Transport in Quantum Hall Systems : Probing

Anyons and Edge Physics

by

Chenjie Wang

B.Sc., University of Science and Technology of China, Hefei, China, 2007

M.Sc., Brown University, Providence, RI, 2010

A dissertation submitted in partial fulfillment of the

requirements for the degree of Doctor of Philosophy

in Department of Physics at Brown University

PROVIDENCE, RHODE ISLAND

May 2012

c© Copyright 2012 by Chenjie Wang

This dissertation by Chenjie Wang is accepted in its present form

by Department of Physics as satisfying the

dissertation requirement for the degree of Doctor of Philosophy.

Date

Prof. Dmitri E. Feldman, Advisor

Recommended to the Graduate Council

Date

Prof. J. Bradley Marston, Reader

Date

Prof. Vesna F. Mitrovic, Reader

Approved by the Graduate Council

Date

Peter M. Weber, Dean of the Graduate School

iii

Curriculum Vitae

Personal Information

Name: Chenjie Wang

Date of Birth: Feb 02, 1984

Place of Birth: Haining, China

Education

• Brown University, Providence, RI, 2007-2012

- Ph.D. in physics, Department of Physics (May 2012)

- M.Sc. in physics, Department of Physics (May 2010)

• University of Science and Technology of China, Hefei, China, 2003-

2007

- B.Sc. in physics, Department of Modern Physics (July 2007)

Publications

1. Chenjie Wang, Guang-Can Guo and Lixin He, Ferroelectricity driven by the

iv

noncentrosymmetric magnetic ordering in multiferroic TbMn2O5: a first-principles

study, Phys. Rev. Lett. 99, 177202 (2007)

2. Chenjie Wang, Guang-Can Guo, and Lixin He, First-principles study of the

lattice and electronic structures of TbMn2O5, Phys. Rev B 77, 134113 (2008)

3. Chenjie Wang and D. E. Feldman, Transport in line junctions of ν = 5/2

quantum Hall liquids, Phys. Rev. B 81, 035318 (2010)

4. Chenjie Wang and D. E. Feldman, Identification of 331 quantum Hall states

with Mach-Zehnder interferometry, Phys. Rev. B 82, 165314 (2010)

5. Chenjie Wang and D. E. Feldman, Rectification in Y-junctions of Luttinger

liquid wires, Phys. Rev. B 83, 045302 (2011)

6. Chenjie Wang and D. E. Feldman, Fluctuation-dissipation theorem for chiral

systems in non-equilibrium steady states, Phys. Rev. B 84, 235315 (2011)

7. Chenjie Wang and D. E. Feldman, in preparation.

v

Acknowledgements

Well, it is time to say goodbye to Brown. How time flies!

Five years ago when I arrived in Providence, a beautiful and quiet small town,

I was curious and uncertain about everything. Now after five years of struggling,

looking back to my graduate life, I find it is more than satisfying. I would like to

thank several people without whom my PhD would be impossible and my life would

be harder.

Without a doubt, my advisor Professor Dima Feldman is the first one to whom

I would like to express my thanks for his guidance, encouragement and most im-

portantly his unique personality influence. Dima’s intelligence and deep insight in

science influence me a lot. “You are not thinking logically.” I still remember these

words he said to me when I was learning quantum many-body physics. These words

were the point where I actually started to think scientifically, and I am still benefiting

from them. Thanks, Dima!

I would like to thank several condensed matter professors, Professor Michael

Kosterlitz, Professor Xinsheng Ling, Professor Brad Marston, Professor Vesna Mitro-

vic and Professor See-Chen Ying for their teaching, helpful discussions and/or being

on my prelim and defense committee. In particular, I thank Professor Marston with

whom I had many fruitful discussions on numerical methodology. Professor Koster-

litz also deserves my special acknowledgements. I joined two study groups that he

vi

supervised, from which I learned much knowledge of the renormalization group and

quantum phase transition. I also thank them for being my recommendation letter

writers.

I am grateful to my current and former officemates Hao Tu, Guang Yang, Wan-

ming Qi, Florian Sabou, Pengyu Liu and Lei Wang, Feifei Li and Dina Obeid. We

all have or had been suffering from the windowless office for many years, with me

the luckiest – my desk is close to the door. There are many other classmates and

friends that I need to thank, without whom my life would have been harder and

of less fun : Feifei Li, Dima’s former student, who helped a lot at the beginning of

my graduate research; Jun He, my former roommate, who offered me many helps

when I was still learning to survive in this country; Congkao Wen for introducing me

to the view of high-energy physics; Chao Li, a nice life companion; Xuqing Huang,

a very good friend who always offered free hospitality and meals when I went to

Boston; Hong Pan, a very good friend with whom I had so many useful discussions

on physics experiments; Bosheng Zhang, a nice friend with whom I had many useful

discussions on life philosophy; and Yana Cheng, Yuzhen Guan, Xin Jia, Mingming

Jiang, Dongfang Li, Wenzhe Zhang, Ilyong Jung and Alex Geringer-Sameth for their

help and encouragement. I would like to thank the administrative staff of the Physics

Department, Barbara Dailey, Sabina Griffin, and Jane Martin and the Chair of the

Physics Department, Prof. James Valles, for their help. It is hard for me to think

of a complete list now, because writing the thesis has squeezed all my energy out. I

am sorry to those who are important to me but not on this list.

Finally my beloved parents. They know nothing about physics, but they have

been supporting my career all the time. I cannot imagine getting a PhD without

their tremendous support and love.

vii

Contents

Curriculum Vitae iv

Acknowledgments vi

1 Introduction 1

1.1 From the Classical Hall Effect to the Quantum Hall effect . . . . . . . 4

1.1.1 Energy Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2 Theories of QHE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2.1 IQHE: Landau Quantization . . . . . . . . . . . . . . . . . . . 12

1.2.2 Laughlin FQHE States . . . . . . . . . . . . . . . . . . . . . . 16

1.2.3 Quasiparticles, Anyons and Anyonic Statistics . . . . . . . . . 18

1.2.4 FQHE at ν = 5/2: Non-Abelian States . . . . . . . . . . . . . 22

1.3 Edge theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.4 Interferometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.4.1 Fabry-Perot Interferometer . . . . . . . . . . . . . . . . . . . . 30

viii

1.4.2 Mach-Zehnder Interferometer . . . . . . . . . . . . . . . . . . 34

1.5 Transport Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

1.5.1 Landauer-Buttiker Approach . . . . . . . . . . . . . . . . . . . 37

1.5.2 Fluctuation-Dissipation Theorem . . . . . . . . . . . . . . . . 40

1.5.3 Keldysh Formalism . . . . . . . . . . . . . . . . . . . . . . . . 41

1.5.4 Example: Tunneling through a Single QPC . . . . . . . . . . . 43

1.6 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2 Identification of 331 Quantum Hall States with Mach-Zehnder

Interferometry 49

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.2 Statistics in the 331 state . . . . . . . . . . . . . . . . . . . . . . . . 54

2.3 Mach-Zehnder interferometer . . . . . . . . . . . . . . . . . . . . . . 56

2.4 Electric current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.5 Shot noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3 Transport in Line Junctions of ν = 5/2 Quantum Hall Liquids 71

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.2 Proposed 5/2 states . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.3 Qualitative discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.4 Calculation of the current . . . . . . . . . . . . . . . . . . . . . . . . 89

3.5 The number of singularities . . . . . . . . . . . . . . . . . . . . . . . 97

3.6 I-V curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

ix

3.6.1 Tunneling into integer edge modes. . . . . . . . . . . . . . . . 107

3.6.2 K = 8 state . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

3.6.3 331 state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

3.6.4 Pfaffian state . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

3.6.5 Reconstructed Pfaffian state . . . . . . . . . . . . . . . . . . 116

3.6.6 Disorder-dominated anti-Pfaffian state . . . . . . . . . . . . . 119

3.6.7 Non-equilibrated anti-Pfaffian state . . . . . . . . . . . . . . . 122

3.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

4 Fluctuation-Dissipation Theorem for Chiral Systems in Nonequi-

librium Steady States 129

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

4.2 Heuristic derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

4.3 Proof of the nonequilibrium FDT . . . . . . . . . . . . . . . . . . . . 137

4.3.1 Chiral systems . . . . . . . . . . . . . . . . . . . . . . . . . . 138

4.3.2 Initial density matrix and Heisenberg current operator . . . . 140

4.3.3 Voltage bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

4.3.4 Main argument . . . . . . . . . . . . . . . . . . . . . . . . . . 146

4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

4.5 Generalization to Multi-Terminal Setup and Heat Transport – Proof

by Fluctuation Relations . . . . . . . . . . . . . . . . . . . . . . . . . 153

4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

5 Summary 164

x

A Multi-component Halperin States 166

A.1 Quasiparticle statistics and edge modes . . . . . . . . . . . . . . . . . 167

A.2 The case of only one flavor allowed to tunnel . . . . . . . . . . . . . . 171

A.2.1 Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

A.2.2 Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

A.3 General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

B Integer Edge Reconstruction 181

C FDT in an Ideal Gas Model 184

xi

List of Tables

1.1 Energy Scales, estimated for a GaAs-AlGaAs system with the width

of the 2D electron gas set to 25 nm and the magnetic field at 5 Tesla

(typical for 5/2 FQHE). Data for ν = 1/3 and ν = 5/2 is from Ref. [34] 12

3.1 The number of conductance singularities for different models in dif-

ferent setups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

3.2 Summary of singularities in the voltage dependence of the differential

conductance GAtun for different 5/2 states. The “Modes” column shows

the numbers of left- and right-moving modes in the fractional edge,

the number in the brackets being the number of Majorana modes.

“A” or “N” in the next column means Abelian or non-Abelian statis-

tics. The “Singularities” shows the number of singularities, including

divergencies (S), discontinuities (D) and cusps (C), i.e., discontinuities

of the first or higher derivative of the voltage dependence of GAtun. The

table refers to the tunneling into a boundary of ν = 5/2 and ν = 2

liquids. The case of weak interaction, Fig. 3.1, is closely related. . . 124

xii

List of Figures

1.1 Classical Hall effect. A current I, carried by electrons, flows along the

x axis of a conducting strip. A Hall voltage VH is measured between

the two edges of the strip after a perpendicular magnetic field B that

points upward is turned on. . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Sketch of the quantum Hall effect. When plateaus occur in the mag-

netic field dependence of the Hall resistance (red), the longitudinal

resistance (blue) vanishes when the temperature is low enough. On

the plateaus, the Hall resistance has quantized value RH = h/e2ν.

IQHE has integer ν while FQHE (not shown) has fractional ν. At

very low magnetic fields, the classical Hall effect takes places. . . . . . 7

1.3 Sketch of the multi-terminal setup to measure Hall resistance and

longitudinal resistance. The Hall voltage is measured between contact

1 and 3, and the longitudinal voltage is measured between contact 1

and 2. A current is maintained between the source S and drain D. . 7

1.4 (a) Landau levels in translationally invariant systems. (b) Broadened

Landau levels in systems with disorder. The shaded tails of each level

are Anderson localized states. . . . . . . . . . . . . . . . . . . . . . . 14

xiii

1.5 Landau levels under adiabatic bending due to a confining potential.

The vertical axis is energy, while the horizontal axis can be either

position or momentum since the two are locked through the relation

x = −kl2 in our semiclassical picture. The dots represent occupied

states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.6 Exchange statistics. Particle 1 moves around particle 2 in a full circle.

In 3D, the path can be contracted back to the starting point without

meeting any particles. In 2D, particle 2 prevents such a contraction. 18

1.7 Shape distortion of an incompressible QHE droplet. Such distortions

are edge excitations. Effectively, the edge excitations form a Fermi

liquid when the droplet is on IQHE plateaus but become a Luttinger

liquid on FQHE plateaus. . . . . . . . . . . . . . . . . . . . . . . . . 27

1.8 Fabry-Perot interferometer. Two quantum point contacts (QPCs) are

introduced in a Hall bar by side gates (gray triangles). Charges flowing

out of source S can either tunnel through QPC1 or QPC2 and arrive

coherently to drain D. Modifying the area encircled by the two pathes

or changing the magnetic field leads to an Aharonov-Bohm oscillation

of the current, measured at drain D. The Aharonov-Bohm oscillations

are affected by localized anyons (red dots) inside the interference loop

in FQHE systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

xiv

1.9 Mach-Zehnder interferometer. Currents flow from source S1 to drain

D2. Similarly to the Fabry-Perot interferometer, two QPCs are in-

troduced. There are two paths for quasiparticles to go from S1 to

D2: S1-QPC1-A-QPC2-D2 and S1-QPC1-B-QPC2-D2. The differ-

ence from the Fabry-Perot interferometer is that quasiparticles stay

inside the interference loop QPC1-B-QPC2-A-QPC1 after a tunnel-

ing event, while in the Fabry-Perot interferometer quasiparticles stay

outside the interference loop after a tunneling event. . . . . . . . . . 34

1.10 Quantum Hall bar with a quantum point contact. . . . . . . . . . . 37

1.11 The Keldysh contour. . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.1 Schematic picture of an anyonic Mach-Zehnder interferometer. Ar-

rows indicate propagation directions of the edge modes on Edge 1

(from source S1 to drain D1) and Edge 2 (from source S2 to drain

D2). Quasiparticles can tunnel between the two edges through two

quantum point contacts, QPC1 and QPC2. . . . . . . . . . . . . . . 52

2.2 Possible states of a Mach-Zehnder interferometer in the 331 state.

Panel (a) shows a general case, eight possible states labeled by topo-

logical charges and the transition rates between them. Arrows show

the allowed transitions at zero temperature. Solid blue lines represent

tunneling events involving quasiparticles of flavor a, and dashed black

lines represent tunneling events involving particles of flavor b. Special

cases with pak = pbk ≡ pk and pbk = 0 are illustrated in Panels (b) and

(c) respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

xv

2.3 Flux-dependence of the tunneling current in the 331 and Pfaffian

states. We set Aa = 1, ua = ub = 1, and δa = δb = 0 for all curves

for the 331 state. Different curves correspond to different values of

γ = Ab/Aa in the 331 state. The curve for the Pfaffian state is plotted

according to Eq. (8) in Ref. [64] with r+11 = r+12 = 1 and Γ1 = Γ2 such

that the maximum matches the maximum of the curve for the 331

state with γ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.4 The maximal e∗ as a function of γ and δ. . . . . . . . . . . . . . . . . 64

3.1 (a) Tunneling between ν = 5/2 and ν = 2 QHE liquids. The edges of

the upper and lower QHE liquids form a line junction. (b) Tunneling

between ν = 5/2 QHE liquid and a quantum wire. In both setups,

contacts C1 and C2 are kept at the same voltage V . . . . . . . . . . . 73

3.2 Tunneling between the fractional QHE channels of the ν = 5/2 edge

and the ν = 2 integer channels. Contacts C1 and C2 are kept at the

same voltage V and the other contacts are grounded. . . . . . . . . . 75

3.3 A bar geometry that can be used to detect the non-equilibrated anti-

Pfaffian state. Solid lines denote Integer QHE edge modes, the dashed

lines denote fractional QHE charged modes and dotted lines denote

Majorana modes. Arrows show mode propagation directions. . . . . 82

xvi

3.4 Illustration of the graphical method. (a) Tunneling between two inte-

ger QHE modes. The left solid line represents the electron spectrum

at the upper edge at zero voltage. The right solid line represents the

spectrum at the lower edge. The dashed lines represent the electron

spectra at the upper edge at different voltages. Black dots repre-

sent occupied states. The momentum mismatch between two edges

∆k > 0. (b) Tunneling between an integer QHE edge and a Pfaf-

fian edge. The right line represents the spectrum of the integer edge.

The left line shows the spectrum of the charged boson mode at the

Pfaffian edge. The unevenly dashed lines (λ lines) represent Majo-

rana fermions. The figure illustrates a tunneling event in which an

electron with the momentum k0 tunnels into the Pfaffian edge and

creates a boson with the momentum k and a Majorana fermion with

the momentum k0 − k. . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.5 A 3-dimensional illustration of the integration volume in the integral

(3.24). The integral (3.24) is taken over the volume under the shaded

surface in the positive orthant. In panel (a), ω < vRi∆k and the ωRi

axis intersects superplane Σ closer to the origin than the plane Ω. In

panel (b) ω > vRi∆k and the order of the intersection points reverses. 95

3.6 (a) Voltage dependence of the differential conductance in the K = 8

state at a fixed momentum mismatch ∆k in the case of tunneling

into the edge between the states with ν = 5/2 and ν = 2. Voltage

is shown in units of ω0 = v2∆k, and the conductance is shown in

arbitrary units. (b) Momentum mismatch dependence of GAtun at a

fixed voltage. ∆k0 = ω/v2. For both curves, we set v3/v2 = 0.8. . . . 108

xvii

3.7 Voltage and momentum mismatch dependence of the tunneling dif-

ferential conductance GA,utun in the 331 state; u is either a or b. We

have chosen the ratios of the edge velocities to be v3/v2 = 0.8 and

v4/v2 = 1.2. The left three panels show the voltage dependence of

GA,utun at a fixed momentum mismatch ∆k for 3 cases of different scal-

ing exponent ranges: (a) 0 < g4 < 1; (c) 1 < g4 < 2; (e) 2 < g4 < 3; we

set g4 = 0.5, 1.5 and 2.5 respectively in the plots. Voltage is shown in

units of ω0 = v2∆k. Panels (b), (d) and (f) show the same three cases

for the momentum mismatch dependence of GA,utun at a fixed ω with

the momentum expressed in units of ∆k0 = ω/v2. The differential

conductance is shown in arbitrary units. . . . . . . . . . . . . . . . . 110

3.8 (a) Voltage dependence of the tunneling differential conductance GAtun

in the Pfaffian state. The reference voltage ω0 = v2∆k. (b) Momen-

tum mismatch dependence of GAtun in the Pfaffian state. The reference

momentum ∆k0 = ω/v2. We set the edge velocity ratios, v3/v2 = 1.2

and vλ/v2 = 0.5. GAtun is shown in arbitrary units. . . . . . . . . . . 115

3.9 The differential conductance GA,λtun in the edge-reconstructed Pfaffian

state. Panels (a) and (b) show the voltage and momentum mismatch

dependence of GA,λtun (in arbitrary units) respectively. The reference

voltage ω0 = v2∆k and the reference momentum mismatch ∆k0 =

ω/v2. We have set vλ/v2 = 0.5, v3/v2 = 0.8, v4/v2 = 1.2 and the

scaling exponent g4 = 1.5 . . . . . . . . . . . . . . . . . . . . . . . . 119

xviii

3.10 Differential conductance GAtun in the non-equilibrated anti-Pfaffian

edge state. All left panels show the voltage dependence of GAtun and

right panels show the momentum mismatch dependence of GAtun, at

different choices of v3/v2 and g3 + g4. In the top four panels, we

have chosen v3/v2 = 0.7, and in the bottom four panels v3/v2 = 1.5.

v4/v2 = 1.2 for all cases. In panels (a), (b), (e) and (f), illustrating

the 0 < g3 + g4 < 2 cases, we set g3 + g4 = 1.5. In panels (c), (d), (g)

and (h), illustrating the g3 + g4 > 2 cases, we set g3 + g4 = 2.5. The

reference voltage ω0 = v2∆k and the reference momentum mismatch

∆k0 = ω/v2. GAtun is shown in arbitrary units. . . . . . . . . . . . . . 120

4.1 Three-terminal setup. A quantum Hall bar is connected to source S at

the voltage V . Charge tunnels into terminal C. The arrows represent

the directions of the chiral edge modes. . . . . . . . . . . . . . . . . . 133

4.2 The same low-frequency current IS flows through both dashed lines. . 139

4.3 Illustration of the bias voltage. δA and δE are applied in the region

between two solid vertical lines. In the example in the figure the region

with δE crosses both the source (shaded) and the gapped QHE region

(white). δφ is constant on the vertical dashed line. δφ = 0 in point Q

and δφ = δV in point P. . . . . . . . . . . . . . . . . . . . . . . . . . 144

4.4 A possible experimental setup. Charge carriers, emitted from the

source, can either tunnel through the constriction Q and continue

towards the drain or are absorbed by the Ohmic contact C. . . . . . . 151

4.5 A non-chiral system. The solid line along the lower edge illustrates

the “downstream mode”, propagating from the source to the drain.

The dashed line shows a counter-propagating “upstream” mode. . . . 152

xix

4.6 Setup for fluctuation relations of a system with r reservoirs interacting

via a subsystem S whose transport channels are chiral and located

at its edges (arrows). Each reservoir is at equilibrium with its own

temperature and chemical potential. Complex structures of S, such

as quantum point contacts, may exist in the dashed circle. . . . . . . 154

C.1 Ideal gas in a reservoir with a tube. . . . . . . . . . . . . . . . . . . . 185

xx

Chapter One

Introduction

2

Quantum mechanical effects in macroscopic systems always fascinate and puzzle

physicists. Superconductivity and quantum Hall effect are among the most remark-

able macroscopic quantum mechanical effects. In this thesis, we study some exotic

phenomena in the quantum Hall effect (QHE), especially the fractional quantum

Hall effect.

The quantum Hall effect was first realized in 2D electron gases constructed in

semiconductor systems [1–3], and recently in the 2D crystal graphene [4, 5], under

a perpendicular uniform magnetic field B. The basic experimental observation of

the quantum Hall effect is the quantization of the Hall resistance Rxy = h/νe2

to a precision of one part in 109, and the vanishing of the dissipative longitudinal

resistance Rxx, at certain magnetic fields. Plateaus appear on the Rxy ∼ B curve.

The dimensionless quantity ν counts the number of filled Landau levels. The so-

called integer quantum Hall effect (IQHE) comes with an integer ν, and the fractional

quantum Hall effect (FQHE) comes with a fractional ν = p/q, with p and q integers.

The most fascinating feature of QHE systems is the existence of topological

order [6, 7] in the gapped bulk which protects gapless modes [8–10] on the edge of

a Hall bar. The vanishing of longitudinal resistance Rxx indicates existence of a

(mobility) gap in the bulk of the 2D electron system. Then all transport channels

are on the 1D edge. In many cases, transport modes are chiral [8, 9], with the

counter-propagating modes located on the opposite sides of the sample. In some

FQHE systems, counter-propagating modes (most probably neutral modes) on the

same side are predicted [11–13].

The topological order of FQHE is more exciting than that of IQHE. Due to

strong electron correlation, elementary excitations in FQHE carry fractional charges

and obey fractional statistics, i.e., they are neither fermions nor bosons. They are

3

called anyons [14]. When an anyon moves around other anyons, non-trivial phases

will be accumulated in the many-body wave function of the system. If there is only a

phase change, these anyons are called Abelian anyons. Beside Abelian anyons, there

also exists, at least theoretically, non-Abelian anyons. Non-Abelian systems change

not only the wave function but also the quantum state when one anyon encircles

another. Currently enormous efforts have been put into the search for non-Abelian

anyons in FQHE, with a focus on the ν = 5/2 FQHE [15,16], because of their possible

application in topological quantum computing [16].

This thesis covers several theoretical studies on the bulk and edge physics of

FQHE, including proposals to detect non-Abelian anyons in the 5/2 FQHE and

predictions of transport features of FQHE edge states. In this chapter, I briefly in-

troduce the quantum Hall effect, heavily on the theoretical side. Accompanied with

the introduction, I gradually unveil the fundamental motivations of my graduate re-

search projects which are presented in the following chapters. In Sec. 1.1, we discuss

the necessity to take quantum mechanics into accounts to explain the experimentally

observed effect, and introduce some early history of the quantum Hall effect. Basic

QHE theories are introduced in Sec. 1.2 (bulk theories and anyons) and Sec. 1.3

(edge theories). The following two sections focus on theoretical predictions of ex-

perimentally observable phenomena, with Sec. 1.4 devoted to detecting anyons by

interferometers and Sec. 1.5 on transport theory of edge states. We give an overview

of the rest of the thesis in Sec. 1.6.

4

1.1 From the Classical Hall Effect to the Quantum

Hall effect

Let us start with the classical Hall effect discovered in 1879 by Edwin Hall. Consider

electron flow on a long strip with a width w and a length l along the x axis (see

Fig. 1.1), giving rise to a current I. Under the effect of a perpendicular magnetic

field B = Bz, trajectories of electrons are bent, resulting in accumulation of net

charges on the edges of the strip. Hence, a Hall voltage VH develops across the strip,

which gives rise to an electric field VH

wy. Electrons flow in a straight line after the

electric force due to the Hall voltage equals the Lorentz force,

q

cv ×B = −qVH

wy (1.1)

with v the velocity of moving electrons and q = −e the electron charge. The current

I = |q|nvw, with n the electron density. So we obtain the Hall resistance,

RH =VHI

=B

nqc. (1.2)

IB

+

−

+

−

+

−

+

−

+

−

+

−

+

−

+

−

+

−

+

−

+

−

+

−

+

−

+

−

+

−

+

−

+

−

VH

x

y

Figure 1.1: Classical Hall effect. A current I, carried by electrons, flows along the x axis ofa conducting strip. A Hall voltage VH is measured between the two edges of the strip after aperpendicular magnetic field B that points upward is turned on.

5

Two points follow the expression for the Hall resistance. First, the sign of the Hall

resistance reflects the sign of the carrier charge. In our case, electron is the current

carrier (q < 0), so we should have a negative Hall resistance RH . Second, the

Hall resistance depends linearly on the magnetic field B. We may write RH in the

following form, ignoring the sign,

RH =h

e21

ν. (1.3)

It has a closer relation to that of the quantum Hall effect. The filling factor ν

ν =nhc

eB=

n

Be/hc=

density of particles

density of magnetic flux quanta. (1.4)

Φ0 ≡ hc/e is the magnetic flux quantum. In the quantum mechanical treatment, we

will see that ν counts the number of occupied Landau levels(Sec. 1.2.1).

As long as the force due to the Hall voltage cancels the Lorentz force, the longi-

tudinal voltage drop Vx along the x axis does not depend on the magnetic field. The

longitudinal transport can be described by the classical Drude theory [17],

eVxl

=mv

τ(1.5)

with τ the relaxation time1 andm the mass of electrons2. So we have the longitudinal

resistance

Rxx =VxI

=m

ne2τ

l

w, (1.6)

a constant for a given sample. The quantity ρxx.= m/ne2τ is the longitudinal

1In the original Drude theory, the relaxation time τ measures the mean time between twocollisions with the ion cores. The band theory, which is quantum mechanical, tells us that theion cores in a perfect periodic structure cannot scatter wave-like electrons. The relaxation time τresults from scattering off imperfections in the crystal, such as impurities and phonons, etc.

2 m will be the effective mass of electrons if there is an underlining crystal structure or ifelectron-electron interaction exists.

6

resistivity. The value of ρxx tells how clean the sample is. A pure sample without

imperfections has ρxx → 0. A more common quantity to describe the purity of a

sample is the mobility µ = 1/ρxxne = eτ/m.

Even before the discovery of QHE, experiments have already contradicted the

above classical picture of Hall effect. First of all, the sign of the Hall resistance

was observed to be either positive or negative in electronic systems, depending on

materials [17]. To explain the positive Hall resistance, we need the concept of holes

from the band theory of electronic structures in the presence of an underlying periodic

crystal potential. Band theory is a quantum mechanical theory. Second, at low

temperatures and high magnetic fields (not as high as those for QHE), it is observed

that the longitudinal resistance Rxx is not a constant, instead it oscillates as the

magnetic field changes. These are the so called Shubnikov - de Haas oscillations.

The facts that electrons are fermions and they form Landau levels in a magnetic

field, both quantum mechanical concepts, are required to explain the Shubnikov - de

Haas oscillations.

Breakdown of the classical picture tells us that quantum mechanics is important

in 2D electronic systems. This importance has been completely uncovered after the

discovery of the QHE. Let us now briefly review the history of the QHE.

In the year 1980, Klaus von Klitzing et.al measured [1] the Hall conductance and

longitudinal resistance of a 2D electron gas sample. The 2D electron gas was trapped

at the interface between a semiconductor and an insulator in a silicon MOSFET

(metal-oxide-semiconductor field effect transistor). Measurements were taken at low

temperatures (∼ 4K) and high magnetic fields (∼ 15T)(See Fig. 1.3 for a sketch of

experimental setups). The sample was very clean for the time of the experiment, with

a mobility ∼ 104 cm2/Vs. They observed that the Hall conductance RH deviated

7

B ∼ 10 Tesla

RH(h/e2)

1

1/2

1/3

1/41/5

Rxx(arb.unit)

Figure 1.2: Sketch of the quantum Hall effect. When plateaus occur in the magnetic field depen-dence of the Hall resistance (red), the longitudinal resistance (blue) vanishes when the temperatureis low enough. On the plateaus, the Hall resistance has quantized value RH = h/e2ν. IQHE hasinteger ν while FQHE (not shown) has fractional ν. At very low magnetic fields, the classical Halleffect takes places.

S D

1 2

3 4

Figure 1.3: Sketch of the multi-terminal setup to measure Hall resistance and longitudinal re-sistance. The Hall voltage is measured between contact 1 and 3, and the longitudinal voltage ismeasured between contact 1 and 2. A current is maintained between the source S and drain D.

from the linear dependence of magnetic field B and developed many plateaus (see

a sketch in Fig. 1.2). The on-plateau Hall resistance was quantized at h/e2ν, with

ν = 1, 2, 3 · · · . At the points where the Hall resistance developed plateaus, the

longitudinal resistance vanished as the temperature approached absolute zero.

This observation completely goes beyond the picture given by the classical theory.

The effect is called the integer quantum Hall effect (IQHE) nowadays. Physicists

immediately understood that quantum mechanics is necessary to explain the IQHE,

and the interplay between the magnetic field and disorder is important. Landau

8

quantization due to the magnetic field and Anderson localization due to disorder

turn out to be the key reasons for the appearance of IQHE. The integer ν turns out

to be the number of the filled Landau levels.

Nature never stops to surprise humans, and humans never know when nature will

surprise us. Two years after the discovery of the IQHE, D. C. Tsui, H. L. Stormer

and A. C. Gossard measured the Hall effect again in a cleaner sample (mobility ∼

0.8×105 cm2/Vs), a GaAs-AlGaAs heterostructure [2]. They found that new plateaus

developed on the curve of the magnetic field dependence of the Hall resistance, with

the on-plateau Hall resistance being quantized at 3h/e2, i.e., ν = 1/3. It is the so

called fractional quantum Hall effect (FQHE). This surprising result goes beyond a

single electron picture of Landau quantization. Electron-electron interaction must

be taken into accounts in a non-mean-field way. One year after the experimental

observation, R. Laughlin wrote down a many-body variational wave function [18]

that explains the existence of the 1/3 plateau.

Soon after the discovery of the 1/3 FHQE, many other FHQE were observed at

ν = 2/5, 3/7, 4/9, 3/5, 4/7, 5/9 · · · in the lowest Landau level and other Landau lev-

els. Laughlin’s theory was soon extended to such states in the form of the hierarchical

construction [19, 20] and later J. Jain’s composite fermion picture [21–23].

Another milestone in the history of FQHE is the observation of even-denominator

FQHE, which was first discovered by R. Willett et. al [24] at filling factor ν = 5/2.

The FQHE states that we mentioned above all have odd denominators, which is

natural in Laughlin’s theory and its generalizations because of Fermi statistics. A

possible explanation for the 5/2 FQHE is the non-Abelian Pfaffian state proposed

by G. Moore and N. Read in 1991 [25], which incorporates a pairing mechanism in to

the construction. It was later discovered that the Moore-Read state can be viewed

9

as a px + ipy pairing superconducting state [26].

Besides the quantization of the Hall conductance in a quantum Hall liquid, other

exciting features include the existence of anyons which emerge as localized excita-

tions, carry fractional charges and obey fractional statistics. Abelian anyons were

first proposed in Laughlin states [18, 27], and Non-Abelian ones were first proposed

in the Moore-Read Pfaffian state [25]. Experimentally, fractional charges have been

confirmed by shot noise measurements [28–30], while it is hard to obtain unam-

biguous experimental signatures for fractional statistics. Recently, there was an

experiment measuring phase slips in Fabry-Perot interferometers that seems to give

convincing results for the existence of Abelian statistics in the 1/3 Laughlin state [31].

However, finding non-Abelian anyons in nature remains an open question (Ref. [31]

has some evidence for non-Abelian statistics too, but less convincing than that for

Abelian statistics). Besides the fact that a search for non-Abelian is intrinsically im-

portant, much interest also comes from possible applications to topological quantum

computing in non-Abelian systems [16]. There are also attempts to find non-Abelian

statistics in other systems such as topological insulators [32, 33].

This is how we have gone from the classical Hall effect to the quantum Hall effect.

Before we go to the theories of the QHE, let us mention the energy scales on which

different QHE plateaus occur.

1.1.1 Energy Scales

Every physical phenomenon has its particular energy scale. Generally speaking,

condensed matter physics focuses on “infrared” phenomena which are destroyed at

higher energy scales than its own, while high energy physics focuses on “ultraviolet”

10

phenomena that can not been seen until we reach that energy scale.

In QHE systems, we require the temperature to be low enough such that thermal

fluctuations do not destroy the QHE. We require disorder to be weak enough too, so

that random potential fluctuations in space cannot destroy the QHE. In other words,

the temperature and disorder cannot exceed the energy scale intrinsic for a particular

QHE. These two requirements point out the reason why QHE was not observed

until technological developments allowed physicists to reach very low temperatures

and obtain very clean samples. Nowadays, mobility can reach ∼ 3 × 107cm2/V s

and temperature can reach ∼ 10mK. Energy scales of QHE are controlled by the

magnetic field. The higher the magnetic field is, the higher the energy scales are,

and so the easier for the QHE to occur. Usually we need a high magnetic field to

observe the QHE.

We list several energy scales in QHE systems related to different phenomena.

They are summarized in Table 1.1. For more theoretical discussions, please see

Sec. 1.2.

1. Subband band gap: Real physical systems are 3D. 2D electron gases are real-

ized by trapping electrons in a narrow quantum well in the third dimension.

Electrons are still able to move freely in the other two dimensions. Modeling

the quantum well as an infinitely deep square potential, we get the subband

energies,

En =~2π2

2mW 2n2, n = 1, 2, 3 · · · . (1.7)

In GaAs heterostructures, the effective mass m = 0.068me with me the bare

mass of electrons. The width of the quantum well is ∼ 25 nm. So we have the

band gap between the first and second subbands is ∆E = E2 − E1 ≈ 26 meV

11

= 310 K.

2. IQHE: The energy scale for IQHE is determined by the cyclotron frequency

ωc. Classically, 2π/ωc is the circling period of a charge particle in a uniform

magnetic field. The cyclotron frequency is

ωc =eB

mc. (1.8)

The energy related to ωc is Ec = ~ωc. The usual experimental magnetic field is

between several Tesla (second Landau level) and 20s Tesla (first Landau level).

For the 5/2 FQHE, a typical magnetic field is 5T, so the Landau level splitting

is about 100K.

3. Zeeman splitting and Coulomb energy: Zeeman splitting for electrons in a mag-

netic field is

EZ = gµBB = ge~B

2me. (1.9)

The bare g factor of electrons is 2, however in GaAs semiconductors g ≈ −0.4

due to spin orbit coupling. For a typical magnetic field at 5 T, EZ ≈ 1.5K.

Certainly, Zeeman energy is related to the spin polarization of a QHE sys-

tem, however it is not the dominant factor for the polarization. Exchange

forces between electrons, due to the Pauli exclusion principle and Coulomb

interaction, dominate instead. The exchange energy can be estimated as the

Coulomb potential energy e2/ǫd with average spatial separation between elec-

trons d = 1/√n and ǫ the dielectric constant. For a typical n ∼ 1011/cm2 and

ǫ ∼ 10 in GaAs systems, we have the Coulomb energy EC = 50 K.

4. Energy gap of the ν = 1/3 State: We give the experimental energy gap of

the Laughlin state at ν = 1/3. The gap is 1 ∼ 10 K [34], depending on the

magnetic field (Certainly while the magnetic field changes, density changes

12

as well to fix the filling factor). A greater energy gap is found in suspended

graphene [35], around 20 K at B = 14 T, mainly due to the smaller dielectric

constant.

5. Energy gap of the 5/2 State: The 5/2 is more fragile than the 1/3 FHQE.

Experimental measurements of the energy gap yield 100 ∼ 500 mK [34].

