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

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. Mitrovi´c, 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 importantly 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-B¨uttiker 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

A conductance Gtun 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

A of the ﬁrst or higher derivative of the voltage dependence of Gtun. 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

2 the plateaus, the Hall resistance has quantized value RH = h/e ν.

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 pa = pb p and pb = 0 are illustrated in Panels (b) and k k ≡ k k (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 r11 = r12 =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 k k...... 85 0 − 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

A arbitrary units. (b) Momentum mismatch dependence of Gtun 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-

A,u ferential conductance Gtun 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

A,u Gtun at a ﬁxed momentum mismatch ∆k for 3 cases of diﬀerent scal-

ing exponent ranges: (a) 0

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

A,u for the momentum mismatch dependence of Gtun at a ﬁxed ω with

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

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

A 3.8 (a) Voltage dependence of the tunneling diﬀerential conductance Gtun

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

A tum mismatch dependence of Gtun in the Pfaﬃan state. The reference

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

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

A,λ 3.9 The diﬀerential conductance Gtun in the edge-reconstructed Pfaﬃan state. Panels (a) and (b) show the voltage and momentum mismatch

A,λ dependence of Gtun (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 A 3.10 Diﬀerential conductance Gtun in the non-equilibrated anti-Pfaﬃan

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

A right panels show the momentum mismatch dependence of Gtun, 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

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

A ∆k0 = ω/v2. Gtun 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 remarkable 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

2 the quantum Hall effect is the quantization of the Hall resistance Rxy = h/νe to a precision of one part in 109, and the vanishing of the dissipative longitudinal resistance R , at certain magnetic fields. Plateaus appear on the R B curve. xx xy ∼ 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 introduce the quantum Hall effect, heavily on the theoretical side. Accompanied with the introduction, I gradually unveil the fundamental motivations of my graduate research 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 experimentally 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 = Bzˆ, 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,

VH which gives rise to an electric ﬁeld w yˆ. Electrons ﬂow in a straight line after the electric force due to the Hall voltage equals the Lorentz force,

q V v B = q H yˆ (1.1) c × − w

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, | |

V B R = H = . (1.2) H I nqc

+ + + + + + + + + + + + + + + + + B I VH y x − − − − − − − − − − − − − − − − −

Figure 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. 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,

h 1 R = . (1.3) H e2 ν

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

nhc n density of particles ν = = = . (1.4) eB Be/hc density of magnetic ﬂux quanta

Φ hc/e is the magnetic ﬂux quantum. In the quantum mechanical treatment, we 0 ≡ 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],

V mv e x = (1.5) l τ

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

resistance V m l R = x = , (1.6) xx I ne2τ w

. 2 a constant for a given sample. The quantity ρxx = m/ne τ is the longitudinal

1In the original Drude theory, the relaxation time τ measures the mean time between two collisions with the ion cores. The band theory, which is quantum mechanical, tells us that the ion cores in a perfect periodic structure cannot scatter wave-like electrons. The relaxation time τ results 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 if electron-electron interaction exists. 6

resistivity. The value of ρxx tells how clean the sample is. A pure sample without imperfections has ρ 0. A more common quantity to describe the purity of a xx → 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 R deviated ∼ H 7

2 RH (h/e )

1/2

1/3 Rxx(arb.unit) 1/4 1/5

B 10 Tesla ∼ Figure 1.2: Sketch of the quantum Hall effect. When plateaus occur in the magnetic field dependence of the Hall resistance (red), the longitudinal resistance (blue) vanishes when the temperature 2 is low enough. On the plateaus, the Hall resistance has quantized value RH = h/e ν. IQHE has integer ν while FQHE (not shown) has fractional ν. At very low magnetic fields, the classical Hall effect takes places.

3 4

S D

1 2

Figure 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. 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 excitations, 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 unambiguous 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 important, 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/Vs ∼ × 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 realized 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, ~2π2 E = n2, n =1, 2, 3 . (1.7) n 2mW 2 ···

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 = E E 26 meV 2 − 1 ≈ 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

eB ω = . (1.8) c mc

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 magnetic ﬁeld is e~B EZ = gµBB = g . (1.9) 2me

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, E 1.5K. Z ≈ Certainly, Zeeman energy is related to the spin polarization of a QHE system, 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 electrons 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 E = 50 K. ∼ C

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

1 q 2 q 2 2 H = (px Ax) +(py Ay) + pz + V (z), (1.10) 2m∗ − c − c h i 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 = Bzˆ. The potential V (z) = 0, if 0 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 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]

Subband Cyclotron Zeeman ν =1/3 ν =5/2 Energy 310K 100K 1.5K 1 10K 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 Schr¨odinger’s equation, one ﬁnds that

~2 2 s π 2 En = 2 N , N =1, 2, 3, (1.11) 2m∗W ··· 2 Nπx χ (z)= sin . (1.12) n W W r

The index N labels the N-th subband.

It is also easy to solve the Schr¨odinger’s equation in the xy plane. We have the eigenenergy and wave function

1 ǫ =(n + )~ω , n =0, 1, 2, (1.13) nk 2 c ··· 1 1 2 2 iky 2 2 (x+kl ) ψnk(x, y)= e Hn(x + kl )e− 2l , (1.14) √Ly

eB ~c with ωc = m∗c the cyclotron frequency, l = eB the magnetic length, and Hn the nth q 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 L L . k is quantized to 2mπ/L and x × y y also restricted by the physical size along the x-direction: 0 < kl2 < L . So we − x 2 have the degeneracy being LxLy/2πl = 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

ǫ/~ωc ǫ/~ωc

DOS DOS (a) (b)

Figure 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.

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 extended, able to conduct electric current. Dramatically, the contribution to the electric 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 conductance, 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 the longitudinal 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

1 ǫ˜ (n + )~ω + V ( kl2), (1.15) nk ≈ 2 c −

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

Fig. 1.5). The group velocity of the wave ψ˜nk(x, y) is

1 ∂ǫ v = k y.ˆ (1.16) k ~ ∂k

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

1 e ∂ǫ e I = ev = dk nk dk = n (µ µ ). (1.17) L k 2π~ ∂k h L − R y n Xnk X Z 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, µ µ = eV , and we have the Hall conductance equal L − R H 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 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. −

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 appearance 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

1 A = Bzˆ r, (1.18) −2 × 17 gives a magnetic ﬁeld Bzˆ. 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

1 m z 2/4 ψm = z e−| | (1.19) √2πl22mm! 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

N 2 m P zi /4 Ψ = (z z ) e− i | | . (1.20) m i − j i

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, the path can be contracted back to the starting point without meeting any particles. In 2D, particle 2 prevents 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

2 2 2 P xii /12l Ψ= dξ dξ∗ (ξ∗ ξ∗) e− i | | (z ξ )Ψ (1.21) i i i − j i − j 3 i i

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 fractional 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 quasiparticles (negatively charged) and quasiholes (positively charged). Let us focus on 19 quasiholes, whose wave functions are of the form

1 Ψ = (z ξ)Ψ (1.22) ξ i − m C(ξ,ξ∗) i Y p 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 (z ξ)mΨ describes a state with a charge e i i − m hole at ξ. However, (z ξ)mΨ Qcan also be viewed as a state with m quasiholes. i i − m So, a single quasiholeQ 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(r ,r , ) changes to a new wave function Ψ(˜ r ,r , ). In 1 2 ··· 1 2 ··· 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 =1, non-Abelian Anyons. (1.25) h | i 6

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, Ψ , Ψ , , Ψ . The relation for the old and new wave function can be described { 1 2 ··· M } by a unitary matrix U rotating the topological Hilbert space

Ψ˜ 1 Ψ1     Ψ˜ 2 Ψ2   = U   . (1.26)  .     .         ˜     ΨM   ΨM          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

e e Φ φ = ∗ A dx =2π ∗ (1.27) AB ~c · e Φ I 0 contains information about the charge of quasiholes. So, we consider a two-quasihole state 1 Ψ = (z ξ )(z ξ )Ψ . (1.28) ξ1,ξ2 i − 1 i − 2 m C(ξ1,ξ1∗,ξ2,ξ2∗) i Y p Let the ﬁrst quasihole make adiabatic circular motion ξ1(t). The Berry phase γ(t) satisﬁes dγ d = i Ψ Ψ , (1.29) dt h ξ1,ξ2 |dt| ξ1,ξ2 i 21 or

∗ dγ = aξ1 dξ1 + aξ1 dξ1∗ (1.30)

with d d ∗ aξ1 = i Ψξ1,ξ2 Ψξ1,ξ2 , aξ1 = i Ψξ1,ξ2 Ψξ1,ξ2 . (1.31) h |dξ1 | i h |dξ1∗ | i

The Berry phase is given by dγ for a speciﬁed path. Compared to Eq. (1.27), aξ1

∗ H and aξ1 can be viewed as eﬀective gauge ﬁelds interacting with quasiholes. It is not hard to obtain that

i ∂ i ∂ ∗ aξ1 = ln C, aξ1 = ln C. (1.32) 2 ∂ξ1 −2 ∂ξ1∗

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

1 2 2 2/m ( ξ1 + ξ2 ) C = D ξ ξ − e 2ml2 | | | | , (1.33) | 1 − 2|

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

iξ i a = 1∗ (1.34) ξ1 4ml2 − 2m(ξ ξ ) 1 − 2 iξ1 i a ∗ = + . (1.35) ξ1 −4ml2 2m(ξ ξ ) 1∗ − 2∗

Hence the Berry phase equals

1 Φ 2π γ =2π + . (1.36) m Φ0 m

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 experiments [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,

2 2 P zi /4 1 Ψ = (z z ) e− i | | Pf( ), (1.37) MR i − j z z ij i j Y −

where Pf is the Pfaﬃan, the square root of the determinant, of the antisymmetric 23 matrix with the entries 1/(z z ). Explicitly i − j

1 1 1 Pf( )= ( ), (1.38) z z A z z z z ··· i − j 1 − 2 3 − 4

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

2 2 P zi /4 1 Ψ= z (z z ) e− i | | Pf( ), (1.39) i i − j z z i ij i j Y Y − 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

2 2 P zi /4 1 1 Ψ= (z z ) e− i | | (z ), (1.40) i − j A 0 z z z z ··· ij 1 2 3 4 Y − −

or increase the angular momentum of a Cooper pair

2 2 P zi /4 (zi η1)(zj η2)+(zi η2)(zj η1) Ψ= (z z ) e− i | | Pf( − − − − ). (1.41) i − j z z ij i j Y −

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,

2 2 3 3 P ( zi wi )/4 (z z ) (w w ) (z w )e− i | | −| | . (1.44) i − j i − j i − j i

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, 1 S = dxdt∂ φ( ∂ + v∂ )φ (1.45) −4πν x ± t x Z 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

ψ eiφ(x,t) (1.46) 1/3 ∼ 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 annihilation operator is given by

ψ eiφ(x,t)/ν . (1.47) e ∼

We can check that ψe and ψe† 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) h ( i − ±

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 excitations. Eﬀectively, the edge excitations form a Fermi liquid when the droplet is on IQHE plateaus but become a Luttinger liquid on FQHE plateaus. the correlation function

1 ψ† (x, t)ψ (0, 0) . (1.49) h 1/3 1/3 i ∼ (δ + i(t x/v))ν ±

At ﬁnite temperatures, the correlation function is

sin πT [δ + i(t x/v)] φ x, t)φ(0, 0) = ν ln ± . (1.50) h ( i − πT

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

1 N S = dxdt (∂ φ K ∂ φ + ∂ φ V ∂ φ ). (1.51) −4π t I IJ x J x I IJ x J IJ=1 Z X

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

1 is positive deﬁnite. The bosonic modes satisfy [φ (x),φ (x′)] = iπK− sign(x x′). I J IJ − For a FQHE liquid with a single right(left)-moving mode, K = 1/ν. The general ± quasiparticle operator is

i l φ ψl = e PI I J . (1.52)

1 It has the charge et K− 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

1 2πl K− l . 1 · · 2

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

1 S = dxdt∂ φ(∂ + v∂ )φ + i dxdtλ(∂ + v ∂ )λ. (1.53) −2π x t x t n x Z Z

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

1 3 S = dxdt∂ φ(∂ + v∂ )φ + i dxdtλ (∂ v ∂ )λ . (1.55) −2π x t x i t − n x i i=1 Z X Z The existence of upstream modes in the anti-Pfaffian state is a huge difference between Pfaffian and anti-Pfaffian. Experimental evidence currently supports the existence 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.

