guiding-center hall viscosity and intrinsic dipole moment ......zograf[2] showed there is a...

$: Guiding-center Hall viscosity and intrinsic dipole moment ......Zograf[2] showed there is a dissipationless viscosity, which is termed \Hall viscosity," in IQH states. Then, in 2009,$
Guiding-center Hall viscosity and intrinsic

dipole moment

of fractional quantum Hall states

YeJe Park

A Dissertation

Presented to the Faculty

of Princeton University

in Candidacy for the Degree

of Doctor of Philosophy

Recommended for Acceptance

by the Department of

Physics

Adviser: F. D. M. Haldane

November 2014

c© Copyright by YeJe Park, 2014.

All rights reserved.

Abstract

The fractional quantum Hall effect (FQHE) is the archetype of the strongly correlated

systems and the topologically ordered phases. Unlike the integer quantum Hall effect (IQHE)

which can be explained by single-particle physics, FQHE exhibits many emergent properties

that are due to the strong correlation among many electrons. In this Thesis, among those

emergent properties of FQHE, we focus on the guiding-center metric, the guiding-center

Hall viscosity, the guiding-center spin, the intrinsic electric dipole moment and the orbital

entanglement spectrum.

Specifically, we show that the discontinuity of guiding-center Hall viscosity (a bulk prop-

erty) at edges of incompressible quantum Hall fluids is associated with the presence of an

intrinsic electric dipole moment on the edge. If there is a gradient of drift velocity due to a

non-uniform electric field, the discontinuity in the induced stress is exactly balanced by the

electric force on the dipole.

We show that the total Hall viscosity has two distinct contributions: a “trivial” contri-

bution associated with the geometry of the Landau orbits, and a non-trivial contribution

associated with guiding-center correlations.

We describe a relation between the intrinsic dipole moment and “momentum polariza-

tion”, which relates the guiding-center Hall viscosity to the “orbital entanglement spec-

trum(OES)”.

We observe that using the computationally-more-onerous “real-space entanglement spec-

trum (RES)” in the momentum polarization calculation just adds the trivial Landau-orbit

contribution to the guiding-center part. This shows that all the non-trivial information is

completely contained in the OES, which also exposes a fundamental topological quantity

γ = c − ν, the difference between the “chiral stress-energy anomaly” (or signed conformal

anomaly) and the chiral charge anomaly. This quantity characterizes correlated fractional

quantum Hall fluids, and vanishes in integer quantum Hall fluids which are uncorrelated.

iii

Acknowledgements

I would like to deeply thank my advisor, Professor F. D. M. Haldane. Without his

invaluable guidance, this work would not have been possible. It was an honor to learn at

first hand from the giant in condensed matter physics.

I would like to thank Professor WooWon Kang who first introduced me to the fractional

quantum Hall effect.

I would like to thank Professor Joseph Maciejko who provided me the chance to work on

fractional quantum Hall nematics and Helium 3B phase.

I would like to thank my talented and enthusiastic colleagues, Yang-Le Wu and Aris

Alexandradinata with whom I had many insightful discussions.

I would like to thank Zlatko Papic and Professor Nicolas Regnault whose matrix product

state generating code was indispensable for my thesis.

I would like to thank Professor Elliot Lieb for his kind words and for introducing me to

quantum entropy.

I would like to thank fellow graduate students : Loren Alegria, Philip Hebda, Dima

Krotov, GhooTae Kim, YoungSuk Lee, RinGi Kim, TaeHee Han, SeHyoun Ahn, JungHo

Kim, HyunCheol Jung, Hans Bantilan, Miroslav Hejna, Ilya Belopolski, HyungWon Kim,

Bo Yang, Ilya Drozdov, Andras Gyenis, Grisha Tarnopolskiy, Mykola Dedushenko and Victor

Mikhaylov for their friendship and for sharing their expertise.

I would like to thank my former roommate YoonSoo Park for his consideration and

kindness.

I would like to thank my loving parents, HyeSun and JangChun. Especially, my mother

is the source of inspiration.

iv

To my parents.

v

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

1 Introduction 1

1.1 Brief history of quantum Hall effect . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Playground for the study of strong correlation . . . . . . . . . . . . . . . . . 2

1.3 Topological phase classification by OES . . . . . . . . . . . . . . . . . . . . . 3

1.4 Physical content in OES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5 Geometric degree of freedom and incompressibility . . . . . . . . . . . . . . . 5

1.6 The aims of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Guiding-center physics 8

2.1 Wavefunctions need not be holomorphic. . . . . . . . . . . . . . . . . . . . . 8

2.2 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Landau-orbit metric and guiding-center metric . . . . . . . . . . . . . 11

2.2.2 Landau-orbit and Guiding-center Hall viscosities . . . . . . . . . . . . 18

2.3 The relationship between intrinsic dipole moment and guiding-center Hall

viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4 Guiding-center spin and Hall viscosity for a droplet . . . . . . . . . . . . . . 26

3 Jack polynomials 31

vi

3.1 Model FQH wavefunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1.1 Definition of Jack polynomials . . . . . . . . . . . . . . . . . . . . . . 31

3.1.2 Laughlin wavefunction as a Jack polynomial . . . . . . . . . . . . . . 35

3.1.3 Moore-Read wavefunction as a Jack polynomial . . . . . . . . . . . . 36

3.2 Mapping Jack polynomials into physical states . . . . . . . . . . . . . . . . . 39

3.2.1 On plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.2 On sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2.3 On cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.3 Numerical results from Jack polynomials . . . . . . . . . . . . . . . . . . . . 47

3.3.1 Occupation number . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.3.2 Luttinger sum rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3.3 Intrinsic dipole moment . . . . . . . . . . . . . . . . . . . . . . . . . 57

4 Entanglement spectrum 61

4.1 Orbital entanglement spectrum (OES) . . . . . . . . . . . . . . . . . . . . . 61

4.1.1 OES and Momentum polarization . . . . . . . . . . . . . . . . . . . . 62

4.1.2 Decomposition of 〈∆ML〉 and 〈∆NL〉 . . . . . . . . . . . . . . . . . . 67

4.2 Momentum polarization from Real-space cut . . . . . . . . . . . . . . . . . . 75

5 Collective excitation 84

5.1 Girvin-MacDonald-Platzman approximation . . . . . . . . . . . . . . . . . . 84

5.2 The relationship between SMA and guiding-center metric . . . . . . . . . . . 89

6 Conclusion 93

A Evaluation of topological spins 95

B Matrix product state (MPS) expansion 98

C Relation between structure factors 101

vii

Bibliography 102

viii

List of Figures

2.1 Edge of Hall fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2 Example occupation profile of 1/3 Laughlin state . . . . . . . . . . . . . . . 23

3.1 Squeezing of a partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2 The locations of Fermi momenta for bosonic and fermonic states . . . . . . . 45

3.3 1/2 Laughlin state occupation profile . . . . . . . . . . . . . . . . . . . . . . 49



3.6 2/2 Moore-Read state occupation profile : 2020. . . . . . . . . . . . . . . . . . 50

3.7 2/4 Moore-Read state occupation profile : 0110. . . . . . . . . . . . . . . . . . 51

3.8 2/4 Moore-Read state occupation profile: 0101. . . . . . . . . . . . . . . . . . 51

3.9 1/2 Laughlin state ∆N(k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54



3.12 2/2 Moore-Read state ∆N(k) : 2020. . . . . . . . . . . . . . . . . . . . . . . . 55



3.15 1/2 Laughlin state dipole moment . . . . . . . . . . . . . . . . . . . . . . . . 58



3.18 2/2 Moore-Read state dipole moment : 2020. . . . . . . . . . . . . . . . . . . 59

ix



4.1 OES of Laughlin 1/3 on cylinder . . . . . . . . . . . . . . . . . . . . . . . . 68

4.2 OES of Laughlin 1/5 on cylinder . . . . . . . . . . . . . . . . . . . . . . . . 69

4.3 OES of Moore-Read 2/4 on cylinder . . . . . . . . . . . . . . . . . . . . . . . 70

4.4 Dipole moments from orbital entanglement spectra . . . . . . . . . . . . . . 74

4.5 The subleading contributions in 〈∆ML〉 of Laughlin 1/3 . . . . . . . . . . . . 81

4.6 The subleading contributions in 〈∆ML〉 of Laughlin 1/5 . . . . . . . . . . . . 82

4.7 The subleading contributions in 〈∆ML〉 of Moore-Read 2/4 . . . . . . . . . . 83

5.1 Excitation spectrum for Coulomb in LLL at ν = 1/3 . . . . . . . . . . . . . 89

x

Chapter 1

Introduction

1.1 Brief history of quantum Hall effect

The integer quantum Hall (IQH) effect was first observed by von Klitzing[32] in 1980. In

the IQHE, the 2D bulk has an energy gap, and so it is insulating. However, as first noted

by Halperin[26], there is an edge excitation at the 1D edge of the bulk which carries electric

charge without dissipation and accounts for the quantized integer Hall conductance. Then,

in 1982, a far more puzzling discovery of the fractional quantum Hall (FQH) effect was

made by Tsui, Stormer and Gossard[57]. It was soon realized that such systems exhibiting

fractionally quantized Hall conductance can support fractionally charged quasi-particles[37]

which obey fractional statistics[1]. In case of the FQH edges, it was shown by Wen[61]

that the edge excitations are described “chiral Luttinger liquid” theory. In 1989, it was

shown by Zhang, Hansson and Kivelson[70] that the topological effective field theory of the

Abelian FQH states is the Abelian Chern-Simons theory. In 1991, Moore and Read[43]

made a prediction that there could be a FQH state at filling ν = 2 + 1/2 which supports

fractionally charged quasi-particles with non-Abelian statistics. In 1995, Avron, Seiler and

Zograf[2] showed there is a dissipationless viscosity, which is termed “Hall viscosity,” in

IQH states. Then, in 2009, Read[50, 52] made a prediction for the Hall viscosity in FQH

1

states with rotational invariance. This was generalized by Haldane[20] relaxing the rotational

invariance. These developments then culminated in the discovery of the “geometric degree

of freedom” (GDOF) in FQH states by Haldane[21] in 2011. The GDOF is essentially the

shape of the correlation hole which was previously assumed to be identical with the shape

of the cyclotron motion. These last two developments in FQHE, the Hall viscosity and the

GDOF, are the main subject of the Thesis.

1.2 Playground for the study of strong correlation

The fractional quantum Hall effect offers an ideal and simple setup to study exotic phenom-

ena that are created due to strong correlation of electrons. With large magnetic field, the

energy gap between two adjacent Landau level becomes sufficiently large so that the effective

Hamiltonian describing the many-particle system becomes solely the Coulomb interaction

projected into a single Landau level (See Haldane in [48]). The single-particle kinetic energy,

i.e. the energy due to the cyclotron motion, becomes identical for all particles in the Lan-

dau level. The remaining degrees of freedom are the guiding-centers of the electrons. The

problem completely reduces into the study of correlations among the guiding-centers.

In spite of the apparent simplicity, the many-body physics of the FQHE offers extremely

diverse physical states. To list some of them, we have the following many-body wavefunc-

tions: Abelian states include Laughlin[37], Haldane-Halperin Hierarchy[18, 30], Halperin

multi-layer[27], Jain composite fermion[29] wavefunctions. Non-Abelian states include Pfaf-

fian(also known as Moore-Read)[43], anti-Pfaffian[38, 39], Gaffnian[55], Read-Rezayi[51]

wavefunctions. This list is incomplete. In 2008, Bernevig and Haldane[4] showed that many

model wavefunctions are, in fact, symmetric Jack polynomials (times the Vandermonde fac-

tor). We will discuss in detail about the Jack polynomials in Ch.3.

There is a further complication that though two wavefunctions are constructed based

on different schemes, they may represent an identical topological phase, and they may be

2

adiabatically deformed (i.e. without closing an energy gap) into each other as argued by

Wen and Zee[63].

1.3 Topological phase classification by OES

Anyway, what do we mean by a topological phase? The traditional classification of phases

of matter was based on symmetries. The distinction between a crystal and a liquid is an

example of such classification. However, in FQHE, we cannot distinguish two FQH states

with the same filling factor ν using the symmetry classification.

It appears that ground states of topological phases contain enough information to distin-

guish between topological phases. The orbital entanglement spectrum (OES) is computed

from the Moore-Read ground state wavefunction by Li and Haldane[40] (We define the orbital

entanglement spectrum in Ch.4). When the OES was plotted against the total momentum

quantum numbers, it revealed the same degeneracy counting as its underlying conformal

field theory (a minimal model times the chiral boson, M(4, 3)⊗ U(1)).

There is also a study on the orbital entanglement spectra of ν = 2/5 states by Regnault,

Bernevig and Haldane[53]. In their work, they compare the OES of three wavefunctions: the

Jain state, Gaffnian state1 and the ground state of the Coulomb interaction in the lowest

Landau level. The degeneracy counting of the Coulomb ground state and the Jain state

matched the degeneracy counting of the two decoupled U(1) boson conformal field theory,

and meanwhile Gaffnian has the almost identical degeneracy counting as the former two but

misses some of them. Therefore, their OES distinguish between the Jain state and Gaffnian

state as different topological phases.

1The Gaffnian state is the ground state of the model three-body interaction Hamiltonian.

3

1.4 Physical content in OES

The orbital entanglement spectrum (OES) apparently contains a rich amount of informa-

tion. However, it is not obvious whether it is related to physical quantities. We will see

in the Thesis that the OES of an incompressible FQH state contains information about the

“guiding-center Hall viscosity,” ηHabcd, which is a part of the total Hall viscosity η′H

abcd,

η′Habcd = ηH

abcd + ηH

abcd (1.1)

The latter part ηHabcd in the total Hall viscosity is called “Landau-orbit Hall viscosity” which

is the contribution found by Avron et al.[2] (We define the two types of Hall viscosity in

Sec.2.2.)

In 2009, it was shown by Haldane[20] that the intrinsic dipole moment of the edge of a

FQH state can be related with the guiding-center Hall viscosity. The intuitive idea is that

the electric force on the intrinsic dipole moment at the edge of the FQH fluid should be

canceled by counter-balancing force. This latter force is the stress due to the guiding-center

Hall viscosity. Thus, by calculating the intrinsic dipole moment, we are able to find the value

of the guiding-center Hall viscosity. (We review this physical argument in Sec.2.3.)

Now, because of the uniform normal magnetic field, the intrinsic dipole moment is also

a total momentum (up to a constant factor e`2B/~). The total momentum of a subsystem

(from the edge of the fluid to some contour within the fluid) is termed as the “momentum

polarization” by Qi[58]. Since the expectation value of the total momentum within the

subsystem can be calculated from the OES, which is essentially the density matrix of the

subsystem, we see that the OES contains information about the guiding-center Hall viscosity,

and this result will be presented in Ch.4.

4

1.5 Geometric degree of freedom and incompressibility

The total Hall viscosity η′Habcd contains two parts. One of the two parts, “Landau-orbit Hall

viscosity” ηHabcd, is the response to the variation of the shape of the Landau-orbit(i.e. cy-

clotron motion), and the latter, “guiding-center Hall viscosity” ηHabcd, is the response to the

variation of the shape of the correlation hole. Each of these two shapes can be parameter-

ized by a 2×2 spatial unimodular metric[21]. The metric associated with the Landau-orbit

shape is called “Landau-orbit metric” and the one associated with the correlation hole shape

is called “guiding-center metric” (see Sec.2.2.1). Haldane realized that the guiding-center

metric is not only a variational parameter but also a dynamical field. This generalization

was the crucial insight from Haldane[21] which led him to finally predict that the gapped

collective modes in FQH states are the dynamical fluctuations of the guiding-center metric.2

We review the relationship between the guiding-center metric and the collective excitation

in Ch.5.

The energy gap in integer quantum Hall effect can be easily understood by solving single-

particle Schrodinger equation as first solved by Landau[36]. However, the presence of an en-

ergy gap for bulk collective modes in the partially filled Landau level is not easily understood.

The necessary condition for the presence of the energy gap was first successfully described

by Girvin, Macdonald and Platzman[16, 17](GMP) using the single-mode approximation of

Feynman[14, 13] projected into the lowest Landau level. This provided a crucial physical

insight that the density fluctuation is gapped within the FQH states.3 Though successful,

the GMP approach was restricted to the lowest Landau level. Because the collective mode

is due to guiding-center degrees of freedom as argued by Haldane[24, 21], we can generalize

2The previous attempts[70, 56] to explain the collective mode energy gap are unsatisfactory because theirenergy gap depends on the bare mass of electron. The bare mass should be irrelevant because the origin ofthe gap in FQHE is the Coulomb interaction.

3This should be contrasted with the collective mode in He 4 where it is gapless.

5

GMP collective mode for an incompressible FQH state in any partially filled Landau level

(see Sec.5.1).4

The complete field theory of the guiding-center metric is missing. As we reported

previously[47], in terms of the guiding-center metric, the density fluctuations of FQH states

correspond to the non-vanishing Gaussian curvature of the metric.[21]

1.6 The aims of the Thesis

There have been attempts to link the Hall viscosity with other physical observable such as the

Hall conductivity [28, 6]. The basic assumption of those calculations is Galilean invariance.

