is defined by the response equation

P = − B

2π(P3 + const.), whereB is an applied magnetic field andP is the induced electric

polarization. Such a fractionalized bulk topological quantum number leads to a fractional

QH conductance ofpqe2

2hon the surface of the fractional TI. In contrast to the usual QH

effect in2 + 1 dimensions, the surface QH effect does not necessarily exhibit edge states

and thus cannot be directly probed by transport measurements. Alternatively, it can only

be experimentally observed through probes which couple to each surface separately, such

as magnetooptical Kerr and Faraday rotation [16, 35]. Generically, a slab of fractional TI

can have different fractional Hall conductance on the top and bottom surfaces, which can


be determined separately by combined Kerr and Faraday measurements, independent of

non-universal properties of the material [37].

7.2 Projective construction and trial wave function

Our approach is inspired by the composite particle, or projective construction of fractional

QH states [9, 130, 131, 132, 133, 134, 135, 136, 137]. The ideais to decompose the

electron with chargee into N fractionally charged, fermionic “partons”, which have a dy-

namics of their own. One considers the case that the partons form a known topological

state, say a topological band insulator. When the partons are recombined to form the phys-

ical electrons, a new topological state of electrons emerges. In the fractional QH case for

example, theν = 13

Laughlin state can be obtained by splitting each electron into N = 3

partons of chargee3. Each parton fills the lowest Landau level and forms a noninteracting

ν = 1 integer QH state. Ignoring the exponential factors, the parton wave function is the

Slater determinant integer QH wave functionΨ(zi) ∝ ∏

i<j (zi − zj), and the electron

wave function is obtained by gluing three partons together,which leads to the Laughlin

wave function [123]Ψ1/3(zi) ∝∏

i<j (zi − zj)3. Similarly, in 3+1 dimensions one can

construct an interacting many-body wave function by gluingpartons which are in aZ2

topological band insulator state. The parton ground state wave functionΨ1(rnsn) is a

Slater determinant describing the ground state of a noninteracting TI Hamiltonian such as

the lattice Dirac model [16], withrnsn, n = 1, . . . , N the position and spin coordinates

of the partons. The electron wave function is obtained by requiring the coordinates of all

Nc partons forming the same electron to be the same [130],

ΨNc(rnsn) = [Ψ1(rnsn)]Nc . (7.1)

Equation (7.1) is the(3+1)-dimensional generalization of the Laughlin wave function, and

serves as a trial wave function for the simplest fractional TI phases we propose.

More generally, we can considerNf different “flavors” of partons, withN (f)c partons

of each flavorf = 1, . . . ,Nf . This decomposition has to satisfy two basic rules. First, to

preserve the fermionic nature of the electron, the total number of partons per electron must


be odd (Fig. 7.1a),

N (1)c +N (2)

c + · · ·+N (Nf )c = odd. (7.2)

Second, ifqf < e is the (fractional) charge of partons of flavorf , the total charge of the

partons must add up to the electron chargee,

N (1)c q1 +N (2)

c q2 + · · ·+N (Nf )c qNf

= e. (7.3)

For instance, theν = 13

Laughlin state described above corresponds toNf = 1, N (1)c = 3,

andq1 = e3, which satisfies both conditions. Here we consider that partons of each flavor

f condense in a (generally different) noninteracting TR invariant TI state with axion angle

θf = π mod2π. This is the analog of having partons condense in various integer QH states

in the fractional QH construction. Finally, the partons have to be bound together to yield

physical electrons. As we will see, this can be done by coupling partons of flavorf to a

SU(N (f)c ) gauge field, which can be interpreted as a “color field” where partons of flavor

f come inN (f)c colors. Since the TI analog of theν = 1

3Laughlin state will involve three

partons coupled to aSU(3) gauge field in3 + 1 dimensions, we dub our partons “quarks”

by analogy with quantum chromodynamics (QCD).

7.3 Effective gauge theory

To obtain a more systematical understanding of the fractional TI, we now deduce its effec-

tive gauge theory by way of a gedanken experiment. We consider subjecting a noninteract-

ing TI to strong electron-electron interactions, and startwith the simplest case ofNf = 1

with Nc odd. The electron being split intoNc quarks of chargeeNc

, the electron operator

will be written as a product ofNc quark operatorsψiα, i = 1, . . . ,Nc. However, the quark

operators act in a Hilbert space which is larger than the physical electron Hilbert space.

We need to remove those states of the quark Hilbert space which are not invariant under

unitary transformations which leave the electron operatorunchanged, i.e.SU(Nc) trans-

formations with quarks in theNc representation. The projection onto the electron Hilbert


Figure 7.1: a) Quark picture of fractional TI with flavor and color degrees of freedom; b)surface fractional QH effect vs transport measurements [Eq. (7.7)]; c) nontrivial vs trivialfractional TI; d) Witten effect as a probe of bulk topology.

space can therefore be implemented by coupling the quarks toa SU(Nc) gauge fieldaµ

with a coupling constantg. Outside the fractional TI, we expect the system to be in the

confined phase, in analogy to quark confinement in QCD, which has onlySU(Nc) singlet

excitations in its low-energy spectrum, i.e. gauge-invariant “baryons”. Quarks of a given

flavor within the baryon are antisymmetric in theirN (f)c color indices; Fermi statistics then

implies that their spins are aligned. In a relativistic theory this would imply that in the

Nf = 1 theory the baryon has spinNc

2. In nonrelativistic lattice models this is not a con-

cern, but even within the context of relativistic continuumfield theories one can obtain

composite spin-12

electrons forNf > 1.

Inside the fractional TI, electron-electron interactionstranslate into complicated in-

teractions among quarks. We consider the case that these interactions lead the quarks to

condense at low energies into a noninteracting TR invariantTI state with axion angleθ, and

that the non-Abelian gauge fieldaµ enters a deconfined phase1. We now show that such a

phase is a fractional TI. A low-energy effective Lagrangianfor Nf = 1 can be conjectured

1This can be achieved either by adding additional colored butelectrically neutral matter or consideringthe special case with only Abelian groups,N (f)

c = 1 for all f .


in the form

L = ψ† (iD0 −Hθ[−iD])ψ + Lint(ψ†, ψ), (7.4)

whereDµ = (D0,−D) = ∂µ + i eNcAµ + igaµ is theU(1)em × SU(Nc) gauge covari-

ant derivative, andHθ = Hθ(p) is the single-particle Hamiltonian for a TR invariant TI

with axion angleθ. Lint represents weak TR invariant residual interactions which do not

destabilize the gapped TI phase, and can thus be safely ignored. The kinetic Yang-Mills

Lagrangian foraµ is generally present but not explicitly written.

Since the quarks are in a gapped TI phase, they can be integrated out to yield an effective

Lagrangian for the gauge fields [16],

Leff =θ

32π2ǫµνλρ tr

(

e

NcFµν + gfµν

)(

e

NcFλρ + gfλρ

)

=θeffe

2

32π2ǫµνλρFµνFλρ +

θg2

32π2ǫµνλρ tr fµνfλρ, (7.5)

wheretr is the trace in theNc representation ofSU(Nc), Fµν = ∂µAν − ∂νAµ andfµν =

∂µaν − ∂νaµ + ig[aµ, aν ] are theU(1)em andSU(Nc) field strengths, respectively, and the

electromagnetic response is governed by an effective axionangle

θeff =θ

Nc

= 0,± π

Nc

,±3π

Nc

,±5π

Nc

, . . . , Nc odd. (7.6)

Equation (7.5) is obtained by replacing theU(1)em “electron” field strengtheFµν in the

U(1)em topological term θ2π

e2

2πE · B = θe2

32π2 ǫµνλρFµνFλρ for noninteracting TI [16] by the

totalU(1)em×SU(Nc) “quark” field strength eNcFµν+gfµν . The crossed terms of the form

trFµνfλρ vanish due to the tracelessness of theSU(Nc) gauge field. In general,θeff can be

obtained from the Adler-Bell-Jackiw anomaly [138, 139], sinceθ corresponds to the phase

of the quark mass [16]. The effective theory can also be obtained for quarks in a trivial

insulator state withθ = 2nπ, n ∈ Z. However, such a state is adiabatically connected to

a θ = 0 vacuum and is a trivial insulator in the bulk, although a fractional θeff can still be

obtained due to pure surface effects. Since the focus of the present work is a fractional TI

state with nontrivial bulk, we always consider quarks withθ = π mod 2π in the following.


7.4 Time-reversal symmetry and quantization of the ax-

ion angle

We are now faced with our initial question of whether the effective theory (7.5),(7.6) breaks

TR invariance. According to the first term in Eq. (7.5), TR invariance would requireθeff to

be quantized in integer multiples ofπ if the minimal electric charge wase [140]. However,

the minimal charge in our theory iseNc, i.e. that of the quarks. Therefore,θeff has to be

quantized in integer multiples ofπN 2c. On the other hand, the second term in Eq. (7.5)

requiresθ to be quantized2 in integer multiples ofπ, which means by Eq. (7.6) thatθeff has

to be quantized in units ofπNc. This latter constraint is consistent with, but stronger than,

the former3, and the values ofθeff allowed by TR invariance are thus correctly given by

Eq. (7.6).

Equations (7.5) and (7.6) constitute a topological field theory which, precisely because

it is topological, is insensitive to small TR invariant perturbations and defines a new stable

phase of matter, the TR invariant fractional TI in3 + 1 dimensions. The effective theory

can also be derived in the multi-flavor caseNf ≥ 1, with N (f)c satisfying rules (7.2) and

(7.3). Considering that quarks of flavorf form a noninteracting TI with axion angleθf =

π mod 2π and integrating them out yields an effective Lagrangian in the form of (7.5),

but with gauge groupU(1)em×∏Nf

f=1 U(N(f)c )/U(1)diag. HereU(1)diag is the overallU(1)

gauge transformation of the electron operator. The electromagnetic axion angleθeff is given

by θeff =(

∑Nf

f=1N (f)

c

θf

)−1

. When θfπ

is odd for each flavor, one can show thatθeff = πp/q

with p, q odd integers.

2For a recent discussion, see Ref. [141, 142].3Color neutral monopoles carryU(1)em magnetic chargeNc

e , and lead toθeff = nπN 2

c

, n ∈ Z. However,

monopoles with smallerU(1)em magnetic charge1e are allowed but also carry color magnetic charge [143],and lead toθeff = nπ

Nc

, n ∈ Z.


7.5 Bulk topology and surface “half” fractional quantum

Hall effect

Important physical properties of the fractional TI can be read off from Eq. (7.5). The

surface of the fractional TI is an axion domain wall with theU(1)em axion angle jumping

from θeff in the fractional TI to0 in the vacuum. Such a domain wall has a surface QH effect

with surface Hall conductanceσH,s =θeff2π

e2

h[16]. Therefore, the surface Hall conductance

of the fractional TI has the general form

σH,s =p

q

e2

2h, p, q odd. (7.7)

For example, in the simplest single-flavor case withθ = π in Eq. (7.5), we haveσH,s =1Nc

e2

2hwith Nc an odd integer, corresponding to half of aν = 1

Ncfractional QH Laughlin

state. The more general result (7.7) corresponds to half of ageneric Abelian fractional QH

state [144, 145, 130].

The fractional axion angle and the associated surface Hall conductance (7.7) are prop-

erties of the bulk topology. It is important to distinguish them from a TI withθeff = ±πand where the surface Dirac fermions form a fractional QH state [146]. In a noninteracting

TI with θeff = −π for example, both the axion domain wall and the surface fractional QH

state contribute toσH,s,

σH,s =

(

−1

2+n

q

)

e2

h=

2n− q

q

e2

2h, (7.8)

with nq

an allowed filling fraction for a fractional QH state in2+1 dimensions. For Abelian

fractional QH statesq is odd, hence the surface Hall conductanceσH,s = 2n−qq

e2

2hhas the

same general form as for the fractional TI [Eq. (7.7)]. As thesimplest example, the Laugh-

lin state with nq= 1

3leads toσH,s = −1

6e2

h(Fig. 7.1c, right) which is the same as for a

genuine fractional TI with bulkP3 = −16

(Fig. 7.1c, left). However, the bulk topology is

very different in both cases. Therefore, surface measurements are not sufficient to deter-

mine the bulk topology and bulk measurements ofP3 are needed. One such measurement

would consist in embedding a monopole with magnetic chargeqm inside the fractional TI


(Fig. 7.1d) and measuring its electric chargeqe induced by the Witten effect [147, 38, 148].

Another possible “experiment” is to measure the ground state degeneracy on topolog-

ically nontrivial spatial 3-manifolds. Consider a fractional TI on a manifoldΣg × I with

Σg a Riemann surface of genusg andI = [0, L] a bounded interval, whereL is the sam-

ple “thickness” and the two copies ofΣg (at each end ofI) are the two bounding sur-

faces. We first discuss contributions to the ground state degeneracy arising solely from the

boundary, and comment on bulk contributions later on. A noninteracting TI with aν = 13

Laughlin state deposited on both surfaces is described by two independent Chern-Simons

theories [9, 149, 150, 151] and has a ground state degeneracyof 3g (mg for ν = 1m

) on

each surface for a total ground state degeneracy of(3g)2 = 32g. The situation is different

for a genuine fractional TI withP3 = ±16

(Fig. 7.1c, left). To study the ground state de-

generacy we set the external electromagnetic fields to zero in Eq. (7.5) and consider the

internalSU(3) θ-term. Assuming that the system stays gapped as we take the limit of

zero thicknessL → 0 where the gauge fieldsaµ on both surfaces become identified, the

system is described by a singleSU(3)k Chern-Simons theory onΣg where the levelk is

the sum of the contributions from both surfaces. If on both surfacesθ goes to the same

value outside the TI, thenk = 0 and the ground state is unique. Ifθ = 0 on one side and

θ = 2π on the other, we have aSU(3)1 Chern-Simons theory with ground state degeneracy

3g 6= 32g [150, 151]. In addition to the boundary contributions of theθ-term to the ground

state degeneracy, the gauge theory in the bulk can have a nontrivial ground state degeneracy

even in the absence of boundaries. For instance, the deconfined phase ofSU(Nc) gauge

theory has a ground state degeneracy ofN 3c onT 3 [152]. The total ground state degener-

acy has in general both bulk and boundary contributions, anddepends on the details of the

gauge group.

7.6 Three classes of parton models

Topological insulators [115, 116] are new states of quantummatter that cannot be adiabat-

ically connected to conventional insulators. They are fully gapped in the bulk but support

gapless boundary modes which are protected by discrete symmetries [128, 153, 129, 154].

Three-dimensional TR invariant TI [27, 29] have attracted agreat deal of attention in the


past few years, for the most part due to their theoretical prediction[30, 22] and subsequent

experimental detection [31, 24, 25] in Bi-based semiconductors, the latter through the ob-

servation of their protected helical surface states. The existence and stability of these sur-

face states is protected by a bulkZ2 topological invariant which corresponds physically to a

quantized magnetoelectric effect [16, 126]. The experimental observation of the quantized

magnetoelectric effect, for instance by way of magnetooptical measurements,[16, 36, 37]

is a key goal in the field which is being actively pursued [118,155, 156]. From a theo-

retical standpoint [16], the quantized magnetoelectric effect is described at energies much

smaller than the energy gap by the addition of a term proportional toE · B (with E and

B the electric and magnetic fields, respectively) to the usualLagrangian for Maxwell elec-

trodynamics, i.e. axion electrodynamics [32]. This term isin fact the Abelian version of

the topologicalθ-term in QCD [157, 158], and its coefficientθ is periodic (under certain

conditions [159]) with period2π. SinceE ·B is odd under TR symmetry, the only values

of θ allowed by TR symmetry are0 or π mod 2π, with θ = 0 for theZ2 trivial insula-

tor andθ = π for theZ2 nontrivial insulator [16]. TI can be described microscopically

by noninteracting, spin-orbit coupled electrons hopping on a lattice, and the axion angle

θ can be computed from a knowledge of the single-particle wavefunctions in momentum

space [16, 126].

The effect of electron-electron interactions on the surface modes of 3D TI has been a

topic of considerable current interest. Weak interactionscan either turn the noninteracting

helical surface state into a weakly interacting helical Fermi liquid with spin-charge coupled

collective modes [160], or drive a transition to a superconducting state [161, 162]. Strong

enough interactions can lead to spontaneous TR symmetry-breaking on the surface and fer-

romagnetic [163, 164, 165, 166] or helical spin density waveorder [167]. In contrast, the

bulk of a TI is fully gapped and thus expected to be perturbatively stable to interactions. On

the other hand, exotic states known as 3D topological Mott insulators [168, 169, 170] have

been theoretically proposed, whereas a topologically nontrivial bandstructure is dynami-

cally generated as a consequence of strong electron-electron interactions. Although these

are strongly interacting states, their mean-field description is still that of a topological band

insulator, and the axion angleθ remains quantized [171, 172] to0 or π mod2π. Another

type of topological Mott insulator has been theoretically proposed [173, 174] in which


spin-charge separation leads to a bulk insulator with a helical liquid of gapless spinons, but

an electromagneticθ-term is not generated because the spinons are electricallyneutral.

In many regards, the 3D TR invariant TI can be viewed as a generalization of the 2D

integer QH effect to 3D. The topologicalZ2 quantization of the bulk axion angleθ in 3D is

the direct analog of the topologicalZ quantization of the bulk Hall conductance in 2D. By

analogy with the relation between the integer QH effect and the fractional QH effect, one is

naturally led to the question whether there can exist a “fractional 3D TI” which preserves

TR symmetry but is characterized by afractional quantized axion angle, i.e. whereθ is

a non-integer, rational multiple ofπ. In 2D and in the case thatSz (the z component

of the electron spin) is conserved, recent theoretical work[14, 127, 175, 176] has shown

that a TR invariant fractional QSH insulator can in principle exist. In this special case of

conservedSz, the QSH insulator is equivalent to two decoupled integer QHsystems with

equal and opposite effective magnetic fields. Both integer QH systems can be driven into

fracitonal QH states by adding electron-electron interactions and appropriately tuning the

effective magnetic fields while keeping them equal and opposite, which yields a fractional

QSH insulator without breaking TR symmetry. On the other hand, this procedure is not

directly applicable to 3D case, where one cannot in general reduce a TR invariant TI to two

decoupled topological states which break TR symmetry in an equal and opposite way4.

In our previous work [180], we introduced a theory of fractional TI in 3D based on a

parton construction [130, 131, 132, 133, 134, 135]. Simply postulating that an electron, un-

der the influence of strong interactions in the underlying lattice Hamiltonian, fractionalizes

into Nc “colors” of partons gives a realization of a fractional TI, as long as each color of

partons forms a topological band insulator. In order to ensure that outside the fractional TI

the partons recombine into electrons we needed to add additional degrees of freedom, anal-

ogous to the statistical gauge fields in the theory of the fractional QH effect [9]. Outside

the TI these gauge fields are confining. As a consequence, the partons, which are charged

4Such a reduction is possible for weak TI [27] which essentially consist of stacked QSH states, inwhich case the two decoupled TR symmetry-breaking topological states would correspond to layered 3DQH phases [177, 178, 179]. We are interested in strong TI [27]which are genuinely 3D. Although in thiscase there is no obvious reduction to decoupled TR symmetry-breaking states, recent work [137] on novel“bulk” 3D fractional QH phases may provide a useful startingpoint.


under the “statistical” gauge fields, can never be observed in isolation but are instead con-

fined into electrons, just like quarks are confined into mesons and baryons in QCD. The

theory of the partons together with the statistical gauge fields should be thought of as a

low-energy effective description of the system. The goal ofthis paper is to show that con-

sistent low-energy theories with the characteristic properties of a fractional TI exist and to

analyze their properties.

There are many choices and questions associated with the additional statistical gauge

fields. Are they Abelian or non-Abelian in nature? By construction they should confine out-

side the fractional TI, but what phase do they realize insidethe fractional TI? In Ref. [180]

we mostly focused on the case of a deconfined gauge field. It is easy to write down decon-

fined models and to demonstrate the theoretical consistencyof fractional TI in principle.

Such models can either be Abelian or non-Abelian. Examples of both types were presented

in our earlier work. One generic feature that all deconfined realizations of fractional TI

share is the presence of additional gapless degrees of freedom. One typically considers

TI phases that are fully gapped in the bulk. In the case of a deconfined realization of a

fractional TI we have to slightly generalize this understanding by demanding that all de-

grees of freedom charged under electromagnetism are gapped, while allowing for gapless

electrically neutral degrees of freedom. In that case, the system is indeed an insulator as

far as electrical transport is concerned. These additionalneutral gapless degrees of free-

dom play a role similar to the low-energy phonons in a band insulator, and do not spoil the

quantization of the electromagnetic response.

An alternative to the deconfined realization of the fractional TI is to put the statistical

gauge fields in a Higgs phase. We will present an explicit model of this kind in this work.

In this case the system is truly gapped with no gapless degrees of freedom. The statistical

gauge group is broken down to a discrete gauge group; the latter is sufficient to ensure that

in physical states the net electron number is integer.

As we will elaborate in more detail in this work, the basic topological feature of the

fractional TI, a fractionalθ angle, is completely robust in that it only depends on the total

number of partons and not on any of the details of the statistical gauge sector — whether

it is Abelian or non-Abelian, deconfined or Higgsed. This universalθ angle characterizes

the quantized magnetoelectric effect in the material together with its physically measurable


surface properties, such as a surface fractional QH effect with half the conductivity of a

typical Laughlin state per surface as well as the corresponding magnetooptical Kerr and

Faraday effects.

There is, however, a second topological feature of the fractional TI thatdoesdepend

on details of the gauge field sector: the ground state degeneracy. Some basic features of

the ground state degeneracy have already been discussed in Ref. [180], but here we will

elaborate on this. It has recently been proven [181] that fora gapped system,θ can only

be fractional if the ground state on the3-torusT 3 is degenerate. This observation makes

it clear that a confined phase is not an option for the fractional TI, as it would result in a

unique ground state in a completely gapped system. In the case of a deconfined realization

of the fractional TI, we will find that the ground state in the Abelian models is unique,

whereas in the non-Abelian models it is typically degenerate. The reason why this is not

in contradiction with the theorem of Ref. [181] can be tracedto the fact that we have

additional gapless degrees of freedom, therefore violating the assumption of Ref. [181]

that the system is gapped. We will demonstrate explicitly how this allows us to avoid the

arguments of Ref. [181]. In the Higgs models, the ground state is degenerate as the theorem

just mentioned requires, and we determine the ground state degeneracy.

We illustrate our findings with the study of three examples. All three models correspond

toNc = 3 partons. Model A is an Abelian, deconfined realization of a fractional TI based

on aU(1)× U(1) statistical gauge field. Model B is a non-Abelian, deconfinedrealization

based onSU(3). Model C is a Higgsed realization where the gauge group of model B is

broken down toZ3.

The structure of the paper is as follows. In Sec. 7.6, we introduce three general classes

of parton-gauge boson effective theories: the AbelianU(1) deconfined models, the non-

Abelian deconfined models, and the Higgs models. We illustrate the general ideas in each

case by a specific example of three colors of partons (Nc = 3), corresponding to three

classes of fractional TI but all with a fractional axion angle θ = π/3. In Sec. 7.7, we

prove the topological quantization of the axion angleθ in fractional multiples ofπ using

the chiral anomaly. This derivation allows us to derive the quantization of the axion angle

in both Abelian and non-Abelian models, whether gapless or gapped. In Sec. 7.8, we

discuss the issue of ground state degeneracy on spatial3-manifolds of nontrivial topology.


field U(1)em U(1)A U(1)Bψ1 e/3 2g −gψ2 e/3 −g 2gψ3 e/3 −g −g

Table 7.1: Gauge charge assignments of parton fields under electromagneticU(1)em andstatisticalU(1)2 = U(1)A×U(1)B gauge groups in the simplest Abelian model forNc = 3.

This is the direct analog of the topological degeneracy in fractional QH states on Riemann

surfaces [149, 150, 151], and can be taken as an indication that fractional TI states exhibit

topological order in the many-body sense. In Sec. 7.9 we briefly speculate on the nature of

the gapless surface states, and summarize the paper in Sec. 7.10.

In this section, we introduce the three basic classes of possible statistical gauge sectors

for an fractional TI, and describe in more detail a specificNc = 3 model in each class.

7.6.1 Deconfined models

As described earlier, one way to realize a fractional TI is todrive the statistical gauge

fields into a deconfined phase. This can be realized either in an Abelian or a non-Abelian

setting with rather different properties. What is common toany deconfined realization of a

fractional TI is the appearance of extra gapless matter. This extra gapless matter is neutral

from the point of view of theU(1)em Maxwell gauge field5. Therefore, the system is still an

insulator, i.e. all degrees of freedom charged underU(1)em are gapped. These additional

gapless degrees of freedom can be considered as soft “phonons”, since they do not enter

the electrically charged sector.

Abelian models

Model A: In the simplest Abelian model of a fractional TI withNc = 3, the electron frac-

tionalizes into three fermionic partonsψi, i = 1, 2, 3 with a statisticalU(1)2 = U(1)A ×5The emergent gapless “photon” arising in the layered 3D fractional QH states considered in Ref. [137]

is a direct example of this.


U(1)B gauge group in addition to theU(1)em Maxwell gauge group. The charge assign-

ments are given in Table 7.1, wheree is the electromagnetic gauge charge andg is the

U(1)2 “statistical” gauge charge. These gauge groups ensure thatthe only gauge-invariant

operator that carries Maxwell charge is the product of the three parton operators, hence

the gauge-invariant electron operator isψ1ψ2ψ3. Indeed, the generators of the weight lat-

tice (see Sec. 7.7.2) corresponding to the representation of Table 7.1 aree1 = (2g,−g),e2 = (−g, 2g), e3 = (−g,−g), and the equation

∑

i niei = 0 requiring an operator

ψn11 ψ

n22 ψ

n33 to be gauge-invariant has the one-parameter family of solutionsn1 = n2 = n3,

i.e. operators of the form(ψ1ψ2ψ3)n of which onlyn = 1 corresponds to an operator with

electromagnetic chargee. For a TR invariant system, all three partons have a real mass. As

we will review in the next section, in the topologically nontrivial phase all three partons

have a real but negative mass.

This model is the simplest of theU(1)Nc/U(1)diag Abelian models described in

Ref. [180]. The generic model starts out with a statisticalU(1)Nc gauge group andNc

partons. Each parton carries chargee/Nc under the MaxwellU(1)em gauge group. Theith

parton carries chargeg under theithU(1) factor of the statistical gauge group, and is neutral

under the remainingU(1) factors. One then takes the quotient of the statistical gauge group

by its diagonalU(1)diag subgroup whose generator is simply the sum of the generatorsof

the individualU(1)i factors. For the particular case ofNc = 3, the gauge group presented

above represents the two remaining gauge group factorsU(1)A = 2U(1)1−U(1)2−U(1)3andU(1)B = −U(1)1 + 2U(1)2 − U(1)3. These are two linear combinations that are

orthogonal toU(1)diag, in the sense that their generators satisfytr tAtem = tr tBtem = 0,

with tA = diag(2g,−g,−g), tB = diag(−g, 2g,−g), andtem = diag(e/3, e/3, e/3), and

tr is the trace in the representation of Table 7.1. For this choice of generators, one ob-

tains the convenient feature that the statistical parton charges are integer multiples ofg.

These generators are however neither orthogonal to each other (tr tAtB 6= 0) nor properly

normalized (tr tA,BtA,B 6= g2). Therefore, if one really starts out with aU(1)3 gauge the-

ory, the Maxwell gauge kinetic termtrFµνFµν would contain a mixedFA

µνFµνB term. The

generators can however be easily orthonormalized, and upondoing so we find precisely

the generatorsH1 andH2 [Eq. (7.14)] of the maximal diagonal subgroupU(1)2 of SU(3)

(see Sec. 7.7.2). The same construction works for general number of “colors”Nc, yielding


once again parton charges which can be chosen to only take values2g and−g under the

various Abelian factors, at the price of non-diagonal kinetic terms. As the gauge kinetic

terms do not impact the topological properties this is of no importance. This allows one to

study a simplified, topologically equivalent version of themodel where one takes the above

charge assignments with standard Maxwell terms (no mixing)and forgets about the fact

that theU(1)Nc−1 gauge group originated fromU(1)Nc in the first place. If desired, a set

of orthonormal generators is provided by the Cartan generatorsH1, . . . , HNc−1 of SU(Nc),

i.e. the generators of its maximal diagonal subgroupU(1)Nc−1. In fact, as far as the elec-

trically charged degrees of freedom are concerned, theU(1)Nc/U(1)diag model discussed

above is equivalent to a HiggsSU(Nc) model whereSU(Nc) is spontaneously broken to

its maximal diagonal subgroupU(1)Nc−1.

Upon integrating out the massive partons and setting the electromagnetic gauge poten-

tial to zero, in the fractional TI phase we are left with a pureU(1)Nc−1 dynamical gauge

theory with θ = π mod 2π. For a continuum theory, there is a single phase, the de-

confined Coulomb phase. This phase corresponds to a free fieldinfrared fixed point at

which the renormalized couplingg vanishes. In other words, in the infrared we have free

Dirac fermions (the partons) and photons (the statistical gauge fields). For a theory de-

fined on the lattice, we obtain two phases, the deconfined Coulomb phase and the confined

phase [182, 183, 184]. The fractional TI phase corresponds to the Coulomb phase, which

is equivalent to the continuum theory in the infrared exceptfor a doubling of the number of

fermion species. The topological properties are the same inboth the continuum and lattice

cases. However, the value (0 orπ mod2π) of theθ angle for the (free) partons is dependent

upon the choice of regularization procedure in the continuum theory, while it is fixed by the

lattice Hamiltonian in the lattice theory [16]. In analogy with the results of Cardy and Ra-

binovici [185] and Cardy [186] forZN gauge theory in3+1 dimensions, one may however

wonder whether the statisticalθ-term with nonzeroθ = π could affect the phase diagram of

the statistical gauge sector, since the gauge theory is now dynamical. This is not so because

aU(1) gauge theory can be viewed as theN → ∞ limit of a ZN gauge theory, and in this

limit the “electric” charges ofZN gauge theory disappear from the spectrum [187] and the

θ-term has no effect on the bulk free energy. Therefore, the Abelian models are simplest to

analyze. For example, the free fields do not allow for a nontrivial ground state degeneracy.


Non-Abelian models

One may question whether a deconfined phase is actually realizable for non-Abelian mod-

els. Although pureSU(Nc) Yang-Mills theory is generally believed to be confining at zero

temperature, adding gapless matter can result in a deconfined phase. For example,N = 4

supersymmetric Yang-Mills (SYM) theory with any gauge group, in particular with gauge

groupSU(Nc), is actually conformally invariant and hence in a deconfinedphase [188]. In

fact, this is the generic case even in non-supersymmetric Yang-Mills theories with enough

gapless matter: for matter in the fundamental representation as well as for matter in two-

index tensor representations it is believed thatSU(m), SO(m) andSp(m) gauge theories

all exhibit a “conformal window”, that is, the gauge theory is in a conformally invariant

and deconfined phase as long as the number of flavorsNf is within a certain finite range.

A conjecture for the exact values ofNf that bound the conformal windows has been put

forward for example in Ref. [189]. The upper end of the conformal window is theoretically

well established: forNf larger than this maximal value the gauge theory loses asymptotic

freedom. It is weakly coupled in the infrared but ill-definedin the ultraviolet. Just below

the upper end of the conformal window it can be established, using perturbation theory,

that at least for largeNc the theory indeed flows to a conformal fixed point [190]. The

lower bound of the conformal window is mainly conjectural based on partial resumma-

tions. There has been a lot of recent activity on numerical studies of the conformal window

using lattice gauge theory [191]. While the precise lower bound of the conformal window

is still up to debate, the existence of a conformal window hasbeen firmly established. In

gauge theories withN = 1 supersymmetry the full conformal window has been mapped

out using the power of holomorphy [192]. TheN = 1 supersymmetricSU(Nc) gauge

theory is for example conformal for32Nc < Nf < 3Nc.