1.2 Theories of QHE

1.2.1 IQHE: Landau Quantization

Let us start with the Hamiltonian for free electrons moving in a 2D space under a

perpendicular uniform magnetic field. We assume that there is an infinitely deep

square well in the third dimension so that motion of electrons in that dimension is

confined. The Hamiltonian is

H =1

2m∗

[

(px −q

cAx)

2 + (py −q

cAy)

2 + p2z

]

+ V (z), (1.10)

where the gauge field A = (Ax, Ay, 0) = (0, Bx, 0) under the Landau gauge and m∗

the effective mass of electrons. q = −e is the electron charge. One may check that

the magnetic field B = ∇×A = Bz. The potential V (z) = 0, if 0 < z < W , and

V (z) = ∞, if z < 0 or z > W , with W the width of the quantum well.

Table 1.1: Energy Scales, estimated for a GaAs-AlGaAs system with the width of the 2D electrongas set to 25 nm and the magnetic field at 5 Tesla (typical for 5/2 FQHE). Data for ν = 1/3 andν = 5/2 is from Ref. [34]

Subband Cyclotron Zeeman ν = 1/3 ν = 5/2Energy 310 K 100K 1.5 K 1 ∼ 10 K 0.1 ∼ 0.5 K

13

The electron motion in the third direction is decoupled from its motion along the

other two directions. Solving the Schrodinger’s equation, one finds that

Esn =

~2π2

2m∗W 2N2, N = 1, 2, 3, · · · (1.11)

χn(z) =

√

2

Wsin

(

Nπx

W

)

. (1.12)

The index N labels the N -th subband.

It is also easy to solve the Schrodinger’s equation in the xy plane. We have the

eigenenergy and wave function

ǫnk = (n +1

2)~ωc, n = 0, 1, 2, · · · (1.13)

ψnk(x, y) =1√Ly

eikyHn(x+ kl2)e−1

2l2(x+kl2)2 , (1.14)

with ωc =eBm∗c

the cyclotron frequency, l =√

~ceB

the magnetic length, andHn the nth

Hermite polynomial. The form of the wave function ψnk tells that electron motion

along the y direction is a plane wave with a wave vector k, and electron behaves like

a Harmonic oscillator along the x direction with oscillations around the point −kl2.

The Harmonic oscillator levels are called Landau levels. Note that the energy ǫnk

only depends on the Landau level index n. This means that each Landau level is

highly degenerate. The degeneracy equals the number of all possible states labeled

by k. Suppose the xy plane has the size of Lx × Ly. k is quantized to 2mπ/Ly and

also restricted by the physical size along the x-direction: 0 < −kl2 < Lx. So we

have the degeneracy being LxLy/2πl2 = BLxLy/Φ0. Φ0 is the flux quantum. So the

degeneracy of each Landau level is in fact the number of flux quanta in the xy plane.

In real systems (MOSFET or GaAs-AlGaAs heterostructures), disorder exists.

Disorder broadens the degenerate Landau levels, as depicted in Fig. 1.4. Further-

14

DOS

ǫ/~ωc

(a)

DOS

ǫ/~ωc

(b)

Figure 1.4: (a) Landau levels in translationally invariant systems. (b) Broadened Landau levelsin systems with disorder. The shaded tails of each level are Anderson localized states.

more, the tail of each broadened Landau level contains localized states, due to the

Anderson localization mechanism. The states in the middle of each level are ex-

tended, able to conduct electric current. Dramatically, the contribution to the elec-

tric conductance from the extended states of each Landau level is e2/h. So if the

Fermi energy lies between the localized states with n Landau levels below, the total

conductance is ne2/h. A small change of the Fermi energy will not change the con-

ductance, leading to the observed quantum Hall plateaus3. Note that a change of

the Fermi energy is equivalent to a change of the magnetic field.

That each Landau level contributes e2/h to the conductance is better understood

through the topological consideration [6] or edge state theory [8] (see Sec. 1.3 and

Sec. 1.5.1). Here we provide a simple argument. Real samples always have edges.

Consider a Hall bar with finite width in the x direction (Do not confuse the current

x direction with the x direction in Fig. 1.1. The current x-axis is the y axis in

Fig. 1.1). There exists a confining potential V (x). We would adiabatically turn

3On-plateau two-terminal measurement of a Hall bar gives the Hall conductance, since thelongitudinal resistance vanishes.

15

on the potential. This adiabatic process will give an eigenstate ψnk(x, y) for each

original eigenstate ψnk(x, y), with the eigenenergy becoming

ǫnk ≈ (n+1

2)~ωc + V (−kl2), (1.15)

since the position and momentum are locked through Landau quantization (see

Fig. 1.5). The group velocity of the wave ψnk(x, y) is

vk =1

~

∂ǫk∂k

y. (1.16)

In the middle of the Hall bar where the confining potential V (x) is almost flat, the

group velocity is 0. So no current is carried by particles occupying these states. At

the edges of the sample, V (x) has nonzero slope, giving a nonzero group velocity. The

two edges have opposite signs of their slopes, so in fact on the two edges the currents

are flowing in opposite directions. If the two ends of the Hall bar are connected to

two reservoirs with different chemical potentials, µL and µR respectively, a nonzero

net current results. We can calculate the total current

I =1

Ly

∑

nk

evk =e

2π~

∑

n

∫

dk∂ǫnk∂k

dk = ne

h(µL − µR). (1.17)

The summation is taken over occupied Landau levels and n is the number of occupied

Landau levels. The chemical potential difference is tuned by applying a voltage bias

at the two ends of the Hall bar, which is actually the Hall voltage, voltage difference

between the two edges. So, µL−µR = eVH , and we have the Hall conductance equal

to ne2/h, with each Landau level contributing e2/h. Compared to Fig. 1.4, the states

on the two edges of the Hall bar are those extended states, while the states at the

center of the Hall bar with zero group velocity are localized in a systematic theory.

16

••••

•••

• • • • • • • • • • • • • • • • • • • • • • •••••

••••

•• • • • • • • • • • • • • • • • • • • • • • •

••

Figure 1.5: Landau levels under adiabatic bending due to a confining potential. The vertical axisis energy, while the horizontal axis can be either position or momentum since the two are lockedthrough the relation x = −kl2 in our semiclassical picture. The dots represent occupied states.

Here are some final words on Landau quantization and edge states. Fig. 1.5 show

the bent Landau levels. We see the particles in the bulk are hard to excite, due to

the ~ωc gap between Landau levels. However, the edge states can be easily excited

to unoccupied states in the same Landau level. This tells that the quantum Hall

system is gapped in the bulk but gapless on its edge. In other words, the quantum

Hall liquid is an insulators in its bulk but becomes metalic on its edge, justifying the

above statement that currents are only carried by edge states.

1.2.2 Laughlin FQHE States

While the Landau quantization and Anderson localization well explain the appear-

ance of IQHE plateaus, electron-electron interaction plays an essential role for FQHE.

The first successful FQHE theory was proposed by R. Laughlin [18] in 1983.

Let us switch to the symmetric gauge, a convenient gauge to explain Laughlin’s

theory. The symmetric gauge

A = −1

2Bz × r, (1.18)

17

gives a magnetic field −Bz. Laughlin’s theory focuses on the lowest Landau level

and ignores Landau level mixing. It also assumes that spin is polarized. Under the

symmetric gauge, eigenstates in the lowest Landau level are written as

ψm =1√

2πl22mm!zme−|z|2/4 (1.19)

where z = (x+ iy) is complex and we have used the magnetic length l as units of z.

Here m is an integer. Under the symmetric, angular momentum is a good quantum

number. The above state has an angular momentum of m~.

R. Laughlin wrote down a many-body wave function to describe the ν = 1/m

FQHE state. The wave function, now called Laughlin wave function, is

Ψm =

N∏

i<j

(zi − zj)me−

∑i |zi|2/4. (1.20)

This wave function is the exact ground state of a system with electron-electron inter-

action being the short-range interaction V (x) = ∇2δ(x). For Coulomb interaction,

numerical calculation shows a big overlap between the exact ground state and Laugh-

lin’s wave function. So Laughlin’s wave function is a good variational wave function

with no variational parameters. The Laughlin state is incompressible, i.e., all bulk

excitations are gapped. The success of Laughlin’s theory is also reflected in the

fact that it predicts the existence of quasiparticles that carry fractional charges [18]

and obey fractional statistics [27]. See the following Section for a discussion about

anyonic excitations.

Original Laughlin’s theory explains only FQHE at filling factors ν = 1/3, 1/5,

· · · . To explain the experimentally observed 2/5, 3/7, · · · states, we have to go to

hierarchical construction of Laughlin wave functions [19,20] or the composite fermion

18

•

•

•2

•1

• •

Figure 1.6: Exchange statistics. Particle 1 moves around particle 2 in a full circle. In 3D, thepath can be contracted back to the starting point without meeting any particles. In 2D, particle 2prevents such a contraction.

approach [21–23]. Here we briefly discuss hierarchical states. The basic idea of

hierarchical states is that quasiparticles or quasiholes can form a Laughlin state by

themselves, since they also feel effective magnetic field. For examples, the following

wave function

Ψ =

∫

∏

i

dξidξ∗i

∏

i<j

(ξ∗i − ξ∗j )2e−

∑i |xii|2/12l2

∏

ij

(zi − ξj)Ψ3 (1.21)

describes an incompressible state with filling factor 2/7, with ξi representing coor-

dinates of quasiholes. In this state, charge e/3 quasiholes condense into their own

Laughlin state on top of the 1/3 state.

1.2.3 Quasiparticles, Anyons and Anyonic Statistics

The existence of localized quasiparticles that carry fractional charge and obey frac-

tional statistics is an interesting feature of FQHE. The existence of these particles,

called anyons, is strongly related to the dimensionality [24].

In Laughlin’s theory, we can write down wave functions for localized quasipar-

ticles (negatively charged) and quasiholes (positively charged). Let us focus on

19

quasiholes, whose wave functions are of the form

Ψξ =1

√

C(ξ, ξ∗)

∏

i

(zi − ξ)Ψm (1.22)

where Ψm is the Laughlin state at filling factor 1/m, ξ is the position of the quasihole,

and C(ξ, ξ∗) is a normalization factor. The quasihole charge is e/m. A simple

argument is that the wave function∏

i(zi − ξ)mΨm describes a state with a charge e

hole at ξ. However,∏

i(zi − ξ)mΨm can also be viewed as a state with m quasiholes.

So, a single quasihole has charge e/m. These fractionally charged quasiparticles are

in fact Abelian anyons. When one moves around another in a full circle, the many-

body wave function acquires a 2π/m phase. Before discussing detailed theories about

these quasiparticles, we discuss why anyons exist in 2D in principle.

The existence of anyons in 2D has its topological origin. Consider particles in

Fig. 1.6. Particle 1 moves around particle 2 in a full circle. After such a motion, the

original wave function Ψ(r1, r2, · · · ) changes to a new wave function Ψ(r1, r2, · · · ). In

3D, the path in Fig. 1.6 can be easily removed by smoothly contracting it back to the

starting point of the path. Hence, Ψ = Ψ in 3D. Bosons and Fermions both satisfy

this relation. In 2D, such a smooth contraction does not exist. Whenever one tries to

contract the path, it will always meet particle 2 which prevents such a contraction.

So, there is no general relation between Ψ and Ψ. We may classify particles in 2D

by specifying the relation between Ψ and Ψ. Below is the classification

Ψ = Ψ, Fermions and Bosons; (1.23)

Ψ = eiθΨ, Abelian Anyons; (1.24)

〈Ψ|Ψ〉 6= 1, non-Abelian Anyons. (1.25)

For Abelian anyons, θ is called the statistical phase. For non-Abelian anyons,

20

Eq. (1.25) means after one particle makes a circle around another and comes back to

its original position, the quantum state changes. This means that the positional

degrees of freedom are not enough to describe a state of a non-Abelian anyon

system. Other quantum numbers need to be specified. These quantum numbers

are called topological quantum numbers. We will have a topological Hilbert space,

Ψ1,Ψ2, · · · ,ΨM. The relation for the old and new wave function can be described

by a unitary matrix U rotating the topological Hilbert space

Ψ1

Ψ2

...

ΨM

= U

Ψ1

Ψ2

ΨM

. (1.26)

The matrix U specifies the non-Abelian statistics.

Let us now discuss the quasiparticles in Laughlin FQHE states. We calculate the

Berry phase when the quasihole makes a circular adiabatic motion ξ(t). If there is a

second quasihole in the circle, the Berry phase will contain two parts, the Aharonov-

Bohm phase and the anyonic statistical phase. The Aharonov-Bohm phase

φAB =e∗

~c

∮

A · dx = 2πe∗

e

Φ

Φ0(1.27)

contains information about the charge of quasiholes. So, we consider a two-quasihole

state

Ψξ1,ξ2 =1

√

C(ξ1, ξ∗1 , ξ2, ξ

∗2)

∏

i

(zi − ξ1)(zi − ξ2)Ψm. (1.28)

Let the first quasihole make adiabatic circular motion ξ1(t). The Berry phase γ(t)

satisfies

dγ

dt= i〈Ψξ1,ξ2|

d

dt|Ψξ1,ξ2〉, (1.29)

21

or

dγ = aξ1dξ1 + aξ∗1dξ∗1 (1.30)

with

aξ1 = i〈Ψξ1,ξ2 |d

dξ1|Ψξ1,ξ2〉, aξ∗1 = i〈Ψξ1,ξ2 |

d

dξ∗1|Ψξ1,ξ2〉. (1.31)

The Berry phase is given by∮

dγ for a specified path. Compared to Eq. (1.27), aξ1

and aξ∗1can be viewed as effective gauge fields interacting with quasiholes. It is not

hard to obtain that

aξ1 =i

2

∂

∂ξ1lnC, aξ∗1 = − i

2

∂

∂ξ∗1lnC. (1.32)

The normalization factor can be calculated using the Plasma analogy [18, 36]. We

obtain

C = D|ξ1 − ξ2|−2/me1

2ml2(|ξ1|2+|ξ2|2), (1.33)

with D being a constant independent of ξ1 and ξ2. Then

aξ1 =iξ∗14ml2

− i

2m(ξ1 − ξ2)(1.34)

aξ∗1 = − iξ14ml2

+i

2m(ξ∗1 − ξ∗2). (1.35)

Hence the Berry phase equals

γ = 2π1

m

Φ

Φ0+

2π

m. (1.36)

The first term is proportional to the flux enclosed by the circle. It is the Aharonov-

Bohm phase. After a comparison with Eq. 1.27, we find that the charge carried by

a quasihole is e∗ = e/m. The second term is a constant, resulting from encircling of

quasihole 2 by quasihole 1. The statistical phase of quasiholes θ = 2π/m. Now we

22

have proved that these quasiparticles are indeed Abelian anyons.

While the existence of fractional charges have been verified by various experi-

ments [28, 29], a direct observation of fractional statistics is still under debate [37].

A recent experiment [31] on observation of phase slips in the Aharonov-Bohm os-

cillation of Fabry-Perot interferometer (see Sec. 1.4) seems a convincing evidence of

fractional statistics.

1.2.4 FQHE at ν = 5/2: Non-Abelian States

While Laughlin’s theory is able to explain most FQHE states with a filling factor ν =

p/q with an odd q, experimentalists discovered new FQHE with even-denominator

filling fractions [24]. The most famous one is the ν = 5/2 FQHE. It is suspected

that the 5/2 FQHE is non-Abelian, described by a state belonging to the university

class [38, 39] of the non-Abelian Moore-Read wave function [25]. Other promising

candidates include the anti-Pfaffian state [12, 13] – particle-hole conjugate of the

Moore-Read Pfaffian state, and the Abelian 331 state. However, the nature of the

5/2 state remains an open question. We discuss some trial wave functions below.

To explain the 5/2 state, G. Moore and N. Read constructed a wave function [25]

at filling 1/2 while assuming that the underlying ν = 2 IQHE is inert,

ΨMR =∏

ij

(zi − zj)2e−

∑i |zi|2/4Pf(

1

zi − zj), (1.37)

where Pf is the Pfaffian, the square root of the determinant, of the antisymmetric

23

matrix with the entries 1/(zi − zj). Explicitly

Pf(1

zi − zj) = A(

1

z1 − z2

1

z3 − z4· · · ), (1.38)

where A is the antisymmetrization operator. Like Laughlin’s wave function, it is

a spin polarized state. This wave function incorporates a pairing mechanism, and

later was found to be equivalent to px+ ipy topological superconductors of composite

fermions [26,40]. Similar to Laughlin states, there exist fractionally charged anyonic

excitations described by wave functions of the form,

Ψ =∏

i

zi∏

ij

(zi − zj)2e−

∑i |zi|2/4Pf(

1

zi − zj), (1.39)

where we conveniently put the quasiparticle at the origin. The quasihole has the

charge e/2. In contrast to Laughlin states, there are two more types of excitation.

We can either add one unpaired electron

Ψ =∏

ij

(zi − zj)2e−

∑i |zi|2/4A(z0

1

z1 − z2

1

z3 − z4· · · ), (1.40)

or increase the angular momentum of a Cooper pair

Ψ =∏

ij

(zi − zj)2e−

∑i |zi|2/4Pf(

(zi − η1)(zj − η2) + (zi − η2)(zj − η1)

zi − zj). (1.41)

The first case describes a neutral Majorana fermion ψ, while in the second we create

two charge-e/4 non-Abelian anyons, denoted as σ. The anyons σ and ψ, together

with the trivial topological charge I, form the non-Abelian part of excitations in

the Moore-Read state, while the Laughlin-like bosonic excitations form the Abelian

part.

24

The non-Abelian part has non-trivial fusion rules, i.e., when two anyons come

close and fuse into a composite particle, the resulting particle is not unique. For

example, two σ anyons can either fuse to ψ or I. The complete fusion rule [16] is

σ × σ = I or ψ, σ × ψ = σ, σ × I = σ (1.42)

ψ × ψ = I, ψ × I = ψ, I × I = I. (1.43)

This nontrivial fusion rule results in a topological Hilbert space for a system with non-

Abelian anyons. Let us consider a system with four σ anyons. The total topological

charge must be trivial, since it is essentially an electronic state. So σ×σ×σ×σ = I.

However, there are two ways to get the final trivial topological charge: (1) the first

two σ anyons fuse to I, and the last two σ fuse to I too; (2) the first two σ fuse to ψ

and the last two σ fuse to ψ too. So there is a two-dimensional topological Hilbert

space that cannot be specified by the positions of anyons, but are specified by the

fusion channels of the anyons. The topological Hilbert space is essentially the fusion

space spanned by fusion channels. The information of how anyons fuse is non-local,

therefore cannot be easily destroyed by local perturbations. So non-Abelian anyons

can be used for fault-tolerant quantum computing [16, 41].

While the Moore-Read state is a promising state for the experimentally observed

5/2 state, there exist other possible candidates. The most likely Abelian candidate

is the 331 state,

∏

i<j

(zi − zj)3∏

i<j

(wi − wj)3∏

ij

(zi − wj)e−

∑i(|zi|2−|wi|2)/4. (1.44)

It is a two-layer state with zi and wi describing particles in the two layers. The

elementary excitations of the 331 state have charge e/4 and obey Abelian statistics.

Physically the two layers can be represented by spin-up and spin-down layers. So

25

it is a spin-unpolarized state. Currently there are experiments supporting a spin

unpolarized 5/2 state [34, 42–44]. There is also a spin-polarized version of the 331

state [45]. Other Abelian candidates include the strong-pairing K = 8 state. In

this state, electrons form Cooper pairs first and the Cooper pairs condense to a 1/8

Laughlin state [45].

Without Landau level mixing, the half filled Landau level has a particle-hole

symmetry. However, the Moore-Read Pfaffian state is not particle-hole symmetric.

Hence, in 2006, two collaborations [12, 13] proposed the anti-Pfaffian state. The

anti-Pfaffian state has the same non-Abelian statistics with a minor difference of the

Abelian part of the statistics from the Pfaffian state. However, the anti-Pfaffian state

has a quite different edge structure from the Pfaffian state (see Sec. 1.3). Currently,

much evidence supports the anti-Pfaffian state [46, 47], but it is not conclusive yet.

All above candidates support charge e/4 excitations but different statistics. It

has been experimentally verified via various methods [30,46,48,49] that elementary

excitations in the 5/2 state indeed have charge e/4.

1.3 Edge theory

We have briefly mentioned edge states while introducing the IQHE. They also exist

in FQHE. Edge states are the only gapless excitations in quantum Hall systems

including both IQHE and FQHE. All of the bulk excitations of a quantum Hall

system have a finite gap. While the edge states of IQHE are Fermi liquids, edge

states of FQHE form chiral Luttinger liquids. This was first pointed out by Xiao-

Gang Wen [9,36]. There are many ways to obtain the chiral Luttinger liquid theory

26

for low-energy edge physics, including the hydrodynamic argument and effective

Chern-Simons gauge theory [36]. In the following we state the effective field theory

for chiral Luttinger liquids.

The simplest 1/3 FQHE has one chiral edge mode, described by the following

action,

S = − 1

4πν

∫

dxdt∂xφ(±∂t + v∂x)φ (1.45)

where v is the propagation speed of the mode and x is the coordinate along the

circumference. The ‘±’ signs denote different chirality of the modes, with ‘+’ for

a right-moving mode and ‘−’ for a left-moving mode. The real bosonic mode φ

satisfies the equal time commutation relation [φ(x), φ(x′)] = ±iπνsign(x− x′). The

charge density fluctuation is ρ(x) = ∂xφ(x)/2π (see Fig. 1.7 for a physical intuition.)

Anyons can be expressed through these edge excitations, with the operator

ψ1/3 ∼ eiφ(x,t) (1.46)

annihilating a charge-1/3 quasiparticle on the edge. This relation is very closely

related to Bosonization relation in one-dimension systems [50]. The electron annihi-

lation operator is given by

ψe ∼ eiφ(x,t)/ν . (1.47)

We can check that ψe and ψ†e satisfy anticommutation relations. At zero temperature,

the bosonic mode has the correlation function

〈φ(x, t)φ(0, 0)〉 = −ν ln[δ + i(t± x/v)] (1.48)

with ± for right- and left-moving modes respectively and δ is infinitesimally positive.

Note that the system is translationally invariant. The quasiparticle operators have

27

Figure 1.7: Shape distortion of an incompressible QHE droplet. Such distortions are edge excita-tions. Effectively, the edge excitations form a Fermi liquid when the droplet is on IQHE plateausbut become a Luttinger liquid on FQHE plateaus.

the correlation function

〈ψ†1/3(x, t)ψ1/3(0, 0)〉 ∼

1

(δ + i(t± x/v))ν. (1.49)

At finite temperatures, the correlation function is

〈φ(x, t)φ(0, 0)〉 = −ν ln(

sin πT [δ + i(t± x/v)]

πT

)

. (1.50)

With these results in hand, we are able to calculate transport properties of FQHE

edges (see Sec. 1.5).

For more complicated hierarchical Abelian FQHE systems, a general edge theory

is given by the K-matrix action

S = − 1

4π

∫

dxdtN∑

IJ=1

(∂tφIKIJ∂xφJ + ∂xφIVIJ∂xφJ). (1.51)

The chiral boson field φI describes gapless edge excitations of the Ith condensate.

The matrix element KIJ is integer valued, a descendent quantity of the bulk topo-

28

logical order. The matrix V describes the interaction between the edge modes, so it

is positive definite. The bosonic modes satisfy [φI(x), φJ(x′)] = iπK−1

IJ sign(x − x′).

For a FQHE liquid with a single right(left)-moving mode, K = ±1/ν. The general

quasiparticle operator is

ψl = ei∑

I lIφJ . (1.52)

It has the charge et ·K−1 · l, where t is called the charge vector of the hierarchical

FQHE. t is determined by the bulk topological order and is integer valued too. For

two quasiparticles, represented by l1 and l2 respectively, their statistical phase is

2πl1 ·K−1 · l2.

The above hierarchical theory does not forbid counter-propagating modes on the

same FQHE edge. A famous example of FQHE with counter propagating modes is

the ν = 2/3 FQHE. In the simplest model of this state, the edge is composed of one

downstream ν = 1 integer edge mode and another upstream ν = 1/3 fractional edge

mode. However, this model does not give the correct Hall conductance 2e2/3h. It

gives 4e2/3h. This contradiction made C. Kane, M. P. A. Fisher and Polchinski [11]

to discuss the effect of disorder on the edge. Disorder results in random tunneling

and eventually leads to an equilibration between the counter-propagating modes.

In the disorder-dominated limit, the fixed-point edge theory consists of two effective

modes, one charged mode, contributing 2e2/3h to the conductance, and an upstream

neutral mode. Experiments has been designed to detect these upstream neutral

modes [47, 51, 52]

While chiral Luttinger liquid theory is commonly used for Abelian FQHE edges,

the edge of non-Abelian FQHE is more interesting. The Moore-Read Pfaffian state

has two edge modes, a 1/2 fractional charge bosonic mode φ and a neutral Majorana

29

fermion mode λ with the same chirality

S = − 1

2π

∫

dxdt∂xφ(∂t + v∂x)φ+ i

∫

dxdtλ(∂t + vn∂x)λ. (1.53)

The Majorana field operator satisfies λ† = λ and vn is its propagating speed. An

electron operator on the edge is expressed as

ψ = λei2φ. (1.54)

The anti-Pfaffian state has one charged bosonic mode and three upstream Majorana

modes, with the action

S = − 1

2π

∫

dxdt∂xφ(∂t + v∂x)φ+ i

3∑

i=1

∫

dxdtλi(∂t − vn∂x)λi. (1.55)

The existence of upstream modes in the anti-Pfaffian state is a huge difference be-

tween Pfaffian and anti-Pfaffian. Experimental evidence currently supports the ex-

istence of neutral modes [47, 51, 52].

1.4 Interferometers

With the above theories of anyons and edge states, one may ask if these theories are

correct and how to verify them experimentally. In the following two subsections, we

discuss predictions from these theories and briefly compare the predictions with the

current experimental results. We discuss QHE interferometers in this subsection, and

edge transport characteristics in the next subsection. Interferometers are the most

direct tools to identify anyons, and transport characteristics provide signatures of

edge physics. Interferometric experiments certainly rely on edge transport, however

30

the focus will be on Aharonov-Bohm oscillations and their modulations by anyonic

statistics. In the next subsection when discussing the transport theory, we focus on

the I-V characteristics and current fluctuations.

1.4.1 Fabry-Perot Interferometer

The Fabry-Perot interferometer in QHE systems is sketched in Fig. 1.8. A voltage

bias is applied at S and all other contacts are grounded, resulting in a net charge

flow from S to D. Two quantum point contacts (QPCs) are introduced by side

gates, so that quasiparticles4 flowing along the lower edge can tunnel to the upper

edge, and eventually arrive at drain D. In the language of optical interferometers,

the incoming quasiparticle “beams” from source S are reflected at QPC1 and QPC2

(two “beam splitters”), and the two reflected “beams” interfere at the upper edge

and hit the “detector” drain D in the end. If the tunneling amplitudes at the two

QPCs are Γ1 and Γ2 respectively, the tunneling current measured at drain D is, to

4In the following, we do not distinguish quasiparticles and quasiholes.

S

D

QPC1 QPC2

Figure 1.8: Fabry-Perot interferometer. Two quantum point contacts (QPCs) are introduced ina Hall bar by side gates (gray triangles). Charges flowing out of source S can either tunnel throughQPC1 or QPC2 and arrive coherently to drain D. Modifying the area encircled by the two pathesor changing the magnetic field leads to an Aharonov-Bohm oscillation of the current, measured atdrain D. The Aharonov-Bohm oscillations are affected by localized anyons (red dots) inside theinterference loop in FQHE systems.

31

the lowest order of tunneling amplitudes,

I ∝ |Γ1 + Γ2|2 = |Γ1|2 + |Γ2|2 + 2|Γ1Γ2| cos(∆φ), (1.56)

where ∆φ is the relative phase difference of Γ1 and Γ2. The tunneling amplitudes

certainly depend on many things, including the side gate voltages on the QPCs,

the source-drain voltage, the temperature and most importantly the type of tunnel-

ing quasiparticles. In IQHE systems, tunneling particles are certainly electrons. In

FQHE systems, there are many types of quasiparticles, for example, the 1/3 Laugh-

lin state has charge e/3, 2e/3, · · · quasiparticles. In some hierarchical states, there

exists quasiparticles with the same electric charge but different topological charges,

i.e., satisfying different fractional statistics, for example, the two-layer 331 state has

two types of charge-e/4 quasiparticles. Usually, tunneling of quasiparticles with the

smallest charge contributes most to the tunneling current. Theoretical argument is

based on the renormalization group analysis. The tunneling operators of quasiparti-

cles with higher charges are less relevant. In the following we discuss Laughlin states

which have quasiparticles of the smallest charge eν, and focus on the ∆φ dependence

of the current I. The voltage and temperature dependence of the current is the topic

of the next subsection.

A change of ∆φ leads to an oscillation of the tunneling current, as is seen from

Eq. (1.56). The phase difference ∆φ contains three parts,

∆φ = φAB + φs + φ0 (1.57)

with φAB the Aharonov-Bohm phase, φs the statistical phase due to braiding of the

tunneling quasiparticles around localized quasiparticles inside the interference loop,

and φ0 a constant phase depending on other details of the system. The statistical

32

phase φs only exists in FQHE systems. The Aharonov-Bohm phase

φAB = 2πe∗

e

BA

Φ0(1.58)

with e∗ the charge carried by tunneling quasiparticles, Φ0 = hc/e the flux quantum,

B the magnetic field, and A the area enclosed by the two interference paths. If we

change the area by δA with a side gate or change the magnetic field by δB, we will

observe the periodic Aharonov-Bohm oscillation of the current as a function of δA

or δB. It is important to notice that the periodicity depends on e/e∗. Hence, in the

same system, the periodicity for ν = 1/3 FQHE is three times that for IQHEs, since

quasiparticles in the 1/3 FQHE carry 1/3 of the electron charge.

The Aharonov-Bohm oscillation will be modulated by fluctuations of the number

of localized quasiparticles inside the interference loop. Suppose there areNL localized

quasiparticles inside the loop. We count charge 2e∗ quasiparticles as two charge e∗

quasiparticles, etc. When a quasiparticle tunnels from the lower edge to the upper

edge, the phase difference ∆φ acquires an additional statistical phase

φs = NLθ (1.59)

with θ the statistical phase, being 2πν for Laughlin states. If NL remains a constant

during the time interval that we change the area A or magnetic field B, the Aharonov-

Bohm oscillation is not modified. However, if NL changes to NL+1 in an experiment,

a phase slip of 2πν will be observed in the Aharonov-Bohm oscillation. This is a

direct evidence of Abelian fractional statistics. The phase slip has been observed

in a recent experiment [31], justifying the existence of Anyons in the 1/3 Laughlin

state.

33

The above picture is ideal, assuming no Coulomb interaction exists between

anyons, either in the bulk or on the edge. In reality, the Coulomb interaction is

important, especially for small devices. Interaction adds another phase to ∆φ. The

period tripling of Aharonov-Bohm oscillation at 1/3 filling is observed experimen-

tally [53–55], however the Coulomb interaction is also able to produce such a period

tripling. We do not discuss that here. For details, please see Ref. [56, 57].

In the 5/2 FQHE state, situation is more complicated [48,58–60]. A big question

for the 5/2 state is whether it is a non-Abelian state. A Fabry-Perot Interferometer

is predicted to have the so-called even-odd effect in its Aharonov-Bohm oscillation

assuming the 5/2 plateau is in the Moore-Read Pfaffian state. The even-odd effect

is expressed as

I ∝

const., if NL is odd

A +B cos(∆φ), if NL is even.(1.60)

When there are an odd number of quasiparticles inside the interference loop, no

interference is seen. When the number is even, an Aharonov-Bohm oscillation can

be seen [48]. However, this even-odd effect is not unique to the Pfaffian state [15].

It is predicted that the Abelian 331 state also exhibits this even-odd effect if a

flavor symmetry exists [15]. Besides the strong interaction between anyons, there

are many difficulties in identifying the nature of the 5/2 state with the Fabry-Perot

interferometer. A Mach-Zehnder Interferometer is another tool to look for anyons

and identify the topological nature of a FQHE system. We discuss the Mach-Zehnder

interferometer below.

34

S2 D2

S1 D1

QPC1 QPC2A

B

Edge 1

Edge 2 QH liquid

Figure 1.9: Mach-Zehnder interferometer. Currents flow from source S1 to drain D2. Similarly tothe Fabry-Perot interferometer, two QPCs are introduced. There are two paths for quasiparticlesto go from S1 to D2: S1-QPC1-A-QPC2-D2 and S1-QPC1-B-QPC2-D2. The difference from theFabry-Perot interferometer is that quasiparticles stay inside the interference loop QPC1-B-QPC2-A-QPC1 after a tunneling event, while in the Fabry-Perot interferometer quasiparticles stay outsidethe interference loop after a tunneling event.

1.4.2 Mach-Zehnder Interferometer

The geometry of a QHE Mach-Zehnder is shown in Fig. 1.9. Similarly to the Fabry-

Perot interferometer, two QPCs are introduced so that quasiparticles can tunnel

from Edge 1 to Edge 2. There are two tunneling paths, forming an interference loop

QPC1-B-QPC2-A-QPC1. So changing the magnetic field or the area enclosed by the

interference loop, we can see Aharonov-Bohm oscillations. These oscillations have

been observed in IQHE at Weizmann Institute of Science in Israel [61].

In the case of fractional statistics, Mach-Zehnder interferometer is significantly

different from Fabry-Perot interferometer. Again, we consider the Laughlin states

as the simplest example. The probability for a quasiparticle to tunnel from Edge 1

to Edge 2 is

Pn = |Γ1|2 + |Γ2|2 + 2|Γ1Γ2| cos(φ0 + φAB + nθ) (1.61)

with θ the statistical phase 2πν, n the number of the quasiparticles inside the loop

QPC1-B-QPC2-A-QPC1. Consider the situation in which the incoming current is

small so that the time separation between two tunneling events is longer than the

quasiparticle travel time from QPC1 to QPC2. Then for a single tunneling event

35

the number of quasiparticles inside the interference loop is fixed and so is the tun-

neling probability Pn. However, after a tunneling event, the tunneling quasiparticle

enters inside the interference loop. The next incoming quasiparticle sees one more

quasiparticle inside the loop. So the tunneling probability becomes Pn+1. The next

incoming quasiparticle sees two more quasiparticle inside the loop and so on. Since

Pn+1/ν = Pn, the procedure restarts after 1/ν tunneling events. The average tunnel-

ing current is [62]

I ∝ 1∑1/ν

i=11Pi

. (1.62)

Inserting Eq. (1.61), one finds that the periodicity of the Aharonov-Bohm oscillations

is the same as in IQHE, i.e., as a function of Φ = BA, the current is periodic with

the period being one flux quantum Φ0. Fluctuations of the number of localized

quasiparticles inside the interference loop cannot affect the Aharonov-Bohm effect

in the Mach-Zehnder interferometers, since the measured current Eq. (1.62) is an

average of all possible topologically equivalent configurations of inner quasiparticles.

To observe a qualitative difference between IQHE and FQHE in a Mach-Zehnder

interferometer, one may measure the current noise and Fano factor. The Fano factor

is the ratio of the noise to the current. Consider a time period t during which a total

charge of Q(t) tunnels from Edge 1 to Edge 2. The average current (1.62) is Q(t)/t.

The line above Q(t) means the average over fluctuations of Q(t). The noise is

S = 2δQ(t)2

t(1.63)

with δQ(t) = Q(t) − Q(t) the fluctuation of Q(t). One can calculate the Fano

factor [63]

F =S

I= 2e

∑1/νn=0 1/P

2n

(

∑1/νn=1 1/Pn

)2 . (1.64)

36

The Fano factor also exhibits Aharonov-Bohm oscillations between νe and e. In

IQHE, no such oscillations exist.

For the 5/2 state, the Mach-Zehnder interferometer is better than the Fabry-Perot

interferometer, because it is able to distinguish the Pfaffian and 331 candidates. We

do not discuss this part here, Chapter 2 is devoted to the Mach-Zenhder interferom-

etry of the 331 states and how it is different for the Pfaffian and 331 states. Readers

can read Chapter 2 and Refs. [64, 65] for details.

1.5 Transport Theory

In this subsection we discuss nonequilibrium transport theory in QHE systems. We

go beyond the linear response transport theory and introduce two basic approaches

to calculate I-V characteristics in QHE systems, the Landauer-Buttiker approach

and Keldysh formalism. While the Landauer-Buttiker approach is generally used

for non-interacting systems, the Keldysh formalism is correct for any systems. How-

ever, the latter can be developed only perturbatively in general. Understanding

nonequilibrium transport properties of QHE is very important, since it is a way

to detect edge states. We can obtain many bulk properties from edge physics due

to the edge-bulk correspondence in topological states of matters. An example of a

Keldysh calculation in FQHE systems with a single QPC at 1/3 filling is provided.