QPC1 QPC2

Figure 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. 31 the lowest order of tunneling amplitudes,

I Γ +Γ 2 = Γ 2 + Γ 2 +2 Γ Γ cos(∆φ), (1.56) ∝| 1 2| | 1| | 2| | 1 2|

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 quasiparticles 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

e∗ BA φAB =2π (1.58) e Φ0

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 are NL 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

const., if NL is odd I (1.60) ∝   A + B cos(∆φ), if NL is even.

 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

Edge 2 QH liquid B

S2 D2

Edge 1 S1 D1 QPC1A QPC2

Figure 1.9: Mach-Zehnder interferometer. Currents ﬂow from source S1 to drain D2. Similarly to the Fabry-Perot interferometer, two QPCs are introduced. There are two paths for quasiparticles to go from S1 to D2: S1-QPC1-A-QPC2-D2 and S1-QPC1-B-QPC2-D2. The diﬀerence from the Fabry-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 outside the 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 P = Γ 2 + Γ 2 +2 Γ Γ cos(φ + φ + nθ) (1.61) n | 1| | 2| | 1 2| 0 AB 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 tunneling 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 tunneling current is [62] 1 I . (1.62) ∝ 1/ν 1 i=1 Pi Inserting Eq. (1.61), one ﬁnds that the periodicityP 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

δQ(t)2 S =2 (1.63) t

with δQ(t) = Q(t) Q(t) the ﬂuctuation of Q(t). One can calculate the Fano − factor [63] 1/ν 2 S n=0 1/Pn F = =2e 2 . (1.64) I 1/ν Pn=1 1/Pn P 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 interferometry 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-Büttiker approach and Keldysh formalism. While the Landauer-Büttiker 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-B¨uttiker Approach

The idea of the Landauer-B¨uttiker 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-B¨uttiker approach expresses 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 aL   = S   . (1.65) bR aR         . The S matrix is energy dependent and unitary S†S = SS† = 1. Then R(E) = . S 2 = S 2, T (E) = S 2 = S 2 =1 R(E). R(E) is the reﬂection coeﬃcient | 11| | 22| | 12| | 21| −

bL aR SD

aL bR

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)

e i(E E′)t/~ Iˆ(t)= dEdE′e − [a† (E)a (E′) b† (E)b (E′)] (1.66) h L L − L L Z

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† (E)a (E′) = δ δ(E E′)f (E) (1.67) h α β i αβ − α

where α, β=L or R and fα(E) is the Fermi-Dirac distribution of terminal α

1 fα(E)= (1.68) (E eVα)/Tα e − +1

with the Boltzmann constant set to 1. Terminal α has the temperature Tα and

5 chemical potential µα = eVα controlled by thevoltage bias . Now we can easily calculate the average current

e I = Iˆ = dET (E)[f (E) f (E)]. (1.69) h i h L − R Z

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. The charge density along the edge is then changed by chemical potentials of terminals, and we assume the 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

e2 G = I/(V V )= T. (1.70) L − R h

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 S (ω)= dteiωt ∆I (t)∆I (0)+∆I (0)∆I (t) (1.71) αβ h α β β α i Z with ∆I = Iˆ Iˆ . We focus on the zero-frequency case and set ω = 0. Using the α −h i scattering matrix and the relation Eq. (1.67), one can obtain that the zero-frequency noise equals

e2 S =2 dE T (E)[f (1 f )+ f (1 f )]+ T (E)[1 T (E)](f f )2 . (1.72) h { L − L R − R − L − R } Z

Again taking the approximation T (E)= T, and assuming that the temperatures are

the same TL = TR = T one derives that

e2 eV S =2 2T T2 + eV coth T(1 T) . (1.73) h 2T −

One may take a limit that V T and the transmission coeﬃcient T 1, then ≫ ≪

e2 S 2 eV T =2eI, (1.74) ≈ h

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 e2 S =4 T T =4GT. (1.75) h

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 equilibrium 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 nonequilibrium 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-B¨uttiker 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 H ( ) = 0. The system with the Hamiltonian H is assumed to be solved 1 −∞ 0 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( βH ) −∞ − 0 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 Oˆ(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

I iH0t/~ iH0t/~ H1 (t)= e H1(t)e− (1.78)

I iH0t/~ iH0t/~ O (t)= e O(t)e− (1.79)

then the evolution operator S(t, ) satisﬁes −∞

∂S(t, ) i~ −∞ = HI (t)S(t, ). (1.80) ∂t 1 −∞

t i R dτHI (τ)/~ The formal solution is S(t, )= Tˆ e− −∞ 1 , with Tˆ 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) h i −∞ −∞ −∞

with S( , t)= S†(t, ). Inserting the formal solution for S(t, ), we have −∞ −∞ −∞

t t i R dτHI (τ)/~ I i R dτHI (τ)/~ O = Tr Tˆ′e −∞ 1 O (t) Tˆ e− −∞ 1 ρ( ) , (1.82) h i −∞ n o

where Tˆ′ 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),

I i R dτHI (τ)/~ I i R dτHI (τ)/~ O = Tr T O (t)e− K 1 ρ( ) = T O (t)e− K 1 , h i K −∞ h K i n o (1.83) where T is time ordering along the Keldysh contour. is a further abbreviation K hi 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

Oˆ1(t1) −∞ • Oˆ (t ) • 2 2 Oˆ3(t3) −∞ • Figure 1.11: The Keldysh contour. the initial state 0 (normally the ground state of H ), and have | i 0

I i R dτHI (τ)/~ O = 0 T O (t)e− K 1 0 . (1.84) h i h | K | i

We usually have to calculate the expectation value perturbatively, by expanding

I over H1 (t) assuming it is small. The nth order term is

( i)n O = − dτ dτ T [OI(t)HI (τ ) HI(τ )] . (1.85) h in ~nn! 1 ··· nh K 1 1 ··· 1 n i ZK

I I All operators H1 (τi) and O (t) are on the Keldysh contour and the integrals are along the contour as well. The quantity T [OI (t)HI(τ ) HI (τ )] is usually known in h K 1 1 ··· 1 n i 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-Büttiker 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

v H = dx[(∂ φ )2 +(∂ φ )2] (1.86) 0 4πν x L x R Z with φL being the upper left-moving mode and φR the lower right-moving mode.

The two ﬁeld operators satisfy [φ (x),φ (x′)] = iνπsign(x x′), [φ (x),φ (x′)] = L L − − R R iνπsign(x x′), and they mutually commute. The charge densities are ∂ φ /2π and − x L

∂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

i[φL(x)+φR(x)] HT = dxδ(x) γe + H.c. , (1.87) Z with γ being the bare tunneling amplitude.

We calculate transport properties of the system by adiabatically turning the tunneling Hamiltonian H . Suppose that at t = , H is oﬀ and the system is in T −∞ T the ground state 0 of H . H is adiabatically turned on. At time t, we calculate | i 0 T the observable with an operator Oˆ. By using the Keldysh formalism, we have

i R dτHT (τ)/~ O = 0 T O(t)e− K 0 (1.88) h i h | K | i

iH0t/~ iH0t/~ with O(t)= e Oeˆ − and

iH0t/~ iH0t/~ i[φL(x,t)+φR(x,t)] HT (t)= e HT e− = dxδ(x) γe + H.c. , Z

where φL/R(x, t) are Heisenberg ﬁeld operators.

Strictly speaking, the initial state 0 is the ground state of the Hamiltonian | i 45

G = H eV N eV N , where V (V ) is the voltage applied to the left(right) 0 − L R − R L L R 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

φ φ νeV t, φ φ + νeV t. (1.89) R → R − L L → L R

Then the tunneling Hamiltonian becomes

i[φL(x,t)+φR(x,t) νeV t] HT (t)= dxδ(x) γe − + H.c. (1.90) Z with V = V V . L − R

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 I . I is proportional to the rate of the change of the − tun tun particle number on one of the edges. So,

dQ i I (t)= R = e [ dx∂ φ /2π,H (t)] (1.91) tun dt − ~ x R T Z i i[φL(0,t)+φR(0,t) νeV t] = e γe − 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

i I = dt T I (t )H (t ) (1.93) tun −~ 1h K tun 0 T 1 i ZK i t0 = dt I (t )H (t ) H (t )I (t ) . (1.94) −~ 1h tun 0 T 1 − T 1 tun 0 i Z−∞ 46

The free bosonic theory (1.86) gives the zero-temperature correlation function,

φ (x, t)φ (0, 0) = ν ln[δ + i(t x/v)] (1.95) h L L i − − φ (x, t)φ (0, 0) = ν ln[δ + i(t + x/v)]. (1.96) h R R i −

So ﬁnally we obtain

0 e 2 1 1 ieνV t ieνV t I = γ dt (e− e ) (1.97) tun ~2 | | (δ it)2ν − (δ + it)2ν − Z−∞ − e 2 2π 2ν 1 = γ eνV − sign(eV ). (1.98) ~2 | | Γ(2ν)| |

This power law in the I-V characteristic is a feature of Luttinger liquid. The exponent 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

So we see the Fano factor S/ I =2ν e , (1.101) | tun| | | 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 interesting, 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-Büttiker 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 identical 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 interferometer 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 beautiful 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 application 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

Edge 2 QH liquid B

S2 D2 Edge 1 S1 D1 QPC1A QPC2

Figure 2.1: Schematic picture of an anyonic Mach-Zehnder interferometer. Arrows 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. 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 experimental data. In particular, the most elegant and conceptually simple approach to detecting non-Abelian anyons is based on Fabry-Pérot interferometry [48,58–60,81]. It was found recently [82–84] that a Fabry-Pérot interferometer may produce identical 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 quasiparticle 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

1 T and are given by hn = lnK− ln . We are only interested in the most relevant quasi-

1 T particle operators. The quasiparticle charge Q = et K− l . Transport through − n n 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

1 T accumulated by particle 1 moving around particle 2 equals θ12 =2πl1K− l2 .

1 The spin-unpolarized 331 state [71] can be described as a bilayer state with ν = 4 in each layer. The K-matrix is

3 1 K =   . (2.1) 1 3     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 φ = π/4. We will 12 −

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 3 2 − K =   . (2.2) 2 4  −    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 l = (0, 1) and l = (1, 1). All 1 2 − 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 3 2 1 1 − T   = W   W , W =   (2.3) 1 3 2 4 1 0    −          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

x two point contacts as Γk, 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 accumulated 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])