As for the Abelian models, the deconfined non-Abelian modelshave additional gapless

degrees of freedom. For example, in the case ofN = 4 SYM theory the extra matter

consists of adjoint fermions and scalars, in addition to thegauge fields. One important

difference compared to the Abelian case is that in the case ofa non-Abelian gauge theory

at a nontrivial fixed point the extra matter is not free, but remains interacting with a fixed

renormalized couplingg∗, whereg∗ is fixed by the requirement that the renormalization


groupβ-function vanishes. In the special case ofN = 4 SYM theory, g∗ is in fact a

free parameter and theβ-function vanishes identically for all values of the coupling [188].

Not only do the statistical gauge fields remain strongly coupled to each other, the partons

remain strongly coupled to the statistical gauge fields. Below we will argue that the value

of the fractionalθ angle only depends on the number of partons the electron fractionalizes

into, and is completely robust even against these strong interactions that remain in the

low-energy effective theory of the partons. In fact, in caseof N = 4 SYM theory this

has been demonstrated explicitly in the extreme limit of very large coupling, employing a

holographic realization of this particular model [193].

Model B: The canonical example in this class, which we will refer to asan example

in various sections, is a model withNc = 3 partons of electric chargee/3 coupled in the

fundamental representation to aSU(3) gauge field with some additional adjoint matter to

drive it into the deconfined phase. Details of this extra matter are largely irrelevant from

a topological point of view, except for the question of the ground state degeneracy onT 3.

N = 4 SYM theory can serve as the canonical example.

One consequence of the residual interactions in the infrared is that sometimes TR sym-

metry is spontaneously broken in the topologically nontrivial phase, that is atθ = π.

Whether this happens or not depends on many details of the statistical sector [194]. For the

purpose of constructing parton models of fractional TI, theimportant lesson to remember is

that, even though spontaneous TR symmetry-breaking does occur sometimes, non-Abelian

gauge theories with unbroken TR symmetry do exist. In the largeNc limit the situation

is slightly better understood [195]: while pureSU(Nc) Yang-Mills theory is believed to

spontaneously break TR symmetry atθ = π, additional gapless matter can prevent this. In

particular,N = 4 SU(Nc) SYM theory is TR invariant atθ = π in the largeNc limit.

One other important difference between the Abelian and the non-Abelian models con-

cerns spin. In model A, each of the three partons belongs to a different one-dimensional

irreducible representation of the statistical gauge groupU(1)2. The requirement that for a

fermionic state the many-body wave function has to be antisymmetric under interchange

of both color and spin indices does not constrain the symmetrization properties of the spin

quantum numbers, and hence the total spin, because gauge invariance does not require a


separate, complete antisymmetrization of the color indices in this case. A spin-1/2 elec-

tron is always possible. In model B however, gauge invariance under theSU(3) group

requires that the many-body wave function be completely antisymmetric in color indices.

Correspondingly the spin indices have to be completely symmetrized, and in model B the

electron would have spin3/2. Because spin rotation invariance is already broken in a

topological band insulator by spin-orbit coupling, this does not create a problem, but it is

certainly an aspect of our non-Abelian models to keep in mind.

General deconfined model

The general deconfined model will have some Abelian (free) factors and some non-Abelian

(interacting) factors. A large class of models of this type was introduced in our earlier

work [180]. The statistical gauge group in this general model is

Nf∏

a=1

U(Nac )/U(1)diag,

and the total number of partons is

Nc =

Nf∑

a=1

Nac .

For every flavora = 1, . . . , Nf we haveNac partons transforming in the fundamental rep-

resentation of the non-AbelianSU(Nac ) factor and carryingU(1)a chargeqa = g/Na

c as

well as electromagnetic chargeqema , while being neutral under all the other gauge groups.

The diagonal subgroupU(1)diag, whose generator is the sum of all theU(1)a generators, is

modded out. The only gauge-invariant operator that carriesnonvanishing electromagnetic

charge is the product of all the partons. To be gauge-invariant under each of the individual

non-Abelian factors, one needs to form baryonic operators out of the partons in the fun-

damental representation of that factor. These individual baryons however will carryU(1)a

chargeg. Only the product of all the individual baryons is gauge-invariant, as this is the

only way to get a gauge-invariant operator whose only statisticalU(1) charge is the charge

under the diagonal subgroupU(1)em, which is removed from the statistical gauge group.


The electromagnetic charge of this gauge invariant operator is

Q =

Nf∑

a=1

Nac q

ema , (7.9)

hence theqema have to be chosen in such a way thatQ = e.

Another alternative one might want to consider is to use models based on orthogonal

or symplectic gauge groups. The gauge-invariant operatorsin such theories are the mesons

qaqbδab andqaqbωab, respectively, whereδab is the Kronecker delta andωab the correspond-

ing antisymmetric invariant tensor for the symplectic group. Both theories have baryons

qa1 · · · qaNcǫa1···aNc . However, in the symplectic theorySp(Nc) whereNc is even, these

baryons are not independent operators but are equivalent toa product of mesons, because

the symplectic tensorωab is antisymmetric. By analogy with the[U(M)×Sp(2k)]1 Chern-

Simons theory of theZk parafermion fractional QH states obtained from a parton construc-

tion [136], these theories have the promise of generating more exotic surface states. How-

ever, if we take the partons to have electric chargeq, not only we obtain baryons of electric

chargeNcq which would suggestq = e/Nc as before, but the mesons are also charged as

we can make mesons from two fundamental quarks. In theSU(Nc) case, mesons are made

from a fundamental quark and an antifundamental antiquark.Therefore, this time we are

forced to identify the mesons with the electrons [136]. Furthermore, we have to assign

the partons chargeq/2, and at the same time take the number of partonsNc to be even so

that the baryons carry an integer multiple of the electron charge, which is consistent with

the fact that symplectic groups are only defined for evenNc. To ensure that this mesonic

electron is a fermion, we need the electron to split into two different partons, a fermion

and a boson, both in the fundamental representation of the gauge group. We would also

get gauge-invariant scalars with the same chargee as the electron, but presumably these

can be made gapped. In this case, the axion angleθ has to be an integer multiple ofNcπ/4

from theNc fermions of chargee/2. This structure is somewhat reminiscent of theZ2 spin

liquid model of a fractional TI put forward in Ref. [181].


field U(1)em Z3

ψ1 e/3 1ψ2 e/3 1ψ3 e/3 1

Table 7.2: Gauge charge assignments of parton fields under electromagneticU(1)em andunbroken statisticalZ3 gauge groups in the simplest Higgs model forNc = 3. The thirdcolumn corresponds to the triality of the representation.

7.6.2 Higgs models

In order to get a completely gapped system that realizes a TI,it is sufficient to add elec-

trically neutral Higgs fields to the deconfined models of Sec.7.6.1, and consider a specific

pattern of spontaneous symmetry breakingG → H with G the original gauge group, such

that the unbroken gauge groupH in the Higgs phase is discrete. One can modify model B

above to realize this possibility:

Model C: A simple Higgs model can be obtained by augmenting the simplest non-

Abelian three-parton model, model B above, by inclusion of two complex scalar Higgs

fields in the adjoint representation ofSU(3).

The center ofSU(3) is Z3 and the adjoint Higgs fields are neutral under this center

symmetry. If one of the scalars acquires a vacuum expectation value, the continuous part

of the gauge group is broken to its maximal diagonal subgroupU(1)2 and one recovers

the fermions and gauge fields of model A together with severalcharged scalars. Giv-

ing generic noncommuting expectation values to both adjoint Higgs scalars completely

breaks the continuous part of the gauge group and generically only the discreteZ3 cen-

ter symmetry is unbroken. Under this discrete gauge group the parton charges are given

in Table 7.6.2. The third column corresponds to the nonzero triality k = 1 mod 3 of

the fundamental representation with charactere2πi/3. In general, for aSU(Nc) → ZNc

Higgs mechanism we want the partons to be in a representationof SU(Nc) with nonzero

Nc-ality k = 1, 2, . . . , Nc − 1 modNc with charactere2πik/Nc 6= 1. This discrete sub-

group of the original continuous statistical gauge group iscompletely sufficient to ensure

that all gauge-invariant operators have charges that are aninteger multiple of the electron

charge. Indeed, the centerZ3 of SU(3) is generated by a single element of the Cartan


subalgebra,H2 ∝ diag(1, 1,−2) in Eq. (7.14). The statistical weight lattice is generated

by the (unnormalized) weight vectorse1 = 1, e2 = 1, e3 = −2. A statistically neutral

operatorψn11 ψ

n22 ψ

n33 must satisfy

∑

i niei = 0 which leads ton1 + n2 = 2n3, hence the

total Maxwell charge ise(n1 + n2 + n3)/3 = n3e ∈ Ze. ψ1ψ2ψ3 is still the simplest

gauge-invariant operator, but as long as the fermions have internal spin states that can be

antisymmetrized,ψ3i or ψ2

i ψj with i, j = 1, 2, 3 and i 6= j would give rise to additional

gauge-invariant operators. In the relativistic continuumtheory the fermions carry spin1/2

andψ3i vanishes identically. As the fundamental degrees of freedom of the theory are the

partons, the presence of these extra bound states carrying integer electron chargee does not

affect the physics. The important requirement is that Gauss’ law, which enforces overall

Z3 neutrality, ensures that the net charge of the whole sample is an integer multiple of the

electron charge.

For Higgs fields that transform trivially under the centerZNcof SU(Nc), as is the case

here, it is known that the confined and Higgs phases are distinct [196]. In the Higgs phase,

the system is completely gapped even in the charge neutral sector because the unbroken

gauge groupZ3 is discrete. However, in the limit of infinite Higgs stiffness κ → ∞ the

system behaves like aZNcgauge theory which has an additional gapless Coulomb phase

separating the (gapped) deconfined and confined phases for large enoughNc [196, 187]. It

it possible that this phase persists for finite but large enough κ [187]. In this case, there

would be two distinct fractional TI phases with the same value of θ, one gapless and the

other fully gapped. We also note that the presence of aθ = π term in the statisticalZNc

gauge theory gives rise to the presence of oblique confined phases with dyon condensation

in addition to the usual confined phase [185, 186], but does not remove the gapped de-

confined and gapless Coulomb phases corresponding to the fractional TI. Finally, one can

also construct Abelian Higgs models in addition to the non-Abelian Higgs model discussed

here, withU(1)Nc−1 broken to a discrete subgroup.


7.7 Fractional quantized magnetoelectric effect

7.7.1 Chiral anomaly

The calculation of the effective axion angleθ via the chiral or Adler-Bell-Jackiw (ABJ)

anomaly [138, 139] was mentioned in our original work [180] following earlier discus-

sions [16], and has been spelled out in more detail elsewhere[197, 154, 193, 176, 198, 199].

This argument can be used to obtain TI in any even spacetime dimension [154, 176, 199],

but for here let us specialize to the case of the 3D TR invariant TI. The goal is to calculate

the contribution to the effectiveθ angle of aU(1) gauge group that arises from integrating

out a fermion of chargeq. As (at least in model A) our partons are charged under more

than one Abelian group, we want to calculate all terms of the form

Sθ =i∑

a,b θabe2

32π2

∫

Md4x ǫµνλρF

aµνF

bλρ, (7.10)

whereM is the (here, Euclidean) spacetime manifold and the labela runs over all the

Abelian groups in the problem, i.e. the statistical gauge groups as well as the Maxwell

gauge group. For example, in model A we havea, b ∈ em, A, B. The non-Abelian

case works similarly as will be discussed below. Let us denote by qai the charge of the

ith parton under theath gauge group in units of the corresponding gauge coupling (e for

the electromagnetic sector,g for the statistical sector). In the Dirac kinetic term for the

parton, one can write a complex mass term with massM as the complex bilinear operator

ψ(ReM + iγ5 ImM)ψ. The TR operator takesM into its complex conjugateM∗, so the

system is only TR invariant ifM is real. In other even spacetime dimensions it is a different

discrete symmetry that takes the role of enforcing a real mass term [154]. OnceM is real

we see that there is aZ2 choice of mass terms:M can be real and positive or real and

negative. The two can not be smoothly deformed into each other in a TR invariant fashion

without crossingM = 0, that is, without closing the gap.

As all that matters in terms of physics of interfaces is the difference inθ, we can choose

the θ-term to be zero in the case thatM is real and positive, which one identifies as the

topologically trivial case. At the classical level, we can always perform a chiral rotation

ψ → eiαγ5ψ to absorb the phase ofM . This chiral rotation is a symmetry of the massless


theory. One can think of the mass as a spurion, i.e. as arisingfrom the expectation value of

a non-dynamical background field, to restore the symmetry inthe massive case by letting

M transform asM → e2iαM . If the phase ofM is originally θ0, with θ0 = π for the

topologically nontrivial TR invariant insulator,M can be made real and positive by a chiral

rotation with angleα = −θ0/2. However, this chiral rotation is not a symmetry of the

quantum effective action, as the path integral measure is not invariant. Performing such

a rotation generates aθ-term of the form given in Eq. (7.10), with a coefficient that is

determined by a triangle diagram with one axial current and twoU(1) currents,

θab =∑

i

qai qbi θ0, (7.11)

i.e. it is determined entirely by the gauge group representations to which the integrated

fermions belong. For the non-Abelian case, the mixedθ-terms (mixed between two gauge

groups) vanish identically due to the tracelessness of the representation matrices. The

diagonalθ-terms are given by a similar formula with the charges replaced by the trace over

the generators of the group in the representation of the partons. A single Dirac fermion

in the fundamental representation contributesθ = θ0. This calculation is robust against

inclusion of interactions [200] as recently discussed in Ref. [199].

From Eq. (7.11) it follows immediately thatNc partons of electric chargee/Nc generate

a θ-term for the Maxwell field withθ = θ0/Nc, that isθ = π/Nc if the partons realize a

TR invariant TI. No reference to the statistical gauge groupis necessary. In the Abelian

case it is advantageous to ensure that no mixedθ-terms involving the Maxwell field and

a statistical gauge field are generated. Such mixed terms give rise to extra contributions

to the effectiveθ-term for the Maxwell field (denoted by “em”) once the statistical gauge

fields are integrated out. The latter is a delicate thing to doin the deconfined phase, but

these mixed terms would alter the topological properties ofthe Maxwell field. Vanishing

of the mixed terms requires that for anya 6= em,

∑

i

qai qemi = 0. (7.12)

If all partons have the same electromagnetic chargeqemi = 1/Nc, this reads∑

i qai = 0,


which is simply the requirement that the electron is gauge-invariant under the statistical

gauge fields. This, by construction, is automatically satisfied in our Abelian models pre-

sented above. Alternatively, Eq. (7.12) follows from the orthogonality of the generators

tr tatem = 0, a = A,B (Sec. 7.6.1). On the other hand, the general non-Abelian models

introduced in Sec. 7.6.1 have, for every gauge group factor labeled bya, a total ofNac

partons with statistical chargeqa = 1/Nac and a Maxwell electric charge that is more or

less unconstrained up to the overall condition Eq. (7.9). Inthis case mixed terms will be

generated. The final value quoted forθ in Ref. [180] is only obtained after integrating out

the statistical gauge fields in this case.

As an explicit example, take the AbelianNc = 3 parton model, model A (Sec. 7.6.1).

Assume that all partons have a topologically nontrivial mass, i.e.θ0 = π. Using the chiral

symmetry to rotate the phase of all three mass terms to a real and positive mass, we generate

via the ABJ anomaly an electromagneticθ-term withθ = Cπ, where

C =∑

i

(qemi )2 =1

3.

No off-diagonalFem ∧ FA or Fem ∧ FB terms are generated, because the corresponding

anomalies∑

i qemi qAi and

∑

i qemi qBi vanish. In the statistical sector, nonzeroθ-terms will

be generated,

θAA =∑

i

(qAi )2π = 6π, θBB =

∑

(qBi )2π = 6π,

θAB = θBA = qAi qBi π = −3π. (7.13)

These extra terms play an important role in the charge quantization considerations below.

We note that the above calculations only depend on the fermion content of the theory, and

therefore can not distinguish between the deconfined (modelA or B) or Higgs (model C)

realizations of this particular fermion content.


7.7.2 Topological quantization of the axion angle

In a topological band insulator, the low-energy description of the electromagnetic response

at energy scales below the gap is in terms of a topological field theory with aSθ ∼ θE ·Bterm [16]. From the point of view of the low-energy theory, the partition function is TR

invariant only ifeiSθ (or e−Sθ in Euclidean signature) is TR invariant which, together with

the Dirac quantization of fluxes appearing inSθ, constraintsθ to take discrete values. In

the fractional TI, after integrating out the massive partons, the low-energy electromagnetic

response still takes the formθE ·B, but with possibly fractional values forθ. The fractional

values ofθ allowed in this case follow from modified flux quantization conditions due to

the presence of the statistical gauge fields.

Quantization in the Abelian models

Consider theθ-term for the Maxwell field,θeffe2

32π2 ǫµνλρFµνFλρ. The quantization condition

on θeff follows directly from the Dirac quantization condition formagnetic charges, and

the argument applies either in Euclidean [140] or Minkowskispacetime [159] (with cer-

tain conditions on the electromagnetic fields in the latter case). We consider the Euclidean

case for simplicity. With periodic boundary conditions, spacetime is topologically equiv-

alent to the4-torusT 4. The minimal value of the spacetime integralSθ of the θ-termiθeffe

2

32π2 ǫµνλρFµνFλρ is obtained when the smallest allowed magnetic monopole is inserted

inside both2-tori T 212 andT 2

34 whereT 4 ∼= T 212 × T 2

34, andT 2µν is the2-torus generated by

directionsµ andν. In Euclidean spacetime, all directions can be taken to be spacelike and

thus all fluxesFµν are magnetic. We obtain

Sθ ≡iθeffe

2

32π2

∫

T 4

d4x ǫµνλρFµνFλρ =iθeffe

2

4π2

∫

T 212

dx1dx2 F12

∫

T 234

dx3dx4 F34,


where the factor of8 comes from the permutations of theǫ-tensor6. If the fundamental

charge ise, the smallest allowed magnetic monopole has magnetic flux

∫

T 2

F =2π

e≡ B0,

andSθ = iθeff , i.e. θeff is periodic with period2π. In the parton model however, the

fundamental charge is nowe/Nc, and the above argument would yield∫

T 2 F = 2πNc

eand

Sθ = iN2c θeff , which means thatθeff would have period2π

N2c. Also, the requirement that

the minimal allowed magnetic Maxwell flux beNcB0 and no longerB0 seems to be in

stark contrast to the real world, whereB0 fluxes have certainly been realized. Both of

those puzzles get resolved by taking into account that in models with fractional charges

interacting with statistical gauge fields, Maxwell magnetic charges can be accompanied by

“color” magnetic charges, i.e. magnetic charges of the statistical gauge fields [201, 202,

149, 150, 151]. We now explain how this increases the periodicity of θeff from 2πN2

cto 2π

Nc. In

order to do that, we must first review how the Dirac quantization condition is modified in

the presence of multiple Abelian gauge fields.

For multiple Abelian gauge fields, the Dirac quantization condition qeqm ∈ 2πZ with

qe, qm the electric and magnetic fluxes, respectively (in the first example, we hadqe = e

andqm = 2πe

), through a closed2-manifold such asT 2, is replaced by the more generic

condition∑

a qae q

am ∈ 2πZ, where again the superscripta labels the various Abelian gauge

group factors, including the Maxwell gauge group. For any given gauge groupa, qae qam

does not have to be an integer multiple of2π. Therefore a minimalB0 flux is allowed,

even though the productqae qam of the Maxwell magnetic charge producing this flux and

the electric charge of a parton would be2π/Nc, as long as there are also color magnetic

fluxes present. This is possible becauseqe for the parton is nonzero for the statistical

gauge groups. Outside the TI, the statistical fields are confined and their magnetic flux

has no physical consequence. Indeed, confinement of the color electric fluxes corresponds

to condensation of the magnetic fluxes [203], so the latter fluctuate wildly and there is no

energy cost associated with them. Therefore, we recover thefact that in a topologically

6The same value applies for all manifolds which are spin, suchasT 4. For a manifold that is not spin theminimal value of the integral ofF ∧ F is half as large [140]. As our parton model contains fermionsandhence can only be formulated on manifolds that are spin, we will restrict ourselves to that case.


trivial insulator aB0 flux is possible. Inside a fractional TI, the statisticalU(1) gauge field

is deconfined and the color magnetic flux does have an effect.

To determine the periodicity ofθ, we need to consider the contribution of all gauge

fields to the actionSθ. We consider the general topological term Eq. (7.10) withM = T 4,

together with the result of the anomaly calculation Eq. (7.11). Recall thatqai denotes the

charge of theith parton under theath gauge group in units of the gauge coupling, so that

qae = eqai for the parton. Since∫

T 2 Fa = qam, one finds

Sθ =iθ04π2

∑

a,b,i

[

(eqai )(eqib)] [

qamqbm

]

=iθ04π2

∑

i

[

∑

a

(qe)ai (qm)

ai

]2

.

The Dirac quantization condition∑

a qae q

am ∈ 2πZ ensures that this isθ0 times a sum of

integers squared, soθ0 has the standard2π periodicity. The Maxwellθ angle isθ0/Nc,

according to Eq. (7.11), and so has periodicity2π/Nc as announced earlier.

We now demonstrate this explicitly on the example of model A.The Dirac quantization

condition, i.e. the condition that the parton wave functionshould be single-valued, allows a

B0 flux for the Maxwell field together with a color magnetic fluxB0/3 for the (say)U(1)B

magnetic field. To see this, we note that the phaseαi by which the wave function of thei

parton changes when taken around a loop enclosing this flux is

αi = 2π

(

eqBiB0

3+ eqemi B0

)

,

which yieldsα1 = 0, α2 = 2π, α3 = 0 and the wave function is indeed single-valued. A

calculation of theθab angles in model A, similar to Eq. (7.13) but with a generalθ0, yields

θem = θ0/3 and θBB = 6θ0, and we obtain witha, b ∈ em, A, B and in Lorentzian

signature

eiSθ = exp

(

i∑

a,b

θab2π

e2

2π

∫

d4xEa ·Bb

)

= exp

[

i

(

6θ0 ×1

32+θ03× 12

)]

= eiθ0,

as announced earlier.θ0 has periodicity2π, and hence the Maxwellθ angle has periodicity

2π/3.


Quantization in the non-Abelian models

In the non-Abelian case, the discussion of charge quantization closely follows the Abelian

case discussed above. The proper quantization condition for magnetic fluxes in that case

is ea · mb ∈ 2πZδab, whereea is a “electric flux vector”,mb is a “magnetic flux vector”,

and the indicesa, b run over the generators of the gauge group [149, 150, 151, 204, 205].

Mathematically,ea is a vector in the weight lattice of the Lie algebra of the gauge group,

and the quantization condition definesmb as a vector in the dual weight lattice, i.e. the

“reciprocal” weight lattice [206].

Let us consider the example of a deconfined non-Abelian gaugetheory model based on

Nc = 3 partons and anSU(3) gauge group (model B). Including electromagnetism, the

total gauge group isSU(3)× U(1)em which has the following Cartan generators,

H1 =g√2λ3 =

g√2

1 0 0

0 −1 0

0 0 0

, H2 =g√2λ8 =

g√6

1 0 0

0 1 0

0 0 −2

,

H3 =

e3

0 0

0 e3

0

0 0 e3

, (7.14)

whereλ3, λ8 are Gell-Mann matrices, and we have explicitly written theSU(3) andU(1)em

gauge couplingsg and e, respectively. H1 andH2 are the two Cartan generators of

SU(3) [206], normalized totr(HaHb) = g2δab, a, b = 1, 2, andH3 is the generator of

U(1)em with all three quarks having the same electric chargee/3. The weight lattice is

generated by the fundamental weights which are

e1 =

(

g√2,g√6,e

3

)

, e2 =

(

− g√2,g√6,e

3

)

, e3 =

(

0,− 2g√6,e

3

)

. (7.15)

We recall thateia is the eigenvalue of the Cartan generatorHi associated with theath com-

mon eigenvector|ea〉 of all three Cartan generators, i.e.Hi|ea〉 = eia|ea〉 [206]. The first

two entries ofea correspond to non-AbelianSU(3) “color” charges, and the last entry cor-

responds to the usualU(1)em electric charge. The dual weight lattice is generated by its


own fundamental weights, which are defined as the “reciprocal lattice basis vectors”mb,

ea ·mb = 2πδab, a, b = 1, 2, 3. (7.16)

The linear system Eq. (7.15), (7.16) is easily solved to yield7

m1 = 2π

(

1√2g,

1√6g,1

e

)

, m2 = 2π

(

− 1√2g,

1√6g,1

e

)

, m3 = 2π

(

0,− 2√6g,1

e

)

.

(7.17)

The allowed magnetic monopoles, i.e. the allowed magnetic flux configurations, are given

by linear combinations of the fundamental dual weightsmb with integer coefficients,

m = n1m1 + n2m2 + n3m3, n1, n2, n3 ∈ Z. (7.18)

We are now in a position to discuss the periodicity ofθeff . The “colorless” magnetic

monopole configuration discussed earlier, which led to a periodicity of 2πN2

c= 2π

9, corre-

sponds to the dual weight vector

m = m1 +m2 +m3 =

(

0, 0,6π

e

)

,

i.e. n1 = n2 = n3 = 1 in Eq. (7.18). However, we now see that this is not the “smallest”

magnetic monopole: we can choose a smaller monopole for which some of theni, i =

1, 2, 3 are zero. In particular, the smallest monopoles have only one ni equal to1. But as

seen in Eq. (7.17), these monopoles will necessarily be “colored”, i.e. they will carry some

amount of non-Abelian magnetic charge. Let us now evaluateSθ for a colored magnetic

monopole, saym = m1, and see how it affects the periodicity ofθeff . Since the monopole

is colored, we cannot simply discard theSU(3) θ-term in the effective action [Eq. (6) in

Ref. [180]]. Denoting byF andf theU(1)em andSU(3) field strengths, respectively, we

7Were it not for the thirdU(1)em entry, the weight lattice would be self-dual, i.e.ea = ma

2π . This wouldbe the case forU(3), or more generally forU(Nc) [141, 142]. We are concerned with the Lie algebra ofSU(Nc)×U(1)em, which is different from that ofU(Nc) because we have chosen a different normalizationfor theSU(Nc) generators than for theU(1)em generator. This difference is important, and is the mathemat-ical reason for the appearance ofθ/Nc rather thanθ/N2

c in theθ-term for theU(1)em gauge field.


have

Sθ =iθeffe

2

32π2

∫

T 4

d4x ǫµνλρFµνFλρ +iθg2

32π2

∫

T 4

d4x ǫµνλρfaµνf

aλρ

=iθ

3

e2

4π2

(

2π

e

)2

+iθg2

4π2

[

(

2π√2g

)2

+

(

2π√6g

)2]

=iθ

3+iθ

2+iθ

6= iθ,

i.e. θ has periodicity2π, which means thatθeff = θ/3 has periodicity2π3

, as claimed

in Ref. [180]. It is readily checked that the other “minimal”monopoles,m = m2 and

m = m3, give the same quantization condition.

Quantization in the Higgs models

The analysis of charge quantization in the Higgs models is very similar to the Abelian case.

The basic statement is as before. A MaxwellB0 flux is possible only if it is accompanied

by color magnetic flux of the unbroken gauge groupH, otherwise onlyNcB0 is possible.

While color magnetic flux before was quantized in integer multiples of a basic unit, it only

now takes a finite number of discrete values. For example, in our model C, magnetic flux

for H = Z3 can only take three different values: three units of color magnetic flux are

equivalent to no color magnetic flux.

7.8 Ground state degeneracy

7.8.1 Ground state degeneracy on the 3-torus

It has recently been shown [181] that a fractionalθ angle in a TR invariant, gapped system

necessarily implies multiple ground states onT 3. However, a deconfined gauge theory may

or may not be gapped, and a unique ground state onT 3 is possible if the system is gap-

less. In general, deconfined gauge theories with continuousgauge groups will be gapless

and deconfined gauge theories with discrete gauge groups will be gapped. In particular,


we will show that the Abelian models in the deconfined phase contain gapless but electro-

magnetically neutral gauge bosons, and have a unique groundstate onT 3. On the other

hand, the non-Abelian models do exhibit a nontrivial groundstate degeneracy onT 3, even

though they avoid the theorem of Ref. [181] due to the extra massless degrees of freedom.

In the Higgs models, the theory obeys all the assumptions of the theorem and there has to

be multiple ground states onT 3. We will see that this is indeed the case. Therefore, various

realizations of a fractional TI with the same value of the Maxwell θ angle can be distin-

guished by their topological ground state degeneracy onT 3. This is similar to the fact that

the Hall conductance, although a topological invariant, isnot sufficient to fully characterize

the topological order in a fractional QH system [149, 150, 151].

Before we discuss our three different models in detail, we must first define precisely

what we mean by ground state degeneracy. As discussed in detail in the previous sections,

in at least two of our three models (the models in the deconfined phase including models

A and B), the theory contains gapless degrees of freedom. In order to arrive at a mean-

ingful definition of topological ground state degeneracy ina gapless system, we need to

study the theory for a finite-size system. Denoting byL the linear system size, the mass-

less degrees of freedom will develop a finite-size gap of order 1/L. In order to focus on

the ground state manifold, we want to study the theory at energies below that finite-size

gap. For a finite-size system, the ground state degeneracy will be generally lifted by non-

perturbative effects [149, 150, 151] corresponding to the tunneling of fractionally charged

partons around noncontractible loops inT 3, which leads to an energy splitting of order

∼ e−mgapL/L wheremgap is the dynamically generated parton mass gap already present at

infinite volume. For finiteL, these states are truly degenerate only in the strictmgap → ∞limit (i.e. for infinitely massive partons), but even at finitemgap we can identify the ground

states by a finite-size scaling analysis of the many-body spectrum.

Abelian models

We first investigate the question of ground state degeneracyonT 3 in the simplest Abelian

three-parton model, model A. When studying the model on aT 3, the ground state is unique.

As the statisticalU(1) fields are free in the infrared, this is simply a question of quantizing


Maxwell electrodynamics. As is well known,U(1)Nc−1 gauge theory onT 3 in the decon-

fined phase is simply equivalent to a set of2(Nc − 1) decoupled three-dimensional (in the

x, y, z directions onT 3) harmonic oscillators, i.e. a harmonic oscillator for eachof theU(1)

gauge fields with two polarizations. For a finite size system,the ground state is unique and

corresponds to each of these harmonic oscillators being in their ground state. The gap to

the first excited state is of order∼ 1/LwhereL is the linear system size, but vanishes in the

L → ∞ limit. According to Ref. [181], a gapped system can not have afractionalθ angle

if it has a unique ground state onT 3. As mentioned, the way our simple Abelian model

avoids this constraint is because it is not gapped (for infinite size). To see how the extra

free photons can circumvent the no-go theorem of Ref. [181],we observe that the argument

of Ref. [181] is essentially a Lorentzian version of the usual Dirac quantization argument.

Consider a gapped scenario where a2-torusT 2 of the spatialT 3 is pierced by a magnetic

flux Bz. In the presence of a unit fluxBz = B0 = 2πe

, the ground-state to ground-state

(G2G) amplitude when inserting the same minimal flux throughthe noncontractible loop

in thez direction picks up a phase given by

eiSθ = exp

(

iθ

2π

e2

2π

∫

d4xEem ·Bem

)

= eiθ.

The time-reversed process picks up a phasee−iθ, therefore for a TR invariant theory we

requireeiθ = e−iθ andθ has to be an integer multiple ofπ.

This argument relies on the fact that the minimal magnetic flux isB0. In a theory of a

Maxwell gauge field alone with partons of chargee/3 a magnetic fluxB0 is not consistent

with single-valuedness of the parton wave function; rather, the minimal flux allowed is3B0.

In this case, the above G2G amplitude picks up a phase factor of ei9θ, hence apparently any

multiple ofπ/9 would be an allowed TR invariant value forθ. Clearly, this situation can not

correspond to nature as we know it since aB0 flux can exist. At this stage, the discussion

is completely parallel to that of Dirac quantization in Sec.7.7.2. As we have seen, a basic

B0 flux is allowed as long as it is accompanied by a magnetic flux ofthe statistical gauge

fields. The Dirac quantization condition implies that the allowed combinations of fluxes

are exactly the ones for whicheiS evaluates toeinπ for integern. The G2G amplitude is TR

invariant despite the fact that the Maxwellθ angle takes the fractional valueπ/3. In fact,


Dirac quantization will ensure that the same happens in all the Abelian models proposed in

Ref. [180]. The no-go theorem is avoided even without groundstate degeneracy onT 3 due

to the presence of extra massless gauge fields.