However, an incompressible FQH state is a topological phase for which such assumption

should not be essential.

In Sec.2.3, we relate the guiding-center Hall viscosity, i.e. the part of the Hall viscosity

due to the guiding-center degrees of freedom, with the intrinsic dipole moment per unit

length along the edge of incompressible FQH states[20, 46]. We do the explicit computation

of intrinsic dipole moments of Laughlin and Moore-Read states on a straight edge using two

numerical methods:

First, in Sec.3.3, we utilize the exact model wavefunctions from the Jack polynomials.

Second, in Sec.4.1, we calculate the intrinsic dipole moment from the “orbital entangle-

ment spectrum” (OES)[40]. Therefore, OES contains enough information to determine the

guiding-center Hall viscosity. The intrinsic dipole moment is essentially the non-vanishing

mean momentum due to the entanglement with the other half of the whole system (also

called “momentum polarization”). For a finite length L of the edge, there is a correction of

order O(L0). This correction is composed of two parts, “topological spin”[69, 58] and a new

4 Further understanding of the long wavelength limit of the GMP collective mode was achieved byconstructing explicit model wavefunctions for these collective modes for Laughlin and Moore-Read states byYang and Haldane[67]. They provided a way to regard the collective mode at the long wavelength limit asa quadraupole excitation consisting of two pairs of quasi-hole and quasi-particle. This is consistent with theexperimental observation for ν = 1/3 made in 2001 that the gap at the long wavelength limit is a roton-pairexcitation[31].

6

topological quantity γ = c − ν which is the difference between signed conformal anomaly

c and chiral charge anomaly ν of the underlying edge theory[46]. We also demystify the

topological spin and the fractional charge by showing that they originate from different cuts

relative to the “root occupation pattern”[4].

The next goal of this Thesis is to show that the computation of Hall viscosity with the

so-called “real-space entanglement spectrum” (RES)[8, 69] merely adds “Landau-orbit Hall

viscosity” which is a rather trivial part of the Hall viscosity due to the cyclotron motion (see

Sec.4.2). For a finite length L, RES also adds the chiral anomaly ν to the O(L0) correction,

and therefore obscures the existence of the new topological quantity γ. We are led to claim

that all the essential information of FQH states are contained in OES.

The last goal of this Thesis is to connect the geometric degree of freedom, i.e. the guiding-

center metric, with the collective excitation of the incompressible FQH states. In Ch.5, we

make an observation that there is the close relation between the long-wavelength limit of

the collective excitation energy in the single-mode-approximation and the energy cost due

to the deformation of the guiding-center metric.

7

Chapter 2

Guiding-center physics

2.1 Wavefunctions need not be holomorphic.

The fractional quantum Hall wavefunctions are usually (and somewhat misleadingly) written

as holomorphic functions. For instance, the celebrated many-body Laughlin wavefunction[37]

is written as

Ψ(m)L ({zi}) =

[∏i<j

(zi − zj)m]

exp

(− 1

4`2B

Ne∑i=1

|zi|2). (2.1)

where `B is the magnetic length, 1 and zi = xi + iyi is the complex coordinate of the i-th

electron (i = 1, 2, . . . Ne). The first factor in the wavefunction is called the Laughlin-Jastrow

(LJ) factor, and the exponent m is an odd integer to satisfy the fermion statistics of the

electrons.2 This is the spin-polarized many-body wavefunction that describes a FQH state

with the filling factor ν = 1/m = 1/3, 1/5, . . . in the lowest Landau level.3

It has the following energetically favorable properties:

1A physical planar sample with the finite total area A is penetrated by the uniform magnetic field B.The magnetic area 2π`2B = A/(B/Φ0) is defined to be the area through which a flux quantum Φ0 = h/epasses through.

2If elementary particles under consideration were bosons with repulsive interaction, we could have abosonic Laughlin state with even integer m. The experimental realization of such system is the rotatingBose-Einstein condensate.[60]

3If we place the electrons on a compact surface, e.g. a sphere, then there are (NΦ + 1) single-particlewavefunctions in the lowest Landau level where NΦ is the total number of flux quanta emanating uniformlythrough the surface. Then, the filling factor is ν = Ne/(NΦ + 1).

8

• The electrons avoid each other as much as possible given the filling factor, i.e. the

wavefunction has a built-in correlation.

• The electronic density is uniform away from the edge.

Based on this wavefunction, many other Abelian model wavefunctions[27, 29] were suggested

to explain FQH states at other filling factors. It also provided a very intuitive way of

understanding the fractionally charged quasi-particle excitations.4

The wavefucntion (2.1) also has a “nice” mathematical property that the Laughlin-

Jastrow factor, which encodes the correlations among electrons, is a holomorphic function

in each zi.5 However, we know that

• There are FQH states in higher Landau levels where the single-particle wavefunctions

are not holomorphic.

• There are FQH states stable against tilting of the applied magnetic field.

• Nematic FQH states with anisotropic response to the external field are possible.

• The equivalent FQH states can be constructed on different manifolds such as a sphere,

cylinder and torus.

The FQH states at fillings ν = 2+1/2, 2+1/3 and 2+2/36 are observed experimentally by

Choi et al [7] and Pan et al [45]7. Also, the FQH states at fillings ν = 2+2/5 and 2+6/13 are

observed experimentally by Kumar et al [33]. The effect of tilting magnetic field on the FQH

states with ν = 2/3 and 3/5 was experimentally studied by Engel et al [9]. The tilting effect

is studied theoretically by Yang et al [68] and Qiu et al [49]. The anisotropic FQH states

4By multiplying∏

i(zi − η), we create a quasi-hole located at η.5i.e. there is no dependence on zi = xi − iyi.6The filling factor ν = 2 +x, where x is some rational number less than 1, means that the lowest Landau

level is filled twice by the spin-up and spin-down electrons, and the second Landau level has a partial fillingx. The Zeeman splitting is much smaller than the Landau level splitting.

7Pan et al also observed a FQH state at ν = 2 + 2/5, and interestingly there was no FQH state at itsparticle-hole conjugate filling ν = 2 + 3/5.

9

are theoretically studied in [44, 42]. The FQH states on a sphere[18, 25], cylinder[54] and

torus[23] were investigated by Haldane and Rezayi.

The assumption in writing the wavefunction as a holomorphic function is the rotational

symmetry which enables one to make a choice of the symmetric gauge for the vector potential

A(r). However, we would like to argue that the rotational symmetry is an unnecessary

constraint which hides the previously unnoticed “geometrical degree of freedom” (GDOF),

i.e. the guiding-center metric[21] that appears in a gauge-invariant approach. In order to

treat the more general situations as mentioned above, we will need to retain what is essential

in the Laughlin wavefunction but avoid making unnecessary assumptions. The essential

property of the Laughlin wavefunction is the correlation of the guiding-centers encoded in

it. We will elaborate this in the next section.

10

2.2 Theoretical Background

In this section, we define the Landau-orbit radius operators Ri and the guiding-center po-

sition operators Ri for each i-th particle. The Landau-orbit radii and the guiding-centers

commute with each other. In a strong magnetic field, the energy gap between the Landau

levels become very large, and the inter-Landau level mixing becomes negligible. Then, the

fractional quantum Hall effect becomes the study of the guiding-center correlation.

We describe the distinct physical origins of the Landau-orbit metric and the guiding-

center metric. Then, the Landau-orbit Hall viscosity and the guiding-center Hall viscosity are

derived as the adiabatic responses to the variation of the Landau orbit metric and the guiding-

center metric respectively, and as the expectation values of area-preserving deformation

(APD) generators. We calculate these quantities for the Laughlin[37] and Pfaffian[43] states.

2.2.1 Landau-orbit metric and guiding-center metric

Consider N electrons with charge −e < 0 living on a 2D plane8 subject to a normal uniform

magnetic field strength B = Bz, B > 0. The i-th electron on the 2D plane has four degrees

of freedom, its coordinate ri and its dynamical momentum πi = pi + eA(ri). Note that

i, j, . . . are indices for electrons, and a, b, . . . are indices for the spatial coordinates. The

coordinate operator can be decomposed into two operators

ri = Ri + Ri, (2.2)

where the first operatorRi is the “guiding-center” of the electron, and the second operator Ri

is the “Landau-orbit radius.” The Landau-orbit radii are defined in terms of the dynamical

8More precisely, we consider a torus with a finite area A and a total number of flux quanta NΦ in thethermodynamic limit NΦ →∞ and A→∞, while keeping 2π`2 = A/NΦ constant.

11

momenta: Rai = εabπb/eB. These operators have the following commutation relations,

[Rai , R

bj] = −iεabδij`2

B (2.3a)

[Rai , R

bj] = iεabδij`

2B (2.3b)

[Rai , R

bj] = 0, (2.3c)

where `2B = ~/eB. This decoupling between Ri and Ri is completely independent of the

choice of a gauge.

Out of these operators, we can form area-preserving deformation (APD) generators[21]

Λab =Ne∑i=1

Λabi =

1

4`2B

Ne∑i=1

{Rai , R

bi} (2.4a)

Λab =Ne∑i=1

Λabi =

1

4`2B

Ne∑i=1

{Rai , R

bi}, (2.4b)

where { , } is an anti-commutation. These satisfy the commutation relations of the special

linear Lie algebra, sl(2,R),

[Λab,Λcd] = + i2(εacΛbd + εadΛbc + a↔ b) (2.5a)

[Λab, Λcd] = − i2(εacΛbd + εadΛbc + a↔ b). (2.5b)

The interacting electrons are described by the following Hamiltonian H which is a sum

of the non-interacting part H0 and the interaction part U ,

H = H0 + U

U =∑i<j

V (ri − rj; ε). (2.6)

12

Note that the Coulomb interaction V also depends on the permittivity tensor εab. The

Fourier-transformed interaction is9

U =1

NΦ

∑q

∑i<j

V (q; ε)eiq·(ri−rj). (2.7)

The most general non-interacting part H0 is

H0 =Ne∑i=1

h(Ri),

where h(r) is a function of r whose constant contours are non-overlapping and closed. The

most general form of the single-particle energy is technically intractable, so we take a model

single-particle energy parameterized by a unimodular symmetric positive-definite 2-tensors

gab which we call the “Landau-orbit metric,”

H0 =Ne∑i=1

h(gabRai R

bi), (2.8)

where h(r) is a monotonically increasing function of r. This form includes, for instance, the

following two examples which break Galilean invariance,

H0 =Ne∑i=1

(gabΛabi )k+1, k ∈ N

H0 =Ne∑i=1

√1 + gabΛab

i ,

The second example is the massive Dirac Hamiltonian of a charged particle subject to a

normal magnetic field.

9The summation normalized as such becomes in the thermodynamic limit NΦ →∞,

1

NΦ

∑q

→∫d2q`2B

2π.

13

If the system is Galilean invariant, the Landau-orbit metric gab is determined by the

effective mass tensor (m−1)ab,

H0 = 12(m−1)ab

Ne∑i=1

πi,aπi,b

= 12~ωc L(g)

L(g) =Ne∑i=1

Li(g) =Ne∑i=1

gabΛabi , (2.9)

where the cyclotron frequency is ωc = eB/|m| and |m| = detm. We also defined the rotation

generator of Landau-orbit radii, L(g). The eigenvalues of Li(g) are sn = n+ 12, n ∈ Z+, and

we call sn the “Landau-orbit spin.”

For the single-particle energy (2.8), we can label the Landau level with the non-negative

integer n from the eigenvalue of Li(g). Note that the system without Galilean invariance

has an unequal energy gap between neighboring Landau levels.

In the strong magnetic field strength limit where Landau level mixing is not allowed,

the Landau-orbit and guiding-center degrees of freedom decouple. Then, the many-particle

ground state |Ψ〉 of the Hamiltonian H in the n-th Landau level can be decomposed as a

tensor product,

|Ψ〉 =

(Ne∏i=1

|n〉L,i)⊗ |Ψ(g)〉G. (2.10)

The vectors with the subscript L (for Landau-orbit) can be acted on only by the Landau-

orbit operators Ri and the vectors with the subscript G (for guiding-center) can be acted on

only by the guiding-center operators Ri. The vector |n〉L,i is the n-th eigenstate of Li(g).

We now discuss how the guiding-center part |Ψ(g)〉G is determined. Since the Landau-

orbit part of |Ψ〉 (i.e. the first factor in the tensor product (2.10)) is fixed, the Hamiltonian

H can be projected into the n-th Landau level. The non-interacting part H0 is a constant

number after the projection. When the interaction U (2.7) is projected into the n-th Landau

14

level, we have

ΠnUΠn =1

2NΦ

∑q

V (q; ε)fn(q)2ρ(q)ρ(−q), (2.11)

where Πn is a formal projection operator into the n-th Landau level, NΦ is the total number of

flux quanta penetrating the QH fluid. Here, we defined the “Landau-orbit form factor” fn(q)

and the guiding-center density operator ρ(q) as follows. Consider the Fourier-transformation

of the density operator ρ0(r),

ρ0(r) =Ne∑i=1

δ(2)(r − ri)

ρ0(q) =Ne∑i=1

∫d2r eiq·rδ(2)(r − ri) =

Ne∑i=1

eiq·ri .

The density operator ρ0(q) is projected into the n-th Landau level by sandwiching the

operator with the vector∏

i|n〉L,i, and this produces the projected density operator ρn(q) as

a product of the form factor and the guiding-center density operator,

ρn(q) := fn(q)ρ(q) (2.12a)

fn(q) := 〈n|eiq·Ri |n〉L,i (2.12b)

ρ(q) :=Ne∑i=1

eiq·Ri . (2.12c)

Here, we used the decomposition of the position ri = Ri +Ri.

If the single-particle energy is of the form (2.8), the Landau-orbit form factor becomes

fn(q) = Ln(

12|q|2g)e−|q|

2g/4 (2.13)

where Ln is a Laguerre polynomial of degree n, and the Landau-orbit metric norm is defined

as |q|2g = gabqaqb`2B (gacgcb = δab ). We can make an alternative definition of the Landau-orbit

15

metric in terms of the Landau-orbit form factor,

gab := s−1n `−2

B ∂qa∂qbfn(q)|q=0. (2.14)

This definition gives us the interpretation of the Landau-orbit metric as the parameter which

determines the shape of the Landau-orbit.

One can re-write the projected interaction (2.11) into a more fundamental expansion

known as Haldane pseudo-potential,[18]

ΠnUΠn =∞∑

M=0

VM(g, n, ε)PM(g) (2.15)

where {VM(g, n, ε) : M = 0, 1, 2, . . . } are the pseudo-potential coefficients, and PM(g) is

the projection operator into a two-particle state with the relative guiding-center angular

momentum M + 12. They are defined explicitly as

VM(g, n, ε) :=1

2Nφ

∑q

V (q; ε)fn(q)2LM(|q|2g)e−|q|2g (2.16a)

PM(g) :=1

Nφ

∑q

LM(|q|2g)e−|q|2g/2ρ(q)ρ(−q). (2.16b)

Here, we introduced a positive-definite symmetric 2-tensor gab which we call the “guiding-

center metric” through the norm |q|g := gabqaqb`2B (gacgcb = δab ). We see that the pseudo-

potential coefficients VM(g, n, ε) are the expansion coefficients of V (q; ε)fn(q)2 using the

Laguerre polynomials {LM(|q|2g) : M = 0, 1, 2, . . . } as the basis. 10 Now, to understand the

action of PM(g), we define the “relative guiding-center rotation generator,”

Lij(g) := 18`2Bgab{Ra

i −Raj , R

bi −Rb

j}. (2.17)

10The Lagurre polynomials Ln(x) satisfy∫∞

0e−xLm(x)Lm′(x) = δmm′ .

16

This operator has a spectrum, “relative guiding-center angular momentum” {M + 12

: M ∈

Z+}. PM(g) has non-vanishing matrix elements for states with a pair of particles with relative

guiding-center angular momentum M + 12.

Instead of using the full expansion as given in (2.15), we can form a model interaction,

Umodel(g) :=

q−1∑M=0

VMPM(g), (2.18)

where {VM : M = 0, 1, 2, . . . , q−1} are positive reals and q is some positive integer. The full

expansion (2.15) does not depend on a particular choice of gab because a different choice gab

merely corresponds to a different Laguerre polynomial basis {LM(|q|2g) : M = 0, 1, 2, . . . }

in the expansion of V (q; ε)fn(q)2. However, the model interaction (2.18) does depend on

gab. The ν = 1q

“Laughlin state” is an exact zero energy state of the model interaction:

Umodel(g)|Ψ(g)〉G = 0. The Laughlin wavefunction is a particular member (gab = gab) of the

family of Laughlin states parameterized by gab in the Galilean invariant system.[21] If the

original projected interaction (2.11) contains the permittivity tensor εab and the Landau-

orbit metric gab that are not related by multiplying a constant, then there is no reason to

prefer the isotropic state with gab = gab.