Fluctuation-dissipation theorems are introduced briefly as a background of Chapter

4.

37

1.5.1 Landauer-Buttiker Approach

The idea of the Landauer-Buttiker approach is to relate transport properties of a

system to its scattering properties. Scattering properties, described by the scattering

matrix S, are to be found with other methods. The Landauer-Buttiker approach ex-

presses transport properties of the system through the S matrix assuming it is known.

The approach usually applies to non-interacting systems in stationary regimes.

We start with the two-terminal setup in the ν = 1 IQHE in Fig. 1.10. The middle

QPC introduces back-scattering of particles between the two edges. A particle from

the left lower edge in the state with the annihilation operator aL(E) is scattered

by the QPC, and then moves either along the left upper edge in the state with

the annihilation operator bL(E) or along the right lower edge in the state with the

annihilation operator bR(E). We assume that the scattering process is elastic, so the

energy is conserved. A similar scattering process occurs for particles coming from

the upper right edge. The scattering processes are described by a scattering matrix

S(E) that relates aL(E) and aR(E) to bL(E) and bL(E),

bL

bR

= S

aL

aR

. (1.65)

The S matrix is energy dependent and unitary S†S = SS† = 1. Then R(E).=

|S11|2 = |S22|2, T (E) .= |S12|2 = |S21|2 = 1−R(E). R(E) is the reflection coefficient

S D

aL

bL

bR

aR

Figure 1.10: Quantum Hall bar with a quantum point contact.

38

and T (E) is the transmission coefficient.

To calculate the current, we need first to write down the current operator. If the

voltage and temperature are much smaller than the Fermi energy, only particles near

the Fermi surface are excited. These particles propagate almost at the same speed,

i.e. the Fermi velocity. Under this assumption, it is found that the current operator

is (here e is the electron charge)

I(t) =e

h

∫

dEdE ′ei(E−E′)t/~[a†L(E)aL(E′)− b†L(E)bL(E

′)] (1.66)

with bL related to aL and aR through Eq. (1.65). Incoming particles from the

source/drain are in equilibrium with the source/drain, so the quantum statistical

average of aL(E) and aR(E) should satisfy

〈a†α(E)aβ(E ′)〉 = δαβδ(E −E ′)fα(E) (1.67)

where α, β=L or R and fα(E) is the Fermi-Dirac distribution of terminal α

fα(E) =1

e(E−eVα)/Tα + 1(1.68)

with the Boltzmann constant set to 1. Terminal α has the temperature Tα and

chemical potential µα = eVα controlled by thevoltage bias5. Now we can easily

calculate the average current

I = 〈I〉 = e

h

∫

dET (E)[fL(E)− fR(E)]. (1.69)

The average is time-independent, meaning that the system is in a steady state. As-

5We always assume there is a back gate so that long-range Coulomb interaction is screened. Thecharge density along the edge is then changed by chemical potentials of terminals, and we assumethe voltage bias only plays the role of the chemical potential.

39

suming that the scale of the energy dependence of the transmission coefficient T (E)

is much larger than the temperatures and voltage difference, we can approximately

set T (E) = T to be a constant. Then we have the conductance

G = I/(VL − VR) =e2

hT. (1.70)

This is the Landauer formula. It agrees with the quantization of the Hall conductance

at e2/h for the ν = 1 IQHE if the transmission coefficient T equals 1.

We can continue with the calculation of the current noise. The noise is defined

as

Sαβ(ω) =

∫

dteiωt〈∆Iα(t)∆Iβ(0) + ∆Iβ(0)∆Iα(t)〉 (1.71)

with ∆Iα = I − 〈I〉. We focus on the zero-frequency case and set ω = 0. Using the

scattering matrix and the relation Eq. (1.67), one can obtain that the zero-frequency

noise equals

S = 2e2

h

∫

dET (E)[fL(1− fL) + fR(1− fR)] +T (E)[1−T (E)](fL − fR)2. (1.72)

Again taking the approximation T (E) = T, and assuming that the temperatures are

the same TL = TR = T one derives that

S = 2e2

h

[

2TT2 + eV coth

(

eV

2T

)

T(1− T)

]

. (1.73)

One may take a limit that V ≫ T and the transmission coefficient T ≪ 1, then

S ≈ 2e2

heV T = 2eI, (1.74)

which is the famous relation between the shot noise (the noise due to the partition

40

at QPC) and the current. The Fano factor S/I is 2e, twice the charge of the current

carriers. We will see below that this relation is still true for strongly correlated FQHE

system, except that the elementary charge e is replaced by a fractional charge e∗.

This is one of the methods used in experiments to detect fractional charges [28–30].

The above single-channel two-terminal case can be generalized to multi-channel

multi-terminal systems, for example a 4-terminal setup for an IQHE system at filling

factor ν = n. For details, readers may consult Ref. [66, 67].

1.5.2 Fluctuation-Dissipation Theorem

We can take the equilibrium limit of Eq. (1.73) by setting V → 0. In this limit, the

noise becomes

S = 4e2

hTT = 4GT. (1.75)

This is the famous Nyquist formula, one of the fluctuation-dissipation theorems.

This formula relates the noise to the conductance that describes dissipation in the

system. In the case of a multi-terminal QHE setup, Eq. (1.75) is generalized as

Sαβ = 2T (Gαβ +Gβα), (1.76)

where Sαβ is the (cross-)noise between the currents at terminal α and terminal β and

Gαβ is the linear response of the current at terminal α to the voltage at terminal β.

Fluctuation-dissipation theorems (FDT) are general relations between equilib-

rium fluctuations of a system and its linear response functions. The system can

be either classical or quantum mechanical. It is also true for stochastic classical

systems, such as a Brownian particle. However, FDTs only hold for equilibrium

41

systems. Research on FDT has recently been focused on its violations in nonequilib-

rium conditions. It became gradually clear that the FDT forms a special case of more

general fluctuation theorems valid for various classes of nonequilibrium systems [68].

Well-known examples are the Jarzynski equality [69] and the Agarwal formula [70].

In Chapter 4, we will discuss a generalized FDT for chiral edges of QHE systems.

The FDT is unique for chiral systems, but is violated in non-chiral systems. Therefore

it is a way to detect the chirality of the QHE edges. As mentioned above, upstream

neutral modes might exist in FQHE systems. Note that a big difference between

the Pfaffian and anti-Pfaffian states is that upstream modes exist on the edge of the

anti-Pfaffian state but not in the Pfaffian state. These upstream modes will destroy

the chiral-system FDT.

1.5.3 Keldysh Formalism

The Landauer-Buttiker approach usually applied for non-interacting systems. To

deal with non-equilibrium transport in strongly interacting systems such as FQHE

systems, we use the Keldysh formalism.

Consider a many-body system with the Hamiltonian

H(t) = H0 +H1(t) (1.77)

where H1(−∞) = 0. The system with the Hamiltonian H0 is assumed to be solved

exactly. Let us consider that at t = −∞ the system has the initial density matrix

ρ(−∞). The density matrix can be an equilibrium Gibbs distribution exp(−βH0)

or a ground state density matrix if the system is at zero temperature. The initial

42

density matrix evolves according to H(t), i~∂tρ(t) = [H(t), ρ(t)]. We are usually

interested in the expectation values Tr(ρ(t)O(t)) of some operators O(t), such as the

electric current or noise, at time t. To take care of the time evolution, it is better to

switch to the interaction picture. Let

HI1 (t) = eiH0t/~H1(t)e

−iH0t/~ (1.78)

OI(t) = eiH0t/~O(t)e−iH0t/~ (1.79)

then the evolution operator S(t,−∞) satisfies

i~∂S(t,−∞)

∂t= HI

1 (t)S(t,−∞). (1.80)

The formal solution is S(t,−∞) = T e−i∫ t

−∞dτHI

1 (τ)/~, with T the time-ordering oper-

ator. We have the expectation value of the operator O(t) expressed as

〈O〉 = Tr[S(−∞, t)OI(t)S(t,−∞)ρ(−∞)] (1.81)

with S(−∞, t) = S†(t,−∞). Inserting the formal solution for S(t,−∞), we have

〈O〉 = Tr(

T ′ei∫ t

−∞dτHI

1 (τ)/~)

OI(t)(

T e−i∫ t

−∞dτHI

1 (τ)/~)

ρ(−∞)

, (1.82)

where T ′ is the reversed time ordering. We can write the above formula in a more

compact way by using the Keldysh contour (See Fig. 1.11),

〈O〉 = Tr

TK

(

OI(t)e−i∫K

dτHI1 (τ)/~

)

ρ(−∞)

= 〈TK(

OI(t)e−i∫K

dτHI1 (τ)/~

)

〉,

(1.83)

where TK is time ordering along the Keldysh contour. 〈〉 is a further abbreviation

meaning taking quantum average with respect to the initial density matrix ρ(−∞).

At zero temperature, we just need to calculate the expectation value with respect to

43

−∞

−∞

•O1(t1)

•O2(t2)

•O3(t3)

Figure 1.11: The Keldysh contour.

the initial state |0〉(normally the ground state of H0), and have

〈O〉 = 〈0|TK(

OI(t)e−i∫K

dτHI1 (τ)/~

)

|0〉. (1.84)

We usually have to calculate the expectation value perturbatively, by expanding

over HI1 (t) assuming it is small. The nth order term is

〈O〉n =(−i)n~nn!

∫

K

dτ1 · · · dτn〈TK [OI(t)HI1 (τ1) · · ·HI

1 (τn)]〉. (1.85)

All operatorsHI1 (τi) and O

I(t) are on the Keldysh contour and the integrals are along

the contour as well. The quantity 〈TK [OI(t)HI1 (τ1) · · ·HI

1 (τn)]〉 is usually known in

principle, since it is a quantum average in the system with a Hamiltonian H0 that

is assumably exactly solved .

1.5.4 Example: Tunneling through a Single QPC

To be concrete, we give an example here. We use the single QPC setup (Fig. 1.10)

again, but the QHE system is in the Laughlin state at filling factor ν. As mentioned

above, the FQHE edge is strongly correlated, so Landauer-Buttiker approach is not

applicable. We use the Keldysh approach to solve the problem perturbatively. We

solve the problem at zero temperature. It is easy to generalize to finite temperatures.

44

Without the QPC, the system has two chiral edges described by the Hamiltonian

H0 =v

4πν

∫

dx[(∂xφL)2 + (∂xφR)

2] (1.86)

with φL being the upper left-moving mode and φR the lower right-moving mode.

The two field operators satisfy [φL(x), φL(x′)] = −iνπsign(x− x′), [φR(x), φR(x

′)] =

iνπsign(x−x′), and they mutually commute. The charge densities are ∂xφL/2π and

∂xφR/2π . The QPC makes it possible for quasiparticles to tunnel between the two

edges. We only consider the tunneling of charge-νe quasiparticles, and tunneling

occurs at x = 0. So with the bosonization relation (1.46), the tunneling Hamiltonian

is written as

HT =

∫

dxδ(x)

γei[φL(x)+φR(x)] +H.c.

, (1.87)

with γ being the bare tunneling amplitude.

We calculate transport properties of the system by adiabatically turning the

tunneling Hamiltonian HT . Suppose that at t = −∞, HT is off and the system is in

the ground state |0〉 of H0. HT is adiabatically turned on. At time t, we calculate

the observable with an operator O. By using the Keldysh formalism, we have

〈O〉 = 〈0|TK(

O(t)e−i∫K

dτHT (τ)/~)

|0〉 (1.88)

with O(t) = eiH0t/~Oe−iH0t/~ and

HT (t) = eiH0t/~HT e−iH0t/~ =

∫

dxδ(x)

γei[φL(x,t)+φR(x,t)] +H.c.

,

where φL/R(x, t) are Heisenberg field operators.

Strictly speaking, the initial state |0〉 is the ground state of the Hamiltonian

45

G = H0 − eVLNR − eVRNL, where VL (VR) is the voltage applied to the left(right)

reservoir. Note that the left(right)-moving edge is in equilibrium with the right(left)

reservoir. It is convenient to switch to the interaction picture of eVL + eVR, so that

φR → φR − νeVLt, φL → φL + νeVRt. (1.89)

Then the tunneling Hamiltonian becomes

HT (t) =

∫

dxδ(x)

γei[φL(x,t)+φR(x,t)−νeV t] +H.c.

(1.90)

with V = VL − VR.

Now we may calculate the current. Without the QPC, the current is I = V νe2/h

due to the quantization of the Hall conductance and vanishing longitudinal resis-

tance. With the QPC, there is an additional tunneling current Itun. The total

current I = V νe2/h − Itun. Itun is proportional to the rate of the change of the

particle number on one of the edges. So,

Itun(t) =dQR

dt= −e i

~[

∫

dx∂xφR/2π,HT (t)] (1.91)

= ei

~

γei[φL(0,t)+φR(0,t)−νeV t] −H.c.

. (1.92)

Knowing the operator, we can now calculate its quantum average by using Eq. (1.88).

To the lowest order in the tunneling amplitudes, we have

Itun = − i

~

∫

K

dt1〈TKItun(t0)HT (t1)〉 (1.93)

= − i

~

∫ t0

−∞dt1〈Itun(t0)HT (t1)−HT (t1)Itun(t0)〉. (1.94)

46

The free bosonic theory (1.86) gives the zero-temperature correlation function,

〈φL(x, t)φL(0, 0)〉 = −ν ln[δ + i(t− x/v)] (1.95)

〈φR(x, t)φR(0, 0)〉 = −ν ln[δ + i(t+ x/v)]. (1.96)

So finally we obtain

Itun =e

~2|γ|2

∫ 0

−∞dt

[

1

(δ − it)2ν− 1

(δ + it)2ν

]

(e−ieνV t − eieνV t) (1.97)

=e

~2|γ|2 2π

Γ(2ν)|eνV |2ν−1sign(eV ). (1.98)

This power law in the I-V characteristic is a feature of Luttinger liquid. The expo-

nent reflects the Luttinger liquid coupling constant, which in the context of QHE is

the filling factor ν. Experiments have been carried out to detect this exponent [10],

but a discrepancy between the theory and experiments still exist. In particular, in

single QPC geometries [30], the experimental I-V curve does not satisfy a simple

power law, indicating incompleteness of the chiral Luttinger theory and complexity

of real quantum Hall edges.

We may also calculate the current noise. The νe2V/h part of the current is

noiseless at zero temperature. All noise comes from the tunneling current, i.e.,

the shot noise due to partition at the QPC. To the lowest order in the tunneling

amplitudes, the zero frequency noise is just

S =

∫

dt〈0| (∆Itun(t)∆Itun(0) + ∆Itun(0)∆Itun(t)) |0〉 (1.99)

= 2e2

~2|γ|2 2π

Γ(2ν)|eνV |2ν−1. (1.100)

So we see the Fano factor

S/|Itun| = 2ν|e|, (1.101)

47

which is twice the fractional charge ν|e|. Hence, shot noise measurements in this

single QPC geometry are used to detect fractional charges [28–30].

1.6 Overview

The above introduction has not been meant to provide complete background knowl-

edge to understand the following chapters, each being a research project that I carried

out with my advisor Prof. Dima Feldman during my graduate study. I hope from

this introduction readers can get a feeling about why the quantum Hall effect is in-

teresting, what basic questions exist, and how we can answer them in principle both

theoretically and experimentally.

In summary, there are two main classes of questions about FQHE:

1. What is the bulk topological order of a particular FQHE, Abelian or Non-

Abelian? What properties, such as fractional charge and fractional statistics,

do the elementary bulk excitations have? How to detect them? etc.

2. What are the edge excitations of a particular FQHE? How many edge modes

are there? Are they chiral? Are they charged or neutral? Are they Luttinger

liquids? Are there Majoranas? How to identify them? etc.

Certainly, there are other questions, such as spin polarization and spin excitations

of a FQHE, isotropy of a FQHE liquid, etc. Among the many FQHEs, of particular

interests is the 5/2 FQHE. It may well be the first non-Abelian state found in nature.

In this thesis, we tackle serval aspects of the two classes of questions.

48

In Chapter 2, we deal with Mach-Zenhder interferometry. We calculate the Fano

factor for the shot noise in a Mach-Zehnder interferometer in the 331 states and

demonstrate that it differs from the Fano factor in the proposed non-Abelian states

(Pfaffian or anti-Pfaffian states) for the 5/2 FQHE. We also calculate the current.

This work provides a theoretical support for distinguishing the Abelian 331 states

from non-Abelian candidates in the 5/2 FQHE in Mach-Zenher interferometers. It

is an advantage of Mach-Zehnder interferometers over Fabry-Perot interferometers.

Chapter 3 considers a long line junction as a tool to detect the edge physics of

the 5/2 FQHE state. We investigate transport properties of two proposed Abelian

states: K=8 and 331 state, and four possible non-Abelian states: Pfaffian, edge-

reconstructed Pfaffian, and two versions of the anti-Pfaffian state. Due to momentum

conservation in the long line junction geometry, there exists qualitative features –

singularities in the I-V curves – which allows one to distinguish different edge models.

In contrast to Chapters 2 and 3, which are based on particular models, Chapter

4 is about a model-independent fluctuation-dissipation theorem that we discovered

for chiral systems. The only assumption is that the system must be chiral. Non-

chiral systems violate the fluctuation-dissipation theorem. We first provide a proof

based on a mixture of Kubo and Landauer-Buttiker formalisms. It deals with a

three-terminal setup and focuses on electric current and noise. We then discuss

another proof based on fluctuation relations, generalizing the three-terminal setup

to a multi-terminal setup and including heat transport as well.

Chapter Two

Identification of 331 Quantum Hall

States with Mach-Zehnder

Interferometry

50

This Chapter is published as Chenjie Wang and D. E. Feldman Phys. Rev. B 82,

165314 (2010).

Abstract of this Chapter: It has been shown recently that non-Abelian states and

the spin-polarized and unpolarized versions of the Abelian 331 state may have identi-

cal signatures in Fabry-Prot interferometry in the quantum Hall effect at filling factor

5/2. We calculate the Fano factor for the shot noise in a Mach-Zehnder interfer-

ometer in the 331 states and demonstrate that it differs from the Fano factor in the

proposed non-Abelian states. The Fano factor depends periodically on the magnetic

flux through the interferometer. Its maximal value is 2×1.4e for the 331 states with a

symmetry between two flavors of quasiparticles. In the absence of such symmetry the

Fano factor can reach 2×2.3e. On the other hand, for the Pfaffian and anti-Pfaffian

states the maximal Fano factor is 2 × 3.2e. The period of the flux dependence of

the Fano factor is one flux quantum. If only quasiparticles of one flavor can tunnel

through the interferometer then the period drops to one half of the flux quantum. We

also discuss transport signatures of a general Halperin state with the filling factor

2 + k/(k + 2).

2.1 Introduction

Two-dimensional electron systems in a strong magnetic field exhibit much beauti-

ful physics and form many states of matter including numerous fractional quantum

Hall liquids [3]. Some of their properties such as fractional charges of elementary

excitations are well understood. Less is known about the statistics of quantum Hall

quasiparticles. We know that gauge invariance requires fractionally charged particles

to be anyons. At the same time, a direct experimental observation of anyonic statis-

51

tics poses a major challenge. This challenge has recently attracted much attention

because of a possibility of non-Abelian anyonic statistics at some quantum Hall filling

factors [16]. In contrast to “tamer” Abelian particles, non-Abelian anyons change

their quantum state after one particle makes slowly a full circle around other anyons.

This property can be used for topological quantum computation [41]. Possible appli-

cation as well as intrinsic interest of such unusual particles have stimulated attempts

to find non-Abelian anyons in nature. In particular, a possibility of non-Abelian

statistics was predicted at filling factors 5/2 and 7/2 [12, 13, 25, 45]. However, the

nature of the quantum Hall states at those filling factors remains an open question

with theoretical proposals including both Abelian and non-Abelian states [45,71–74].

Numerical simulations [38,39,75,76] with small systems of 8-20 electrons provide

support to non-Abelian Pfaffian and anti-Pffafian states. This support is however

not unanimous, see, e.g., Ref. [77]. At the same time, recent experiments [42–44]

suggest an unpolarized state at ν = 5/2. Zero spin polarization is incompatible with

either Pfaffian or anti-Pfaffian states. Some of the experiments can be understood

in terms of disorder-generated skyrmions [78] in a Pfaffian state. However, such

an explanation does not apply to the most recent optical experiment [44] and the

simplest interpretation of the existing limited experimental data is in terms of zero

polarization [34, 79]. The simplest unpolarized state is the Halperin 331 state [45].

Note that the results of the tunneling experiment [46] are compatible with both 331

and anti-Pfaffian states. In a closely related problem of quantum Hall bilayers at

filling factor 5/2, numerics supports the existence of both 331 and Pfaffian states,

separated by a phase transition [80]. Thus, it is important to find a way to identify

and distinguish from each other the 331, Pfaffian and anti-Pfaffian states.

The key difference lies in non-Abelian quasiparticle statistics of the Pfaffian and

anti-Pfaffian states versus Abelian statistics in the 331 state. In order to deter-

52

S2 D2

S1 D1

QPC1 QPC2A

B

Edge 1

Edge 2 QH liquid

Figure 2.1: Schematic picture of an anyonic Mach-Zehnder interferometer. Arrows indicate prop-agation directions of the edge modes on Edge 1 (from source S1 to drain D1) and Edge 2 (fromsource S2 to drain D2). Quasiparticles can tunnel between the two edges through two quantumpoint contacts, QPC1 and QPC2.

mine the statistics of anyons at ν = 5/2 and 7/2 several experiments were proposed

and some of them have been or are being currently implemented (for a review, see

Ref. [15]). Despite those efforts the statistics in the 5/2- and 7/2-states remains an

open question. One of the issues concerns ambiguities in the interpretation of the ex-

perimental data. In particular, the most elegant and conceptually simple approach to

detecting non-Abelian anyons is based on Fabry-Perot interferometry [48,58–60,81].

It was found recently [82–84] that a Fabry-Perot interferometer may produce iden-

tical interference and Coulomb blockade patterns in transport experiments with the

non-Abelian Pfaffian and anti-Pfaffian states and the spin-polarized [45, 72–74] and

unpolarized [71] versions of the Abelian 331 state. Similarity between the patterns

is only present in the case of the exact or approximate symmetry between two quasi-

particle flavors in the 331 states and one may hope to remove such symmetry by

some perturbation or a change in the conditions of the experiment [83]. However, in

the absence of an established theory of 5/2- and 7/2-states, it is hard to tell whether

the flavor symmetry is or is not present and if a particular perturbation would make

it possible to distinguish Abelian and non-Abelian states. Thus, it is desirable to

have another approach to interferometry such that Abelian states would never mimic

non-Abelian.

53

In this paper we show that in an anyonic Mach-Zehnder interferometer [61–65,85],

signatures are different for the Pfaffian and anti-Pfaffian states on the one hand and

the polarized and unpolarized 331 states on the other hand. This conclusion is

true both in the presence and absence of the flavor symmetry. We calculate the low-

temperature zero-frequency noise in the interferometer in the limit of weak tunneling

through the device. In such case the noise is related to the current as S = 2e∗I,

where 2e∗ is the Fano factor. The Fano factor exhibits a periodic dependence on

the magnetic flux. Its maximal value as a function of the flux was calculated for the

Pfaffian and anti-Pfaffian states in Ref. [65] and equals 2e∗ = 2× 3.2e, where e is an

electron charge. We show that in the 331 states with flavor symmetry the maximal

value of the Fano factor is 2e∗ = 2× 1.4e. In the absence of the symmetry the Fano

factor can reach the maximal value of 2× 2.3e in the 331 states. Thus, the maximal

Fano factor gives an unambiguous way to distinguish the Abelian 331 quantum Hall

liquids from the proposed non-Abelian states. We also calculate the electric current

through the interferometer as a function of the flux and voltage. The results for the

331 state differ from the case of the Pfaffian state but the difference between the two

I-V curves is small.

The paper is organized as follows. First, we briefly discuss the 331 states. Next,

we review the structure of the anyonic Mach-Zehnder interferometer in Sec. 2.3.

We calculate the zero-temperature current in Sec. 2.4 and zero-temperature zero-

frequency shot noise in Sec. 2.5. We summarize our results in Sec. 2.6. The Appendix

A contains a detailed discussion of general Halperin states at filling factor ν =

2 + k/(k + 2).

54

2.2 Statistics in the 331 state

A review of different proposed states for the filling factor 5/2 can be found in Ref. [45].

Here we summarize the properties of the 331 states [71–74].

Two different variants of the 331 state are described in the literature. The spin-

unpolarized version was introduced in Ref. [71]. It can be understood as a bilayer

state with two spin components playing the role of the layers. The filling factor

in each layer is 1/4. The spin-polarized version [45, 72–74] can be described in

terms of the condensation of the charge-2e/3 quasiparticles on top of the Laughlin

ν = 1/3 state. The two states differ in many respects but have the same key

features: the topological order and the statistics of quasiparticles [45]. Since anyonic

interferometry is only sensitive to the quasiparticle statistics, it cannot distinguish

the two states. Below we describe the statistics with the help of the K-matrix

formalism, Ref. [36]. We only focus on the half-filled Landau level. Integer edge

channels formed in the lower completely filled Landau levels are unimportant for

our problem since the transport through the interferometer is dominated by the

tunneling of fractionally charged excitations on top of the half-filled Landau level.

The K-matrix formalism encodes the information about quasiparticles in terms

of a matrix K and a charge vector t. Each elementary excitation is described by

a vector ln with integer components. In the 331 states all edge modes propagate

in the same direction. In such case the scaling dimensions of quasiparticle creation

and annihilation operators are independent of the interactions between the modes

and are given by hn = lnK−1lTn . We are only interested in the most relevant quasi-

particle operators. The quasiparticle charge Q = −etnK−1lTn . Transport through

the Mach-Zehnder interferometer depends on statistical phases accumulated by the

55

wave function when one particle makes a full circle around another. The phase

accumulated by particle 1 moving around particle 2 equals θ12 = 2πl1K−1lT2 .

The spin-unpolarized 331 state [71] can be described as a bilayer state with ν = 14

in each layer. The K-matrix is

K =

3 1

1 3

. (2.1)

The charge vector t=(1,1). The two most relevant quasiparticles are characterized

by the l-vectors l1 = (1, 0) and l2 = (0, 1). Both particles have charge e/4. This

elementary excitation charge agrees with experiments [30, 46, 48]. When a particle

makes a circle around an identical particle it accumulates the phase φ11 = φ22 =

3π/4. The mutual statistical phase of two different particles is φ12 = −π/4. We will

also need to know what phases are accumulated when a charge-e/4 quasiparticle q0

moves around a composite anyon built from several e/4-quasiparticles q1, . . . , qk. In

the Abelian 331 state such statistical phase is simply the sum of mutual statistical

phases of q0 with each of the qm particles.

The spin-polarized 331 state [45] is formed by the condensation of the charge-

2e/3 quasiparticles on top of the Laughlin ν = 1/3 state. This state is characterized

by the K-matrix

K =

3 −2

−2 4

. (2.2)

and the charge vector t = (1, 0). Calculations of the electric charges and statistical

phases are the same as above. The two most relevant elementary excitations carry

charges e/4 again and are characterized by vectors l1 = (0, 1) and l2 = (1,−1). All

statistical phases are the same as in the spin-unpolarized 331 state.

56

In fact, the two versions of the 331 state are topologically equivalent [36], since

the two K-matrices satisfy

3 1

1 3

= W

3 −2

−2 4

W T , W =

1 1

1 0

(2.3)

and the charge vectors satisfy

(1, 1) = (1, 0)W T (2.4)

This explains why the two states have the same quasiparticle charges and braiding

statistics.

2.3 Mach-Zehnder interferometer

Figure 2.1 shows a sketch of the Mach-Zehnder geometry. Because of the bulk energy

gap, the low-energy physics is determined by chiral edge modes. Charge flows along

Edge 1 from source S1 to drain D1 and along Edge 2 from source S2 to drain D2.

Quasiparticles tunnel between the two edges at quantum point contacts QPC1 and

QPC2. If one keeps S1 at a positive voltage V and the other source and drains

are grounded then there is a net quasiparticle flow into Edge 2 and a net tunneling

current. The current is measured at drain D2.

As discussed above, there are two flavors of charge-e/4 quasiparticles in the 331

state. Let us denote their topological charges (or flavors) as a and b. Since the most

relevant quasiparticle operators create particles of these two types, we consider only

the tunneling of e/4-quasiparticles with flavors a and b below. We focus on the limit

57

of small tunneling amplitudes between the edges. In such case the problem can be

accessed with perturbation theory. We denote the small tunneling amplitudes at the

two point contacts as Γxk, where k = 1, 2 is the number of the point contact and

x = a, b is the topological charge of the tunneling quasiparticle. The tunneling rate

from Edge 1 to Edge 2 can be found from the Fermi golden rule. It depends on the

tunneling amplitudes and on the phase difference for the quasiparticles which follow

from S1 to D2 through QPC1 and QPC2. The latter consists of two contributions:

the Aharonov-Bohm phase due to the external magnetic field and the statistical

phase accumulated by a quasiparticle making a full circle around the “hole” in the

interferometer. The statistical phase is determined by the total topological charge

that tunneled previously between the edges. Indeed, the total topological charge

of Edge 2 can only change during tunneling events at QPC1 and QPC2. Charge

exchange with the Fermi-liquid drain D2 and source S2 cannot affect the topological

charge of the edge. Taking into account that the edges are chiral we see that all

topological charge that previously tunneled into Edge 2 accumulates inside the loop

QPC1-A-QPC2-B-QPC1. We will denote that loop as L below. Certainly, the accu-

mulated topological charge only assumes a discrete set of values and hence changes

quasiperiodically as a function of time. We find the following tunneling rate from

edge 1 to edge 2 for a quasiparticle of flavor x (cf. Ref. [65])

w+x,d = r1(|Γx

1|2 + |Γx2|2) + (r2Γ

x1Γ

x∗2 e

iφmag+iφxd + c.c.), (2.5)

where r1(V, T, x) and r2(V, T, x) depend on the voltage, temperature and quasipar-

ticle flavor, d is the topological charge trapped in the interferometer before the

tunneling event, φxd the statistical phase discussed in the previous section, and the

Aharonov-Bohm phase φmag = πΦ/(2Φ0) is expressed in terms of the magnetic flux

Φ through the loop L and the flux quantum Φ0 = hc/e. r1,2 cannot be calculated

58

without a detailed understanding of the edge physics. Fortunately, we will not need

such a calculation to determine the main features of the current and noise in the

interferometer. Eq. (2.5) assumes that the two tunneling amplitudes are small and

hence the mean time between two consecutive tunneling events is much longer than

the time spent by a tunneling quasiparticle between the point contacts. At a nonzero

temperature, quasiparticles are allowed to tunnel from Edge 2 to Edge 1 which has a

higher potential. The corresponding tunneling rate w−x,d+x = exp[−eV/(4kBT )]w+

x,d

is connected to w+, Eq. (2.5), by the detailed balance principle. Here d + x is the

topological charge of the fusion of anyons with topological charges d and x, i.e.,

the topological charge of Edge 2 before the tunneling event. In what follows we

concentrate on the limit of low temperatures and neglect w−.

Our main focus will be on the situation with the exact or approximate flavor sym-

metry, i.e., we assume that the tunneling amplitudes Γxk and coefficients rk(V, T, x) in

(2.5) do not depend on the flavor x. Since Γ’s and r’s only enter the transition rates

in the combinations r1(V, T, x)|Γx1,2|2 and r2(V, T, x)Γx

1Γx∗2 , the results do not change

if the tunneling amplitudes depend on the flavor but the above combination are the

same for x = a and b. As discussed in the introduction, Fabry-Perot interferometry

cannot distinguish the 331 states with flavor symmetry from non-Abelian states. We

will see below that no such issue exists for Mach-Zehnder interferometry. We will

also briefly discuss signatures of the 331 states without flavor symmetry in the shot

noise in a Mach-Zehnder interferometer.

59

(0, 0) (1, 0) (2, 0) (3, 0)

(4, 0) (0, 1) (1, 1) (2, 1)

pa

0 pa

3 pa

6

pa

4 pa

7 pa

2

pb

0 pb

7 pb

6

pb

4 pb

3 pb

2

pb

5

pa

5

pb

1

pa

1(a)

(0, 0)

(4, 0)(0, 1)

(1, 0)(2, 0)

(1, 1)(2, 1)

(3, 0)2p0

2p4

p3

p7

2p6

2p2

p5

p1 (0, 0)(1, 0)

(2, 0)

(4, 0)(3, 0)(0, 1)

(1, 1)

(2, 1)

pa

0

pa

3

pa

1

pa

6

pa

4

pa

7

pa

5

pa

2

(b) (c)

Figure 2.2: Possible states of a Mach-Zehnder interferometer in the 331 state. Panel (a) shows ageneral case, eight possible states labeled by topological charges and the transition rates betweenthem. Arrows show the allowed transitions at zero temperature. Solid blue lines represent tun-neling events involving quasiparticles of flavor a, and dashed black lines represent tunneling eventsinvolving particles of flavor b. Special cases with pak = pbk ≡ pk and pbk = 0 are illustrated in Panels(b) and (c) respectively.

60

2.4 Electric current

We are now in the position to calculate the tunneling current I. From now on, we

will only use the language of the spin-unpolarized 331 state in which flavor a can be

understood as the l-vector a = (1, 0) and b as b = (0, 1). From Sec. 2.2 we know

how to evaluate the statistical phase in (2.5). The tunneling rate depends on the

topological charge d = (m,n) trapped in the interferometer. Each tunneling event

of a particle with flavor a changes d → (m + 1, n). If a particle of flavor b tunnels

then d → (m,n + 1).

Any anyon has a trivial mutual statistical phase 2πk with an electron [86]. Com-

bining this condition with the knowledge of the electron charge one finds that the

l-vectors (1, 3) and (3, 1) describe electrons. Thus, the topological charges d and

d′ = d+n1(1, 3)+n2(3, 1) can be viewed as identical since anyons accumulate iden-

tical topological phases moving around charges d and d′. One can easily see that in

a 331 state, the trapped topological charge falls into one of eight equivalence classes

which mark eight possible topological states of the area enclosed by the loop L.

Fig. 2.2(a) shows the 8 states and possible transitions between them due to anyon

tunneling between the edges at zero temperature. The transition rates shown in

Fig. 2.2 are given by the equation

pxk = Ax[1 + ux cos(πΦ/(2Φ0) + πk/4 + δx)] (2.6)

where x = a or b, and k = 0, 1, . . . , 7. The parameters Ax = r1(|Γx1 |2 + |Γx

2 |2),

ux = 2|r2Γx1Γ

x2 |/[r1(|Γx

1 |2 + |Γx2 |2)] and δx = arg(r2Γ

x1Γ

x∗2 ). In the presence of the

flavor symmetry the diagram in Fig. 2.2(a) simplifies to Fig. 2.2(b) with six instead

of eight vertexes. Two pairs of vertexes in Fig. 2.2 (a) are merged into single vertexes

61

0 1 2

0.1

0.2

0.3 I

Φ/Φ0

γ=1

γ=0.25

γ=0.05

γ=0

Pfaffian

Figure 2.3: Flux-dependence of the tunneling current in the 331 and Pfaffian states. We setAa = 1, ua = ub = 1, and δa = δb = 0 for all curves for the 331 state. Different curves correspondto different values of γ = Ab/Aa in the 331 state. The curve for the Pfaffian state is plottedaccording to Eq. (8) in Ref. [64] with r+11 = r+12 = 1 and Γ1 = Γ2 such that the maximum matchesthe maximum of the curve for the 331 state with γ = 1.

in Fig. 2.2 (b). This is legitimate due to numerous equalities between tunneling rates.

In Fig. 2.2 (b), we use the notation pk = pak = pbk. We will also denote A = Aa = Ab,

u = ua = ub and δ = δa = δb in the presence of the flavor symmetry. Fig. 2.2(c)

illustrates another simple limit in which pbk = 0.