Non-Abelian models

A deconfinedSU(Nc) gauge theory onT 3 hasN3c degenerate ground states [152]. There

are two complementary ways of establishing this result, oneby keeping the fundamental

partons in the spectrum, and the other by sending the parton mass to infinity,mgap → ∞,

and considering the pureSU(Nc) gauge theory. For concreteness we will discuss the case

Nc = 3. In the presence of fundamental quarks, one can construct a topological symmetry

algebra consisting of operatorsUa, a = 1, 2, 3 which insert a2π flux of theU(1)em gauge

field through theath noncontractible loop ofT 3, and operatorsTa which move a parton

around theath noncontractible loop. In the TR invariant case, we have[Ta, Tb] = [Ua, Ub] =

0, a, b = 1, 2, 3, but [152]

TaUb = e−2πi/3δabUbTa, (7.19)

which simply means that partons can pick up a nontrivialU(1)em Aharonov-Bohm phase

because they are fractionally charged. The operatorTa moves a parton around a closed

loop and is clearly a symmetry of the partition function. TheoperatorUa is also a sym-

metry of the partition function because theU(1)em phasee2πi/3 acquired by the partons is

also an element ofZ3 = 1, e2πi/3, e4πi/3, the center ofSU(3), and can thus be gauged

away. The existence of an algebra of operators which commutewith the Hamiltonian

but not among themselves implies the degeneracy of the energy eigenstates. The sym-

metry algebra Eq. (7.19) is somewhat reminiscent of the topological symmetry algebra

in the fractional QH states [149, 150, 151]. A representation of the algebra Eq. (7.19)

can be constructed by first constructing a representation ofthe Abelian subalgebra gen-

erated by theTa, i.e. by taking a set of states which diagonalizes theTa simultaneously,

Ta|η〉 = eiηa |η〉 with η = (η1, η2, η3). Pick a particular noncontractible loopa. By ap-

plyingUa to |η〉 repeatedly and using the commutation relations (7.19), onecan show that


Figure 7.2:SU(Nc) gauge transformationΩ on a noncontractible loopCx in the spatialmanifold [Eq. (7.21)], periodic up to an elemente2πik/Nc of the centerZNc

of SU(Nc),and under which the global Wilson loopW (Cx) transforms nontrivially asW (Cx) →e−2πik/NcW (Cx).

|η〉 = U3a |η〉, Ua|η〉, U2

a |η〉 is a 3-dimensional representation of the subalgebra gener-

ated byT1, T2, T3 and theUa corresponding to this specific loop. For generalNc, we would

obtain theNc-dimensional representation|η〉 = UNca |η〉, Ua|η〉, U2

a |η〉, . . . , UNc−1a |η〉.

The analysis can then be repeated by starting from this enlarged set of states and applying

theUa corresponding to the remaining noncontractible loops. At the end of this process one

finds that the dimension of the representation of the full symmetry algebra (7.19), which

is the same as the ground state degeneracy, isNb1(M)c whereb1(M) = dimH1(M,R), the

first Betti number [207] of the spatial manifoldM , corresponds to the number of noncon-

tractible loops inM . It is the dimension of the first homology groupH1(M,R) of M with

real coefficients. ForT 3 we haveb1(M) = 3.

Alternatively, this result can be obtained by studying the pureSU(Nc) gauge theory

which is the low-energy effective description at energies much less than the parton mass

gapmgap. In this language, the deconfined phase onT 3 corresponds to the condensation of

spatial Wilson loops, i.e. a spatial version of the condensation of the Polyakov loop [208]


(temporal Wilson loop). If all fields are in the adjoint (suchas is the case once the partons

have been integrated out),SU(Nc) gauge theory formulated on a spacetime 4-manifoldMwith nontrivial first homology groupH1(M) 6= 0 develops a globalZNc

symmetry [209,

210] originating from the center ofSU(Nc). This global symmetry is generated by large

gauge transformations, where the gauge parameter is not periodic onT 3 but only periodic

up to an element of the center. An order parameter for spontaneous breaking of this center

symmetry is a spatial Wilson loop along a noncontractible loopCa,

W (Ca) = trP exp

(

ig

∮

Ca

a

)

, (7.20)

whereg is the Yang-Mills gauge coupling,a is the statisticalSU(Nc) gauge potential,P

indicates path-ordering along the loop, and the trace is in the fundamental representation.

The fact thatW (Ca) transforms nontrivially under the center ofSU(Nc) is easily seen

by regularizing the theory on a lattice. Denote lattice sites by a triplet of integersn =

(nx, ny, nz), with na = 1, . . . , Na, a = x, y, z, and periodic boundary conditions on the

link variablesUn,µ = Un+Naa,µ, µ = t, x, y, z, in all spatial directionsa. Consider the

following family of ZNc⊂ SU(Nc) local gauge transformationsΩ(mx)

n parameterized by

an integermx,

Ω(mx)n

=

1, nx = 1, . . . , mx, . . . , Nx,

e2πik/Nc , nx = mx +Nx,(7.21)

with k = 1, . . . , Nc. Although the gauge transformation Eq. (7.21) is not periodic Ω(mx)n 6=

Ω(mx)n+Nxx

, it still is a valid gauge transformation because the usual gauge-invariant operators∑

trU are invariant underany local gauge transformation, including multivalued ones

(recall that the plaquette variableU ≡ Un,µUn+µ,νU†n+ν,µU

†n,ν transforms in the adjoint

asUΩ−→ ΩnUΩ

†n

for an arbitrary local gauge transformationΩn). Another way to say

this is that althoughΩ(mx)n is not periodic as aSU(Nc) gauge transformation, it is periodic

as aSU(Nc)/ZNcgauge transformation. Since the gauge fields transform trivially under

ZNcbecause they are in the adjoint, such a gauge transformationpreserves the periodic

boundary conditions on the gauge fields.


We can now show that Eq. (7.20) transforms nontrivially under Ω(mx)n . The global

Wilson loop Eq. (7.20) has a natural lattice regularization,

W (Ca) = tr∏

n∈Ca

Un,a.

Denotingn⊥ ≡ (ny, nz), we see thatW (Cx) transforms underΩ(mx)n as

W (Cx) = trU(1,n⊥),xU(2,n⊥),x · · ·U(Nx,n⊥),x

= trU(mx,n⊥),xU(mx+1,n⊥),x · · ·U(mx+Nx−1,n⊥)

Ω−→ tr(1 · U(mx,n⊥),x · 1)(1 · U(mx+1,n⊥),x · 1) · · · (1 · U(mx+Nx−1,n⊥) · e−2πik/Nc)

= e−2πik/NcW (Cx), (7.22)

using the periodicity of the trace to shift the base point of the loop from(1,n⊥) to (mx,n⊥)

and the fact that the link variablesUn,µ transform asUn,µΩ−→ ΩnUn,µΩ

†n+µ. When the theory

is quantized, expectation values of operators such asW (Ca) are computed by averaging

over all SU(Nc) gauge transformations that are periodicΩn = Ωn+Naa. Note that the

transformation law Eq. (7.22) is independent of the local data mx specifying where the

discontinuity of the gauge functionΩ(mx)n occurs. Therefore, although such local data can

be lost by performing theSU(Nc) gauge averaging, the global data (i.e. the parameter

k) is not averaged out andW (Ca) can develop a nonzero expectation value. This is why

the transformation law Eq. (7.22) is in fact a global symmetry. As a result, spontaneous

breaking of this symmetry〈W (Ca)〉 6= 0 does not violate Elitzur’s theorem [211].

The Wilson loop Eq. (7.20) can be interpreted as the semiclassical process of creating

a heavy parton-antiparton pair atx = 0 and annihilating them again atx = L/2 along the

noncontractible loopCa, or in other words the worldline of a virtual parton taken around

the loop once. Therefore, we recover essentially the same physics as when we explicitly

considered the fundamental partons in the previous approach of establishing the ground

state degeneracy. The Polyakov loop is the finite-temperature version of Eq. (7.20), where

the spatial manifold need not have a nontrivial first homology but the relevant noncon-

tractible loop in that case is the periodic imaginary time directionτ ∈ [0, β], with β = 1/T


the inverse temperature. In this case, the logarithm of the Polyakov loop is the negative

of the free energy associated with a single parton. In a confined phase, this free energy is

infinite as it takes an infinite amount of energy to separate a single color charge from its

color charge conjugate, and the Polyakov loop has zero expectation value. Center symme-

try is unbroken. On the other hand, in a deconfined phase the Polyakov loop typically has

a nonzero expectation value and center symmetry is spontaneously broken. In our case we

are considering a zero-temperature theory and the spatial Wilson loop Eq. (7.20) character-

izes G2G amplitudes at zero temperature. As eachW (Ca) carries charge underZNc, there

areNc different values thatW (Ca) can take for each noncontractible loopCa. Therefore,

there will beN3c degenerate ground states onT 3 andN b1(M)

c ground states for a general

spatial manifoldM without boundary.

We remark that this ground state degeneracy was not requiredby the theorem of

Ref. [181]. Whether thisN b1(M)c degeneracy actually occurs depends on details of the sys-

tem beyond the number of partons and the gauge group. In particular, if the extra gapless

neutral matter added in order to drive the non-Abelian statistical gauge field into a decon-

fined phase is in thefundamentalrepresentation of the gauge group, there is no global

center symmetry and we expect that the ground state would be unique. Another example

where spontaneous breaking of the globalZNcsymmetry (and hence presumably the cor-

responding ground state degeneracy) does not necessarily occur isN = 4 SYM theory.

As long as one imposes periodic boundary conditions for the adjoint fermions along all

noncontractible loops, the potential for the corresponding Wilson loop is flat due to super-

symmetry. The ground state expectation value of the Wilson loop is one more modulus in

the theory. One can tune its expectation value arbitrarily,hence we can obtain a deconfined

phase withZNceither broken or unbroken.

Higgs models

Because the Higgs models of Sec. 7.6.2 are completely gapped, they must have multiple

ground states onT 3 to be consistent with the theorem of Ref. [181]. As we necessarily

have unbroken discrete gauge groups, this is indeed ensured. Discrete gauge groups give

rise to degenerate ground states that differ by the value of the Wilson loop of the discrete

gauge field around noncontractible loops, i.e. discrete global fluxes. For example, in our


model C with itsZ3 unbroken gauge group, there are three different discrete fluxes per

noncontractible loop of the spatial manifoldT 3, which label the various ground states for a

total ground state degeneracy of33 = 27. A genericSU(Nc) Higgs model on a boundary-

less spatial manifoldM with all fields in the adjoint, and whereSU(Nc) is spontaneously

broken to its centerZNcwill have a ground state degeneracy ofN

b1(M)c . This is the three-

dimensional version of theN2 ground state degeneracy onT 2 in the deconfined phase of

ZN gauge theory in2 + 1 dimensions [212]. A similar mechanism is also responsible for

the ground state degeneracy in theZ2 spin liquid model put forward in Ref. [181] as an

explicit realization of a fully gapped fractional TI.

7.8.2 Ground state degeneracy on 3-manifolds with boundaries

On a3-manifold with boundaries, such as the cartesian productM = Σ× I of a Riemann

surfaceΣ and an intervalI discussed in our previous work [180], we need to also consider

the effect of the Chern-Simons terms induced on the boundary. To be specific, let us focus

on theM = Σ × I case with boundary∂M = Σ ∪ Σ. The nontrivialθ-term of the

statistical gauge fields in the bulk induces a Chern-Simons term on the boundary due to

the axion domain wall [16] betweenθ 6= 0 inside the fractional TI andθ = 0 outside the

fractional TI. For the case ofθ = π, the corresponding Chern-Simons term has level1/2.

The fact that this term is not gauge-invariant as a purely(2+1)-dimensional theory does not

matter, because the gauge noninvariance of the boundary theory is compensated by the bulk.

This is made clear by writing theθ-term as a manifestly gauge-invarianttr ǫµνλρfµνfλρ

term [213, 214], wheref is the statistical field strength. The statistical gauge field also has

a kinetic (Yang-Mills) term∼ − 1g2tr fµνf

µν . The details of the ground state degeneracy

will depend on the low-energy dynamics of the gauge fields. Wehave already shown that

the gauge fields need to be in a deconfined phase to have a fractional TI. In the following,

we will further assume that the deconfined phase occurs at weak couplingg ≪ 1, so that

we can consider the gauge fields as being essentially free. The situation is less clear in the

case of a deconfined but strongly coupled theory such asN = 4 SYM theory. With this

assumption, spatially-varying gauge field configurationsaµ(x) will cost a nonzero energy

∼ 1/L for a system of linear sizeL. Therefore, in the ground state the gauge field has to be


independent of the spatial coordinates,aµ(x, t) = aµ(t). In particular, this means that the

gauge fields on both copies ofΣ at each end of the intervalI = [−L/2, L/2] are identified,

aµ(z = −L/2) = aµ(z = L/2), where we denote byz the coordinates alongI and byx, y

the coordinates onΣ. Denoting byωCS = a ∧ da + ig 23a ∧ a ∧ a the Chern-Simons form

and using the fact [207] thattr f ∧ f = dωCS with f = da + iga ∧ a the statistical field

strength, we have

g2

8π2

∫

M×R

θ(z) tr f ∧ f = − g2

8π2

(

∆θz=−L2

∫

Σ×R

ωCS +∆θz=L2

∫

Σ×R

ωCS

)

,

with ∆θz=±L2≡ θ(z = ±L/2 + ǫ) − θ(z = ±L/2 − ǫ), for ǫ a positive infinitesimal,

andR denotes time. Sincez = ±L/2 correspond to the boundary between a fractional TI

and the vacuum which is a trivial insulator, we generically have∆θz=±L2= (2k± + 1)π

wherek+, k− ∈ Z. The resulting theory is that of a single Chern-Simons gaugefield on

Σ × R with integer levelk ≡ k+ + k− + 1. This Chern-Simons theory gives a additional

contribution to the ground state degeneracy when∂M 6= 0.

In the Abelian case, we obtain a sum ofU(1)k Chern-Simons terms on the boundary

(see Sec. A.9). However, the ground state degeneracy is not simply the product of the

ground state degeneracy for eachU(1)k Chern-Simons term, because large gauge transfor-

mations which mix severalU(1) factors give additional constraints on the ground state

Hilbert space [150, 151]. For the AbelianU(1)2 theory (Sec. 7.6.1) corresponding to

Nc = 3, the ground state degeneracy is12(k + 1)(k + 2) for Σ = T 2 [150, 151], which

reproduces the familiar threefold degeneracy of theν = 1/3 fractional QH state on the

torus fork = 1. For theZNcHiggs models, the Maxwell term of theZNc

gauge theory

itself defines a topological field theory and the ground statedegeneracy isN2gc whereg is

the genus ofΣ [131, 132, 133, 134, 135].

In the non-Abelian case, we obtain a non-Abelian Chern-Simons term onΣ [180]. The

ground state degeneracy for aSU(Nc)k Chern-Simons theory onΣ is equal to the number

of conformal blocks of the level-k SU(Nc) Wess-Zumino-Witten (WZW) conformal field

theory [215]. This number has been determined for any gauge groupG, levelk and genusg

of Σ [216, 217], but the answer is particularly easy for the special case of the torus,g = 1.

In this case the ground state degeneracy at levelk is given by1/(Nc − 1)!∏Nc−1

j=1 (k + j).


For the special case ofk = 1 we are mostly interested in this simply evaluates toNc. For

Nc = 3, the ground state degeneracy is12(k + 1)(k + 2) in complete agreement with the

answer found in Ref. [150, 151]. This is the same as the AbelianU(1)2 theory.

The boundary Chern-Simons theory involves only the gauge field componentsax, ay

with x, y the coordinates onΣ. In addition to the ground state degeneracy associated with

these boundary degrees of freedom, there is also a contribution to the ground state degen-

eracy coming from the bulk, i.e. a “remnant” of the physics considered in Sec. 7.8.1. For

the Abelian models, the bulk is in the Coulomb phase and does not contribute any addi-

tional factor to the ground state degeneracy (Sec. 7.8.1). We therefore only discuss the

non-Abelian and Higgs models. In the non-AbelianSU(Nc) case, there is an additional

contribution due to center symmetry breaking (as in Sec. 7.8.1) associated withaz. If the

ZNccenter symmetry is spontaneously broken due to a nonzero expectation value of the

Wilson lineW (I) ≡ trPeig∫Idz az on the intervalI, this contributes an additional factor of

Nc to the ground state degeneracy. A similar factor ofNc arises in the case of the Higgs

model with a discrete unbrokenZNcgauge group. To obtain the full ground state degen-

eracy, this factor multiplies the boundary ground state degeneracy we obtained from the

Chern-Simons theory onΣ. To demonstrate that the Wilson lineW (I) is a gauge invariant

observable even in the case thatI is an open interval and not a closed loop, let us focus

on the Abelian case to simplify the discussion. The non-Abelian case is similar. Under

a gauge transformationaz → az + g−1∂zλ, the Wilson lineW (I) transforms by a factor

ei∆λ where∆λ ≡ λ(z = L/2) − λ(z = −L/2) is the difference between the two val-

uesλ takes on the two endsz = ±L/2 of the intervalI. The only gauge transformations

that are allowed in this theory are the ones such that∆λ vanishes modulo2π. Therefore,

W (I) is a legitimate gauge invariant observable. The reason why∆λ has to vanish modulo

2π on Σ × I is very similar to the reason why the Wilson loopW (Ca) [Eq. (7.20)] is a

gauge invariant observable for noncontractible loopsCa. Let us briefly review that case.

Even for a noncontractible loopCz, one could imagine performing a gauge transformation

which sets∮

Czdz az = 0, e.g. for a constantaz = a this can be achieved by choosing

λ = −ga(z + L/2). This would imply thatW (Cz) is not a gauge invariant observable,

even for noncontractible loops. However, this gauge transformation is not allowed because

of its action on the wave functions of charged particles. Fora unit charge particle, the wave


functionψ atz = −L/2 would transform asψ(z = −L/2) → ψ(z = −L/2), but the wave

function atz = L/2 (identified withz = −L/2 for the closed loopCz) would transform as

ψ(z = L/2) → ψ(z = L/2)e−igaL. For generica this gauge transformation changes the

boundary conditions imposed on the matter particles and hence changes the theory. Ifa is

an integer multiple of2π/gL however, the boundary conditions on the charged fields are

unchanged, we stay within the same theory and so this gauge transformation is allowed.

This is the reason why the variable∮

Czdz az is periodic and only its complex exponen-

tial W (Cz) is single valued. Applying the same logic on the interval, one reaches a very

similar conclusion. The gauge transformationλ = −ga(z + L/2) that would set∫

Idz az

to zero changes the boundary conditions imposed on charged matter fields. The boundary

conditions are part of the definition of the theory onΣ× I, even ifI is not a closed loop. A

gauge transformation that changes boundary conditions isnot a redundancy of the theory

and so should not be allowed. The set of gauge transformations which leaves the bound-

ary conditions on the matter fields unchanged impose a periodic identification on∫

Idz az.

HenceW (I) is a gauge invariant observable for the theory onΣ× I, and thus can acquire

a nonzero expectation value.

7.9 Gapless surface states

So far we have considered effective gauge theories for systems with periodic boundary con-

ditions in all spatial directions, i.e. the3-torusT 3, or for systems with boundaries but where

the boundary is gapped since it is described by a Chern-Simons term. In these two cases,

the fermionic partons are gapped everywhere including on the boundary, which allows us

to integrate them out. The Chern-Simons terms break TR symmetry on the boundary and

are absent if TR symmetry is preserved everywhere. In the latter case, we expect that the

fractional TI should support gapless surface states since each color of partons condenses

(at the mean-field level) into a topological band insulator state, which does support gapless

surface states. The question therefore arises: what is the nature of the gapless surface state

of a fractional TI?

Since we do not at present have a microscopic model of fractional TI, i.e. a model of

interacting electrons, it is at present difficult to answer this question. Since as we have seen,


several different effective gauge theories can give rise tothe same quantized fractionalθ

angle, we expect a variety of gapless surface states with properties largely dependent on

the details of the microscopic model.

From the point of view of the effective gauge theories discussed in this work, the gap-

less surface states consist of a helical liquid of partons interacting with a three-dimensional

gauge field. An effective theory for the(2 + 1)-dimensional surface can be obtained

by integrating out the bulk gauge fluctuations. A similar calculation was performed re-

cently [174] for the surface helical spinon liquid in spin-charge separated topological Mott

insulators [173], using a perturbative approach. In this case, the bulk consists of a decon-

fined U(1) gauge field. Therefore, the results of Ref. [174] should apply qualitatively

for the models of Sec. 7.6.1, i.e. the deconfinedU(1)Nc−1 models. Indeed, since the

U(1)Nc−1 gauge theories are deconfined at weak couplingg ≪ 1 (g is the parton-gauge

boson coupling), we expect that perturbation theory ing should be reliable. Furthermore,

Ref. [174] shows that due to the three-dimensional nature ofthe gauge fluctuations, pertur-

bation theory is better controlled than in two dimensions. However, since the microscopic

degrees of freedom in a fractional TI are electrons which correspond to gauge-invariant

“baryons” in the parton gauge theory, one should only calculate correlation functions of

gauge-invariant operators. Ref. [174] finds that perturbation theory at 1-loop gives only a

logarithmic modification of the tree-level result for the2kF surface spin-spin correlation

function, i.e. 〈S+(r)S−(0)〉 ∼ 1/r2 → 1/(r2 ln kF r). We expect that the2kF surface

current-current correlation function of the Abelian deconfined fractional TI, i.e. its surface

electromagnetic response, should also only exhibit logarithmic modification compared to

the noninteracting helical Fermi liquid. In the fully gapped Higgs models,g is an irrelevant

coupling because the bulk gauge fluctuations are massive. Weexpect that the electromag-

netic response of the gapless surface state should be the same as that of the noninteracting

helical Fermi liquid, up to corrections that are irrelevantat low energies. For the deconfined

non-Abelian models, since most known examples of these (such asN = 4 SYM theory)

occur at nonzero couplingg = g∗, we expect that the gapless surface state will be a strongly

correlated version of the helical Fermi liquid and it is difficult to guess what its properties

will be. We conjecture that such a state is a “helical non-Fermi liquid”, and holographic

realizations of fractional TI [193] may be a useful tool to compute its properties.


7.10 Conclusion

In this work, we considered a variety of gauge theories in3 + 1 dimensions and discussed

the conditions they must fulfill to be consistent low-energydescriptions of a fractional TI. A

fractional TI was defined phenomenologically in previous work [180, 181] as a TR invariant

state of interacting electrons which exhibits a quantized fractional axion angleθ in its low-

energy electromagnetic response. We considered AbelianU(1) models and non-Abelian

models. In both cases, the confined phase is not an option for afractional TI because

there would be no fractionally charged excitations in the spectrum, and the existence of

fractionally charged states is necessary for a fractionalθ angle to be consistent with TR

symmetry. This leaves us with two options: a deconfined (or Coulomb) phase and a Higgs

phase. The deconfined phase of AbelianU(1) models is a theory of noninteracting, gapless

gauge bosons. The gaplessness of the gauge bosons does not affect the quantization ofθ

because they are electrically neutral. We showed the fractional quantization ofθ explicitly

using the Adler-Bell-Jackiw chiral anomaly, which did not require the assumption that the

gauge bosons should be gapped. Achieving a deconfined phase in non-Abelian models is

more difficult, but non-Abelian models with sufficient electrically neutral gapless matter,

such asN = 4 SYM theory, are known to realize deconfined phases. These arehowever

usually strongly coupled phases. However, the chiral anomaly still holds in the case of

non-Abelian gauge groups, and we could again show explicitly the fractional quantization

of θ. Higgs models in which the AbelianU(1) or non-AbelianSU(N) groups were broken

down to a discreteZN group were shown to lead to fractional TI as well, with a fullygapped

spectrum in this case.

We investigated the ground state degeneracy of these effective gauge theories on spa-

tial 3-manifolds of nontrivial topology. On the three-torusT 3, the deconfined Abelian

U(1) models have a unique ground state. Indeed, this correspondssimply to quantizing

several independent flavors of Maxwell electrodynamics in abox with periodic bound-

ary conditions in all three directions. The deconfined non-Abelian models can have a

nontrivial ground state degeneracy onT 3 due to the fact that the first homology group

H1(T3,Z) = Z× Z× Z is nontrivial, corresponding to the existence of three inequivalent


noncontractible loops inT 3. Whether there are multiple ground states or not in the decon-

fined phase of aSU(N) non-Abelian model depends on whether the centerZN of SU(N)

is spontaneously broken in the ground state or not. This is a question of dynamics which

depends on the details of the model. Higgs models with residual ZN gauge group can be

viewed as a subset of the previous case. Indeed,SU(N) gauge theories with all fields in

the adjoint representation develop the centerZN as a global symmetry which can be spon-

taneously broken in the ground state. Our Higgs models consist of adding adjoint Higgs

fields to pureSU(N) gauge theory such thatSU(N) is spontaneously broken to its cen-

ter. Wilson loops around noncontractible loops will still transform nontrivially underZN ,

and their acquiring nonzero expectation values means spontaneous breaking of this global

ZN symmetry and multiple degenerate ground states. We restricted our consideration of

3-manifoldsM with boundary∂M to the caseM = Σ× I with Σ a Riemann surface (say,

in thex, y directions) andI an interval (in thez direction). In this case, a Chern-Simons

term was induced on the boundary∂M = Σ∪Σ, and in the ground state the Chern-Simons

gauge fields on both copies ofΣ were identified. The resulting Chern-Simons theory had

integer level and its contribution to the total ground statedegeneracy (bulk and surface)

could be computed using standard methods.

Finally, we briefly commented on what one should expect for the electromagnetic re-

sponse properties of the gapless surface states of a fractional 3D TI based on the general

characteristics of the effective gauge theories discussedhere. We expect that deconfined

AbelianU(1) models should give at most a logarithmically modified version of the nonin-

teracting helical Fermi liquid, while the fully gapped Higgs models should only give cor-

rections that are irrelevant at low energies. The deconfined, strongly coupled non-Abelian

models should give rise to the most interesting case. We conjecture that the gapless surface

states of non-Abelian fractional TI are “helical non-Fermiliquid” states and suggest that

holographic methods [193] should be a promising way to studytheir properties.

Chapter 8

Conclusion

This dissertation reported some theoretical studies on 2D and 3D time-reversal invariant

topological insulators which were largely motivated by theexperimental realization of the

former in inverted HgTe quantum wells in 2007 and of the latter in Bi-based compounds in

2008.

The first four chapters provided answers to the questions raised in the introductory

Chapter 1. The nonlocal transport measurements reported and theoretically analyzed in

Chapter 2 constituted strong evidence that transport in inverted HgTe quantum wells does

indeed proceed by robust edge channels, as predicted by Bernevig, Hughes, and Zhang. The

Landauer-Buttiker theory of edge transport reported in this work was able to explain the

paradoxical observation of a finite resistance caused by voltage probes, as well as provide

evidence for the helical nature of the edge channels. The explanation of the different role

played by incoherent scattering in the quantum Hall and quantum spin Hall states paved

the way for the study reported in Chapter 3 of an impurity spinon the edge of the quantum

spin Hall insulator. Kondo scattering from the fluctuating impurity spin provided a possible

explanation for the experimentally observed increased suppression of the conductance with

decreasing temperature. In addition, we found that strong Coulomb interactions can lead

to the formation of an unconventional insulating phase withfractionally charged emergent

excitations. The associated quantized fractional Fano factor of 1/2 was found to be a direct

measure of theZ2 invariant of the quantum spin Hall state. Chapter 4 showed that the

cusplike magnetoconductance peak observed in experimentson HgTe quantum wells could

130

CHAPTER 8. CONCLUSION 131

be explained by a disorder effect, provided that the disorder strength was on the order of the

bulk insulating gap. Chapter 5 proposed a novel way of manipulating spins in condensed

matter systems in a purely quantum-mechanical way, by making use of the helical property

of the quantum spin Hall edge states.

The last two chapters focused on 3D topological insulators.Chapter 6 proposed an op-

tical rotation experiment to measure the topological magnetoelectric effect in a universal

way, independent of material properties such as dielectricconstant and magnetic perme-

ability. Although the topological magnetoelectric effecthas not yet been observed at the

time of this writing, we believe that its observation remains the prime goal of experimen-

tal studies on 3D topological insulators. Several experimental groups around the world

are currently involved in this pursuit at the experimental frontier of the field of topologi-

cal insulators. Chapter 7 introduced the concept of fractional 3D topological insulator and

showed that it is consistent with time-reversal invariance. This was an effort to reconcile

two notions of topological order — that of Thouless and that of Wen — in time-reversal in-

variant systems, in the same way that the quantum Hall effectwith its integer and fractional

realizations does so for time-reversal breaking systems. Although we did not write down a

microscopic Hamiltonian which realizes the fractional 3D topological insulator phase, re-

cent theoretical work has discussed 2D lattice models with nearly flat topologically nontriv-

ial bands [218, 219] as well as numerical evidence for fractional 2D topological insulators,

both time-reversal breaking [220, 221, 222] and time-reversal invariant [223, 224]. Fur-

thermore, a recently proposed microscopic model for a 3D topological insulator with flat

bands [225] paves the way for the construction of a microscopic model of a fractional 3D

topological insulator. Such studies lie at the theoreticalfrontier of the field of topological

insulators.

The experimental discovery of the integer quantum Hall effect in semiconductor het-

erostructures led to the theoretical discovery of topologyin condensed matter physics.

Given the elegant simplicity of the theory of this effect, itis surprising to note that fifty years

of solid state physics passed before its discovery. Nevertheless, it tremendously deepened

our understanding of quantum matter, revealing unexpectedconnections between it and

topics as diverse as elementary particle physics, conformal field theory, noncommutative

CHAPTER 8. CONCLUSION 132

geometry, and characteristic classes. It seems as though time-reversal invariant topolog-

ical insulators will add a new chapter to this story. Likewise, given that the partition of

time-reversal invariant band insulators into topologically distinct classes can be illustrated

using a simple4 × 4 Hamiltonian matrix, it is striking that this result escapednotice for

decades, despite the development of increasingly sophisticated band structure calculation

techniques. Although conceptually similar to the quantum Hall effect, topological insula-

tors build bridges of their own between condensed matter physics and topics in high-energy

physics and mathematics as diverse as the theta-vacuum, axions,K-theory, and Clifford al-

gebras. It is our hope that the study of topological insulators will encourage theorists and

experimentalists alike to revisit classic physics with newquestions, so that we might use

the past to imagine the future.

Appendix A

Theoretical methods

A.1 Weak-coupling renormalization group equations

We model the impurity on the edge of the QSH system by a singleS = 1/2 local spin

coupled by exchange interaction to a one-dimensional helical liquid with electron-electron

interactions. We also consider that the impurity leads to a local two-particle backscattering

term. The system is then described by the Hamiltonian

H = H0 +HK +H2.

H0 is the Hamiltonian of the helical liquid in the absence of theimpurity and reads in the

continuum limit

H0 =− ivF

∫

dx(

ψ†R+∂xψR+ − ψ†

L−∂xψL−)

+ vF

∫

dx[

g4(

(ρR+)2 + (ρL−)

2)

+ g2(

ρR+ρL− +H.c.)

]

,

whereψ†λσ(x) with λ = R,L creates, respectively, a right and left moving electron with

helicity σ = ± at positionx, andρλσ =: ψ†λσψλσ : are the corresponding normal-ordered

densities.vF is the Fermi velocity of the edge electrons, andg4 andg2 are the (dimension-

less) forward and dispersion scattering amplitudes [61]. In HgTe QW the wave vector at

which the edge dispersion enters the bulk is usually much smaller thanπ/2 such that the

133

APPENDIX A. THEORETICAL METHODS 134

umklapp term can be ignored.HK is the Kondo Hamiltonian,

HK = J‖a(

S− : ψ†R+(0)ψL−(0) : + S+ : ψ

†L−(0)ψR+(0) :

)

+ JzaSz

(

ρR+(0)− ρL−(0))

,

(A.1)

whereS± = Sx ± iSy andSz are the spin operators for the impurity located atx = 0, and

a is the size of the impurity. Note thatJ‖,z have units[E] and the continuum field operators

ψR,L(x) have units[L]−1/2 so that a factor of length is necessary to have the correct units for

the Hamiltonian. Essentially, if we start from a lattice model for the Kondo problem [226],

H =∑

k

ǫkc†kσckσ +

J

NS ·∑

kk′

c†kασαβck′β ,

then the continuum limit is

H → L

∫

dk

2πǫkc

†kσckσ + L2 J

NS ·∫

dk

2π

∫

dk′

2πc†kασ

αβck′β

≡∫

dk

2πǫkψ

†kσψkσ + JaS ·

∫

dk

2π

∫

dk′

2πψ†kασ

αβψk′β,

wherea = L/N is the lattice constant andψk =√Lck are the continuum field operators.