Therefore, we see that the particular form of the model interaction (i.e. the set of

numbers {VM : M = 0, 1, . . . , q − 1}) determines the correlation among particles; it tells

us what relative guiding-center momenta are energetically unfavorable. Meanwhile, the

guiding-center metric determines the shape of the correlation hole.

Given the family of states {|Ψ(g)〉G : gab} which minimize Umodel(g) and are parameter-

ized by gab, the equilibrium guiding-center metric is finally determined by minimizing the

correlation energy,

EG(g) = 〈Ψ(g)|ΠnUΠn|Ψ(g)〉G. (2.19)

Note that the guiding-center metric describes an emergent geometry of the correlated elec-

trons while Landau-orbit metric directly comes from the Landau-orbit form factor. The

17

guiding-center metric may vary on the length scale much larger than `B. Furthermore, it

was proposed by Haldane[20] to be a dynamical field that describes the gapped collective

mode of the incompressible FQH fluid.

2.2.2 Landau-orbit and Guiding-center Hall viscosities

In the last section, we described the definition of the equilibrium values of the Landau-orbit

metric and the guiding-center metric. Here, we want to deform the metrics preserving their

determinants, and find the Hall viscosities as the response of the incompressible FQH state

without assuming Galilean and rotation invariances.

The APD generators preserve the determinant of the metric gab and gab. To see this

(let’s focus on guiding-centers first), define the unitary operator U(α) parameterized by a

real symmetric 2-tensor αab,

U(α) = exp iαabΛab. (2.20)

Then, this unitary operator deforms the metric gab into g′ab by group conjugation but leaves

the determinant unchanged,

Lij(g′) = U(α)†Lij(g)U(α), det g′ = det g.

If αab is infinitesimal, then the variation in the metric is

δgab = −gacεcdαdb + a↔ b. (2.21)

Suppose that we have an incompressible FQH state |Ψ〉 of the form (2.10) whose guiding-

center metric minimizes the correlation energy (2.19). Then, we can define a deformed state

|Ψ(α)〉 = U(α)|Ψ〉.

18

We can find the generalized force by the adiabatic response associated with the variation

αab,

F ab = − ∂EG(g)

∂αab

∣∣∣∣α=0

+ Γabcdαcd.

The first term vanishes because the correlation energy is minimized for the equilibrium

guiding-center metric gab, and the second term is

Γabcd = −~ Im〈∂αabΨ(α)|∂αcdΨ(α)〉|α=0

= −i~ 〈Ψ|[Λab,Λcd]|Ψ〉.

Dividing by the area A occupied by the QH fluid,11 we find a 4-tensor ηabcdH which we identify

as the guiding-center Hall viscosity tensor with raised indices,

ηabcdH = − 1

AΓabcd =

~2π`2

B

i

NΦ

〈Ψ|[Λab,Λcd]|Ψ〉. (2.22)

In the active transformation, the deformation of the metric corresponds to the following

mapping of Ri,

Rai → U(α)†Ra

iU(α)

= Rai − iαbc[Λbc, Ra

i ] +O(α2)

= Rai + εabαbcR

ci .

Thus, we identify εacαcb as the analog of the derivative of the displacement vector ∂bua in

the classical elasticity theory.[35] With this, we further identify the symmetric strain tensor

uab in terms of εacαcb:

uab = 12(gacε

cdαdb + a↔ b). (2.23)

11The area A is defined by the relation, A = 2π`2BNΦ

19

Then, the guiding-center Hall viscosity tensor is

ηHacbd = ηaecfH εebεfd. (2.24)

We can use the commutation relations of the guiding-center APD generators to expand the

4-tensor ηabcdH in terms of a symmetric 2-tensor ηabH ,

ηabcdH = 12(εacηbdH + εadηbcH + a↔ b) (2.25a)

ηabH = − ~2π`2

B

1

NΦ

〈Ψ|Λab|Ψ〉 (2.25b)

The quantity 〈Ψ|Λab|Ψ〉 contains both super-extensive (∝ N2) and extensive (∝ N) terms.

The former contribution comes from the uniform background number-density ν/2π`2B

(ν = N/NΦ). This super-extensive term should be subtracted so that the guiding-center

Hall viscosity is regularized. The extensive term does not vanish only if the electrons develop

correlation.

Now, consider the Landau-orbit degree of freedom. After replacing Λab with Λab and

Lij(g) with L(g), the same argument works. The Landau-orbit Hall viscosity tensor is

ηHacbd = ηaecfH εebεfd (2.26a)

ηabcdH = 12(εacηbdH + εadηbcH + a↔ b) (2.26b)

ηabH =~

2π`2B

1

NΦ

〈Ψ|Λab|Ψ〉 (2.26c)

The Landau-orbit Hall viscosity does not need regularization. The sign difference between

(2.25b) and (2.26c) originates from the commutation relations of Landau-orbit and guiding

center APD generators, cf.(2.5). This Landau-orbit Hall viscosity exists whether or not the

electrons are correlated. If the single-particle energy is of the form (2.8), then the Landau-

20

orbit Hall viscosity tensor can be expressed in terms of Landau-orbit spin,

ηabH =~

2π`2B

νsngab (2.27)

This is the Hall viscosity first discussed by Avron, Seiler and Zograf in the Galilean invariant

system.[2]

2.3 The relationship between intrinsic dipole moment

and guiding-center Hall viscosity

In this section, we relate the intrinsic dipole moment of the incompressible FQH state with

its guiding-center Hall viscosity.

Let’s clarify what we mean by the intrinsic dipole moment. Consider electrons on a

cylinder through whose surface an uniform magnetic field B passes. We confine the electrons

by an external electric potential V (y) that depends on y, one of the two spatial coordinates,

x and y. Then, single-particle states |φm〉 are labeled by guiding-centers ym = 2πm`2B/L,

m ∈ Z + 12

(`2B = ~/eB). Given a many-particle state |Ψ〉, we can calculate its occupation-

number profile which is the set of the expectation values of occupation-number operators

nm for each index m. For instance, consider an IQH state in the first Landau level, |Ψ1〉,

filling the upper-half plane with a “Fermi momentum” at y = 0 (See Fig.2.1a). Then, its

occupation profile is {. . . , n−3/2, n−1/2, n1/2, n3/2, . . . } = {. . . , 0, 0, ν, ν, . . . } where the filling

factor ν = 1. In the continuum limit L → ∞, the occupation profile for this uncorrelated

state is a step function in y : n(y) = ν θ(y).

Now, let’s consider as an example of a correlated state, the Laughlin ν = 13

state[37],

|Ψ1/3〉. As before, suppose the Fermi momentum is at y = 0 (when the circumference

L is finite, it is not obvious where the Fermi momentum is. This will be clarified later,

Sec.3.2.3). For a given L, we can obtain a occupation profile. For L = 15`B, we have the

21

x

y

vx

¶y vx¹0

(a) A straight edge (b) An arbitrary edge

Figure 2.1: The gray area represents the Hall fluid. In general, the drift velocity depends onthe distance from the edge. Each edge follows an equipotential line.

occupation profile in Fig.2.2. Unlike the uncorrelated state |Ψ1〉, the occupation profile of the

correlated state |Ψ1/3〉 deviates from the filling factor ν = 13

near the edge. In the continuum

limit, the occupation profile becomes n(y) ∝ y(ν−1−1) as predicted by chiral boson theory[61]

(cf. Sec.3.3.1). We see that the correlation among the electrons develops an extra “intrinsic

dipole moment” at the edge by “pulling them inward” (this corresponds to the fact that FQH

model wavefunctions are spanned by states obtainable by “squeezing” the “root state”[4],

See Sec.3.1).

Because an incompressible FQH state is a topological phase, the straightness of the edge

should not be essential. Therefore, we consider an edge of an arbitrary shape as in Fig.2.1b

on a flat 2D plane. We denote the line element along the edge by dLa. We relate the intrinsic

dipole moment dpa per a line element dLa by introducing a dimensionless symmetric 2-tensor

Qab,

dpa = −eQabεbcdLc. (2.28)

The electric charge −e is negative, and εab = εab is the Levi-Civita anti-symmetric tensor,

εxy = −εyx = 1. Throughout the Thesis, we distinguish covariance and contravariance of

indices, and we use the Einstein summation convention. In general, the electric field which

22

æ

æ

æ

æ

æ

æ

æ

ææ æ

ææ æ æ æ æ æ æ æ æ æ æç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç

0 2 4 6 80.0

0.1

0.2

0.3

0.4

0.5

0.6

y_m �

2 Π m

L

n_m

Figure 2.2: Occupation profile (•) for Laughlin ν = 1/3 state and the uniform occupationprofile nm = ν (◦) on a cylinder with circumference L = 15 (`B = 1).

derives from the Coulomb interaction and the confining potential is not constant but depends

on the distance from the edge. The gradient of the electric field coupled with the intrinsic

dipole moment results in an electric force,

dFel,a = dpb∂aEb. (2.29)

If the edge is to be stable, this electric force should be balanced. What should this

counter-balancing force be? The counter-balancing force against the electric force on the

intrinsic dipole comes from the guiding-center Hall viscosity. Here, we review the physical

argument[20], and then we will provide two kinds of numerical proofs, first utilizing the

exact model wavefunctions in Sec.3.3 and secondly utilizing the orbital entanglement spectra

in Sec.4.1.

Firstly, we note that pressure is absent. An incompressible FQH state is a topologi-

cal quantum phase. In the bulk, all excitations are separated by an energy gap, and its

low-energy effective description is the Chern-Simons Lagrangian[70] with a vanishing Hamil-

23

tonian. Because the incompressible state has no phonons to mediate the effect of external

force, the bulk pressure vanishes entirely[20].

We should take into account only the guiding-center part of the total Hall viscosity

because the “trivial” Landau-orbit Hall viscosity ηHabcd is present whether or not the electrons

are correlated (this part of the Hall viscosity will be discussed together with RES in Sec.4.2).

When the electrons develop correlations among themselves, there arises the additional non-

trivial guiding-center Hall viscosity ηHabcd, concurrently with the intrinsic dipole moment.

The non-uniform electric field near the edge results in a non-vanishing gradient of the

drift velocity va = εabEb/B. Then, the edge experiences a dissipationless stress σba due to

the guiding-center Hall viscosity proportional to the gradient of the drift velocity,

σab = −ηH acbd ∂cv

d = −ηaecfH εebεfd ∂cvd. (2.30)

where in the second equality, we raised the two lower indices of ηHacbd using Levi-Civita

symbols. Note that ηabcdH is anti-symmetric under the exchange of the two pairs of indices

(ab)↔ (cd), and symmetric under the exchange of two indices a↔ b or c↔ d (cf. Sec.2.2.2).

Such 4-tensor can be expanded in terms of a symmetric 2-tensor ηabH ,

ηabcdH = 12(εacηbdH + εadηbcH + a↔ b).

With this expansion, the expression for the stress tensor becomes

σab = 12B−1(εacεebη

efH + δcbη

afH + c↔ f) ∂cEf (2.31)

From the stress, we find the viscous force dFvisc,a on a line element dLa,[35]

dFvisc,a = σbaεbcdLc. (2.32)

24

We make two physical assumptions to reduce the viscous force equation further. The first

assumption is that the magnetic field is static so that the Maxwell’s equation gives εab∂aEb =

0. This implies the 2-tensor ∂aEb is symmetric under the exchange of the indices, a↔ b. The

second assumption is that the line element dLa of the edge is directed along the equipotential

line so that EadLa = 0. From these two assumptions, the viscous force reduces to

dFvisc,a = (B−1ηbcHεcddLd)∂aEb. (2.33)

Then, from the requirement that the net force on the line element vanishes, dFel,a+dFvisc,a =

0, we obtain the relationship between the intrinsic dipole moment tensor Qab and the guiding-

center Hall viscosity 2-tensor ηabH ,

ηabH = eBQab. (2.34)

Thus, if we know the guiding-center Hall viscosity tensor ηabH , we also know the intrinsic

dipole moment dpa along the static equipotential edge,

dpa = −B−1ηabH εbcdLc. (2.35)

This relationship (2.35) between the intrinsic dipole moment and the guiding-center Hall

viscosity tensor was derived from a local balance of forces. Therefore, we expect the rela-

tionship to hold for an edge of any smooth arbitrary shape reflecting the topological nature

of an incompressible FQH state.

To give a specific example, consider the situation depicted in Fig.2.1a for which ∂yvx

is the only non-vanishing component of the velocity gradient. Then, the stress expression

reduces to

σyy = −ηyyH∂yEyB

. (2.36)

25

The viscous force per a line element dLx on the edge is given by

dFvisc,y = ηyyH∂yEyB

dLx. (2.37)

The electric force on the dipole for this situation is

dFel,y = −eQyydLx∂yEy. (2.38)

Vanishing of the net force gives us,

dpy

dLx=ηyyHB. (2.39)

The left-hand side of (2.39) can be numerically calculated from occupation profiles (for

instance, Fig.2.2). This will be done in Sec.3.3 and Sec.4.1. The right-hand side of (2.39)

can be analytically calculated as the expectation value of the “area-preserving deformation

generators”. This calculation is described in the next section, Sec.2.4.

2.4 Guiding-center spin and Hall viscosity for a droplet

In the last section, we derived the two kinds of Hall viscosity without assuming Galilean and

rotational symmetry. Furthermore, there was no assumption about the shape of the QH fluid

(it could take any shape as in Fig.2.1b). In this section, we take the shape of the QH fluid

to be a “droplet,” and then we extract a quantity called the “guiding-center spin” which is

an emergent spin associated with a “composite boson.” Then, we express the guiding-center

Hall viscosity in terms of the guiding-center spin.

Suppose we have a “droplet” of the incompressible ν = p/q FQH state |Ψp/q〉 of the form

(2.10) which is a condensate of “composite bosons.” Suppose its Landau-orbit metric gab and

guiding-center metric gab take their equilibrium values.

A “composite boson” is made of p particles (which can be either fermion or boson) with

q flux quanta. The droplet contains N “elementary” particles so that there are N = N/p

26

composite particles. The droplet is penetrated by NΦ = qN flux quanta. If we exchange

two composite bosons, the state acquires a phase ξp from the particle statistics(ξ = 1 for

the bosonic particle and ξ = −1 for the fermionic particle) and the Aharonov-Bohm phase

(−1)pq. The composite object is a boson, and so these two phases should cancel ξp ×

(−1)pq = 1. This imposes a condition on possible combinations of the integers p and q. The

model incompressible FQH states under our consideration all satisfy this condition : bosonic

Laughlin states with p = 1 and even q, fermionic Laughlin states with p = 1 and odd q,

bosonic Moore-Read state with p = 2 and q = 2, and fermionic Moore-Read state with p = 2

and q = 4

The “droplet” means that it is an eigenstate of the “guiding-center rotation generator”

L(g) = gabΛab (2.40a)

L(g)|Ψp/q〉 = (12pqN2 + sN)|Ψp/q〉. (2.40b)

where the second equation defines the rational number s. (In the language of the wave-

functions in the symmetric gauge, the eigenvalue of L(g) is the sum, the total power of all

zi = xi + iyi in a monomial plus 12N .) Note that the first term is the guiding-center angular

momentum from the uniform occupation profile nm = p/q,

12pqN2 =

NΦ−1∑m=0

(m+ 12)nm+1/2.

As an analogue of the usual decomposition Jz = Lz + Sz (the total angular momentum is

the sum of orbital angular momentum and the spin), we may regard the extensive term sN

as the spin part of the total angular momentum from N composite bosons. We call s the

“guiding-center spin.”

Let’s calculate s for Laughlin 1/3 state as an example. Since the Laughlin state is a Jack

polynomial[4] with the proper normalization factors (cf. Sec.3.2), its guiding-center angular

momentum can be calculated from the “root occupation profile” {n0m+1/2 : m ∈ Z+} =

27

ν 12

13

14

22

24

26

s −12−1 −3

2−1 −2 −3

−sq

14

13

38

12

12

12

Table 2.1: Guiding center spin s. The first row lists the filling factor ν for model statesdiscussed in the text. The second row lists the corresponding guiding-center spins, and thethird row lists their dipole moments per unit length in units of e/4π.

{n01/2, n

03/2, n

05/2, n

07/2, . . . } = {1, 0, 0, 1, 0, . . . }. Its root occupation profile is a repetition of

the pattern (1, 0, 0). The guiding-center angular momentum is then

NΦ−1∑m=0

(m+ 12)n0

m+1/2 = 32N2 − N

Comparing this with (2.40b), we deduce s = −1 for the Laughlin 1/3 state. Take Moore-Read

2/4 state as the second example. It has the root occupation profile {n0m+1/2 : m ∈ Z+} =

{1, 1, 0, 0, 1, 1, 0, 0, . . . }, and this corresponds to the guiding-center angular momentum,

NΦ−1∑m=0

(m+ 12)n0

m+1/2 = 82N2 − 2N .

Thus, the guiding-center spin of the Moore-Read 2/4 state is s = −2.

The guiding-center spins of Laughlin 1/q state for q = 2, 3, 4 and Moore-Read 2/q state

for q = 2, 4, 6 are listed in the Table.2.1. Note that the guiding-center spin vanishes for

uncorrelated uniform states.