The transition rates can be used to write down a kinetic equation for the proba-

bilities fd of different topological charges d trapped in the interferometer. In terms

of the distribution function fd the average tunneling current between the edges

I =e

4

∑

d

fd(w+a,d + w+

b,d − w−a,d − w−

b,d), (2.7)

where d goes over the eight possible states in Fig. 2.2(a). The distribution function

62

satisfies the steady state equation

0 =dfddt

=∑

x=a,b

(fd−xw+x,d−x + fd+xw

−x,d+x)

−∑

x=a,b

fd(w+x,d + w−

x,d), (2.8)

and the normalization condition∑

d fd = 1. In Eq. (2.8), (d + x) means the topo-

logical charge of the fusion of anyons with topological charges d and x. The fusion

of x and d− x has topological charge d. In the absence of the flavor symmetry, the

tunneling current can be found with a lengthy but straightforward calculation from

Eqs. (2.7) and (2.8). In the presence of the flavor symmetry, the low-temperature

current can be easily calculated from Eqs. (2.7,2.8) in the picture with six distinct

topological charges, Fig. 2.2 (b). One obtains the following result

I =eA

2

1− u2 + u4

8(1− cos(2πΦ/Φ0 + 4δ))

1− (34− 1

4√2)u2 + u4

16

[

(1− 1√2)(1− cos(2πΦ/Φ0 + 4δ))− 1√

2sin(2πΦ/Φ0 + 4δ)

]

(2.9)

This formula is quite similar to the expression for the current in the Pfaffian state

(cf. Eq. (8) in Ref. [64]). The similarity originates from the similarity of the diagram

Fig.2.2(b) with a corresponding diagram in the Pfaffian state [64]. They have the

same topology with six vertexes including two “cross-roads”. Still, in contrast to the

Fabry-Perot case, the expressions for the current are not identical for the 331 and

Pfaffian states. Similar to the Pfaffian state [64], the expression for the current for the

opposite voltage sign can be obtained by changing both the overall sign of the current

and the sign before sin(2πΦ/Φ0 + 4δ) in the denominator. Thus, the I-V curve is

asymmetric just like in the Pfaffian case. The current depends periodically on the

magnetic flux with the period Φ0 in accordance with the Byers-Yang theorem [87].

We do not include an analytical expression for the current in the absence of the

flavor symmetry since it is lengthy. I(Φ) is plotted in Fig. 2.3 for the Pfaffian and

63

331 states for different values of γ = Ab/Aa at u = 1 which maximizes the visibility

of the Aharonov-Bohm oscillations.

Another simple limit corresponds to the situation with pbk = 0 (Fig. 2.2(c)). In

that case, I(Φ) has a reduced period Φ0/2. The period reduction can be understood

from the diagram Fig. 2.2(c). The system can return to its initial state only after

eight tunneling events instead of four in Fig. 2.2 (b). This means a transfer of the

charge 2e in each cycle of tunneling events. Such “Cooper pair” charge agrees with

the “superconductor” periodicity.

2.5 Shot noise

Shot noise measurements have helped to determine the charges of elementary exci-

tations at several quantum Hall filling factors including 5/2 [28–30]. In the Mach-

Zehnder geometry, zero-frequency noise contains also information about the quasi-

particle statistics. Below we calculate the noise in the 331 states in the limit of weak

tunneling. We assume that the temperature is much lower than the voltage bias.

In such case it is possible to neglect the temperature and perform calculations at

T = 0.

Shot noise is defined as the Fourier transform of the current-current correlation

function,

S(ω) =

∫ +∞

−∞〈I(0)I(t) + I(t)I(0)〉 exp(iωt)dt. (2.10)

Below we consider ω = 0 only. In the weak tunneling limit, the noise can be expressed

as S = 2e∗I, where 2e∗ is known as the Fano factor. e∗ can be understood as an

effective charge tunneling through the interferometer. We will see that the Fano

64

0.0

0.5

1.0

Γ

0

2

4

6

∆

0.5

1.0

1.5

2.0

emax*

Figure 2.4: The maximal e∗ as a function of γ and δ.

factor is different in the Pfaffian state and the 331 states with or without the flavor

symmetry. In the Mach-Zehnder interferometer, the Fano factor 2e∗(Φ) exhibits

oscillations as a function of the magnetic flux. For Laughlin states, the maximal e∗

can never exceed 1.0e, while in the Pfaffian state e∗ can be as large as 3.2e, Ref. [65].

We will see that the maximal Fano factor is lower in the 331 state than in the Pfaffian

state.

A zero-frequency shot noise can be conveniently connected with the fluctuation of

the charge transmitted through the interferometer over a long time t (cf. Ref. [65]):

S/2 = limt→∞

〈δQ2(t)〉/t, (2.11)

where δQ(t) is the fluctuation of the charge Q(t) that has tunneled into the loop

L during the time t. The average current I = limt→∞〈Q(t)〉/t. We will use the

generating function method developed in Ref. [65] (see also Ref. [63]) to calculate

the zero-frequency shot noise and the Fano factor. Suppose for certainty that the

65

initial state inside the loop L has topological charge (0, 0). Our results will not

depend on this choice of the initial topological charge. The electric charge that had

tunneled through the interferometer, Q(t), is zero at t = 0, Q(0) = 0. The system

evolves over a time period t according to a kinetic equation with the transition rates

from the diagram Fig. 2.2(a). The probability of finding the interferometer in the

state with the topological charge d on Edge 2 and the transmitted electric charge

Q(t) = k e4will be denoted as fd,k(t) below. For a given d, only certain k’s are possible

(e.g., if d = (0, 1) then k can only be 4n + 1 with an integer n). The distribution

function satisfies the kinetic equation

dfd,k(t)

dt=∑

x=a,b

(fd−x,k−1w+x,d−x + fd+x,k+1w

−x,d+x)

−∑

x=a,b

fd,k(w+x,d + w−

x,d). (2.12)

Let us now define the generating function

fd(s, t) =∑

k

fd,k(t)sk, (2.13)

where the summation extends over all possible k’s for a given d. One can express

the average transmitted charge 〈Q(t)〉 = e∑

k kfd,k(t)/4 and its fluctuation as

〈Q(t)〉 = e

4

(

d

ds

∑

d

fd(s, t)

)∣

∣

∣

∣

∣

s=1

(2.14)

and

〈δQ2(t)〉 =(e

4

)2(

d

dssd

ds

∑

d

fd(s, t)

)∣

∣

∣

∣

∣

s=1

− 〈Q(t)〉2, (2.15)

where the angular brackets stay for the average with respect to the distribution

function fd,k(t). It is easy to see that fd(1, t) equals the probability fd(t) introduced

66

in Sec. 2.4. The kinetic equation (2.12) then reduces to

d

dtfd(s, t) =

∑

x=a,b

(sfd−xw+x,d−x +

1

sfd+xw

−x,d+x)

−∑

x=a,b

fd(w+x,d + w−

x,d) (2.16)

The above equation can be rewritten in a matrix form, ~f(s, t) =M(s)~f(s, t), where

M(s) is a time-independent 8 × 8 matrix. We can solve the above linear equation

and obtain

fd(s, t) =

7∑

i=0

ξd,i(s)eλi(s)tPi(t), (2.17)

where λi(s) are eigenvalues of M(s), ξd,i constants and Pi(t) polynomials, Pi = 1 for

nondegenerate eigenvalues λi. The solution can be further simplified with the help

of the Rohrbach theorem [88]. Indeed, from (2.16) we see that M(s) has negative

diagonal elements, non-negative off-diagonal elements, and the sum of the elements

of each column is zero at s = 1. In such situation the Rohrbach theorem applies.

According to the theorem, the eigenvalue of M(s = 1) with the maximal real part is

nondegenerate and equals zero. All other eigenvalues have negative real parts. For s

close to 1, the maximal eigenvalue λ0(s) should also be nondegenerate by continuity

of the eigenvalues as functions of s. Thus, the λ0(s) term in Eq. (2.17) dominates at

large time t and P0(t) = 1.

According to (2.14), (2.15) and (2.17) together with the conservation of proba-

bility∑

d fd(1, t) = 1, we have

I =e

4λ′0(1)

S = (e/4)2 [λ′′0(1) + λ′0(1)]

e∗ =e

4[1 + λ′′0(1)/λ

′0(1)] (2.18)

67

The derivatives of λ0(s) can be evaluated in the following way. Define a function

G(s, λ) = det(M(s)− λE) (2.19)

where det is the determinant of a matrix and E is the identity matrix. We know that

λ0(s) satisfies the equation G(s, λ0(s)) = 0. Then by differentiating this equation

one gets

λ′0(1) = −Gs/Gλ|(s,λ)=(1,0)

λ′′0(1) = −(GssG2λ +GλλG

2s − 2GsλGsGλ)/G

3λ|(s,λ)=(1,0), (2.20)

where Gs and Gλ are the first derivatives of G(s, λ) with respect to s and λ, and

Gss, Gsλ and Gλλ are the three second derivatives of G(s, λ). Then we can easily

evaluate the shot noise and the Fano factor.

When the symmetry between the two flavors of quasiparticles exists the expres-

sion for the Fano factor at zero temperature simplifies:

λ′0(1) = 16A/(4 +

3∑

k=0

p2k+1

p2k+4)

e∗ =eλ′0(1)

2

16

1

4A

∑

k

p2k+1

p22k+4

− 1

16A2(∑

k

p2k+1

p2k+4)2

+1

2A2

[

1 +1

4(p3p6

+p7p2)(p1p4

+p5p0)

]

(2.21)

where the identity pk+pk+4 = 2A is used to simplify the expressions. Like the current,

the Fano factor e∗ is a periodic function of the magnetic flux Φ with the period Φ0.

Numerical analysis of Eq. (2.21) shows that maximal e∗ is 1.4e, greater than 1.0e

for Laughlin states but smaller than 3.2e for the Pfaffian state. The maximal Fano

factor is achieved at u = 1. This value of u corresponds to Γ1 = Γ2 and r1 = r2.

68

As can be seen from a renormalization group picture, the latter relation between ri

is satisfied at low voltages and temperatures, eV, T < hv/a, where v is the velocity

of the slowest edge mode and a the distance between the point contacts along the

edges of the interferometer. The values of Γk can be controlled with gate voltages.

Note that u = 1 is the maximal possible value of u. Indeed, u > 1 would result in

negative probabilities (2.6) at some values of the magnetic flux.

An interesting situation emerges in the special limit when only particles of one

flavor a or b can tunnel (i.e., Aa = 0 or Ab = 0). The interferometer can be tuned

to that limit with the following approach: one keeps the filling factor ν = 5/2 in the

gray (green online) region in Fig. 2.1 and ν = 0 in the white region. In addition, a

narrow region with a filling factor 7/3 is created along the edges. In such situation,

quasiparticles tunnel through the 5/2-liquid between the interfaces of the ν = 5/2

and ν = 7/3 regions at the tunneling contacts. Since the interface contains only one

edge mode, only one type of quasiparticles can tunnel (for a detailed discussion of

the interface mode see Appendix A).

We will assume for certainty that pbk = 0. This case is illustrated in Fig. 2.2 (c).

The calculation of the zero-temperature current and Fano factor greatly simplifies in

that limit. One finds expressions resembling the results for the Laughlin states [65]:

I =2e

∑7k=0 1/pk

; (2.22)

e∗ = 2e

∑

1/p2k[∑

1/pk]2. (2.23)

Combining the above equations with Eq. (2.6) one can see that in the special case of

only one type of quasiparticles allowed to tunnel, the current and noise are periodic

functions of the magnetic flux with the “superconducting” period Φ0/2. This is

69

certainly compatible with the Bayers-Yang theorem [87]. The numerator of the

fraction in Eq. (2.23) is always smaller or equal to the denominator. They become

equal when one of the probabilities pk approaches zero. In that case the Fano factor

is maximal and e∗ = 2e. As is clear from Eq. (2.6), this can happen only at u = 1.

Note that the period and maximal Fano factor are also Φ0/2 and 2×2e in the K = 8

Abelian state [45].

Finally, let us discuss the most general case with nonzero Aa and Ab and no

flavor symmetry. Generally, when the flavor symmetry is absent, e∗(Φ) is a periodic

function with the period Φ0. Our numerical results show that the maximal e∗ is

achieved at ua = ub = 1 for any choice of Aa, Ab, δa and δb. We calculated the

maximal e∗ ≡ e∗max(γ, δ) as a function of the tunneling amplitude ratio γ = Ab/Aa

and phase difference δ = δb − δa. We find that the maximal value of e∗max = 2.3e is

achieved at γ = 0.08 and δ = −0.03. The dependence of e∗max on γ and δ is shown in

Fig. 2.4. We see that at γ = 0, e∗max is 2e, increases to 2.3e at small positive γ and

drops to 1.4e at γ = 1.

In a general case, the current and noise depend on several parameters and the

system may not be tuned to the regime with the maximal Fano factor 2 × 2.3e. At

the same time, such tuning is possible in the flavor-symmetric case, in the case when

only one flavor tunnels and in the Pfaffian state. In all those cases one just needs to

achieve u = 1 which corresponds to Γ1 = Γ2. Making Γ1,2 equal is straightforward:

one just has to make sure that the current is the same when only QPC1 or only

QPC2 is open. Thus, in the absence of the flavor symmetry, the identification of the

331 states simplifies by operating the interferometer in the regime in which only one

quasiparticle flavor can tunnel.

70

2.6 Summary

We have calculated the current and noise in the Mach-Zehnder interferometer in the

331 state and compared the results with those for the Pfaffian state. Note that the

transport behavior is essentially the same in the Pfaffian and anti-Pfaffian states.

The current dependence on the magnetic flux turns out to be quite similar for the

Pfaffian and 331 states. In both states the I-V curves are asymmetric. The states

can be unambiguously distinguished with a shot noise measurement. In the Pfaffian

state the maximal Fano factor is 2 × 3.2e. In the 331 state the Fano factor cannot

exceed 2×2.3e. If the flavor symmetry is present than the maximal Fano factor drops

to 2 × 1.4e. The difference of the predicted maximal Fano factors well exceeds the

current experimental accuracy of 15 percent [89]. An interesting situation emerges,

if quasiparticles of only one flavor can tunnel through the interferometer. In that

case the current and noise are periodic functions of the magnetic flux with the period

Φ0/2. In a general case the period is Φ0. We show that it is possible to tune the

interferometer to the regime with the period Φ0/2 in the 331 state.

Chapter Three

Transport in Line Junctions of

ν = 5/2 Quantum Hall Liquids

72

This Chapter is published as Chenjie Wang and D. E. Feldman, Phys. Rev. B 81,

035318 (2010).

Abstract of this chapter: We calculate the tunneling current through long-line

junctions of a ν=5/2 quantum Hall liquid and (i) another ν=5/2 liquid, (ii) an

integer quantum Hall liquid, and (iii) a quantum wire. Momentum-resolved tunneling

provides information about the number, propagation directions, and other features of

the edge modes and thus helps distinguish several competing models of the 5/2 state.

We investigate transport properties of two proposed Abelian states: K=8 and 331

state, and four possible non-Abelian states: Pfaffian, edge-reconstructed Pfaffian,

and two versions of the anti-Pfaffian state. We also show that the nonequilibrated

anti-Pfaffian state has a different resistance from other proposed states in the bar

geometry.

3.1 Introduction

One of the most interesting aspects of the quantum Hall effect (QHE) is the pres-

ence of anyons which carry fractional charges and obey fractional statistics. In many

quantum Hall states, elementary excitations are Abelian anyons [3]. They accu-

mulate non-trivial statistical phases when move around other anyons and can be

viewed as charged particles with infinitely long solenoids attached. A more interest-

ing theoretical possibility involves non-Abelian anyons [16]. In contrast to Abelian

QHE states, non-Abelian systems change not only their wave functions but also

their quantum states when one anyon encircles another. This property makes non-

Abelian anyons a promising tool for quantum information processing [41]. However,

their existence in nature remains an open question.

73

C1 C2

C3 C4

(a)

ν = 2

ν = 5/2

C1 C2

C3 C4

(b)

ν = 5/2

quantum wire

Figure 3.1: (a) Tunneling between ν = 5/2 and ν = 2 QHE liquids. The edges of the upper andlower QHE liquids form a line junction. (b) Tunneling between ν = 5/2 QHE liquid and a quantumwire. In both setups, contacts C1 and C2 are kept at the same voltage V .

It has been proposed that non-Abelian anyons might exist in the QHE liquid at

the filling factor ν = 5/2, Ref. [25]. Possible non-Abelian states include different

versions of Pfaffian and anti-Pfaffian states [12, 13, 45]. At the same time, Abelian

candidate wave functions such as K = 8 and 331 states were also suggested [45,

72] for ν = 5/2. Different models predict different quasiparticle statistics but the

same quasiparticle charge q = e/4, where e < 0 is an electron charge. Since the

experiments [30, 46, 48] have been limited to the determination of the charge of the

elementary excitations, the correct physical state remains unknown.

Several methods to probe the statistics in the 5/2 state were suggested but nei-

ther was successfully implemented so far. This motivates further investigations of

possible ways to test the statistics. The definition of exchange statistics involves

quasiparticle braiding. Hence, interferometry is a natural choice. An elegant and

74

conceptually simplest interferometry approach involves an anyonic Fabry-Perot in-

terferometer [58–60, 81, 90]. Its practical implementation faces difficulties due in

part to the fluctuations of the trapped topological charge [91, 92]. A very recent

Fabry-Perot experiment might have shown a signature of anyonic statistics [93].

However, interpretation of such experiments is difficult [94] and must take into ac-

count sample-specific factors such as Coulomb blockade effects. [57,95] An approach

based on a Mach-Zehnder interferometer [61–65] is not sensitive to slow fluctuations

of the trapped topological charge but just like the Fabry-Perot interferometry it

cannot easily distinguish Pfaffian and anti-Pfaffian states. On the other hand, the

structure of edge states contains full information about the bulk quantum Hall liquid

and thus a tunneling experiment with a single quantum point contact might be suffi-

cient [45]. Unfortunately, even in the case of simpler Laughlin states the theory and

experiment have not been reconciled for this type of measurements [10]. Besides, the

scaling behavior of the tunneling I −V curve is non-universal and depends on many

factors such as edge reconstruction [96] and long range Coulomb interactions. An

approach based on two-point-contact geometry [97] identifies different states through

their universal signatures in electric transport. This comes at the expense of the ne-

cessity to measure both current and noise. Recently an approach based on tunneling

through a long narrow strip of the quantum Hall liquid was proposed [98]. This ap-

proach, however, has the same limitation as the Fabry-Perot geometry: interference

is smeared by the quasiparticle tunneling into and from the strip. In this paper we

analyze a related approach with tunneling through a long narrow line junction of

quantum Hall liquids and a line junction of a ν = 5/2 quantum Hall liquid and a

quantum wire. Since only electrons tunnel in such geometry, the interference picture

is not destroyed by quantum fluctuations.

Fig. 3.1 shows sketches of our setups. Electrons tunnel from the ν = 5/2 fractional

75

C1 C2C5 C6

C3 C4

ν = 5/2

ν = 2

Figure 3.2: Tunneling between the fractional QHE channels of the ν = 5/2 edge and the ν = 2integer channels. Contacts C1 and C2 are kept at the same voltage V and the other contacts aregrounded.

QHE state to the ν = 2 or ν = 1 integer QHE state through a line junction in the

weak tunneling regime (Fig. 3.1a)) at near zero temperature. A similar setup has

already been realized in the integer QHE regime [99]. Fig. 3.1b) illustrates a setup

with electron tunneling between the edge of the ν = 5/2 liquid and a one-channel

quantum wire. The most important feature in these setups is the conservation of

both energy and momentum in each tunneling event [99–101]. The two conservation

laws lead to singularities in the I − V curve. Each singularity emerges due to one of

the edge modes on one side of the junction. Thus, the setups allow one to count the

modes and distinguish different proposed states since they possess different numbers

and types of edge modes with different propagation directions and velocities. In

particular, these setups are able to distinguish different Abelian and non-Abelian

states.

The edge of the 5/2 state includes both a fractional 5/2 edge and two integer

quantum Hall channels. In the setups Fig. 3.1, electrons tunnel both into the frac-

tional and integer channels on the edge. However, our calculations are also relevant

for a setup in which tunneling occurs into the fractional 5/2 edge only. Such situation

can be achieved in the way illustrated in Fig. 3.2, similar to experiments [102–104].

In the setup Fig. 3.2, a voltage difference is created between integer and fractional

quantum Hall channels on the same edge and tunneling occurs between the integer

76

and fractional channels. Our results also apply to the setup considered in Ref. [105].

In that setup, tunneling occurs into an edge separating ν = 2 and ν = 5/2 quantum

Hall liquids.

The paper is organized as follows. We review several models of the 5/2 state and

their corresponding edge modes in Sec. 3.2. Sec. 3.3 contains a qualitative discussion

of the momentum resolved tunneling. We describe our technical approach in Sec. 3.4.

The number of conductance singularities allows one to distinguish different models.

This number is computed in section 3.5. Detailed calculations of the I − V curve

for each edge state are given in Sec. 3.6 in the limit of weak interactions between

fractional and integer edge channels. Our results are summarized in Sec. 3.7. We

discuss effects of possible reconstruction of integer QHE modes in Appendix B.

3.2 Proposed 5/2 states

Numerical experiments [38,39,75,76] generally support a spin-polarized state for the

quantum Hall liquid with ν = 5/2. Below we review the simplest spin-polarized

candidate states, including the abelian K = 8 state, a version of the 331 state,

and non-abelian Pfaffian and anti-Pfaffian states. In all those states, the lowest

Landau level is fully filled with both spin-up and spin-down electrons which form

two integer QHE liquids, while in the second Landau level electrons form a spin-

polarized ν = 1/2 fractional QHE liquid. Our approach can be easily extended to

spin-unpolarized states. In the following, we focus on the 1/2 fractional QHE liquid

and its edge. The lowest Landau level contributes two more edge channels.

The K = 8 state can be understood as a quantum Hall state of Cooper pairs. The

77

331 state is formed by the condensation of the charge-2e/3 quasiparticles on top of

the Laughlin ν = 1/3 state. A different version of the 331 state is also known [106].

Since that version is not spin-polarized, we do not consider it below.

The abelian K = 8 and 331 states [45] can be described by Ginzburg-Landau-

Chern-Simons effective theories [9], with the Lagrangian density given by

L = − ~

4π

∑

IJµν

KIJaIµ∂νaJλǫµνλ, (3.1)

where µ, ν = t, x, y are space-time indices. The K-matrix describes the topological

orders of the bulk, and its dimension gives the number of layers in the hierarchy.

The U(1) gauge field aIµ describes the quasiparticle/quasihole density and current

in the Ith hierarchical condensate. This effective bulk theory also determines the

theory at the edge, where the U(1) gauge transformations are restricted. The edge

theory, called chiral Luttinger liquid theory, has the Lagrangian density

Ledge = − ~

4π

∑

IJ

(∂tφIKIJ∂xφJ + ∂xφIVIJ∂xφJ). (3.2)

The chiral boson field φI describes gapless edge excitations of the Ith condensate,

and VIJ is the interaction between the edge modes. We see that the dimension of

the K-matrix gives the number of the edge modes. In the K = 8 state, electrons

first pair into charge-2e bosons, then these bosons condense into a ν = 1/8 Laughlin

state. Hence, the K-matrix is a 1 × 1 matrix whose only element equals 8, and so

there is only one right-moving edge mode. The 331 state is characterized by

K =

3 −2

−2 4

(3.3)

which has two positive eigenvalues, so there are two right-moving modes at the edge.

78

This state should be contrasted with the spin-unpolarized version of the 331 state,

whose K-matrix has entries equal to 3 and 1 only. The same name is used for the

two states since they have the same topological order [45].

The Pfaffian state [25] can be described by the following wave function for the

1/2 fractional QHE liquid

ΨPf = Pf(1

zi − zj)∏

i<j

(zi − zj)2e−

∑i |zi|2, (3.4)

in which zn = xn + iyn is the coordinate of the nth electron in units of the magnetic

length lB, and Pf is the Pfaffian of the antisymmetric matrix 1/(zi − zj). At the

edge, there is one right-moving charged boson mode and one right-moving neutral

Majorana fermion mode. The edge action assumes the form (3.45). In the presence of

edge reconstruction, the action changes [45]. In the reconstructed edge state, there

are one right-moving charged and one right-moving neutral boson mode, and one

left-moving neutral Majorana fermion mode. The edge action becomes Eq. (3.47).

The anti-Pfaffian state [12,13] is the particle-hole conjugate of the Pfaffian state,

i.e., the wave function of the anti-Pfaffian state can be obtained from the Pfaffian

wave function through a particle-hole transformation [107], given in Ref. [13]. There

are two versions of the anti-Pfaffian edge states. One possibility is a non-equilibrated

edge. In that case tunneling between different edge modes can be neglected and the

modes do not equilibrate. The action contains two counter-propagating charged bo-

son modes and one left-moving neutral Majorana fermion mode Eq. (3.55). The

other version is the disorder-dominated state, in which there are one right-moving

charged boson mode and three left-moving neutral Majorana fermion modes of ex-

actly the same velocity, Eq. (3.54). As discussed below, only limited information

about the latter state can be extracted from the transport through a line junction

79

since momentum does not conserve in tunneling to a disordered edge.

We see from the above discussion that different proposed edge states have differ-

ent numbers and types of modes. This important information can be used to detect

the nature of the 5/2 state as discussed in the rest of this paper.

3.3 Qualitative discussion

In this section we discuss some details of the setup. We also provide a qualitative

explanation of the results of the subsequent sections in terms of kinematic constraints

imposed by the conservation laws.

Our setups are shown in Fig. 3.1. The long uniform junction couples the edge of

the upper ν = 5/2 fractional QHE liquid with the edge of the lower ν = 2 or ν = 1

integer QHE liquid. Such a system with two sides of the junction having different

filling factors can be realized experimentally in semiconductor heterostructures with

two mutually perpendicular 2D electron gases (2DEG) [100,108]. Properly adjusting

the direction and magnitude of the magnetic field one can get the desired filling

factors [100]. Depending on the direction of the magnetic field, the upper and lower

edge modes in Fig. 3.1a) can be either co- or counter-propagating. In Sec. 3.7, we

will also briefly discuss the tunneling between two 5/2 states. This situation can

be realized by introducing a barrier in a single 2DEG [99]. We will see however

that the second setup is less informative than the first one. Finally, we will consider

tunneling between a 5/2 edge and a uniform parallel one-channel quantum wire.

Such setup can come in two versions: a) tunneling into a full 5/2 edge that includes

both fractional and integer modes and b) tunneling into a fractional edge between

80

ν = 2 and ν = 5/2 QHE liquids [105]. A closely related setup is illustrated in Fig.

3.2. There the tunneling occurs between different modes of the same edge.

Below we will use the language referring to tunneling between two QHE liquids,

a 5/2 liquid and an integer ν = 2 QHE liquid. This language can be easily translated

to the quantum wire situation. In contrast to the integer QHE edge, a quantum wire

contains counter-propagating modes. However, the energy and momentum conser-

vation, together with the Pauli principle, generally restrict tunneling to only one of

those modes.

The Hamiltonian assumes the following general structure:

H = H5/2 +Hint +Htun, (3.5)

where the three contributions denote the Hamiltonians of the 5/2 edge, the integer

edge and the tunneling term. The latter term expresses as

Htun =

∫

dxψ†(x)∑

n

Γn(x)ψn(x) + H.c., (3.6)

where x is the coordinate on the edge, ψ†(x) is the electron creation operator at the

integer edge, ψn are electron operators at the fractional QHE edge and Γn(x) are

tunneling amplitudes. Several operators ψn correspond to different edge modes. We

assume that the system is uniform. This imposes a restriction

Γn(x) ∼ exp(−i∆knx), (3.7)

where ∆kn should be understood as the momentum mismatch between different

modes. In order to derive Eq. (3.7) we first note that in a uniform system |Γm(x)|

cannot depend on the coordinate. Next, we consider the system with the tunneling

81

Hamiltonian H ′tun = ψ†(x0)Γm(x0)ψm(x0) + ψ†(x0 + a)Γm(x0 + a)ψm(x0 + a) + H.c.

The current can depend on a only and not on x0 - otherwise different points of the

junction would not be equivalent. Applying the second order perturbation theory

in Γm to the calculation of the current one finds that Γm(x0)Γ∗m(x0 + a) must be a

constant, independent of x0. Using the limit of small a one now easily sees that the

phase of the complex number Γm(x) is a linear function of x. This proves Eq. (3.7).

We assume that the same voltage V is applied to both contacts at the upper

ν = 5/2 edge in Fig. 3.1, so that all right-moving and left-moving modes at the

upper edge are in equilibrium with the chemical potential µ1 = eV . The lower edge

is grounded, i.e., the chemical potential at the lower edge µ2 = 0.

∆kn may depend on the applied voltage V since the width of the line junction

may change when the applied voltage changes. We will neglect that dependence in

the case of the setup with the tunneling between two QHE liquids; more specifically,

we will assume that both liquids are kept at a constant charge density and the

tunneling between them is weak. In the case of the tunneling between a QHE liquid

and a quantum wire we will assume that the charge density is kept constant in 2DEG

but can be controlled by the gate voltage in the one-dimensional wire. The Fermi-

momentum kF in the quantum wire depends on the charge density and any change

of kF results in an equal change of all ∆kn. Thus, we will assume a setup with

two 2DEG in the discussion of the voltage dependence of the tunneling current at

fixed ∆kn. The setup with a quantum wire will be assumed in the discussion of the

dependence of the current on kF at a fixed low voltage. In all cases we will assume

that the temperature is low.

In our calculations we will use the Luttinger liquid model for the edge states [36].

It assumes a linear spectrum for each mode and neglects tunneling between different

82

ν = 5/2C1 C2

V

Figure 3.3: A bar geometry that can be used to detect the non-equilibrated anti-Pfaffian state.Solid lines denote Integer QHE edge modes, the dashed lines denote fractional QHE charged modesand dotted lines denote Majorana modes. Arrows show mode propagation directions.

modes on the same edge. These assumptions are justified in the regime of low

energy and momentum. Thus, we expect that the results for the tunneling between

two 2DEG are only qualitatively valid at high voltage.

Our main assumption is that both energy and momentum conserve in each tun-

neling event. This means that we neglect disorder at the edges. This assump-

tion needs a clarification in the case of the disorder-dominated anti-Pfaffian state

because its formation requires edge disorder. We will assume that for that state

only neutral modes couple to disorder and one can neglect disorder effects on the

charged mode. For completeness, we include a discussion of the momentum resolved

tunneling into the non-equilibrated anti-Pfaffian state. However, a much simpler

experiment is sufficient to detect that state. One just needs to measure the conduc-

tance of the 5/2-liquid in the bar geometry illustrated in Fig. 3.3. Indeed, in the

non-equilibrated anti-Pfaffian state, disorder is irrelevant. Each non-equilibrated

edge has three charged Fermi-liquid modes propagating in one direction and another

Luttinger-liquid charged mode (and a neutral mode) propagating in the opposite di-

rection. In the bar geometry, the lower edge carries the current 3e2V/h. The upper

charged mode carries the current e2V/(2h) in the same direction. Hence, the total

current is 7e2V/(2h) and the conductance is 7/2 and not 5/2 conductance quanta.

83

Our discussion assumes an ideal situation with no disorder. In a large system even

weak disorder, irrelevant in the renormalization group sense, might result in edge

equilibration. Nevertheless, if the QHE bar is shorter than the equilibration length

the nature of the state can be probed by the conductance measurement in the bar

geometry.

Before presenting the calculations we will discuss a qualitative picture. Unless

otherwise specified we consider ∆kn > 0. As seen from the calculations in the

following section, the particle-hole symmetry for Luttinger liquids implies that the

tunneling current at negative ∆kn can be found from the relation Itun(V,∆k) =

−Itun(−V,−∆k). We assume that tunneling is weak and hence only single electron

tunneling matters. One can imagine two types of electron operators on the edge: one

type of operators simply creates an electron in one of the integer or fractional chan-

nels. The second type of operators creates and destroys electrons in different edge

channels of the same edge. Generally, operators of the second type are less relevant

than operators of the first type and we will neglect them (see, however, a discussion

in Appendix B for the case of reconstructed integer edge channels). An exception is

the K = 8 state. Only electrons pairs can tunnel into the fractional K = 8 edge.

As we will see in section 3.6, the most relevant single-electron operator transfers

two electron charges into the fractional edge and removes one electron charge from

a co-propagating integer edge. For simplicity of our qualitative discussion, in this

section we will disregard that operator and concentrate instead on the two-electron

tunneling operator into the fractional edge. Such operator is most relevant in the

setup Fig 3.2.

We will use another simplifying assumption in this section: we will neglect inter-

action between different integer and fractional modes. This assumption is not crucial

as discussed in section 3.5 and we make it solely for simplicity. We will find the total

84

number of singularities both for strongly and weakly interacting edges. At the same

time, the current can be found analytically in the case of weak interactions, Sec. 3.6.

At the lower edge there are two edge modes for spin-up and -down electrons.

At the upper edge there are two spin-up and -down integer modes and one or more

modes corresponding to the ν = 1/2 edge. Spin is conserved during the tunneling

process. Thus, we have three contributions to the tunneling current: (A) tunneling

between the upper spin-down fractional edge modes and the lower spin-down integer

edge mode; (B) tunneling between the upper spin-down integer edge mode and the

lower spin-down integer edge mode; (C) tunneling between the upper spin-up integer

edge mode and the lower spin-up integer edge mode. We use only the lowest order

perturbation approximation so these contributions are independent. Thus, the total

tunneling current is Itun = IAtun + IBtun + ICtun. Contributions (B) and (C) are similar

since the Zeeman energy is small compared to the Coulomb interaction under typical

magnetic fields. Thus, we will only consider spin-down electrons below.

All edge modes are chiral Luttinger liquids with the spectra of the form E =

±vα(k−kFα), where ±vα is the edge mode velocity, the sign reflects the propagation

direction. We will first consider case (B) (case (C) is identical), tunneling between

two integer Fermi-liquid edge modes. Denote the upper edge velocity as v1 and the

lower edge velocity as v2. If an electron of momentum k from the upper edge tunnels

into the lower edge or vice verse, energy and momentum conservation gives

v1(k − kF1)− ω = −v2(k − kF2), (3.8)

where ω = −eV/~ (in the rest of this paper, we will refer to both ω and V as the

applied voltage). The tunneling happens only when (k− kF1)(k− kF2) < 0, i.e., one

of the two states is occupied and the other is not. Eq. (3.8) is easy to solve directly

85

(a)

k

E/~

slope = v1slope = −v2

∆k

ω > 0

ω < 0

(b)

k

E/~

slope = v3slope = −v2

slope = vλ

k

k0

∆k

ω > 0

ω < 0

Figure 3.4: Illustration of the graphical method. (a) Tunneling between two integer QHE modes.The left solid line represents the electron spectrum at the upper edge at zero voltage. The right solidline represents the spectrum at the lower edge. The dashed lines represent the electron spectra at theupper edge at different voltages. Black dots represent occupied states. The momentum mismatchbetween two edges ∆k > 0. (b) Tunneling between an integer QHE edge and a Pfaffian edge. Theright line represents the spectrum of the integer edge. The left line shows the spectrum of thecharged boson mode at the Pfaffian edge. The unevenly dashed lines (λ lines) represent Majoranafermions. The figure illustrates a tunneling event in which an electron with the momentum k0tunnels into the Pfaffian edge and creates a boson with the momentum k and a Majorana fermionwith the momentum k0 − k.

86

but a graphical approach is more transparent. Fig. 3.4(a) shows the spectra in the

energy-momentum space, where the left line describes the upper edge mode and the

right line describes the lower edge mode, and the intersection point represents the

solution of Eq. (3.8). The black dots represent occupied states. We see that when

ω = 0, both states at the intersection point are unoccupied, therefore no tunneling

happens. When ω increases, the left line moves down. For a small ω, there is still

no tunneling. After ω reaches the value of v1∆k = v1(kF2 − kF1) and the state from

the right line at the intersection point becomes occupied, an electron from the lower

edge can tunnel into the upper edge. This results in a positive contribution to the

tunneling current. Since the tunneling happens only at the intersection point and

the tunneling density of states (TDOS) is a constant in Fermi liquids, the current will

remain constant for ω > v1∆k. For a negative ω, the situation is similar. Before ω

reaches the value −v2∆k, i.e., |ω| < v2∆k, no tunneling happens. When |ω| > v2∆k,

an electron from the upper edge can tunnel into the lower edge and a negative voltage-

independent tunneling current results. Thus, the IBtun −V characteristics is a sum of

two step functions, with two jumps at ω = −v2∆k and v1∆k. The positions of the

two jumps provide the information about the edge mode velocities. The differential

conductance GBtun is simply a combination of two δ-functions of ω.