Henceψk has units[L]1/2, which means that the Fourier transformψ(x) =∫

(dk/2π)eikxψk

has units[L]−1/2.

Finally,H2 is the two-particle backscattering term,

H2 = λ2a2ψ†

R+(0)ψ†R+(ξ)ψL−(0)ψL−(ξ) + H.c.,

where point splitting with a short-distance cutoffξ is required by the Pauli exclusion prin-

ciple, andλ2 is the two-particle backscattering amplitude and has units[E]. Here the short-

distance cutoff is given by the penetration of the edge stateinto the bulkξ, in the same

way that the magnetic lengthℓB provides a short-distance cutoff for chiral edge theories

in the QH regime [62]. The HamiltonianH2 is manifestly nonlocal, and upon expanding


ψ(ξ) ≃ ψ(0) + ξ∂xψ(0), taking theξ → 0 limit gives the local interaction term

H2 = λ2a2ξ2ψ†

R+(0)∂xψ†R+(0)ψL−(0)∂xψL−(0) + H.c.

The Hamiltonian can be bosonized in the standard way [227, 228] using the chiral

vertex operatorsψ†R+,L−(x) = 1√

2πξ: e∓i2

√πφR+,L−(x) : . A single nonchiral boson field

φ = φR+ + φL− is sufficient to describe the helical liquid, reflecting the fact that it has the

same number of degrees of freedom as a spinless fermion. The Hamiltonian in bosonized

form reads

H = H0 +J‖a

2πξ

(

S− : e−i2

√πφ(0) : + H.c.

)

− Jza√πSzΠ(0)

+λ22π2

(

a

ξ

)2

: cos 4√πφ(0) : ,

whereH0 is the usual Tomonaga-Luttinger Hamiltonian [61]

H0 =v

2

∫

dx

[

KΠ2 +1

K(∂xφ)

2

]

, (A.2)

with K =√

π+g4−g2π+g4+g2

the Luttinger parameter andv = vF

√

(

1 + g4π

)2 −(

g2π

)2the renor-

malized edge state velocity.

We now derive the renormalization group equations (RGE) Eq.(3.4) for the flow of the

coupling constantsJ‖ andJz. We start from the Kondo Hamiltonian (A.1)

HK = J‖(S−ψ†RψL + S+ψ

†LψR) + JzSz(ψ

†RψR − ψ†

LψL). (A.3)

whereψR,L ≡ ψR+,L−(0), and we have rescaled the couplingsJ‖, Jz by 1/a. In the

bosonization treatment, all the fermion bilinears in this Hamiltonian are considered to be

normal ordered as far as their boson representation is concerned. In this section, we will

also neglect the factors of1/√ξ in the bosonization formulae for the chiral vertex operators

(ξ is the short-distance cutoff) since the scaling dimensionsof the operators including their

anomalous dimensions appear explicitly in the operator product expansions (OPE), and it

is those dimensions which control the RG flow of the couplings. In other words, when


writing the(0 + 1)-dimensional OPE

Oi(τ)Oj(τ′) ∼

∑

k

ckij|τ − τ ′|∆i+∆j−∆k

Ok(τ′),

where∆ is the scaling dimension (and the sum is over the members of the operator algebra),

we actually mean

Oi(τ)Oj(τ′) ∼

(

1

ξ

)∆engi +∆eng

j −∆engk ∑

k

ckij

(

ξ

|τ − τ ′|

)∆i+∆j−∆k

Ok(τ′),

which explicitly respects dimensional analysis, where∆eng is the engineering dimension.

We use boson fields on the infinite line [227] such that[φR(τ), φL(τ′)] = i

4, hence the

fermions properly anticommute and we do not need Klein factors. Since the number of

degrees of freedom of the helical liquid is the same as that ofa spinless fermion, we only

need a single species of bosons. The normal-ordered fermionbilinears are [227]

: ψ†RψL : =

1

2π: e−i2

√πφ : ,

: ψ†LψR : =

1

2π: ei2

√πφ : ,

: ψ†RψR − ψ†

LψL : = : ρR : − : ρL : =: jx : = − Π√π,

whereφ ≡ φR + φL is the nonchiral boson, and we work in imaginary timeτ = it. As

mentioned previously, the Hamiltonian (A.3) becomes in bosonized form

HK(Π, φ) = J‖1

2π

(

S− : e−i2

√πφ : + S+ : e

i2√πφ :

)

− Jz1√πSzΠ,

where we have omitted the free part. The canonical momentum is given in this case by

Π =∂L∂∂tφ

=1

vK

(

∂tφ+2√πJzSz

)

,

whereL is the Lagrangian density andK is the Luttinger parameter. FromH = Π∂tφ−L


it can be shown that the corresponding Euclidean action is

SK[φ] =

∫ β

0

dτ

[

J‖1

2π

(

S− : e−i2

√πφ : + S+ : e

i2√πφ :

)

− Jzi

vK√πSz∂τφ

]

,

with β the inverse temperature.

A.1.1 Operator algebra

In view of performing the OPE, we first define the members of theoperator algebra to be

O1(τ) =1

2π

(

S− : e−i2

√πφ : + S+ : e

i2√πφ :

)

,

O2(τ) = − i

vK√πSz∂τφ,

such that the total action reads

S[φ] = S0[φ] +∑

i

gi

∫ β

0

dτ Oi(τ),

where

S0[φ] =1

βK

∑

iωn

|ωn||φ(iωn)|2

is the conformally invariant Tomonaga-Luttinger action and g1 ≡ J‖ andg2 ≡ Jz are the

couplings. The coefficients of the OPE

Oi(τ)Oj(τ′) ∼

∑

k

ckij|τ − τ ′|∆i+∆j−∆k

Ok(τ′) (A.4)

enter the RGE [229, 230]

dgkdℓ

= (1−∆k)gk − π∑

ij

ckijgigj +O(g3), (A.5)

where the first term corresponds to tree level and the second term is the one-loop contribu-

tion. We will first calculate the scaling dimensions∆i of the operators, and then perform


the OPE to calculate the coefficientsckij . While RGE in the weak coupling limit are often

derived using the Wilson momentum-shell RG [228], for field theories in1 + 1 and0 + 1

dimensions conformal invariance makes it more convenient to work in real space and use

the OPE [229, 230].

A.1.2 Scaling dimensions

The scaling dimension∆i of an operatorOi(τ) is defined as [230]

〈Oi(τ)O†i (τ

′)〉 ∼ 1

|τ − τ ′|2∆i,

where for the scaling dimension, the numerical prefactors in the operators are not important.

We have

〈O1(τ)O†1(τ

′)〉 ∼〈S−(τ)S+(τ′)〉〈 : e−i2

√πφ(τ) : : ei2

√πφ(τ ′) : 〉

+ 〈S+(τ)S−(τ′)〉〈 : ei2

√πφ(τ) : : e−i2

√πφ(τ ′) : 〉.

For aSz eigenstate of the impurity spin withSz componentm, we have

〈S±(τ)S∓(τ′)〉 = S(S + 1)−m2 ±m sgn(τ − τ ′).

The spin-spin correlators have no dynamics and contribute nothing to the scaling dimen-

sions. We have

〈 : e±i2√πφ(τ) : : e∓i2

√πφ(τ ′) : 〉 = e4π〈φ(τ)φ(τ

′)〉 =1

|τ − τ ′|2K ,

since the boson correlator is

〈φ(τ)φ(τ ′)〉 = K

2πln

1

|τ − τ ′|


as we will show now. We have

〈φ(τ)φ(0)〉 = 1

β

∑

iωn

e−iωnτG0(iωn), (A.6)

whereG0(iωn) = K/2|ωn| is obtained from the unperturbed action

S0[φ] =1

βK

∑

iωn

|ωn||φ(iωn)|2

≡ 1

2

1

β

∑

iωn

φ(iωn)∗G−1

0 (iωn)φ(iωn). (A.7)

Note that theωn = 0 term does not appear in the action because it is zero, therefore we

have to omit then = 0 term in the Fourier transform Eq. (A.6). We use the sum

∞∑

n=1

e−αn

n= − ln(1− e−α),

so we get

〈φ(τ)φ(0)〉 = K

4πln

1

2− 2 cos(2πτ/β)=K

4πln

1

4 sin2(πτ/β),

where we can drop the factor of 4 in front of the sinus squared,which is just a constant

(the propagator is a logarithm, so this does not matter for the purposes of the OPE). We

therefore obtain

〈φ(τ)φ(0)〉 = K

4πln

1

sin2(πτ/β).

At zero temperatureβ → ∞, sin2 x→ x2 and we obtain

〈φ(τ)φ(τ ′)〉 = K

4πln

1

(τ − τ ′)2=K

2πln

1

|τ − τ ′| .

Once more, we have dropped a constantπ which we could have absorbed into some short-

time cutoffτc in the numerator of the argument of the logarithm, i.e.ln(τc/|τ |). Again, this


does not matter for the purposes of the OPE. Thus

〈O1(τ)O†1(τ

′)〉 ∼ 1

|τ − τ ′|2K ,

and we read off∆1 = K.

ForO2, we have

〈O2(τ)O†2(τ

′)〉 ∼ 〈Sz(τ)Sz(τ′)〉〈∂τφ(τ)∂τ ′φ(τ ′)〉.

Again,〈Sz(τ)Sz(τ′)〉 = m2 has no dynamics. We have

〈O2(τ)O†2(τ

′)〉 ∼ 〈∂τφ(τ)∂τ ′φ(τ ′)〉 = ∂τ∂τ ′〈φ(τ)φ(τ ′)〉 ∼1

|τ − τ ′|2 ,

hence∆2 = 1.

A.1.3 Operator product expansions

We now calculate the OPE. In this case, the numerical prefactors are important but we need

only retain the terms which are most singular in the limit|τ − τ ′| → 0. The impurity spin

operators have a nonsingular product and can be taken at equal timesSα(τ)Sβ(τ′) → SαSβ.

The first OPE is

O1(τ)O1(τ′) ∼ 1

4π2

(

S− : e−i2

√πφ(τ) : + S+ : e

i2√πφ(τ) :

)

×(

S− : e−i2√πφ(τ ′) : + S+ : e

i2√πφ(τ ′) :

)

.

First consider the product

: e±i2√πφ(τ) : : e±i2

√πφ(τ ′) : =: e±i2

√π[φ(τ)+φ(τ ′)] : e−4π〈φ(τ)φ(τ ′)〉 ∼ |τ − τ ′|2K ,


which is nonsingular sinceK > 0, hence we drop this term. Then we have

: e±i2√πφ(τ) : : e∓i2

√πφ(τ ′) : =: e±i2

√π[φ(τ)−φ(τ ′)] : e4π〈φ(τ)φ(τ

′)〉

=1

|τ − τ ′|2K : e±i2√π[φ(τ)−φ(τ ′)] : .

We Taylor expand

: e±i2√π[φ(τ)−φ(τ ′)] : =: e±i2

√π[(τ−τ ′)∂τ ′φ(τ

′)+...] : ,

hence we have

O1(τ)O1(τ′) ∼ 1

4π2

1

|τ − τ ′|2K(

S+S− : 1 + i2√π(τ − τ ′)∂τ ′φ(τ

′) + . . . :

+S−S+ : 1− i2√π(τ − τ ′)∂τ ′φ(τ

′) + . . . :)

.

The termS+, S− = 2(S2x + S2

y) is a constant since the HamiltonianHK = J‖S‖ ·σ‖ + JzSzσz is invariant underO(2) rotations in thexy plane. In the second term, us-

ing [S+, S−] = 2Sz we have

O1(τ)O1(τ′) ∼ const. +

i

π√π

1

|τ − τ ′|2K−1Sz∂τ ′φ(τ

′),

hence the OPE ofO1 with itself is

O1(τ)O1(τ′) ∼ const.− vK

π

1

|τ − τ ′|2K−1O2(τ

′). (A.8)

The OPE ofO2 with itself is

O2(τ)O2(τ′) ∼ − 1

π(vK)2S2z : ∂τφ(τ) : : ∂τ ′φ(τ

′) : .


By Taylor expanding, we obtain

: ∂τφ(τ) : : ∂τ ′φ(τ′) : =: ∂τ ′φ(τ

′) + (τ − τ ′)∂2τ ′φ(τ′) + . . . : : ∂τ ′φ(τ

′) :

=: (∂φ)2 : + (τ − τ ′) : (∂2φ)∂φ : + . . . ,

where it is clear that there are no singular terms. Thereforewe have

O2(τ)O2(τ′) ∼ const. (A.9)

Finally, we have the OPE ofO2 with O1,

O2(τ)O1(τ′) ∼ − i

2πvK√π

(

SzS−∂τφ(τ) : e−i2

√πφ(τ ′) : + SzS+∂τφ(τ) : e

i2√πφ(τ ′) :

)

.

We first need to prove the following result,

∂τφ(τ) : eiαφ(τ ′) : ∼ −iαK

2π

1

τ − τ ′: eiαφ(τ

′) : . (A.10)

First expand the exponential

∂τφ(τ) : eiαφ(τ ′) : = ∂τφ(τ) : 1 +

∞∑

n=1

(iα)n

n!φ(τ ′)n : .

We note that

: φ(τ ′)n : =: φ(τ ′) : n,

where we neglect infinite constants, since all fields are at the same point. Then consider

applying Wick’s theorem,

: ∂τφ(τ) : : φ(τ′) : n = : ∂τφ(τ)φ(τ

′)φ(τ ′)n−1 : +: ∂τφ(τ)φ(τ′)φ(τ ′)φ(τ ′)n−2 : + . . .

= n〈∂τφ(τ)φ(τ ′)〉 : φ(τ ′)n−1 :

= −nK2π

1

τ − τ ′: φ(τ ′)n−1 : ,


where we have used

〈∂τφ(τ)φ(τ ′)〉 = ∂τ 〈φ(τ)φ(τ ′)〉 = ∂τ

[

K

4πln

1

(τ − τ ′)2

]

= −K

2π

1

τ − τ ′,

hence we obtain

∂τφ(τ) : eiαφ(τ ′) : ∼ −K

2π

1

τ − τ ′

∞∑

n=1

(iα)n

(n− 1)!: φ(τ ′)n−1 : ,

where we have neglected the first term∂τφ(τ) = ∂τ ′φ(τ′) + (τ − τ ′)∂2τ ′φ(τ

′) + . . . which

is nonsingular. Pulling out a factor ofiα, this is just the exponential again and we obtain

Eq. (A.10). Using this result, and the identitySzS± = ±12S±, we obtain

O2(τ)O1(τ′) ∼ − 1

(2π)2v

1

τ − τ ′

(

S− : e−i2

√πφ(τ ′) : + S+ : e

i2√πφ(τ ′) :

)

,

hence the desired OPE is

O2(τ)O1(τ′) ∼ − 1

2πv

1

τ − τ ′O1(τ

′). (A.11)

One can also show thatO1(τ)O2(τ′) has exactly the same OPE. Note that Eq. (A.8), (A.9)

and (A.11) are consistent with Eq. (A.4) and the scaling dimensions∆1 = K, ∆2 = 1.

From these equations we can extract the coefficients

c211 = −vKπ, c112 = c121 = − 1

2πv.

The RGE (A.5) thus take the form

dJ‖dℓ

= (1−K)J‖ +1

vJ‖Jz,

dJzdℓ

= vKJ2‖ .

Upon rescaling the couplingsJ‖ → K−1/2J‖ andJz → vJz, we obtain the RGE in the

form of Eq. (3.4),dJ‖dℓ

= (1−K)J‖ + JzJ‖,dJzdℓ

= J2‖ . (A.12)


In this section we worked in units such that the density of statesν = 1, but the factor ofν

in Eq. (3.4) can be restored by a further rescalingJz → νJz andJ‖ → νJ‖.

A.2 Weak-coupling Kubo formula calculation of the edge

conductance

The finite-temperature Euclidean action corresponding to the Tomonaga-Luttinger Hamil-

tonian (A.2) is

S0 =

∫ β

0

dτ

∫

dxv

2K

[

1

v2(∂τφ)

2 + (∂xφ)2

]

, (A.13)

where1∫

dx ≡∫∞−∞ dx. The nonlocal conductivityσxx(x, x′;ω) defined as

jex(x, ω) =

∫ L

0

dx′ σxx(x, x′;ω)Ex(x

′, ω), (A.14)

wherejex = −ejx is the electric charge current, is given in linear response by the Kubo

formula [226],

σxx(x, x′;ω) =

ie2

ω + iδΠR

xx(x, x′;ω),

where the retarded current-current correlation function is

ΠRxx(x, x

′; t, t′) = −iθ(t− t′)〈[jx(x, t), jx(x′, t′)]〉.

We consider a uniform electric fieldEx(x′, ω) = Ex(ω). In 1D it is given byEx = V/L

whereV is the applied bias andL is the length of the region to which we apply the elec-

tric field. To establish the formalism, we first compute the conductance of the unperturbed

helical liquid without the Kondo impurity, and afterwards compute the conductance in the

presence of the impurity. In the absence of the impurity, theproblem is translationally

1Here we use the “field-theoretic” formulation [227] in whichφ(x) is defined on a infinite regionR withthe boundary conditionsφ(x → ±∞) = 0. This does not contradict the fact that the electric field is nonzeroonly in a region of finite sizeL because the correlation functions are calculated at equilibrium, in the absenceof the electric field. Note however that the ensuing commutation relations between the chiral fieldsφR,L and,most notably, the mode expansions, will in general differ from the finite-size formulationx ∈ [0, L] withperiodic boundary conditionsφ(x+ L) = φ(x).


invariant. If the current-current correlation function istranslationally invariant, we have∫

dx′ ΠRxx(x, x

′;ω) =∫

dxΠRxx(x, ω) andjx(x, ω) = jx(ω) is independent ofx. Further-

more, in 1D the current densityjex = I is the current itself. Hence we obtain

I(ω) = G(ω)V (ω),

where

G(ω) =ie2

(ω + iδ)L

∫ L

0

dxΠRxx(x, ω)

is the ac conductance. We are interested in the dc conductance,

G = limω→0

ie2

(ω + iδ)L

∫ L

0

dxΠRxx(x, ω), (A.15)

where

ΠRxx(x, ω) =

∫ ∞

−∞dt eiωtΠR

xx(x, t),

andΠRxx(x, t) = −iθ(t)〈[jx(x, t), jx(0, 0)]〉. This retarded function is obtained by ana-

lytic continuation of the Euclidean correlatorΠxx(x, τ) = −〈Tτ jx(x, τ)jx(0, 0)〉. At finite

temperature we have

ΠRxx(x, ω) = lim

iωn→ω+iδΠxx(x, iωn),

where

Πxx(x, iωn) =

∫ β

0

dτ eiωnτΠxx(x, τ),

andωn = 2πn/β, n ∈ Z are discrete bosonic Matsubara frequencies. At zero temperature

β → ∞, the frequencyωn → ω becomes a continuous variable and we have

Πxx(x, iω) =

∫ ∞

0

dτ eiωτΠxx(x, τ),

and all following expressions can be converted to zero temperature by settingβ → ∞ and

ωn → ω.

Taking theω → 0 limit is equivalent to takingiωn → 0, and sinceω+iδ → iωn appears


already in Eq. (A.15), we might as well work directly with theMatsubara frequencies,

G = − limωn→0

ie2

L

1

iωn

∫ L

0

dx

∫ β

0

dτ eiωnτ 〈Tτjx(x, τ)jx(0, 0)〉

= − limωn→0

e2

ωnL

∫ L

0

dx

∫ β

0

dτ eiωnτ 〈Tτjx(x, τ)jx(0, 0)〉.

According to the bosonization prescription in imaginary timeτ = it, the fermion number

current isjx(x, τ) = −i∂τφ(x, τ)/√π, hence

G = limωn→0

e2

πωnL

∫ L

0

dx

∫ β

0

dτ eiωnτ 〈Tτ∂τφ(x, τ)∂τφ(0, 0)〉.

It is easily found from the action (A.13) that theφ propagatorD(x, τ ; x′, τ ′) =

〈Tτφ(x, τ)φ(x′, τ ′)〉 satisfies the differential equation

(

1

vK∂2τ +

v

K∂2x

)

D(x, τ ; x′, τ ′) = −δ(x− x′)δ(τ − τ ′).

By translational invariance the propagator depends onx−x′ andτ−τ ′ only. For an infinite

system (see Eq. (A.13)), we impose the boundary conditionD(x − x′ → ±∞) = 0. In

imaginary time, the boundary condition is periodic with period β such that

D(x− x′, τ − τ ′) =1

β

∑

iωn

e−iωn(τ−τ ′)D(x− x′, iωn),

and we get(

ω2n

vK− v

K∂2x

)

D(x− x′, iωn) = δ(x− x′),

which shows thatD has a jump of magnitude−K/v at x = x′. From these boundary

conditions we obtain

D(x− x′, iωn) =K

2|ωn|e−|ωn||x−x′|/v.

Note the clear nonanalytic dependence on the frequency|ωn|. We can obtain the current-

current correlator〈Tτ∂τφ(x, τ)∂τφ(0, 0)〉 by differentiatingD, where the contact terms


coming from taking the derivative of theθ-function in theτ -ordered product do not con-

tribute to the conductance and are ignored2. Hence we obtain

〈Tτ∂τφ(x, τ)∂τ ′φ(x′, τ ′)〉 = ∂τ∂τ ′D(x− x′, τ − τ ′).

Denoting

G(x, iωn) =

∫ β

0

dτ eiωnτ 〈Tτ∂τφ(x, τ)∂τφ(0, 0)〉,

then from the double derivative∂τ∂τ ′ we obtainG(x, iωn) = ω2nD(x, iωn). Since we

take the limit ofωn → 0 from above (we consider physical, positive driving frequencies),

|ωn| = ωn and

G = limωn→0

e2ωn

πL

∫ L

0

dxD(x, iωn).

For small frequencies, we have

∫ L

0

dxD(x, iωn) =Kv

2ω2n

(

1− e−|ωn|L/v) ≃ KL

2|ωn|,

Hence theωn andL cancel, and we are left with

G =Ke2

2π=Ke2

h, (A.16)

upon restoring~ = h/2π. Note that this is valid for an infinite helical Luttinger liquid in

the absence of Fermi liquid leads [71, 72]. The presence of Fermi liquid leads contacting a

finite segment of helical Luttinger liquid of lengthL introduces a infrared cutoff frequency

ωL = ~vF/L in the theory. The present calculation is thus only valid forfrequencies above

this cutoff, in which case the helical Luttinger liquid dominates the physics (L is large). For

frequencies below this cutoff, e.g. for the true dc conductance of a real system contacted

to Fermi liquid leads, the transport is dominated by the Fermi liquid leads and there is a

crossover to the noninteracting valueG = e2/h atω . ωL [71, 72, 232].

In the presence of the impurity, the current-current correlation function is not trans-

lationally invariant, but the dc current is still independent of x. Indeed, the continuity

2They actually cancel the diamagnetic part of the response [231] which we have not included explicity.


equation

−iωρ(x, ω) + ∂xjx(x, ω) = 0,

implies that∂xjx(x, ω = 0) = 0 so thatjx(x, ω = 0) = jx independent ofx. Hence

generally we can calculate the current at an arbitrary point,

jex(ω) =

∫ L

0

dx′ σxx(0, x′;ω)Ex(x

′, ω),

so that the ac conductance is given by

G(ω) =ie2

(ω + iδ)L

∫ L

0

dx′ ΠRxx(0, x

′;ω).

Finally, for impurity problems where translation invariance is broken only at a single point

x = 0, we can choose to apply the electric field only at this point [Eq. (A.22)]. The

conductance involves then only the local current-current correlation function [Eq. (A.23)].

Effective action for x = 0

Since only the local current-current correlation functionatx = 0 is needed for the computa-

tion of the conductance in the presence of the impurity, we will integrate out the Tomonaga

wavesφ away fromx = 0 and obtain an effective action forx = 0. This can be done

exactly since the perturbation due to the impurity is localized atx = 0, and therefore does

not couple to the Tomonaga waves away fromx = 0. We will present two derivations of

this (0 + 1)-dimensional effective action.

Heuristic derivation [233].—Because of periodic boundary conditions in imaginary

timeφ(x, τ + β) = φ(x, τ), we have the expansion

φ(x, τ) =1

β

∑

iωn

e−iωnτφ(x, iωn),


with ωn = 2πn/β, n ∈ Z. The Tomonaga-Luttinger action (A.13) can then be rewritten as

S0 =1

β

∑

iωn

∫

dxv

2K

(

ω2n

v2φ(x,−iωn)φ(x, iωn) + ∂xφ(x,−iωn)∂xφ(x, iωn)

)

(A.17)

Since the perturbation acts only atx = 0, to simplify the problem we treat the degrees of

freedom forx 6= 0 as classical and consider quantum fluctuations only forx = 0. In other

words, we can say that forx 6= 0, we expand the field asφ(x) = φcl(x) + δφ(x) where

the classical fieldφcl(x) satisfies the classical equation of motion, and integrate out the

fluctuationsδφ(x) which contribute only a constant to the effective action forφ(0) since

the actionS0 is quadratic. Hence we can neglect these fluctuations altogether and treat

φ(x 6= 0) as classical.

The field equation is easily obtained from Eq. (A.17) andδS/δφ = 0,

(

− ω2n

vK+

v

K∂2x

)

φ(x, iωn) = 0, (A.18)

which is obviously the equation for the propagator without the contact term. Under the

boundary conditionsφ(x→ ±∞, iωn) = 0 we obtain

φ(x, iωn) = φ(iωn)e−|ωn||x|/v, (A.19)

which is taken to be our quantum field, withφ(iωn) ≡ φ(0, iωn) the fluctuating variable.

Since this solution is symmetric underx ↔ −x, we can replace∫

dx → 2∫∞0dx in

Eq. (A.17). Integrating by parts∂xφ∂xφ, we obtain

S =1

β

∑

iωn

v

K

∫ ∞

0

dx φ(x,−iωn)

(

ω2n

v2− ∂2x

)

φ(x, iωn)

− 1

β

∑

iωn

v

Kφ(x,−iωn)∂xφ(x, iωn)

∣

∣

∣

∞

0.

The first term vanishes becauseφ(x, iωn) satisfies the field equation (A.18). The last term is

the boundary term coming from the integration by parts. Substituting the solution (A.19),

only φ(iωn) remains (the field vanishes at the upper boundaryx → ∞ because of the


boundary conditions) and the effective action becomes [71,72]

Seff =1

βK

∑

iωn

|ωn||φ(iωn)|2.

where we have used the fact that sinceφ(τ) ≡ φ(0, τ) is real, thenφ(−iωn) = φ(iωn)∗.

Derivation through auxiliary fields.—The same result can be obtained somewhat more

rigorously by using auxiliary fields [234]. Consider the following representation of the

delta “functional”,

δ[φ(0, τ)− φ0(τ)] =

∫

Dλ(τ) ei∫ β0 dτ λ(τ)[φ(0,τ)−φ0(τ)],

whereλ(τ) is an auxiliary field. We insert a resolution of unity

1 =

∫

Dφ0 δ[φ(0, τ)− φ0(τ)],

in the path integral for the partition function,

Z =

∫

Dφ0DλDφ e−

∫ β0 dτ(

L0(φ,∂φ)+iλ(τ)[φ(0,τ)−φ0(τ)])

,

where

L0(φ, ∂φ) =

∫

dxv

2K

[

1

v2(∂τφ)

2 + (∂xφ)2

]

,

is the Lagrangian. Going to momentum space

D(x, τ) =1

β

∑

iωn

∫

dq

2πe−iωnτeiqxD(q, iωn),

the propagator is

D(q, iωn) =vK

ω2n + v2q2

,

and the action is

S0[φ] =1

2

1

β

∑

iωn

∫

dq

2πφ(−q,−iωn)D(q, iωn)

−1φ(q, iωn).


We have

∫ β

0

dτ λ(τ)φ(0, τ) =1

β

∑

iωn

∫

dq

2πλ(−iωn)φ(q, iωn),

∫ β

0

dτ λ(τ)φ0(τ) =1

β

∑

iωn

λ(−iωn)φ0(iωn),

therefore

Z =

∫

Dφ0DλDφ

×e− 12

1β

∑iωn

∫dq2π

φ(−q,−iωn)D(q,iωn)−1φ(q,iωn)+i 1β

∑iωn

λ(−iωn)[∫

dq2π

φ(q,iωn)−φ0(iωn)].

The integral being Gaussian, we can integrate outφ,

Z =

∫

Dφ0Dλ e− 1

21β

∑iωn

∫dq2π

λ(−iωn)D(q,iωn)λ(iωn) e−i 1β

∑iωn

λ(−iωn)φ0(iωn),

but∫

dq

2πD(q, iωn) =

∫ ∞

−∞

dq

2π

vK

ω2n + v2q2

=K

2|ωn|,

using contour integration. We can finally integrate out the auxiliary field λ,

Z =

∫

Dφ0 e− 1

βK

∑iωn

|ωn||φ0(iωn)|2 =

∫

Dφ0 e−S0[φ0],

where

S0[φ0] =1

βK

∑

iωn

|ωn||φ0(iωn)|2,

is our effective(0+1)-dimensional action for the impurity site. As will be shown later, this

effective action is nonlocal in the time domain (we drop the subscript onφ0 for simplicity),

1

βK

∑

iωn

|ωn||φ(iωn)|2 =π

2β2K

∫ β

0

dτ

∫ β

0

dτ ′(

φ(τ)− φ(τ ′))2

sin2 π(τ − τ ′)/β, (A.20)

and therefore cannot be written as a simple quantum mechanics problem with a Hermitian

single-particle Hamiltonian. In fact, Eq. (A.20) corresponds to a finite temperature version


of the Caldeira-Leggett action for quantum dissipative tunnelling, with φ ≡ φ(x = 0)

playing the role of the coordinateq of a quantum mechanical particle,φ(x 6= 0) playing

the role of the bath degrees of freedom, and1/K playing the role of the friction coefficient

η [235].

Potential barrier at x = 0

As a first exercise, we will compute the conductance in the presence of a classical magnetic

impurity, corresponding to the local perturbation

δH =

∫

dxU(x)ψ†(x)ψ(x),

with a magnetic potentialU(x) localized aroundx = 0. We neglect forward scattering (i.e.

theψ†RψR, ψ†

LψL terms) and consider only2kF backscattering (i.e. theψ†RψL,ψ†

LψR terms).

We will then be able to essentially read off the results for the Kondo and two-particle

backscattering terms from the calculation for a magnetic potential. Upon bosonization we

obtain the following contribution to the effective action,

δSeff = u1

∫ β

0

dτ cos 2√πφ(τ),

whereu1 is the2kF component of the Fourier transform ofU(x). Therefore our effective

actionS = S0 + δSeff becomes

S =1

βK

∑

iωn

|ωn||φ(iωn)|2 + u1

∫ β

0

dτ cos 2√πφ(τ), (A.21)

which is known as the boundary sine-Gordon model. This modelappears in various con-

texts, such as quantum diffusion in the presence of a periodic potential [236, 237], resis-

tively shunted Josephson junctions [238, 239], and a potential impurity in an interacting

one-dimensional electron gas [71, 72].