From the fact that |Ψp/q〉 is the eigenstate of L(g), we can calculate its expectation value

of the regularized guiding-center APD generator δΛab,

〈Ψp/q|δΛab|Ψp/q〉 = 12gabsN. (2.41)

28

Inserting this into (2.25b), we obtain the regularized guiding-center Hall viscosity tensor,

ηabH = − ~4π`2

B

s

qgab. (2.42)

In general, the guiding-center metric may depend on the spatial coordinates on the length

scale much larger than `B while the guiding-center spin remains quantized,

ηabH (r) = − ~4π`2

B

s

qgab(r). (2.43)

From (2.35), we obtain the expression of the intrinsic dipole moment per a line element

dLa in terms of the guiding-center spin and the number of flux quanta in a composite boson,

Bdpa =~

4π`2B

s

qgabεbcdL

c. (2.44)

For a line element dLx,

Bdpy = − ~4π`2

B

s

qgyydLx. (2.45)

The expected intrinsic dipole moments dpy are listed in Table.2.1 in the unit e/4π (gab = δab).

This will be verified numerically in Sec.3.3 and Sec.4.1.

We recover the Hall viscosity discussed in other works[52, 6] if we impose inessential

rotational invariance gab = gab. In this case, the usual angular momentum Lz is a good

quantum number.

Lz = gab(Λab − Λab), δLz = gab(Λ

ab − δΛab),

where δLz is the regularized angular momentum subtracting the contribution from the uni-

form density. The expectation value δLz for the model states |Ψp/q〉 divided by the number

29

of composite bosons N gives

N−1〈Ψp/q|δLz|Ψp/q〉 = ps− s,

which is the total spin per composite boson. For 1/q Laughlin states, the guiding-center

spin is s = 12(1− q), and the total spin per composite boson is 1

2q. This coincides with what

was called “orbital spin” by Wen and Zee,[62] and later by Read and Rezayi.[52] In such

rotational invariant system, the sum of the Landau-orbit and guiding-center Hall viscosities

becomes

ηabH + ηabH =~

4π`2B

(νs− s

q

)gab.

This is the Hall viscosity discussed by Read and Rezayi.[52] Note this is valid only in the

rotational invariant system, and it misses the separation of two types of Hall viscosity.

30

Chapter 3

Jack polynomials

3.1 Model FQH wavefunctions

In this section, we first present the definition of a Jack polynomial (Sec.3.1.1). Then, we iden-

tify some Jack polynomials with “unnormalized” Laughlin and Moore-Read wavefunctions,

Sec.3.1.2 and Sec.3.1.3.[4] Then, in Sec.3.2, we obtain the physical many-body states |Ψ〉

from the Jack polynomials Jαλ0 after mapping monomials mλ into normalized non-interacting

states |{nm(λ)}〉 on three different manifolds: plane, sphere and cylinder. In Sec.3.3, we

will use these exact model wavefunctions on the cylinder to calculate the intrinsic dipole

moment.

3.1.1 Definition of Jack polynomials

Given a positive integer N , i.e. the number of particles, let us define a partition λ =

(λ1, λ2, . . . , λN) to be a set of N non-negative integers λi which satisfy λi ≥ λj for any pair

i < j.1 Let us define the multiplicity nm(λ) of a non-negative integer m for the partition λ

to be the number of occurrences of m within the set {λ1, λ2, . . . , λN}. The vector space of

the symmetric polynomials of N independent variables {z1, z2, . . . , zN} may be spanned by

1This differs from the usual definition of partition in mathematics because we allow λi to be zero.

31

the monomials mλ which are defined as

mλ(z1, . . . , zN) :=∏m

1

nm(λ)!

∑τ∈SN

N∏j=1

zλ(j)τ(j) , (3.1)

where SN is the set of all permutations of {1, . . . , N}. The monomial mλ can be interpreted

as a non-interacting wavefunction of N bosons occupying single-particle states labeled by λi,

i = 1, 2, . . . , N . Then, the multiplicity nm(λ) can be interpreted as the “occupation number”

of the single particle state labeled by an integer m.

Now, let us define the Laplace-Beltrami operator which acts on the vector space of the

symmetric polynomials:

HLB(α) :=∑i

(zi∂

∂zi

)2

+1

α

∑i<j

zi + zjzi − zj

(zi∂

∂zi− zj

∂

∂zj

). (3.2)

The important feature about the Laplace-Beltrami operator is that when we apply this

operator on a monomial mλ0 labeled by a partition λ0, then we obtain a superposition

of monomials mλ labeled by partitions λ “squeezed” from λ0. One squeezing operation

corresponds to changing λj → λj − 1 and λk → λk + 1 for a pair (j, k) with j < k. Let

us define an ordering < of partitions in terms of the squeezing. Given two partitions λ and

µ, we denote µ < λ, i.e. µ is dominated by λ, if µ can be obtained from λ after several

squeezing operations. See Fig.3.1 for an example. 2

Now, the Jack polynomials Jαλ0 are symmetric polynomials which are the eigenstates

of the Laplace-Beltrami operator HLB(α). The Jack polynomial Jαλ0 is labeled by a single

partition λ0 which is called the “root partition” because it is expanded by monomials mλ

where λ are partitions squeezed from λ0:

Jαλ0 = mλ0 +∑λ<λ0

aλ0,λ(α)mλ. (3.3)

2This does not give a total ordering but only a partial ordering.

32

Figure 3.1: Squeezing of a partition λ0 = (6, 0, 0, . . . ). This image is taken from [64]

As an example, the 3-particle 1/2 Laughlin state is a Jack polynomial labeled by a partition

λ0 = (4, 2, 0). In terms of the occupation numbers nm(λ0), we have n4 = n2 = n0 = 1

while nm = 0 for m 6= 0, 2, 4. The 3-particle 1/2 Laughlin state is a superposition of

monomials labeled by partitions λ squeezed from λ0. These squeezed partitions include λ =

(4,1,1), (3,3,0), (3,2,1), (2,2,2). If we translate these into occupation numbers, we have

{n0n1n2n3n4} = {02001}, {10020} {01110}, {00300}, respectively.

Since Jαλ0 is, by definition, an eigenstate of HLB(α) and the application of HLB(α) on a

monomial mλ produces mµ with µ ≤ λ, we can derive a recursion relation which gives the

coefficient aλ0,µ(α) from the coefficients aλ0,λ(α) with λ > µ.

Finally, the (unnormalized) N -particle bosonic FQH model wavefunction at filling ν =

p/q is the Jack polynomial

Jα(p,q)

λ0(p,q)(z1, z2, . . . zN),

with a negative constant α,

αp,q = −p+ 1

q − 1, (3.4)

33

where p, q ∈ N, and (p+1) and (q−1) are coprime.[4] The root partition λ0(p, q) labeling the

Jack polynomial Jα(p,q)

λ0(p,q) is required to be “(p, q,N)-admissible.” Given p and q, a partition

λ is (p, q,N)-admissible if it satisfies the following two conditions. First, the occupation

numbers nm(λ) obey the “generalized Pauli exclusion”: there are no more than p particles

in q consecutive orbitals, or equivalently

q∑j=1

nm+j−1(λ) ≤ p, ∀m ≥ 0. (3.5)

Secondly, the partition λ minimizes |λ| which is defined as

|λ| :=∑m

mnm(λ). (3.6)

A Jack polynomial for a fermionic model FQH wavefunction at the filling ν = p/(p+ q)

is obtained from the symmetric Jack Jαλ0 by multiplying a Vandermonde factor.[5] For p = 1,

the Jack polynomial corresponds to Laughlin wavefunctions, and for p = 2, it corresponds

to Moore-Read wavefunctions.

For each Jack polynomial Jαλ0, there is a recursion relation for the rational expansion

coefficients aλ0,λ(α) with aλ0,λ0(α) = 1.[5] This recursion relation allows us to generate model

quantum Hall states with a large number of particles. This is the advantage of knowing that

the model wavefunctions are Jack polynomials. For the Thesis, we used Jacks with 14 and

15 particles for ν = 1/2 bosonic Laughlin state and ν = 1/3 fermionic Laughlin state.[37]

For ν = 1/4 bosonic Laughlin state, we used a Jack with 11 particles. We used Jacks with 18

and 20 particles for ν = 2/2 bosonic Moore-Read state and ν = 2/4 fermionic Moore-Read

state. The MR states we used are in topologically trivial sectors: the MR 2/2 state has the

root occupation numbers 2020...202 and the MR 2/4 state has the root occupation numbers

11001100...110011. [40]

34

3.1.2 Laughlin wavefunction as a Jack polynomial

In this section we want to show that the bosonic Laughlin wavefunction Ψ(q)L at filling ν = 1/q,

with q even, is a Jack polynomial.[4]

Ψ(q)L =

N∏i<j

(zi − zj)q = Jα1,r

λ0(1,q), α1,q = − 2

q − 1. (3.7)

To see that the Laughlin state is indeed a Jack polynomial, first note that the polynomial

contains a monomial mλ0 where

mλ0 = zq(N−1)1 z

q(N−2)2 · · · z0

N + (permutations of particle indices)

= zλ0

11 z

λ02

2 · · · zλ0NN + (· · · ).

We see that the occupation numbers nm(λ0) for this monomial are

{nm(λ0)} = {1q−1︷︸︸︷

0 0 · · · 0 1

q−1︷︸︸︷0 0 · · · 0 1 0 0 · · · }

Thus, the occupation numbers satisfy the generalized Pauli exclusion with one particle in

q consecutive orbitals. We cannot squeeze further while satisfying the exclusion. All other

monomials in the bosonic Moore-Read wavefunction are labeled by partitions squeezed from

λ0. Thus, λ0 is the (1, q, N)-admissible root partition.

To show that the Laughlin wavefunction is an eigenstate of the Laplace-Beltrami operator

HLB(α1,q), we note that it is annihilated by DL,qi

DL,qi ψ

(q)L = 0, (3.8)

35

where the differential operator DL,qi is defined as

DL,qi =

∂

∂zi− q

∑j 6=i

1

zi − zj. (3.9)

Then, it is also annihilated by

HLB(α1,q)− E0 =∑i

ziDL,−1i ziD

L,qi (3.10)

E0 =1

12qN(N − 1)[N + 1 + 3q(N − 1)].

Therefore, Ψ(q)L is an eigenstate ofHLB(α1,q). Thus, Ψ

(q)L is indeed the Jack polynomial J

α1,q

λ0(1,q)

for q even.

Note that as one particle at z1 approaches another particle at z2 = z1+δ, the wavefunction

Ψ(q)L vanishes as δq. We say that it satisfies “(1, q) clustering property.”

Note that the Laughlin state may be written as a correlator of vertex operators V (z)

constructed out of chiral boson fields ϕ(z).

3.1.3 Moore-Read wavefunction as a Jack polynomial

In this section, we want to show that the N -particle bosonic Moore-Read wavefunction ΨBMR

at filling ν = 2/2 is a Jack polynomial,[4]

ΨBMR = Pf

(1

zi − zj

)Ψ

(1)L = J

α2,2

λ0(2,2), α2,2 = −3 (3.11)

The first factor which is called a Pfaffian, and the second one is the Vandermonde factor

Ψ(1)L . A Pfaffian of an anti-symmetric matrix Aij is the square-root of its determinant,

Pf(A) = (detA)1/2. detA vanishes if A is odd dimensional. In our case, Aij = (zi − zj)−1.

Therefore, the Moore-Read wavefunction defined above does not vanish only for even number

of particles. The Paffian factor is anti-symmetric under the exchange of particles: zi ↔ zj.

36

Therefore, ΨBMR is a bosonic wavefunction as expected. In order to obtain a fermionic

Moore-Read wavefunction ΨFMR at ν = 2/4, we multiply by a Vandermonde factor Ψ

(1)L :

ΨFMR = ΨB

MRΨ(1)L .

The bosonic Moore-Read wavefunction ΨBMR has the clustering property such that when

positions z1 and z2 of two particles coincide at Z = z1 = z2, the wavefunction does not

vanish. Meanwhile, when a third particle is brought to zk+1 = Z + δ where k = 2, then

the wavefunction vanishes as δq where q = 2. Thus, we say that the bosonic Moore-Read

wavefunction satisfies the “(2, 2) clustering property.”

To see that the bosonic Moore-Read state ΨBMR is indeed a Jack polynomial, we need

to show that it is an eigenstate of the Laplace-Beltrami operator HLB(α) for some constant

α. The differential equation derives from the condition that the Pfaffian is a correlator of

Majorana fields ψ(z) in the minimal model M(4, 3).[3, 15] We elaborate this point.

The minimal model M(4, 3) contains three primary fields,

I(z) := φ(1,1), hI := h1,1 = 0 (3.12a)

ψ(z) := φ(2,1), hψ := h2,1 =1

2(3.12b)

σ(z) := φ(1,2), hσ := h1,2 =1

16. (3.12c)

The Pfaffian is written as a correlator of ψ(z) fields with both asymtotic states given as |I〉,

Pf

(1

zi − zj

)= 〈ψ(z1)ψ(z2) . . . ψ(zN)〉CFT. (3.13)

Being a minimal model M(4, 3), for each of the three primary fields, there are null fields,

i.e. those that behave like the primary fields with vanishing norm. The identity field I(z)

has a null field at level 1 × 1 = 1, the Majorana fermion field ψ(z) has a null field at level

2× 1 = 2, and the spin field σ(z) has a null field at level 1× 2 = 2. This can be seen from

the Kac determinant formula.

37

Let us consider the null field χ(z) which is a level-2 descendant of ψ(z).

χ(z) = (L−2ψ)(z)− 3

2(2hψ + 1)(L2−1ψ)(z), (3.14)

where Ln are the Virasoro generators. For M(4, 3), the central charge is 1/2, and thus the

Virasoro algebra is,

[Ln, Lm] = (n−m)Ln+m +1/2

12n(n2 − 1)δn+m,0. (3.15)

Being a null field, when the field χ(z) is inserted into a correlator, then the correlator

vanishes.

〈χ(z1)ψ(z2) . . . ψ(zN)〉CFT = 0.

When inside a correlation function, the Virasoro generators Ln can be written as differential

operators Ln,

L−n → L−n =N∑i=2

{(n− 1)hψ(zi − z1)n

− 1

(zi − z1)n−1∂zi

}. (3.16)

Thus, the constraint can be re-written (up to an overall constant factor) into a second-order

differential equation,

DMR1 :=

N∑i=2

{∂2z1− Aψz1 − zi

∂zi −Bψ

(z1 − zi)2

}DMR

1 〈ψ(z1)ψ(z2) . . . ψ(zN)〉CFT = 0. (3.17)

where we defined the two constants as follows,

Aψ =2

3(2hψ + 3), Bψ = hψAψ. (3.18)

38

Now, generalizing the particle index on the differential operator, DMR1 → DMR

i , we define a

differential operator DMR ,

DMRi :=

∑j 6=i

{∂2zi− Aψzi − zj

∂zj −Bψ

(zi − zj)2

}

DMR :=N∑i=1

DMRi . (3.19)

Now, by direct calculation, we can show that DMR becomes the Laplace-Beltrami operator

H(α2,2) after a transformation as following,

DMR → Ψ(1)L DMR(Ψ

(1)L )−1 = H(α2,2) + const. (3.20)

The details of this calculation can be found in Estienne et al.[11] This shows that ΨBMR is an

eigenstate of the Laplace-Beltrami operator,

H(α2,2)ΨBMR = EΨB

MR. (3.21)

The root occupation numbers of the bosonic Moore-Read wavefunction is[4]

{nm(λ0)} = {20202020 · · · }

i.e. λ0 is (2, 2, N)-admissible. Thus, we see ΨBMR is indeed the Jack polynomial J

α2,2

λ0(2,2).

3.2 Mapping Jack polynomials into physical states

A Jack polynomial with variables {z1, z2, . . . , zN} knows only about the “clustering

property,”[4] and it becomes physical only after we map monomials mλ spanning the Jack

into states in a Landau Level depending on the geometry where the Hall fluid is placed on,

such as a cylinder, sphere or plane. We map each zmj for m ∈ Z+ in the monomial into a

39

single particle wavefunction :

zmj → w(m)|m〉, (3.22)

where |m〉 is a geometry-dependent normalized single particle wavefunction with quantum

number m in the lowest Landau Level (we may work within other Landau level) and w(m) is

the inverse of the geometry-dependent normalization factor. Then, the monomial mλ maps

to a physical N -particle wavefunction |Ψλ〉:

mλ → |Ψλ〉 =

∏Nj=1w(λj)∏m

√nm(λ)!

|{nm(λ)}〉, (3.23)

where {nm(λ)} is the set of occupation numbers corresponding to the partition λ, and

|{nm(λ)}〉 is the normalized many-particle state with nm particles in the orbital with quan-

tum number m. Finally, the Jack polynomial maps to a physical model quantum Hall state

|Ψαλ0〉 (without overall normalization):

Jαλ0→ |Ψα

λ0〉 =

∑λ≤λ0

aλ0,λ(α)|Ψλ〉. (3.24)

3.2.1 On plane

Here, we want to derive w(m) that appears in (3.23) when we map the monomials into

non-interacting many-particle states on the plane.