+ x 2 x 2 x x iφmag+iφ w = r ( Γ + Γ )+(r Γ Γ ∗e xd + c.c.), (2.5) x,d 1 | 1| | 2| 2 1 2

where r1(V,T,x) and r2(V,T,x) depend on the voltage, temperature and quasiparticle 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− = exp[ eV/(4k T )]w x,d+x − B 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-

x metry, i.e., we assume that the tunneling amplitudes Γk 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

x 2 x x in the combinations r (V,T,x) Γ and r (V,T,x)Γ Γ ∗, the results do not change 1 | 1,2| 2 1 2 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-Pérot 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

a (a) p1

b (4, 0) (0, 1)p1 (1, 1) (2, 1) a a a p4 p7 p2 b b b p4 p3 p2

b b b p0 p7 p6 a a a p0 p3 p6 (0, 0) (1, 0)b (2, 0) (3, 0) p5

a p5

(b) (c) a a p5 p0 p1 (0, 0) a a 2p4 p7 2p2 p2 (2, 1) (1, 0) p3 (4, 0) (1, 1) (0, 1) (2, 1) (1, 1) (2, 0) (1, 0) (3, 0) (0, 0) (2, 0) a a 2p0 p3 2p6 p7 (0, 1) (3, 0) p6

5 (4, 0) p a a p4 p1

Figure 2.2: Possible states of a Mach-Zehnder interferometer in the 331 state. Panel (a) shows a general case, eight possible states labeled by topological 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 a b b involving particles of ﬂavor b. Special cases with pk = pk pk and pk = 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 identical 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

x x x x pk = A [1 + u cos(πΦ/(2Φ0)+ πk/4+ δ )] (2.6) where x = a or b, and k = 0, 1,..., 7. The parameters Ax = r ( Γx 2 + Γx 2), 1 | 1 | | 2 | x x x x 2 x 2 x x x u = 2 r Γ Γ /[r ( Γ + Γ )] and δ = arg(r Γ Γ ∗). In the presence of the | 2 1 2 | 1 | 1 | | 2 | 2 1 2 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

I 0.3 γ=1 γ=0.25 γ=0.05 γ=0 Pfaffian 0.2

0.1

Φ/Φ0 0 1 2

Figure 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 r11 = r12 =1 and Γ1 =Γ2 such that the maximum matches the 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.

a b a b In Fig. 2.2 (b), we use the notation pk = pk = pk. We will also denote A = A = A , u = ua = ub and δ = δa = δb in the presence of the ﬂavor symmetry. Fig. 2.2(c)

b illustrates another simple limit in which pk = 0.

The transition rates can be used to write down a kinetic equation for the probabilities fd of diﬀerent topological charges d trapped in the interferometer. In terms of the distribution function fd the average tunneling current between the edges

e + + I = f (w + w w− w− ), (2.7) 4 d a,d b,d − a,d − b,d Xd

where d goes over the eight possible states in Fig. 2.2(a). The distribution function 62 satisﬁes the steady state equation

dfd + 0= = (fd xwx,d x + fd+xwx,d− +x) dt − − xX=a,b + f (w + w− ), (2.8) − d x,d x,d xX=a,b and the normalization condition d fd = 1. In Eq. (2.8), (d + x) means the topological charge of the fusion of anyonsP 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

2 u4 eA 1 u + (1 cos(2πΦ/Φ0 +4δ)) I = − 8 − 2 1 ( 3 1 )u2 + u4 (1 1 )(1 cos(2πΦ/Φ +4δ)) 1 sin(2πΦ/Φ +4δ) − 4 − 4√2 16 − √2 − 0 − √2 0 h (2.9) i 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-Pérot 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.

b Another simple limit corresponds to the situation with pk = 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 excitations 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)= Iˆ(0)Iˆ(t)+ Iˆ(t)Iˆ(0) exp(iωt)dt. (2.10) h i Z−∞ 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

2.0 1.5 * emax1.0 0.5 6 0.0 4

0.5 ∆ Γ 2

1.0 0

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 = lim δQ2(t) /t, (2.11) t →∞h i

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 →∞h i 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

e Q(t)= k 4 will 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) + = (fd x,k 1wx,d x + fd+x,k+1wx,d− +x) dt − − − xX=a,b + f (w + w− ). (2.12) − d,k x,d x,d xX=a,b

Let us now deﬁne the generating function

k fd(s, t)= fd,k(t)s , (2.13) Xk where the summation extends over all possible k’s for a given d. One can express the average transmitted charge Q(t) = e kf (t)/4 and its ﬂuctuation as h i k d,k P e d Q(t) = fd(s, t) (2.14) h i 4 ds ! d s=1 X

and 2 2 e d d 2 δQ (t) = s fd(s, t) Q(t) , (2.15) h i 4 ds ds ! −h i d s=1 X 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 + 1 fd(s, t)= (sfd xwx,d x + fd+xwx,d− +x) dt − − s xX=a,b + f (w + w− ) (2.16) − d x,d x,d xX=a,b

˙ 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 7 λi(s)t fd(s, t)= ξd,i(s)e Pi(t), (2.17) i=0 X 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 probability d fd(1, t) = 1, we have P e I = λ′ (1) 4 0 2 S =(e/4) [λ0′′(1) + λ0′ (1)] e e∗ = [1 + λ′′(1)/λ′ (1)] (2.18) 4 0 0 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′ (1) = G /G 0 − s λ|(s,λ)=(1,0) 2 2 3 λ′′(1) = (G G + G G 2G G G )/G , (2.20) 0 − ss λ λλ s − sλ s λ λ|(s,λ)=(1,0)

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 expression for the Fano factor at zero temperature simpliﬁes:

3 p2k+1 λ0′ (1) = 16A/(4 + ) p2k+4 Xk=0 2 eλ0′ (1) 1 p2k+1 1 p2k+1 2 e∗ = 2 2 ( ) 16 (4A p2k+4 − 16A p2k+4 Xk Xk 1 1 p p p p + 1+ ( 3 + 7 )( 1 + 5 ) (2.21) 2A2 4 p p p p 6 2 4 0 )

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).

b We will assume for certainty that pk = 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]:

2e I = 7 ; (2.22) k=0 1/pk P 2 1/pk e∗ =2e . (2.23) [ 1/p ]2 P k Combining the above equations with Eq.P (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 Φ /2 and 2 2e in the K =8 0 × 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

b a maximal e∗ e∗ (γ, δ) as a function of the tunneling amplitude ratio γ = A /A ≡ max b a and phase diﬀerence δ = δ δ . We ﬁnd that the maximal value of e∗ = 2.3e is − max achieved at γ =0.08 and δ = 0.03. The dependence of e∗ on γ and δ is shown in − max

Fig. 2.4. We see that at γ = 0, emax∗ 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 presence 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 interesting 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

(a)

C1 ν = 5/2 C2

C3 ν = 2 C4

(b)

C1 ν = 5/2 C2

C3 C4 quantum wire

Figure 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 .

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 interferometer [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 account 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 approach, 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

C5 C1 C2 C6

ν = 5/2

ν = 2

C3 C4

Figure 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.

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 fractional 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

~ = K a ∂ a ǫµνλ, (3.1) L −4π IJ Iµ ν Jλ IJµνX 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

~ = (∂ φ K ∂ φ + ∂ φ V ∂ φ ). (3.2) Ledge −4π t I IJ x J x I IJ x J XIJ

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

3 2 − K =   (3.3) 2 4  −    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

2 1 2 P zi Ψ = Pf( ) (z z ) e− i | | , (3.4) Pf z z i − j i j i

in which zn = xn + iyn is the coordinate of the nth electron in units of the magnetic length l , and Pf is the Pfaﬃan of the antisymmetric matrix 1/(z z ). At the B i − j 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 exactly 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 conservation, 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) Γn(x)ψn(x) + H.c., (3.6) n Z X 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

Γ (x) exp( i∆k x), (3.7) n ∼ − n

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 Γ (x) | m | cannot depend on the coordinate. Next, we consider the system with the tunneling 81

Hamiltonian Htun′ = 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)Γm∗ (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

C1 ν = 5/2 C2

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 modes and 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 tunneling event. This means that we neglect disorder at the edges. This assumption 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) = I ( V, ∆k). We assume that tunneling is weak and hence only single electron − tun − − 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 interaction 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

A B C tunneling current is Itun = Itun + Itun + Itun. 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 = v (k k ), where v is the edge mode velocity, the sign reﬂects the propagation ± α − F α ± α 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

v (k k ) ω = v (k k ), (3.8) 1 − F 1 − − 2 − F 2 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 k )(k k ) < 0, i.e., one − F 1 − F 2 of the two states is occupied and the other is not. Eq. (3.8) is easy to solve directly 85

(a) E/~ ω < 0 slope = −v2 slope = v1

ω > 0

∆k

(b) E/~ ω < 0 slope = −v2 slope = v3

ω > 0

k k0 k slope = vλ ∆k

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 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 represent occupied states. The momentum mismatch between two edges ∆k> 0. (b) Tunneling between an integer QHE edge and a Pfaffian 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 Majorana 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. − 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 v ∆k = v (k k ) and the state from 1 1 F 2 − F 1 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 v ∆k, i.e., ω < v ∆k, no tunneling happens. When ω > v ∆k, − 2 | | 2 | | 2 an electron from the upper edge can tunnel into the lower edge and a negative voltage- independent tunneling current results. Thus, the IB V characteristics is a sum of tun − two step functions, with two jumps at ω = v ∆k and v ∆k. The positions of the − 2 1 two jumps provide the information about the edge mode velocities. The diﬀerential

B conductance Gtun 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 k > 0 and linear spectrum. − F 2 Thus, the relation between the momentum and energy remains the same as in the

Fermi liquid case. Hence, the IA V curve has singularities at ω = v ∆k and tun − − 2 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 k k) 0 −

and reaches the left line at k>kF 3 (otherwise the tunneling is not possible since the momentum change (k k ) of the Bose mode must be positive). Both energy − F 3 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

intersects the left line at k = kF 3, 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>kF 3 (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 IA V curve. For negative ω, it tun − 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 = v ∆k, the − 2 tunneling process involving a zero-momentum Majorana fermion becomes possible.

Thus, ω = v ∆k is the negative threshold voltage in the IA V curve. We see − 2 tun − 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 IA V characteristics and they are closely related to tun − 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 expression, 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

A B C current can be separated into three independent parts, Itun = Itun + Itun + Itun. The calculation of IB and IC is essentially the same. So in the following, we will only

A B consider Itun and Itun.

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]

1 = (t, x) ∂ φ (∂ + v ∂ )φ L Lfrac − 4π x 1 t 1 x 1 1 ∂ φ ( ∂ + v ∂ )φ , (3.9) − 4π x 2 − t 2 x 2 − Htun 90 with the tunneling Hamiltonian density

n n = γ Ψ†(x)Ψ (x)+ γ Ψ†(x)Ψ (x) + H.c.., (3.10) Htun A 2 frac B 2 1 n X

n where Ψ1 is the electron operator for the integer QHE mode of the upper edge, Ψfrac

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 [φ (x),φ (x′)] = iσ πδ sign(x x′), with σ = +1 and σ = 1. The i j j ij − 1 2 − Lagrangian density for the fractional QHE edge depends on the state and will Lfrac 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

n tunneling amplitudes. It will be convenient for us to assume that γA and γB are independent of the coordinate and absorb the factors exp( i∆k x) into the electron − n 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

σj iφj +ikF,jx Ψj = e , where kF,j represents the Fermi momentum. The corresponding

electron density ρj =(∂xφj + kF,j)/2π. In the fractional edge, there may be several

n relevant electron operators Ψfrac. In our calculations, only the most relevant electron 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

B conductance Gtun(ω) is just a combination of two δ-functions. Therefore, the shape of the voltage dependence of the total diﬀerential conductance Gtun is determined

A by Gtun(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-

n n iµ1t/~ nient to switch to the interaction representation with Ψ Ψ e− , Ψ frac → frac 1 → iµ1t/~ iµ2t/~ Ψ e− and Ψ Ψ e− , where µ = eV and µ = 0. This introduces time- 1 2 → 2 1 2 dependence into the tunneling operators (cf. Ref. [109]). The electron operator

n Ψfrac(x) can be written in a bosonized form according to the chiral Luttinger liquid

n i PI (lI φI +lI kF,I x) i PI (lI φI +lI kF,I x) theory, Ψfrac(x)= e , or λ(x)e , if a Majorana mode λ(x) exists.