Conductance from the Kubo formula

In this section we will calculate the equilibrium conductance toO(u21) using the Kubo

formula. Instead of considering a constant electric field ona segment of lengthL, we

consider an abrupt potential which has a step precisely at the impurity site (but the overall

potential difference between left and right “leads” is the same). The electric field is thus

localized on the impurity,

Ex(x′, ω) = V (ω)δ(x′), (A.22)

hence the conductance is given by

G(ω) = σxx(0, 0;ω) =ie2

ω + iδΠR(0, ω), (A.23)

whereΠR ≡ ΠRxx. Since what we callG is actually only the real part of the dc conductance,

we have

G = − limω→0

e2

ωImΠR(0, ω). (A.24)

For 1D systems, analytic continuation in the frequency domain is more difficult than in the

time domain. We will therefore rather calculate the Matsubara correlatorΠ(0, τ) in the

time domain, and obtain the frequency-dependent retarded functionImΠR(0, ω) directly

from the Matsubara function (see Sec. A.2.1). The current-current correlator is

Π(0, τ) =1

π〈Tτ∂φ(τ)∂φ(0)〉,

where we use the short-hand notation∂φ(τ) ≡ ∂τφ(τ). ToO(u21), we have

〈Tτ∂φ(τ)∂φ(0)〉 = 〈∂φ(τ)∂φ(0)〉0 +u212

∫ β

0

dτ1

∫ β

0

dτ2

×〈∂φ(τ)∂φ(0) cos 2√πφ(τ1) cos 2√πφ(τ2)〉0,conn

+O(u41), (A.25)


where the connected 4-point correlator is

〈∂φ(τ)∂φ(0) cos 2√πφ(τ1) cos 2√πφ(τ2)〉0,conn

= 〈∂φ(τ)∂φ(0) cos 2√πφ(τ1) cos 2√πφ(τ2)〉0

−〈∂φ(τ)∂φ(0)〉0〈cos 2√πφ(τ1) cos 2

√πφ(τ2)〉0,

and we denote the (time-ordered) path integral average withS0[φ] as〈· · · 〉0 for simplicity.

In Sec. A.2.2 we show that the linear term〈∂φ(τ)∂φ(0) cos 2√πφ(τ1)〉0 vanishes due to the

neutrality condition [Eq. (A.35)], and derive the second-order term, i.e. the 4-point function

Eq. (A.36). The first term in Eq. (A.25) gives the unperturbedconductanceKe2/h which

we have already obtained, hence we evaluate only the second term.

Using the notation

P (τ1 − τ2) = e4π〈φ(τ1)φ(τ2)〉0 , f(τ − τ1) = 〈∂φ(τ)φ(τ1)〉,

and noting that we neglect theδ-function terms which arise when the derivative is pulled

out of a time-ordered product (as explained previously), weobtain

Π(0, τ) = −K2u21

4

1

β

∑

iωn

e−iωnτ [2P (0)− P (iωn)− P (−iωn)],

whereP (0) = P (iωn = 0). Theφ propagator was derived previously from Eq. (A.6) and

(A.7). Keeping the short-time cutoffτc explicitly, we have

〈φ(τ)φ(0)〉0 =K

2πln

πτc/β

sin(π|τ |/β) ,

at finite temperature, while at zero temperatureβ → ∞, we have

〈φ(τ)φ(0)〉0 β→∞−→ K

2πlnτc|τ | . (A.26)


We therefore have

〈Tτ : ei2√πφ(τ1) : : e−i2

√πφ(τ2) : 〉0 = e4π〈φ(τ1)φ(τ2)〉0

=

(

πτc/β

sin(π|τ1 − τ2|/β)

)2K

= P (τ1 − τ2). (A.27)

The functionP (τ) is thus given by

P (τ) =

(

πτc/β

sin π|τ |/β

)2K

= θ(τ)P+(τ) + θ(−τ)P−(τ),

hence we can extract

P±(τ) =

(

πτc/β

sin π(±τ)/β

)2K

,

which will be necessary for the analytic continuation. SinceP (τ) is even and is periodic

with periodβ, it is easy to show thatP (−iωn) = P (iωn) and we thus have

Π(0, iωn) = −K2u21

2[P (0)− P (iωn)],

with the corresponding (imaginary part of the) retarded function

ImΠR(0, ω) = −K2u21

2[ImPR(0)− ImPR(ω)].

Since we do not know whatP (iωn) is, we need to evaluateImPR(ω) using the analytic

continuation formula in the time domain, Eq. (A.30) (see Sec. A.2.1). We have

ImPR(ω) = 12

(

πτcβ

)2K ∫ ∞

−∞dt eiωt

[

1

sin2K(πit/β)− 1

sin2K(π(−it)/β)

]

.

We remark that the term in the square brackets is odd int. Since we are interested in the dc

limit, we expandeiωt ≃ 1+ iωt, and thus the first term in the expansion vanishes. Thus we


see thatImPR(0) = 0 and

ImΠR(0, ω) = 12K2u21 ImPR(ω). (A.28)

We obtain

ImPR(ω → 0) = 2ω sin πK

(

πτcβ

)2K (β

π

)2 ∫ ∞

0

dx x sinh−2K x,

were we used1

i2K− 1

(−i)2K = e−iπK − eiπK = −2i sin πK.

The integral is given by

∫ ∞

0

dx x sinh−2K x =π

4

Γ(12)Γ(K)

Γ(12+K)

csc πK,

hence

ImPR(ω → 0) =ω

2π

Γ(12)Γ(K)

Γ(12+K)

(πτc)2KT 2(K−1).

From Eq. (A.28) and (A.24), we thus obtain

∆G ≡ G−G0 = −(

Ke2

h

)

12K

Γ(12)Γ(K)

Γ(12+K)

(u1Λ

)2(

T

Λ

)2(K−1)

, (A.29)

whereG0 = K e2

h, we have restored~ andΛ = ~/πτc is a high-energy cutoff.

In the presence of Fermi liquid leads, Eq. (A.29) would rather read

∆G = −(

KLe2

h

)

12KL

Γ(12)Γ(KW )

Γ(12+KW )

(u1Λ

)2(

T

Λ

)2(KW−1)

,

for ~v/L ≪ T < Λ, whereKL is the Luttinger parameter of the leads andKW is that

of the wire [65]. This is because if we treated the boundary conditions correctly for ax-

dependentK(x) parameter, Eq. (A.28) would readImΠR(0, ω) = 12K2

Lu21 ImPR(ω), i.e.

theff prefactor becomes long-ranged in the dc limit and probes thephysics of the leads,

whereas theP correlator corresponds to2kF backscattering and as such depends only on


short-distance physics and probes only the physics of the wire.

This was for a classical magnetic impurity. For the Kondo problem, the calculation is

very similar, and we only need to perform the substitutions

u21 →2

3π2

(

a

ξ

)2

S(S + 1)J2‖ , KL → 1 (for Fermi liquid leads), KW → K,

where the factorS(S + 1) comes from the evaluation of the impurity spin correlators as

we now discuss. The thermodynamic average of a functionO(Si) of the impurity spin

operatorsSi is given by

〈O(Si)〉 =

S∑

m=−S

〈m|O(Si)|m〉

S∑

m=−S

1

,

where theSz eigenstates|m〉 such thatSz|m〉 = m|m〉 form a complete set of states for the

impurity spin and the impurity spin has no dynamics (HS = 0). It is not hard to show by

using theSU(2) commutation relations[Si, Sj] = iǫijkSk that

〈m|TτS±(τ1)S∓(τ2)|m〉 = S(S + 1)−m2 ±m sgn(τ1 − τ2),

〈m|TτSz(τ1)Sz(τ2)|m〉 = m2.

Using the following sums,

S∑

m=−S

1 = 2S + 1,

S∑

m=−S

m = 0,

S∑

m=−S

m2 =1

3S(S + 1)(2S + 1),

we find

〈S±(τ1)S∓(τ2)〉 =2

3S(S + 1), 〈Sz(τ1)Sz(τ2)〉 =

1

3S(S + 1)(2S + 1),

where we see explicitly that the spin correlators have no dynamics. In this calculation,

we neglect the forward scattering contribution proportional to 〈SzSz〉, as it must vanish by

unitarity (the conductance cannot exceed the unitarity limit G = e2/h per edge).


The Kubo formula result for Kondo scattering in the weak coupling limit is thus

∆GK

e2/h= −Γ(1

2)Γ(K)

Γ(12+ K)

1

3π2S(S + 1)

(

J‖Λ

)2(a

ξ

)2(T

Λ

)2(K−1) [

1 + 2νJz ln

(

Λ

T

)]

,

whereΛ = ~/πτc = ~v/πξ with vτc = ξ, andK = K(1 − νJz/2K)2 as explained in

Sec. 3.3. However we have

(

J‖Λ

)2(a

ξ

)2

= π4(νJ‖)2,

whereν = a/πv is the (renormalized) density of states. Hence we obtain

∆GK

e2/h= −Γ(1

2)Γ(K)

Γ(12+ K)

π2

3S(S + 1)(νJ‖)

2

(

T

Λ

)2(K−1) [

1 + 2νJz ln

(

Λ

T

)]

.

The calculation for the correction due to two-particle backscattering is very similar to the

case of a magnetic potential; the only difference is the scaling dimension of the cosine

operator.

A.2.1 Analytic continuation in the time domain

The Matsubara Green function for theφ boson is

D(τ) = −〈Tτφ(τ)φ(0)〉 = θ(τ)D+(τ) + θ(−τ)D−(τ),

which definesD±(τ). With the spectral functionA(ω) = −2 ImDR(ω), we have the

Lehmann representation

D(iωn) =

∫

dω′

2π

A(ω′)

iωn − ω′ = −∫

dω′

π

ImDR(ω′)

iωn − ω′ ,

and

D(τ) = −∫

dω′

πImDR(ω′)

1

β

∑

iωn

e−iωnτ

iωn − ω′ .

We need to calculate the Matsubara sum. We need to pick a function which has poles at


Figure A.1: Integration contour for evaluating Matsubara sum.

the (bosonic) Matsubara frequenciesiωn. For τ > 0, we picknB(−z) which has residues

−1/β and forτ < 0, we picknB(z) which residues1/β. (nB(z) ≡ (eβz − 1)−1 is the

Bose function.) We choose the contour as in Fig. A.1a. The contour encloses the poles

clockwise, so we have

∫

dznB(−z)e−zτ

z − ω′ = −2πi∑

iωn

(

− 1

β

)

e−iωnτ

iωn − ω′ , τ > 0,

∫

dznB(z)e

−zτ

z − ω′ = −2πi∑

iωn

(

1

β

)

e−iωnτ

iωn − ω′ , τ < 0,

so we can write

1

β

∑

iωn

e−iωnτ

iωn − ω′ =

1

2πi

∫

dznB(−z)e−zτ

z − ω′ , τ > 0,

− 1

2πi

∫

dznB(z)e

−zτ

z − ω′ , τ < 0.


Now we deform the contour as in Fig. A.1b, pushing the vertical lines to the left and to the

right. Our choice of different functionsnB(±z) ensures that the integral on the right and

left infinite arcs vanishes. The only contribution is the small circle which encloses the pole

at z = ω′ counterclockwise. We thus have

1

β

∑

iωn

e−iωnτ

iωn − ω′ =

e−ω′τ

e−βω′ − 1, τ > 0,

− e−ω′τ

eβω′ − 1, τ < 0,

so we obtain

D+(τ) =

∫

dω′

πImDR(ω′)

e−ωτ ′

1− e−βω′ ,

D−(τ) =

∫

dω′

πImDR(ω′)

e−ωτ ′

eβω′ − 1.

UsingnB(−ω) + nB(ω) = −1, we have

D+(τ)−D−(τ) =

∫

dω

πImDR(ω)e−ωτ ,

such that settingτ = it, we have

12[D+(it)−D−(it)] =

∫

dω

2πImDR(ω)e−iωt.

Taking the inverse Fourier transform, we have

ImDR(ω) = 12

∫ ∞

−∞dt[D+(it)−D−(it)]e

iωt. (A.30)

Equation (A.30) allows us to perform analytic continuationfrom the Matsubara Green func-

tions directly in the time domain.

More generally, the time-ordered functionDT (t) defined as

iDT (t) = 〈Ttφ(t)φ(0)〉 =tr e−βKTtφ(t)φ(0)

tr e−βK, (A.31)


with K ≡ H − µN and

DT (t) = θ(t)D>(t) + θ(−t)D<(t), (A.32)

has a branch cut on the real axis (we consider the complext-plane now), with a disconti-

nuity across the cut given

DT (t + iǫ)−DT (t− iǫ) = D<(t)−D>(t), (A.33)

for t real andǫ > 0 (see Fig. A.2). The sums in the trace in Eq. (A.31) are assumedto be

absolutely convergent if the terms in each sum are of the forme−A with ReA > 0 [240].

Then

iD>(t) =e−βKeitHφ(0)e−itHφ(0)

tr e−βK,

converges absolutely forRe(β−it) > 0 andRe(it) > 0 (there are two sums in the Lehmann

representation because we need to insert two complete sets of states), and

iD<(t) =e−βKe−itHφ(0)eitHφ(0)

tr e−βK,

converges absolutely forRe(β + it) > 0 andRe(−it) > 0. Hence

D>(t) is analytic for− β < Im t < 0,

D<(t) is analytic for0 < Im t < β.

From its definition on the real axis [Eq. (A.32)], we extend the definition ofDT (t) to the

complext-plane as shown in Fig. A.2, withDT (t) = D<(t) in the upper half-plane and

DT (t) = D>(t) in the lower half-plane. So in particular we have

DT (t+ iǫ) = D<(t) andDT (t− iǫ) = D>(t),

so Eq. (A.33) follows. From the definition of the Matsubara Green function, we have

D±(τ) = −tr e−βKe±τKφ(0)e∓τKφ(0)

tr e−βK,


Figure A.2: Analytic structure of the time-ordered functionDT (t) in the complext-plane.The function has a cut on the real axis and is analytic elsewhere for−β < Im t < β.

from which we immediately see that

iD>(t) = −D+(it) andiD<(t) = −D−(it).

As a result, the retarded and advanced functions can be written as

DR,A(t) = ±θ(±t)[D>(t)−D<(t)]

= ±iθ(∓t)[D+(it)−D−(it)],

with the upper sign (lower sign) corresponding to the retarded (advanced) function. The

spectral function follows,

A(t) = i[DR(t)−DA(t)] = D−(it)−D+(it),


so we obtain

ImDR(ω) = −12

∫

dt eiωtA(t)

= 12

∫

dt eiωt[D+(it)−D−(it)],

and we recover Eq. (A.30). The imaginary part of the retardedfunction can thus be directly

obtained from the knowledge of the Matsubara in the (imaginary) time domain. This is

useful for one-dimensional systems where it is more efficient to work in the time domain

than in the frequency domain.

A.2.2 Bosonic correlators

In this section we calculate the correlators of the masslessfree boson with vertex operators.

Here〈· · · 〉 ≡ 〈· · · 〉0 means the correlators evaluated in the free theoryS0[φ], Eq. (A.7).

The correlators calculated here are not the connected ones.The disconnected parts have to

be subtracted to get〈· · · 〉conn.

Three-point function

We want to calculate〈φ(τ)φ(τ ′)eiαφ(τ1)〉. We have

〈φ(τ)φ(τ ′)eiαφ(τ1)〉 = 〈φ(τ)φ(τ ′)〉+∞∑

n=1

(iα)n

n!〈φ(τ)φ(τ ′)φ(τ1)n〉. (A.34)

We can now use Wick’s theorem. The(n + 2)-point function vanishes unlessn = 2p + 2

is even. We have

〈φ(τ)φ(τ ′)φ(τ1)2p+2〉 = 〈φ(τ)φ(τ ′)〉(2p+ 1)!!〈φ(τ1)φ(τ1)〉p+1

+(2p+ 2)〈φ(τ)φ(τ1)〉(2p+ 1)〈φ(τ ′)φ(τ1)〉×(2p− 1)!!〈φ(τ1)φ(τ1)〉p

= (2p+ 1)!!〈φ(τ1)φ(τ1)〉p[

〈φ(τ)φ(τ ′)〉〈φ(τ1)φ(τ1)〉+(2p+ 2)〈φ(τ)φ(τ1)〉〈φ(τ ′)φ(τ1)〉

]

.


Substituting back in the sum in Eq. (A.34) and using

(2p+ 1)!!

(2p+ 2)!=

1

2p+1(p+ 1)!,

the sum gives exponentials back, and we have

〈φ(τ)φ(τ ′)eiαφ(τ1)〉 =[

〈φ(τ)φ(τ ′)〉 − α2〈φ(τ)φ(τ1)〉〈φ(τ ′)φ(τ1)〉]

e−12α2〈φ(τ1)φ(τ1)〉.

However, the bosonic correlator is [241]

〈φ(z, z)φ(0, 0)〉 = − 1

4πln

(

zz + ξ2

R2

)

,

with z = τ + ix andz = τ − ix, whereξ → 0 is a short-distance (ultraviolet) cutoff and

R → ∞ is a large-distance (infrared) cutoff. The exponential factor in the correlator then

becomes

e−2πα2 ln(R2/ξ2) → 0,

in the thermodynamic limitR2/ξ2 → ∞, unlessα = 0, which is just a statement of the

neutrality condition [230]. Hereα = 2√π so the correlator vanishes,

〈φ(τ)φ(τ ′) cos 2√π(τ1)〉 = 0. (A.35)

Four-point function

We now calculate the 4-point function,

〈φ(τ)φ(τ ′)eiαφ(τ1)eiβφ(τ2)〉 = 〈φ(τ)φ(τ ′)eiα[φ(τ1)+βφ(τ2)]〉,

where the equality is valid because in the path integral average, the operators are justc-

numbers (we have a boson theory) and we can move them at will. (Wick’s theorem takes

care of the product of normal-ordered vertex operators.) Wecan again use Wick’s theorem

as for the 3-point function withαφ(τ1) + βφ(τ2) as our “third” field (since it is just a linear

combination of fields, Wick’s theorem applies also to this composite object). In this case


we get

〈φ(τ)φ(τ ′)eiαφ(τ1)eiβφ(τ2)〉 =[

〈φ(τ)φ(τ ′)〉

−〈φ(τ)[αφ(τ1) + βφ(τ2)]〉〈φ(τ ′)[αφ(τ1) + βφ(τ2)]〉]

×e−12〈[αφ(τ1)+βφ(τ2)][αφ(τ1)+βφ(τ2)]〉,

and the exponential factor becomes

e−12〈[αφ(τ1)+βφ(τ2)][αφ(τ1)+βφ(τ2)]〉 =

(

ξ2

zz + ξ2

)−αβ/4π

e−18π

(α+β)2 ln(R2/ξ2).

ForR2/ξ2 → ∞, the correlator survives only ifα + β = 0 (neutrality condition). Hence

we takeβ = −α and get

〈φφ′eiα[φ1−φ2]〉 = eα2〈φ1φ2〉

[

〈φφ′〉 − α2(

〈φφ1〉 − 〈φφ2〉)(

〈φ′φ1〉 − 〈φ′φ2〉)

]

, (A.36)

in obvious notation. The〈φφ′〉 disappears in the connected correlator.

A.3 Coulomb gas description of the strong coupling regime

In this section, we derive the low-energy effective theory around the strong coupling fixed

point corresponding to the insulating phaseK < Kc = 1/4. Using a semiclassical instan-

ton calculation, we show that the effective theory is equivalent to a classical 1D Coulomb

gas. The coordinate of a particle in the Coulomb gas corresponds to the (imaginary) time

at which an instanton pumps chargee/2 across the cut between the two semi-infinite he-

lical liquids. We also derive the RG equation for the half-charge tunneling amplitudet in

the strong coupling limit, as well as the correct scaling exponent for the low temperature

behavior of the conductance.


A.3.1 Euclidean action

Let us recall the action we are working with. The free part is

S0[φ] =1

βK

∑

iωn

|ωn||φ(iωn)|2.

Sinceφ(τ) is a real scalar field, we haveφ∗(ωn) = φ(−ωn) so we get

S0[φ] =2

βK

∞∑

n=1

ωn|φ(iωn)|2,

since we have bosonic frequenciesωn = 2nπ/β. By Fourier transformation we have

ωnφ(iωn) =∫ β

0dτ eiωnτ i∂τφ(τ), hence we can write

S0[φ] =2

βK

∞∑

n=1

∫ β

0

dτ

∫ β

0

dτ ′ e−iωn(τ−τ ′)φ(τ)i∂τ ′φ(τ′).

From the geometric sum

∞∑

n=1

e−iωn(τ−τ ′) =e−iπ(τ−τ ′)/β

2i sin π(τ − τ ′)/β,

we can write

S0[φ] =1

βK

∫ β

0

dτ

∫ β

0

dτ ′ φ(τ)e−iπ(τ−τ ′)/β

sin π(τ − τ ′)/β∂τ ′φ(τ

′).

However the action should be realS0 = ReS0, hence we obtain

S0[φ] =1

βK

∫ β

0

dτ

∫ β

0

dτ ′ φ(τ)[cot π(τ − τ ′)/β]∂τ ′φ(τ′).

We now integrate by parts onτ ′. Sincecot πτ/β is periodic inτ with periodβ and the bo-

son field obeys periodic boundary conditionsφ(τ+β) = φ(τ), the boundary term vanishes.


Usingcot′ x = −1/ sin2 x, we obtain

S0[φ] = − π

β2K

∫ β

0

dτ

∫ β

0

dτ ′φ(τ)φ(τ ′)

sin2 π(τ − τ ′)/β.

However, we have

∫ β

0

dτ

∫ β

0

dτ ′φ2(τ)

sin2 π(τ − τ ′)/β= −

∫ β

0

dτ φ2(τ)

∫ τ−β

τ

ds

sin2 πs/β

=β

π

∫ β

0

dτ φ2(τ)

[

cotπ(τ − β)

β− cot

πτ

β

]

= 0,

because of the periodicity of the cotangent. By symmetry thesame is true forφ2(τ ′), hence

we can write

S0[φ] =π

2β2K

∫ β

0

dτ

∫ β

0

dτ ′(φ(τ)− φ(τ ′))2

sin2 π(τ − τ ′)/β. (A.37)

This was for the free part. Now, we also have the cosine potential, i.e. the two-particle

backscattering term,

S2[φ] = λ2

∫ β

0

dτ cos 4√πφ(τ).

To the action we add an ultraviolet regulator [61]

S ′0[φ] ≡

M

2

∫ β

0

dτ(∂τφ)2,

with a large massM , which will suppress the high-energy/short-time contribution to the

path integral with a factore−M2

∫dτ(∂φ)2 ∼ e−M

∑ω ω2φ2

ω/2. Indeed, the free action has the

same short-time behavior,

∫ β

0

dτ

∫ β

0

dτ ′(φ(τ)− φ(τ ′))2

sin2 π(τ − τ ′)/β

|τ−τ ′|≪β−→∫ β

0

dτ

∫ β

0

dτ ′(∂τ ′φ)

2(τ − τ ′)2(

πβ

)2

(τ − τ ′)2∝∫ β

0

dτ(∂τφ)2,


hence the full action reads

S[φ] =π

2β2K

∫ β

0

dτ

∫ β

0

dτ ′(φ(τ)− φ(τ ′))2

sin2 π(τ − τ ′)/β

+M

2

∫ β

0

dτ(∂τφ)2 + λ2

∫ β

0

dτ cos 4√πφ(τ).

A.3.2 Semiclassical instanton calculation

In order to perform a saddle-point approximation to the pathintegral for largeλ2, we first

rescale timeτ → τ =√λ2τ , φ(τ) → φ(τ), sin π(τ − τ ′)/β → sin π(τ − τ ′)/β

√λ2. We

obtain

S0[φ] → S0[φ] = (√

λ2)−2

∫ β√λ2

0

dτ

∫ β√λ2

0

dτ ′(

φ(τ )− φ(τ ′))2

sin2 π(τ − τ ′)/β√λ2.

Since we are concerned with the low-temperature regime, we will always consider thatβ

is very large. Therefore the sine function is in the linear region,

sin π(τ − τ ′)/β√

λ2 ≃1√λ2

sin π(τ − τ ′)/β,

hence we obtain

S0[φ] ≃∫ β

√λ2

0

dτ

∫ β√λ2

0

dτ ′(

φ(τ)− φ(τ ′))2

sin2 π(τ − τ ′)/β.

This means that the free action is unchanged (forβ → ∞, β√λ2 = β = ∞ anyway). We

will omit the integration bounds with the understanding that we always consider largeβ (in

the vicinity of zero temperature). However, the regulator and the cosine term are affected

by the rescaling,

S2[φ] → S2[φ] =√

λ2

∫

dτ cos 4√πφ(τ),

and

S ′0[φ] → S ′

0[φ]M

2√λ2

∫

dτ

(

∂φ(τ )

∂(τ /√λ2)

)2

=√

λ2M

2

∫

dτ(∂τ φ)2,


hence the rescaled action reads (we now writeφ asφ for simplicity)

S[φ] = S0[φ] +√

λ2

(∫

dτM

2(∂τφ)

2 +

∫

dτ cos 4√πφ

)

.

Sinceλ2 is very large, it dominates over the kinetic partS0, hence we neglectS0 and

perform a saddle-point approximation to the path integral∫

Dφ e−√λ2F [φ] by minimizing

the functionalF [φ] =∫

dτ L(φ, ∂τφ). We have

δ

δφ

∫

dτ L(φ, ∂τφ) = 0 ⇒ ∂τ

(

∂L

∂∂τφ

)

=∂L

∂φ,

and we obtain the Euler-Lagrange equation

Md2φ

dτ 2=dV (φ)

dφ, (A.38)

which is obviously the sine-Gordon equation, whereV (φ) = cos 4√πφ is the “inverted”

potential. We wish to consider an instanton solution corresponding to a soliton solution

of the sine-Gordon equation. We want to tunnel between nearby minima of the cosine

potential, say betweenφ = ±√π/4. We thus search for a “kink” solution withφ(τ =

±∞) = ±√π/4. The anti-kink corresponds toφ(τ = ±∞) = ∓√

π/4.

We multiply the sine-Gordon equation (A.38) by∂τφ and integrate from−∞ to τ . We

obtainM

2

(

dφ

dτ

)2

= V (φ)− V (−√π/4),

assuming thatdφ(−∞)/dτ = 0 which is required if the kink is to have a finite action. The

cosine potential givesdφ

dτ= ±

√

2

M

√

1 + cos 4√πφ.

We pick the positive slope which corresponds to the kink (theanti-kink has negative slope).

This equation can be integrated. We fix the origin of the kink (whereφ = 0) at τ = τ0, and


integrate fromτ0 to τ , meaning fromφ = 0 to φ = φ(τ):

∫ φ(τ)

0

dφ√

1 + cos 4√πφ

=

√

2

M

∫ τ

τ0

dτ =

√

2

M(τ − τ0).

Performing the integral, we obtain the kink solution,

φ(τ) =1√πarctan

(

tanh 2

√

π

M(τ − τ0)

)

.

One can check that it satisfies the sine-Gordon equation, theboundary condition on the

slopedφ(±∞)/dτ ∝ sech(x = ±∞) = 0 and on the endpointsφ(±∞) = ±√π/4. The

latter is easy to check sincetanh(±∞) = ±1 andarctan(±1) = ±π/4.

The action for this instanton can be calculated by substituting back the solution into the

action functional,

Sinst ≡ S[φ(τ)] =√

λ2

∫ ∞

−∞dτ

[

M

2(∂τ φ)

2 + cos 4√πφ

]

=√

λ2

∫ ∞

−∞dτ M(∂τ φ)

2 + infinite negative const.,

where we have extended the range of integration from0 < τ < β to −∞ < τ < ∞ since

the length of the intervalβ√λ2 is assumed to be much larger than the width of the instanton

∼√M . The infinite constant comes from the finite negative value ofthe potential minima

V (±√π/4) = −1 (it shows up from usingcos 4

√πφ = M

2(∂τφ)

2−1) but can be gotten rid

of by shifting the potential up by a constant, so it is ignored. The integral is easily evaluated

and gives

Sinst = 2

√

λ2M

π, (A.39)

which is manifestly finite. By symmetry the anti-kink has thesame action. The tunneling

ratet between nearby minima of the cosine is thus given by

t = e−Sinst,


which is the fugacity of the instanton gas. This semiclassical tunneling amplitude is ac-

tually renormalized by small fluctuation effects containedin the determinant arising from

integrating out the Gaussian fluctuationsδφ(τ) about the classical solutionφ(τ) = φ(τ) +

δφ(τ), but we will not consider these effects here (we are interested only in the leading or-

der temperature dependence of the tunneling amplitude which is a tree-level property while

these effects are one-loop effects).

A.3.3 Dilute instanton gas approximation

Sinceλ2 is large, we consider a rarefied instanton gas (few instantonevents sincet ≪ 1).

Direct tunneling across two valleys (e.g. from−√π/4 to 3

√π/4) is possible, but the

action for this event is much larger so the corresponding tunnel amplitude is exponentially

suppressed with respect tot. The leading order effect comes from several “sequential”

tunneling events.

We thus consider a multi-instanton configuration, i.e. an instanton gas configuration

where the boson fieldφ(τ) is of the form

φ(τ) =∑

i

φqi(τ − τi), (A.40)

where the “charge” indexqi = ±1 indicates a kink (qi = +1) or anti-kink (qi = −1). There

is aneutrality requirement∑

i

qi = 0,

arising from the periodic boundary condition to be satisfiedby the boson fieldφ(0) = φ(β)

in the partition function (the number of anti-kinks has to equal the number of kinks if the

field is to return to its initial value). Hencei runs from 1 to2p wherep > 0 is the number

of kink/anti-kink pairs.

Before we substitute the field configuration (A.40) into the action, we make the further

approximation ofneglecting the width of the kink, i.e. we take the limit√

M/4π → 0. The

derivative of the field isdφ

dτ=∑

i

dφqi

dτ.


The slopedφ/dτ of the kink is strongly peaked at its originτ = τ0. In the limit√M → 0,

it becomes a delta function

lim√M→0

dφ

dτ= aδ(τ − τ0),

where the value of the constanta is easily found by integration,

a =

∫ ∞

−∞dτ aδ(τ − τ0) =

∫ ∞

−∞dτ

dφ

dτ= φ(∞)− φ(−∞) =

√π

2,

from the boundary conditionφ(±∞) = ±√π/4. Hence the derivative of the field is

dφ

dτ=

√π

2

∑

i

qiδ(τ − τi), (A.41)

since the anti-kink (qi = −1) is simply−φ. So we see thatdφ/dτ is none but the charge

density of the instanton gas (which makes sense since in bosonizationjx ∝ ∂τφ is the

current density which corresponds to the density of tunneling events).

From the knowledge of the derivative∂τφ we can compute the action of the instanton

gas. We first know that

S ′0[φ] + S2[φ] =

∑

i

Sinst = 2pSinst,

corresponding to the action ofp noninteracting kink/anti-kink pairs. The nonlocal kinetic

(free) partS0[φ] gives rise to a long-range interaction between the instantons (i.e.|τ−τ ′| ≫width of the instanton∼

√M ). We substitute Eq. (A.41) in Eq. (A.37) and consider the


low-temperatureβ → ∞ limit. We obtain

S0[φ] =1

2πK

∫

dτ

∫

dτ ′(φ(τ)− φ(τ ′))2

(τ − τ ′)2

= − 1

πK

∫

dτ

∫

dτ ′φ(τ)φ(τ ′)

(τ − τ ′)2

= − 1

πK

∫

dτ

∫

dτ ′ ∂τφ(τ)∂τ ′φ(τ′) ln |τ − τ ′|

= − 1

4K

∑

ij

qiqj ln |τi − τj |

= − 1

2K

∑

i<j

qiqj ln |τi − τj |,

where we used the identity [242]

∫

dx dx′f(x)f(x′)

(x− x′)2=

∫

dx dx′ f ′(x)f ′(x′) ln |x− x′|,

which is easily proved by integrating by parts. Putting everything together, the partition

function becomes

Z =

∫

Dφ(τ) e−S0[φ]−S′0[φ]−S2[φ]

=

∞∑

p=1

t2p

(

2p∏

i=1

′∑

qi=±

)

∫ β

0

dτ2p

∫ τ2p−δ

0

dτ2p−1 · · ·∫ τ2−δ

0

dτ1

× exp

(

1

2K

∑

i<j

qiqj ln|τi − τj |

δ

)

, (A.42)

where we have added a short-time cutoffδ, i.e. the instanton gas is a gas of ‘hard rods’ with

diameterδ (the long-range logarithmic interaction is valid only for distances larger than the

cutoff |τ − τ ′| > δ). The prime on the sum indicates that the neutrality condition∑

i qi = 0

should be obeyed, and the instanton fugacityt = e−Sinst is also the tunneling amplitude.