First, consider the single-particle states labeled by non-negative integers m in the lowest

Landau level n = 0,

〈r||n = 0〉L ⊗ |m〉G = (2π2mm!)−1/2zme−zz/4`2B , (3.25)

where the subscript L denotes the “Landau-orbit” degree of freedom, and G denotes the

“guiding-center” degree of freedom. This is the notation introduced previously in (2.10).

40

Now, suppose there are N particles in the lowest Landau level with complex coordinates

{z1, z2, . . . , zN}. The monomial mλ with N coordinates {zi} was defined as

mλ(z1, . . . , zN) :=∏m

1

nm(λ)!

∑τ∈SN

N∏i=1

zλiτ(i).

First, we map each zmi for m ∈ Z+ into a single-particle state as

zmi 7→ (2π2mm!)1/2|n = 0〉L ⊗ |m〉G. (3.26)

Then, the monomial mλ maps to a non-interacting many-particle state as

mλ 7→(∏

m

√N !

nm(λ)!

)[N∏i=1

w(λi)

]|{nm(λ)}〉, (3.27)

where

|{nm(λ)}〉 :=

(N∏i=1

|ni = 0〉L)⊗ |{nm(λ)}〉G (3.28)

is the normalized many-particle state occupying the single-particle states with the quantum

numbers m = λ1, λ2, . . . , λN , and the geometry dependent factor w(m) is defined as

w(m) = (2π2mm!)1/2 for m = 0, 1, 2, . . . (3.29)

Now, this mapping from a monomial to a non-interacting many-particle state can be easily

generalized to higher Landau levels.

3.2.2 On sphere

Here, we want to derive w(m) that appears in (3.23) when we map the monomials into

non-interacting many-particle states on the sphere.

On the sphere, the single-particle states carry quantum numbers which are eigenvalues

of the rotation generators L2 and Lz. If the monopole strength, i.e. the total number of

41

flux through the surface of the sphere is NΦ = 2S, then the possible eigenvalues of L2 are

S(S + 1), (S + 1)(S + 2), . . . . The number S can be either an integer or a half-integer so

that 2S + 1 is an integer. The single-particle states in the lowest Landau level (i.e. having

the eigenvalue L2 = S(S + 1)) on the sphere are the monopole harmonics[65, 66, 18]

YSSm(u, v) :=

[2S + 1

4π

(2S

S −m

)]1/2

(−1)S−m(uv

)m(uv)S

=〈(u, v)|S,m〉, (3.30)

where the spinor variables (u, v) satisfy |u|2 + |v|2 = 1.

The azimuthal angular momentum quantum number m can be

m = S, S − 1, . . . ,−S, (3.31)

and so there are 2S+1 monopole harmonics (single-particle states) in the lowest Landau level.

3 Now, we describe the mapping between the monomial mλ and the physical normalized

many-particle state on the sphere. First, we map each zm′

i for m′ = S−m ∈ {0, 1, 2, . . . , 2S}

into a single-particle state

zm′

i 7→[

2S + 1

4π

(2S

m′

)]−1/2

(−1)m′ |S, S −m′〉. (3.32)

Then, the monomial mλ maps to a non-interacting many-particle state as

mλ 7→(

2S∏m′=0

√N !

nm′(λ)!

)[N∏i=1

w(λi)

]|{nm′(λ)}〉, (3.33)

where |{nm′(λ)}〉 is the normalized many-particle state occupying the single-particle states

with the quantum numbers {m = S − λi : i = 1, 2, . . . , N}, and the geometry dependent

3There is one more single-particle state than the total number of flux quanta NΦ. This is widely calledas the “shift.” This additional single-particle state originates from the coupling of the Landau-orbit spins = 1/2 with the Gaussian curvature of the sphere.

42

factor w(m′) is defined as

w(m′) =

[2S + 1

4π

(2S

m′

)]−1/2

(−1)m′

for m′ = 0, 1, 2, . . . , 2S. (3.34)

Now, this mapping from a monomial to a non-interacting many-particle state can be easily

generalized to higher Landau levels.

3.2.3 On cylinder

We want to map Jack polynomials into many-body FQH wavefunctions on the cylinder. This

is the geometry which we used to calculate physical quantities such as the intrinsic dipole

moment in Sec.3.3.

We can consider cylinders periodic with circumferences of different lengths L along the

edge direction x and infinite in the direction y, i.e. we use the Landau gauge. In the

lowest Landau level (n = 0), the normalized single-particle states φk(r) are labeled by the

momentum ~k along x direction:

〈r||n = 0〉L ⊗ |m〉R = φk(r) =e−(k`B)2/2

(π)1/4(`BL)1/2zke−(y/`B)2/2, z = ei(x−iy), (3.35)

where we wrote write the wave-vector as k = 2πm/L. If the underlying constituent particles

of a quantum Hall state are bosons, then the allowed values of m are {m ∈ Z} and if they

are fermions, {m ∈ Z + 1/2}. This choice of assigning the momentum quantum numbers

allows the many-particle wavefunctions to carry zero total momentum.

Now, we would like to map Jack polynomials into physical states on the cylinder. How-

ever, the mapping (3.22) is not uniquely defined: consider the bosonic case. For m′ ∈ Z+,

if the term zm′

i in a monomial is mapped to the single-particle state with definite mo-

mentum k = (2π/L)m′, then another mapping that maps zm′

to a state with momentum

k = (2π/L)(m′ +M) is also possible for any fixed integer M . This arbitrariness is removed

when we choose one of the Fermi momenta to be k = 0, and the first occupied state to have

43

a momentum k = (2π/L)m0. With this mapping, each zm′

i for m′ ∈ Z+ maps to

zm′

i 7→ (π)1/4(`BL)1/2 exp

{1

2

[2π`B(m′ +m0)

L

]2}|n = 0〉L ⊗ |m′ +m0〉R, (3.36)

i.e. m′ = m−m0. Then, the monomial mλ maps to a non-interacting many-particle state as

mλ 7→(∞∏

m′=0

√N !

nm′(λ)!

)[N∏i=1

w(λi)

]|{nm′(λ)}〉, (3.37)

where the geometry-dependent factor w(m′) is

w(m′) = (π)1/4(`BL)1/2 exp

{1

2

[2π`B(m′ +m0)

L

]2}, (3.38)

and

|{nm′(λ)}〉 :=

(N∏i=1

|ni = 0〉L)⊗ |{nm′(λ)}〉G, (3.39)

is the many-particle state occupying the single-particle states with quantum numbers {m =

λi +m0 : i = 1, 2, . . . , N}. This can easily be generalized to higher Landau levels.

We want to see how m0 is determined. For instance, consider a Laughlin ν = 1/q state,

the number m0 is fixed by its chiral boson edge theory : its first non-zero occupation occurs

at the momentum k = πq/L. This can be seen by Fourier-transforming the electron Green’s

function in the chiral boson theory [61]

G(x− y) ∝ (sin (π(x− y − iη)/L))−q , η → 0+.

From this, we can obtain the expectation value of the occupation number operator of the

Laughlin state

〈nm〉 ∝(m+ q/2− 1)!

(q − 1)!(m− q/2)!.

44

Figure 3.2: N = 2 and N = 4 root occupations for (a) ν = 1/2 and (b) ν = 1/3 Laughlinstates with two different circumferences L and 2L. The two bold arrows denote two Fermimomenta. For ν = 1/2, there are 4 = 2 · 2 and 8 = 2 · 4 states respectively between the twoFermi momenta. For ν = 1/3, there are 6 = 3 · 2 and 12 = 3 · 4 states respectively.

For ν = 1/2, the occupation behaves as 〈nm〉 ∝ m. The first occupied state corresponds

to m = 1. Thus, a factor z0i in a monomial mλ should be mapped to the single-particle

state with k = (2π/L)m0 = 2π/L. For ν = 1/3, 〈nm〉 ∝ (m + 1/2)(m − 1/2). The first

occupied state corresponds to m = 3/2, so we should map z0i to the single-particle state with

k = (2π/L)m0 = (2π/L)(3/2). For ν = 1/4, 〈nm〉 ∝ (m + 1)m(m− 1). In general, we have

m0 = q/2 for ν = 1/q Laughlin state.

This implies that when a N = pN particle quantum Hall state with the filling factor

ν = p/q is put on a cylinder, then between the two Fermi momenta there are qN orbitals.

For example, for ν = 1/2 and ν = 1/3 Laughlin states, we have the following (Fig.3.2) root

momentum occupations (i.e. the occupations of the state corresponding to the monomial

mλ0) for two circumferences L and 2L.

In order to have the two edges not interact with each other, we need to take a limit

N → ∞ first, and then take L → ∞. In practice, we can have only finite N , and this

restricts the largest available L for the fixed N . If L increased further than this value, then

the Jack polynomial becomes a wavefunction of Calogero-Sutherland model with its two

45

edges interacting strongly [54]. If L is too small, then the Jack becomes a charge-density-

wave state. For fixed L, the occupation numbers converge to some limits as the number of

particles N increases.

46

3.3 Numerical results from Jack polynomials

3.3.1 Occupation number

At zero temperature, N non-interacting electrons in an IQH fluid fill up the states from

m = 1/2 to m = N/2 with 〈nm〉 = 1. In the case of FQH fluid with filling factor ν, not only

the range of momenta of occupied states changes so as to satisfy 〈nm〉 ≈ ν but also 〈nm〉

deviates from ν appreciably near the Fermi momenta. This variation of 〈nm〉 gives rise to

the intrinsic dipole moment. We analyze the occupation numbers.

Given a ground state wavefunction |Ψαλ0〉 that we obtain from a Jack polynomial Jαλ0

by a

mapping described in the preceding section, we can calculate the occupation number (i.e. the

expectation value of the occupation number operator nm) for each momentum k = 2πm/L.

We evaluate

〈nm〉0 ≡〈Ψα

λ0|nm|Ψα

λ0〉

〈Ψαλ0|Ψα

λ0〉 =

∑λ≤λ0

aλ0,λ(α)2〈Ψλ|nm|Ψλ〉∑λ≤λ0

aλ0,λ(α)2〈Ψλ|Ψλ〉,

where λ ≤ λ0 means that the sum is over all partitions that are obtainable by squeezing

from the root partition λ0. Note that 〈n0〉0 = 0 for ν = 1/2 Laughlin state, 〈n1/2〉0 = 0 for

ν = 1/3 Laughlin state, and so on.

The occupation numbers are calculated for the model wavefunctions and are plotted in

Fig.3.3, 3.4, 3.5, 3.6 and 3.7. The first two plots which are occupations of Laughlin 1/2 and

1/3 states have total numbers of particles N = 14 and 15. The next plot is that of Laughlin

1/4 state with a total number of particles N = 11. The last two plots are occupations of

Moore-Read 2/2 and 2/4 states have total numbers of particles N = 18 and 20. These plots

show only half of the occupation profile because the other half can be obtained by mirror

symmetry. The occupation numbers are plotted as a function of the momentum k rather

than the quantum number m,

n(k) = 〈nm〉0, k = 2πLm.

47

The figures contain data for several values of L on the same plot.

Each occupation plot seems to follow a smooth profile that might appear in the limit

L→∞. This observation allows us to observe how well these wavefunctions of finite numbers

of particles agree with the behavior of n(k) near k = 0 described by the chiral boson theory.

In each occupation profile plot, we calculate the linear fit of log n(k) versus log k with those

momenta k = (2π/L)m0 , the first non-vanishing occupation numbers. We observe that

they are quite linear, and their linear fit coefficient is the exponent r in n(k) ∝ kr as k → 0.

For each Laughlin 1/2, 1/3, 1/4 state, the exponent is calculated to be 0.963, 1.853, 2.722

respectively, while the expected exponents are 1, 2 and 3. For each Moore-Read 2/2 and 2/4

state, the exponent is calculated to be 1.076 and 1.879 while the expected exponents are 1

and 2.

48

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

0.5

k = 2πmL

n(k)

1/2 Laughlin state occupation numbers

N = 14N = 15

Figure 3.3: ν = 1/2 Laughlin state density profile : Red dots for N = 14 and blue dots forN = 15. It plots data obtained with different L = 15 to 24 with increments by 0.5 (in unitsof `B). The data points for N = 14 are shifted up by 1/10. The horizontal lines are 1/2 and1/2+1/10. The linear fit of log(k = 2π/L) versus log n(k) gives log n(k) = 0.963 log k−0.010with the norm of residues 0.003

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

0.33

k = 2πmL

n(k)


N = 14N = 15

Figure 3.4: ν = 1/3 Laughlin state density profile : Red dots for N = 14 and blue dots forN = 15. L = 12.5 to 22 with increments by 0.5. The data points for N = 14 are shiftedup by 1/10. The horizontal lines are 1/3 + 1/10. The linear fit of log(k = 3π/L) versuslog n(k) gives log n(k) = 1.853 log k − 0.609 with the norm of residues 0.017.

49

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

0.25

k = 2πmL

n(k)


N = 11

Figure 3.5: ν = 1/4 Laughlin state density profile : Blue dots for N = 11. L = 12.5 to 22with increments by 0.5. The horizontal line is 1/4. The linear fit of log(k = 4π/L) versuslog n(k) gives log n(k) = 2.722 log k − 0.530 with the norm of residues 0.028

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

k = 2πmL

n(k)

2/2 Moore-Read occupation numbers

N = 18N = 20

Figure 3.6: ν = 2/2 Moore-Read state density profile: Red dots for N = 18 and blue dotsfor N = 20. L = 13 to 20 with increments by 0.5. The data points for N = 18 are shiftedup by 1/10. The horizontal lines are 1 and 1+1/10. The linear fit of log(k = 2π/L) versuslog n(k) gives log n(k) = 1.076 log k + 0.598 with the norm of residues 0.002.

50

0 1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

0.5

k = 2πmL

n(k)

2/4 Moore-Read occupation numbers : 01100110...

N = 18N = 20

Figure 3.7: ν = 2/4 Moore-Read state density profile: Red dots for N = 18 and blue dotsfor N = 20. L = 16 to 19.5 with increments by 0.5. The data points for N = 18 are shiftedup by 1/10. The horizontal lines are 1/2 and 1/2+1/10. The linear fit of log(k = 3π/L)versus log n(k) gives log n(k) = 1.879 log k − 0.054 with the norm of residues 0.002.

0 1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

0.5

k = 2πmL

n(k)

2/4 Moore-Read occupation numbers : 0101010...

N = 18N = 19

Figure 3.8: ν = 2/4 Moore-Read state density profile with a quasi-hole at the Fermi surface:Red dots for N = 18 and blue dots for N = 19. L = 16 to 19.5 with increments by 0.5. Thedata points for N = 18 are shifted up by 1/10. The horizontal lines are 1/2 and 1/2+1/10.

51

3.3.2 Luttinger sum rule

Given the occupation numbers, we can also verify the that they satisfy the Luttinger sum

rule[41]. For a one dimensional system with “Fermi surface” singularities in the occupation

numbers n(k) at “Fermi points” ki, this states that

N

L=

∫dk

2πn(k) =

∫dk

2πn0(k), (3.40)

where in a Luttinger liquid (the 1D analog of a Fermi liquid), n0(k) is a integer topological

index that is constant in regions ki < k < ki+1 and counts the number of occupied bands

below the Fermi level with momentum or Bloch index k. (From a “modern” viewpoint, the

Luttinger theorem is an early example of the identification of a topological index n0(k) that

remains invariant as the actual n(k) is continuously modified by the interactions in the Fermi

liquid that conserve the existence of the singularity at the Fermi surface.) In the fractional

quantum Hall effect in the L → ∞ limit of the cylinder geometry, this generalizes to n0(k)

= ν(k), the filling factor in the region ki`2B < y < ki+1`

2B.

The applicability of the Luttinger theorem to the fractional quantum Hall fluid[19, 59]

is immediately visible in the Jack polynomial description: the “root” configuration of, e.g.,

the ν = 13

Laughlin state is . . . 000|010010010 . . . 010010010|000 . . . with a mean occupation

of ν = 13

between the Fermi points marked as “|”. This is a uniform filling ν in the

thermodynamic limit. The “squeezing” of pairs of “1”’s together in the full Jack configu-

ration preserves this mean filling in the interior of strips much wider than `B, creating the

dipoles near the Fermi points, and preserving the Luttinger sum rule. The Luttinger sum

rule is the integral form of the differential relation dN = (L/2π)∑

i ∆νidki, where ∆νi =

ν(k = k+i )− ν(k = k−i ) is the chiral anomaly of the Fermi point[19].

52

We define the function ∆N(k) which is the integration of the difference between the

actual occupation number and the uniform occupation number from 0 to k is

∆N(k)

L=

∫ k

0

dk′

2π(n(k′)− ν).