In order to pay special attention to momentum mismatches, we deﬁne

Ψn (x) Ψ˜ n (x)ei PI lI kF,I x. (3.11) frac ≡ frac

ikF,jx Similar deﬁnitions are also made for the integer QHE modes, Ψj(x) = e Ψ˜ j(x). Thus, the density of the tunneling Hamiltonian can be rewritten in the interaction

picture as

n iωt i∆kn x n = γ e − 2f Ψ˜ †(x)Ψ˜ (x) Htun A 2 frac n X iωt i∆k21x ˜ ˜ + γBe − Ψ2†(x)Ψ1(x) + H.c., (3.12)

where ∆kn = k l k , ∆k = k k and ω = (µ µ )/~ = eV/~. 2f F,2 − I I F,I 21 F,2 − F,1 2 − 1 − It is worth to mentionP 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

˜ n ˜ because the pair charge doubles, and Ψfrac and Ψ2 should be understood as bosonic operators that annihilate electron pairs.

The operator for the tunneling current density is given by

dρ e j(t, x)= e 2 = [ρ (x),H ], (3.13) dt i~ 2 tun

where ρ (x) is the electron density of the lower edge, and H = dx (x) is the 2 tun Htun tunneling Hamiltonian. Expanding the commutator in Eq. (3.13)R we get

e n iωt i∆kn x n j(t, x)= γ e − 2f Ψ˜ † (x)Ψ˜ (x) i~{ A 2 frac n X iωt i∆k21x + γ e − Ψ˜ †(x)Ψ˜ (x) H.c. . (3.14) B 2 1 − }

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 I (t)= 0 S( , t)IS(t, ) 0 , (3.15) tun h | −∞ −∞ | i

where 0 is the initial state, the operator I = dxj(t, x) and h | R t S(t, ) = Texp( i Hdt′/~) −∞ − Z−∞

is the evolution operator. To the lowest order in the tunneling amplitudes, the tunneling current reduces to

i t I (t)= dxdx′ dt′ 0 [j(t, x), (t′, x′)] 0 . (3.16) tun −~ h | Htun | i Z Z−∞ 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-

i j i terms, proportional to γ (γ )∗ with i = j, or γ γ∗ . There are only contributions A× A 6 A× B proportional to γi 2 or γ 2. Thus, without loss of generality we can assume that | A| | B| only one of the tunneling amplitudes is nonzero and write

e iωt i∆kx j (t, x)= (γe − Ψ˜ † (t, x)Ψ˜ (t, x) H.c.). (3.17) αβ i~ α β −

The operators Ψ˜ α and Ψ˜ β represent electron operators on two sides of the junction. 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

2 t αβ e γ iω∆t i∆k∆x I = | | dxdx′ dt′(e − c.c.) tun − ~2 − Z Z−∞ [G (∆t, ∆x) G ( ∆t, ∆x)] (3.18) × αβ − αβ − −

with ∆t = t t′, ∆x = x x′ and − −

Gαβ(∆t, ∆x)

= 0 Ψ˜ † (t, x)Ψ˜ (t′, x′)Ψ˜ (t, x)Ψ˜ † (t′, x′) 0 , (3.19) h | α α β β | i

and we used the fact that 0 Ψ˜ † (t, x)Ψ˜ (t′, x′) 0 = 0 Ψ˜ (t, x)Ψ˜ † (t′, x′) 0 h | α/β α/β | i h | α/β α/β | i and the translational invariance for chiral Luttinger liquids. Eq. (3.18) can be sim- pliﬁed as 2 αβ e γ iωτ i∆ky I = L | | dydτ(e − c.c.)G (τ,y), (3.20) tun − ~2 − αβ Z 94 where L is the length of the junction.

Let there be N right-moving and M left-moving modes in total at both edges. In the chiral Luttinger liquid theory a general expression for the correlation function is

N τ gRi G (τ,y)= l2 c αβ B δ + i(τ y/v ) i=1 Ri Y − M τ gLi c , (3.21) × δ + i(τ + y/v ) i=1 Li Y

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 v < v < < v and v < v < < v . R1 R2 ··· RN L1 L2 ··· LM ˜ ˜ The scaling dimension of the tunneling operator Ψα† (t, x)Ψβ(t, x) is g =1/2( i gRi +

i gLi). P P Using the Fourier transformation

+ g 1 1 ∞ iωt ω − = dω e− | | θ(ω), (3.22) (δ + it)g Γ(g) Z−∞ 95

(a): ωvRi∆k

ωRi ωRi Σ Ω Σ Ω ωLi ωLi

Figure 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.

we integrate out τ and y in Eq. (3.20). Then we obtain

e γ 2 Iαβ = 4π2L | | [dω dω ] tun − ~2 Ri Li Z ωRi ωLi δ(ω ωRi ωLi)δ(∆k + ) × − − − vRi vLi n X X X X (ω ω, ∆k ∆k) − ↔ − ↔ − o gRi 1 θ(ωRi) gLi−1 θ(ωLi) ωRi − ωLi , (3.23) × | | Γ(gRi) | | Γ(gLi) Y Y

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,

∞ αβ gRi 1 gLi 1 Itun =A [dωRidωLi]i 2 ωRi − ωLi − ≥ | | | | 0 i 2 i 2 Z Y≥ Y≥ ω ωRi ωLi gL1 1 ∆k − ×|v − − vRR − vLR | R1 i 2 i1 i 2 i1 X≥ X≥ ω ω ω θ( ∆k Ri Li ) × v − − vRR − vLR R1 i 2 i1 i 2 i1 X≥ X≥ ω ωRi ωLi gR1 1 + ∆k − ×|v − vRL − vLL | L1 i 2 i1 i 2 i1 X≥ X≥ ω ω ω θ( + ∆k Ri Li ) × v − vRL − vLL L1 i 2 i1 i 2 i1 X≥ X≥ (ω ω, ∆k ∆k), (3.24) − ↔ − ↔ − with

2 2 4π e γ RL gR1+gL1 1 − A = L 2 | | (v11 ) (3.25) − ~ Γ(gRi)Γ(gLi) v v v v vRR = RiQR1 , vLL = Li L1 , i 2, (3.26) i1 v v i1 v v ≥ Ri − R1 Li − L1 RL vRivL1 LR vLivR1 vi1 = , vi1 = , i 1. (3.27) vRi + vL1 vLi + vR1 ≥

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

αβ to Itun. The integration is taken over the volume in the positive orthant of the

(M + N 2)-dimensional space spanned by ωRi,ωLi i 2 under both of the following − { } ≥ superplanes

ω ω ω Σ: Ri + Li = ∆k, (3.28) vRR vLR v − i 2 i1 i 2 i1 R1 X≥ X≥ ω ω ω Ω: Ri + Li = + ∆k. (3.29) vRL vLL v i 2 i1 i 2 i1 L1 X≥ X≥ 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

N M P gRi+P gLi 1 ω i=2 i=1 − Iαβ ∆k . (3.30) tun ∼ v − R1

Now let us consider the ω -intercepts of the two superplanes, Σ =(ω/v ∆k)vRR Ri Ri R1− i1 and Ω =(ω/v + ∆k)vRL, i 2. We ﬁnd that Ri L1 i1 ≥

ΣRi < ΩRi, when ω < vRi∆k;

ΣRi > ΩRi, when ω > vRi∆k. (3.31)

Thus, when ω passes v ∆k, the shape of the (M + N 2)-dimensional integration Ri − volume changes, as is illustrated in Fig. 3.5 for the 3D case. This volume change leads to a singularity in the I V curve. The precise nature of the singularities tun − depends on the model and will be discussed in the following section. For the ωLi- intercepts, Σ = (ω/v ∆k)vLR is always smaller than Ω = (ω/v + ∆k)vLL, Li R1 − i1 Li L1 i1 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.

αβ Similar behavior of Itun(ω) manifests itself when ω < 0, with singularities at ω = v ∆k. Thus, each mode contributes a singularity. − Li

3.5 The number of singularities 98 3.2 = 1 instead of 2, Fig. ν ent setups. , 3.1 , Fig. 3.1 2 Fig. / =5 ν 26 15 24 8 15 3 8 Boundary of The number of conductance singularities for different models in differ Table 3.1: StateK=8 331 PfaffianEdge-reconstructed PfaffianNon-equilibrated anti-Pfaffian 10 3 and 2 3 61 strong 18 interaction strong interaction 18 34 10 10 13 4 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 discussed 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 the K = 8 fractional edge depends on the setup. For the setup Fig. 3.1, the most relevant operator creates an electron pair on the fractional K = 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 operators. 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 fractional 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 conductance. 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 information 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 Tˆ0, 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

2 1 2 0 I V × − = V and the contribution of the next most relevant operators I ∼ ∼ 2 2 2 2 V × − = V 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 ofR 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 Tˆ1 = Tˆ0λ∂xλ is generated from Tˆ0 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/~ = ∆kv , where ∆k is the momentum mismatch between the integer − l 105

QHE mode and the ‘spectator’ mode and vl denote edge mode speeds. The strong

singularities due to the operator Tˆ0 occur at the same voltages. However, Tˆ0 does

not contain Majorana fermions and hence Tˆ0 does not generate a singularity at eV/~ = ∆kv , where v is the speed of the Majorana fermion. On the other − M M

hand, Tˆ1 contains a Majorana fermion and hence is responsible for an additional ‘weak’ singularity at eV/~ = ∆kv . Since there are two integer edge modes, we − M 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’ singularities. 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:

ˆ T / , = ψd, / ψu,† / 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 Tˆ describes electron tunneling between lower and upper integer

edge modes. The combination T ′ = λ exp( iφ ) describes charge redistribution ± n 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 Tˆ 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

2 2 B 4π e γB v1v2 Itun = L 2 | | − ~ (v1 + v2) [θ(ω v ∆k ) θ( ω v ∆k )]. (3.33) × − 1 21 − − − 2 21

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

B B picture, Itun is indeed a combination of two step functions and so Gtun is just a combination of two δ-functions. The two singularities, positive and negative thresholds, appear at ω = v ∆k and v ∆k . 1 21 − 2 21 108

K = 8 state (a) (b) 6 x103 90 (arb.) (arb.) A tun A tun 60 G 4 G

2 30

0 0 ω/ω 0 -3 -2 -1 0 1 2 -2 -1 0 1 2∆k/∆k0

Figure 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 A units. (b) Momentum mismatch dependence of Gtun at a fixed voltage. ∆k0 = ω/v2. For both curves, we set v3/v2 =0.8.