A.3.4 Tree-level renormalization

We now perform renormalization of the partition functionat tree levelwhich is the leading

nonvanishing order in the beta functionβ(t) for the tunneling amplitudet. Consider an

action with reduced cutoff,

Zbδ(t) =∞∑

p=1

t2p

(

2p∏

i=1

′∑

qi=±

)

∫ bβ

0

dτ2p

∫ τ2p−bδ

0

dτ2p−1 · · ·∫ τ2−bδ

0

dτ1

× exp

(

1

2K

∑

i<j


δ

)

,

with b = 1 + dℓ. We now perform a scale transformationτi = τi/b to restore the cutoff to

δ. The integration measure transforms as

dτ2p · · · dτ1 =∂(τ2p, . . . , τ1)

∂(τ2p, . . . , τ1)dτ2p . . . dτ1 = b2p dτ2p . . . τ1,

hence

∫ bβ

0

dτ2p

∫ τ2p−bδ

0

dτ2p−1 · · ·∫ τ2−bδ

0

dτ1 = b2p∫ β

0

dτ2p

∫ τ2p−δ

0

dτ2p−1 · · ·∫ τ2−δ

0

dτ1.

The exponential becomes

exp

(

1

2K

∑

i<j


δ

)

exp

(

1

2K

∑

i<j

qiqj ln b

)

,

where the last exponential factor can be written as

exp

(

1

2K

∑

i<j

qiqj ln b

)

= b1

2K

∑i<j qiqj .

However, by neutrality of the Coulomb plasma we have

0 =

(

∑

i

qi

)2

=∑

ij

qiqj =∑

i 6=j

qiqj +∑

i

q2i = 2∑

i<j

qiqj +∑

i

1,


hence∑

i<j qiqj = −12

∑

i 1 = −p, so that we haveb1

2K

∑i<j qiqj = b−p/2K . Therefore, the

renormalized action is

Zδ =

∞∑

p=1

t2pb2p(1−1/4K)

(

2p∏

i=1

′∑

qi=±

)

∫ β

0

dτ2p

∫ τ2p−δ

0

dτ2p−1 · · ·∫ τ2−δ

0

dτ1

× exp

(

1

2K

∑

i<j


δ

)

≡ Zδ(t),

where we can now identity the renormalized tunneling amplitude by comparison with the

original partition function,

t = tb1−1/4K .

Expanding(1+ dℓ)1−1/4K = 1+ (1− 1/4K)dℓ andt = t+ (dt/dℓ)dℓ, we obtain the RGE

for the tunneling amplitude,dt

dℓ=

(

1− 1

4K

)

t.

The square of the tunneling amplitude obviously followsdt2

dℓ= 2

(

1− 14K

)

t2. Cutting off

the RG flow at a scale set by the temperatureT , we obtain the renormalized tunneling

strengtht2 as a function of temperature,

t2(T ) = t2(Λ)

(

T

Λ

)2(1/4K−1)

,

whereΛ ∼ EF is some high-energy scale of the order of the bandwidth, andt2(Λ) is the

“bare” tunneling amplitude at this energy, i.e.t2 = e−2Sinst with Sinst given by Eq. (A.39)

andλ2 taken to be the bare two-particle backscattering amplitude. Since the conductance

proceeds by tunneling, we have

G(T ) ∝ t2(T ) ∝ T 2(1/4K−1).


A.3.5 Mapping to dual boundary sine-Gordon model

To get an intuitive understanding of the low-energy physics, we want to show the equiv-

alence of the Coulomb gas partition function with the partition function of the boundary

sine-Gordon model for the dual bosonθ. By rephrasing the cosine term as a tunneling term

in the fermion language, we can identify what physical processes restore the conductance

at finite temperature.

We conjecture that the Coulomb gas partition function just derived can be written as a

functional integral over the dual bosonθ(τ) ≡ θ(x = 0, τ),

Z =

∫

Dθ e−Sdual[θ],

where

Sdual[θ] =K

β

∑

iωn

|ωn||θ(iωn)|2 + t

∫ β

0

dτ cosα√πθ(τ), (A.43)

and we have to findα andt in terms of the fugacityt = e−Sinst of the Coulomb gas model

Eq. (A.42). From the unperturbed action (the first term), theθ propagator is

〈θ(τ)θ(τ ′)〉 = − 1

2πln |τ − τ ′|1/K . (A.44)

We expand the partition function in powers oft. We have

Z

Z0= 1 +

∞∑

n=1

(−t)nn!

∫ β

0

dτn · · ·∫ β

0

dτ1〈T cosα√πθ(τ1) · · · cosα

√πθ(τn)〉,

where〈· · · 〉 =∫

Dθ e−S0[θ] · · · /Z0, with Z0 =∫

Dθ e−S0[θ] andS0[θ] the unperturbed

action. Only terms withn even survive the average, so that we have

Z

Z0

= 1 +∞∑

n=1

(−t)2n(2n)!

∫ β

0

dτ2n · · ·∫ β

0

dτ1〈T cosα√πθ(τ1) · · · cosα

√πθ(τ2n)〉.

We have explicitly written the time-ordering operatorT to stress the fact that the2n-point

correlator is automatically time-ordered because it was defined as a path integral average


(Feynman-Kac formula). We rewrite the integral over[0, β]2n as an integral over a2n-

simplex in the standard way3 [226],

∫ β

0

dτ2n · · ·∫ β

0

dτ1〈TA(τ1) · · ·A(τ2n)〉

= (2n)!

∫ β

0

dτ2n

∫ τ2n

0

dτ2n−1 · · ·∫ τ2

0

dτ1 〈A · · ·A〉,

where the last operator product is not time-ordered. The factorial is cancelled and the

partition function becomes

Z

Z0= 1 +

∞∑

n=1

t2n∫ β

0

dτ2n

∫ τ2n

0

dτ2n−1 · · ·∫ τ2

0

dτ1〈cosα√πθ(τ1) · · · cosα

√πθ(τ2n)〉.

Writing the cosine as

cosα√πθ(τi) =

12

∑

qi=±eiαqi

√πθ(τi),

we have

〈cosα√πθ(τ1) · · · cosα√πθ(τ2n)〉

=1

22n

(

2n∏

i=1

∑

qi=±

)

〈 : eiαq1√πθ(τ1) : · · · : eiαq2n

√πθ(τ2n) : 〉.

We now use the result〈 : eA1 : · · · : eAN : 〉 = exp∑N

i<j〈AiAj〉 for expectation values of

products of vertex operators [230] and theθ propagator Eq. (A.44) to obtain

〈cosα√πθ(τ1) · · · cosα√πθ(τ2n)〉 =

1

22n

(

2n∏

i=1

′∑

qi=±

)

exp

(

α2

2K

2n∑

i<j

qiqj ln |τi − τj |)

,

3Special thanks to T. T. Ong for pointing this out.


where the primed sum indicates the neutrality condition∑

i qi = 0, which is verified be-

cause the2n-point correlator of vertex operators: eiαqi√πθ(τi) : vanishes otherwise. Sub-

stituting back into the expression for the partition function, we find

Z

Z0= 1 +

∞∑

n=1

(

t

2

)2n(

2n∏

i=1

′∑

qi=±

)

∫ β

0

dτ2n

∫ τ2n

0

dτ2n−1 · · ·∫ τ2

0

dτ1

× exp

(

α2

2K

2n∑

i<j

qiqj ln |τi − τj |)

,

which is the same partition function as that of the Coulomb gas Eq. (A.42), apart from (1)

a multiplicative factorZ0 which is just a shift of free energy, (2) the first termZ/Z0 = 1

which corresponds to a configuration with no instantons (we could have included it in

Eq. (A.42)), and (3) the presence of a short-time cutoffδ which we should incorporate in

the present expression. We see that the identification of thetwo partition functions requires

t = 2t = 2e−Sinst andα2 = 1,

which implies that our “tunneling” action should read

Stunnel = t

∫ β

0

dτ cos√πθ(τ).

The dimension of this operator is easily obtained,

〈 : cos√πθ(τ) : : cos

√πθ(τ ′) : 〉 ∼ eπ〈θ(τ)θ(τ

′)〉 =1

|τ − τ ′|1/2K =1

|τ − τ ′|2∆ ,

meaning that∆ = 1/4K, which leads to the expected RGE

dt

dℓ=

(

1− 1

4K

)

t.

From the scaling dimension of the tunneling operator, we cansee that it corresponds to

tunneling the “square root” (or half) of an electron (ordinary single-particle tunneling cor-

responds to the operatorcos 2√πθ [71, 72]). In the next section we derive the form of


the current operator in the strong coupling regime and confirm the half-charge tunneling

picture.

A.3.6 Current operator in strong coupling regime

Suppose we would like to calculate the expectation value of the electric currentje = −ejat equilibrium,

〈je(τ)〉eq =ie√π〈∂τφ〉 =

ie

2

⟨

∑

i

qiδ(τ − τi)

⟩

=

∞∑

p=1

t2p

(

2p∏

i=1

′∑

qi=±

)

∫ β

0

dτ2p

∫ τ2p−δ

0

dτ2p−1 · · ·∫ τ2−δ

0

dτ1

×ie2

∑

i

qiδ(τ − τi) exp

(

1

2K

∑

i<j


δ

)

,

where we have used Eq. (A.41). Consider now a modified form of the action Eq. (A.43),

S[θ, a] =K

β

∑

iωn

|ωn||θ(iωn)|2 + 2t

∫ β

0

dτ cos[√πθ(τ) + 1

2ea(τ)]. (A.45)

The Coulomb gas representation of this action now reads

Z[a]

Z0

= 1 +∞∑

p=1

t2p

(

2p∏

i=1

′∑

qi=±

)

∫ β

0

dτ2p

∫ τ2p−δ

0

dτ2p−1 · · ·∫ τ2−δ

0

dτ1

× exp

(

1

2K

∑

i<j


δ+ie

2

∑

i

qia(τi)

)

.

We observe that

1

Z0

δZ[a]

δa(τ)=

∞∑

p=1

t2p

(

2p∏

i=1

′∑

qi=±

)

∫ β

0

dτ2p

∫ τ2p−δ

0

dτ2p−1 · · ·∫ τ2−δ

0

dτ1

× exp

(

1

2K

∑

i<j


δ+ie

2

∑

i

qia(τi)

)

ie

2

∑

i

qiδ(τ − τi),


hence we see that

〈je(τ)〉eq = lima→0

1

Z0

δZ[a]

δa(τ). (A.46)

Note that the factor of12

is required in the boson action Eq. (A.45) to correctly matchthe

prefactor in the instanton representation of the current Eq. (A.41). If we calculate the right-

hand side directly from the action Eq. (A.45), as we take the limit a(τ) → 0 we obtain

〈je(τ)〉 = te〈sin√πθ(τ)〉, hence we can identify

j = −t sin√πθ, je = te sin√πθ, (A.47)

as the current operators in the strong coupling regime, whereas the usual single-particle

tunneling operator is

iet(ψ†1ψ2 − h.c.) ∝ et sin 2

√πθ,

where the indices1, 2 refer to two disconnected semi-infinite Luttinger liquids [71, 72].

Hence we can already expect from Eq. (A.47) that in our case the excitations which tunnel

through the impurity are the “square root” of an electron, orhalf-charged quasiparticles.

A.4 Keldysh calculation of the shot noise

As the dual bosonθ is expected to transform in some way under a (local) gauge transforma-

tion, the currents Eq. (A.47) are clearly not gauge invariant. It is meaningful to calculate the

linear response of these non gauge invariant currents to an appliedelectromagnetic field,

because the piece of the current which restores gauge invariance gives a contribution to the

response which is nonlinear in the applied electromagneticfield, and therefore neglected in

a linear response calculation. However, if one is to calculate the fully nonlinear, nonequi-

librium current, then the gauge invariant (kinematical) current must be used. To obtain the

shot noise which is the noise in the presence of a finite current, we need to use the gauge

invariant current. The gauge invariant current is obtainedsimply bynot letting a → 0 in

Eq. (A.46),

〈je(τ)〉 =1

Z0

δZ[a]

δa(τ)= te sin[

√πθ(τ) + 1

2ea(τ)], (A.48)


which can be interpreted as “taking the square root” of the current operator considered by

Kane and Fisher [71, 72] for single-particle tunneling:

〈j1P〉 ∝ sin[2√πθ + ea(τ)].

In this section we want to derive an expression for the nonequilibrium noise spectrum in

the strong coupling regime at low temperatures and finite bias.

A.4.1 Basics of noise

We first review the basic ideas of noise. The noise spectrum measures the amount of

fluctuations in the current operator,

S(ω) ≡∫

dt eiωt〈∆I(t),∆I(t′)〉,

whereI(t) is the current operator in the Heisenberg picture and∆I(t) ≡ I(t)− 〈I〉 where

we assume steady-state such that the average current〈I〉 does not depend on time. We have

〈∆I(t),∆I(t′)〉 = 〈I(t), I(t′)〉 − 2〈I〉2 = 〈I(t), I(t′)〉conn,

and the zero-frequency noise is thus given by

S(ω = 0) =

∫

dt 〈I(t), I(t′)〉conn.

Classical shot noise: Schottky formula

Consider a mesoscopic device connected to external leads. There are two sources of noise.

Thermalnoise occurs due to the thermal fluctuations of the occupation numbers in the

leads, exists at zero bias, and vanishes at zero temperature. Shotnoise is the excess noise

in the presence of a finite current and is a nonequilibrium process. It is due to the fact that

current is carried by discrete charges.

The expression for the noise in a classical conductor is simply derived in the following

way. Consider that the current proceeds by random “tunneling” events in which a particle


with chargee is transported from one lead to the other. If we assume that the tunneling

events are uncorrelated, the probability for havingN tunneling events during timeτ is

given by the Poisson distribution,

PN(τ) =〈N〉NN !

e−〈N〉.

It is well-known that for the Poisson distribution the variance and mean are equal,

〈N2〉 − 〈N〉2 = 〈N〉. (A.49)

If we define the current as the random variableI ≡ eN/τ , the average current is given by

〈I〉 = e〈N〉/τ. (A.50)

The zero-frequency noise is then given by

S(ω = 0) =

∫ τ

0

dt(

〈I(t), I(t′)〉 − 2〈I〉2)

= 2

∫ τ

0

dt(

〈I2〉 − 〈I2〉)

= 2τe2(〈N2〉 − 〈N〉2)/τ 2.

Using Eq. (A.49) and (A.50), this is just

S(ω = 0) = 2e〈I〉, (A.51)

which is the Schottky formula. In the presence of a finite current, we thus see that the noise

contains information about the charge of the tunneling particles.

Equilibrium/thermal/Johnson-Nyquist noise

The zero-frequency thermal noise is

S(ω = 0) = 4GkBΘ,


whereG is the equilibrium conductance andΘ is the temperature. A simple derivation

follows below [243]. The current through a noninteracting two-terminal device with trans-

mission coefficientT (E) and twofold spin degeneracy is given by

I =2e

h

∫

dE T (E)[fL(E)− fR(E)],

whereE is the energy of a tunneling electron andfL, fR are the electron distribution

functions in the left and right leads, respectively. For simplicity, consider the classical limit

(the conclusions are unchanged in the Fermi-Dirac case [243]) where(E − µ)/kBΘ ≫ 1,

f(E) = (eβ(E−µ) + 1)−1 → e−β(E−µ).

Then one has

fL(E)− fR(E) = e−β(E−µ−eVL) − e−β(E−µ−eVR) = (eβeVL − eβeVR)e−β(E−µ).

Now assume the limit of small biasVL,R ≪ kBΘ,

fL(E)− fR(E) → βe(VL − VR)e−β(E−µ),

hence the equilibrium conductance is given by

G = limVL−VR→0

I

VL − VR=

2e2

kBΘh

∫

dE T (E)e−β(E−µ).

The idea is that although the net current in equilibrium is zero I = IL + IR = 0 since

IL = −IR, the noise produced by each stream|IL| = |IR| adds:

IS = |IL|+ |IR| = 2|IL| 6= 0.

At equilibriumVL = VR = 0, the current injected by the left lead is

I =2e

h

∫

dE T (E)fL(E) →2e

h

∫

dE T (E)e−β(E−µ),


Hence the total “noise current”IS is

IS = 2|IL| =4e

h

∫

dE T (E)e−β(E−µ) =2GkBΘ

e.

Thermal noise is the “shot noise” produced by this current. Using the Schottky relation

Eq. (A.51) withI = IS, we have

S(ω = 0) = 4GkBΘ,

hence we see that the thermal noise contains the same information as the conductance, i.e.

it does not contain any information about the charge of the tunneling particle.

Quantum shot noise: thermal/shot noise crossover

A quantum mechanical treatment yields [73]

S(ω = 0) =4e2

h

[

2kBΘT 2 + eV coth

(

eV

2kBΘ

)

T (1− T )

]

,

whereT is the transmission coefficient. We consider the limit of weak tunnelingT ≪ 1

and thus neglect the termsO(T 2). We obtain

S(ω = 0) = 2e〈I〉 coth(

eV

2kBΘ

)

.

The thermal noise is recovered at low biaseV ≪ kBΘ,

S(ω = 0) → 4GkBΘ, eV ≪ kBΘ,

whereG = limV→0〈I〉/V is the equilibrium conductance, and we usedcoth x ≃ 1/x for

x ≪ 1. In the high bias regimeeV ≫ kBΘ, the Schottky relation for the shot noise is

recovered,

S(ω = 0) → 2e〈I〉, eV ≫ kBΘ,

usingcothx→ 1 for x≫ 1.


A.4.2 Schwinger-Keldysh action

We now go back to our initial problem. Since we are considering a nonequilibrium situ-

ation, we need to consider a real-time action defined on the Schwinger-Keldysh contour

C [244, 245],

S[θ, A] = S0[θ] + 2t

∫

Cdt cos[

√πθ(t) + 1

2eA(t)],

where we denote the tunneling amplitude byt ≡ t to avoid confusion with the real time

variablet, andA(t) is the vector potential in real time,A(t) = −V t. In this case the current

is given by〈I(t)〉 = 1Z

δZδiA(t)

. We can make a consistency check of this formula by noticing

that in the presence of the usual electromagnetic coupling term

δS[θ, A] =

∫

dt je(t)A(t),

the above-mentioned functional derivative does indeed give rise to the expectation value of

the current,

〈I(t)〉 = 1

Z

δ

δiA(t)

∫

Dθ eiS0[θ]+i∫dt je(t)A(t) = 〈je(t)〉.

We want to derive a relation [73] between the zero-frequencynoise spectrum and the

nonequilibrium current,

S(ω = 0) = 2e∗|〈I〉| coth(

e∗V

2kBT

)

, (A.52)

wheree∗ is the charge of the tunneling quasiparticles. This relation holds for the interedge

tunneling of quasiparticles in the fractional QH regime (inthe weak tunneling regime) and

has been used experimentally to confirm the fractional charge e∗ = νe of the quasiparti-

cles [246, 247] whereν is the filling factor. The goal is to show that it also holds in our case

and thate∗ = e/2 in the strong coupling regime. The calculation proceeds in two steps:

a) calculating the nonequilibrium current〈I〉, b) calculating the zero-frequency noise spec-

trum S(ω = 0). By comparing the obtained results with Eq. (A.52), we can read out the

quasiparticle chargee∗.


A.4.3 Nonequilibrium current

In the Hamiltonian representation we have [73]

〈I(t)〉 = 12

∑

η

〈TCI(tη)e−i∫C dt1 H′(t1)〉, (A.53)

where the current operator is in the interaction representation with respect to the perturba-

tionH ′ defined as

H ′(t1) = −2t : cos[√πθ(t1) +

12eA(t1)] : .

In Eq. (A.53), TC is the contour-ordering operator on the Schwinger-KeldyshcontourC,

andη = ±1 denotes the forward (+) or backward (−) part of the contour. Since the current

〈I(t)〉 is a one-point correlation function, it does not matter on which branch of the contour

we choose to evaluate it, and we choose the symmetric combination 〈I(t)〉 = 12

∑

η〈I(tη)〉.At low temperaturesT ≪ T ∗

2 , the tunneling is weak and we calculate the current to

O(t2). Since the current operatorI is alreadyO(t), we need only expand the evolution

operatore−i∫C dt1 H′(t1) to first order inH ′. The integration over the two branches of the

Schwinger-Keldysh contour is written as

∫

Cdt1 F (t1) =

(∫ ∞

−∞+

∫ −∞

∞

)

dt1 F (t1) =∑

η1

η1

∫ ∞

−∞dt1 F (t

η11 ),

wheret±1 indicates a time argument on the± branch of the contour. Using Eq. (A.48) for

the current, we have

〈I(t)〉 =iet2

2

∑

ηη1

η1

∫ ∞

−∞dt1 sin

e

2[A(tη)−A(tη11 )]〈TC : ei

√πθ(tη) : : e−i

√πθ(t

η11 ) : 〉

= −iet2

2

∑

ηη1

η1

∫ ∞

−∞dt1 sin

eV

2(t− t1)e

πDηη1 (t−t1),

whereDηη1(t − t1) = 〈TCθ(tη)θ(tη11 )〉 is the Keldysh Green function of the dual boson

θ. We see that the current does not depend on the timet since we can perform a change of


variablest− t1 → t1. We subsequently relabelt1 ≡ t for simplicity and write

〈I〉 = −iet2

2

∑

ηη1

η1

∫ ∞

−∞dt sin

eV t

2eπDηη1 (t).

The causal and anti-causal Green functionsD++,D−− are functions of|t| only,

D±±(t) =1

2πKln

(

πτc/β

sin π(±i|t|)/β

)

,

and thus even undert → −t, hence they do not contribute to the integral which becomes

(with η1 → −η)

〈I〉 = iet2

2

∑

η

η

∫ ∞

−∞dt sin

eV t

2eπDη,−η(t).

The Green functionsD±∓ are given by

D+−(t) = D<(t) =1

2πKln

(

πτc/β

sin π(−it)/β

)

,

D−+(t) = D>(t) =1

2πKln

(

πτc/β

sin πit/β

)

.

The integral can then be computed explicitly and one obtains

〈I〉 = −2et2(

πτcβ

)1/2Kβ

π

21/2K−2

Γ(1/2K)

∣

∣

∣

∣

Γ

(

1

4K+iβeV

4π

)∣

∣

∣

∣

2

sinh

(

βeV

4

)

, (A.54)

whereΓ is the usual Gamma function.

A.4.4 Nonequilibrium noise

The nonequilibrium noise spectrumS(ω) is given by the Fourier transform of the nonequi-

librium current-current correlator

S(t, t′) = 〈I(t), I(t′)〉,


where, is the anticommutator. Again, in the Keldysh formalism we have

S(t, t′) =∑

η

〈TCI(tη)I(t′−η)e−i∫Cdt1 H′(t1)〉.

Since we want to calculate the noise spectrum toO(t2) as well, and each current operator

isO(t), we have to expand the evolution operator toO(1) only,

e−i∫Cdt1 H′(t1) → 1.

We thus obtain

S(t, t′) = e2t2∑

η

〈TC : sin[√πθ(tη) + 1

2eA(tη)] : : sin[

√πθ(t′−η) + 1

2eA(t′η)] : 〉

+O(t4)

=e2t2

2

∑

η

coseV (t− t′)

2eπDη,−η(t−t′) = S(t− t′).

The zero-frequency noise spectrum is then

S(ω = 0) =

∫ ∞

−∞dt S(t) =

e2t2

2

∑

η

∫ ∞

−∞cos

eV t

2eπDη,−η(t).

The integral is very similar to the one done previously, and we obtain

S(ω = 0) = 2e2t2(

πτcβ

)1/2Kβ

π

21/2K−2

Γ(1/2K)

∣

∣

∣

∣

Γ

(

1

4K+iβeV

4π

)∣

∣

∣

∣

2

cosh

(

βeV

4

)

. (A.55)

A.4.5 Schottky relation and fractional Fano factor

Taking the ratio of Eq. (A.54) and (A.55), we obtain

S(ω = 0) = e|〈I〉| coth(

eV

4kBT

)

, (A.56)


which has the same form as Eq. (A.52) with a quasiparticle charge

e∗ =e

2,

hence a noise measurement should be able to confirm the fractional charge of the tunneling

quasiparticles.

Equation (A.56) contains the full thermal/shot noise crossover. The Johnson-Nyquist

thermal (equilibrium) noise is recovered at low biaseV ≪ kBT ,

S(ω = 0) → 4GkBT, eV ≪ kBT,

whereG = limV→0〈I〉/V is the equilibrium conductance, and we usedcoth x ≃ 1/x for

x ≪ 1. The quasiparticle chargee∗ cancels out in this expression, so we need to consider

a large enough biaseV > kBT . In the high bias regimeeV ≫ kBT , the Schottky relation

for the shot noise is recovered,

S(ω = 0) → 2e∗|〈I〉|, eV ≫ kBT,

usingcothx→ 1 for x≫ 1.

A.5 Mean-field description of half-charge tunneling

In this section we use a mean-field description to show that the fractionalized tunneling cur-

rent in the insulating regimeK < 1/4 corresponds to the Goldstone-Wilczek current [69]

for (1 + 1)D Dirac fermions in an instanton background in the time domain. We consider

the vicinity of the zero temperature insulating fixed point forK < 1/4. At the fixed point

the boson density fieldφ is classical and pinned at some odd multiple of√π/4 since the

two-particle backscattering coupling constantλ2 has flown to infinity.

The discreteφ → φ+√π/2 symmetry and TR symmetry are spontaneously broken at

T = 0. This can be understood from the fact that the(0 + 1)D two-particle backscattering

problem can be mapped to a classical 1D Coulomb gas problem with long-range interac-

tions (Sec. A.3.3). In the presence of short-range interactions, due to the Mermin-Wagner


theorem there can only be spontaneous breaking of these (discrete) symmetries atTeff = 0

whereTeff is the effective temperature of the classical problem. Indeed,d = 1 is the lower

critical dimension for spontaneous breaking of a discrete symmetry for systems with short-

range forces. In the presence of long-range forces however,one can have spontaneous

symmetry breaking below a finiteTeff = Tc, which corresponds to a finiteK = Kc = 1/4

in the original problem. In the present case, the RGE tell us that any nonzeroλ2 will lead

to spontaneous symmetry breaking forK < Kc.

This means that the TR symmetry-breaking mass order parametersO1 ∼ cos 2√πφ and

O2 ∼ sin 2√πφ are also classical at zero temperature [21]. At finite temperatures,λ2 is

finite and TR symmetry is restored by fluctuations of the orderparameters, but at very low

temperatures these fluctuations are small and it is reasonable to perform a mean field de-

composition of the point-split two-particle backscattering operatorψ†R(0)ψ

†R(a)ψL(a)ψL(0).

The two-particle backscattering operatorψ†R(0)ψ

†R(a)ψL(a)ψL(0) + H.c. with a some

small distance of the order of the lattice constant is decomposed as

−〈ψ†R(0)ψL(a)〉ψ†

R(a)ψL(0)− 〈ψ†R(a)ψL(0)〉ψ†

R(0)ψL(a)

+〈ψ†R(0)ψL(0)〉ψ†

R(a)ψL(a) + 〈ψ†R(a)ψL(a)〉ψ†

R(0)ψL(0) + H.c.,

where the pairing terms〈ψ†R(0)ψ

†R(a)〉 and 〈ψL(a)ψL(0)〉 vanish. The point splitting of

the four-fermion operator is important: the mean field decomposition of a naive operator

product vanishes (since the naive operator product is itself zero by the Pauli principle). At

this stage, the point splitting can be formally removed fromthe two-fermionoperators, al-

though we need to keep it to evaluate the two-pointcorrelators. By translational invariance

we have

ψ†R(0)ψL(0)

(

2〈ψ†R(0)ψL(0)〉 − 〈ψ†

R(0)ψL(a)〉 − 〈ψ†R(a)ψL(0)〉

)

+H.c.,

where the last two terms are equal as we will see. We evaluate the correlators at zero tem-

perature where the density fieldφ is classical (pinned) but the phase fieldθ is fluctuating.


The correlator thus contains a free boson average overθ while keepingφ fixed,

〈ψ†R(0)ψL(a)〉 =

1

2πξ

⟨

e−i√π[φ(0)−θ(0)]ei

√π[−φ(a)−θ(a)]

⟩

=1

2πξ

⟨

ei√π[θ(0)−θ(a)]

⟩

e−i√π[φ(0)+φ(a)],

where the point splitting can be removed in the last factor,e−i√π[φ(0)+φ(a)] ≃ e−i2

√πφ(0).

Theθ correlator is

⟨

ei√π[θ(0)−θ(a)]

⟩

= exp(

−π2〈[θ(0)− θ(a)]2〉

)

= eπ〈θ(a)θ(0)〉 = exp

[

− 1

4Kln

(

a2 + ξ2

ξ2

)]

.

The mean field decomposition thus reads

1

πξψ†RψLe

−i2√πφ

1− exp

[

− 1

4Kln

(

a2 + ξ2

ξ2

)]

+H.c.

For the point splitting witha to make sense in a theory which is already regularized in the

ultraviolet by a short-distance cutoffξ, we have to consider the limitξ/a ≪ 1 in which

ln(a2+ξ2

ξ2) ≃ ln(a2/ξ2) ≫ 1. As a result, the last term vanishes and we are left simply with

1

πξψ†RψLe

−i2√πφ +H.c.

The mean field backscattering operator is thus

H2 = gψ†R

(

cos 2√πφ(0, t)− i sin 2

√πφ(0, t)

)

ψL +H.c. = gΨ†ma(0, t)σaΨ,

whereg = λ2/πξ,m1 = cos 2√πφ,m2 = sin 2

√πφ andΨ is the Dirac spinor [Eq. (A.57)].

Since the superconducting order parameters〈ψ†Rψ

†R〉 and〈ψLψL〉 are destroyed by the fluc-

tuations of the Josephson phase fieldθ, which unlikeφ is not pinned, the mean-field La-

grangian can be written as a mass term for Dirac fermions in1 + 1 dimensions,

δL = gΨ(m1 + iγ5m2)Ψ,

wherem1(0, τ) = cos 2√πφ(τ) ∼ O1 the normal mass andm2(0, τ) = sin 2

√πφ(τ) ∼ O2


the axial mass. The Dirac field is

Ψ =

(

ψR↑

ψL↓

)

, (A.57)

andγ0 = σ1, γ1 = iσ2 andγ5 = iγ0γ1 = −iσ3 are the Dirac matrices. The mass terms

are localized in space atx = 0 but “slowly-varying” in time (classical instanton gas). Asis

well-known, the instanton background induces a topological Goldstone-Wilczek fermion

current [69] given by

〈jµ〉 = 1

2πǫµνǫab

ma∂νmb

|m|2 ,

with µ, ν = t, x anda, b = 1, 2. A single instanton event pumps a quantizede/2 charge

along the edge, which is just the effect of rotating the “θ” mass domain wall parameter by

π in the Qi-Hughes-Zhang fractional charge proposal [70]. The electric current is just

j = −e〈jx〉 = 1

2π(m2∂tm1 −m1∂tm2) =

e√π∂tφ(0, t),

which is the standard bosonization formula. As a result, thecharge pumped by a single

instanton configuration in whichφ jumps, say, from−√π/4 to

√π/4, is

∆Qinst =

∫ t2

t1

dt j =e√π[φ(t2)− φ(t1)] =

e

2.

The instanton gas representation at low temperatures thus corresponds to the tunneling

of half-charged particles. The effect studied here can alsobe seen as the time-domain

counterpart to the statice/2 charge induced by a spatial magnetic domain wall on the QSH

edge [70].

A.6 Phase space derivation of theT 6 behavior in the non-

interacting case

The purpose of this section is to explain the∝ −T 6 behavior of the conductance in the

noninteracting regimeK = 1. In this case, we do not need to rely on bosonization and we


can use the fermion language. Consider the two-particle backscattering process

R, k;R, k′ −→ L, p;L, p′, (A.58)

whereR,L indicates a right/left mover, andk, k′, p, p′ are the momenta. The two-particle

backscattering Hamiltonian is

H2 = λ2ψ†R(0)ψ

†R(a)ψL(0)ψL(a) + H.c.,

where a point-splitting with a short-distance cutoffa is required by the exclusion principle.