For finite L, the integration is approximated by the sum,

∆N (k) =m−1∑

m′=0 or 1/2

〈nm′〉0 + 12〈nm〉0 − νm, (3.41)

where the summation is over integers m′ = 0, 1, 2, . . . ,m−1 for a bosonic state, and it is over

half-integers m′ = 12, 3

2, 5

2, . . . ,m − 1 for a fermionic state. If the Luttinger’s theorem holds

this should vanish as k gets larger. Because we are limited by the finite size, we calculate

∆N(k) only up to the center of the fluid. ∆N(k) is plotted against k. Each plot includes

data from a range of circumferences L. See Fig . 3.9, 3.10, 3.11, 3.12 and 3.13. We observe

Luttinger’s theorem indeed holds in presence of interactions among particles.

53

0 1 2 3 4 5 6−0.2

−0.1

0

0.1

0.2

0.3

k = 2πmL

∆N

(k)

1/2 Laughlin state Luttinger sum

N = 14N = 15

Figure 3.9: ∆N(k) for ν = 1/2 Laughlin state : Red dots for N = 14 and blue dots forN = 15. It plots data obtained with L = 15 to 24 with increments by 0.5. The data pointsfor N = 14 are shifted up by 1/10.

0 2 4 6 8 10−0.2

−0.1

0

0.1

0.2

0.3

k = 2πmL

∆N

(k)


N = 14N = 15

Figure 3.10: ∆N(k) for ν = 1/3 Laughlin state : Red dots for N = 14 and blue dots forN = 15. L = 12.5 to 22 with increments by 0.5. The data points for N = 14 are shifted upby 1/10.

54

0 2 4 6 8 10−0.2

−0.15

−0.1

−5 · 10−2

0

5 · 10−2

0.1

0.15

0.2

k = 2πmL

∆N

(k)


N = 11

Figure 3.11: ∆N(k) for ν = 1/4 Laughlin state: Blue dots for N = 11. L = 12.5 to 22 withincrements by 0.5.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

k = 2πmL

∆N

(k)

2/2 Moore-Read Luttinger sum

N = 18N = 20

Figure 3.12: ∆N(k) for ν = 2/2 Moore-Read state : Red dots for N = 18 and blue dots forN = 20. L = 13 to 20 with increments by 0.5. The data points for N = 18 are shifted upby 1/10.

55

0 1 2 3 4 5 6 7 8−0.2

−0.1

0

0.1

0.2

0.3

k = 2πmL

∆N

(k)

2/4 Moore-Read Luttinger sum : 01100110...

N = 18N = 20

Figure 3.13: ∆N(k) for ν = 2/4 Moore-Read state : Red dots for N = 18 and blue dots forN = 20. L = 16 to 19.5 with increments by 0.5. The data points for N = 18 are shifted upby 1/10.

0 1 2 3 4 5 6 7 8−0.2

−0.1

0

0.1

0.2

0.3

k = 2πmL

∆N

(k)

2/4 Moore-Read Luttinger sum : 0101010...

N = 18N = 19

Figure 3.14: ∆N(k) for ν = 2/4 Moore-Read state with a qausi-hole at the Fermi surface :Red dots for N = 18 and blue dots for N = 19. L = 16 to 19.5 with increments by 0.5. Thedata points for N = 18 are shifted up by 1/10.

56

3.3.3 Intrinsic dipole moment

Here, we calculate the intrinsic dipole moment of FQH states due to variations in occupation

numbers near the edge. The boundary is along the direction x, and there exists the intrinsic

dipole moment py proportional to L. We define a function py(k) which is the intrinsic dipole

moment integrated from the boundary y = 0 to y = k`2B,

py(k)

L= −e

∫ k

0

dk′

2πk′`2

B(n(k′)− ν).

For finite L, the integration is approximated by the sum,

py(k)

L= −2π`2

Be

L2× m−1∑

m′=0 or 1/2

m′〈nm′〉0 + 12m〈nm〉0 − 1

2νm2

(3.42)

where the summation is over integers if the state is bosonic or half-integers if fermionic.

The last term in the bracket subtracts the contribution from the uniform density. Because

a quantum Hall fluid is uniform within its bulk, we expect the dipole moment to converge

to a value as k gets large. We also multiply py(k)/L by (−e/4π)−1 so that it becomes a

dimensionless quantity which is predicted to be −s/q, the guiding-center spin divided the

number of flux quanta attached to each composite boson. See Fig.3.15, 3.16, 3.17, 3.18 and

3.19. We observe that all intrinsic dipole moments approach expected values as we integrate

up to the center of the fluids. The expected values are listed in Table.2.1. Thus, we confirm

the relationship (2.44) between the guiding-center spin and the intrinsic dipole moment holds

not only for a droplet but also for the straight edge.

57

0 1 2 3 4 5 6

0

0.1

0.2

0.3

0.4

0.5

0.25

k = 2πmL

py(k)/L

1/2 Laughlin state dipole moment per unit length

N = 14N = 15

Figure 3.15: p(k)/L in units of −e/4π for ν = 1/2 Laughlin state. Calculated from Fig.3.3.Red dots for N = 14 and blue dots for N = 15. It plots data obtained with L = 15 to 24with increments by 0.5. The data points for N = 14 are shifted up by 1/10. The horizontallines are 1/4 and 1/4+1/10.

0 2 4 6 8 10

0

0.2

0.4

0.6

0.33

k = 2πmL

py(k)/L


N = 14N = 15

Figure 3.16: p(k)/L in units of −e/4π for ν = 1/3 Laughlin state. Calculated from Fig.3.4.Red dots for N = 14 and blue dots for N = 15. L = 12.5 to 22 with increments by 0.5. Thedata points for N = 14 are shifted up by 1/10. The horizontal lines are 1/3 and 1/3+1/10.58

0 2 4 6 8 10

−0.2

0

0.2

0.4

0.6

0.8

k = 2πmL

py(k)/L


N = 11

Figure 3.17: p(k)/L in units of −e/4π for ν = 1/4 Laughlin state. Calculated from Fig.3.5.Blue dots for N = 11. L = 12.5 to 22 with increments by 0.5. The horizontal line is 3/8.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5−0.2

0

0.2

0.4

0.6

0.8

0.5

k = 2πmL

py(k)/L

2/2 Moore-Read dipole moment per unit length

N = 18N = 20

Figure 3.18: p(k)/L in units of −e/4π for ν = 2/2 Moore-Read state. Calculated fromFig.3.6. Red dots for N = 18 and blue dots for N = 20. L = 13 to 20 with increments by0.5. The data points for N = 18 are shifted up by 1/10. The horizontal lines are 1/2 and1/2+1/10.

59

0 1 2 3 4 5 6 7 8

−0.2

0

0.2

0.4

0.6

0.8

1

0.5

k = 2πmL

py(k)/L

2/4 Moore-Read dipole moment per unit length : 01100110...

N = 18N = 20

Figure 3.19: p(k)/L in units of −e/4π for ν = 2/4 Moore-Read state. Calculated fromFig.3.7. Red dots for N = 18 and blue dots for N = 20. L = 16 to 19.5 with increments by0.5. The data points for N = 18 are shifted up by 1/10. The horizontal lines are 1/2 and1/2+1/10.

0 1 2 3 4 5 6 7 8−0.2

0

0.2

0.4

0.6

0.8

1

0.5

k = 2πmL

py(k)/L

2/4 Moore-Read dipole moment per unit length : 0101010...

N = 18N = 19

Figure 3.20: p(k)/L in units of −e/4π for ν = 2/4 Moore-Read state with a quasi-hole atthe Fermi surface. Calculated from Fig.3.8. Red dots for N = 18 and blue dots for N = 19.L = 16 to 19.5 with increments by 0.5. The data points for N = 18 are shifted up by 1/10.The horizontal lines are 1/2 and 1/2+1/10. 60

Chapter 4

Entanglement spectrum

4.1 Orbital entanglement spectrum (OES)

In the last section, we calculated the intrinsic dipole moments using the exact model wave-

functions. The intent of this section is to show that the intrinsic dipole moment is nothing

other than the O(L2) part of the momentum polarization that is calculated from the orbital

entanglement spectrum.

We first review the definition of the orbital entanglement spectrum(OES)[40] in Sec.4.1.1.

Then, we describe how to calculate the total net momentum quantum number(“momentum

polarization”) for a subsystem on a cylinder. Before presenting the results of the momentum

polarization calculation, we note some general properties of OES of the model wavefunctions.

First, the chirality of their OES can be explained by the fact the model wavefunctions derive

from Jack polynomials. Secondly, we derive the minimum change in the momentum quantum

number as a function of the change in the particle number within a subsystem from the

manipulation of root occupation numbers.

In Sec.4.1.2, we relate the momentum polarization with the intrinsic dipole moment, and

we present the actual data. We show that the momentum polarization can be decomposed

into three distinct parts: a O(L2) part which is the intrinsic dipole moment and two O(L0)

61

parts which are topological terms. One of the two topological terms known as “topological

spin” is calculated solely from the root occupation numbers. The other topological term

γ = c− ν is identified as a purely FQHE quantity which vanishes for IQHE.

In Sec.4.2, we show that the momentum polarization calculated from the RES merely

adds a trivial Landau-orbit contribution to the one calculated from the OES.

4.1.1 OES and Momentum polarization

We review the orbital entanglement spectrum. OES was first introduced by Li and

Haldane.[40] They noted that the OES of the Moore-Read state displays a gapless spectrum

whose degeneracy matches the corresponding conformal field theory (the tensor product of

the minimal model M(4, 3) and chiral U(1) boson). They also diagonalized the second-

Landau-level-projected Coulomb interaction at half-filling, and they found that the low-lying

orbital entanglement spectrum of the ground state is essentially identical with that of the

Moore-Read state with the rest of the spectrum separated by an “entanglement gap.”

We first discuss the mathematical definition of the entanglement spectrum. Let H be a

Hilbert(or Fock) space. Now, we divideH into two subspacesHL andHR, i.e. H = HL⊗HR.

Then, any state |Ψ〉 in H can be expressed as

|Ψ〉 =∑ij

wij|ψiL〉 ⊗ |ψjR〉, (4.1)

where {|ψiL〉 ∈ HL} and {|ψjR〉 ∈ HR} are some orthonormal bases for the subspaces. Now,

it is possible to choose the orthonormal bases {|ΨiL〉 ∈ HL} and {|Ψj

R〉 ∈ HR} related to the

original bases by unitary transformations such that the matrix wij is diagonalized and all

the eigenvalues are non-negative,

|Ψ〉 =∑r

e−(1/2)ξr |ΨrL〉 ⊗ |Ψr

R〉. (4.2)

62

This change of basis is the Schmidt decomposition. The set of numbers {ξr} are known as the

entanglement spectrum (or the pseudo-energies). If |Ψ〉 is normalized, then the entanglement

spectrum satisfies∑

r e−ξr = 1.

The entanglement spectrum also appears in the density matrix of a subsystem. The

density matrix of the full system is ρ = |Ψ〉〈Ψ|. Then, taking a partial trace over the

subsystem R, we obtain the density matrix

ρL = TrR ρ =∑r

e−ξr |ΨrL〉〈Ψr

L|. (4.3)

The same entanglement spectrum appears if we trace out the other subsystem.

ρR = TrL ρ =∑r

e−ξr |ΨrR〉〈Ψr

R|. (4.4)

We now specialize to the orbital entanglement spectrum of FQHE. In FQHE, the degrees

of freedom associated with the Landau-orbit and those associated with the guiding-center

decouple, and the many-body wavefunction is a direct product of the Landau-orbit part and

the guiding-center part. (See the discussion around (2.10)) In our present case, the spatial

manifold is a cylinder with circumference L,1 and the single-particle states in a single Landau

level are labeled by the guiding-centers ym. The momentum quantum numbers k = 2πm/L

along x direction are related by ym = k`2B. Given Ne electrons, the guiding-center part of

the many-body state |Ψν〉 with a filling factor ν is found by placing the Ne electrons on

NΦ = ν−1Ne single-particle states. We then divide the whole system into two subsystems L

and R depending on whether guiding-center momentum quantum number is either positive or

negative. This is known as the “orbital cut.” The location of the cut (the zero momentum)

does not matter in principle as long as it belongs to a “vacuum sector” if the number of

particles is sufficiently large (We clarified what we mean by vacuum sector in Appendix.A).

1We hope that there is no confusion using the same letter L for the circumference and for the label ofthe subsystem L.

63

However, in order to minimize the finite size effect, we should choose the cut to be located

near the middle of the fluid as much as possible.

For instance, consider the Laughlin 1/3 state |Ψ1/3〉 with N = 4 particles on the cylinder.

It is derived from a Jack polynomial with the root occupation numbers,

{n0m} = 010010010010.

Because it is a Jack, it is a superposition of 16 states |{nm}〉 with occupation numbers {nm}

that are obtained by squeezing the root occupation numbers {n0m}:

|Ψ1/3〉 = w1|010010010010〉

+w2|001100010010〉

+w3|001010100010〉

+w4|000111000010〉...

+w16|000011110000〉. (4.5)

The orbital cut corresponds to dividing the total system into subsystems L and R, and

writing each state as a tensor product as follows,

|Ψ1/3〉 = w1|010010〉L ⊗ |010010〉R (2,−6)

+w2|001100〉L ⊗ |010010〉R (2,−6)

+w3|001010〉L ⊗ |100010〉R (2,−5)

+w4|000111〉L ⊗ |000010〉R (3,−4− 12)

...

+w16|000011〉L ⊗ |110000〉R (2,−2). (4.6)

64

This is a special case of the general expression (4.1). On the right of each state in (4.6),

we wrote the two quantum numbers, (NL,ML), the number of particles in the subsystem L

and the total guiding-center quantum number of the particles of the subsystem L. For the

root state, there are N0L = 2 = N0

R particles in the subsystem L. Assigning zero to the cut,

the total guiding-center momentum quantum number is M0L = −(3

2+ 9

2) = −6 = −M0

R for

the subsystem L. These are called the “natural” values of NL, NR, ML and MR. However,

within the subsystem L, it can contain other non-negative number NL of particles as we can

see in the fourth line of (4.6). Also, other total guiding-center momentum quantum number

ML is possible as long as it satisfies ML +MR = 0 as we can see in the last line of (4.6). The

entanglement spectrum can be obtained by performing the Schmidt decomposition in each

block matrix labeled by NL and ML.

Returning to a general model FQH state, after performing the Schmidt decomposition,

the many-body state may be written as

|Ψ〉 =∑

ML,NL,r′

e−(1/2)ξr′,NL,ML |Ψr′,NL,ML

L 〉 ⊗ |Ψr′,NR,MR

R 〉, (4.7)

where {r′} are the remaining labels of states.

Now, the chirality of the entanglement spectrum manifests itself when we note that the

model FQH state derives from a Jack polynomial so that the many-particle state is spanned

by the states obtainable by squeezing operation. Hence, it is a superposition of states with

ML ≥M0L (thus, MR ≤M0

R),

|Ψ〉 =∑

ML≥M0L,NL,r

′

e−(1/2)ξr′,NL,ML |Ψr′,NL,ML

L 〉 ⊗ |Ψr′,NR,MR

R 〉. (4.8)

Thus, the change in total guiding-center quantum number ∆ML = ML −M0L is always non-

negative for any pseudo-energies ξr (See Fig.4.1, 4.2 and 4.3). We also define ∆NL = NL−N0L.

65

Now, we can calculate the expectation value of ∆ML

〈∆ML〉 =

∑r′,NL,ML

∆MLe−ξr′,NL,ML∑

r′,NL,MLe−ξr′,NL,ML

. (4.9)

We call 〈∆ML〉 the “momentum polarization.”

Before presenting the momentum polarization calculations, we note that the lower bound

of ∆ML is determined by ∆NL. For instance, consider the following root state of the Laughlin

1/3 state,

|010 . . . 010010〉L ⊗ |010010 . . . 010〉R.

The underlined particle in the subsystem R carries the guiding-center momentum quantum

number 3/2. By squeezing the two underlined particles, we can pull this extra particle from

the subsystem R into the subsystem L, and obtain a state with ∆NL = 1 and ∆ML = 3/2,

|010 . . . 000111〉L ⊗ |000010 . . . 010〉R.

By squeezing, we can pull two particles from the subsystem R which carry the guiding-center

momentum quantum numbers 3/2 and 9/2,

|010 . . . 010010〉L ⊗ |010010 . . . 010〉R.

Then, we obtain a state in the subsystem L with ∆NL = 2 and ∆ML = 3/2 + 9/2. When

∆NL = −1, it corresponds to the absence of an electron with the guiding-center momentum

quantum number −3/2, and thus ∆ML = 3/2. (See Fig.4.1.) For 1/q Laughlin ground

states, with these observations, we can express the quantum number ∆ML measured with

respect to the “vacuum” cut as

∆ML =q

2(∆NL)2 +

∞∑m=0

mb†mbm, (4.10)

66

where the boson number operator b†mbm in the second term can take any non-negative integer

values, and it describes additional increments in ∆ML when the squeezing between a particle

in the left subsystem and another in the right subsystem does not cause any further change

of particle numbers in each subsystem. This is exactly the free chiral boson Hamiltonian.[61]

By the same method, we can deduce that for 2/4 Moore-Read ground state, ∆ML takes

a specific form

∆ML =2

2(∆NL)2 +

∞∑m=0

mb†mbm +∞∑

m=1/2

mf †mfm

(−1)∆NL = (−1)∑m f†mfm , (4.11)

where the second term is the chiral boson contribution, and the last term is the chiral

Majorana fermion contribution. The fermion momenta are half-integers, and the fermion

occupation numbers are either 0 or 1. The second line is a constraint on the total number

of Majorana fermions. For example, the minimum change of the total quantum number is

∆ML = 1 + 1/2 because ∆NL = 1 requires at least one Majorana fermion. (See Fig.4.3.)