This graphical method can also be used to analyze case (A). Consider the K = 8

state in the setup Fig. 3.2 as the simplest example. For the K = 8 state, only

electron pairs can tunnel through the junction since single electrons are gapped.

This does not create much difference for the further analysis. It is convenient to

use bosonization language for the description of the K = 8 edge. All elementary

excitations are bosons with positive momenta k − kF2 > 0 and linear spectrum.

Thus, the relation between the momentum and energy remains the same as in the

Fermi liquid case. Hence, the IAtun − V curve has singularities at ω = −v2∆k and

87

ω = v3∆k, where v3 is the velocity at the K = 8 fractional edge. However, the

current is no longer a constant above the thresholds because of a different TDOS.

We will see below that the current exhibits universal power-law dependence on the

voltage bias near the thresholds.

In the Pfaffian state, case (A) involves three modes: a charged boson mode φ3 and

a neutral Majorana fermion mode λ from the upper edge, and the Fermi-liquid mode

from the lower edge. They have velocities v3, vλ and v2 respectively. Any tunneling

event involves creation of a Majorana fermion. The spectrum of the Majorana mode

is linear: E = vλk > 0. The total energy and momentum of the three modes should

be conserved. As usual, we denote the momentum mismatch between the upper

and lower edges as ∆k. Fig. 3.4(b) demonstrates the graphical approach for the

Pfaffian state. The left line represents the spectrum of the charged boson at the

upper edge and the right line describes the spectrum of the lower edge. Consider a

tunneling process such that an electron from the lower edge tunnels into the upper

edge. This may happen at a positive applied voltage. In this process the electron

emits a Majorana fermion and creates excitations of the charged boson mode at the

upper edge. The energy and momentum of the electron are the sums of the energies

and momenta of the charged boson and Majorana modes. The unevenly dashed lines

of slope vλ in Fig. 3.4(b) represent the Majorana fermion. We will call them λ-lines.

Different λ-lines start at different occupied states on the right line and correspond to

different momenta of the electron at the lower edge. One can visualize the tunneling

process in the following way: an electron with the momentum k0 from the right line

slides along the λ-line (emitting a Majorana fermion with the momentum k0 − k)

and reaches the left line at k > kF3 (otherwise the tunneling is not possible since

the momentum change (k − kF3) of the Bose mode must be positive). Both energy

and momentum are conserved in such picture. Because the Majorana fermion has

88

a positive momentum the λ-line points downward and leftward. When ω is positive

and small enough, all the states at the intersections of the left line with the λ-

lines have k < kF3, thus, no tunneling happens. At ω = vλ∆k, the highest λ-line

intersects the left line at k = kF3, so the tunneling becomes possible and contributes

a positive current. Thus ω = vλ∆k is the positive threshold voltage. When ω reaches

v3∆k, the intersection point of the right and left lines corresponds to k > kF3 (an

‘empty state’) at the upper edge and a filled state at the lower edge. The tunneling

process involving those two states and a zero-momentum Majorana fermion becomes

possible. This results in another singularity in the IAtun −V curve. For negative ω, it

is expected that a Majorana fermion and an excitation of the charged boson mode

combine into an electron and tunnel into the lower edge. The same analysis as above

shows that there is no current when ω is negative and small. When ω = −v2∆k, the

tunneling process involving a zero-momentum Majorana fermion becomes possible.

Thus, ω = −v2∆k is the negative threshold voltage in the IAtun − V curve. We see

three singularities in the tunneling current in agreement with the presence of three

modes.

For all other proposed fractional states, the graphical method also works but

becomes more complicated, so we will not discuss them in detail here. The above

discussion, based only on the conservation of energy and momentum, confirms that

singularities appear in the IAtun − V characteristics and they are closely related to

the number and nature of the edge modes. In the following section, we discuss the

calculations based on the chiral Luttinger liquid theory.

The calculations below involve the velocities of the charged and neutral edge

modes. We generally expect charged modes to be faster. Indeed, in the chiral

Luttinger liquid theory the kinetic energy and the Coulomb interaction enter in the

same form, quadratic in the Bose-fields. Since the Coulomb contribution exists only

89

for the charged mode, it is expected to have a greater velocity.

3.4 Calculation of the current

We now calculate the tunneling current. In this section we derive a general expres-

sion, valid for all models. In the next two sections it will be applied to the six models

discussed above.

As mentioned above, to the lowest order of the perturbation theory the tunneling

current can be separated into three independent parts, Itun = IAtun + IBtun + ICtun. The

calculation of IB and IC is essentially the same. So in the following, we will only

consider IAtun and IBtun.

We will use below the bosonization language which can be conveniently applied

to all modes except Majorana fermions. Thus, we will not explicitly discuss Majo-

rana modes in this section. However, all results can be extended to the situation

involving Majorana fermions without any difficulty. Indeed, in the lowest order of the

perturbation theory only the two-point correlation function of the Majorana fermion

operators is needed. It is the same as for ordinary fermions and the case of ordinary

fermions can be easily treated with bosonization.

We consider the Lagrangian density [36]

L =Lfrac(t, x)−1

4π∂xφ1(∂t + v1∂x)φ1

− 1

4π∂xφ2(−∂t + v2∂x)φ2 −Htun, (3.9)

90

with the tunneling Hamiltonian density

Htun =∑

n

γnAΨ†2(x)Ψ

nfrac(x) + γBΨ

†2(x)Ψ1(x) + H.c.., (3.10)

where Ψ1 is the electron operator for the integer QHE mode of the upper edge, Ψnfrac

annihilate electrons at the 1/2-edge, Ψ2 is the electron operator at the lower edge;

Bose-fields φj(x) (j = 1, 2) represent the right/left-moving integer edge modes of

velocities vj at the upper/lower QHE liquid. The Bose-fields satisfy the commutation

relation [φi(x), φj(x′)] = iσjπδijsign(x − x′), with σ1 = +1 and σ2 = −1. The

Lagrangian density for the fractional QHE edge Lfrac depends on the state and will

be discussed in detail later. Eq. (3.9) does not include interaction between the

inter and fractional QHE modes. Our analysis can be extended to include such

interactions (section 3.5). However, a full analytical calculation of the I − V curve

(Sec. 3.6) is only possible, if it is legitimate to neglect such interactions.

We assume that the line junction is infinitely long and the system is spatially

uniform. As discussed above this restricts possible coordinate dependence of the

tunneling amplitudes. It will be convenient for us to assume that γnA and γB are

independent of the coordinate and absorb the factors exp(−i∆knx) into the electron

creation and annihilation operators. The tunneling amplitudes are also assumed

to be independent of the applied voltage V . In the tunneling Hamiltonian density

(3.10), Ψj(x) is the corresponding electron operator of the integer mode φj(x) with

Ψj = eσj iφj+ikF,jx, where kF,j represents the Fermi momentum. The corresponding

electron density ρj = (∂xφj + kF,j)/2π. In the fractional edge, there may be several

relevant electron operators Ψnfrac. In our calculations, only the most relevant elec-

tron operators will be considered, in the sense of the renormalization group theory.

Generally, tunneling between integer QHE modes is more relevant than tunneling

into the fractional ν = 1/2 edge mode. However, as is clear from the above dis-

91

cussion, for weak interactions between integer and fractional modes, the tunneling

conductance GBtun(ω) is just a combination of two δ-functions. Therefore, the shape

of the voltage dependence of the total differential conductance Gtun is determined

by GAtun(ω). Thus, we focus on tunneling into the fractional channel. In the case

of strong interaction, the analysis of the present section has to be slightly modified

(Sec. 3.5).

Since the upper and lower edges have different chemical potentials, it is conve-

nient to switch to the interaction representation with Ψnfrac → Ψn

frace−iµ1t/~, Ψ1 →

Ψ1e−iµ1t/~ and Ψ2 → Ψ2e

−iµ2t/~, where µ1 = eV and µ2 = 0. This introduces time-

dependence into the tunneling operators (cf. Ref. [109]). The electron operator

Ψnfrac(x) can be written in a bosonized form according to the chiral Luttinger liquid

theory, Ψnfrac(x) = ei

∑I (lIφI+lIkF,Ix), or λ(x)ei

∑I (lIφI+lIkF,Ix), if a Majorana mode λ(x)

exists.

In order to pay special attention to momentum mismatches, we define

Ψnfrac(x) ≡ Ψn

frac(x)ei∑

I lIkF,Ix. (3.11)

Similar definitions are also made for the integer QHE modes, Ψj(x) = eikF,jxΨj(x).

Thus, the density of the tunneling Hamiltonian can be rewritten in the interaction

picture as

Htun =∑

n

γnAeiωt−i∆kn

2fx؆

2(x)Ψnfrac(x)

+ γBeiωt−i∆k21xΨ†

2(x)Ψ1(x) + H.c., (3.12)

where ∆kn2f = kF,2 −∑

I lIkF,I , ∆k21 = kF,2 − kF,1 and ω = (µ2 − µ1)/~ = −eV/~.

It is worth to mention that in the K = 8 state, electron pairs and not electrons

92

tunnel through the junction, thus in the first term of Eq. (3.12) ω should be doubled

because the pair charge doubles, and Ψnfrac and Ψ2 should be understood as bosonic

operators that annihilate electron pairs.

The operator for the tunneling current density is given by

j(t, x) = edρ2dt

=e

i~[ρ2(x), Htun], (3.13)

where ρ2(x) is the electron density of the lower edge, and Htun =∫

dxHtun(x) is the

tunneling Hamiltonian. Expanding the commutator in Eq. (3.13) we get

j(t, x) =e

i~∑

n

γnAeiωt−i∆kn

2fx؆

2(x)Ψnfrac(x)

+ γBeiωt−i∆k21xΨ†

2(x)Ψ1(x)− H.c.. (3.14)

The current can now be calculated with the Keldysh technique. We assume that

the tunneling was zero at t = −∞ and then gradually turned on. Both edges were

in their ground states at t = −∞. At zero temperature, the current is given by the

expression

Itun(t) = 〈0|S(−∞, t)IS(t,−∞)|0〉, (3.15)

where 〈0| is the initial state, the operator I =∫

dxj(t, x) and

S(t,−∞) = T exp(−i∫ t

−∞Hdt′/~)

is the evolution operator. To the lowest order in the tunneling amplitudes, the

tunneling current reduces to

Itun(t) = − i

~

∫

dxdx′∫ t

−∞dt′〈0|[j(t, x),Htun(t

′, x′)]|0〉. (3.16)

93

After a substitution of Eqs. (3.12) and (3.14) into Eq. (3.16), we can compute the

tunneling current since we know all the electron correlation functions from the chiral

Luttinger liquid theory.

In the lowest order perturbation theory the current does not contain any cross-

terms, proportional to γiA×(γjA)∗ with i 6= j, or γiA×γ∗B . There are only contributions

proportional to |γiA|2 or |γB|2. Thus, without loss of generality we can assume that

only one of the tunneling amplitudes is nonzero and write

jαβ(t, x) =e

i~(γeiωt−i∆kxΨ†

α(t, x)Ψβ(t, x)− H.c.). (3.17)

The operators Ψα and Ψβ represent electron operators on two sides of the junc-

tion. For brevity, we have dropped subscripts of the momentum mismatch ∆k and

tunneling amplitude γ. Using Eq. (3.16), the tunneling current can be expressed as

Iαβtun =− e|γ|2~2

∫

dxdx′∫ t

−∞dt′(eiω∆t−i∆k∆x − c.c.)

× [Gαβ(∆t,∆x)−Gαβ(−∆t,−∆x)] (3.18)

with ∆t = t− t′, ∆x = x− x′ and

Gαβ(∆t,∆x)

= 〈0|Ψ†α(t, x)Ψα(t

′, x′)Ψβ(t, x)Ψ†β(t

′, x′)|0〉, (3.19)

and we used the fact that 〈0|Ψ†α/β(t, x)Ψα/β(t

′, x′)|0〉 = 〈0|Ψα/β(t, x)Ψ†α/β(t

′, x′)|0〉

and the translational invariance for chiral Luttinger liquids. Eq. (3.18) can be sim-

plified as

Iαβtun = −Le|γ|2

~2

∫

dydτ(eiωτ−i∆ky − c.c.)Gαβ(τ, y), (3.20)

94

where L is the length of the junction.

Let there be N right-moving andM left-moving modes in total at both edges. In

the chiral Luttinger liquid theory a general expression for the correlation function is

Gαβ(τ, y) = l2B

N∏

i=1

(

τcδ + i(τ − y/vRi)

)gRi

×M∏

i=1

(

τcδ + i(τ + y/vLi)

)gLi

, (3.21)

where vRi and vLi denote the velocities of the ith right- and left-moving modes,

τc is the ultraviolet cutoff and lB is the magnetic length. This expression relies

on the fact that the quadratic Luttinger liquid action can always be diagonalized

and represented as the sum of the actions of non-interacting chiral modes. All

the velocities vRi/vLi and scaling exponents gRi/gLi depend on the details of the

Hamiltonian and this dependence is discussed separately for each state in Sec. 3.6.

We choose the convention that vR1 < vR2 < · · · < vRN and vL1 < vL2 < · · · < vLM .

The scaling dimension of the tunneling operator Ψ†α(t, x)Ψβ(t, x) is g = 1/2(

∑

i gRi+∑

i gLi).

Using the Fourier transformation

1

(δ + it)g=

∫ +∞

−∞dω e−iωt |ω|g−1

Γ(g)θ(ω), (3.22)

95

ωRi

ωLi

Σ Ω

ωRi

ωLi

Σ Ω

(a): ω < vRi∆k (b): ω > vRi∆k

Figure 3.5: A 3-dimensional illustration of the integration volume in the integral (3.24). Theintegral (3.24) is taken over the volume under the shaded surface in the positive orthant. In panel(a), ω < vRi∆k and the ωRi axis intersects superplane Σ closer to the origin than the plane Ω. Inpanel (b) ω > vRi∆k and the order of the intersection points reverses.

we integrate out τ and y in Eq. (3.20). Then we obtain

Iαβtun = −4π2Le|γ|2~2

∫

[dωRidωLi]

×

δ(ω −∑

ωRi −∑

ωLi)δ(∆k −∑ ωRi

vRi+∑ ωLi

vLi)

− (ω ↔ −ω,∆k ↔ −∆k)

×∏

|ωRi|gRi−1 θ(ωRi)

Γ(gRi)

∏

|ωLi|gLi−1θ(ωLi)

Γ(gLi), (3.23)

where we absorbed the cutoff τc and the magnetic length lB into the tunneling

amplitude γ for brevity. The two δ-functions represent the energy and momentum

conservation. Integrating out ωR1 and ωL1 by using the two δ-functions we obtain

96

our general expression for the tunneling current,

Iαβtun =A

∫ ∞

0

[dωRidωLi]i≥2

∏

i≥2

|ωRi|gRi−1∏

i≥2

|ωLi|gLi−1

× | ωvR1

−∆k −∑

i≥2

ωRi

vRRi1

−∑

i≥2

ωLi

vLRi1

|gL1−1

× θ(ω

vR1−∆k −

∑

i≥2

ωRi

vRRi1

−∑

i≥2

ωLi

vLRi1

)

× | ωvL1

+∆k −∑

i≥2

ωRi

vRLi1

−∑

i≥2

ωLi

vLLi1

|gR1−1

× θ(ω

vL1+∆k −

∑

i≥2

ωRi

vRLi1

−∑

i≥2

ωLi

vLLi1

)

− (ω ↔ −ω,∆k ↔ −∆k), (3.24)

with

A = −L 4π2e|γ|2~2∏

Γ(gRi)Γ(gLi)(vRL

11 )gR1+gL1−1 (3.25)

vRRi1 =

vRivR1

vRi − vR1, vLLi1 =

vLivL1vLi − vL1

, i ≥ 2, (3.26)

vRLi1 =

vRivL1vRi + vL1

, vLRi1 =vLivR1

vLi + vR1, i ≥ 1. (3.27)

Let us discuss the above expression in general before applying it to the six models.

We first consider ω > 0. In that case only the first term in Eq. (3.24) contributes

to Iαβtun. The integration is taken over the volume in the positive orthant of the

(M+N−2)-dimensional space spanned by ωRi, ωLii≥2 under both of the following

superplanes

Σ :∑

i≥2

ωRi

vRRi1

+∑

i≥2

ωLi

vLRi1

=ω

vR1

−∆k, (3.28)

Ω :∑

i≥2

ωRi

vRLi1

+∑

i≥2

ωLi

vLLi1

=ω

vL1+∆k. (3.29)

97

If ω < vR1∆k then the integration volume is 0 and so is the tunneling current. The

tunneling only appears when ω > vR1∆k, thus, we see that vR1∆k is the positive

threshold voltage. It is easy to see that the asymptotic behavior of the tunneling

current at ω & vR1∆k is

Iαβtun ∼(

ω

vR1−∆k

)

∑Ni=2 gRi+

∑Mi=1 gLi−1

. (3.30)

Now let us consider the ωRi-intercepts of the two superplanes, ΣRi = (ω/vR1−∆k)vRRi1

and ΩRi = (ω/vL1 +∆k)vRLi1 , i ≥ 2. We find that

ΣRi < ΩRi, when ω < vRi∆k;

ΣRi > ΩRi, when ω > vRi∆k. (3.31)

Thus, when ω passes vRi∆k, the shape of the (M +N − 2)-dimensional integration

volume changes, as is illustrated in Fig. 3.5 for the 3D case. This volume change

leads to a singularity in the Itun − V curve. The precise nature of the singularities

depends on the model and will be discussed in the following section. For the ωLi-

intercepts, ΣLi = (ω/vR1 −∆k)vLRi1 is always smaller than ΩLi = (ω/vL1 +∆k)vLLi1 ,

so no extra singularities emerge. Thus, we see that on the positive voltage branch,

the tunneling current has N singularities in one to one correspondence with the

right-moving modes.

Similar behavior of Iαβtun(ω) manifests itself when ω < 0, with singularities at

ω = −vLi∆k. Thus, each mode contributes a singularity.

3.5 The number of singularities

98

Table 3.1: The number of conductance singularities for different models in different setups.

Boundary of ν = 5/2 Fig. 3.1, Fig. 3.1, ν = 1 instead of 2, Fig. 3.2State and 2 strong interaction strong interactionK=8 2 15 8 3331 6 24 15 8Pfaffian 3 18 10 4Edge-reconstructed Pfaffian 10 61 34 13Non-equilibrated anti-Pfaffian 3 18 10 4

99

The analysis of the preceding section allows us to determine the numbers of

the conductance singularities in each model for different setups. Below we consider

the K = 8, 331, Pfaffian, edge-reconstructed Pfaffian and non-equilibrated anti-

Pfaffian states. The special case of the disorder-dominated anti-Pfaffian state will

be considered in section 3.6.6.

We will need the information about the number of channels and most relevant

tunneling operators. This information is discussed in detail in Sec. 3.6. Here we just

summarize relevant facts.

We first consider the edge between ν = 5/2 and ν = 2 states, where only fractional

modes exist. The K = 8 fractional edge contains a single Bose mode. The 331

edge has two bosonic modes. The Pfaffian edge contains a charged boson and a

neutral Majorana fermion. The edge-reconstructed Pfaffain and non-equilibrated

anti-Pfaffian states are characterized by two Bose modes and a Majorana fermion.

The edge between ν = 5/2 and ν = 0 regions has two additional integer QHE

edge modes with opposite spin orientations.

What operators are more relevant depends on the interaction strength as dis-

cussed in the next section (see Sec. 3.6.7). Unless the interaction is very strong, the

relative importance of different tunneling operators is the same as in the absence

of interaction of different edge modes. Below we will assume that the set of most

relevant operators is the same as for non-interacting modes. Since we consider weak

tunneling, only operators which transfer one electron charge will be included. We

will have to consider 2-electron operators for the K = 8 edge between ν = 5/2 and

ν = 2 regions and for the K = 8 state in the setup Fig. 3.2 since single-electron

tunneling is impossible in those cases.

100

Thus, the choice of the most relevant tunneling operator into theK = 8 fractional

edge depends on the setup. For the setup Fig. 3.1, the most relevant operator creates

an electron pair on the fractionalK = 8 edge and removes an electron from an integer

edge channel with the same spin orientation. In the setup Fig. 3.2, the most relevant

operator transfers an electron pair.

In the 331 state there are two most relevant tunneling operators in the fractional

edge. In the bosonization language, both of them are products of exponents of Bose

operators representing two edge channels. The only tunneling operator in the Pfaffian

case is the product of a Bose-operator and a Majorana fermion creation/annihilation

operator. The reconstructed Pfaffian state has three most relevant tunneling opera-

tors. Two of them express via Bose-modes only. The third operator contains also a

Majorana fermion. The most important tunneling operator for the non-equilibrated

anti-Pfaffian state does not depend on the Majorana fermion.

The above list takes into account only operators that transfer charge into frac-

tional edge modes. In the setup Fig. 3.1, two operators for the tunneling of spin-up

and -down electrons to the integer edge modes must be added. Many more tunneling

operators are possible if the integer modes on the edge undergo reconstruction. The

reconstruction effects are discussed in Appendix B.

Each tunneling operator contributes two or more singularities into the total con-

ductance. As is clear from the preceding section, the number of the singularities

coincides with the number of Bose-modes in the expression for the operator. If

the operator contains a Majorana fermion there is an additional singularity. These

conclusions are based on the form of the Green function (3.21). As discussed in

the previous section, the expression (3.21) can be obtained by diagonalizing the

Luttinger liquid Hamiltonian for interacting edge modes. Hence, the number of

101

Bose-modes in the relevant tunneling operator depends on the details of inter-mode

interactions. If all modes interact strongly then after diagonalization each tunneling

operator contains the same number of Bose modes; this number equals the total

number of Bose-channels including all integer QHE channels. If, on the other hand,

the interaction between fractional modes and different integer modes is negligible

then the operators of tunneling into the integer edge modes contain only informa-

tion about the integer edge channels; the tunneling operators into the fractional

modes are independent of the two integer modes on the 5/2 edge.

We are now in the position to count the singularities in different setups. The

results are summarized in Table 3.1.

Let us first consider tunneling from a single spin-down channel (‘spectator’ mode)

into a boundary between ν = 5/2 and ν = 2 states (cf. Ref. [105] for the Pfaffian and

non-equilibrated anti-Pfaffian states). There are only two modes (the K = 8 mode

and the ‘spectator’ mode). Hence, there are 2 singularties. For the 331 state, there

are 3 modes and 2 tunneling operators. The number of the singularities 2×3 = 6. The

Pfaffian state is characterized by three modes and one tunneling operator. There are

3 singularities. The reconstructed Pfaffian state has one Majorana mode, two Bose

modes plus a ‘spectator’ Bose mode. One tunneling operator expresses in terms of all

four modes. The other two tunneling operators do not contain a Majorana operator.

Thus, we find 2 × 3 + 4 = 10 singularities. Finally, the most relevant operator for

the non-equilibrated anti-Pfaffian state does not depend on the Majorana fermion.

The remaining three modes result in 3 singularities.

Let us now turn to the setup Fig. 3.2. We assume strong interaction between all

modes. For the K = 8 state, we get (1 operator) × (3 modes) = 3 singularities; for

the 331 state, we get 2×4 = 8 singularities; for the Pfaffian state, the number of the

102

singularities is 1× 4 = 4; for the reconstructed Pfaffian state we find 2× 4 + 5 = 13

singularities; the non-equilibrated anti-Pfaffian state is characterized by 1 × 4 = 4

singularities.

Next, we consider the setup Fig. 3.1. We first assume that there is no interaction

between integer and fractional modes. With the exception of the K = 8 state the

number of the singularities due to the tunneling into fractional edge channels remains

the same as for the tunneling into the edge between ν = 5/2 and 2. One has, however,

to add 4 more singularities due to the tunneling of spin-up and -down electrons into

two integer edge channels. Tunneling into the K = 8 fractional edge is described by

an operator which expresses in terms of three Bose modes. Thus, the total number

of the singularities for the K = 8 state becomes 3 + 4 = 7.

In the case of strong interaction in the same setup Fig. 3.1, the number of

singularities increases. There are two types of single-electron tunneling operators:

tunneling into integer and fractional QHE modes. The first group includes more

relevant operators [36], cf. Sec. 3.6. There are two operators in that group: one

for spin-up and one for spin-down electrons. Each of them is responsible for N

singularities, where N is the total number of Bose modes (including 2 ‘spectator’

modes on the lower edge). We will call those singularities ‘strong’. Thus, we have

2 × 5 = 10 strong singularities for the K = 8 state; 2 × 6 = 12 strong singularities

for the 331 state; 2 × 5 = 10 strong singularities for the Pfaffian state; 2 × 6 = 12

strong singularities for the edge-reconstructed Pfaffian state and 2 × 6 = 12 strong

singularities for the non-equilibrated anti-Pfaffian state.

Clearly, these numbers alone are not enough to distinguish the states. Additional

information comes from transport singularities due to the next most relevant tunnel-

ing operators. They are responsible for additional ‘weak’ singularities. In the K = 8,

103

331 and non-equilibrated anti-Pfaffian states such operators describe tunneling into

the fractional modes. Those next most relevant operator were discussed above (see

also section 3.6) and do not contain Majorana fermions. Let us find the total number

of ‘weak’ and ‘strong’ singularities. In the K = 8 state we get 10+1× 5 = 15 singu-

larities; in the 331 state the answer is 12 + 2 × 6 = 24; and in the non-equilibrated

anti-Pfaffian state the answer is 12 + 1× 6 = 18.

The situation is more complicated in the Pfaffian and edge-reconstructed Pfaffian

states. Just like in the previous three cases we need to take into account tunneling

into the fractional edge. This adds 1× (5+1) = 6 ‘weak’ singularities in the Pfaffian

case and 2× 6 + 7 = 19 ‘weak’ singularities for the edge reconstructed state. There

are, however, several additional ‘weak’ singularities for both states. They emerge

from tunneling into integer edge channels.

To understand their origin, we need to have a look at the scaling dimensions of

the tunneling operators. Interaction between co-propagating modes has no effect on

scaling dimensions of the operators [36]. Interaction between counter-propagating

modes may change scaling dimensions. Below we will assume that either 1) all Bose

modes are co-propagating or 2) the upper and lower edges in Fig. 3.1a) are counter-

propagating but the interaction between the two edges is weak. Thus, we will use

the same scaling dimensions as for non-interacting modes.

The most relevant operators T0, describing tunneling into integer edge channels,

have scaling dimension 1, Ref. [36]. The next most relevant operators, describing

tunneling into the fractional edge have dimension 2 for both models, Ref. [45]. This

allows us to calculate how the current scales at low voltages V , Ref. [36]. We take

the square of the renormalized amplitude of the tunneling operator at the energy

scale V . The renormalized amplitude is ∼ V 2d, where d is the scaling dimension.

104

Then we divide it by V 2 to reflect the integration over time and coordinate in the

expression for the current Eq. (3.20). The contribution of the most relevant operators

I ∼ V 2×1−2 = V 0 and the contribution of the next most relevant operators I ∼

V 2×2−2 = V 2 in agreement with Section 3.6.

Now let us consider operators which describe the interaction of the Majorana

mode λ and an integer QHE Bose-mode φ. The conservation of the topological

charge excludes operators, linear in λ. Taking into account that λ2 = 1 and that

φ can enter only in the form of a derivative, we find the most relevant interaction

term in the action: Q =∫

dxdtλ∂xλ∂xφ, where x is the coordinate along the edge.

The scaling dimension of the operator Q equals 1. In order to understand the effect

of Q on low-energy transport, let us perform a renormalization group procedure.

It should stop at the energy scale E ∼ eV . At that scale, different contributions

to the current can be obtained from the squares of the renormalized amplitudes of

the contributions to the action describing different tunneling processes (since the

action contains integrations over t and x, we will also need to multiply by V 2 to

reflect rescaling, cf. Ref. [110]). At the scale eV the operator Q is suppressed by the

prefactor c ∼ eV/∆, where ∆ is the energy gap. The prefactor reflects the scaling

dimension of the operator Q. Thus, the renormalized action contains the term cQ.

Similarly, the contribution to the action, proportional to T0, acquires a prefactor,

proportional to 1/V .

The renormalization group flow generates numerous operators. In particular,

the operator T1 = T0λ∂xλ is generated from T0 and Q. As is clear from the above

analysis, it enters the action with the prefactor ∼ c/V ∼ 1. Hence, its contribution

to the current scales as V 2 and has the same order of magnitude as for the operators

describing tunneling into fractional edges. This contribution to the current is singular

whenever eV/~ = −∆kvl, where ∆k is the momentum mismatch between the integer

105

QHE mode and the ‘spectator’ mode and vl denote edge mode speeds. The strong

singularities due to the operator T0 occur at the same voltages. However, T0 does

not contain Majorana fermions and hence T0 does not generate a singularity at

eV/~ = −∆kvM , where vM is the speed of the Majorana fermion. On the other

hand, T1 contains a Majorana fermion and hence is responsible for an additional

‘weak’ singularity at eV/~ = −∆kvM . Since there are two integer edge modes, we

discover two additional ‘weak’ singularities.

The above argument completes our discussion of the Pfaffian state. In the edge-

reconstructed Pfaffian state there is another mechanism for additional ‘weak’ singu-

larities. The quadratic part of the action of the fractional edge channels in that state

is given by Eq. (3.47). Let us consider the following four tunneling operators:

T↑/↓,± = ψd,↑/↓ψ†u,↑/↓λ exp(±iφn), (3.32)

where ψu/d,↑/↓ are annihilation operators for spin-up/down (↑ / ↓) electrons on the

upper (u) and lower (d) edges. λ is the Majorana fermion, φn the bosonic neutral

mode. The operator T describes electron tunneling between lower and upper integer

edge modes. The combination T ′ = λ exp(±iφn) describes charge redistribution

between different fractional modes. As is clear from the expressions under Eq. (3.47),

T ′ is a product of annihilation and creation operators for electrons in fractional

channels. The scaling dimension of the operators T is the same as for the operators

describing tunneling into fractional edge modes. Since we have 4 operators and 7

modes, we get 28 additional ‘weak’ singularities.

The total number of ‘weak’ and ‘strong’ singularities is summarized in Table 3.1.

A very similar analysis applies to the tunneling between ν = 5/2 and ν = 1

106

states. The results are shown in Table 3.1.

We focused above only on the number of the singularities due to Majorana-

fermion and single-Boson excitations at ω = vl∆kn. All ‘strong’ singularities must be

in this class. All singularities due to the tunneling into fractional edge modes must

also be in this class. We were not able to exclude additional ‘weak’ singularities

at ω = ul∆kinteger, where ∆kinteger is the momentum mismatch for integer modes

and ul is the speed of a collective excitation. Such singularities might be found

if one takes into account contributions to the action, cubic in Bose-fields. If such

additional weak singularities are present it will be easy to separate them from the

rest of the singularities. Indeed, the ratios of all vl for bosonic modes can be found

from the positions of ‘strong’ singularities. Comparison with the positions of ‘weak’

singularities allows then extracting the ratios of all momentum mismatches ∆km and

the speed of the Majorana fermion. After that it is straightforward to check if any

singularities due to collective excitations of bosonic modes are present.

The total number of singularities is the same for the Pfaffian and edge-reconstructed

Pfaffian states. However, the number of strong singularities is different for those

models in the setup Fig. 3.1 with strong inter-mode interactions. Thus, the models

can be distinguished just from the number of the singularities in that setup. At

the same time, that number is greater than in other setups and thus requires higher

resolution for its detection. The number of the singularities alone is not enough to

distinguish different models in other setups. One also needs information about the

nature of the singularities (divergence, cusp or discontinuity of the conductance).

The next section discusses the nature of the singularities for the setup Fig. 3.1 with

weak interactions and the setup from Ref. [105].

107

3.6 I-V curves

In this section we study the setup Fig. 3.1 and focus on the regime of weak interaction

with integer QHE modes. More specifically, we neglect interactions of fractional

modes with integer modes (including ‘spectator’ modes on the lower edge) and the

interaction among different integer modes. Our calculations also apply to the setup

Ref. [105], i.e., tunneling into an edge between ν = 2 and ν = 5/2 states. In contrast

to other cases, the I − V can be analytically computed in the regimes, considered

below.

3.6.1 Tunneling into integer edge modes.

Now using the general expression, Eq. (3.24), we discuss the properties of the tun-

neling current Itun and conductance Gtun in detail. First, let us consider the simplest

case, tunneling between two integer edge modes. Following Eq. (3.24), it is easy to

derive that

IBtun =− L4π2e|γB|2v1v2~2(v1 + v2)

× [θ(ω − v1∆k21)− θ(−ω − v2∆k21)]. (3.33)

where v1 and v2 are velocities of the upper and lower edge modes respectively, ∆k21 is

the momentum mismatch between the two modes. As expected from the qualitative

picture, IBtun is indeed a combination of two step functions and so GBtun is just a com-

bination of two δ-functions. The two singularities, positive and negative thresholds,

appear at ω = v1∆k21 and −v2∆k21.

108

-3 -2 -1 0 1 20

2

4

6

(a)

K = 8 state

GA tu

n (

arb

.)

ω/ω 0

x103

-2 -1 0 1 2

0

30

60

90(b)

G

A tun (

arb

.)

∆k/∆k0

Figure 3.6: (a) Voltage dependence of the differential conductance in the K = 8 state at a fixedmomentum mismatch ∆k in the case of tunneling into the edge between the states with ν = 5/2and ν = 2. Voltage is shown in units of ω0 = v2∆k, and the conductance is shown in arbitraryunits. (b) Momentum mismatch dependence of GA

tun at a fixed voltage. ∆k0 = ω/v2. For bothcurves, we set v3/v2 = 0.8.

In the following subsections, we will discuss IAtun and GAtun as functions of both

voltage ω and momentum mismatch ∆k for six proposed fractional QHE states.

3.6.2 K = 8 state

We distinguish two situations: tunneling into an edge between ν = 2 and ν = 5/2

states and tunneling into a 5/2 edge with both integer and fractional modes. We need

to distinguish those regimes since they are characterized by different most relevant

operators, transfering charge into the fractional K = 8 mode.

Boundary between ν = 2 and ν = 5/2 states

In the fractional edge of the K = 8 state [45], there is one right-moving boson mode

φ3 with the Lagrangian density

Lfrac = −2~

π∂xφ3(∂t + v3∂x)φ3. (3.34)

109

Only electron pairs are allowed to tunnel into the edge. The electron pair annihilation

operator is Ψfrac = ei8φ3 , and the charge density ρfrac = e∂xφ3/π. The pair correlation

function is 〈0|Ψ†frac(t, x)Ψfrac(0, 0)|0〉 = 1/[δ+ i(t− x/v3)]

8, i.e., the scaling exponent

g3 = 8. In the integer edge, the operator Ψ2, Eq. (3.10), should also be understood

as the pair annihilation operator with 〈0|Ψ†2(t, x)Ψ2(0, 0)|0〉 = 1/[δ + i(t + x/v2)]

4

and g2 = 4. Substituting the scaling exponents and edge velocities into Eq. (3.24),

we obtain

IAtun = −L 8π2e|γA|2~2Γ(8)Γ(4)

(v2v3v2 + v3

)11(ω

v3−∆k2f )

3

(ω

v2+∆k2f )

7[

θ(ω − v3∆k2f )− θ(−ω − v2∆k2f)]

, (3.35)

Just like in the case of the tunneling current IBtun between two integer QHE edges,

there are two threshold voltages, the positive threshold ω = v3∆k2f and the negative

one ω = −v2∆k2f . However, in contrast to IBtun, the tunneling current IAtun increases

smoothly as the voltage passes the thresholds. At ω & v3∆k2f , the tunneling current

IAtun behaves as ∼ (ω − v3∆k2f)3, and at ω . −v2∆k2f , IAtun ∼ (ω + v2∆k2f)

7. Thus

IAtun follows power laws near the thresholds. The exponents in the scaling laws for the

current near the thresholds provide information about states. However, inter-edge

Coulomb interactions may change these exponents and make them non-universal.

When |ω| ≫ v2∆k2f and v3∆k2f , IAtun will asymptotically behave like ∼ ω10 for

both positive and negative voltages. We plotted the differential conductance GAtun =

∂IAtun/∂ω as a function of ω at fixed ∆k2f , and a function of ∆k2f at fixed ω in

Fig. 3.6.

110

-2 -1 0 1 2

0

5

10

15

GA

,u

tun (

arb

.)

ω/ω 0

(a) 0<g4<1

-1 0 1

0

5

10

15(b) 0<g

4<1

GA

,u

tun (

arb

.)