We thus see that the interaction is nonlocal in space. In the continuum limit, we can expand

ψ(a) ≃ ψ(0) + a∂xψ(0) and we obtain

H2 = λ2ψ†R(0)∂xψ

†R(0)ψL(0)∂xψL(0) + H.c.,

where we have absorbed the factor ofa2 in the coupling constantλ2. Fourier transforming,

we obtain

H2 = λ2∑

k1k2k3k4

k2k4(c†Rk1

c†Rk2cLk3cLk4 +H.c.).

The scattering rate for the process of Eq. (A.58) is given by the Fermi Golden rule,

Γk,k′→p,p′ =2π

~|〈L, p;L, p′|H2|R, k;R, k′〉|2δ(εk + εk′ − εp − εp′).

The matrix element between the two-particle states|R, k;R, k′〉 = c†Rkc†Rk′|0〉 and

|L, p;L, p′〉 = c†Lpc†Lp′|0〉 is easily obtained,

〈L, p;L, p′|H2|R, k;R, k′〉 = −λ2(k − k′)(p− p′),

and the contribution to the inverse lifetime of the single-particle state|R, k〉 due to this

process is given by

1

τk∝∑

k′pp′

Γk,k′→p,p′nF (εk′)[1− nF (εp)][1− nF (εp′)],


where we have include the Fermi occupation numbersnF (ε) = (eβ(ε−µ) + 1)−1 to account

for the exclusion principle. At this stage we can already understand where theT 6 behavior

comes from. We sum over three variablesk′, p, p′. There is no constraint from momentum

conservation because the backscattering occurs at a singlepoint in space. There is one

constraint coming from energy conservation (theδ function in the scattering rateΓ) hence

we really sum over only two independent variables, for example εp andεp′. Each of these

variables contributes a phase space factorT/EF , hence so far we haveT 2. However the

matrix element squared contains(k−k′)2(p−p′)2 and since all momenta are confined to the

vicinity of the Fermi points, this contributes(k−kF )4 which translates into(ε−εF )4 ∼ T 4

at finite temperature, because the spectrum is linearε ∝ k (more generally, it can always

be linearized in the vicinity of the Fermi points). Hence we see that the helical nature of

the system suppresses the phase space for backscattering (which is another manifestation

of its robustness to disorder) because of the nonlocal nature of the backscattering.

We can do the calculation directly to verify our conjecture.We consider the linear

dispersionεk = ±~vFk. The right-movers haveεk,k′ = ~vF (k, k′) and the left-movers

haveεp,p′ = −~vF (p, p′), hence the energy-conservingδ function is

δ(εk + εk′ − εp − εp′) =1

~vFδ(k + k′ + p + p′),

which imposes the constraintk′ = −(k + p+ p′), and we have

1

τk∝∑

pp′

(2k + p+ p′)2(p− p′)2nF (ε−(k+p+p′))[1− nF (εp)][1− nF (εp′)].

Instead of doing the calculation at finiteT , we considerT = 0 but withk−kF > 0 slightly

away from the Fermi surface. Since the dispersion is linear,k − kF ∝ ε − εF and we can

then make the substitutionk − kF → T to estimate the finiteT result. Converting sums to

integrals and consideringT = 0, we have

1

τk∝∫

−∞dp

∫

−∞dp′ (2k + p+ p′)2(p− p′)2θ(kF + k + p+ p′)θ(−p− kF )θ(−p′ − kF ),

sincenF (εk) = nF (~vFk) = θ(kF − k) at T = 0 with εF = ~vFkF . The domain of


integration is easily drawn and we have

1

τk∝∫ −kF

−k

dp

∫ −kF

−(kF+k+p)

dp′ (2k + p+ p′)2(p− p′)2 =11

90(k − kF )

6 ∼ T 6,

which confirms the estimate.

A.7 Transport in a spin Aharonov-Bohm ring

A.7.1 S-matrix analysis

We wish to obtain an expression for theS-matrix S relating outgoingb to incominga

current amplitudes,

(

bl

br

)

= S(

al

ar

)

with S =

(

r t′

t r′

)

, (A.59)

whereal andbl (ar andbr) arepL × 1 (pR × 1) column vectors of the current amplitudes

outside the QSH region in the left (right) lead (see Fig. 5.1), andpL (pR) is the number of

propagating channels at the Fermi energy in the left (right)lead. The matrixS therefore

has dimensions(pL + pR) × (pL + pR) and the submatricesr, r′ and t, t′ are reflection

and transmission matrices, respectively. The two-terminal conductanceG from left to right

is given by the Landauer formula [89]G = e2

htr tt†. We assume that phase coherence

is preserved throughout the sample so thatS can be obtained by combiningS-matrices

for different portions of the device coherently [89]. We define the(pL,R + 2) × (pL,R +

2) scattering matricesSL, SR for the left (L) and right (R) FM/QSH junctions (e.g. see

Fig. 5.2(a) for the left junction),

(

bl

bl′

)

= SL

(

al

al′

)

,

(

br′

br

)

= SR

(

ar′

ar

)

, (A.60)

wherel′ (r′) is the QSH region immediately to the right (left) of the left(right) junction,

such thatal′ , ar′ andbl′ , br′ are the 2-component spinors of edge state current amplitudes.


They are related through the geometric AB phaseϕ (different for each spin polarization)

and the dynamical phaseλ = 2kF ℓ (identical for both spin polarizations) whereℓ is the

distance travelled by the edge electrons from left to right junction andkF is the edge state

Fermi wave vector,

ar′↑,↓

al′↑,↓

= eiλ/2e∓iϕ/2

bl′↑,↓

br′↑,↓

, (A.61)

where the upper sign forϕ corresponds to spin up. Using Eq. (A.60) and (A.61), we can

write(

e−iλ/2Φ†al′

br

)

= SR

(

eiλ/2Φbl′

ar

)

, (A.62)

where we defineΦ ≡ e−iϕσz/2. Using the first equality in Eq. (A.60) together with

Eq. (A.62), we can eliminate the intermediate amplitudesal′ , bl′ and obtain relations be-

tween the left lead amplitudesal, bl and the right lead amplitudesar, br, which gives usS[Eq. (A.59)]. The2×2 transmission matrixt, i.e. the lower left block ofS, is then obtained

in the form

t = tRStL, (A.63)

wheretL andtR are the2× pL andpR × 2 transmission matrices for the left and right junc-

tions, respectively (i.e. the lower left blocks ofSL, SR following the notation of Eq. (A.59)),

andS is a2 × 2 matrix defined in Eq. (5.1). The effective spin density matricesρL, ρR of

the FM leads used in Eq. (5.2) are defined asρL = tLt†L andρR = t†RtR.

If the arms of the ring are asymmetric, the dynamical phaseλ is generally different for

each arm and we haveλbottom − λtop ≡ δ 6= 0. In this case, one can show that Eq. (5.2)

still holds, but with the substitutions

ρL(θL) → RδρL(θL)R−1δ = ρL(θL + δ),

r′L(θL) → Rδr′L(θL)R

−1δ = r′L(θL + δ),

whereRδ ≡ e−iσzδ/2 rotates the spin about thez axis by an angleδ. In other words, a

phase asymmetry is equivalent to a rigid flux-independent rotation of the electron spin, and


simply shifts the conductance pattern by a constant angleδ:

G(φ, θ ≡ θR − θL) → G(φ, θR − (θL + δ))

= G(φ, θ − δ). (A.64)

A.7.2 Scattering at the junction

In order to solve the 1D scattering problem at the FM/QSH interface, we first observe that

the number of degrees of freedom is equal on either side of thejunction. If the Fermi level

εF is chosen such that both spin subbands in the FM leads are occupied, there are four

propagating modes on each side of the junction (two spins andtwo chiralities). The QSH

spin statesφQSH(±) areσz eigenstates while the FM spin statesφFM(±)(θ) are eigenstates of

n · σ and depend explicitly onθ. The Schrodinger equation for the junction is then solved

by the following scattering ansatz,

ψ(+)σ (x) =

φ<(+)σ√v<σeik

<σ x +

∑

σ′ rσ′σφ<(−)

σ′√v<σ′

e−ik<σ′x, x < 0,

∑

σ′ tσ′σφ>(+)

σ′√v>σ′

eik>

σ′x, x > 0,

for a right-moving scattering state, and with similar expressions for a left-moving scattering

stateψ(−)σ . Spin is denoted byσ, chirality by ± and side of the junction by<,>. The

propagating modes are explicitly normalized to unit flux such that rσ′σ and tσ′σ are the

desired reflection and transmission matrices. Requiring the continuity ofψ(±)σ andvxψ

(±)σ

at the interfacex = 0 (with vx ≡ ∂H/∂kx the velocity operator), we obtain a system of

linear equations for the sixteen matrix elementsrL, tL, r′L, t

′L constitutingSL. As illustrated

in Fig. 5.1, the magnetization angle is set to zero in the leftlead and toθ in the right lead

and we obtainr′L(0) andrR(θ) in Eq. (5.1).


A.7.3 Tight-binding model

The effective tight-binding model describing HgTe QW is defined as [19, 54]

H =∑

i

c†iVici +∑

ij

(

c†iTijeiAijcj + h.c.

)

, (A.65)

whereTij = Txδj,i+x + Tyδj,i+y is the nearest-neighbor hopping matrix,Aij =e~c

∫ j

idr ·A

is the Peierls phase withA the electromagnetic vector potential, andVi, Tx andTy are4×4

matrices containing thek ·p parameters and the effective Zeeman term. The4×4 matrices

Tx, Ty andVi used in the tight-binding Hamiltonian (A.65) are given by

Tx =

D+ − iA2

− i∆e

20

− iA2

D− 0 − i∆h

2

− i∆e

20 D+

iA2

0 − i∆h

2iA2

D−

,

Ty =

D+A2

∆e

20

−A2

D− 0 −∆h

2

−∆e

20 D+

A2

0 ∆h

2−A

2D−

, (A.66)

and

Vi = (C − 4D − εF + Eg(i))14×4 + (M − 4B)12×2 ⊗ σz +HeffZ‖ +Heff

Z⊥,

whereD± ≡ D ± B andA, B, C, D, M , ∆e, ∆h arek · p parameters [54], and1n×n

denotes then× n unit matrix. The Fermi energyεF is uniform throughout the device. The

gate potentialEg(i) is different in the QSH and lead regions [Fig. 5.4(a)], and isused to

tune the central region into the QSH insulating regime. The in-planeHeffZ‖ and out-of-plane

HeffZ⊥ effective Zeeman terms, which are used to mimick the injection of spin-polarized


carriers from a FM layer (Fig. 5.1), are given by [54]

HeffZ‖ = g‖µB

0 0 Beff− 0

0 0 0 0

Beff+ 0 0 0

0 0 0 0

,

HeffZ⊥ = µBB

effz

gE⊥ 0 0 0

0 gH⊥ 0 0

0 0 −gE⊥ 0

0 0 0 −gH⊥

, (A.67)

whereBeff± = Beff

x ± iBeffy , Beff = (Beff

x , Beffy , B

effz ) is some effective magnetic field the role

of which is to induce a spin polarization in the leads,µB is the Bohr magneton, andg‖ and

gE⊥, gH⊥ are the in-plane and out-of-planeg-factors, respectively.

A.8 Magnetooptical Kerr/Faraday rotation in thick film

geometry

In this section, we give some details of the calculation of the Kerr and Faraday angles in

the thick film geometry of Fig. 6.1 (repeated in Fig. A.3), fornormal incidence. Defining

E as in Ref. [35],

E =

(

2αE

H

)

, D =

(

D

2αB

)

, (A.68)

we have

E(z, t) =

E+1 e

i(k1z−ωt) + E−1 e

i(−k1z−ωt), z < 0,

E+2 e

i(k2z−ωt) + E−2 e

i(−k2z−ωt), 0 < z < L,

E+3 e

i(k3z−ωt), z > L.


Figure A.3: Incident, reflected, and transmitted light for Kerr/Faraday rotation experimenton thick topological insulator film of thicknessL.

Since we are at normal incidence,E is entirely in thexy plane and the only boundary

condition atz = 0 andz = L is the continuity ofE . This gives us two relations,

E+1 + E

−1 = E

+2 + E

−2 ,

E+2 e

ik2L + E−2 e

−ik2L = E+3 e

ik3L.

Using the fact that for a plane waveE = ck× (−iσy)D wherec is the speed of light in the

medium in which a given plane wave propagates as well as the constitutive relations [35]

D = ME, we obtain two more equations,

T12(E+1 − E

−1 ) = E

+2 − E

−2 ,

T23(eik2LE

+2 − e−ik2LE

−2 ) = eik3LE+

3 ,


where we define a “transfer matrix”Tij ≡ cicjM−1

j Mi. After some algebra, we obtain

eik3LE+3 = SFE

+1 ,

with

SF =(

I+ T23Q∗12[P∗

12]−1)−1 T23

(

Q12 +Q∗12[P∗

12]−1P12

)

, (A.69)

where we define

P12 ≡ cos k2L · I+ i sin k2L · T12,

Q12 ≡ cos k2L · T12 + i sin k2L · I.

Note that Eq. (A.69) contains all the multiple reflection effects. Using Eq. (A.68), we can

extract the relation between the transmittedEt ≡ E+3 and incidentEin ≡ E+

1 electric fields,

Et = S11F Ein +

c1ε12α

S12F z × Ein,

where we have removed the global phase factoreik3L which is irrelevant, because we care

only about therelativephase between thex andy components (i.e. the polarization). Con-

sider a wave initially polarized in thex direction,Eyin = 0. Then the Faraday rotation will

be given by the ratio betweeny andx components of the transmitted electric fields,

tan θF =Ey

t

Ext

=c1ε12α

S12F

S11F

=1

2α

S12F

S11F

, (A.70)

where we have used the fact thatc1ε1 = 1 in vacuum. The Kerr angle can be extracted in a

similar way,

tan θK =Ey

r

Exr

=c1ε12α

S12K

S11K

=1

2α

S12K

S11K

, (A.71)

whereSK is defined as

SK = [P∗12]

−1 (SF − P12) .


In general there can be additional “trivial” integer QH layers at the interfaces. We would

like to be able to measure the bulk axion angle independentlyof these trivial layers. To

account for them, we consider the general case that the substrate has a trivial axion an-

gle θ = 2pπ with p integer (see Fig. A.3). Generally, Eq. (A.70) and (A.71) depend in a

complicated way on the materials constants of both the topological insulator and the sub-

strate,c2, ε2 andc3, ε3, respectively, as well as on the topological insulator film thickness

L and the photon frequencyω. However, the results simplify at the “magic frequencies”

ω = 2πnc/√ε2µ2L andω = 2π(n + 1

2)c/

√ε2µ2L, n ∈ Z, corresponding to reflectivity

minima and maxima, respectively, allowing us to form combination of measured angles

that are independent of the material propertiesε2, c2 (or µ2), ε3, andc3 (or µ3).

A.9 Surface Chern-Simons theory for an Abelian fractio-

nal 3D topological insulator

In this section, we consider the AbelianU(1) × U(1) fractional topological insulator on

the spatial 3-manifoldM = T 2 × I with I = [−L/2, L/2] in thez direction. Because the

bulk 3D statistical gauge fields are free, we can integrate them out explicitly to obtain an

effective action for the gauge fields on the 2D boundary∂M = T 2 ∪ T 2. It is a Maxwell-

Chern-Simons theory with two coupled gauge fieldsa+µ anda−µ corresponding to the two

copies ofT 2. In the long-wavelength limitq ≪ 1/L the two gauge fields become identified

a+µ = a−µ ≡ αµ and the level of the Chern-Simons term forαµ is the sum of that for the two

surfaces, i.e. it is integer.

The spatial manifold isM = T 2 × I with I = [−L/2, L/2]. We can always choose the

generators ofU(1)×U(1) to satisfytr tatb = δab, a, b = 1, 2. The action in imaginary time

is

S3D[aµ] =

∫

d4x

(

1

4g2faµνf

aµν −

iθ(z)

32π2ǫµνλρf

aµνf

aλρ

)

,

with faµν = ∂µa

aν − ∂νa

aµ, a = 1, 2 theU(1)× U(1) “statistical” field strength and

∂zθ =∑

η=±1

(2kη + 1)πδ(z − ηL/2),


with kη ∈ Z. To derive an effective 2D action on∂M = T 2 ∪ T 2, we introduce a Lagrange

multiplier which constrains the gauge field to live on∂M . Then we integrate out the bulk

gauge fieldaaµ. (The approach is essentially the same as that used in Sec. A.2 to derive an

effective action for thex = 0 degrees of freedom in the helical liquid.) In other words, we

introduce a “resolution of unity”

1 =

∫

D a+µ D a−µ δ[aaµ(x, L/2)− a+,a

µ (x)]δ[aaµ(x,−L/2)− a−,aµ (x)],

in the partition function, whereaη,aµ , η = ± are two auxiliary gauge fields which are defined

only on the 2D surfacex = (x0 = t, x1, x2), with a+,aµ living on the 2-torus atz = L/2 and

a−,aµ living on the 2-torus atz = −L/2. We represent the functional delta function as

∏

η

δ[aaµ(x, ηL/2)−aη,aµ (x)] =

∫

D j+µ D j−µ exp

(

i

∫

d3x jη,aµ (x)[aaµ(x, ηL/2)− aη,aµ (x)]

)

where jη,aµ is a Lagrange multiplier which implements the constraint that aη,aµ (x) =

aaµ(x, ηL/2). The idea is to integrate out firstaµ, and then the Lagrange multiplierjηµ,

to get an effective actionSeff2D in terms of the 2D gauge fieldsaηµ alone. In other words, the

partition function is

Z =

∫

D a+µ D a−µ D j+µ D j−µ Daµ

× exp

(

−S3D[aµ] + i

∫

d3x jη,aµ (x)[aaµ(x, ηL/2)− aη,aµ (x)]

)

≡∫

D a+µD a−µ e−Seff

2D[a+µ ,a−µ ].

We want to see whether these gauge fields will be identified or not, i.e. do we get two

Chern-Simons theories or only one?

First of all, since theθ-term is a total derivative it contributes only to the boundary

piece. Therefore if we writeSeff2D = SMax

2D + Sθ2D we immediately know that

Sθ2D = −i

∑

η

kη +12

4π

∫

d3x ǫµνλaη,aµ ∂ν a

η,aλ ,


i.e. we obtain two decoupled Chern-Simons terms of half-odd-integer levelkη + 12. To

computeSMax we need to integrate out the 3D bulk gauge fluctuations, whichwe can do

exactly in the Abelian case because the gauge bosons are noninteracting. We can gauge-fix

the Maxwell term in the usual way [248] by adding a12g2ξ

(∂µaaµ)

2 term to the Lagrangian.

In the Feynman gaugeξ = 1 the 3D gauge boson propagator is

Dabµν(k, kz) =

g2δabδµν

k2 + k2z, (A.72)

with k = (k0 = ω, k1, k2). Because the Lagrange multipliersjη,aµ (k) =∫

d3x e−ikν xν jη,aµ (x)

are independent ofkz, we have to sum over allkz to obtain an effective 2D propagator. The

effective 2D Maxwell propagator for the 2D gauge fieldsaη,aµ is therefore

Dab,ηη′

µν (k) = g2δabδµν1

L

∑

kz

e−ikz(η−η′)L/2

k2 + k2z,

with |k| =√

ω2 + k2. Because the fieldsaη,aµ must satisfy some sort of boundary condition

at z = ±L/2 (either Dirichlet or Neumann),kz is a discrete variable,kz = nπ/L, n ∈ Z.

Performing the discrete sum overkz, we obtain

Dabµν(q) =

g2δabδµν|q| sinh |q|L

(

cosh |q|L 1

1 cosh |q|L

)

,

where we denoteq ≡ k for simplicity. The inverse propagator is

[D−1]abµν(q) =1

g2Lδabδµν

|q|L|q| sinh |q|L

(

cosh |q|L −1

−1 cosh |q|L

)

.

To obtain the final form ofSeff2D, we need to integrate out the Lagrange multipliersjη,aµ which

simply amounts to inverting the2 × 2 matrix propagator (A.72). Since now all quantities


are 2D, we can drop all the tildes for simplicity, and obtain

Seff2D[a

+µ , a

−µ ] =

1

2g2Lz

∫

d3q

(2π)3|q|Lz

sinh |q|Lzδµν

×(

a+,aµ a−,a

µ

)

−q

(

cosh |q|Lz −1

−1 cosh |q|Lz

)(

a+,aν

a−,aν

)

q

− i∑

η

kη +12

4π

∫

d3x ǫµνλaη,aµ ∂νa

η,aλ , (A.73)

with q ≡ (ω,q) and|q| =√

ω2 + q2. We denotedLz ≡ L for clarity.

Consider fixing the scaling dimension of the gauge fields by the Chern-Simons term.

The latter is therefore marginal and contains one power ofq. The effective Maxwell term

in Eq. (A.73) contains higher powers of|q|. Let us expand it to quadratic order in|q|Lz,

Seff2D[a

+µ , a

−µ ] =

1

2g2Lz

∫

d3q

(2π)3δµν

×(

a+,aµ a−,a

µ

)

−q

(

1 + 13q2L2

z −1 + 16q2L2

z

−1 + 16q2L2

z 1 + 13q2L2

z

)(

a+,aν

a−,aν

)

q

− i∑

η

kη +12

4π

∫


η,aλ .

We now consider the long-wavelength, low-energy limit|q| ≪ 1/Lz. In this limit, the

quadratic Maxwell termsq2L2z are irrelevant and the leading term is

Seff2D[a

+µ , a

−µ ] =

1

2g2Lz

∫

d3q

(2π)3δµν

(

a+,aµ a−,a

µ

)

−q

(

1 −1

−1 1

)(

a+,aν

a−,aν

)

q

=1

2g2Lz

∫

d3x (a+,aµ − a−,a

µ )2.

This term contains no derivatives ofaη,aµ but simply implements a constraint. The equations


of motion read

0 =δSeff

2D

δaa,+µ

=∂L

∂aa,+µ

= 2(aa,+µ − aa,−µ ),

0 =δSeff

2D

δaa,−µ

=∂L

∂aa,−µ

= −2(aa,+µ − aa,−µ ),

which imply thataa,+µ = aa,−µ and the gauge fields on the two 2-tori are identified. There-

fore, in the limit|q| ≪ 1/Lz the Chern-Simons term in Eq. (A.73) becomes

Sθ2D = −i

∑

η

kη +12

4π

∫


η,aλ = −i k

4π

∫

d3x ǫµνλαaµ∂να

aλ,

with αaµ ≡ aa,+µ = aa,−µ andk =

∑

η(kη +12) = k+ + k− + 1 is the effective Chern-Simons

level, which is integer.

Bibliography

[1] L. D. Landau and E. M. Lifshitz.Statistical Physics. Pergamon Press, Oxford, 1980.

[2] K. von Klitzing, G. Dorda, and M. Pepper. New method for high-accuracy determi-

nation of the fine-structure constant based on quantized Hall resistance.Phys. Rev.

Lett., 45:494, 1980.

[3] D. C. Tsui, H. L. Stormer, and A. C. Gossard. Two-dimensional magnetotransport

in the extreme quantum limit.Phys. Rev. Lett., 48:1559, 1982.

[4] E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V.Ramakrishnan. Scaling

theory of localization: Absence of quantum diffusion in twodimensions.Phys. Rev.

Lett., 42:673, 1979.

[5] X. L. Qi, Y.-S. Wu, and S. C. Zhang. General theorem relating the bulk topological

number to edge states in two-dimensional insulators.Phys. Rev. B, 74:045125, 2006.

[6] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs. Quantized Hall

conductance in a two-dimensional periodic potential.Phys. Rev. Lett., 49:405, 1982.

[7] B. Simon. Holonomy, the quantum adiabatic theorem, and Berry’s phase.Phys. Rev.

Lett., 51:2167, 1983.

[8] S. C. Zhang, T. H. Hansson, and S. Kivelson. Effective-field-theory model for the

fractional quantum Hall effect.Phys. Rev. Lett., 62:82, 1989.

[9] S. C. Zhang. The Chern-Simons-Landau-Ginzburg theory of the fractional quantum

Hall effect. Int. J. Mod. Phys. B, 6:25, 1992.

207

BIBLIOGRAPHY 208

[10] V. Ambegaokar, B. I. Halperin, D. R. Nelson, and E. D. Siggia. Dynamics of super-

fluid films. Phys. Rev. B, 21:1806, 1980.

[11] M. P. A. Fisher and D.-H. Lee. Correspondence between two-dimensional bosons

and a bulk superconductor in a magnetic field.Phys. Rev. B, 39:2756, 1989.

[12] C. L. Kane and E. J. Mele.Z2 topological order and the quantum spin Hall effect.

Phys. Rev. Lett., 95:146802, 2005.

[13] C. L. Kane and E. J. Mele. Quantum spin Hall effect in graphene.Phys. Rev. Lett.,

95:226801, 2005.

[14] B. A. Bernevig and S. C. Zhang. Quantum spin Hall effect.Phys. Rev. Lett.,

96:106802, 2006.

[15] F. D. M. Haldane. Model for a quantum Hall effect withoutLandau levels:

Condensed-matter realization of the “parity anomaly”.Phys. Rev. Lett., 61:2015,

1988.

[16] X. L. Qi, T. L. Hughes, and S. C. Zhang. Topological field theory of time-reversal

invariant insulators.Phys. Rev. B, 78:195424, 2008.

[17] H. Min, J. E. Hill, N. A. Sinitsyn, B. R. Sahu, L. Kleinman, and A. H. MacDon-

ald. Intrinsic and Rashba spin-orbit interactions in graphene sheets.Phys. Rev. B,

74:165310, 2006.

[18] Y. Yao, F. Ye, X. L. Qi, S. C. Zhang, and Z. Fang. Spin-orbit gap of graphene:

First-principles calculations.Phys. Rev. B, 75:041401(R), 2007.

[19] B. A. Bernevig, T. L. Hughes, and S.C. Zhang. Quantum spin Hall effect and topo-

logical phase transition in HgTe quantum wells.Science, 314:1757, 2006.

[20] M. Konig, S. Wiedmann, C. Brune, A. Roth, H. Buhmann, L. W. Molenkamp, X. L.

Qi, and S. C. Zhang. Quantum spin Hall insulator state in HgTequantum wells.

Science, 318:766, 2007.

BIBLIOGRAPHY 209

[21] C. Wu, B. A. Bernevig, and S. C. Zhang. Helical liquid andthe edge of quantum

spin Hall systems.Phys. Rev. Lett., 96:106401, 2006.

[22] H. J. Zhang, C. X. Liu, X. L. Qi, X. Dai, Z. Fang, and S. C. Zhang. Topological

insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface.

Nature Phys., 5:438, 2009.

[23] C. X. Liu, X. L. Qi, H. J. Zhang, X Dai, Z. Fang, and S. C. Zhang. Model Hamilto-

nian for topological insulators.Phys. Rev. B, 82:045122, 2010.

[24] Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. Grauer, Y. S. Hor,

R. J. Cava, and M. Z. Hasan. Observation of a large-gap topological-insulator class

with a single Dirac cone on the surface.Nature Phys., 5:398, 2009.

[25] Y. L. Chen, J. G. Analytis, J.-H. Chu, Z. K. Liu, S.-K. Mo,X. L. Qi, H. J. Zhang,

D. H. Lu, X. Dai, Z. Fang, S. C. Zhang, I. R. Fisher, Z. Hussain,and Z.-X. Shen. Ex-

perimental realization of a three-dimensional topological insulator, Bi2Te3. Science,

325:178, 2009.

[26] D. Hsieh, Y. Xia, L. Wray, D. Qian, A. Pal, J. H. Dil, J. Osterwalder, F. Meier,

G. Bihlmayer, C. L. Kane, Y. S. Hor, R. J. Cava, and M. Z. Hasan.Observation of

unconventional quantum spin textures in topological insulators. Science, 323:919,

2009.

[27] L. Fu, C. L. Kane, and E. J. Mele. Topological insulatorsin three dimensions.Phys.

Rev. Lett., 98:106803, 2007.

[28] J. E. Moore and L. Balents. Topological invariants of time-reversal-invariant band

structures.Phys. Rev. B, 75:121306(R), 2007.

[29] R. Roy. Topological phases and the quantum spin Hall effect in three dimensions.

Phys. Rev. B, 79:195322, 2009.

[30] L. Fu and C. L. Kane. Topological insulators with inversion symmetry.Phys. Rev.

B, 76:045302, 2007.

BIBLIOGRAPHY 210

[31] D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z. Hasan. A

topological Dirac insulator in a quantum spin Hall phase.Nature, 452:970, 2008.

[32] F. Wilczek. Two applications of axion electrodynamics. Phys. Rev. Lett., 58:1799,

1987.

[33] R. D. Peccei and H. R. Quinn.CP conservation in the presence of pseudoparticles.

Phys. Rev. Lett., 38:1440, 1977.

[34] Z. Wang, X. L. Qi, and S. C. Zhang. Equivalent topological invariants of topological

insulators.New J. Phys., 12:065007, 2010.

[35] A. Karch. Electric-magnetic duality and topological insulators. Phys. Rev. Lett.,

103:171601, 2009.

[36] W.-K. Tse and A. H. MacDonald. Giant magneto-optical Kerr effect and univer-

sal Faraday effect in thin-film topological insulators.Phys. Rev. Lett., 105:057401,

2010.

[37] J. Maciejko, X. L. Qi, H. D. Drew, and S. C. Zhang. Topological quantization in

units of the fine structure constant.Phys. Rev. Lett., 105:166803, 2010.

[38] X. L. Qi, R. Li, J. Zang, and S. C. Zhang. Inducing a magnetic monopole with

topological surface states.Science, 323:1184, 2009.

[39] F. D. M. Haldane and L. Chen. Magnetic flux of “vortices” on the two-dimensional

Hall surface.Phys. Rev. Lett., 53:2591, 1984.

[40] R. Li, J. Wang, X. L. Qi, and S. C. Zhang. Dynamical axion field in topological

magnetic insulators.Nature Phys., 6:284, 2010.

[41] J. Wang, R. Li, S. C. Zhang, and X. L. Qi. Topological magnetic insulators with

corundum structure.e-print arXiv:1008.2666, 2010.

[42] X. G. Wen. Quantum Field Theory of Many-Body Systems: From the Origin of

Sound to an Origin of Light and Electrons. Oxford University Press, Oxford, 2004.

BIBLIOGRAPHY 211

[43] S. C. Zhang. Topological states of quantum matter.Physics, 1:6, 2008.

[44] C. Xu and J. Moore. Stability of the quantum spin Hall effect: effects of interactions,

disorder, andZ2 topology.Phys. Rev. B, 73:045322, 2006.

[45] C. W. J. Beenakker and H. van Houten. Quantum transport in semiconductor nanos-

tructures.Solid State Phys., 44:1, 1991.

[46] M. Buttiker. Absence of backscattering in the quantumHall effect in multiprobe

conductors.Phys. Rev. B, 38:9375, 1988.

[47] M. Buttiker. Four-terminal phase-coherent conductance. Phys. Rev. Lett., 57:1761,

1986.

[48] C. Brune, A. Roth, E. G. Novik, M. Konig, H. Buhmann, E.M. Hankiewicz,

W. Hanke, J. Sinova, and L. W. Molenkamp. Evidence for the ballistic intrinsic

spin Hall effect in HgTe nanostructures.Nature Phys., 6:448, 2010.

[49] E. M. Hankiewicz, L. W. Molenkamp, T. Jungwirth, and J. Sinova. Manifestation of

the spin Hall effect through charge-transport in the mesoscopic regime.Phys. Rev.

B, 70:241301(R), 2004.

[50] Y. Meir and N. S. Wingreen. Landauer formula for the current through an interacting

electron region.Phys. Rev. Lett., 68:2512, 1992.

[51] H. M. Pastawski. Classical and quantum transport from generalized Landauer-

Buttiker equations.Phys. Rev. B, 44:6329, 1991.

[52] H. M. Pastawski. Classical and quantum transport from generalized Landauer-

Buttiker equations. II. Time-dependent resonant tunneling. Phys. Rev. B, 46:4053,

1992.

[53] L. P. Kadanoff and G. Baym.Quantum Statistical Mechanics. Benjamin, Menlo

Park, 1962.