A A In the following subsections, we will discuss Itun and Gtun 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

2~ = ∂ φ (∂ + v ∂ )φ . (3.34) Lfrac − π x 3 t 3 x 3 109

Only electron pairs are allowed to tunnel into the edge. The electron pair annihilation

i8φ3 operator is Ψ˜ frac = e , and the charge density ρfrac = e∂xφ3/π. The pair correlation

8 function is 0 Ψ˜ † (t, x)Ψ˜ (0, 0) 0 =1/[δ + i(t x/v )] , i.e., the scaling exponent h | frac frac | i − 3

g3 = 8. In the integer edge, the operator Ψ2, Eq. (3.10), should also be understood

4 as the pair annihilation operator with 0 Ψ˜ †(t, x)Ψ˜ (0, 0) 0 = 1/[δ + i(t + x/v )] h | 2 2 | i 2

and g2 = 4. Substituting the scaling exponents and edge velocities into Eq. (3.24), we obtain

2 2 A 8π e γA v2v3 11 ω 3 Itun = L 2 | | ( ) ( ∆k2f ) − ~ Γ(8)Γ(4) v2 + v3 v3 −

ω 7 ( + ∆k2f ) θ(ω v3∆k2f ) θ( ω v2∆k2f ) , (3.35) v2 − − − − h i

B Just like in the case of the tunneling current Itun between two integer QHE edges,

there are two threshold voltages, the positive threshold ω = v3∆k2f and the negative one ω = v ∆k . However, in contrast to IB , the tunneling current IA increases − 2 2f tun tun

smoothly as the voltage passes the thresholds. At ω & v3∆k2f , the tunneling current IA behaves as (ω v ∆k )3, and at ω . v ∆k , IA (ω + v ∆k )7. Thus tun ∼ − 3 2f − 2 2f tun ∼ 2 2f A Itun 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 v ∆k and v ∆k , IA will asymptotically behave like ω10 for | | ≫ 2 2f 3 2f tun ∼ A both positive and negative voltages. We plotted the diﬀerential conductance Gtun =

A ∂Itun/∂ω as a function of ω at ﬁxed ∆k2f , and a function of ∆k2f at ﬁxed ω in Fig. 3.6. 110

331 state

(a) 0

10 G 10 G

5 5

0 0 -2 -1 0 1 2 ω/ω 0 -1 0 1 ∆k/∆k0

6 G 6 G

3 3

0 0 -2 -1 0 1 2 ω/ω 0 -1 0 1 ∆k/∆k0

(e) 2

5 5

0 0 -2 -1 0 1 2 ω/ω 0 -1 0 1 ∆k/∆k0

Figure 3.7: Voltage and momentum mismatch dependence of the tunneling differential conduc- A,u tance Gtun in the 331 state; u is either a or b. We have chosen the ratios of the edge velocities A,u to be v3/v2 =0.8 and v4/v2 =1.2. The left three panels show the voltage dependence of Gtun 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 is shown in units of ω0 = v2∆k. Panels (b), (d) and (f) show the same three cases for the momentum A,u mismatch dependence of Gtun 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

i8φ3 iφ1+iφ2 i8φ3+2iφ2 e − is present and is more relevant then the pair tunneling operator e , 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

4π2e γ 2 I = L | | v − ~28! 12 v8 (ω/v + ∆k)8, ω< v ∆k − 23 2 − 2   0, v2∆k<ωv3∆k    where v = v v /(v + v ), v = v v / v v and v = v v /(v + v ). Here we 12 1 2 1 2 13 1 3 | 3 − 1| 23 2 3 2 3 assume v3 > v1. 112

When v1 > v3 the tunneling current is

4π2e γ 2 I = L | | v8 v tun − ~28! 23 13 [(ω/v ∆k)8 − 3 −  8 8 8  v12/v23(ω/v1 ∆k) ], ω< v2∆k  − − −    0, v2∆k<ωv1∆k  − −   

In both cases three singularities are found.

3.6.3 331 state

The 331 state [45] has the edge Lagrangian density

~ = (3∂ φ ∂ φ 2∂ φ ∂ φ 2∂ φ ∂ φ Lfrac − 4π t 3 x 3 − t 3 x 4 − t 4 x 3

+4∂tφ4∂xφ4 + Vmn∂xφm∂xφn). (3.38) m,n=3,4 X

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,

˜ a i3φ3 i2φ4 ˜ b iφ3+i2φ4 Ψfrac = e − and Ψfrac = e . Before applying Eq. (3.24) to the calculation ˜ a of the tunneling current, one needs to compute the correlation functions of Ψfrac and Ψ˜ b . Since the Lagrangian density is quadratic, we can rewrite it in terms of frac Lfrac 113

˜ ˜ two decoupled ﬁelds φ3 and φ4, such that

~ = ∂ φ˜ (∂ + v ∂ )∂ φ˜ . (3.39) Lfrac −4π x n t n x x n n=3,4 X

φ˜ and φ˜ are linear combinations of φ and φ , with 0 φ˜ (x, t)φ˜ (0, 0) 0 = ln[δ + 3 4 3 4 h | n n | i − i(t + x/vn)], where the velocities are

1 v = 4V +4V +3V 3,4 16 33 34 44 (1 + x2) 4V +4V V , (3.40) ∓ ×| 33 34 − 44| p and x =2√2(V +2V )/(4V +4V V ) is an interaction parameter. Note that 44 34 33 34 − 44 v3 is smaller than v4. It is easy to prove that both v3 and v4 are positive, so φ˜3 and φ˜4 are right-moving. In the limit of strong interaction, (V )2 V V , v approaches 34 → 33 44 3 0. The two-point correlation functions of those operators can be expressed as

u u 0 Ψ˜ † (x, t)Ψ˜ (0, 0) 0 h | frac frac | i 1 = u u , (3.41) [δ + i(t x/v )]g3 [δ + i(t x/v )]g4 − 3 − 4

where u = a, b and the scaling exponents

u 3 1 σu2√2x g = − sign(4V33 +4V34 V44); (3.42) 3,4 2 ∓ 2√1+ x2 − the sign factors σ = +1, σ = 1. It is worth to notice that the sum of gu and gu a b − 3 4 is always 3.

˜ a There are two tunneling operators in the action. They are proportional to Ψfrac ˜ b and Ψfrac. 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

2 2 A,u 4π e γA g4 g3 ω 2 Itun = L 2 | | v24v23( + ∆k2f ) − ~ Γ(g3)Γ(g4) v2

B(1,g4,g3), ω>v4∆k2f  v34(ω/v3 ∆k2f )  B( − ,g4,g3), v3∆k2f <ω

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 v = v v /(v v ). 34 3 4 4 − 3

A,a A,b Consider any of the two contributions Itun or Itun , 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

zg4 , z 0 B(z,g ,g ) ∼ (3.44) 4 3 ∼   (1 z)g3 + const, z 1 − ∼  A,u g4 1 Thus, when ω & v ∆k , the diﬀerential conductance G (ω/v ∆k ) − is 3 2f tun ∼ 3 − 2f

singular at ω = v3∆k2f , if g4 < 1. Hence, the diﬀerential conductance diverges near

A,u the threshold. Similarly, Gtun is singular at ω = v4∆k2f , if g3 < 1, i.e., g4 > 2. Hence, the shape of the GA,u ω is quite different at different values of g and tun ∼ 3 A,u g4, i.e., different interaction strengths x. Fig. 3.7 shows the dependence of Gtun on ω and ∆k2f in 3 different cases: g4 < 1, 1 < g4 < 2 and g4 > 2. The total

A A,a A,b diﬀerential conductance Gtun = Gtun + Gtun has two sets of singularities originating 115

Pfaffian state (a) (b)

4 4 (arb.) (arb.) A tun 3 A tun 3 G G

2 2

1 1

0 0

ω/ω 0 -2 -1 0 1 2 -2 -1 0 1 2 ∆k/∆k0

A Figure 3.8: (a) Voltage dependence of the tunneling differential conductance Gtun in the Pfaffian A state. The reference voltage ω0 = v2∆k. (b) Momentum mismatch dependence of Gtun in the Pfaffian state. The reference momentum ∆k0 = ω/v2. We set the edge velocity ratios, v3/v2 =1.2 A and vλ/v2 =0.5. Gtun is shown in arbitrary units. from the two individual contributions to the current. The shape of the curve of

A a b a b a b Gtun(ω) depends on the relative values of γA v.s. γA, ∆k2f v.s. ∆k2f , and g4 v.s. g4. 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]

2~ = ∂ φ (∂ + v ∂ )φ + iλ(∂ + v ∂ )λ (3.45) Lfrac −4π x 3 t 3 x 3 t λ x

where φ3 is the right-moving charged boson mode and λ is the neutral Majorana

fermion mode. The most relevant electron operator is Ψ˜ frac = λ exp(i2φ3). Its correlation function G =1/[(δ + i(t x/v ))2(δ + i(t x/v ))] equals the product of − 3 − λ 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

2π2e γ 2 IA = L | A| v tun − ~2 2λ 2 2 v23(ω/v2 + ∆k2f ) , ω>v3∆k2f  2 2  v3λ(ω/vλ ∆k2f ) , vλ∆k2f <ω

A 3.2) and the appearance of a discontinuity for Gtun at ω = v3∆k (see Fig. 3.8). On

A the negative voltage side, Gtun 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

~ = [2∂ φ (∂ + v ∂ )φ + ∂ φ (∂ + v ∂ )φ Lfrac − 4π x c t c x c x n t n x n +2v ∂φ φ ]+ iλ(∂ v ∂ )λ, (3.47) nc c n t − λ x

where φc is a charged mode and φn is a neutral mode. There are three most relevant

λ electron operators Ψ± = exp(i2φ iφ ) and Ψ = λ exp(i2φ ). Thus, we need frac c ± n frac c to consider three tunneling operators, proportional to these three electron operators. 117

As discussed in the previous section they generate three independent contributions to

A the tunneling current Itun. We ﬁrst discuss the current contributions which originate

from the tunneling terms containing Ψfrac± . For these two contributions, the situation is quite similar to the 331 state because the Majorana fermion does not enter the

operators Ψfrac± . 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 φ ,φ { c n} to two free fields φ˜ , φ˜ with velocities v , v respectively. Then the two-point { 3 4} { 3 4} correlation function 0 Ψ±† (x, t)Ψ± (, 0, 0) 0 can be calculated as we did for 331 h | frac frac | i A, state. With Eq. (3.24) we then obtain the same form of the tunneling current Itun± as in Eq. (3.43), but with different tunneling amplitudes, momentum mismatches, edge velocities and scaling exponents. The edge velocities are

1 v = v + v (v v )√1+2x2 , (3.48) 3,4 2 c n ∓ c − n

and scaling exponents are 3 1+4σx gσ = , (3.49) 3,4 2 ∓ 2√1+2x2

+ where σ = +1 for the case of Ψ and σ = 1 for Ψ− ; the interaction parameter frac − frac x = v /(v v ). It is assumed that (v v ) is positive. Indeed, we expect the nc c − n c − n 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

A, diﬀerent shapes of the Gtun± 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.