∆k/∆k0

-2 -1 0 1 2

0

3

6

9

ω/ω 0

(c)1<g4<2

GA

,u

tun (

arb

.)

-1 0 1

0

3

6

9

∆k/∆k0

(d) 1<g4<2

GA

,u

tun (

arb

.)

331 state

-2 -1 0 1 2

0

5

10

15

ω/ω 0

(e) 2<g4<3

GA

,u

tun (

arb

.)

-1 0 1

0

5

10

15

∆k/∆k0

(f) 2<g4<3

GA

,u

tun (

arb

.)

Figure 3.7: Voltage and momentum mismatch dependence of the tunneling differential conduc-tance GA,u

tun in the 331 state; u is either a or b. We have chosen the ratios of the edge velocities

to be v3/v2 = 0.8 and v4/v2 = 1.2. The left three panels show the voltage dependence of GA,utun at

a fixed momentum mismatch ∆k for 3 cases of different scaling exponent ranges: (a) 0 < g4 < 1;(c) 1 < g4 < 2; (e) 2 < g4 < 3; we set g4 = 0.5, 1.5 and 2.5 respectively in the plots. Voltage isshown in units of ω0 = v2∆k. Panels (b), (d) and (f) show the same three cases for the momentum

mismatch dependence of GA,utun at a fixed ω with the momentum expressed in units of ∆k0 = ω/v2.

The differential conductance is shown in arbitrary units.

111

Boundary between ν = 5/2 and ν = 0

The action remains the same, Eq. (3.34). However, an electron tunneling operator

ei8φ3−iφ1+iφ2 is present and is more relevant then the pair tunneling operator ei8φ3+2iφ2 ,

considered above. It transfers only one electron into the 5/2 edge. Two electrons

go into the fractional K = 8 channel and one electron is removed from the spin-up

integer channel on the 5/2 edge.

Our calculations give

I =− L4π2e|γ|2~28!

v12

×

−v823(ω/v2 +∆k)8, ω < −v2∆k

0, −v2∆k < ω < v1∆k

v813(ω/v1 −∆k)8, v1∆k < ω < v3∆k

v823(ω/v2 +∆k)8, ω > v3∆k

(3.36)

where v12 = v1v2/(v1 + v2), v13 = v1v3/|v3 − v1| and v23 = v2v3/(v2 + v3). Here we

assume v3 > v1.

112

When v1 > v3 the tunneling current is

Itun = −L4π2e|γ|2~28!

v823v13

×

−[(ω/v3 −∆k)8

−v812/v823(ω/v1 −∆k)8], ω < −v2∆k

0, −v2∆k < ω < v3∆k

(ω/v3 −∆k)8, v3∆k < ω < v1∆k

(ω/v3 −∆k)8

−v812/v823(ω/v1 −∆k)8, ω > v1∆k

(3.37)

In both cases three singularities are found.

3.6.3 331 state

The 331 state [45] has the edge Lagrangian density

Lfrac =− ~

4π(3∂tφ3∂xφ3 − 2∂tφ3∂xφ4 − 2∂tφ4∂xφ3

+ 4∂tφ4∂xφ4 +∑

m,n=3,4

Vmn∂xφm∂xφn). (3.38)

Both modes φ3 and φ4 are right-moving, and the real symmetric matrix V represents

intra-edge interactions. There are two most relevant electron operators in this model,

Ψafrac = ei3φ3−i2φ4 and Ψb

frac = eiφ3+i2φ4. Before applying Eq. (3.24) to the calculation

of the tunneling current, one needs to compute the correlation functions of Ψafrac and

Ψbfrac. Since the Lagrangian density Lfrac is quadratic, we can rewrite it in terms of

113

two decoupled fields φ3 and φ4, such that

Lfrac = − ~

4π

∑

n=3,4

∂xφn(∂t + vn∂x)∂xφn. (3.39)

φ3 and φ4 are linear combinations of φ3 and φ4, with 〈0|φn(x, t)φn(0, 0)|0〉 = − ln[δ+

i(t + x/vn)], where the velocities are

v3,4 =1

16

(

4V33 + 4V34 + 3V44

∓√

(1 + x2)× |4V33 + 4V34 − V44|)

, (3.40)

and x = 2√2(V44 +2V34)/(4V33 +4V34 − V44) is an interaction parameter. Note that

v3 is smaller than v4. It is easy to prove that both v3 and v4 are positive, so φ3 and φ4

are right-moving. In the limit of strong interaction, (V34)2 → V33V44, v3 approaches

0. The two-point correlation functions of those operators can be expressed as

〈0|Ψu†frac(x, t)Ψ

ufrac(0, 0)|0〉

=1

[δ + i(t− x/v3)]gu3 [δ + i(t− x/v4)]g

u4

, (3.41)

where u = a, b and the scaling exponents

gu3,4 =3

2∓ 1− σu2

√2x

2√1 + x2

sign(4V33 + 4V34 − V44); (3.42)

the sign factors σa = +1, σb = −1. It is worth to notice that the sum of gu3 and gu4

is always 3.

There are two tunneling operators in the action. They are proportional to Ψafrac

and Ψbfrac. These tunneling operators are responsible for two contributions to the

114

current. Based on Eq.(3.41) and Eq. (3.24), both contributions have the form

IA,utun = −L 4π2e|γA|2

~2Γ(g3)Γ(g4)vg424v

g323(

ω

v2+∆k2f )

2

×

B(1, g4, g3), ω > v4∆k2f

B(v34(ω/v3−∆k2f )

v24(ω/v2+∆k2f ), g4, g3), v3∆k2f < ω < v4∆k2f

0, −v2∆k2f < ω < v3∆k2f

−B(1, g4, g3), ω < −v2∆k2f ,

(3.43)

where u = a or b. We omitted the index u in the scaling exponents g3 and g4, in

the tunneling amplitude γA, and in the momentum mismatch ∆k2f in Eq. (3.43).

B(z, g4, g3) is the incomplete Beta function, v23 = v2v3/(v2+v3), v24 = v2v4/(v2+v4)

and v34 = v3v4/(v4 − v3).

Consider any of the two contributions IA,atun or IA,b

tun , Eq. (3.43). We see expected

singularities marked by the edge velocities, with two singularities on the positive

voltage side and one on the negative voltage side. The incomplete Beta function

B(z, g4, g3) has the following asymptotic behaviors

B(z, g4, g3) ∼

zg4 , z ∼ 0

(1− z)g3 + const, z ∼ 1

(3.44)

Thus, when ω & v3∆k2f , the differential conductance GA,utun ∼ (ω/v3 − ∆k2f)

g4−1 is

singular at ω = v3∆k2f , if g4 < 1. Hence, the differential conductance diverges near

the threshold. Similarly, GA,utun is singular at ω = v4∆k2f , if g3 < 1, i.e., g4 > 2.

Hence, the shape of the GA,utun ∼ ω is quite different at different values of g3 and

g4, i.e., different interaction strengths x. Fig. 3.7 shows the dependence of GA,utun

on ω and ∆k2f in 3 different cases: g4 < 1, 1 < g4 < 2 and g4 > 2. The total

differential conductance GAtun = GA,a

tun +GA,btun has two sets of singularities originating

115

-2 -1 0 1 2

0

1

2

3

4

(a)

Pfaffian state

GA tu

n (

arb

.)

ω/ω 0 -2 -1 0 1 2

0

1

2

3

4

(b)

GA tu

n (

arb

.)

∆k/∆k0

Figure 3.8: (a) Voltage dependence of the tunneling differential conductance GAtun in the Pfaffian

state. The reference voltage ω0 = v2∆k. (b) Momentum mismatch dependence of GAtun in the

Pfaffian state. The reference momentum ∆k0 = ω/v2. We set the edge velocity ratios, v3/v2 = 1.2and vλ/v2 = 0.5. GA

tun is shown in arbitrary units.

from the two individual contributions to the current. The shape of the curve of

GAtun(ω) depends on the relative values of γaA v.s. γbA, ∆k

a2f v.s. ∆kb2f , and g

a4 v.s. gb4.

Thus, momentum-resolved tunneling allows one to extract considerable information

about the details of the edge theory.

3.6.4 Pfaffian state

The Pfaffian state has the edge Lagrangian density [16]

Lfrac = −2~

4π∂xφ3(∂t + v3∂x)φ3 + iλ(∂t + vλ∂x)λ (3.45)

where φ3 is the right-moving charged boson mode and λ is the neutral Majorana

fermion mode. The most relevant electron operator is Ψfrac = λ exp(i2φ3). Its

correlation function G = 1/[(δ+ i(t− x/v3))2(δ+ i(t− x/vλ))] equals the product of

the correlation function of the Majorana fermion and the correlation function of the

exponent of the Bose-field. The velocity of the charged mode exceeds the Majorana

fermion velocity, vλ < v3. A straightforward application of the results of the previous

116

section yields the tunneling current

IAtun = −L2π2e|γA|2~2

v2λ

×

v223(ω/v2 +∆k2f)2, ω > v3∆k2f

v23λ(ω/vλ −∆k2f )2, vλ∆k2f < ω < v3∆k2f

0, −v2∆k2f < ω < vλ∆k2f

−v223(ω/v2 +∆k2f )2, ω < −v2∆k2f

(3.46)

where v2λ = vλv2/(v2+vλ), v23 = v2v3/(v3+v2) and v3λ = v3vλ/(v3−vλ). Singularities

appear again, two of them on the positive voltage side and one on the negative voltage

side, quite similar to the results for the 331 state. However, the Pfaffian state can be

distinguished from the 331 state by a different total number of singularities (Table

3.2) and the appearance of a discontinuity for GAtun at ω = v3∆k (see Fig. 3.8). On

the negative voltage side, GAtun behaves in the same way as in the 331 state, i.e., it

is a linear function of ω.

3.6.5 Reconstructed Pfaffian state

The reconstructed Pfaffian state [45] has the Lagrangian density

Lfrac =− ~

4π[2∂xφc(∂t + vc∂x)φc + ∂xφn(∂t + vn∂x)φn

+ 2vnc∂φcφn] + iλ(∂t − vλ∂x)λ, (3.47)

where φc is a charged mode and φn is a neutral mode. There are three most relevant

electron operators Ψ±frac = exp(i2φc ± iφn) and Ψλ

frac = λ exp(i2φc). Thus, we need

to consider three tunneling operators, proportional to these three electron operators.

117

As discussed in the previous section they generate three independent contributions to

the tunneling current IAtun. We first discuss the current contributions which originate

from the tunneling terms containing Ψ±frac. For these two contributions, the situation

is quite similar to the 331 state because the Majorana fermion does not enter the

operators Ψ±frac. We diagonalize the bosonic part of the effective action (3.47) into the

form of Eq. (3.39). This requires a transformation from the original fields φc, φn

to two free fields φ3, φ4 with velocities v3, v4 respectively. Then the two-point

correlation function 〈0|Ψ±†frac(x, t)Ψ

±frac(, 0, 0)|0〉 can be calculated as we did for 331

state. With Eq. (3.24) we then obtain the same form of the tunneling current IA,±tun

as in Eq. (3.43), but with different tunneling amplitudes, momentum mismatches,

edge velocities and scaling exponents. The edge velocities are

v3,4 =1

2

(

vc + vn ∓ (vc − vn)√1 + 2x2

)

, (3.48)

and scaling exponents are

gσ3,4 =3

2∓ 1 + 4σx

2√1 + 2x2

, (3.49)

where σ = +1 for the case of Ψ+frac and σ = −1 for Ψ−

frac; the interaction parameter

x = vnc/(vc − vn). It is assumed that (vc − vn) is positive. Indeed, we expect the

charged mode to be faster than the neutral mode. Thus, for repulsive interactions

x is always positive. Similar to the 331 state, different values of x give significantly

different shapes of the GA,±tun curve, e.g., divergence may appear for certain values of

x. All three cases discussed in the subsection on the 331 state could also emerge in

the edge-reconstructed Pfaffian state.

Now let us turn to the tunneling operator, proportional to Ψλfrac. In this case

all four modes participate in the tunneling process. The correlation function of the

field Ψλfrac is the product of the correlation function of two Majorana fermions and

118

the Bose part. The correlation function for Majorana fermions is the same as for

ordinary fermions, 1/[δ + i(t + x/vλ)]. The Bose part has the same structure as in

Eq. (3.41) with the scaling exponents

gλ3,4 = 1∓ 1√1 + 2x2

, (3.50)

where different signs correspond to indices 3 and 4. Again, by using Eq. (3.24) we

obtain the following contribution to the tunneling current:

IA,λtun = −L 4π2e|γλA|2

~2Γ(g3 + 1)Γ(g4)v2λ sign(ω)

×[

vg33λvg44λ(

ω

vλ+∆kλ2f )

2B(f(ω), g4, g3 + 1)

− vg323vg424(

ω

v2+∆kλ2f )

2B(g(ω), g4, g3 + 1)]

, (3.51)

where

f(ω) =

(ω/v3−∆kλ2f

)v34

(ω/vλ+∆kλ2f

)v4λ, v3 <

ω∆kλ

2f

< v4

1, ω∆kλ

2f

< −vλ or > v4

0, −vλ < ω∆kλ

2f

< v3

(3.52)

and

g(ω) =

(ω/v3−∆kλ2f

)v34

(ω/v2+∆kλ2f

)v24, v3 <

ω∆kλ

2f

< v4

1, ω∆kλ

2f

< −v2 or > v4

0, −v2 < ω∆kλ

2f

< v3

(3.53)

The dependence of GA,λtun on the voltage ω and momentum mismatch ∆kλ2f is illus-

trated in Fig. 3.9. There are no divergencies for any gλ4 . All singularities appear

as voltage thresholds or discontinuities of the derivative of Gλtun(ω). The Majorana

119

-2 -1 0 1 2

0.0

0.1

0.2

0.3

0.4

(a)

Reconstructed Pfaffian state

G

A,λ

tun (

arb

.)

ω/ω 0 -3 -2 -1 0 1 2

0.0

0.1

0.2

(b)

G

A, λ

tun (

arb

.)

∆k/∆k0

Figure 3.9: The differential conductance GA,λtun in the edge-reconstructed Pfaffian state. Panels

(a) and (b) show the voltage and momentum mismatch dependence of GA,λtun (in arbitrary units)

respectively. The reference voltage ω0 = v2∆k and the reference momentum mismatch ∆k0 = ω/v2.We have set vλ/v2 = 0.5, v3/v2 = 0.8, v4/v2 = 1.2 and the scaling exponent g4 = 1.5

fermion mode is responsible for the negative voltage threshold (we assume that the

Majorana is slower than the integer QHE mode at the opposite side of the junction).

Thus, in the edge reconstructed Pfaffian state, three sets of singularities can be

observed. Each set corresponds to one of the three most relevant electron operators.

One set contains more singularities than the other two. That extra singularity is due

to the neutral Majorana fermion mode.

3.6.6 Disorder-dominated anti-Pfaffian state

The very name of this state shows that the momentum-resolved tunneling can only

have limited utility in this case. Indeed, momentum conservation assumes that

disorder can be neglected and this assumption fails for the state under consideration

[12,13]. In the disorder-dominated anti-Pfaffian state, the amplitudes of the electron

tunneling operators are expected to be random. Thus, one expects that interference

between different tunneling sites is irrelevant for the total tunneling current since

the disorder average of the product of two tunneling amplitudes from two different

120

-2 -1 0 1 2

0

3

6

9(a) v

2>v

3, 0<g

3+g

4<2

GA tu

n (

arb

.)

ω/ω 0 -2 -1 0 1

0

3

6

9(b) v

2>v

3, 0<g

3+g

4<2

GA tu

n (

arb

.)

∆k/∆k0

-2 -1 0 1 2

0

1

2

3

ω/ω 0

(c) v2>v

3, g

3+g

4>2

GA tu

n (

arb

.)

-2 -1 0 1

0

1

2

3

∆k/∆k0

(d) v2>v

3, g

3+g

4>2

GA tu

n (

arb

.)

Non-equilibrated Anti-Pfaffian state

-3 -2 -1 0 1 2 3

0

3

6

9

ω/ω 0

(e) v2<v

3, 0<g

3+g

4<2

GA tu

n (

arb

.)

-1 0 1

0

3

6

9

∆k/∆k0

(f) v2<v

3, 0<g

3+g

4<2

GA tu

n (

arb

.)

-3 -2 -1 0 1 2 3

0.0

0.5

1.0

1.5

ω/ω 0

(g) v2<v

3, g

3+g

4>2

GA tu

n (

arb

.)

-1 0 1

0.0

0.5

1.0

1.5

∆k/∆k0

(h) v2<v

3, g

3+g

4>2

GA tu

n (

arb

.)

Figure 3.10: Differential conductance GAtun in the non-equilibrated anti-Pfaffian edge state. All

left panels show the voltage dependence of GAtun and right panels show the momentum mismatch

dependence of GAtun, at different choices of v3/v2 and g3+g4. In the top four panels, we have chosen

v3/v2 = 0.7, and in the bottom four panels v3/v2 = 1.5. v4/v2 = 1.2 for all cases. In panels (a),(b), (e) and (f), illustrating the 0 < g3 + g4 < 2 cases, we set g3 + g4 = 1.5. In panels (c), (d), (g)and (h), illustrating the g3 + g4 > 2 cases, we set g3 + g4 = 2.5. The reference voltage ω0 = v2∆kand the reference momentum mismatch ∆k0 = ω/v2. G

Atun is shown in arbitrary units.

121

points is zero. Hence, the leading contribution to the current is the same as for

the tunneling through a single quantum point contact. Nevertheless, momentum

resolved tunneling might be possible for electron pairs. This happens, if disorder

only couples to neutral modes and does not affect the charged mode. As we see

below, the momentum resolved tunneling current of pairs is the same as for the

K = 8 state (Sec. 3.6.2).

In the disorder-dominated anti-Pfaffian edge state, there are 3 left-moving SO(3)-

symmetric Majorana modes and one right-moving charged mode, with the Lagrangian

density [12, 13]

Lfrac = −2~

4π∂xφc(∂t + vc∂x)φc + i

3∑

n

[λn(∂t − vλ∂x)λn]. (3.54)

There are three electron operators corresponding to the three Majorana fermions,

Ψnfrac = λne

iφc , n = 1, 2, 3. Their products yield pair operators. We focus on the

pair operator exp(2iφc) which contains no information about neutral modes. One

can easily verify that its correlation function is the same as the correlation function

of the pair operator in the K = 8 state. Hence, all results can be taken without

modifications from our discussion of the K = 8 state. Certainly, the total tunneling

current includes also a single-electron part. One may expect that it is greater than

the momentum-resolved contribution due to the pair tunneling since the tunneling

amplitude is greater for single electrons than for pairs.

122

3.6.7 Non-equilibrated anti-Pfaffian state

The non-equilibrated anti-Pfaffian edge has the Lagrangian density [12]

Lfrac =− ~

4π[∂xφc1(∂t + vc1∂x)φc1

+ 2∂xφc2(−∂t + vc2∂x)φc2 + 2v12∂xφc1∂xφc2]

+ iλ(∂t − vλ∂x)λ. (3.55)

Again the action can be rewritten in terms of two linear combinations of the Bose

fields φc1 and φc2: a free left-moving mode φ3 and a right-moving mode φ4 with

velocities v3 and v4 respectively. From the renormalization group, we find that

the most relevant electron operators depend on the interaction strength parameter

x = v12/(vc1 + vc2). Below we will only consider x < 2/3. The action (3.55) is only

stable for x < 1/√2 and hence we ignore a small region 2/3 < x < 1/

√2 in the

parameter space. For x < 2/3, the most relevant electron operator is Ψfrac = eiφc1 .

The expression for the tunneling current IAtun and, in particular, the asymptotic

behavior near singularities depends on the relative values of v2 and v3, the velocities

123

of the two left-moving modes. If v2 > v3 we obtain the following tunneling current

IAtun =− L4π2e|γA|2

~2Γ(g3)Γ(g4)vg3+g4−134 sign(ω)

×

v24g3

∣

∣

∣

ωv4

−∆k2f

∣

∣

∣

g3∣

∣

∣

ωv3

+∆k2f

∣

∣

∣

g4−1

F (1, 1− g4, 1 + g3,v24(ω/v4−∆k2f )

v23(ω/v3+∆k2f )),

ω > v4∆k2f or ω < −v2∆k2fv23g4

∣

∣

∣

ωv4

−∆k2f

∣

∣

∣

g3−1∣∣

∣

ωv3

+∆k2f

∣

∣

∣

g4F (1, 1− g3, 1 + g4,

v23(ω/v3+∆k2f )

v24(ω/v4−∆k2f )),

−v2∆k2f < ω < −v3∆k2f

0,

otherwise

(3.56)

where the scaling exponents equal

g3,4 =1

2√1− 2x2

∓ 1

2(3.57)

and F is the hypergeometric function.

For the interaction strength we focus on, 0 < x < 2/3, we always have 0 <

g3 < 1 and 1 < g4 < 2. Asymptotically, IAtun ∼ (ω − v4∆k2f )g3 when ω & v4∆k2f .

Thus, ω = v4∆k2f corresponds to a divergency of the differential conductance. If

ω . −v3∆k2f then the tunneling current is asymptotically equal to (ω + v3∆k2f )g4.

When ω ≈ −v2∆k2f , we have IAtun ∼ (ω + v2∆k2f )g3+g4−1. Hence, when g3 + g4 < 2,

the differential conductance diverges at −v2∆k2f , while for g3 + g4 > 2 only a cusp

is present as is shown in Fig. 3.10.

124

Table 3.2: Summary of singularities in the voltage dependence of the differential conductance GAtun

for different 5/2 states. The “Modes” column shows the numbers of left- and right-moving modes inthe fractional edge, the number in the brackets being the number of Majorana modes. “A” or “N”in the next column means Abelian or non-Abelian statistics. The “Singularities” shows the numberof singularities, including divergencies (S), discontinuities (D) and cusps (C), i.e., discontinuities ofthe first or higher derivative of the voltage dependence of GA

tun. The table refers to the tunnelinginto a boundary of ν = 5/2 and ν = 2 liquids. The case of weak interaction, Fig. 3.1, is closelyrelated.

State Modes Statistics SingularitiesK=8 1R A 2C331 2R A 4C+2S or 5C+SPfaffian 2R(1) N 2C+DEdge-reconstructed Pfaffian 1L(1) + 2R N 8C+2S or 9C+SNon-equilibrated anti-Pfaffian 2L(1) + 1R N C+2S or 2C+S

If v2 < v3 then the tunneling current is

IAtun = −L 4π2e|γ|2~2Γ(g3)Γ(g4)

vg323vg424

∣

∣

∣

ω

v2+∆k2f

∣

∣

∣

g3+g4−1

sign(ω)

×

B(v34(ω/v4−∆k2f )

v23(ω/v2+∆k2f ), g3, g4),

ω∆k2f

> v4 or < −v3

B(1, g3, g4), −v3 < ω∆k2f

< −v2

0, otherwise

(3.58)

In this case the behavior near ω = v4∆k2f is the same as above. The behavior near

ω = −v3∆k2f and ω = −v2∆k2f is also the same as above but these singularities

appear now in the opposite order since v2 < v3. The differential conductance is

shown in Fig. 3.10.

3.7 Discussion

We have found the number of the transport singularities in different models and

setups, Table 3.1. We also determined the nature of the singularities for the tunneling

into the boundary of the ν = 5/2 and ν = 2 states, Table 3.2. The information from

125

Tables 3.1 and 3.2 allows one to distinguish different models of the 5/2 state.

The results listed in Table 3.2 are also relevant for the transport in the setup

Fig. 3.1 in the case of weak interactions. Only the case of the K = 8 state should be

reconsidered as discussed in Sec. 3.6.2. The same types and numbers of singularities

will be found in both versions of the setup, Fig. 3.1a) and Fig. 3.1b). In the

second case, the control parameter is not voltage bias but the momentum mistmatch

between the quantum wire and the QHE edge.

Certainly, in setup Fig. 3.1, only the total tunneling current

Itun =∑

i

IA,itun + IBtun + ICtun (3.59)

and the total tunneling differential conductance Gtun can be measured, thus, singu-

larities originating from all three contributions to the current will be seen. Here, IB,Ctun

describe tunneling into the integer edge modes. However, these last two contribu-

tions to the current (3.59) always exhibit the same behavior for a weakly interacting

system. They simply give rise to 4 delta-function conductance peaks.

Let us briefly discuss tunneling between two identical ν = 5/2 states. A signifi-

cant difference from the previous discussion comes from the symmetry of the system.

The symmetry considerations yield the identity Itun(ω) = −Itun(−ω). In contrast

to our previous discussion, it is no longer possible to read the propagation direction

of the modes from the I − V curve as there is no difference between positive and

negative voltages.

The tunneling current through a line junction between two 5/2 states expresses

126

as

Itun =IAtun(∆k, ω) + IAtun(−∆k, ω)

+ IBtun + ICtun + IFtun, (3.60)

where IFtun is the tunneling current between two fractional QHE edges, IAtun stays for

tunneling between integer QHEmodes on one side of the junction and fractional QHE

modes on the other side of the junction, and IB,Ctun describe tunneling between integer

QHE modes on different sides of the junction. Since the tunneling operator between

two fractional edge modes is less relevant than the other tunneling operators, the

contribution IFtun is smaller than the other contributions. All remaining contributions

have already been calculated above.

We considered several different setups. While calculations are similar for all of

them, they offer different advantages and disadvantages for a practical realization. In

the setups Fig. 3.1, the main contribution to the current comes from the tunneling

into integer edge states and additional singularities due to the fractional edge modes

are weaker. In the setup shown in Fig. 3.2, all singularities are due to the tunneling

into fractional quantum Hall modes only. However, controlling momentum difference

between integer and fractional edges in the setup Fig. 3.2 would require changing

the distance between the fractional and integer edge channels. This may potentially

result in different patterns of edge reconstruction for different momentum differences

and make the interpretation of the transport data difficult. A recent paper [105]

considers momentum-resolved tunneling into a 5/2 edge in another related geometry:

Electrons tunnel into an edge between ν = 2 and ν = 5/2 QHE liquids. This allows

bypassing the problem of tunneling into integer edge modes. At the same time, it

might be more difficult to create a geometrically straight edge in such a setup than on

the edge of a sample whereas momentum-resolved tunneling depends on momentum

127

conservation and hence on straight edges. Our results apply to all above setups

including that of Ref. [105]. In contrast to our paper, Ref. [105] only considers two

candidate states: Pfaffian and non-equilibrated anti-Pfaffian. As discussed above,

non-equilibrated anti-Pfaffian state can be probed with a conductance measurement

in a bar geometry since its conductance is 7e2/(2h) in contrast to other candidate

states. In this paper, we show how the Pfaffian state can be distinguished from

several other proposed states which have the same conductance in the bar geometry.

We assumed that the temperature is low. A finite temperature would smear the

singularities. To understand the thermal smearing we recall that singularities are

obtained at ~∆k = |eV/vm|, where vm is an edge mode velocity. A finite temperature

can be viewed as a voltage uncertainty of the order of kT . Thus, the width of

the smeared singularity is δk ∼ kT/[~vm]. This suggests that the total number

of singularities that can be resolved is of the order of ∆k/δk ∼ eV/kT . The lowest

available temperatures in this type of experiments are under 10 mK [111]. eV cannot

exceed the energy gap for neutral excitations. While there is no data for this gap, it

is expected to be lower than the gap for charged excitations. The latter exceeds 500

mK in high-quality samples [112]. This suggests that N ∼ 10 singularities could be

resolved in a state-of-art experiment. Hence, as the discussion in Appendix B shows,

our approach is restricted to the systems with no or only few additional channels

due to the reconstruction of the integer edges. Recent observations of the fractional

QHE in graphene [113,114] may potentially drastically increase relevant energy gaps

and the number of singularities that could be resolved.

In conclusion, we considered the electron tunneling into ν = 5/2 QHE states

through a line junction. Momentum resolved tunneling can distinguish several pro-

posed candidate states. The number of singularities in the I−V curve tells about the

number of the modes on the two sides of the junction. The nature and propagation

128

directions of the modes can be read from the details of the I − V curve.

Chapter Four

Fluctuation-Dissipation Theorem

for Chiral Systems in

Nonequilibrium Steady States

130

Part of this Chapter is published as Chenjie Wang and D. E. Feldman, Phys. Rev.

B 84, 235315 (2011).

Abstract of this chapter: We establish a fluctuation dissipation theorem for both

electric and heat currents in chiral systems in non-equilibrium states. We first con-

sider a three-terminal system with a chiral edge channel connecting the source and

drain terminals. Charge can tunnel between the chiral edge and a third terminal. The

third terminal is maintained at a different temperature and voltage than the source

and drain. We prove a general relation for the current noises detected in the drain

and third terminal using a mixture of Landauer-Buttiker and Kubo formalisms. The

relation has the same structure as an equilibrium fluctuation-dissipation relation with

the nonlinear response ∂I/∂V in place of the linear conductance. The result applies

to a general chiral system and can be useful for detecting upstream modes on quan-

tum Hall edges. Then, we generalize the result to a multi-terminal setup and for heat

transport as well. The proof the generalized result is based on fluctuation relations,

exact relations for non-equilibrium systems.

4.1 Introduction

Fluctuation-dissipation theorems (FDT) [68] establish a beautiful and useful con-

nection between response functions and noise. They have a long history beginning

with the Einstein relations and Nyquist formula and culminating in Kubo’s linear

response theory. The standard FDT applies in thermal equilibrium only and much

attention has been focused on its violations in nonequilibrium conditions. It became

gradually clear that the FDT forms a special case of more general fluctuation theo-

rems valid for various classes of nonequilibrium systems [68]. Well-known examples

131

are the Jarzynski equality [69] and the Agarwal formula [70].

The foundations of the linear response theory and the FDT are the Gibbs dis-

tribution and causality. According to the causality principle, there is a fundamental

asymmetry between the past and the future since the future depends on the past

but the past is not influenced by future events. This imposes crucial restrictions on

response to any perturbations. In this paper we address chiral systems 1 which pos-

sess a similar asymmetry between left and right so that what happens on the right

affects what later happens on the left but not vice versa. Obviously, this may only be

possible if excitations can propagate just in one direction. Such chiral transport can

occur in topological states of matter, a primary example being a low-temperature

2D electron gas in the conditions of the quantum Hall effect [36] (QHE). The gas is

gapped in the bulk and its low-energy physics is determined by 1D chiral edge excita-

tions. In the simplest QHE states all edge modes have the same chirality and hence

the current flows in one direction only, e.g., clockwise. We show that in such chiral

systems a Nyquist-type formula (4.1) holds for the low-frequency current noise and

nonlinear conductance even far from equilibrium. This far-from-equilibrium FDT is

different from a more general Agarwal formula [70] for non-chiral systems which con-

nects quantities that do not generally have an obvious physical meaning and cannot

be easily extracted from experiment.

Our results apply beyond QHE. As usual in statistical mechanics, the simplest

example of a chiral system comes from the physics of ideal gases. Consider a large

reservoir filled with an ideal gas. A narrow tube with smooth walls and an open

end is connected to the reservoir. Molecules can leave the reservoir through the

tube. The projections of their velocities on the tube axis cannot change. Hence,

1Consider a system whose Hamiltonian has a time-dependent contribution Ht =∫ y

−∞dxh(x, t),

where the integration extends to the left of point y. In a chiral system, local observables to theright of point y do not depend on the form of h(x, t) for any initial conditions.

132

they can only move from the reservoir to the open end of the tube and the system is

chiral. Imagine now that molecules can escape through the walls of the tube with the

probability depending on their position and velocity. A relation similar to Eq. (4.1)

can then be derived for the particle flux through the walls and the fluctuations of the

fluxes through the walls and the open end of the tube. We discuss that relation in

Appendix C. The gas example is one-dimensional. Chiral systems are also possible

in 2D. Indeed, topological states of matter with gapless chiral excitations on a 2D

surface of a 3D system are possible (e. g., Ref. [115] and related systems). In such

systems, charge can propagate in both directions along one of the coordinate axes

but only in one direction along the second axis. Moreover, chiral models can emerge

beyond conventional condensed matter physics. For example, statistical mechanics

has been used to describe traffic [116]. A chiral model describes traffic on a network

of one-way streets with no parking as long as no traffic jams form.

Chiral edge states in QHE are of particular interest. It was proposed that non-

Abelian anyons exist in QHE at some filling factors [16], such as 5/2 and 7/2. If the

prediction is true this will have major implications for fundamental physics and quan-

tum information technology [16]. However, the nature of the 5/2 state remains an

open question. Competing theories predict both Abelian and non-Abelian statistics

[see Ref. [45] for a review of proposed states]. Some of the proposed states have chiral

edges and others do not. In particular, all published proposals for Abelian states are

chiral2. Thus, testing chirality of QHE edges is important in this context [47,97] and

the theorem (4.1) will be useful for that purpose. On the other hand, it is generally

believed that the edges of the Laughlin states at ν = 1/(2p + 1) are chiral. This

expectation is supported by the chiral Luttinger liquid model [36] (CLL). However,

CLL faces major challenges from experiment (for a review, see Ref. [89]). For exam-

2Pairs of counter-propagating modes, which may emerge from edge reconstruction, are likely tobe localized by disorder.

133

D S

C

IU

IL

IT

IS

V

Figure 4.1: Three-terminal setup. A quantum Hall bar is connected to source S at the voltage V .Charge tunnels into terminal C. The arrows represent the directions of the chiral edge modes.

ple, it cannot explain observed quasiparticle transmission through an opaque barrier

without bunching into electrons [117, 118]. Thus, it is important to test major as-

sumptions of CLL. One of them is chirality. Our theorem can be used for testing

that assumption. Eq. (4.1) has already been verified [119] in the limiting cases of

T ≫ V and V ≫ T .

A nonequilibrium FDT can be formulated for chiral systems in various geometries.

Below we fist focus on the simplest geometry illustrated in Fig. 4.1 (A multi-terminal

setup will be considered in Sec. 4.5). We consider a quantum Hall bar connected

to the source (S) and drain (D) terminals. The impedance between the bar and

the outside world is small. Long-range Coulomb forces are screened by a gate (this

also ensures the absence of bulk currents [120]). Excitations propagate from the

right to the left on the lower edge and in the opposite direction on the upper edge.

The system size is much greater than the magnetic length; we assume that the

chiral edges are far apart and do not influence each other. A third terminal C is

connected to the lower edge through a tunneling contact. The details of the contact

are unimportant and our results apply no matter how high or low the tunneling

current IT into terminal C is. A voltage bias V is applied between the source and

C. The temperature T of the source and drain reservoirs may be different from the

134

temperature of reservoir C. Our results do not depend on the latter temperature

or the nature of conductor C. V and T should be much lower than the QHE gap

(otherwise, QHE is absent and the system is not chiral). We consider the current

noise in terminal C, SC =∫

dt〈∆IT(t)∆IT(0) + ∆IT(0)∆IT(t)〉, and in the drain,

SD =∫

dt〈∆ID(t)∆ID(0) + ∆ID(0)∆ID(t)〉, where ID is the electric current in the

drain and ∆I = I − 〈I〉, and derive the relation

SD = SC − 4T∂IT∂V

+ 4GT, (4.1)

whereG is the Hall conductance of the quantum Hall bar without a tunneling contact.

This is the main result of the paper.

A similar formula in a different geometry was obtained in Refs. [121–123] for the

exactly solvable CLL model. As seen below, Eq. (4.1) holds independently of the

integrability of a model and does not rely on CLL. Only chirality matters. This point

is disguised in the model [121–123] since the same solvable Hamiltonian describes a

chiral system with tunneling between quantum Hall edges and a nonchiral quantum

wire with an impurity.

In the next section we give a simple heuristic derivation. Section 4.3 contains a

full quantum proof of the fluctuation-dissipation theorem. We discuss experimental

implications of the three-terminal setup results in Section 4.4. In Section 4.5, we gen-

eralize the results to a multi-terminal setup and consider heat transport as well. This

section is based on a different method – fluctuation relations from non-equilibrium

physics. Appendix C contains a derivation of Eq. (4.1) in an ideal gas model.

135

4.2 Heuristic derivation

Equation (4.1) does not contain the Planck constant and so we first give its heuristic

classical derivation. The current ID = IL−IU is composed of the current IL, entering

the drain along the lower edge, and the current IU on the upper edge. These currents

are uncorrelated and hence the noise in the drain is the sum of the noises of IU and

IL:

SD = SU + SL. (4.2)

The noise on the upper edge is the same as in the symmetric situation without

tunneling into C. In the latter case the noise SD is given by the Nyquist formula.