4.1.2 Decomposition of 〈∆ML〉 and 〈∆NL〉

We now describe the relationship between the momentum polarization 〈∆ML〉 with the

dipole moment py. The total momentum Px of the subsystem L is, by definition, related to

〈∆ML〉

Px =2π~L〈∆ML〉.

Because the guiding-centers are defined as ym = 2πm`2B/L = k`2

B, we can relate the total

momentum Px to the dipole moment py as

py =−e`2

B

~Px =

−2π`2Be

L〈∆ML〉. (4.12)

67

Figure 4.1: OES of 1/3 Laughlin state on cylinder with circumference 16`B at truncationlevel 16 with the vacuum cut. Note that ∆ML ≥ 0. Note that the sectors with ∆NL = ±1have the minimum value of ∆ML = 1.5 and the sectors with ∆NL = ±2 have the minimumvalue of ∆ML = 6. All sectors with different ∆NL have the same degeneracy counting :1,1,2,3,5,7,11,. . .

We now show that this dipole moment contains the one we calculated using occupation

numbers in Sec.3.3. We can re-write the total guiding-center quantum number ML as

ML =∑m∈L

mnm, (4.13)

68

Figure 4.2: OES of 1/5 Laughlin state on cylinder with circumference 16`B at truncationlevel 16 with the vacuum cut. Note that ∆ML ≥ 0. Note that the sectors with ∆NL = ±1have the minimum value of ∆ML = 2.5 and the sectors with ∆NL = ±2 have the minimumvalue of ∆ML = 10. All sectors with different ∆NL have the same degeneracy counting :1,1,2,3,5,7,11,. . .

69

Figure 4.3: OES of 2/4 Moore-Read state on cylinder with circumference 16`B at truncationlevel 12 with the vacuum cut. Note that ∆ML ≥ 0. Note that the sectors with ∆NL = ±1have the minimum value of ∆ML = 1.5 and the sectors with ∆NL = ±2 have the minimumvalue of ∆ML = 4. The degeneracy counting of the even ∆NL sector is 1,1,3,5,10,16,. . . andthe degeneracy counting of the odd ∆NL sector is 1,2,4,7,13,. . .

70

where m’s are the guiding-center quantum numbers that belong to the subsystem L, and

nm’s are the electron occupation number operators. Meanwhile, M0L is just a number that

depends on the root occupation numbers {n0m} of the model FQHE state

M0L =

∑m∈L

mn0m. (4.14)

Then, the expectation value 〈∆ML〉 is written as

〈∆ML〉 = TrL [∆MLρL]

=

(∑m∈L

m〈nm〉0 − ML

)− (M0

L − ML), (4.15)

where ML = −νm2F/2 is the total guiding-center quantum number for the subsystem L with

the uniform number density ν. The first term is the intrinsic dipole moment calculated

previously in (3.42). We denote the second term as

hα = M0L − ML. (4.16)

hα depends on the location of the cut. This quantity is called the topological spin by other

authors.[58, 69] Similarly, we define N0L =

∑m∈L n

0m and NL = ν|mF |. Then, 〈∆NL〉 can be

written as

〈∆NL〉 = TrL[∆NLρL] (4.17)

=

(∑m∈L

〈nm〉0 − NL

)− (N0

L − NL). (4.18)

The first term vanishes by Luttinger’s theorem.

We denote the second term as

qα = N0L − NL. (4.19)

71

This expression gives us the fractional charges associated with the topological sector α.

We can calculate hα and qα for different model FQH states only using the root occupation

numbers in Appendix.A.

In 〈∆ML〉, the most dominant term is proportional to the squared circumference L2. We

define the sub-leading term as

γ

24− hα = 〈∆ML〉+

1

2

(L

2π`B

)2s

q(4.20)

In order to calculate this sub-leading term, we need a large system. We generated orbital

entanglement spectra for 1/3 Laughlin state, 1/5 Laughlin state and 2/4 Moore-Read state

using the “matrix product state (MPS)” program developed by Estienne et al [10] (The

first implementation of MPS to FQHE was by Zaletel et al [69]). Each state contains 100

particles. Their accuracy is limited by the so-called “truncation level” (which we call plevel

in the figures). As the truncation level increases the approximation to the exact state gets

better (see appendix.B). We plot the 〈∆ML〉 against different values of circumference in

Fig.4.4. The sub-leading term γ/24 is also plotted in Fig.4.5, 4.6 and 4.7. The numerical

calculation is consistent with the prediction[22] that γ may be expressed as

γ = c− ν (4.21)

where c is the total signed central charge c − c of the underlying edge theory: c = 1 for

Laughlin states and c = 3/2 for 2/4 Moore-Read state.

The theoretical derivation of this result will be presented elsewhere.[22] It is the anomaly

of the signed Virasoro algebra,[22] with generators Lm, which are the Fourier components of

the momentum density; this survives as a universal algebra, with no renormalization, despite

the breaking of Lorentz and conformal invariance when the various linearly-dispersing modes

acquire different propagation speeds. Note that integer quantum Hall states, where the effect

is due to simple filling of Landau levels by the Pauli principle (and which are not topologically

72

ordered) do not exhibit a gapless “orbital” entanglement spectrum of the type discussed here,

and have c − ν = 0. The anomaly c appears in (4.21) as a “Casimir momentum,” which

is a feature of chiral theories: this remains universal so long as translational invariance is

unbroken, while the Casimir energy (the origin of the finite-size correction in non-chiral cft)

becomes non-universal once Lorentz invariance is lost.

73

0 100 200 300 400 500 600 700 8000

1

2

3

4

5

L2

〈∆M

L〉

Dipole moment per length

10|01, plevel = 12

100|001, plevel = 12

110|011, plevel = 12

Figure 4.4: Dots represent 〈∆ML〉 of 1/3 Laughlin (red), 1/5 Laughlin (blue) and 2/4Moore-Read (green) states for different values of circumference L calculated from the orbitalentanglement spectra with the truncation level equal to 12. Here, each orbital cut is avacuum cut, hα = 0. Each line represents the dominant value L2/(8π2`2

B) × (−s/q) where−s/q = 1/3, 2/5, 1/2 respectively.

74

4.2 Momentum polarization from Real-space cut

The “orbital-cut” entanglement spectrum only has a gapless spectrum when it is applied

to states with topological order. In particular, it does not show a gapless spectrum when

applied to integer quantum Hall states, which are not topologically-ordered (they do not

exhibit a topological ground-state degeneracy when constructed on surfaces on genus > 0,

which is the defining property of “topological order”). Dubail et al.[8] perceived this feature

as a defect of the orbital-cut method, and introduced a modified “real-space” entanglement

spectrum for quantum Hall states as a remedy. (However, it should be noted that the absence

of a gapless orbital-cut entanglement spectrum in the trivial integer QHE case is consistent

with Li and Haldane’s claim[40] that a gapless spectrum is a characteristic property of a

topologically-ordered state.)

In the high field limit, quantum Hall states in Landau levels become an unentangled

product of the state of the guiding-centers Ri and the Landau orbit (cyclotron motion) radii

Ri. Each Landau level is characterized by a form-factor

fn(q) = 〈n|eiq·R|n〉L = 1− Λabn qaqb`

2B +O(q4), (4.22)

where |n〉L is the n-th Landau level single-particle state. If only a single Landau level is

occupied, the electronic state is a simple product of the guiding center state used in the “or-

bital cut” with a trivial completely-symmetric state of the Landau-orbit radii, characterized

by a form factor f(q) = f(qx, qy). (This is the type of state for which the “real-space cut”

was constructed in [8].) In the “Landau gauge”, the wavefunctions φn,m(x, y) have a profile

|φn,m(x, y)|2 =1

L

∫ ∞−∞

dqy2π

fn(0, qy)eiqy(y−ym), (4.23)

75

where ym = 2πm`2B/L, m ∈ Z + 1/2. The real-space cut at y = 0 is based on the partition

PLn,m =

∫ 0

−∞dy

∫ L

0

dx|φn,m(x, y)|2, PLn,m + PR

n,m = 1. (4.24)

Note also that

∑m>0

mPLn,m −

∑m<0

mPRn,m =

∑m

m(PLn,m − θ(m))

=Λyyn L

2

(2π`B)2+

1

24+O(L−1). (4.25)

where for Galilean-invariant Landau levels with an effective mass tensor mgab (with det g =

1),

Λabn =

1

2sng

ab. (4.26)

In order to obtain the dipole moment from the real-space cut[8], we first double the

single-particle Hilbert space H1 on a cylinder into two subspaces H1L and H1R where a new

“pseudospin” index that takes values “R” and “L” has been introduced:

H1 7→ H1L ⊗H1R. (4.27)

If a function f(r) belongs to H1X , then f(r) = 0 if r 6∈ X where X can be either the

subsystem L or R. We choose the line x = 0 to be the boundary along the translational

invariant direction so that the guiding-center remains as a good quantum number. Now,

consider the Fock space H. Denote a vacuum state with no particle by |vac〉. We create a

particle with the guiding-center m in n-th Landau level by c†n,m. This creation operator can

be decomposed as

c†n,m = un,mc†n,m,L + vn,mc

†n,m,R (4.28a)

|un,m|2 = PLn,m, |vn,m|2 = PR

n,m, (4.28b)

76

where the physical state satisfies the constraint

(vn,mcn,m,L − un,mcn,m,R)|Ψ〉 = 0, (4.29)

so all occupied orbitals have a pseudospin which is fully-polarized in the “physical” direction.

For notational convenience, we concentrate on a single Landau level and drop the index n.

Given a Slater determinant state |{nm}〉 labeled by occupation numbers nm,

|{nm}〉 =∏m

(c†m)nm|vac〉

=∏m

(umc

†m,L + vmc

†m,R

)nm|vac〉 (4.30)

the product of creation operators can be expanded. Then, we obtain

|{nm}〉 =∑

α,β:NL+NR=N

Aαβ({nm})|ΨLα〉 ⊗ |ΨR

β 〉 (4.31)

where |ΨXα 〉 are Slater determinant states belonging to the Fock space HX (X = L, R) and

Aαβ({nm}) is a product of um and vm . With this expansion, and after translating the

partition λ into the occupation numbers {nm}, the mapping of a Jack polynomial into a

model FQH state |Ψ〉 in (3.24) becomes

|Ψ〉 =∑α,β

NL+NR=N

∑{nm}≤{n0

m}

a{nm}Aαβ({nm})|ΨLα〉 ⊗ |ΨR

β 〉 (4.32)

We can further Schmidt-decompose the model FQH state |Ψ〉. However, if our objective

is only to calculate the diagonal operators such as ML and NL, the information we gathered

from the orbital cut is enough. Consider the expectation value of the operator nLm = c†m,Lcm,L

〈nLm〉′ = TrL[c†m,Lcm,Lρ′L] (4.33)

77

where ρ′L is the normalized density matrix for the subsystem L, and we placed an apostrophe

on the bracket 〈...〉′ to distinguish the real-space cut expectation value with the orbital cut

expectation value 〈...〉. For all guiding-centers m′ such that m′ 6= m, the factors PLm′ and

PRm′ appear in pairs in the expectation value, and add to one. From this observation, we see

that the expectation value simplifies to

〈nLm〉′ = PLm〈nm〉0 (4.34)

Using this expression, in the expectation value of ∆ML,

〈∆ML〉′ =∑m

m〈nLm〉′ −M0L

=∑m

m(PLm − θ(m))〈nm〉0

+∑m<0

m〈nm〉0 −M0L. (4.35)

The first term is an additional term that appears when we consider the real-space cut.

The second term is the expectation value of ∆ML with the orbital cut that we calculated

previously. In the first term, PLm → 1 for m � 0, and the summand vanishes. Meanwhile,

as m → 0, which is the location of the real-space cut, we are deep into the bulk so that

〈nm〉0 = ν. Thus, in the thermodynamic limit, the expectation value 〈∆ML〉′ becomes

〈∆ML〉′ = ν∑m

m(PLm − θ(m)) + 〈∆ML〉 (4.36)

The first term was already considered in (4.25).

For simplicity, we now assume Galilean-invariant Landau orbits, so Λyyn = 1

2sng

yy, where

sn = n + 12

is the Landau-orbit spin. If we further include the contributions from the filled

78

Landau levels 0,1,...,n− 1, then 〈∆ML〉′ is

〈∆ML〉′ =1

2

(L

2π`B

)2(

n∑n′=0

sn′νn′ gyy − s

qgyy

)

+(ν ′ + γ)

24+O(L−1), (4.37)

where νn′ = cn′ = 1 for n′ < n and νn = ν. We also defined ν ′ =∑n

n′=0 νn′ and γ = cn − νn.

We explicitly wrote the two metrics gab and gab since they need not coincide as noted before

[21]. There are topological contributions from each cut: we get n/24 from n filled Landau

levels and ν/24 from the partially filled Landau level as a result of the real-space cut. We get

γ/24 from the variation of orbital occupations near the physical edge. The normal vector of

the surface of the fluid at the physical edge is reversed from the normal vector at the real-

space cut. We note here that the Landau-orbit spins sn′ (n′ = 0, . . . , n) are positive while the

guiding-center spin s is negative. The general expression for the total Hall viscosity tensor

η′abH (the sum of the Landau-orbit ant guiding-center contributions) is

η′abcdH =12

(η′acH εbd + η′bdH εac + η′bcH ε

ad + η′adH εbc)

(4.38a)

η′abH =eB

2π

(∑n

Λabn νn −

1

2

s

qgab

), (4.38b)

Using this expression for the Hall viscosity, we can write the momentum polarization in a

fully covariant tensor form as

〈∆ML〉′ = ~−1η′abH εacεbdLcLd

2π`2B

+(ν ′ + γ)

24(4.39)

The O(L2) term gives the Hall viscosity, which is now the sum of two terms: one is

derived from the Landau-orbit form factors, weighted by the Landau level occupation, and

the other is the guiding-center contribution derived from the orbital cut.

79

We note the the “real-space cut” involves far greater computational effort than the “or-

bital cut”, but at least as far as the “momentum polarization” is concerned, merely adds

trivial contributions to the Hall viscosity and topological terms e.g., (c− ν) + ν = c. Clearly

all the non-trivial topological and entanglement information of the topologically-ordered

states is fully present in the “orbital-cut”. From this viewpoint, we are tempted to conclude

that use of the “real-space cut” is an unnecessary use of computational resources that merely

serves to conceal the structures of the “orbital cut” entanglement spectrum by convoluting

them with the form-factor of the Landau orbits.

80

0 100 200 300 400 500 600 700 800−2

0

2

4

6

·10−2

L2

γ/2

4

Topological term of 1/3 Laughlin state

10|01, plevel= 12

100|1, plevel= 12

10|01, plevel= 13

Figure 4.5: The plot of sub-leading term γ/24 for 1/3 Laughlin state. Red dots: vacuumcut with truncation level 12. Blue dots: quasi-hole cut with truncation level 12. Green dots:vacuum cut with truncation level 13. The horizontal line represents 1/36.

81

0 100 200 300 400 500 600 700 800

0

2

4

6

8

·10−2

L2

γ/2

4

Topological term of 1/5 Laughlin state

100|001, plevel= 13

1000|01, plevel= 12

10000|1, plevel= 12

100|001, plevel= 16

Figure 4.6: The plot of sub-leading term γ/24 for 1/5 Laughlin state. Red dots: vacuumcut with truncation level 13. Blue dots: one quasi-hole cut with truncation level 12. Greendots: two quasi-hole cut with truncation level 12. Black dots: vacuum cut with truncationlevel 16. The horizontal line represents 1/30.

82

0 100 200 300 400 500 600 700 800

0

2

4

6

8

·10−2

L2

γ/2

4

Topological term of 2/4 Moore-Read state

110|011, plevel= 11

1100|11, plevel= 9

11001|1, plevel= 9

110|011, plevel= 12

Figure 4.7: The plot of sub-leading term γ/24 for Moore-Read 2/4. Red dots: vacuum cutwith truncation level 11. Blue dots: one quasi-hole cut with truncation level 9. Green dots:isolated fermion cut with truncation level 9. Black dots: vacuum cut with truncation level12. The horizontal line represents 1/24.

83

Chapter 5

Collective excitation

The purpose of this chapter to understand the connection between the gapped collective

excitation in the incompressible quantum Hall states and the fluctuation of the guiding-

center metric defined in Sec.2.2.1.

In Sec.5.1, we first discuss the collective excitation energy obtained using the single-

mode-approximation implemented by Girvin, MacDonald and Platzman.