λ Now let us turn to the tunneling operator, proportional to Ψfrac. In this case all four modes participate in the tunneling process. The correlation function of the

λ ﬁeld Ψfrac 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

1 gλ =1 , (3.50) 3,4 ∓ √1+2x2

where diﬀerent signs correspond to indices 3 and 4. Again, by using Eq. (3.24) we

obtain the following contribution to the tunneling current:

2 λ 2 A,λ 4π e γA Itun = L 2 | | v2λ sign(ω) − ~ Γ(g3 + 1)Γ(g4)

g3 g4 ω λ 2 v3λv4λ( + ∆k2f ) B(f(ω),g4,g3 + 1) × vλ h g3 g4 ω λ 2 v23v24( + ∆k2f ) B(g(ω),g4,g3 + 1) , (3.51) − v2 i

where

λ (ω/v3 ∆k )v34 − 2f ω λ , v3 < λ < v4 (ω/vλ+∆k2f )v4λ ∆k2f  f(ω)=  ω (3.52)  1, ∆kλ < vλ or > v4  2f −  ω 0, vλ < ∆kλ < v3  − 2f    and

λ (ω/v3 ∆k )v34 − 2f ω λ , v3 < λ < v4 (ω/v2+∆k2f )v24 ∆k2f  g(ω)=  ω (3.53)  1, ∆kλ < v2 or > v4  2f −  ω 0, v2 < ∆kλ < v3  − 2f    A,λ λ The dependence of Gtun on the voltage ω and momentum mismatch ∆k2f is illus-

λ trated in Fig. 3.9. There are no divergencies for any g4 . All singularities appear

λ as voltage thresholds or discontinuities of the derivative of Gtun(ω). The Majorana 119

Reconstructed Pfaffian state (a) (b)

0.4 (arb.) (arb.) 0.2 λ λ tun A, tun 0.3 A, G G

0.2 0.1 0.1

0.0 0.0

ω/ω 0 -2 -1 0 1 2 -3 -2 -1 0 1 2 ∆k/∆k0

A,λ Figure 3.9: The diﬀerential conductance Gtun in the edge-reconstructed Pfaﬃan state. Panels A,λ (a) and (b) show the voltage and momentum mismatch dependence of Gtun (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

Non-equilibrated Anti-Pfaffian state

(a) v >v , 0v , 0

6 G 6 G

3 3

0 0 -2 -1 0 1 2 ω/ω 0 -2 -1 0 1 ∆k/∆k0 (c) v >v , g +g >2 (d) v >v , g +g >2 3 2 3 3 4 3 2 3 3 4 (arb.) (arb.) A tun A tun

2 G 2 G

1 1

0 0 -2 -1 0 1 2 ω/ω 0 -2 -1 0 1 ∆k/∆k0 (e) v

A tun 6 A tun 6 G G

3 3

0 0 -3 -2 -1 0 1 2 3ω/ω 0 -1 0 1 ∆k/∆k0 (g) v 2 (h) v 2 1.5 2 3 3 4 1.5 2 3 3 4 (arb.) (arb.) A tun A tun

G 1.0 G 1.0

0.5 0.5

0.0 0.0 -3 -2 -1 0 1 2 3ω/ω 0 -1 0 1 ∆k/∆k0

A Figure 3.10: Differential conductance Gtun in the non-equilibrated anti-Pfaffian edge state. All A left panels show the voltage dependence of Gtun and right panels show the momentum mismatch A dependence of Gtun, 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 2 cases, we set g3 + g4 =2.5. The reference voltage ω0 = v2∆k A and the reference momentum mismatch ∆k0 = ω/v2. Gtun 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]

2~ 3 = ∂ φ (∂ + v ∂ )φ + i [λ (∂ v ∂ )λ ]. (3.54) Lfrac −4π x c t c x c n t − λ x n n X There are three electron operators corresponding to the three Majorana fermions,

˜ n iφc Ψfrac = λne , 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]

~ = [∂ φ (∂ + v ∂ )φ Lfrac − 4π x c1 t c1 x c1 +2∂ φ ( ∂ + v ∂ )φ +2v ∂ φ ∂ φ ] x c2 − t c2 x c2 12 x c1 x c2 + iλ(∂ v ∂ )λ. (3.55) t − λ x

Again the action can be rewritten in terms of two linear combinations of the Bose

ﬁelds φc1 and φc2: a free left-moving mode φ˜3 and a right-moving mode φ˜4 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

iφc1 parameter space. For x< 2/3, the most relevant electron operator is Ψfrac = e .

A The expression for the tunneling current Itun 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

2 2 A 4π e γA g3+g4 1 − Itun = L 2 | | v34 sign(ω) − ~ Γ(g3)Γ(g4) g3 g4 1 v24 ω ω − v24(ω/v4 ∆k2f ) ∆k2f + ∆k2f F (1, 1 g4, 1+ g3, − ), g3 v4 − v3 − v23(ω/v3+∆k2f )   ω > v4∆k2f or ω < v2∆k2f  −  g 1 g  3 4 v (ω/v +∆k )  v23 ω ∆k − ω + ∆k F (1, 1 g , 1+ g , 23 3 2f ),  g4 v4 2f v3 2f 3 4 v24(ω/v4 ∆k )  − − − 2f ×   v2∆k2f <ω< v3∆k2f  − −  0,     otherwise    (3.56)

where the scaling exponents equal

1 1 g3,4 = (3.57) 2√1 2x2 ∓ 2 −

and F is the hypergeometric function.

For the interaction strength we focus on, 0

g < 1 and 1 < g < 2. Asymptotically, IA (ω v ∆k )g3 when ω & v ∆k . 3 4 tun ∼ − 4 2f 4 2f

Thus, ω = v4∆k2f corresponds to a divergency of the diﬀerential conductance. If

ω . v ∆k then the tunneling current is asymptotically equal to (ω + v ∆k )g4 . − 3 2f 3 2f A g3+g4 1 When ω v ∆k , we have I (ω + v ∆k ) − . Hence, when g + g < 2, ≈ − 2 2f tun ∼ 2 2f 3 4 the diﬀerential conductance diverges at v ∆k , while for g + g > 2 only a cusp − 2 2f 3 4 is present as is shown in Fig. 3.10. 124

A Table 3.2: Summary of singularities in the voltage dependence of the differential conductance Gtun 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 statistics. The “Singularities” shows the number of singularities, including divergencies (S), discontinuities (D) and cusps (C), i.e., discontinuities of A the first or higher derivative of the voltage dependence of Gtun. 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.

State Modes Statistics Singularities K=8 1R A 2C 331 2R A 4C+2Sor5C+S Pfaffian 2R(1) N 2C+D Edge-reconstructed Pfaffian 1L(1) + 2R N 8C+2S or 9C+S Non-equilibratedanti-Pfaffian 2L(1)+1R N C+2Sor2C+S

If v2 < v3 then the tunneling current is

2 2 g3+g4 1 A 4π e γ g3 g4 ω − Itun = L 2 | | v23v24 + ∆k2f sign(ω) − ~ Γ(g3)Γ(g4) v2

v34(ω/v4 ∆k2f ) ω B( − ,g3,g4 ), > v4 or < v3 v23(ω/v2+∆k2f ) ∆k2f −  ω  B(1,g3,g4), v3 < < v2 (3.58) ×  − ∆k2f −  0, otherwise    In this case the behavior near ω = v4∆k2f is the same as above. The behavior near ω = v ∆k and ω = v ∆k is also the same as above but these singularities − 3 2f − 2 2f

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

A,i B C Itun = Itun + Itun + Itun (3.59) i X

and the total tunneling diﬀerential conductance Gtun can be measured, thus, singu-

B,C larities originating from all three contributions to the current will be seen. Here, Itun describe tunneling into the integer edge modes. However, these last two contributions 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 I (ω) = I ( ω). In contrast tun − tun − 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

I =IA (∆k,ω)+ IA ( ∆k,ω) tun tun tun − B C F + Itun + Itun + Itun, (3.60)

F A where Itun is the tunneling current between two fractional QHE edges, Itun stays for tunneling between integer QHE modes on one side of the junction and fractional QHE

B,C modes on the other side of the junction, and Itun 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

F contribution Itun 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/v , where v is an edge mode velocity. A ﬁnite temperature | m| m can be viewed as a voltage uncertainty of the order of kT . Thus, the width of the smeared singularity is δk kT/[~v ]. This suggests that the total number ∼ m 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 proposed 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 consider 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-Büttiker 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 quantum 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 connection 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 theorems 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 distribution 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 possess 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,

1 y Consider a system whose Hamiltonian has a time-dependent contribution Ht = −∞ dxh(x, t), where the integration 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. R 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 quantum 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 to be localized by disorder. 133

IU V DS IL IS

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 assumptions 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, S = dt ∆I (t)∆I (0) + ∆I (0)∆I (t) , and in the drain, C h T T T T i S = dt ∆I (t)∆I (0) +R ∆I (0)∆I (t) , where I is the electric current in the D h D D D D i D drain andR ∆I = I I , and derive the relation −h i

∂I S = S 4T T +4GT, (4.1) D C − ∂V where G 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 generalize 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 I = I I is composed of the current I , entering D L − U L 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, I = I I , where I is the low-frequency part of the current, emitted from L S − T S the source. Thus, S = S + S 2S , (4.3) L C S − ST

where the cross-noise S = dt ∆I (t)∆I (0)+∆I (0)∆I (t) and the noise of the ST h T S S T i emitted current equals one halfR 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, I = I(GV + I<,I>) , where the brackets denote T h i

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 h − i ∼ −

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 I we can write I = J(GV + I<) , where S T h i 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

∂J(GV ) ∂IT SST = IT,ωI ω + IT, ωIω = IωI ω + I ωIω =2T , (4.5) h − − i G∂V h − − i ∂V

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 < ~ω T . In such case the assumption of the Gaussian distribution cutoﬀ ≪

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 S and S . We assume that V T 100mK. These D C ∼ ≈ are realistic parameters for noise experiments. The injected current I e2V/h S ∼ 137 since G e2/h. We assume that the transmission probability to conductor C is of ∼ the order of 1/2. Thus, I I . Then the second and third terms on the right T ∼ S hand side of Eq. (4.1) are of the order of e2[kT/h] e2[eV/h]. The noise S can ∼ C 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 h i 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 S e2/∆t e2[eV ]/h and has the ∼ C ∼ ∼ 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

28 2 of the order of 10− A /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 = T and V = 0 then the system is in equilibrium. If T T T and V T then C − C ≪ ≪ 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

y Hamiltonian has a time-dependent contribution Ht = dxh(x, t), where the in- −∞ tegration extends to the left of point y. In a chiral system,R 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

V DSA

B C x

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 = )I (t), where ρ is the initial density matrix, is the same −∞ S when the Heisenberg operators

t t 1 I (t) = [T exp( i V (t)dt)]− I ( )[T exp( i V (t)dt)], (4.6) S − S −∞ − Z−∞ Z−∞ where I ( ) is a Schr¨odinger operator, are deﬁned in terms of the Hamiltonians S −∞ V = H and V (t) = H + H (t) = H. The latter can be true for a general ρ( ) 0 0 I −∞ 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)[∂ φ∂ φ t x − 2 v(∂xφ) ], where v is the velocity of the edge excitations, the charge density q = e∂ φ/(2π) and the current operator I = vq. From the equation of motion, ∂ ∂ φ x S t x − 2 v∂xφ = 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( )= ρ ρ factorizes into a product of the initial density matrix −∞ H C

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 Schr¨odinger 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 Schrödinger 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 representation 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 sufficient to assume that the final steady state depends only on the states−∞of the reservoirs and not on the initial state of the finite central part of the system. In such case it is most convenient to perform calculations for the initial state whose density matrix factorizes. Certainly, the latter assumption itself is completely standard and can be easily tested experimentally by comparing steady states prepared from different initial conditions at the same temperatures and chemical potentials of the reservoirs. 142

Heisenberg representation instead and rely on special properties of the matrix elements 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,

ρ = ρ n n (4.7) H,C H,Cn| H,Cih H,C| X and

ρ( )= ρ n n , (4.8) −∞ n| ih | X where

ρn = ρHn′ ρCn′′ (4.9) and

n = n′ n′′ , (4.10) | i | Hi| Ci

where the states n′ and n′′ are selected from the Hilbert spaces of the QHE system | Hi | Ci and conductor C respectively. ρ is a Gibbs distribution, ρ exp( E /T H Hn ∼ − n − e V N /T ), where N is the number of electrons in the QHE subsystem, T the | | n n

temperature of the source and drain reservoirs and En are the eigenenergies of the eigenstates n of the quantum Hall subsystem before the tunneling contact was | Hi turned on, i.e., n are eigenstates of the Hamiltonian H with particle numbers | Hi H

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 n and n are time-independent. Thus, they are no | Hi | Ci 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 n . The chirality property means that the same restriction applies | Ci 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 n with the same N . Again, the chirality property | Hi n 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 O (t ) m = exp(i[E E ](t t )) n O (t ) m . The same relation would h | H 1 | i n − m 1 − 2 h | H 2 | i 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 I (t) m = exp(i[E E ]t) n I (0) m . (4.11) h | S | i n − m h | S | i