Thus, SU is one half of the equilibrium Nyquist noise, SU = 2GT . In order to

evaluate SL we note that in a steady state there is no charge accumulation on the

lower edge and hence the low-frequency component of the current, absorbed by the

drain, IL = IS − IT, where IS is the low-frequency part of the current, emitted from

the source. Thus,

SL = SC + SS − 2SST, (4.3)

where the cross-noise SST =∫

dt〈∆IT(t)∆IS(0)+∆IS(0)∆IT(t)〉 and the noise of the

emitted current equals one half of the Nyquist noise because of chirality,

SS = SU = 2GT. (4.4)

We are left with the calculation of the cross-noise. The tunneling current depends

on the average emitted current GV and its fluctuations Iω. We assume that the

central part of the lower edge has a relaxation time τ . It is convenient to separate

the fluctuations of IS into fast, I>, and slow, I<, parts. I< contains only frequencies

below 1/τ . An instantaneous value of the tunneling current IT(t) depends on the

136

emitted current within the time interval τ . I< does not exhibit time-dependence

within such time interval. Hence, it enters the expression for the tunneling current

in the combination GV + I< only, IT = 〈I(GV + I<, I>)〉, where the brackets denote

the average with respect to the fluctuations of IS. According to the Nyquist formula

for the emitted current, its harmonics with different frequencies have zero correlation

functions: 〈IωI−ω′〉 ∼ δ(ω−ω′). For the sake of the heuristic argument we will assume

a Gaussian distribution of IS and hence independence of its high and low frequency

fluctuations (no such assumptions are needed in a general proof). After averaging

with respect to the fast fluctuations of IS we can write IT = 〈J(GV + I<)〉, where

J is obtained by averaging over I>. I< corresponds to a narrow frequency window

and can be neglected in comparison with GV , i.e., IT = J(GV ). For the calculation

of the cross-noise we expand J(GV + I<) to the first order in I< and obtain

SST = 〈IT,ωI−ω + IT,−ωIω〉 =∂J(GV )

G∂V〈IωI−ω + I−ωIω〉 = 2T

∂IT∂V

, (4.5)

where we used the Nyquist formula for the fluctuations Iω of IS. A combination of

Eqs. (4.2-4.5) yields the desired result (4.1) for nonequilibrium steady states. Note

that only low-frequency particle current conserves along the edge in our screened

conductor and hence it is plausible to expect that IT depends only on harmonics Iω

with ~ω < ~ωcutoff ≪ T . In such case the assumption of the Gaussian distribution

and independence for I<,> is not needed since the same results can be obtained from

the lowest order expansion in I<,>.

The above argument easily generalizes to other geometries with many terminals

and/or tunneling between QHE edges.

One can estimate noises SD and SC. We assume that V ∼ T ≈ 100mK. These

are realistic parameters for noise experiments. The injected current IS ∼ e2V/h

137

since G ∼ e2/h. We assume that the transmission probability to conductor C is of

the order of 1/2. Thus, IT ∼ IS. Then the second and third terms on the right

hand side of Eq. (4.1) are of the order of e2[kT/h] ∼ e2[eV/h]. The noise SC can

be estimated from the fact that its physical meaning corresponds to the ratio of the

fluctuation 〈∆Q2〉 of the transmitted charge to the time ∆t over which the charge

was transmitted to C. The probability of a single tunneling event during the time

interval ∆t ∼ h/eV is of the order of 1/2. Thus SC ∼ e2/∆t ∼ e2[eV ]/h and has the

same order of magnitude as the second and third terms on the right hand side of Eq.

(4.1). Finally, SD has the same order of magnitude. This corresponds to the noises

of the order of 10−28A2/Hz. This is a very small number but even lower noises are

measured in the state of the art experiments in the field.

Counter-propagating edge modes break Eq. (4.1). Imagine that an “upstream”

neutral modes propagates in the direction, opposite to that of the charged mode.

Each tunneling event into C creates a neutral excitation that brings energy ∼ V

back to the source and heats it. This increases the noise, generated by the source,

and raises the effective temperature above T in Eq. (4.1). We would like to emphasize

that the bulk of the source remains at the temperature T since the heat capacity of

the source is large. However, the bulk plays relatively little role in noise generation

in our system due to a relatively low resistance of a massive bulk conductor.

4.3 Proof of the nonequilibrium FDT

We now turn to a full quantum derivation of Eq. (4.1). To simplify notations we

will omit the Boltzmann constant and ~ from the equations below. As is clear from

the above classical argument, the drain potential has no effect on the noises. It

138

will be convenient to assume that the source and drain potentials are equal. It will

also be convenient to assume that at the initial moment of time t = −∞ there was

no tunneling or other interactions between conductor C and the QHE subsystem

containing the quantum Hall bar and the source and drain reservoirs. Thus, the

system is initially described by the Hamiltonian H0 = HC + HH, where HC and

HH denote the Hamiltonians of the two subsystems: conductor C and the QHE

subsystem respectively. At later times the Hamiltonian includes an interaction term,

H = H0 + HI(t). One of the effects of the interaction is charge tunneling between

the QHE subsystem and conductor C. We assume that the interaction HI becomes

time-independent well before the moment of time t = 0 and the system is in its

nonequilibrium steady state at t = 0. The steady state depends on the voltage bias

V , temperature T and the temperature TC of conductor C. All these energy scales

must be lower than the QHE gap. Otherwise, the system is unlikely to allow a chiral

description. We do not assume any special relation between the energy scales. If

T = TC and V = 0 then the system is in equilibrium. If T−TC ≪ T and V ≪ T then

the system is close to equilibrium. Otherwise, the system is far from equilibrium.

Our main result (4.1) applies in all those cases.

4.3.1 Chiral systems

A general definition of a chiral system is the following: Consider a system whose

Hamiltonian has a time-dependent contribution Ht =∫ y

−∞ dxh(x, t), where the in-

tegration extends to the left of point y. In a chiral system, local observables to the

right of point y do not depend on the form of h(x, t) for any initial conditions.

In what follows, we will not need the most general definition. Instead, we will

focus on one particular observable, current IS, emitted from the source to the lower

139

x

z

A

B

D S

C

V

Figure 4.2: The same low-frequency current IS flows through both dashed lines.

edge. We are only interested in the low-frequency regime. In that limit, the precise

choice of the point, where the current IS is measured, is unimportant due to the

charge conservation. The same low-frequency current flows in all points of the lower

edge between the source and conductor C. Let us select a point A in the gapped

region in the bulk of the QHE liquid and a point B below the QHE bar (Fig. 4.2).

The same current IS must flow through any line, connecting A and B. This remains

true, even if the line does not cross the lower edge and instead goes through the

source (and the boundary between the source and the QHE liquid). Thus, IS can be

defined both in terms of the edge and source physics.

The chirality assumption means that the average emitted current IS(t) does not

depend on the presence of conductor C for any initial conditions. In other words,

the expression Trρ(t = −∞)IS(t), where ρ is the initial density matrix, is the same

when the Heisenberg operators

IS(t) = [T exp(−i∫ t

−∞V (t)dt)]−1IS(−∞)[T exp(−i

∫ t

−∞V (t)dt)], (4.6)

where IS(−∞) is a Schrodinger operator, are defined in terms of the Hamiltonians

V = H0 and V (t) = H0 +HI(t) = H . The latter can be true for a general ρ(−∞)

140

only if the Heisenberg operators IS(t) are the same in the presence and absence of

conductor C. Note that this property is satisfied in the chiral Luttinger liquid model.

To the right of conductor C, the CLL action assumes the form L = m~/(4π)[∂tφ∂xφ−

v(∂xφ)2], where v is the velocity of the edge excitations, the charge density q =

e∂xφ/(2π) and the current operator IS = vq. From the equation of motion, ∂t∂xφ−

v∂2xφ = 0, we find that the electric current on the right of conductor C, IS = IS(t +

x/v), depends only on the initial conditions on the right and is not affected by the

form of HI(t). The same property is satisfied in any other chiral conformal theory

and in many other situations. For example, the chirality property of the operator

IS survives, if any changes are introduced into the above CLL action to the left

of the point, where IS is measured. The chirality assumption also holds for QHE

edges with several modes of the same chirality but breaks down, generally, if counter-

propagating modes are present, as e.g., in the anti-Pfaffian state [12,13] proposed at

ν = 5/2.

Edge reconstruction [96] may result in “net chiral” edges that are not chiral. For

example, a pair of counter-propagating integer QHE modes can emerge on a ν = 1/3

edge. In general, this breaks chirality. However, in practice, disorder is likely to

localize such mode pairs and restore chirality.

4.3.2 Initial density matrix and Heisenberg current operator

At the time t = −∞ there is no interaction between conductor C and the QHE

subsystem that includes the source, drain and 2D electron gas. Hence, the initial

density matrix ρ(−∞) = ρHρC factorizes into a product of the initial density matrix

ρC of conductor C and the initial density matrix ρH of the QHE subsystem. Each

of them corresponds to an independent Gibbs distribution determined by an appro-

141

priate reservoir. At later times the subsystems interact, the steady state depends on

both reservoirs, and the factorization property no longer holds in the Schrodinger

representation. Thus, it will be convenient for us to perform calculations in terms

of the initial density matrix because of its simpler structure 3. This means that we

will use the Heisenberg formalism so that all time dependence is placed into the

operators of observables. The chirality property will allow us to extract considerable

information about the matrix elements of the Heisenberg operator IS(t) and prove

Eq. (4.1). We would like to emphasize that the Hamiltonian has a time-dependent

piece HI(t) and this piece enters the definition of all Heisenberg operators. The

presence of that piece is crucial for the difference between the density matrices in

the Heisenberg and Schrodinger representations. We will omit the time argument

in ρH and ρC. It will be always understood that these are initial density matrices

at t = −∞. In all calculations below, ρ is also taken at t = −∞. Certainly, in the

Heisenberg representation, the density matrix does not depend on time.

Our approach resembles the Keldysh formalism, where all correlation functions

are also expressed in terms of the initial density matrix. In the Keldysh technique,

if the interaction is adiabatically turned on the initial density matrix describes free

particles and hence factorizes into a product of single-particle density matrices. A

difference from our approach consists in the application of the interaction represen-

tation in the Keldysh perturbation theory. The average of any properly time-ordered

product of creation and annihilation operators is known exactly in the interaction

representation. This allows development of a diagrammatic technique. We use the

3Strictly speaking, the assumption of factorization at t = −∞ is not necessary. It is sufficientto assume that the final steady state depends only on the states of the reservoirs and not on theinitial state of the finite central part of the system. In such case it is most convenient to performcalculations for the initial state whose density matrix factorizes. Certainly, the latter assumptionitself is completely standard and can be easily tested experimentally by comparing steady statesprepared from different initial conditions at the same temperatures and chemical potentials of thereservoirs.

142

Heisenberg representation instead and rely on special properties of the matrix ele-

ments of the operator IS in the basis, in which the initial density matrix is diagonal.

Any density matrix is Hermitian and can be diagonalized. Hence,

ρH,C =∑

ρH,Cn|nH,C〉〈nH,C| (4.7)

and

ρ(−∞) =∑

ρn|n〉〈n|, (4.8)

where

ρn = ρHn′ρCn′′ (4.9)

and

|n〉 = |n′H〉|n′′

C〉, (4.10)

where the states |n′H〉 and |n′′

C〉 are selected from the Hilbert spaces of the QHE system

and conductor C respectively. ρH is a Gibbs distribution, ρHn ∼ exp(−En/T −

|e|V Nn/T ), where Nn is the number of electrons in the QHE subsystem, T the

temperature of the source and drain reservoirs and En are the eigenenergies of the

eigenstates |nH〉 of the quantum Hall subsystem before the tunneling contact was

turned on, i.e., |nH〉 are eigenstates of the Hamiltonian HH with particle numbers

Nn. We do not make assumptions about ρC. Our proof applies as long as the initial

density matrix ρ(−∞) factorizes and the initial density matrix of the QHE subsystem

ρH is given by the Gibbs distribution. In practice, the initial density matrix ρC is also

likely to be a Gibbs distribution. To avoid a possibility of confusion, we emphasize

that all states in the bases |nH〉 and |nC〉 are time-independent. Thus, they are no

longer eigenstates of the time-dependent Hamiltonian after the interaction HI has

been turned on.

143

If the interaction HI(t) is never turned on then IS(t) acts in the Hilbert space

of the QHE subsystem and hence its nonzero matrix elements are always diagonal

in the basis of |nC〉. The chirality property means that the same restriction applies

to nonzero matrix elements of IS(t) even after the interaction HI(t) has been turned

on since IS(t) must be the same in the presence and absence of the interaction

HI(t). The emitted current operator commutes with the number N of the particles

in the quantum Hall subsystem since it describes particle transfer between the source

and the edge. Thus, in the absence of the interaction HI(t), it has nonzero matrix

elements only between states |nH〉 with the same Nn. Again, the chirality property

means that the same restriction on nonzero matrix elements applies even after the

interaction has been turned on. Before the interaction between the quantum Hall

bar and subsystem C has been turned on, it is easy to write the time-dependence for

matrix elements of any operator acting in the Hilbert space of the QHE subsystem:

〈n|OH(t1)|m〉 = exp(i[En − Em](t1 − t2))〈n|OH(t2)|m〉. The same relation would

apply at all times, if the interaction HI(t) were never turned on. The chirality

property means that the emitted current operator IS(t) exhibits exactly the same

time-dependence at any times, if the interaction HI(t) is turned on and if it is not.

This applies both before and after the tunneling between two subsystems has been

turned on. Setting t2 = 0 in the above relation, we obtain

〈n|IS(t)|m〉 = exp(i[En − Em]t)〈n|IS(0)|m〉. (4.11)

4.3.3 Voltage bias

A standard way to include voltage bias in mesoscopic systems is based on the

Landauer-Buttiker formalism: One assumes that the tunneling term is initially ab-

144

QHE S

δE

Q Pδφ = 0 δφ = δV

Figure 4.3: Illustration of the bias voltage. δA and δE are applied in the region between two solidvertical lines. In the example in the figure the region with δE crosses both the source (shaded) andthe gapped QHE region (white). δφ is constant on the vertical dashed line. δφ = 0 in point Q andδφ = δV in point P.

sent and then turned on and that the lower edge is initially at equilibrium with the

reservoir with the chemical potential V . We will use a mixed Kubo-Landauer for-

malism to determine the response of IT to a small change δV of the voltage bias. It

will allow us to reduce the problem of nonlinear response to V to the linear response

to δV . In the mixed Kubo-Landauer language, an additional electromotive force δV

is generated by an infinitesimal time-dependent vector potential δA.

We assume that different contacts are connected with infinite reservoirs at dif-

ferent electrochemical potentials. Their difference determines the voltage bias V :

the source electrochemical potential is V and the potential of conductor C is 0. A

small change of the bias δV can be introduced with an electric field described by a

time-dependent vector potential, δE = −1/cdδA/dt. The electric field is applied in

a finite part of the source terminal (Fig. 4.3) and cannot affect chemical potentials

of the infinite reservoirs. The chemical potentials determine electric potentials of the

reservoirs because of charge neutrality. The distribution of charges certainly changes

in the middle of the conductor in the presence of δA, so the electrostatic potential

φ also changes. However, the potential difference between the reservoirs does not.

145

The magnetic field must be time-independent inside the sample as required by the

restrictions on e.m.f. sources in the circuit theory. In other words, δB = curlδA = 0

inside the conductor. Hence, the integral of δE does not depend on the choice of a

path inside the conductor at fixed positions of its ends. If a path PQ begins in the

infinite source reservoir and ends on the opposite side from the region with the field

δE (Fig. 4.3) then∫ Q

PdrδE = δV . Hence, δA = ct× gradδφ, where δφ = δV in the

source reservoir and δφ = 0 far on the left in the quantum Hall region (Fig. 4.3). As

a consequence, the vector potential δA can be gauged out inside the conductor at

the expense of changing the electrostatic potential φ→ φ+δφ. This means a change

of δV in the electrochemical potential of the source reservoir and no change in the

potential of conductor C. Thus, one can see that the Kubo formalism is equivalent

to the Landauer-Buttiker approach in the presence of infinite electrically neutral

reservoirs.

δA generates a correction to the Hamiltonian: δH = −∫

d3rδAj/c, where j is

the current density. Consider an arbitrary surface of constant δφ in the region with

nonzero δE (Fig. 4.3). Similar to the discussion of Fig. 4.2, in the low-frequency

limit, the total current through any such surface is the same and equals the total

current through the source

I = IU − IS, (4.12)

the signs in front of IU,S reflecting our conventions about current directions, Fig. 4.1.

This allows rewriting

δH = −IδA/c, (4.13)

146

where dδA/dt = cδV . The same approach can be used to describe small changes of

the drain potential but they are irrelevant for our purposes.

In what follows it will be convenient to consider the case of δA oscillating with a

low frequency ω, δA = cδV sinωt/ω.

4.3.4 Main argument

We now give a full quantum derivation of Eq. (4.1). The arguments leading to Eqs.

(4.2-4.4) do not change compared to Section 4.2 and we concentrate on Eq. (4.5).

The cross-noise can be expressed as

2SST =

∫

dt〈IT(0)IS(t) + h.c.〉(exp(iωt) + exp(−iωt)) =∫

dt∑

mn

(exp(iωt) + exp(−iωt))[〈m|IT(0)|n〉〈n|IS(t)|m〉ρm(−∞)

+ 〈n|IS(t)|m〉〈m|IT(0)|n〉ρn(−∞)], (4.14)

where a low frequency ω < 1/τ (τ is the relaxation time, Section 4.2), ρ(−∞) is the

initial density matrix and IT,S are Heisenberg operators [see Eq. (4.6)]. As usual,

introducing a small nonzero frequency allowed us to write the expression in terms

of IS,T and not ∆IS,T = IS,T − 〈IS,T〉. Eq. (4.14) gives the noise at t = 0, when the

system is in a steady state. As discussed in Section 4.3.2 it is convenient to use the

Heisenberg representation in which the density matrix is not the steady state density

matrix ρ(t = 0) but the initial ρ(t = −∞) since IS(t) exhibits remarkable properties

in such representation. Inserting the time dependence (4.11) of the matrix elements

147

〈n|IS(t)|m〉, one finds

2SST = 2π∑

mn

[ρm(−∞)+ρn(−∞)]〈m|IT(0)|n〉〈n|IS(0)|m〉[δ(En−Em+ω)+δ(Em−En+ω)].

(4.15)

Next, we need to compute RT = ∂IT/∂V . As discussed in Section 4.3.3, this is

a linear response problem with respect to δV . Similar to Ref. [124], RT is given by

the same Kubo formula as in equilibrium. Indeed,

〈IT(t = 0)〉 = Tr[ρ(−∞)SA(−∞, 0)IsTSA(0,−∞)], (4.16)

where IsT is a Schrodinger operator, SA(t2, t1) the evolution operator, SA(0,−∞) =

T exp(−i∫ 0

−∞HA(t)dt), the Hamiltonian HA(t) = H − IδA/c and H = HC +HH +

HI(t). The expansion to the first order in δA yields

RT × δV =i

∫ 0

−∞dtTr[ρ(−∞)S(−∞, t)δHsS(t, 0)IsTS(0,−∞)

− S(−∞, 0)IsTS(0, t)δHsS(t,−∞)], (4.17)

where δHs is the Schrodinger operator (4.13) and S(b, a) = T exp(−i∫ b

adt[HC+HH+

HI(t)]). Substituting (4.13) in the above equation we see that the response of IT to

δV expresses as the sum of the responses of IT to the perturbations ISδA/c and and

−IUδA/c. The latter response is zero since the edges are far apart and perturbations

on the upper edge have no effect on the lower edge. With this in mind, we rewrite

148

the nonlinear response to V in the form

RT =∂IT/∂V = i limω→0

∫ 0

−∞dt∑

mn

exp(iωt)− exp(−iωt)2iω

× [〈n|IS(t)|m〉〈m|IT(0)|n〉ρn(−∞)− 〈m|IT(0)|n〉〈n|IS(t)|m〉ρm(−∞)].

(4.18)

In the above equation we absorbed evolution operators into the Heisenberg current

operators.

It is convenient to combine the above response with the response RS of IS to

the perturbation δV IT sin(ωt)/ω in the Hamiltonian. Certainly, that response is

zero because of chirality. Indeed, we consider a perturbation, acting on the left

of the point, where IS is measured. We get an expression of the same structure

as above with the indices S and T exchanged. In a steady state we expect that

〈IS(0)IT(t)〉 = 〈IS(−t)IT(0)〉. This allows rewriting RS in the form

RS =i limω→0

∫ +∞

0

dt∑

mn

exp(iωt)− exp(−iωt)2iω

× [〈n|IS(t)|m〉〈m|IT(0)|n〉ρn(−∞)− 〈m|IT(0)|n〉〈n|IS(t)|m〉ρm(−∞)]. (4.19)

We next compute RT = RT +RS:

RT =2π limω→0

∑

mn

δ(En − Em + ω)ρn(−∞)− ρm(−∞)

2ω

× [〈n|IT(0)|m〉〈m|IS(0)|n〉+ 〈m|IT(0)|n〉〈n|IS(0)|m〉], (4.20)

where we used the time-dependence (4.11). The above equation contains the initial

density matrix at time t = −∞ and the Heisenberg current operators (4.6) at time

t = 0. Finally we apply the results of Section 4.3.2 for IS(t) and ρ(−∞). We notice

149

that nonzero matrix elements 〈m|IS(0)|n〉 correspond to Nn = Nm and |nC〉 = |mC〉.

Hence, in the limit of low frequencies in Eq. (4.20), [ρn(−∞) − ρm(−∞)]/ω =

−ρCndρHn/dEn = ρn(−∞)/T , where we used the factorization property (4.9) which

is only valid for the initial density matrix. Comparison of Eqs. (4.15) and (4.20) at

small ω establishes Eq. (4.5).

The above calculation relies on the structure of the initial density matrix ρ(−∞).

This does not mean that the steady state depends on minor details of the initial

state. Only the temperatures and chemical potentials of the large reservoirs are

important. Those temperatures and potentials remain the same in the initial and

steady state. If, on the other hand, one of the reservoirs is not large then the steady

state does not depend on the initial density matrix of that reservoir. This can be

easily seen from Eq. (4.1) in the limit of a small reservoir attached to conductor

C. Indeed, in that case, IT = SC = 0 in a steady state since conductor C cannot

accumulate charge. Thus, Eq. (4.1) reduces to SD = 4GT . This is a usual Nyquist

formula, valid for a system in thermal equilibrium at the temperature T and a

uniform chemical potential. Obviously, the steady state is indeed an equilibrium

state with the temperature T , if conductor C is attached to a finite reservoir. In this

example, the final state does not depend on the initial density matrix ρC.

A similar argument does not work for a finite source reservoir. Indeed, the

derivation of Eq. (4.1) relies on the assumption that IU is uncorrelated with IS. If

the source reservoir is not large then the assumption is violated in the steady state

and IU = IS instead.

150

4.4 Discussion

The focus of the preceding section is on QHE, but similar non-equilibrium FDT

apply in many other systems. The simplest example of a chiral system, based on an

ideal gas, is considered in Appendix C. Our results can also be generalized beyond

1D, for example, for the surface transport in a 3D stack of QHE systems.

The geometry of Fig. 4.1 allows only electron tunneling to conductor C. FDT’s,

similar to (4.1), can also be derived in other geometries, where fractionally charged

anyons tunnel: One can consider tunneling between two edges of the same QHE

liquid.

Eq. (4.1) does not contain the temperature TC of conductor C. This, certainly,

does not mean that the properties of the system do not depend on it. The current

IT and the noises SD and SC are all affected by the temperature of C. The general

relation (4.1), however, remains the same. We would like to emphasize that our

derivation does not contain any assumptions about the character of the dependence

of IT and SC on the temperature and voltage. An interesting situation is possible,

if conductor C is chiral and TC 6= T . One can then derive two equations of the

structure (4.1) with two different temperatures in them.

Our main result, Eq. (4.1), applies in chiral systems and can be used for an exper-

imental test of chirality. A convenient measurement setup is illustrated in Fig. 4.4.

Several mechanisms break chirality and can lead to the violation of Eq. (4.1). One

mechanism involves long range forces in the 2D electron gas. Our discussion assumed

that a gate screens long-range Coulomb interaction. This allowed us to assume that

the tunneling Hamiltonian HI does not depend on the voltage bias and the bias man-

ifests itself in the 2D electron gas only through the chemical potential of the lower

151

D S

C

IU

ILQ

IS

V

Figure 4.4: A possible experimental setup. Charge carriers, emitted from the source, can eithertunnel through the constriction Q and continue towards the drain or are absorbed by the Ohmiccontact C.

edge. Without screening, HI may depend explicitly on the voltage and this must be

taken into account at the calculation of ∂IT/∂V . Strong interaction of edge modes

with non-chiral bulk modes may also break chirality.

The most interesting mechanism of chirality breaking involves “upstream” modes

[47, 97], Fig. 4.5. In the simplest example, two charged modes carry charge in the

opposite directions. Let us imagine that the two chiral channels do not interact and

all charge tunneling into C occurs due to particles, populating the upstream channel,

directed from D to S. Then the noise in the drain is the same as in the absence of

C, in contradiction with Eq. (4.1).

This discussion neglects a possible heating effect. To illustrate it, let us assume

that the upstream mode is neutral. It cannot carry charge but carries energy. In gen-

eral, the tunneling operator into C includes a product of operators creating charged

and neutral excitations on the edge. A neutral excitation of the energy ∼ V travels

to the source and heats it. This affects noise, generated by the source, and leads to

the violation of Eq. (4.1). The details of the interaction of a neutral quasiparticle

152

D S

C

IU

IL

IT

IS

V

Figure 4.5: A non-chiral system. The solid line along the lower edge illustrates the “downstreammode”, propagating from the source to the drain. The dashed line shows a counter-propagating“upstream” mode.

and the source are poorly understood theoretically. The experiment suggests that

the heating effect will be strong [52]. Thus, large deviations from Eq. (4.1) can be

expected in the presence of “upstream” neutral modes.

Our only assumption about V , T and TC was that they are much lower than the

QHE gap. Otherwise, a chiral description is unlikely to apply. If the system is chiral

we make no assumptions about the relation between V and T . Nevertheless, our

main focus was on the regime with V ∼ T . Eq. (4.1) greatly simplifies and becomes

less interesting in the opposite limits V ≫ T and T ≫ V . In the former case, let us

set T to zero. Then Eq. (4.1) reduces to SD = SC. This relation reflects noiseless

character of the emitted current. In the opposite limit, let us assume that V = 0 and

T = TC. Then the equilibrium FDT applies. ∂IT/∂V is now linear response. Hence,

SC = 4T∂IT/∂V . Finally, Eq. (4.1) reduces to SD = 4GT . This simple relation

reflects the fact that the lower edge is in thermal equilibrium on both sides of the

contact with C.

153

4.5 Generalization to Multi-Terminal Setup and

Heat Transport – Proof by Fluctuation Rela-

tions

In this section, we provide another proof by using the formalism of fluctuation rela-

tions [125, 126] and extend the FDT to a setup with arbitrary number of terminals.

Besides charge transport, we also study heat transport. Fluctuation relations pro-

vide rigorous predictions of non-equilibrium fluctuations and reduce to the standard

FDTs in the equilibrium limit [68]. Many fluctuation relations, classical or quantum

mechanical, have been found for isolated, closed or open systems [127–130]. We ap-

ply the energy and particle exchange fluctuation relations in quantum open systems

in a steady state [129, 130] to derive the chiral-system FDT. The system (Fig. 4.6)

consists of a chiral subsystem and r reservoirs. The reservoirs are in equilibrium

separately. When the system is in a steady state, electric current Ii and heat current

Ji flow into reservoir i. Our main result is that the zero-frequency cross noise S1i

(S1i) between I1 (J1) and Ii (Ji) with i 6= 1, r is related to the response of Ii (Ji) to

the electrostatic potential V1 (temperature T1) in reservoir 1, through

S1i = −T1∂Ii∂V1

(4.21)

S1i = −(T1)2 ∂Ji∂T1

. (4.22)

The Boltzmann constant kB is set to 1 throughout the paper. Note that the lower

edge of the setup is separated from the upper part of the system. S1i is in fact the

cross correlation between Ii and the current that flows out of reservoir 1 through

the upper edge. Thus, Eq. (4.21), similarly Eq. (4.22), is essentially a relation for

the upper chiral part. We would like to emphasize that this FDT is correct for any

154

β i,µ i

βj , µ

j

β1µ1

βrµr

S

Figure 4.6: Setup for fluctuation relations of a system with r reservoirs interacting via a subsystemS whose transport channels are chiral and located at its edges (arrows). Each reservoir is atequilibrium with its own temperature and chemical potential. Complex structures of S, such asquantum point contacts, may exist in the dashed circle.

chiral systems in steady states.

Below, we first discuss our setup in details. We state the exchange fluctuation

relation derived in Ref. [129] in the language of our setup, and incorporate chirality

into description. Then we use it to derive the chiral-system FDT. Finally we comment

on the results. Our derivation uses a language of QHE systems, but can be easily

adjusted for other chiral systems such as the ideal-gas setup considered in Ref. [131].

The setup (Fig. 4.6) is composed of a subsystem S and r large reservoirs. The

edge of S supports one or several low-energy modes, all with the same chirality. These

modes form transport channels of the system. We consider a thought process that the

subsystem and reservoirs are initially decoupled, then an interaction V(t) that allows

particle and energy exchanges is turned on during a time interval 0 ≤ t ≤ T , and

finally the interaction is turned off at t ≥ T . At t ≤ 0, reservoir i is at equilibrium

with an inverse temperature βi = 1/Ti and a chemical potential µi = qVi. Vi is the

electrostatic potential of the i-th reservoir, and q is the unit charge of particles. We

assume there is only one species of particles being transported. Temperatures and

155

chemical potential differences are smaller than the bulk energy gap of S, so that

transport does not happen in the bulk. The initial state of S is irrelevant, since

we will consider a non-equilibrium steady state that is determined by properties of

the large reservoirs. Hence, we will regroup S with one of the reservoirs [129], for

example the r-th reservoir. The interaction V(t) becomes a constant V0 when fully

turned on during τ ≤ t ≤ T − τ . We assume that τ ≪ T and T is longer than a

relaxation time so that a steady state dominates the thought process.

When the system is in a steady state, the interaction V0 between S and the

reservoirs are schematically shown in Fig. 4.6. Reservoir 1 and r are strongly coupled

to S so that their incoming channels are disconnected and spatially separated from

their outgoing channels. The incoming and outgoing currents of each of the two

reservoirs will not exchange information through the reservoirs, as we will assume

long-range interaction is screened and backscattering is rare in the reservoirs. Short-

range interaction results in equilibration of the incoming particles and heats up or

cools down a portion of the reservoir near the incoming channels. However, this

heating effect has little influence on the outgoing current as long as the spatial

separation between incoming and outgoing channels is longer than an equilibration

length. Other reservoirs can be either strongly or weakly coupled to S, which is

irrelevant to our proof. Complex structures of S, such as quantum point contacts

and strong short-range interactions may exist in the dashed circle in Fig. 4.6. The

lower edge is separated from others. The above assumptions and arguments conclude

that the transport in the lower edge is independent of that in the upper part of the

system.

After a thought process described above, we will observe energy and particles

number changes in each reservoir. Two quantum measurements are needed to observe

156

the changes4. Let Hi and Ni be the Hamiltonian and particle number operator of

the i-th reservoir respectively. In particular, Hr describes the combination of the

subsystem and r-th reservoir. Particle number is conserved in the absence of V(t)

in each reservoir, i.e., [Hi,Ni] = 0. An initial joint quantum measurement of Hi

and Ni is performed at t = 0, so that quantum state of the system collapses to

a common eigenstate |ψn〉, with Hi|ψn〉 = Ein|ψn〉 and Ni|ψn〉 = Nin|ψn〉. Then

the state |ψn〉 evolves according to the evolution operator U(t; +) determined by the

Hamiltonian H(t; +) =∑

i Hi+V(t). The “+” sign represents the clockwise chirality

of the subsystem. When the system evolves at t = T , A second joint measurement

is taken, leading to a collapse of the system to state |ψm〉, with Hi|ψm〉 = Eim|ψm〉

and Ni|ψm〉 = Nim|ψm〉. In this process, we observe energy change ∆Ei,mn = Eim −

Ein and particle number change ∆Ni,mn = Nim − Nin in the i-the reservoir. The

probability to observe such a process is P [m,n] = |〈ψm|U(T ; +)|ψn〉|2ρn, where

ρn =∏

i e−βi[Ein−µiNin−Φ+

i] is the initial Gibbs distribution to find the system in

state |ψn〉 and Φ+i = is the initial grand potential of the i-th reservoir. If we repeat

the above precess for many times, we obtain a distribution function of energy and

particle changes

P [∆E,∆N; +] =∑

mn

∏

i

δ(∆Ei −∆Ei,mn)

× δ(∆Ni −∆Ni,mn)|〈ψm|U(T ; +)|ψn〉|2ρn, (4.23)

where the vector ∆E = ∆Ei and ∆N = ∆Ni. Total energy and particles are

conserved in each process, such that∑

i∆Ei =∑

i∆Ni = 0.

The above precesses are called forward processes in the formalism of fluctuation

relations. We also need to study backward processes, the time reversal of forward pro-

4In experiments, currents are continuously monitored. However, the two-measurement schemeis enough to calculate zero-frequency noises. For continuous measurement scheme, see Ref. [130].

157

cesses. A fluctuation relation relates the distribution function for forward processes

to that for backward processes. In our case, it is equivalent to describe a backward

process of the system as a forward process of its twin system which has an opposite

chirality. The twin system has counter-clockwisely propagating edge modes, with its

chirality denoted by a “−” sign. Its time evolution operator U(t;−) is determined by

the Hamiltonian H(t;−) = ΘH(T − t; +)Θ−1, where Θ is the time-reversal operator.

The i-th reservoir has a Hamiltonian ΘHiΘ−1. One may find that Θ|ψm〉(Θ|ψn〉)

is a common eigenstate of ΘHiΘ−1 and Ni = ΘNiΘ

−1, with eigenvalues Eim(Ein)

and Nim(Nin). Performing two quantum measurements at t = 0 and t = T for each

process and repeating for many times, one finds the distribution function

P [∆E,∆N;−] =∑

mn

∏

i

δ(∆Ei −∆Ei,nm)

× δ(∆Ni −∆Ni,nm)|〈ψn|Θ−1U(T ;−)Θ|ψm〉|2ρm, (4.24)

with ρm =∏

i e−βi[Eim−µiNim−Φ−

i ] the probability to find the twin system initially in

state Θ|ψm〉. One may prove that Φ+i = Φ−

i .

An important property of the evolution operators is [129]

Θ−1U(T ;−)Θ = U †(T ; +). (4.25)

Combining Eq. (4.23), (4.24) and (4.25), we recover the fluctuation relation

P [∆E,∆N; +]

P [−∆E,−∆N;−]=∏

i

eβi(∆Ei−µi∆Ni). (4.26)

In experimental systems, chirality is originated from external magnetic field or spon-

taneous time reversal symmetry breaking, such as QHE and ferromagnetism. Ex-

tension of magnetic field or ferromagnetic materials into the reservoirs will not affect

158

the fluctuation relation.

Given a distribution P [∆E,∆N; ν] with ν = ±, we are able to calculate the

heat currents Jνi = limT →∞〈(∆Ei−µi∆Ni)〉ν/T and their correlation functions, i.e.,

noises. The triangular brackets means taking average with respect to P [∆E,∆N; ν].

The limit T → ∞ is taken so that steady-state quantities are obtained. It is reason-

able to assume that all physical quantities are finite under this limit. The particle

transfer currents Jνi = limT →∞〈∆Ni〉ν/T and their noises Sij can be computed sim-

ilarly. It is convenient to define a cumulant generating function

Q(x,y; ν) = limT →∞

1

T ln

∫

∏

i

(d∆Eid∆Ni)

× e−∑

i xi(∆Ei−µi∆Ni)−∑

i yi∆NiP [∆E,∆N; ν]

, (4.27)

where the vectors x = xi and y = yi. Noticing that P [∆E,∆N; ν] is normalized,

we have Q(0, 0;β,µ; ν) = 0. The currents and noises then can be obtained by taking

derivatives of Q(x,y; ν) over xi or yi and finally setting x = y = 0. So, the currents

Jνi = −∂xi

Q(0, 0; ν), Jνi = −∂yiQ(0, 0; ν), (4.28)

and the noises

Sνij = ∂xixj

Q(0, 0; ν), Sνij = ∂yiyjQ(0, 0; ν). (4.29)

The fluctuation relation (4.26) leads to a symmetry of the generating function

Q(x,y;β,µ; +) = Q(β − x,−βµ+ xµ− y;β,µ;−), (4.30)

where β = βi, µ = µi, βµ = βiµi and xµ = xiµi. We have written down

explicitly the dependence of Q on β and µ.