BIBLIOGRAPHY 212

[54] M. Konig, H. Buhmann, L. W. Molenkamp, T. L. Hughes, C. X. Liu, X. L. Qi, and

S. C. Zhang. The quantum spin Hall effect: theory and experiment. J. Phys. Soc.

Jpn, 77:031007, 2008.

[55] S. Sanvito, C. J. Lambert, J. H. Jefferson, and A. M. Bratkovsky. General Green’s-

function formalism for transport calculation with spd Hamiltonians and giant mag-

netoresistance in Co-and Ni-based magnetic multilayers.Phys. Rev. B, 59:11936,

1999.

[56] K. Kazymyrenko and X. Waintal. Knitting algorithm for calculating Green functions

in quantum systems.Phys. Rev. B, 77:115119, 2008.

[57] S. Datta.Electronic Transport in Mesoscopic Systems. Cambridge University Press,

Cambridge, 1995.

[58] M. Konig. Spin-related transport phenomena in HgTe-based quantum well struc-

tures. PhD thesis, University of Wurzburg, 2007.

[59] A. Furusaki and N. Nagaosa. Kondo effect in a Tomonaga-Luttinger liquid. Phys.

Rev. Lett., 72:892, 1994.

[60] M. Garst, P. Wolfle, L. Borda, J. von Delft, and L. Glazman. Energy-resolved in-

elastic electron scattering off a magnetic impurity.Phys. Rev. B, 72:205125, 2005.

[61] T. Giamarchi.Quantum Physics in One Dimension. Clarendon Press, Oxford, 2003.

[62] X. G. Wen. Edge transport properties of the fractional quantum Hall states and

weak-impurity scattering of a one-dimensional charge-density wave. Phys. Rev. B,

44:5708, 1991.

[63] D. Meidan and Y. Oreg. Multiple-particle scattering inquantum point contacts.Phys.

Rev. B, 72:121312(R), 2005.

[64] V. J. Emery and S. Kivelson. Mapping of the two-channel Kondo problem to a

resonant-level model.Phys. Rev. B, 46:10812, 1992.

BIBLIOGRAPHY 213

[65] D. L. Maslov. Transport through dirty Luttinger liquids connected to reservoirs.

Phys. Rev. B, 52:R14368, 1995.

[66] A. Schiller and K. Ingersent. Exact results for the Kondo effect in a Luttinger liquid.

Phys. Rev. B, 51:4676, 1995.

[67] D.-H. Lee and J. Toner. Kondo effect in a Luttinger liquid. Phys. Rev. Lett., 69:3378,

1992.

[68] P. Nozieres. A “Fermi-liquid” description of the Kondo problem at low temperatures.

J. Low Temp. Phys., 17:31, 1974.

[69] J. Goldstone and F. Wilczek. Fractional quantum numbers on solitons.Phys. Rev.

Lett., 47:986, 1981.

[70] X. L. Qi, T. L. Hughes, and S. C. Zhang. Fractional chargeand quantized current in

the quantum spin Hall state.Nature Phys., 4:273, 2008.

[71] C. L. Kane and M. P. A. Fisher. Transmission through barriers and resonant tunneling

in an interacting one-dimensional electron gas.Phys. Rev. B, 46:15233, 1992.

[72] C. L. Kane and M. P. A. Fisher. Transport in a one-channelLuttinger liquid. Phys.

Rev. Lett., 68:1220, 1992.

[73] T. Martin. Noise in mesoscopic physics.e-print arXiv:cond-mat/0501208, 2005.

[74] L. J. Cooper, N. K. Patel, V. Drouot, E. H. Linfield, D. A. Ritchie, and M. Pepper.

Resistance resonance induced by electron-hole hybridization in a strongly coupled

InAs/GaSb/AlSb heterostructure.Phys. Rev. B, 57:11915, 1998.

[75] C. X. Liu, T. L. Hughes, X. L. Qi, K. Wang, and S. C. Zhang. Quantum spin Hall

effect in inverted type-II semiconductors.Phys. Rev. Lett., 100:236601, 2008.

[76] I. Knez, R. R. Du, and G. Sullivan. Finite conductivity in mesoscopic Hall bars of

inverted InAs/GaSb quantum wells.Phys. Rev. B, 81:201301, 2010.

BIBLIOGRAPHY 214

[77] I. Knez, R. R. Du, and G. Sullivan. Evidence for helical edge modes in inverted

InAs/GaSb quantum wells.e-print arXiv:1105.0137, 2011.

[78] A. Roth, C. Brune, H. Buhmann, L. W. Molenkamp, J. Maciejko, X. L. Qi, and S. C.

Zhang. Nonlocal transport in the quantum spin Hall state.Science, 325:294, 2009.

[79] M. Buttiker. Edge-state physics without magnetic fields. Science, 325:278, 2009.

[80] G. Tkachov and E. M. Hankiewicz. Ballistic quantum spinHall state and enhanced

edge backscattering in strong magnetic fields.Phys. Rev. Lett., 104:166803, 2010.

[81] D. N. Sheng, Z. Y. Weng, L. Sheng, and F. D. M. Haldane. Quantum spin Hall effect

and topologically invariant Chern numbers.Phys. Rev. Lett., 97:036808, 2006.

[82] M. Onoda, Y. Avishai, and N. Nagaosa. Localization in a quantum spin Hall system.

Phys. Rev. Lett., 98:076802, 2007.

[83] H. Obuse, A. Furusaki, S. Ryu, and C. Mudry. Two-dimensional spin-filtered chiral

network model for theZ2 quantum spin Hall effect.Phys. Rev. B, 76:075301, 2007.

[84] J. Li, R.-L. Chu, J. K. Jain, and S.-Q. Shen. TopologicalAnderson insulator.Phys.

Rev. Lett., 102:136806, 2009.

[85] D. Li and J. Shi. Dorokhov-Mello-Pereyra-Kumar equation for the edge transport of

a quantum spin Hall insulator.Phys. Rev. B, 79:241303(R), 2009.

[86] J. Maciejko, C. X. Liu, Y. Oreg, X. L. Qi, C. Wu, and S. C. Zhang. Kondo effect in

the helical edge liquid of the quantum spin Hall state.Phys. Rev. Lett., 102:256803,

2009.

[87] D. G. Rothe, R. W. Reinthaler, C. X. Liu, L. W. Molenkamp,S. C. Zhang, and E. M.

Hankiewicz. Fingerprint of different spin-orbit terms forspin transport in HgTe

quantum wells.New J. Phys., 12:065012, 2010.

[88] R. Winkler. Spin-Orbit Coupling Effects in Two-Dimensional Electron and Hole

Systems. Springer-Verlag, Berlin, 2003.

BIBLIOGRAPHY 215

[89] D. K. Ferry and S. M. Goodnick.Transport in Nanostructures. Cambridge Univer-

sity Press, Cambridge, 1997.

[90] S. Y. Wu, J. Cocks, and C. S. Jayanthi. General recursiverelation for the calcula-

tion of the local Green’s function in the resolvent-matrix approach. Phys. Rev. B,

49:7957, 1994.

[91] M. P. Lopez Sancho, J. M. Lopez Sancho, and J. Rubio. Quick iterative scheme for

the calculation of transfer matrices: application to Mo(100). J. Phys. F, 14:1205,

1984.

[92] B. Zhou, H.-Z. Lu, R.-L. Chu, S.-Q. Shen, and Q. Niu. Finite size effects on helical

edge states in a quantum spin Hall system.Phys. Rev. Lett., 101:246807, 2008.

[93] P. A. Lee and T. V. Ramakrishnan. Disordered electronicsystems.Rev. Mod. Phys.,

57:287, 1985.

[94] G. Bergmann. Weak localization in thin films.Phys. Rep., 107:1, 1984.

[95] M. R. Zirnbauer. Super Fourier analysis and localization in disordered wires.Phys.

Rev. Lett., 69:1584, 1992.

[96] Y. Takane. Quantum electron transport in disordered wires with symplectic symme-

try. J. Phys. Soc. Jpn., 73:1430, 2004.

[97] Y. Takane. Conductance of disordered wires with symplectic symmetry: Compari-

son between odd- and even-channel cases.J. Phys. Soc. Jpn., 73:2366, 2004.

[98] I. Zutic, J. Fabian, and S. Das Sarma. Spintronics: Fundamentals and applications.

Rev. Mod. Phys., 76:323, 2004.

[99] Y. Aharonov and D. Bohm. Significance of electromagnetic potentials in the quan-

tum theory.Phys. Rev., 115:485, 1959.

[100] D. Frustaglia, M. Hentschel, and K. Richter. Quantum transport in nonuniform mag-

netic fields: Aharonov-Bohm as a spin switch.Phys. Rev. Lett., 87:256602, 2001.

BIBLIOGRAPHY 216

[101] M. Hentschel, H. Schomerus, D. Frustaglia, and K. Richter. Aharonov-Bohm

physics with spin. I. Geometric phases in one-dimensional ballistic rings. Phys.

Rev. B, 69:155326, 2004.

[102] D. Frustaglia, M. Hentschel, and K. Richter. Aharonov-Bohm physics with spin.

II. Spin-flip effects in two-dimensional ballistic systems. Phys. Rev. B, 69:155327,

2004.

[103] D. Frustaglia and K. Richter. Spin interference effects in ring conductors subject to

Rashba coupling.Phys. Rev. B, 69:235310, 2004.

[104] S. Murakami. Quantum spin Hall effect and enhanced magnetic response by spin-

orbit coupling.Phys. Rev. Lett., 97:236805, 2006.

[105] A. Shitade, H. Katsura, J. Kunes, X. L. Qi, S. C. Zhang,and N. Nagaosa. Quantum

spin Hall effect in a transition metal oxide Na2IrO3. Phys. Rev. Lett., 102:256403,

2009.

[106] A. D. Stone and A. Szafer. What is measured when you measure a resistance? –The

Landauer formula revisited.IBM J. Res. Dev., 32:384, 1988.

[107] J. C. Slonczewski. Conductance and exchange couplingof two ferromagnets sepa-

rated by a tunneling barrier.Phys. Rev. B, 39:6995, 1989.

[108] S. Datta and B. Das. Electronic analog of the electro-optic modulator.Appl. Phys.

Lett., 56:665, 1990.

[109] Y. Aharonov and A. Casher. Topological quantum effects for neutral particles.Phys.

Rev. Lett., 53:319, 1984.

[110] G. Usaj. Edge states interferometry and spin rotations in zigzag graphene nanorib-

bons.Phys. Rev. B, 80:081414(R), 2009.

[111] P. J. Mohr, B. N. Taylor, and D. B. Newell. CODATA recommended values of the

fundamental physical constants: 2006.Rev. Mod. Phys., 80:633, 2008.

BIBLIOGRAPHY 217

[112] T. H. Hansson, V. Oganesyan, and S. L. Sondhi. Superconductors are topologically

ordered.Ann. Phys., 313:497, 2004.

[113] X. L. Qi and S. C. Zhang. The quantum spin Hall effect andtopological insulators.

Phys. Today, 63:33, 2010.

[114] J. E. Moore. Topological insulators: The next generation. Nature Phys., 5:378, 2009.

[115] M. Z. Hasan and C. L. Kane. Colloquium: Topological insulators.Rev. Mod. Phys.,

82:3045, 2010.

[116] X. L. Qi and S. C. Zhang. Topological insulators and superconductors.e-print

arXiv:1008.2026, 2010.

[117] M.-C. Chang and M.-F. Yang. Optical signature of topological insulators.Phys. Rev.

B, 80:113304, 2009.

[118] A. D. LaForge, A. Frenzel, B. C. Pursley, T. Lin, X. Liu,J. Shi, and D. N. Basov.

Optical characterization of Bi2Se3 in a magnetic field: Infrared evidence for mag-

netoelectric coupling in a topological insulator material. Phys. Rev. B, 81:125120,

2010.

[119] N. P. Butch, K. Kirshenbaum, P. Syers, A. B. Sushkov, G.S. Jenkins, H. D. Drew,

and J. Paglione. Strong surface scattering in ultrahigh mobility Bi 2Se3 topological

insulator crystals.Phys. Rev. B, 81:241301(R), 2010.

[120] J. N. Hancock, private communication.

[121] J. Zhe, private communication.

[122] L. D. Landau and E. M. Lifshitz.Electrodynamics of Continuous Media, 2nd Edition.

Pergamon Press, Oxford, 1984.

[123] R. B. Laughlin. Anomalous quantum Hall effect: an incompressible quantum fluid

with fractionally charged excitations.Phys. Rev. Lett., 50:1395, 1983.

BIBLIOGRAPHY 218

[124] S. C. Zhang and J. P. Hu. A four-dimensional generalization of the quantum Hall

effect. Science, 294:823, 2001.

[125] B. A. Bernevig, C. H. Chern, J. P. Hu, N. Toumbas, and S. C. Zhang. Effective

field theory description of the higher dimensional quantum Hall liquid. Ann. Phys.,

300:185, 2002.

[126] A. M. Essin, J. E. Moore, and D. Vanderbilt. Magnetoelectric polarizability and

axion electrodynamics in crystalline insulators.Phys. Rev. Lett., 102:146805, 2009.

[127] M. Levin and A. Stern. Fractional topological insulators. Phys. Rev. Lett.,

103:196803, 2009.

[128] A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig.Classification of topo-

logical insulators and superconductors in three spatial dimensions. Phys. Rev. B,

78:195125, 2008.

[129] A. Kitaev. Periodic table for topological insulatorsand superconductors.AIP Conf.

Proc., 1134:22, 2009.

[130] J. K. Jain. Incompressible quantum Hall states.Phys. Rev. B, 40:8079, 1989.

[131] X. G. Wen. Edge excitations in the fractional quantum Hall states at general filling

fractions.Mod. Phys. Lett. B, 5:39, 1991.

[132] X. G. Wen. Non-Abelian statistics in the fractional quantum Hall states.Phys. Rev.

Lett., 66:802, 1991.

[133] B. Blok and X. G. Wen. Many-body systems with non-Abelian statistics.Nucl.

Phys. B, 374:615, 1992.

[134] X. G. Wen. Theory of the edge states in fractional quantum Hall effects.Int. J. Mod.

Phys. B, 6:1711, 1992.

[135] X. G. Wen. Projective construction of non-Abelian quantum Hall liquids.Phys. Rev.

B, 60:8827, 1999.

BIBLIOGRAPHY 219

[136] M. Barkeshli and X. G. Wen. Effective field theory and projective construction for

Zk parafermion fractional quantum Hall states.Phys. Rev. B, 81:155302, 2010.

[137] M. Levin and M. P. A. Fisher. Gapless layered three-dimensional fractional quantum

Hall states.Phys. Rev. B, 79:235315, 2009.

[138] S. L. Adler. Axial vector vertex in spinor electrodynamics. Phys. Rev., 177:2426,

1969.

[139] J. S. Bell and R. Jackiw. A PCAC puzzle:π0 → γγ in the sigma model.Nuovo

Cimento A, 60:47, 1969.

[140] E. Witten. OnS-duality in Abelian gauge theory.Sel. Math. New Ser., 1:383, 1995.

[141] A. Kapustin and E. Witten. Electric-magnetic dualityand the geometric Langlands

program.e-print arXiv:hep-th/0604151, 2006.

[142] S. Gukov and E. Witten. Gauge theory, ramification, andthe geometric Langlands

program.e-print arXiv:hep-th/0612073, 2006.

[143] E. Corrigan and D. Olive. Colour and magnetic monopoles. Nucl. Phys. B, 110:237,

1976.

[144] F. D. M. Haldane. Fractional quantization of the Hall effect: A hierarchy of incom-

pressible quantum fluid states.Phys. Rev. Lett., 51:605, 1983.

[145] B. I. Halperin. Statistics of quasiparticles and the hierarchy of fractional quantized

Hall states.Phys. Rev. Lett., 52:1583, 1984.

[146] Y. Ran, H. Yao, and A. Vishwanath. Composite fermions and quantum Hall stripes

on the topological insulator surface.e-print arXiv:1003.0901, 2010.

[147] E. Witten. Dyons of chargeeθ/2π. Phys. Lett. B, 86:283, 1979.

[148] G. Rosenberg and M. Franz. Witten effect in a crystalline topological insulator.

Phys. Rev. B, 82:035105, 2010.

BIBLIOGRAPHY 220

[149] X. G. Wen. Vacuum degeneracy of chiral spin states in compactified space.Phys.

Rev. B, 40:7387, 1989.

[150] X. G. Wen and Q. Niu. Ground-state degeneracy of the fractional quantum Hall

states in the presence of a random potential and on high-genus Riemann surfaces.

Phys. Rev. B, 41:9377, 1990.

[151] X. G. Wen and A. Zee. Topological degeneracy of quantumHall fluids. Phys. Rev.

B, 58:15717, 1998.

[152] M. Sato. Topological discrete algebra, ground-statedegeneracy, and quark confine-

ment in QCD.Phys. Rev. D, 77:045013, 2008.

[153] A. P. Schnyder, S., A. Furusaki, and A. W. W. Ludwig. Classification of topological

insulators and superconductors.AIP Conf. Proc., 1134:10, 2009.

[154] S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Ludwig.Topological insula-

tors and superconductors: tenfold way and dimensional hierarchy. New J. Phys.,

12:065010, 2010.

[155] A. B. Sushkov, G. S. Jenkins, D. C. Schmadel, N. P. Butch, J. Paglione, and H. D.

Drew. Far-infrared cyclotron resonance and Faraday effectin Bi2Se3. Phys. Rev. B,

82:125110, 2010.

[156] G. S. Jenkins, A. B. Sushkov, D. C. Schmadel, N. P. Butch, P. Syers, J. Paglione,

and H. D. Drew. Terahertz Kerr and reflectivity measurementson the topological

insulator Bi2Se3. Phys. Rev. B, 82:125120, 2010.

[157] R. Jackiw and C. Rebbi. Vacuum periodicity in a Yang-Mills quantum theory.Phys.

Rev. Lett., 37:172, 1976.

[158] C. G. Callan, R. F. Dashen, and D. J. Gross. The structure of the gauge theory

vacuum.Phys. Lett. B, 63:334, 1976.

[159] M. M. Vazifeh and M. Franz. Quantization and2π periodicity of the axion action in

topological insulators.Phys. Rev. B, 82:233103, 2010.

BIBLIOGRAPHY 221

[160] S. Raghu, S. B. Chung, X. L. Qi, and S. C. Zhang. Collective modes of a helical

liquid. Phys. Rev. Lett., 104:116401, 2010.

[161] L. Santos, T. Neupert, C. Chamon, and C. Mudry. Superconductivity on the sur-

face of topological insulators and in two-dimensional noncentrosymmetric materi-

als. Phys. Rev. B, 81:184502, 2010.

[162] A. Cortijo. A superconducting instability on the surface of a topological insulator.

e-print arXiv:1012.2008, 2010.

[163] C. Xu. Time-reversal symmetry breaking at the edge states of a three-dimensional

topological band insulator.Phys. Rev. B, 81:020411, 2010.

[164] C. Xu. Quantum critical points of helical Fermi liquids. Phys. Rev. B, 81:054403,

2010.

[165] K.-S. Kim and T. Takimoto. Nambu-Eliashberg theory for multi-scale quantum crit-

icality: Application to ferromagnetic quantum criticality on the surface of three-

dimensional topological insulators.e-print arXiv:1011.4851, 2010.

[166] P. Ghaemi and S. Ryu. Competing orders in the Dirac-like electronic structure and

the non-linear sigma model with topological terms.e-print arXiv:1012.5840, 2010.

[167] J.-H. Jiang and S. Wu. Spin susceptibility and helicalmagnetic order at the edge of

topological insulators due to Fermi surface nesting.e-print arXiv:1012.1299, 2010.

[168] Y. Zhang, Y. Ran, and A. Vishwanath. Topological insulators in three dimensions

from spontaneous symmetry breaking.Phys. Rev. B, 79:245331, 2009.

[169] M. Kargarian, J. Wen, and G. A. Fiete. Competing exotictopological insulator

phases in transition-metal oxides on the pyrochlore lattice with distortion.e-print

arXiv:1101.0007, 2011.

[170] M. Kurita, Y. Yamaji, and M. Imada. Topological insulators from spontaneous

symmetry breaking induced by electron correlation on pyrochlore lattices.e-print

arXiv:1012.0655, 2010.

BIBLIOGRAPHY 222

[171] Z. Wang, X. L. Qi, and S. C. Zhang. Topological order parameters for interacting

topological insulators.Phys. Rev. Lett., 105:256803, 2010.

[172] V. Gurarie. Single-particle Green’s functions and interacting topological insulators.

Phys. Rev. B, 83:085426, 2011.

[173] D. Pesin and L. Balents. Mott physics and band topologyin materials with strong

spin-orbit interaction.Nature Phys., 6:376, 2010.

[174] W. Witczak-Krempa, T. P. Choy, and Y. B. Kim. Gauge fieldfluctuations in three-

dimensional topological Mott insulators.Phys. Rev. B, 82:165122, 2010.

[175] X.-J. Liu, X. Liu, L. C. Kwek, and C. H. Oh. Fractional spin Hall effect in atomic

Bose gases.Phys. Rev. B, 79:165301, 2009.

[176] A. Karch, J. Maciejko, and T. Takayanagi. Holographicfractional topological insu-

lators in2 + 1 and1 + 1 dimensions.Phys. Rev. D, 82:126003, 2010.

[177] B. I. Halperin. Possible states for a three-dimensional electron gas in a strong mag-

netic field.Jpn. J. Appl. Phys. Suppl., 26:1913, 1987.

[178] M. Kohmoto, B. I. Halperin, and Y.-S. Wu. Diophantine equation for the three-

dimensional quantum Hall effect.Phys. Rev. B, 45:13488, 1992.

[179] L. Balents and M. P. A. Fisher. Chiral surface states inthe bulk quantum Hall effect.

Phys. Rev. Lett., 76:2782, 1996.

[180] J. Maciejko, X. L. Qi, A. Karch, and S. C. Zhang. Fractional topological insulators

in three dimensions.Phys. Rev. Lett., 105:246809, 2010.

[181] B. Swingle, M. Barkeshli, J. McGreevy, and T. Senthil.Correlated topological insu-

lators and the fractional magnetoelectric effect.e-print arXiv:1005.1076, 2010.

[182] K. G. Wilson. Confinement of quarks.Phys. Rev. D, 10:2445, 1974.

[183] A. M. Polyakov. Compact gauge fields and the infrared catastrophe.Phys. Lett. B,

59:82, 1975.

BIBLIOGRAPHY 223

[184] T. Banks, R. Myerson, and J. Kogut. Phase transitions in Abelian lattice gauge

theories.Nucl. Phys. B, 129:493, 1977.

[185] J. L. Cardy and E. Rabinovici. Phase structure ofZp models in the presence of aθ

parameter.Nucl. Phys. B, 205:1, 1982.

[186] J. L. Cardy. Duality and theθ parameter in Abelian lattice models.Nucl. Phys. B,

205:17, 1982.

[187] A. Ukawa, P. Windey, and A. H. Guth. Dual variables for lattice gauge theories and

the phase structure ofZ(N) systems.Phys. Rev. D, 21:1013, 1980.

[188] N. Seiberg. Supersymmetry and non-perturbative betafunctions. Phys. Lett. B,

206:75, 1988.

[189] D. D. Dietrich and F. Sannino. Walking in theSU(N). Phys. Rev. D, 75:085018,

2007.

[190] T. Banks and A. Zaks. On the phase structure of vector-like gauge theories with

massless fermions.Nucl. Phys. B, 196:189, 1982.

[191] T. DeGrand. Lattice studies of QCD-like theories withmany fermionic degrees of

freedom.e-print arXiv:1010.4741, 2010.

[192] N. Seiberg. Electric-magnetic duality in supersymmetric non-Abelian gauge theo-

ries. Nucl. Phys. B, 435:129, 1995.

[193] C. Hoyos-Badajoz, K. Jensen, and A. Karch. A holographic fractional topological

insulator.Phys. Rev. D, 82:086001, 2010.

[194] E. Vicari and H. Panagopoulos. Theta dependence ofSU(N) gauge theories in the

presence of a topological term.Phys. Rep., 470:93, 2009.

[195] E. Witten. Theta dependence in the largeN limit of four-dimensional gauge theories.

Phys. Rev. Lett., 81:2862, 1998.

BIBLIOGRAPHY 224

[196] E. Fradkin and S. H. Shenker. Phase diagrams of latticegauge theories with Higgs

fields. Phys. Rev. D, 19:3682, 1979.

[197] P. Hosur, S. Ryu, and A. Vishwanath. Chiral topological insulators, superconductors,

and other competing orders in three dimensions.Phys. Rev. B, 81:045120, 2010.

[198] S. Ryu, J. E. Moore, and A. W. W. Ludwig. Electromagnetic and gravitational

responses and anomalies in topological insulators and superconductors.e-print

arXiv:1010.0936, 2010.

[199] M. Mulligan. Interactions and the theta term in one-dimensional gapped systems.

e-print arXiv:1010.6094, 2010.

[200] G. ’t Hooft, inRecent Developments in Gauge Theories, edited by G. ’t Hooftet al.

(Plenum Press, New York, 1980).

[201] G. ’t Hooft. Magnetic charge quantization and fractionally charged quarks.Nucl.

Phys. B, 105:538, 1976.

[202] J. Preskill, inMagnetic Monopoles, edited by R. Carrigan and P. Trower (Plenum

Press, New York, 1983).

[203] G. ’t Hooft. On the phase transition towards permanentquark confinement.Nucl.

Phys. B, 138:1, 1978.

[204] F. Englert and P. Windey. Quantization condition for ’t Hooft monopoles in compact

simple Lie groups.Phys. Rev. D, 14:2728, 1976.

[205] P. Goddard, J. Nuyts, and D. I. Olive. Gauge theories and magnetic charge.Nucl.

Phys. B, 125:1, 1977.

[206] H. Georgi.Lie Algebras in Particle Physics, 2nd Edition. Perseus Books, Reading,

1999.

[207] M. Nakahara. Geometry, Topology and Physics. Institute of Physics Publishing,

London, 1990.

BIBLIOGRAPHY 225

[208] A. M. Polyakov. Thermal properties of gauge fields and quark liberation.Phys. Lett.

B, 72:477, 1978.

[209] E. Witten. Anti-de Sitter space, thermal phase transition, and confinement in gauge

theories.Adv. Theor. Math. Phys., 2:505–532, 1998.

[210] O. Aharony and E. Witten. Anti-de Sitter space and the center of the gauge group.

J. High Energy Phys., 11:018, 1998.

[211] S. Elitzur. Impossibility of spontaneously breakinglocal symmetries.Phys. Rev. D,

12:3978, 1975.

[212] M. Barkeshli and X. G. Wen. Phase transitions inZN gauge theory and twistedZN

topological phases.e-print arXiv:1012.2417, 2010.

[213] D. Gaiotto and E. Witten. Janus configurations, Chern-Simons couplings, and the

theta-angle inN = 4 super Yang-Mills theory.J. High Energy Phys., 06:097, 2010.

[214] K.-T. Chen and P. A. Lee. Topological insulator and theθ-vacuum in a system with-

out boundaries.e-print arXiv:1012.2084, 2010.

[215] E. Witten. Quantum field theory and the Jones polynomial. Commun. Math. Phys.,

121:351, 1989.

[216] E. P. Verlinde. Fusion rules and modular transformations in 2D conformal field

theory.Nucl. Phys. B, 300:360, 1988.

[217] R. Dijkgraaf and E. P. Verlinde. Modular invariance and the fusion algebra.Nucl.

Phys. Proc. Suppl. B, 5:87, 1988.

[218] E. Tang, J. W. Mei, and X. G. Wen. High temperature fractional quantum Hall states.

Phys. Rev. Lett., 106:236802, 2011.

[219] K. Sun, Z. C. Gu, H. Katsura, and S. Das Sarma. Nearly-flat bands with nontrivial

topology.Phys. Rev. Lett., 106:236803, 2011.

BIBLIOGRAPHY 226

[220] T. Neupert, L. Santos, C. Chamon, and C. Mudry. Fractional quantum Hall states at

zero magnetic field.Phys. Rev. Lett., 106:236804, 2011.

[221] D. N. Sheng, Z. C. Gu, K. Sun, and L. Sheng. Fractional quantum Hall effect in the

absence of Landau levels.Nature Commun., 2:389, 2011.

[222] N. Regnault and B. Andrei Bernevig. Fractional Chern insulator. e-print

arXiv:1105.4867, 2011.

[223] T. Neupert, L. Santos, S. Ryu, C. Chamon, and C. Mudry. Fractional topo-

logical liquids with time-reversal symmetry and their lattice realization.e-print

arXiv:1106.3989, 2011.

[224] M. O. Goerbig. From fractional Chern insulator to a fractional quantum spin Hall

effect.e-print arXiv:1107.1986, 2011.

[225] Y. Li and C. Wu. Three-dimensional topological insulators with Landau levels.e-

print arXiv:1103.5422, 2011.

[226] G. D. Mahan. Many-Particle Physics, 3rd Edition. World Scientific, New York,

2000.

[227] R. Shankar. Bosonization: how to make it work for you incondensed matter.Acta

Phys. Pol. B, 26:1835, 1995.

[228] N. Nagaosa. Quantum Field Theory in Strongly Correlated Electronic Systems.

Springer, Berlin, 1999.

[229] J. L. Cardy.Scaling and Renormalization in Statistical Physics. Cambridge Univer-

sity Press, Cambridge, 1996.

[230] P. Di Francesco, P. Mathieu, and D. Senechal.Conformal Field Theory. Springer,

New York, 1997.

[231] R. Shankar. Solvable model of a metal-insulator transition. Int. J. Mod. Phys. B,

4:2371, 1990.

BIBLIOGRAPHY 227

[232] D. L. Maslov and M. Stone. Landauer conductance of Luttinger liquids with leads.

Phys. Rev. B, 52:R5539, 1995.

[233] E.-A. Kim, private communication.

[234] A. Furusaki and N. Nagaosa. Single-barrier problem and Anderson localization in a

one-dimensional interacting electron system.Phys. Rev. B, 47:4631, 1993.

[235] A. O. Caldeira and A. J. Leggett. Quantum tunnelling ina dissipative system.Ann.

Phys. (N.Y.), 149:374, 1983.

[236] A. Schmid. Diffusion and localization in a dissipative quantum system.Phys. Rev.

Lett., 51:1506, 1983.

[237] M. P. A. Fisher and W. Zwerger. Quantum Brownian motionin a periodic potential.

Phys. Rev. B, 32:6190, 1985.

[238] U. Eckern, G. Schon, and V. Ambegaokar. Quantum dynamics of a superconducting

tunnel junction.Phys. Rev. B, 30:6419, 1984.

[239] M. P. A. Fisher. Quantum phase slips and superconductivity in granular films.Phys.

Rev. Lett., 57:885, 1986.

[240] G. Baym and N. D. Mermin. Determination of thermodynamic Green’s functions.

J. Math. Phys., 2:232, 1961.

[241] A. M. Tsvelik. Quantum Field Theory in Condensed Matter Physics. Cambridge

University Press, Cambridge, 1995.

[242] G. Yuval and P. W. Anderson. Exact results for the Kondoproblem: one-body theory

and extension to finite temperature.Phys. Rev. B, 1:1522, 1970.

[243] R. Landauer. Johnson-Nyquist noise derived from quantum mechanical transmis-

sion. Physica D, 38:226, 1989.

[244] J. Schwinger. Brownian motion of a quantum oscillator. J. Math. Phys., 2:407, 1961.

BIBLIOGRAPHY 228

[245] L. V. Keldysh. Diagram technique for nonequilibrium processes.Zh. Eksp. Teor. Fiz.,

47:1515, 1964 [Sov. Phys. JETP, 20:1018, 1965].

[246] R. de Picciotto, M. Reznikov, M. Heiblum, V. Umansky, G. Bunin, and D. Mahalu.

Direct observation of a fractional charge.Nature, 389:162, 1997.

[247] L. Saminadayar, D. C. Glattli, Y. Jin, and B. Etienne. Observation of thee/3 frac-

tionally charged Laughlin quasiparticle.Phys. Rev. Lett., 79:2526, 1997.

[248] N. Nagaosa.Quantum Field Theory in Condensed Matter Physics. Springer, Berlin,

1999.

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