4.3.3 Voltage bias

A standard way to include voltage bias in mesoscopic systems is based on the Landauer-B¨uttiker formalism: One assumes that the tunneling term is initially ab- 144

QHE S δE

Q P δφ = 0 δφ = δV

Figure 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. 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 formalism 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 different 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 drδE = δV . Hence, δA = ct gradδφ, where δφ = δV in the P × source reservoir andR 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-Büttiker 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 constantR 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 = I I , (4.12) U − S

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 sin ωt/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

2S = dt I (0)I (t)+ h.c. (exp(iωt) + exp( iωt)) = ST h T S i − Z dt (exp(iωt) + exp( iωt))[ m I (0) n n I (t) m ρ ( ) − h | T | ih | S | i m −∞ mn Z X + n I (t) m m I (0) n ρ ( )], (4.14) h | S | ih | T | i n −∞

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 I and not ∆I = I I . Eq. (4.14) gives the noise at t = 0, when the S,T S,T S,T −h S,Ti 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 I (t) exhibits remarkable properties −∞ S in such representation. Inserting the time dependence (4.11) of the matrix elements 147

n I (t) m , one ﬁnds h | S | i

2S =2π [ρ ( )+ρ ( )] m I (0) n n I (0) m [δ(E E +ω)+δ(E E +ω)]. ST m −∞ n −∞ h | T | ih | S | i n− m m− n mn X (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,

I (t = 0) = Tr[ρ( )S ( , 0)Is S (0, )], (4.16) h T i −∞ A −∞ T A −∞ where Is is a Schr¨odinger operator, S (t , t ) the evolution operator, S (0, )= T A 2 1 A −∞ 0 T exp( i HA(t)dt), the Hamiltonian HA(t) = H IδA/c˜ and H = HC + HH + − −∞ −

HI(t). TheR expansion to the ﬁrst order in δA˜ yields

0 R δV =i dtTr[ρ( ) S( , t)δHsS(t, 0)Is S(0, ) T × −∞ { −∞ T −∞ Z−∞ S( , 0)Is S(0, t)δHsS(t, ) ], (4.17) − −∞ T −∞ } where δHs is the Schr¨odinger operator (4.13) and S(b, a) = Texp( i b dt[H +H + − a C H

HI(t)]). Substituting (4.13) in the above equation we see that the responseR of IT to

δV expresses as the sum of the responses of IT to the perturbations ISδA/c˜ and and I δA/c˜ . The latter response is zero since the edges are far apart and perturbations − U 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

0 exp(iωt) exp( iωt) RT =∂IT/∂V = i lim dt − − ω 0 2iω → mn Z−∞ X [ n I (t) m m I (0) n ρ ( ) m I (0) n n I (t) m ρ ( )]. × h | S | ih | T | i n −∞ −h | T | ih | S | i m −∞ (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 δVIT 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 I (0)I (t) = I ( t)I (0) . This allows rewriting R in the form h S T i h S − T i S

+ ∞ exp(iωt) exp( iωt) RS =i lim dt − − ω 0 2iω → 0 mn Z X [ n I (t) m m I (0) n ρ ( ) m I (0) n n I (t) m ρ ( )]. (4.19) × h | S | ih | T | i n −∞ −h | T | ih | S | i m −∞

We next compute RT = RT + RS:

ρn( ) ρm( ) RT =2π lim δ(En Em + ω) −∞ − −∞ ω 0 − 2ω → mn X [ n I (0) m m I (0) n + m I (0) n n I (0) m ], (4.20) × h | T | ih | S | i h | T | ih | S | i

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 I (t) and ρ( ). We notice S −∞ 149 that nonzero matrix elements m I (0) n correspond to N = N and n = m . h | S | i n m | Ci | Ci Hence, in the limit of low frequencies in Eq. (4.20), [ρ ( ) ρ ( )]/ω = n −∞ − m −∞ ρ dρ /dE = ρ ( )/T , where we used the factorization property (4.9) which − Cn Hn n n −∞ 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 T = T . One can then derive two equations of the C 6 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 experimental 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 manifests itself in the 2D electron gas only through the chemical potential of the lower 151

IU V DS IL IS Q

Figure 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.

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 general, 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

IU V DS IL IS

IT C

Figure 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. 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 relations [125, 126] and extend the FDT to a setup with arbitrary number of terminals.

Besides charge transport, we also study heat transport. Fluctuation relations provide 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 J ﬂow into reservoir i. Our main result is that the zero-frequency cross noise i S1i (S ) between I (J ) and I (J ) with i =1,r is related to the response of I (J ) to 1i 1 1 i i 6 i i the electrostatic potential V1 (temperature T1) in reservoir 1, through

∂Ii 1i = T1 (4.21) S − ∂V1 2 ∂Ji S1i = (T1) . (4.22) − ∂T1

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. is in fact the S1i 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 , µ i j , µ βi j

S β1 βr µ1 µr

Figure 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. 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 (t) that allows V particle and energy exchanges is turned on during a time interval 0 t , and ≤ ≤ T ﬁnally the interaction is turned oﬀ at t . At t 0, reservoir i is at equilibrium ≥ T ≤

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 (t) becomes a constant when fully V V0 turned on during τ t τ. We assume that τ and is longer than a ≤ ≤ T − ≪ T T relaxation time so that a steady state dominates the thought process.

When the system is in a steady state, the interaction between S and the V0 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 and be the Hamiltonian and particle number operator of Hi Ni the i-th reservoir respectively. In particular, describes the combination of the Hr subsystem and r-th reservoir. Particle number is conserved in the absence of (t) V in each reservoir, i.e., [ , ] = 0. An initial joint quantum measurement of Hi Ni Hi and is performed at t = 0, so that quantum state of the system collapses to Ni a common eigenstate ψ , with ψ = E ψ and ψ = N ψ . Then | ni Hi| ni in| ni Ni| ni in| ni the state ψ evolves according to the evolution operator U(t; +) determined by the | ni Hamiltonian (t;+) = + (t). The “+” sign represents the clockwise chirality H i Hi V of the subsystem. WhenP the system evolves at t = , A second joint measurement T is taken, leading to a collapse of the system to state ψ , with ψ = E ψ | mi Hi| mi im| mi and ψ = N ψ . In this process, we observe energy change ∆E = E Ni| mi im| mi i,mn im − E and particle number change ∆N = N N in the i-the reservoir. The in i,mn im − in probability to observe such a process is P [m, n] = ψ U( ;+) ψ 2ρ , where |h m| T | ni| n + βi[Ein µiNin Φi ] ρn = i e− − − is the initial Gibbs distribution to ﬁnd the system in state ψQ and Φ+ = is the initial grand potential of the i-th reservoir. If we repeat | ni i the above precess for many times, we obtain a distribution function of energy and particle changes

P [∆E, ∆N;+] = δ(∆E ∆E ) i − i,mn mn i X Y δ(∆N ∆N ) ψ U( ;+) ψ 2ρ , (4.23) × i − i,mn |h m| T | ni| n

where the vector ∆E = ∆E and ∆N = ∆N . Total energy and particles are { i} { i}

conserved in each process, such that i ∆Ei = i ∆Ni = 0. P P 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 scheme is 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 − − 1 the Hamiltonian (t; )=Θ ( t;+)Θ− , where Θ is the time-reversal operator. H − H T − 1 The i-th reservoir has a Hamiltonian Θ Θ− . One may find that Θ ψ (Θ ψ ) Hi | mi | ni 1 1 is a common eigenstate of Θ Θ− and = Θ Θ− , with eigenvalues E (E ) Hi Ni Ni im in and N (N ). Performing two quantum measurements at t = 0 and t = for each im in T process and repeating for many times, one finds the distribution function

P [∆E, ∆N; ]= δ(∆E ∆E ) − i − i,nm mn i X Y 1 2 δ(∆N ∆N ) ψ Θ− U( ; )Θ ψ ρ , (4.24) × i − i,nm |h n| T − | mi| m

− βi[Eim µiNim Φi ] with ρm = i e− − − the probability to ﬁnd the twin system initially in + state Θ ψ Q. One may prove that Φ = Φ−. | mi i i

An important property of the evolution operators is [129]

1 Θ− U( ; )Θ = U †( ;+). (4.25) T − T

Combining Eq. (4.23), (4.24) and (4.25), we recover the ﬂuctuation relation

E N P [∆ , ∆ ; +] βi(∆Ei µi∆Ni) = e − . (4.26) P [ ∆E, ∆N; ] i − − − Y 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 ± ν heat currents Ji = lim (∆Ei µi∆Ni) ν/ and their correlation functions, i.e., T →∞h − i T noises. The triangular brackets means taking average with respect to P [∆E, ∆N; ν].

The limit is taken so that steady-state quantities are obtained. It is reason- T →∞ able to assume that all physical quantities are ﬁnite under this limit. The particle

˜ν ˜ transfer currents Ji = lim ∆Ni ν/ and their noises Sij can be computed sim- T →∞h i T ilarly. It is convenient to deﬁne a cumulant generating function

1 Q(x, y; ν) = lim ln (d∆Eid∆Ni) T →∞ i T Z Y P xi(∆Ei µi∆Ni) P yi∆Ni e− i − − i P [∆E, ∆N; ν] , (4.27) × where the vectors x = x and y = y . Noticing that P [∆E, ∆N; ν] is normalized, { i} { i} 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

J ν = ∂ Q(0, 0; ν), J˜ν = ∂ Q(0, 0; ν), (4.28) i − xi i − yi and the noises ν ˜ν Sij = ∂xixj Q(0, 0; ν), Sij = ∂yiyj Q(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 = β , µ = µ , βµ = β µ and xµ = x µ . We have written down { i} { i} { i i} { i i} 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∆E1′ d∆N1′ P1[∆E1′ , ∆N1′ ; ν] Z P [∆E ∆E′ , ∆N ∆N ′ , ∆E′, ∆N′; ν], (4.31) × 2 1 − 1 1 − 1

where P1[∆E1′ , ∆N1′ ; ν] is the probability to transport ∆E1′ energy and ∆N1′ par-

ticles into reservoir 1 through the lower edge, and P2[∆E1′′, ∆N1′′, ∆E′, ∆N′; ν] is

the probability for the upper part to make changes (∆E1′′, ∆N1′′) 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 µ . In the setup with “ ” chirality, similarly we have P only depends on β and µ r − 1 1 1

while P2 does not depend on β1 and µ1.