159

With the symmetry (4.30) and expressions (4.28) and (4.29), it is able to prove

the standard FDT [129] for equilibrium states. However, in order to prove the

chiral-system steady-state FDT, special properties of the distribution function and

cumulant generating function resulted from chirality are needed. As is discussed,

the lower edge is independent of the upper part of the system. In other words, the

thought process described above contains two statistically independent processes.

Mathematically, this means the distribution function can be written as

P [∆E,∆N; ν] =

∫

d∆E ′1d∆N

′1 P1[∆E

′1,∆N

′1; ν]

×P2[∆E1 −∆E ′1,∆N1 −∆N ′

1,∆E′,∆N′; ν], (4.31)

where P1[∆E′1,∆N

′1; ν] is the probability to transport ∆E ′

1 energy and ∆N ′1 par-

ticles into reservoir 1 through the lower edge, and P2[∆E′′1 ,∆N

′′1 ,∆E′,∆N′; ν] is

the probability for the upper part to make changes (∆E ′′1 ,∆N

′′1 ) in reservoir 1 and

(∆E′,∆N′) in other reservoirs. As discussed in the introduction, chirality induces

causality. In the setup with “+” chirality, energy and particles at the lower edge

flow out of reservoir r and into reservoir 1, thus its transport depends only on βr

and µr. Meanwhile, reservoir r receive energy and particles from the upper part but

does not provide feedbacks, so transport in the upper part does not depend on βr

and µr. Thus, P1 only depends on βr and µr while P2 does not depend on βr and

µr. In the setup with “−” chirality, similarly we have P1 only depends on β1 and µ1

while P2 does not depend on β1 and µ1.

In terms of the cumulant generating function, Eq. (4.31) results in thatQ(x,y;β,µ; ν)

is split to two terms Q1 and Q2, corresponding to P1 and P2 respectively. The

chirality-induced causality leads to that Q1 only depends on βr and µr while Q2 does

not depend on βr and µr for the system with “+” chirality, and Q1 only depends on

160

β1 and µ1 while Q2 does not depend on β1 and µ1 for the system with “−” chirality.

We have verified such a division of Q and dependence of Q1 and Q2 in non-interacting

Bose or Fermi systems, where the exact form of the cumulant generating function Q

exists [132–134].

We are now ready to prove the chiral-system FDT. The proof becomes straight-

forward after the above preparations. Let us start with particle transport. Taking

derivatives on both sides of Eq. (4.30) over yi and µi, we have

∂yiQ(0,y;β,µ; +) =− ∂yiQ(β,−βµ− y;β,µ;−),

∂µiQ(0,y;β,µ; +) =− βi∂yiQ(β,−βµ− y;β,µ;−)

+ ∂µiQ(β,−βµ− y;β,µ;−). (4.32)

with x being set to 0. To be clear, we stress that ∂µiQ(β,−βµ− y;β,µ;−) is

the partial derivative of Q(x,y;β,µ;−) over its variable µi with x → β and y →

−βµ − y finally. Other derivatives have the same meaning. Continuing to take

derivatives over yj and µj on the two sides of Eqs. (4.32) and setting y = 0 in the

end, together with the definitions of currents (4.28) and noises (4.29), it is easy to

find

Tj∂J+

i

∂µj+ Ti

∂J+j

∂µi= −S+

ij + TiTj∂µiµjQ(β,−βµ;β,µ;−). (4.33)

A term ∂µiµjQ(0, 0;β,µ; +) that appears in the derivation is set to zero, since

Q(0, 0;β,µ; ν) = 0. Note that the last term in Eq. (4.33) is a quantity of the

system with “−” chirality, while other terms are quantities of the system with “+”

chirality. When the system is in equilibrium, it has been shown that the last term,

which does not have a clear physical interpretation, is zero [125, 129]. Introducing

electric current Ii = qJ+i and noise Sij = q2S+

ij for the system with “+” chirality, we

arrive at the standard FDT Sij = −T (Gij +Gji) with T = Ti = Tj the temperature

161

of the whole system, and Gij = ∂Ii/∂Vj the linear conductance.

If the system is in a non-equilibrium steady state, the last term of Eq. (4.33)

is usually nonzero. However, the chirality helps to eliminate that term if we con-

sider the cross noise S+1j of J+

1 and J+j with j 6= 1. This can be seen by writing

∂µ1µjQ(β,−βµ;β,µ;−) as the sum of derivatives of Q1 and Q2. Considering the

“−” chirality, Q1 depends only on µ1 while Q2 does not depend on µ1, so the last

term of Eq. (4.33) disappear. Moreover, if j 6= r, the first term of Eq. (4.33) ∂J+1 /∂µj

is also zero, since in the system with “+” chirality J+1 only depends on βr and µr.

Hence we obtain

S1j = −T1∂Ij∂V1

. (4.34)

The physical meaning of this formula is that the cross noise between the currents

in reservoir 1 and j is connected to the response of the current in reservoir j to the

voltage at reservoir 1, regardless of the non-equilibrium nature of the system. If

j = r, the term ∂J+1 /∂µr is not zero, instead equals G/q2, with G the conductance

of a two-terminal setup. The reason is that ∂J+1 /∂µr is all the response of the lower

edge, so replacing the upper part with a single chiral edge is irrelevant. Note G is

a topologically protected constant, being νq2/h in QHE systems with ν the filling

factor.

Similar results can be obtained for heat currents and noises if we take derivatives

of Eq. (4.30) over xi and βi while keeping yi and µi fixed. For the systems with “+”

chirality and j 6= 1, r, we obtain

S1j = −(T1)2 ∂Jj∂T1

(4.35)

where the label “+” is dropped since all quantities belongs to the system with “+”

162

chirality. For j = r, there is an additional term ∂J1/∂βr = −(Tr)2∂J1/∂Tr on the

right hand side of Eq. (fdt-heat). ∂J1/∂Tr is the thermal Hall conductance of a two-

terminal Hall bar, equal to κπ2Tr/3h with κ counting the number of chiral modes

including both neutral and charge modes.

We stress that all above response functions are the responses of currents in a

non-equilibrium steady states, as well as the noises. The FDT (4.34) and (4.35) are

correct for all chiral systems, as long as Ti and Vi are smaller than the energy gap

in the bulk. The same FDTs in the system with “−” chirality can be obtained. For

non-chiral systems, for examples QHE systems with counter-propagating modes at

the edges, the last term of Eq. (4.33) is nonzero in non-equilibrium states, which

breaks the chiral-system FDT.

To verify the chiral-system FDT, experimentalists need to measure cross noises,

which is hard in practice. A three-terminal setup (the r = 3 case of Fig. 4.6), as is

considered in Ref. [131], helps us to avoid cross noise measurements. Due to particle

number conservation, the electric currents I3 = −I1−I2. Hence, S3 = S1+S2+2S12,

where Si is the auto-correlation noise and S12 is the cross noise. Since the edges

connected to reservoir 1 are always in equilibrium with it, we have S1 = 2GT1. Then

by using the chiral-system FDT, we have

S3 = S2 − 2T1∂I2∂V1

+ 2GT1, (4.36)

agreeing with the one found in Ref. [131]5. All the quantities this equation can be

measured, and the equation can be tested.

No such simple relation exists for heat transport in the three-terminal setup, if

5Note that there is a factor-of-2 difference between the definition of noises in the current paperand that in Ref. [131].

163

different voltages are applied in the reservoirs. Voltage differences, i.e. electromotive

forces, do work on the system. The work finally turns into Joule heat. So J3 6=

−J1 − J2 in general. If all the reservoirs are kept at the same voltage, with the

thermoelectric voltages compensated as well, we have J3 = −J1 − J2 and a similar

equation as Eq. (4.36).

4.6 Conclusion

In conclusion, we established a non-equilibrium FDT (4.1) for chiral systems, both

close (V ≪ T ) and far (V > T ) from equilibrium. The result does not apply to non-

chiral conductors and can be helpful in the search of counter-propagating modes on

quantum Hall edges [47, 97]. We further generalize the theorem to a multi-terminal

setup and heat transport as well through the formalism of exchange fluctuation

relations in quantum open systems.

Chapter Five

Summary

165

In summary, we have presented theoretical analysis of two different aspects of the

transport in the 5/2 FHQE and a proof of fluctuation dissipation theorems for general

chiral systems in non-equilibrium steady states.

The two projects on 5/2 FHQE, one focusing on probing anyons and the other

on probing the edge structure, are based on particular models. We predict exper-

imentally observable phenomena to identify the nature of the exotic 5/2 FQHE.

Certainly, the actual 5/2 state may differ from the models that we study. However,

the probing methods that we study, interferometry and momentum-resolved tun-

neling transport, are of great importance. Our theoretical investigations provide a

better understanding of these methods.

The fluctuation dissipation theorems, both for electric and heat transport, are

correct for all chiral Hamiltonian systems. The theorem is model independent, with

chirality being the only assumption. We provide two approaches to prove the the-

orem, one based on a mixture of Kubo and Landauer-Buttiker formalisms and the

other based on the fluctuation relations that are correct for general non-equilibrium

systems.

Appendix A

Multi-component Halperin States

167

In this appendix we consider a general multi-component Halperin state [72] with the

filling factor ν = 2 + k/(k + 2). In the first section of the appendix we describe

the states. In the second section we calculate the current and noise in the situation

in which quasiparticles of one flavor dominate transport through the interferometer.

This is the case of main interest for k > 2. In the third section we consider a general

situation. Below we ignore the lowest filled Landau level and concentrate on the

fractional quantum Hall effect in the second Landau level. Indeed, the most relevant

tunneling operators involve only the fractional edge modes.

A.1 Quasiparticle statistics and edge modes

The multi-component Halperin state [72] with the filling factor ν = 2 + k/(k + 2)

can be described by a k × k matrix (Kmult)ij = 1 + 2δij . All components of the

charge vector t equal 1 and the most relevant quasiparticles are described by vectors

li = (0, . . . , 0, 1, 0, . . . , 0), where the number 1 stays in position i. The charge of the

elementary excitations is q = e/(k + 2). The state of the interferometer is described

by the numbers n1, . . . , nk of the trapped quasiparticles of each of the k types. The

statistical phase, accumulated by a particle of type lm going around the hole in the

interferometer, is

θ = 2π∑

p

K−1mpnp = π[nm − n/(k + 2)], (A.1)

where n =∑

np. Hence, the tunneling probability for a particle of type m is

p = Am[1 + um cos(2πΦ

(k + 2)Φ0+ δm + πnm − πn

k + 2)], (A.2)

where Am, um and δm are real constants.

168

An alternative description of the same topological state of matter can be for-

mulated in terms of single-component hierarchical states. The starting point is the

ν = 1/3 Laughlin state. Condensation of quasiparticle pairs on top of the Laugh-

lin state gives rise to the ν = 2/(2 + 2) hierarchical state. Condensation of its

quasiparticle pairs results in the ν = 3/(3 + 2) state and so on. The appropriate

K-matrix Kh has the following nonzero elements: K11 = 3, Knn = 4 (k ≥ n > 1),

Kn,n−1 = Kn−1,n = −2. The K-matirx Kmult expresses via Kh as Kmult =WKhWT ,

where Wij = 1, if i + j ≤ (k + 1), and Wij = 0, if i + j > k + 1. The charge vector

reads t = (1, 0, . . . , 0). The l-vectors of the most relevant quasiparticle excitations

are l = (1,−1, 0, . . . , 0), (0, 1,−1, 0, . . . , 0), . . . , (0, . . . , 0, 1,−1), (0, . . . , 0, 1).

In the main part of the article we considered k = 2 and focused on the flavor-

symmetric case. This was justified for k = 2 for two reasons. First, in the flavor-

symmetric case, the Fabry-Perot interferometry cannot distinguish the Pfaffian and

331 states. Second, for the unpolarized 331 state the role of the flavors is played

by the electron spin and an approximate symmetry between two spin projections

in quantum Hall systems may give rise to the flavor symmetry. At k 6= 2 neither

reason applies. Indeed, the Fabry-Perot interferometry can distinguish the states

with k 6= 2 from proposed non-Abelian states [15] and k > 2 different flavors cannot

be reduced to different spin projections. In the absence of the flavor symmetry

one expects different tunneling probabilities for different quasiparticle types. As

explained below, we expect, in general, that for one quasiparticle type the tunneling

probability is much higher than for the other quasiparticles.

To understand why this happens we need to identify edge channels in a ν =

k/(k+2) quantum Hall liquid. For this purpose we need to diagonalize the first term

in the edge action [36], Lk = 14π

∫

dtdx[Kh,ij∂tφi∂xφj − Vij∂xφi∂xφj ]. Note that we

use the language of hierarhical states. The diagonalization can be accomplished with

169

the new variables Φn = (n+2)φn− (n+1)φn+1, n = 1, . . . , k−1 and Φk = (k+2)φk.

The action assumes the form

Lk =1

4π

∫

dtdx[

k∑

n=1

K0n∂tΦn∂xΦn − Vij∂xΦi∂xΦj], (A.3)

where K0n = 2/[(n + 1)(n + 2)]. The operator ψn = exp(iΦn) creates an excitation

with one electron charge. The coefficients K0n and the charge created by ψn are

independent of k. This allows for an easy description of the interface between the

ν = k/(k + 2) and ν = (k − 1)/(k + 1) liquids. Indeed, the action for the interface

assumes the following form

L =1

4π

∫

dtdx[

k∑

n=1

K0n∂tΦ

kn∂xΦ

kn −

k−1∑

n=1

K0n∂tΦ

k−1n ∂xΦ

k−1n ]

+ electrostatic interaction + interchannel tunneling, (A.4)

where superscripts k and k−1 refer to the different sides of the interface. The minus

sign before the second term in the action reflects the opposite propagation directions

for the edge states of the two adjacent liquids. The operators exp(iΦpn) create equal

charges for both values of p = k, k − 1. The prefactors in front of ∂tΦpn∂xΦ

pn are

opposite for different values of p. Hence, according to the criterion of stability of edge

states [135,136], the modes Φkn and Φk−1

n gap each other out for each n < k. Thus, the

low-energy interface action reduces to L = 1/(4π)∫

dtdx[K0k∂tΦ

kk∂xΦ

kk − V (∂xΦ

kk)

2]

and contains a single mode Φkk.

Let us now consider a system whose filling factor changes in a step-wise manner:

as one moves across the edge of the quantum Hall bar, the filling factor first changes

from 0 to 1/3, then to 2/(2 + 2), then to 3/(3 + 2) etc. The filling factor in the

innermost part of the sample is k/(k+2). The edge action can still be written in the

170

form (A.3). Different modes Φn correspond to spatially separated interfaces between

consecutive regions with different filling factors. A quantum point contact in such

systems brings close to each other two innermost edge channels, corresponding to

the mode Φk. Clearly, any tunneling process can involve only those two interfaces.

Hence, only one type of the quasiparticles with the l-vector (0, . . . , 0, 1) is allowed to

tunnel. By shrinking the regions with intermediate filling factors 0 < ν < k/(k + 2)

one can deform the above system into a quantum Hall bar with a single interface

between the ν = 0 and ν = k/(k+2) regions. Still, it is clear that the innermost edge

channel corresponds to the field Φk. The tunneling probability rapidly decreases as

the path, traveled by the tunneling particle, grows. Hence, the maximal tunneling

probability corresponds to quasiparticles of type (0, . . . , 0, 1) and it is reasonable to

neglect all other tunneling processes.

Certainly, this is an approximation. In general, the tunneling probabilities (C.1)

depend on the parameters Am, um and δm. Note that one phase δm can be excluded

by absorbing it into the Aharonov-Bohm phase due to the magnetic flux Φ. Two

more parameters can be excluded by tuning gate voltages at the point contacts. Still

3(k − 1) fitting parameters remain. An expression with a large number of fitting

parameters is of limited use. Fortunately, the model with only one type of tunneling

quasiparticles becomes exact, if the edges of the Mach-Zehnder interferometer corre-

spond to the interface between ν = k/(k + 2) and ν = (k − 1)/(k + 1) liquids. This

can be accomplished if in addition to the gray (green online) region with the filling

factor 2 + k/(k + 2) in Fig. 2.1 and the white region with the filling factor 0 one

creates a strip with the filling factor 2 + (k − 1)/(k + 1) along the edges. As shown

in the next section, it is possible to derive an expression for the Fano factor without

any fitting parameters in that case. This situation will be our main focus below.

171

A.2 The case of only one flavor allowed to tunnel

In this section we will omit the index m in Eq. (A.2) since it can assume only one

value. We will also set δm = 0 since it can be absorbed in the Aharonov-Bohm phase.

We will focus on the limit of low temperatures so that quasiparticles only tunnel from

the edge with the higher potential to the edge with the lower potential. The tunneling

probability depends only on the number n of the trapped quasiparticles and reads

pn = A[1 + u cos(2πΦ

(k + 2)Φ0+πn(k + 1)

k + 2)], (A.5)

The number of different probabilities pn depends on the parity of k. For odd k, pn =

pn+k+2 and hence we can define n modulo k+2. For even k, pn = pn+2(k+2) 6= pn+k+2

and n should be defined modulo 2(k + 2). Hence, the number of different values of

pn equals nmax = (k + 2)[3 + (−1)k+2]/2. In what follows we will see a considerable

difference between even and odd k.

It will be convenient to use the following parametrization of Eq. (A.5)

pn = B[v exp(−iφ0) + exp(inθ)][v exp(iφ0) + exp(−inθ)], (A.6)

where B = A[1 −√1− u2]/2, v = [1 +

√1− u2]/u, φ0 = 2πΦ/[(k + 2)Φ0] and

θ = π(k + 1)/(k + 2). All exponents exp(inθ) are nmaxth roots of 1.

A.2.1 Current

We now use Eq. (A.6) for the calculation of the current. The current I = limt→∞Nq/t,

where q = e/(k+2) is a quasiparticle charge and N is the large number of tunneling

172

events over a long time period t. It is easy to find the average time tn between

the tunneling events which change the number of the trapped quasiparticles from

n− 1 to n and from n to n+ 1: tn = 1/pn. Taking into account that there are only

nmax different values of pn, one finds the average time of nmax consecutive tunneling

events:

t = nmax limt→∞

t/N =

nmax∑

n=1

1/pn. (A.7)

Hence,

I =nmaxq∑

1pn

. (A.8)

We next evaluate the sum t =∑

1/pn. Similar sums were evaluated numerically

in the main part of the article. For an arbitrary k we need an analytic expression.

Two different expressions hold for even and odd k. We will only discuss the details

for even k as the case of odd k can be considered in a similar way.

Using the parametrization (A.6) one gets

t =

∑

n |Qn(v exp(−iφ0))|2B|P (v exp(−iφ0))|2

, (A.9)

where P (z) =∏

l(z + exp(ilθ)) and Qn(z) = P (z)/(z + exp(inθ)).

P (z) can be found from the basic theorem of algebra. Indeed, the roots of that

polynomial are known and equal − exp(ilθ). There is only one such polynomial

of power nmax = 2(k + 2) with the coefficient 1 before znmax . This polynomial is

P (z) = znmax − 1. From this one gets Qn(z) =∑nmax−1

l=0 zl(−1)l+1 exp(−iθn[l + 1]).

The sum∑

n |Qn|2 can now be calculated by first performing the summation over n

173

and reduces to∑

n |Qn|2 = nmax(v2nmax − 1)/(v2 − 1). Finally, for even k

te =nmax(v

2nmax − 1)

B(v2 − 1)[v2nmax − 2vnmax cosnmaxφ+ 1]; (A.10)

Ie =eB(v2 − 1)

k + 2

v4k+8 − 2v2k+4 cos 4πΦΦ0

+ 1

v4k+8 − 1. (A.11)

For odd k, a similar calculation yields

Io =eB(v2 − 1)

k + 2

v2k+4 + 2vk+2 cos 2πΦΦ0

+ 1

v2k+4 − 1. (A.12)

The expressions look similar for even and odd k but there is a significant dif-

ference between them. Indeed, for odd k, the current is a periodic function of the

magnetic flux with the period Φ0. For even k the period is two times shorter. Such

superconducting periodicity reflects Cooper pairing of composite fermions.

A.2.2 Noise

The noise is defined as S = 2 limt→∞(Q2 − Q2)/t, where Q is the total charge

transmitted during the time period t. Let us set t = mt, where m is a large integer.

Such choice of t corresponds on average to N = mnmax tunneling events. The total

time required for N tunneling events can be expressed as τ = mt+ δτ , where δτ is a

fluctuation. The total charge transferred through the interferometer after N events

is exactly Nq. In a good approximation, the charge transferred over the time t = mt

is then Q = Nq− Iδτ , where I is the average current (A.11,A.12). Substituting this

expression in the definition of the noise one gets S = 2I2δτ 2/t. The calculation of

δτ 2 is straightforward. One finds: δτ 2 = mδt2, where δt2 =∑nmax

n=1 1/p2n. Finally, the

174

Fano factor

e∗ =S

2I= nmaxq

∑

1/p2n(∑

1/pn)2. (A.13)

Some restrictions on e∗ are evident from elementary inequalities. Obviously,

e∗ ≤ nmaxq. The inequality of quadratic and arithmetic means also implies that

e∗ ≥ q. The first inequality sets different upper limits on the effective charge e∗ for

even and odd k. For odd k, e∗ < e. For even k, e∗ < 2e. This difference agrees

with the difference of the magnetic field dependences of the current, discussed above.

Both upper limits can be reached as we will see below.

As in the calculation of the current, we focus on the case of even k and only

give the final answer for odd k. We need to find δt2. One can easily see that in the

notation of the previous subsection

δt2 =

∑nmax−1n=0 |Qn(v exp(−iφ0))|4B2|P (v exp(−iφ0))|4

=nmax

B2|P |4nmax−1∑

l,m,p,q=0

(−v)l+m+p+q exp(iφ0[p+ q − l −m])

×∑

s

δ(p+ q − l −m− nmaxs), (A.14)

where s is an integer and the discrete delta function δ(0) = 1, δ(a 6= 0) = 0. The

remaining summation is tedious but straightforward. For even k we obtain

e∗e =e

k + 2(

w − 1

w2k+4 − 1)2[

d

dwwd

dw

w2k+5 − 1

w − 1+

(4k + 7− wd

dw)2w4k+7 − w2k+4

w − 1+

2wk+2 cos(4πΦ

Φ0

)(2k + 3− wd

dw)d

dw

w2k+4 − 1

w − 1], (A.15)

where w = [1 +√1− u2]2/u2.

175

A similar calculation for odd k yields

e∗o =e

k + 2(w − 1

wk+2 − 1)2[

d

dwwd

dw

wk+3 − 1

w − 1+

(2k + 3− wd

dw)2w2k+3 − wk+2

w − 1−

2w(k+2)/2 cos(2πΦ

Φ0)(k + 1− w

d

dw)d

dw

wk+2 − 1

w − 1], (A.16)

The expressions are rather similar for even and odd k but their periodicity as a

function of the magnetic flux is different just like for the current.

For a general u, the above expressions are complicated. They simplify in an

important limit case considered below. By tuning the gate voltages at the tunneling

contacts it is always possible to make equal the tunneling amplitudes at the two

contacts. The desired situation can be achieved by selecting such gate voltages that

the total currents would be the same when only one contact is open no matter which

one it is. As discussed in section 2.5, this corresponds to u = 1 at a sufficiently low

temperature and voltage bias. The expressions for the Fano factor greatly simplify

in that limit. For even k,

e∗e = e8(k + 2)2 + 1 + [4(k + 2)2 − 1] cos 4πΦ

Φ0

6(k + 2)2. (A.17)

For odd k,

e∗o = e2(k + 2)2 + 1− [(k + 2)2 − 1] cos 2πΦ

Φ0

3(k + 2)2. (A.18)

As a function of the flux, e∗e oscillates between e[23+ 1

3(k+2)2] and 2e. e∗o oscillates

between e[13+ 2

3(k+2)2] and e. The maximal values of the Fano factor are thus 2×e at

odd k and 2× 2e at even k. The minimal values of the Fano factor uniquely identify

states with different k.

176

A.3 General case

Here we address the situation when all quasiparticle flavors are allowed to tunnel.

As discussed above, this situation is less interesting than the case of just one flavor

allowed to tunnel. Indeed, the expressions for the current and noise depend on

numerous fitting parameters and thus are less useful than the results (A.17,A.18).

Besides, in a general case it would be difficult to tune the system to reach its maximal

possible Fano factor. Instead, there is an interval of relevant Fano factors and this

makes the identification of a state more difficult. Nevertheless, we discuss below how

one can calculate the transport properties for a general k when all flavors are allowed

to tunnel.

The current can be found from the system of equations of the form Eq. (2.7,2.8).

fl in these equations are the probabilities to find the trapped topological charge l in

the interferometer. Thus, it is important to understand how many different values

of the topological charge are possible.

Different topological charges correspond to different np in tunneling probabilities

(C.1). Different np may however describe the same topological state. This happens,

if all tunneling probabilities (C.1) are the same for a certain set of np and another

set of np +∆p. The probabilities are equal, if for each m

π∆m −π∑

p∆p

k + 2= 2πrm, (A.19)

where rm is an integer. One finds from (A.19) that ∆m = 2rm +∑

p ∆p/(k + 2).

177

Adding up such equations for all m, one finds that∑

p∆p = (k+2)∑

p rp and hence

(∆1, . . . ,∆k) =k∑

m=1

rmdm, (A.20)

where dm = (1, . . . , 1, 3, 1, . . . , 1) and the number 3 stays in position m. Eq. (A.20)

means that adding or subtracting a vector dm from the l-vector n = (n1, . . . , nk) does

not change the topological charge. dm can be understood as l-vectors of electrons.

If we choose rk = 1 and rp = −1 for one p < k, then adding the l-vector (A.20)

to n increases nk → nk + 2 and decreases np → np − 2. Setting rk = −1 and rp = 1

corresponds to the operation nk → nk − 2, np → np + 2. This operation involves

adding to n one vector dm and subtracting another vector dn. Performing several

such operations, one can always reduce all np with p < k to zeroes and ones. Choosing

rk = k+1 and rp = −1 for all p < k corresponds to the operation nk → nk+2(k+2),

np → np. The operation involves 2(k+1) additions and subtractions of dm’s. Setting

rk = −(k + 1) and rp = 1 for all p < k corresponds to nk → nk − 2(k + 2), np → np.

Hence, by adding and subtracting vectors dm one can always reduce l-vectors to

topologically equivalent vectors of the form (s1, . . . , sk), where sp = 0, 1 at p < k

and 0 ≤ sk < 2(k+ 2). We will call such vectors allowable. The total number of the

allowable vectors is 2k(k + 2).

We have established that any l-vector can be reduced to the allowable form by

an even number of additions and subtractions of vectors dm. Moreover, one can

get exactly one allowable vector Ne(n) from each n by an even number of additions

and subtractions. Indeed, an even number of additions and subtractions does not

change the parity of each component of the l-vector. This fixes sp, p < k. The

residue (∑

nm) mod 2(k + 2) is also invariant with respect to an even number of

subtractions and additions. This fixes sk. As a consequence, no other allowable

178

vectors can be obtained from an allowable vector by an even number of additions

and subtractions.

Next, note that No(n) = Ne(n + d1) is the only allowable vector that can be

obtained from n by an odd number of additions and subtractions. Indeed, if another

allowable vector Mo(n) could be obtained from n by an odd number of additions

and subtractions then No(n) and Mo(n) could be obtained from each other by an

even number of additions and subtractions: one first gets n from Mo(n) and then

No(n) from n. We have already proved that this is impossible.

Thus, exactly one allowable vector Ne(n) can be obtained from n by an even

number of additions and subtractions and exactly one allowable vector No(n) can be

obtained by an odd number of additions and subtractions. Note that Ne(n) 6= No(n)

since the parity of each component sp changes after each addition or subtraction and

hence the parities of the components of Ne(n) and No(n) are opposite. Note also

that Ne(n) = No(No(n)) and No(n) = No(Ne(n)). Thus, the set of allowable

vectors consists of 2k−1(k + 2) pairs of the form (V,No(V)). Different pairs have

no common elements. l-vectors from different pairs cannot be obtained from each

other by any number of additions and subtractions. If an l-vector can be reduced

to the allowable vector V by an even number of additions and subtractions it can

also be reduced to No(V) by an odd number of additions and subtractions. Let

us now select one arbitrary vector from each pair of allowable vectors (V,No(V)).

We get a set ג of 2k−1(k + 2) allowable vectors. An arbitrary l-vector (n1, . . . , nk)

is topologically equivalent to one of the vectors in .ג Hence, there are 2k−1(k + 2)

different topological charges.

Introducing a probability distribution fl for different trapped topological charges

l, one can compute the current from the system of equations (2.7,2.8). Even for

179

k = 2 the result is complicated and in the main part of the article we showed it only

in the graphical form (Figs. 4.3, 2.4) except for the flavor-symmetric case when it

simplifies considerably.

A simplification is also possible in the flavor-symmetric case for a general k. In

that case one can remove the indexm from the constants Am, um and δm since they do

not depend on the flavor. Let us set n = (∑

np) mod 2(k+2) and S =∑

(np mod 2),

i.e., let n denote the total number of trapped quasiparticles modulo 2(k + 2) and S

show how many np’s are odd. Note that n and S have the same parity. The number

of possible pairs (n, S) equals C = (k + 1)(k + 2). Let us introduce a distribution

function fn,S. A simplification of the steady state equations results from the fact

that one can write a closed set of equations for fn,S:

0 =dfn,Sdt

= −A[1− u cos(2πΦ

(k + 2)Φ0+ δ − πn

k + 2)]S

+A[1 + u cos(2πΦ

(k + 2)Φ0

+ δ − πn

k + 2)](k − S) × fn,S

+fn−1,S−1(k − S + 1)A[1 + u cos(2πΦ

(k + 2)Φ0+ δ − π(n− 1)

k + 2)]

+fn−1,S+1(S + 1)A[1− u cos(2πΦ

(k + 2)Φ0+ δ − π(n− 1)

k + 2)], (A.21)

where the notation implies that fn,S = 0 for S < 0 and S > k. The current can then

be expressed as

I =∑

fn,SA[1− u cos(2πΦ

(k + 2)Φ0+ δ − πn

k + 2)]S

+A[1 + u cos(2πΦ

(k + 2)Φ0+ δ − πn

k + 2)](k − S) (A.22)

180

System (A.21) contains C = (k+1)(k+2) equations, much fewer than (k+2)2k−1

in a general case. Moreover, one can reduce the total number of equations to no

more than k/2 + 1. Indeed, Eq. (A.21) allows one to express fn,S via fn−1,S±1.

fn−1,S±1 can be expressed via fn−2,S′, where S ′ = S − 2, S, S + 2, etc. After 2(k+ 2)

steps, one expresses fn,S via the values of the distribution function fn−2(k+2),S, where

S = 0, . . . , k. Since n is defined modulo 2(k+2), this means that a closed system can

be obtained for k+1 variables fn,S for any fixed n. At least k/2 of those variables are

zeroes since n and S have the same parity for any nonzero fn,S. This leaves no more

than k/2 + 1 equations. After the system is solved, one can immediately compute

fn+1,S from fn,S with Eq. (A.21), then express fn+2,S via fn+1,S and so on.

Appendix B

Integer Edge Reconstruction

182

In the appendix we determine the number of the conductance singularities in the

setup Fig. 3.1 in the presence of additional integer edge modes due to the recon-

struction of the integer QHE edge. As an example, we consider the 331 state. The

situation is similar for other states.

We assume strong interaction between all modes. Additional modes due to edge

reconstruction appear in pairs of counter-propagating modes so that the total Hall

conductance is not affected. Let there be n = (n↑+n↓) additional modes, where n↑/↓

denotes the number of additional modes with the spin pointing up/down. We need

to consider two types of operators: 1) most relevant additional tunneling operators

create one electron charge in one of the additional modes; 2) operators that add

one electron charge to one of the integer modes and transfer one electron charge

between two other integer modes with the same spin. The operators of the second

group are less relevant than the operators of the first group but their contribution

to the current can be comparable with the contribution of the operators describing

tunneling into fractional modes (cf. Sec. 3.5).

We find n new operators of the first type. The number of the operators of the

second type equals

m = (n↑ + 1)n↑(n↑ − 1)/2 + (n↓ + 1)n↓(n↓ − 1)/2

+(n↑ + 1)n↓(n↓ + 1) + (n↓ + 1)n↑(n↑ + 1). (B.1)

The total number of the modes equals n + 6. Hence, each tunneling operator is

responsible for n+ 6 singularities and their total number is (4 + n +m)(n + 6). At

large n this number grows as n4. Such growth of the number of the singularities

183

limits the utility of the proposed approach when n is large since it may be difficult

to resolve the singularities.

Appendix C

FDT in an Ideal Gas Model

185

SD

SS

ITSC

Figure C.1: Ideal gas in a reservoir with a tube.

In this appendix we address a non-equilibrium FDT for an ideal gas system, briefly

discussed in the introduction.

We consider a large reservoir filled with an ideal gas of non-interacting molecules

at the temperature T and chemical potential µ. Molecules can leave the reservoir

through a narrow tube with smooth walls (Fig. C.1). Collisions with the tube sur-

face are elastic and do not change the velocity projection on the tube axis. Thus,

molecules only move from the reservoir to the open end of the tube and the system

is chiral. Imagine now that molecules can escape through a hole in the wall of the

tube. We derive a relation similar to (4.1):

SD = SC − 4T∂IT∂µ

+ SS, (C.1)

where IT is the particle current through the hole in the tube wall, and SD =∫

dt〈∆ID(t)∆ID(0)+∆ID(0)∆ID(t)〉, SS =∫

dt〈∆IS(t)∆IS(0)+∆IS(0)∆IS(t)〉, and

SC =∫

dt〈∆IC(t)∆IC(0)+∆IC(0)∆IC(t)〉 are respectively the particle current noises

at the open end of the tube, at the opposite end of the tube, and at the hole (Fig. C.1).

The noise SS can be determined from the measurement of SD in the geometry without

a hole in the tube wall.

The simplest proof of Eq. (C.1) is a direct calculation along the lines of Ref. [137].

186

The calculation is especially simple in the case of an ideal classical gas which should

be understood as a Fermi gas with a high negative chemical potential in order to

use the above reference. Quantum Fermi- and Bose-gases are also easy to consider.

The current and noise can be expressed as sums of contributions from different small

energy intervals. Let f = 1/[exp(E−µ/T )+1] be the Fermi distribution function

for a particular energy and TE the transmission coefficient through the tube wall for

that energy (TE may depend on the channel number, if there are many channels).

According to Ref. [137], in a Fermi gas, the contribution from a corresponding energy

window to the current, tunneling through the walls, is proportional to TEf , the

contribution to SS is determined by 2f(1 − f), SC by 2TEf(1 − TEf) and SD by

2(1 − TE)f(1 − (1 − TE)f). A combination of these contributions gives the desired

theorem (C.1).

One can also generalize our QHE proof. This approach, certainly, is harder than

a direct calculation. The situation simplifies for a degenerate Fermi gas whose par-

ticles can tunnel outside the tube only for energies, close to the Fermi level. Such

gas can be mapped onto a model of charged particles whose mutual interaction is

completely screened by the gate. The electric current and noise of such charged par-

ticles equal their mass current and noise up to a trivial coefficient. The connection

of the chemical potential and voltage bias is obvious. Such model would describe

left-moving electrons in a quantum wire in the language of the Landauer-Buttiker

formalism. Its low-energy effective Hamiltonian is related to the integer QHE edge

physics. Tunneling through the tube walls plays exactly the same role as the tun-

neling into conductor C in the QHE setting. The only important difference from a

QHE setting, Fig. 4.1, is the absence of the upper edge. Thus, the derivation from

the paper can be repeated with only one modification: SU should be set to zero

in Eq. (4.2). A small modification involves then Eq. (4.4): now SS simply equals

187

the noise at the open end of the tube in the absence of the hole in its side. That

quantity must be substituted instead of 4GT in Eq. (4.1). Nothing else changes in

that equation. [57]

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