In terms of the cumulant generating function, Eq. (4.31) results in that Q(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

β and µ while Q does not depend on β and µ for the system with “ ” chirality. 1 1 2 1 1 −

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 straightforward 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

∂ Q(0, y; β, µ;+) = ∂ Q(β, βµ y; β, µ; ), yi − yi − − − ∂ Q(0, y; β, µ;+) = β ∂ Q(β, βµ y; β, µ; ) µi − i yi − − − + ∂ Q(β, βµ y; β, µ; ). (4.32) µi − − − with x being set to 0. To be clear, we stress that ∂ Q(β, βµ y; β, µ; ) is µi − − − the partial derivative of Q(x, y; β, µ; ) over its variable µ with x β and y − i → → 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 ˜+ ∂J˜+ ∂Ji j ˜+ Tj + Ti = Sij + TiTj∂µiµj Q(β, βµ; β, µ; ). (4.33) ∂µj ∂µi − − −

A term ∂µiµj Q(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 I = qJ˜+ and noise = q2S˜+ for the system with “+” chirality, we i i Sij ij arrive at the standard FDT = T (G + G ) with T = T = T the temperature Sij − ij ji i j 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 consider the cross noise S˜+ of J˜+ and J˜+ with j = 1. This can be seen by writing 1j 1 j 6 ∂ Q(β, βµ; β, µ; ) as the sum of derivatives of Q and Q . Considering the µ1µj − − 1 2 “ ” chirality, Q depends only on µ while Q does not depend on µ , so the last − 1 1 2 1 term of Eq. (4.33) disappear. Moreover, if j = r, the ﬁrst term of Eq. (4.33) ∂J˜+/∂µ 6 1 j + is also zero, since in the system with “+” chirality J1 only depends on βr and µr. Hence we obtain

∂Ij 1j = T1 . (4.34) S − ∂V1

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

˜+ 2 j = r, the term ∂J1 /∂µr is not zero, instead equals G/q , with G the conductance ˜+ of a two-terminal setup. The reason is that ∂J1 /∂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 =1,r, we obtain 6

2 ∂Jj S1j = (T1) (4.35) − ∂T1

where the label “+” is dropped since all quantities belongs to the system with “+” 162 chirality. For j = r, there is an additional term ∂J /∂β = (T )2∂J /∂T on the 1 r − r 1 r right hand side of Eq. (fdt-heat). ∂J1/∂Tr is the thermal Hall conductance of a two-

2 terminal Hall bar, equal to κπ Tr/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 I = I I . Hence, S = S +S +2S , 3 − 1 − 2 3 1 2 12 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

∂I2 S3 = S2 2T1 +2GT1, (4.36) − ∂V1 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 paper and 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 J = 3 6 J J in general. If all the reservoirs are kept at the same voltage, with the − 1 − 2 thermoelectric voltages compensated as well, we have J = J J and a similar 3 − 1 − 2 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 experimentally 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 theorem, one based on a mixture of Kubo and Landauer-B¨uttiker 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 (K ) = 1+2δ . All components of the × mult ij ij 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

1 θ =2π K− n = π[n n/(k + 2)], (A.1) mp p m − p X where n = np. Hence, the tunneling probability for a particle of type m is P 2πΦ πn p = Am[1 + um cos( + δm + πnm )], (A.2) (k + 2)Φ0 − k +2

where Am, um and δm are real constants. 168

An alternative description of the same topological state of matter can be formulated 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 K has the following nonzero elements: K = 3, K =4(k n > 1), h 11 nn ≥ T Kn,n 1 = Kn 1,n = 2. The K-matirx Kmult expresses via Kh as Kmult = WKhW , − − − where W = 1, if i + j (k + 1), and W = 0, if i + j>k + 1. The charge vector ij ≤ ij 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-Pérot 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 = 2 neither 6 reason applies. Indeed, the Fabry-Pérot interferometry can distinguish the states

with k = 2 from proposed non-Abelian states [15] and k > 2 different flavors cannot 6 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], L = 1 dtdx[K ∂ φ ∂ φ V ∂ φ ∂ φ ]. Note that we k 4π h,ij t i x j − ij x i x j use the language of hierarhical states.R The diagonalization can be accomplished with 169 the new variables Φ =(n +2)φ (n +1)φ , n =1,...,k 1 and Φ =(k +2)φ . n n − n+1 − k k The action assumes the form

1 k L = dtdx[ K0∂ Φ ∂ Φ V˜ ∂ Φ ∂ Φ ], (A.3) k 4π n t n x n − ij x i x j n=1 Z X

0 where Kn = 2/[(n + 1)(n + 2)]. The operator ψn = exp(iΦn) creates an excitation

0 with one electron charge. The coeﬃcients Kn 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

k k 1 − 1 0 k k 0 k 1 k 1 L = dtdx[ K ∂ Φ ∂ Φ K ∂ Φ − ∂ Φ − ] 4π n t n x n − n t n x n n=1 n=1 Z X X + 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

p for the edge states of the two adjacent liquids. The operators exp(iΦn) create equal charges for both values of p = k,k 1. The prefactors in front of ∂ Φp ∂ Φp are − t n x n opposite for diﬀerent values of p. Hence, according to the criterion of stability of edge

k k 1 states [135,136], the modes Φn and Φn− gap each other out for each n

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

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 correspond 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

2πΦ πn(k + 1) pn = A[1 + u cos( + )], (A.5) (k + 2)Φ0 k +2

The number of different probabilities pn depends on the parity of k. For odd k, pn = p and hence we can define n modulo k + 2. For even k, p = p = p n+k+2 n n+2(k+2) 6 n+k+2 and n should be defined modulo 2(k + 2). Hence, the number of different values of p equals n =(k + 2)[3 + ( 1)k+2]/2. In what follows we will see a considerable n max − difference between even and odd k.

It will be convenient to use the following parametrization of Eq. (A.5)

p = B[v exp( iφ ) + exp(inθ)][v exp(iφ ) + exp( inθ)], (A.6) n − 0 0 −

where B = A[1 √1 u2]/2, v = [1 + √1 u2]/u, φ = 2πΦ/[(k + 2)Φ ] and − − − 0 0

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: t =1/p . Taking into account that there are only − n n

nmax diﬀerent values of pn, one ﬁnds the average time of nmax consecutive tunneling events: nmax

t¯ = nmax lim t/N = 1/pn. (A.7) t →∞ n=1 X Hence,

nmaxq I = 1 . (A.8) pn P

We next evaluate the sum t¯ = 1/pn. Similar sums were evaluated numerically in the main part of the article. ForP 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

Q (v exp( iφ )) 2 t¯= n | n − 0 | , (A.9) B P (v exp( iφ )) 2 P | − 0 | where P (z)= l(z + exp(ilθ)) and Qn(z)= P (z)/(z + exp(inθ)). Q 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 − nmax of power nmax = 2(k + 2) with the coeﬃcient 1 before z . This polynomial is

P (z) = znmax 1. From this one gets Q (z) = nmax−1 zl( 1)l+1 exp( iθn[l + 1]). − n l=0 − − The sum Q 2 can now be calculated by ﬁrstP performing the summation over n n | n| P 173 and reduces to Q 2 = n (v2nmax 1)/(v2 1). Finally, for even k n | n| max − − P 2nmax nmax(v 1) t¯e = − ; (A.10) B(v2 1)[v2nmax 2vnmax cos n φ + 1] − − max

eB(v2 1) v4k+8 2v2k+4 cos 4πΦ +1 I = − − Φ0 . (A.11) e k +2 v4k+8 1 − For odd k, a similar calculation yields

eB(v2 1) v2k+4 +2vk+2 cos 2πΦ +1 I = − Φ0 . (A.12) o k +2 v2k+4 1 −

The expressions look similar for even and odd k but there is a signiﬁcant difference 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

2 2 The noise is deﬁned as S = 2 limt (Q Q¯ )/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

2 2 2 2 nmax 2 δτ is straightforward. One ﬁnds: δτ = mδt , where δt = n=1 1/pn. Finally, the P 174

Fano factor 2 S 1/pn e∗ = = n q . (A.13) 2I max ( 1/p )2 P n P

Some restrictions on e∗ are evident from elementary inequalities. Obviously,

e∗ n q. The inequality of quadratic and arithmetic means also implies that ≤ max 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

nmax 1 4 − Q (v exp( iφ )) δt2 = n=0 | n − 0 | B2 P (v exp( iφ )) 4 P | − 0 | nmax 1 n − = max ( v)l+m+p+q exp(iφ [p + q l m]) B2 P 4 − 0 − − | | l,m,p,qX=0 δ(p + q l m n s), (A.14) × − − − max s X where s is an integer and the discrete delta function δ(0) = 1, δ(a = 0) = 0. The 6 remaining summation is tedious but straightforward. For even k we obtain

2k+5 e w 1 2 d d w 1 e∗ = ( − ) [ w − + e k +2 w2k+4 1 dw dw w 1 − − d w4k+7 w2k+4 (4k +7 w )2 − + − dw w 1 − 4πΦ d d w2k+4 1 2wk+2 cos( )(2k +3 w ) − ], (A.15) Φ − dw dw w 1 0 −

where w = [1 + √1 u2]2/u2. − 175

A similar calculation for odd k yields

k+3 e w 1 2 d d w 1 e∗ = ( − ) [ w − + o k +2 wk+2 1 dw dw w 1 − − d w2k+3 wk+2 (2k +3 w )2 − − dw w 1 − − 2πΦ d d wk+2 1 2w(k+2)/2 cos( )(k +1 w ) − ], (A.16) Φ − dw dw w 1 0 −

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,

8(k + 2)2 + 1 + [4(k + 2)2 1] cos 4πΦ Φ0 e∗ = e − . (A.17) e 6(k + 2)2

For odd k, 2(k + 2)2 +1 [(k + 2)2 1] cos 2πΦ Φ0 e∗ = e − − . (A.18) o 3(k + 2)2

2 1 As a function of the ﬂux, ee∗ oscillates between e[ 3 + 3(k+2)2 ] and 2e. eo∗ oscillates between e[ 1 + 2 ] and e. The maximal values of the Fano factor are thus 2 e at 3 3(k+2)2 × 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 p ∆p π∆m =2πrm, (A.19) − kP+2

where rm is an integer. One ﬁnds from (A.19) that ∆m = 2rm + p ∆p/(k + 2). P 177

Adding up such equations for all m, one ﬁnds that p ∆p =(k +2) p rp and hence P P k

(∆1,..., ∆k)= rmdm, (A.20) m=1 X

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 r = 1 and r = 1 for one p

adding to n one vector dm and subtracting another vector dn. Performing several

such operations, one can always reduce all np with p

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

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

residue ( nm) mod 2(k + 2) is also invariant with respect to an even number of

subtractionsP 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 N (n) = N (n) e 6 o 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

k 1 vectors consists of 2 − (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)).

k 1 (of 2 − (k + 2) allowable vectors. An arbitrary l-vector (n1,...,nk ג We get a set

k 1 (Hence, there are 2 − (k + 2 .ג is topologically equivalent to one of the vectors in 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 index m 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 trappedP quasiparticles modulo 2(Pk + 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:

df 2πΦ πn 0= n,S = A[1 u cos( + δ )]S dt −{ − (k + 2)Φ0 − k +2 2πΦ πn +A[1 + u cos( + δ )](k S) fn,S (k + 2)Φ0 − k +2 − } × 2πΦ π(n 1) +fn 1,S 1(k S + 1)A[1 + u cos( + δ − )] − − − (k + 2)Φ0 − k +2 2πΦ π(n 1) +fn 1,S+1(S + 1)A[1 u cos( + δ − )], (A.21) − − (k + 2)Φ0 − k +2

where the notation implies that fn,S = 0 for S < 0 and S>k. The current can then be expressed as

2πΦ πn I = fn,S A[1 u cos( + δ )]S { − (k + 2)Φ0 − k +2 X 2πΦ πn +A[1 + u cos( + δ )](k S) (A.22) (k + 2)Φ0 − k +2 − } 180

k 1 System (A.21) contains C =(k+1)(k+2) equations, much fewer than (k+2)2 − 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 reconstruction 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

SS SD

SC IT

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 surface 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):

∂I S = S 4T T + S , (C.1) D C − ∂µ S

where IT is the particle current through the hole in the tube wall, and SD = dt ∆I (t)∆I (0)+∆I (0)∆I (t) , S = dt ∆I (t)∆I (0)+∆I (0)∆I (t) , and h D D D D i S h S S S S i RS = dt ∆I (t)∆I (0)+∆I (0)∆I (t) areR respectively the particle current noises C h C C C C i at theR 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 S is determined by 2f(1 f), S by 2T f(1 T f) and S by S − C E − E D 2(1 T )f(1 (1 T )f). A combination of these contributions gives the desired − E − − E 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 particles 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 particles 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-Büttiker 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 tunneling 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] Bibliography

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