In Sec.5.2, we examine the long-wavelength limit of the collective excitation energy, and

compare it with the energy cost due to the variation of the guiding-center metric. We

observe these two quantities are connected by a 4-tensor Gabcd called the “guiding-center

shear modulus tensor.”[24]

5.1 Girvin-MacDonald-Platzman approximation

Girvin, MacDonald and Platzman (GMP) adopted the single-mode-approximation (SMA)

of Feynman to estimate the excitation energy of an incompressible FQH ground state.[16]

Feynman first used SMA to explain the excitations of He-4 superfluid ground state.[12, 13, 14]

He argued that one can obtain an approximate excited state carrying the momentum k by

multiplying the translationally invariant ground state |Ψ0He-4〉 with the Fourier-transformed

84

density operator ρ0(k) =∑

i eik·ri (See (2.12a)). Explicitly

|ΨHe-4(k)〉 := S(k)−1/2N−1/2He ρ0(k)|Ψ0

He-4〉,

where NHe is the number of particles, and the structure factor S(k) plays the role of the

normalization factor,

S(k) := N−1He 〈Ψ0

He-4|ρ0(−k)ρ0(k)|Ψ0He-4〉

Using this approximation, Feynman was able to produce the “roton” spectrum first

predicted by Landau.[34] Near the long-wavelength limit k ≈ 0, this Feynman single-

mode-approximation exhibits a gapless phonon spectrum. Thus, Helium-4 superfluid is a

compressible quantum fluid. This is in stark contrast with incompressibility of the FQH

states.

In order to explain the existence of the energy gap for phonon in FQHE, GMP modi-

fied Feynman’s approach by projecting the density operator ρ0(k) into the lowest Landau

level[16],

ρ0(k)→Ne∑i=1

exp

[ik

∂

∂zi

]exp

[iik∗

2zi

].

This procedure of projecting the density operator is valid only for the lowest Landau level

and for the choice of the symmetric gauge (so that the single-particle states are analytic

functions except the Gaussian factor). Instead of following GMP, we will follow Haldane’s

generalization[24] which is valid for any single Landau level and which is not dependent on

the choice of gauge.

We first write the coordinate ri = Ri+Ri as a sum of the guiding-center and the Landau-

orbit radius. Then, the projection of the density operator into a single Landau level labeled

by n ∈ Z+ corresponds to sandwiching the density operator ρ0(k) with the eigenvectors |n〉i,L

85

of the Landau-orbit rotation generator Li(g) defined in (2.9). This gives the guiding-center

density operator ρ(k) times the Landau-level-dependent form factor fn(k),

ρ0(k)→ fn(k)ρ(k)

ρ(k) =Ne∑i=1

eik·Ri . (5.1)

Then, given the FQH ground state |Ψ〉, we can find the excited state within the single-mode-

approximation by acting with the guiding-center density operator,

|Ψk〉 = S(k; g)−1/2ρ(k)|Ψ〉, (5.2)

where we defined the guiding-center structure factor S(k; g) in terms of the guiding-center

density operator,

S(k; g) := N−1Φ 〈Ψ(g)|ρ(−k)ρ(k)|Ψ(g)〉G. (5.3)

The relation between S(k; g) and the usual structure factor S(k) is discussed in Appendix.C.

We remind readers that the model FQH ground state |Ψ〉 in a partially filled Landau level

was written as a tensor product of the Landau-orbit part and the guiding-center part (see

Sec.2.2.1),

|Ψ〉 =

(Ne∏i=1

|n〉L,i)⊗ |Ψ(g)〉G, (5.4)

where the subscript L stands for “Landau-orbit” and G stands for “guiding-center.” The

guiding-center part of the many-particle FQH state is |Ψ(g)〉G which is dependent on the

guiding-center metric gab.

Now, we want to estimate the excitation energy for the SMA state (5.2). The two-body

interaction Hamiltonian was projected into a single Landau level, and the residual two-body

86

interaction H is written in terms of the guiding-center density operators,

H =

∫d2q`2

B

4πvn(q)ρ(q)ρ(−q), (5.5)

where vn(q) := V (q)fn(q)2 is the product of the Fourier-transformation of the two-body

interaction V (ri − rj) and the Landau-orbit form factor fn(q) (see Sec.2.2.1). With the

assumption that |Ψ(g)〉G is the ground state of H with the energy E0, we estimate the

excitation energy in SMA by evaluating

∆(k) = S(k; g)−1〈Ψ(g)|ρ(−k)(H − E0)ρ(k)|Ψ(g)〉G. (5.6)

With the assumption that the excitation energy is invariant under parity transformation,

∆(k) = ∆(−k), (5.7)

we may write the excitation energy as

∆(k) = (2S(k; g))−1〈Ψ(g)| [ρ(−k), [H, ρ(k)]] |Ψ(g)〉G. (5.8)

To evaluate this quantity we make use of the commutation relation of the guiding-center

density operators ρ(q):

[ρ(q), ρ(q′)] = 2i sin(12q × q′`2

B)ρ(q + q′), (5.9)

where q × q′ := εabqaq′b. This is called Girvin-MacDonald-Platzman (GMP) algebra.[21]1

GMP found a similar algebra by projecting the usual density operator(2.12a) into the lowest

Landau level.[16]

1This is a generalization of the original GMP algebra.

87

Using the GMP algebra, we express (5.8) as[24]

∆(k) =S(k; g)−1

∫d2q`2

B

4πvn(q) S(q,k; g)

{2 sin

(12q × k `2

B

)}2, (5.10)

where we defined S(k, q; g) in terms of S(k; g) as

S(q,k; g) := 12(S(q + k; g) + S(q − k; g)− 2S(q; g)). (5.11)

Thus, the equation (5.10) suggests that the excitation energy ∆(k) for the collective mode

is completely determined by vn(q) and the guiding-center structure factor S(q).

In Fig.5.1, we plotted the exact energy spectrum for the system of N = 9 and 10 particles

on the sphere given the Coulomb interaction projected into the lowest Landau level. In this

figure, we also plotted two SMA excitation spectra obtained using (5.10) on the Coulomb

ground state and the Laughlin state. Note that although the Laughlin state is not an exact

ground state of the Coulomb interaction, it produces a qualitatively very accurate excitation

spectrum when used in (5.10).

88

Figure 5.1: The energy spectrum for Coulomb interaction projected into the lowest Landaulevel with N = 9, 10 particles on sphere. The green circles are the data points from the exactdiagonalization of the projected interaction. The empty diamonds are the SMA spectrumobtained using the exact Coulomb ground state in (5.10). The filled diamonds are the SMAspectrum obtained using the Laughlin 1/3 state in (5.10).

5.2 The relationship between SMA and guiding-center

metric

In this section, we would like to make the connection between the single-mode-approximation

discussed in the previous section and the guiding-center metric which was discussed in Sec.

2.2.1. The claim made by Haldane[24, 67] is that the SMA excitations in the long-wavelength

limit correspond to the dynamic fluctuation of the guiding-center metric.

89

We first examine the long-wavelength limit of the SMA excitation energy ∆(k). For small

λ, the equation (5.11) becomes

S(q, λk; g) =12λ2kakb∂qa∂qbS(q; g) +O(λ2). (5.12)

Then, from this, we can obtain the long-wavelength limit of (5.10),

S(λk; g)∆(λk) =12λ4kakbkckd`

4B

∫d2q`2

B

4πvn(q)∂qa∂qbS(q; g)εecqeε

fdqf +O(λ6)

:=12λ4kakbkckd`

4B G

abcd (5.13)

where we defined a 4-tensor Gabcd. From this equation, we see that it is necessary to have

the guiding-center structure S(λk; g) to vanish as λ4 for small λ in order to have a finite

energy gap.

We now examine the dependence on the guiding-center metric gab of the correlation energy

E(g). We remind readers that the correlation energy E(g) was defined to be the expectation

value of the the two-body interactions H,

E(g) =

∫d2q`2

B

4πvn(q)〈Ψ(g)|ρ(q)ρ(−q)|Ψ(g)〉G. (5.14)

Writing the correlation energy E(g) in terms of the guiding-center structure factor S(k; g),

we have

E(g) =

∫d2q`2

B

4πvn(q)(S(q; g)− S∞). (5.15)

Thus, we see that the correlation energy E(g) depends on the metric gab through the guiding-

center structure factor S(q; g). In (5.15), we regularized the correlation energy by subtract-

ing the constant contribution at large q using the fact that the limiting value of S(q; g) at

90

large q is[16]2

S∞ := limλ→∞

S(λq; g) = ν − ν2. (5.16)

Now, suppose the minimum value of E(g) is attained by some metric gab.3 Then, consider

the small variation δgab about this equilibrium metric, g′ab = gab + δgab. We parametrize δgab

by a real symmetric tensor αab

δgab = −gacεcdαdb + (a↔ b). (5.17)

The guiding-center part of the many-body state |Ψ(g)〉G changes under the variation of the

metric as

|Ψ(g′)〉G = U(α)|Ψ(g)〉G, (5.18)

where U(α) is a unitary operator

U(α) = exp i∑i

αabΛabi . (5.19)

Thus, for the deformed guiding-center metric, the guiding-center structure factor can be

written as

S(k; g′) = N−1Φ 〈Ψ(g)|U(α)†ρ(−k)U(α)U(α)†ρ(k)U(α)|Ψ(g)〉G. (5.20)

Using the commutation relation [U(α), Rai ] = U(α)εabαbcR

ci , we find

S(k; g′) = N−1Φ 〈Ψ(g)|ρ(−k′)ρ(k′)|Ψ(g)〉G = S(k′; g), (5.21)

where k′a = ka + kbεbcαca.

2If the particles are not fermions but are bosons, then S∞ = ν + ν2.[24]3The existence of such minimum requires that the FQH state is incompressible.

91

Now, Haldane[24] proves an identity (called the “self-duality”) satisfied by S(k; g),

S(k; g)− S∞ = −∫d2q`2

B

4πeik×q`

2B(S(q; g)− S∞), (5.22)

where k × q = εabkaqb. Using (5.21) and (5.22), we have

S(k; g′)− S(k; g)

=−∫d2q`2

B

4πeik×q`

2B(S(q; g)− S∞)

[exp

(ikaε

abαbcεcdqd`

2B

)− 1]

=− kaεabαbc∂kcS(k; g) + 12(kaε

abαbc)(kdεdeαef )∂kc∂kf S(k; g) +O(α3).

Thus, the variation of the correlation energy (5.15) is

E(g′)− E(g) = 12αbcαef

∫d2k`2

B

4πvn(k)∂kc∂kf S(k; g)εabkaε

dekd +O(α3)

= 12αacαbdG

abcd +O(α3), (5.23)

where the term proportional to one αab vanishes from the assumption the equilibrium metric

gab minimizes the correlation energy. Comparing (5.13) and (5.23), we observe the occurrence

of the same 4-tensor Gabcd suggesting a close relation between the long-wavelength limit of

the SMA excitation and the fluctuation of the guiding-center metric.

From (5.23) and from the fact that εabαbc may be interpreted as the derivative of the

displacement vector ∂aua in elasticity theory[35](see Sec.2.2.2), we may identify Gabcd as a

modulus tensor[24]. Since it is a response to the variation of the guiding-center metric, it

may be called the “guiding-center shear modulus tensor.”

92

Chapter 6

Conclusion

We showed that the intrinsic dipole moment along the edges of the incompressible FQH

fluids can be expressed in terms of electric charge e, guiding center spin s, number of fluxes

per a composite boson q, confirming the prediction made in the previous work.[20] This

provides another sum rule for the FQH fluids in addition to the Luttinger sum rule.[41] For

incompressible FQH states, the electric force on the intrinsic dipole moment is balanced the

stress given by the gradient of the flow velocity times the guiding-center Hall viscosity.

We also related the the edge dipole moment to the expectation value of the momentum

(or “momentum polarization”[58]) of the entanglement spectrum. In the high-field limit,

when the guiding-center and Landau-orbit degrees of freedom become unentangled with each

other, the dipole moment and the related Hall viscosity separate cleanly into independent

parts respectively coming from the non-trivial correlated guiding-center degrees of freedom

of the FQH state, and the trivially-calculable one-body properties of the Landau orbits. The

“orbital cut” entanglement spectrum introduced by Li and Haldane[40] contains only infor-

mation on the guiding-center degrees of freedom, and allows the guiding-center contribution

to the Hall viscosity of the FQH fluid to be found as a bulk geometric property, and also

gives the topological quantity γ = c − ν, the difference between the (signed) “conformal

anomaly” (or “chiral stress-energy anomaly”[22] c = c− c, and the chiral charge anomaly ν,

93

which are the two fundamental quantum anomalies of the FQH fluids. It is useful to note

that γ is insensitive to completely-filled Landau levels, and vanishes identically in integer

quantum Hall states, which do not exhibit topological-order.

We also examined the equivalent calculation in the “real-space” entanglement spectrum

described by Dubail et al.,[8] which adds information about the Landau orbit to provide the

combined guiding-center plus Landau-orbit contribution to the Hall viscosity and c rather

than γ. However since the “real-space entanglement” method involves much extra computa-

tional complexity, and convolutes the non-trivial Landau-orbit-independent correlated guid-

ing center data with the essentially trivial (and Landau-level-dependent) Landau-orbit form

factor data, we concluded that there were no advantages to use of the “real-space” as opposed

to “orbital” entanglement spectrum. Indeed, since the Landau-orbit form factor is essen-

tially unrelated to the FQH correlations, and can be chosen as an additional (and arbitrary)

ingredient to convert orbital entanglement data into a “real-space” form, its use may actually

serve to conceal the essential features of the guiding-center entanglement. The “real-space”

spectrum may also be thought of operationally as the use of an essentially ad-hoc function

PLm (4.24) that can be arbitrarily chosen to “smear out” a sharp orbital cut between cylinder

orbitals m and m+ 1, which breaks both guiding-center indistinguishability (by introducing

“pseudo-spin” labels “L” and “R”) and reducing the full 2D translational symmetry (the

parallel to the cylinder axis in the N → ∞ limit, or equivalently, full rotational symmetry

in the spherical geometry) to 1D axial translational symmetry. It interpolates continuously

between two completely-well-defined limits of guiding-center entanglement: the “orbital cut”

which preserves guiding-center indistinguishability while breaking 2D translational symme-

try down to 1D translational symmetry, and the “particle cut” which divides the guiding

centers into two distinguishable groups, but preserves full 2D translational symmetry.

We also observed the close relation between the energy cost due to the deformation of the

guiding-center metric and the excitation energy spectrum in the single-mode-approximation.

94

Appendix A

Evaluation of topological spins

In this section, we calculate the quantities hα and qα for Laughlin 1/3 and 1/5 states and

Moore-Read 2/4 state. These quantities are defined in (4.16) and (4.19) respectively.

The root state of the 4-particle 1/3 Laughlin state is

|{n0m}〉 = |010010010010〉.

We can consider three topologically distinct cuts corresponding to quasi-particle, vacuum

and quasi-hole sectors respectively,

|01001〉L ⊗ |0010010〉R hp = 1/6 qp = 1/3

|010010〉L ⊗ |010010〉R hI = 0 qI = 0

|0100100〉L ⊗ |10010〉R hh = 1/6 qh = −1/3 .

The root state of the 4-particle 1/5 Laughlin state is

|{n0m}〉 = |00100001000010000100〉.

95

In this case, there are five topologically distinct cuts corresponding to two and one quasi-

particles, vacuum, one and two quasi-hole sectors respectively,

|00100001〉L ⊗ |000010000100〉R h2p = 2/5 q2p = 2/5

|001000010〉L ⊗ |00010000100〉R hp = 1/10 qp = 1/5

|0010000100〉L ⊗ |0010000100〉R hI = 0 qI = 0

|00100001000〉L ⊗ |010000100〉R hh = 1/10 qh = −1/5

|001000010000〉L ⊗ |10000100〉R h2h = 2/5 q2h = −2/5

The root state of the 8-particle 2/4 Moore-Read ground state is

|{n0m}〉 = |0110011001100110〉.

We have the four topologically distinct cuts corresponding to isolated fermion, quasi-particle

pair, vacuum and quasi-hole pair sectors respectively,

|011001〉L ⊗ |1001100110〉R hψ = 1/2 qψ = 0

|0110011〉L ⊗ |001100110〉R h2p = 1/4 q2p = 1/2

|01100110〉L ⊗ |01100110〉R hI = 0 qI = 0

|011001100〉L ⊗ |1100110〉R h2h = 1/4 q2h = −1/2 .

The root state of the 8-particle 2/4 Moore-Read state with two separated quasi-holes at

the two Fermi momenta is1

|{n0m}〉 = |01010101010101010〉.

1In this case, the two fermi surfaces are located at the orbitals as it was for the bosons in Fig.3.2.

96

There are two topologically distinct cuts,

|01010101〉L ⊗ |010101010〉R hp = 1/16 qp = 1/4

|010101010〉L ⊗ |10101010〉R hh = 1/16 qh = −1/4 .

We see that hα is exactly the quantity called “topological spin” by other authors.[58, 69] qα

are the fractional charge of the elementary excitations.

97

Appendix B

Matrix product state (MPS)

expansion

We briefly discuss how the MPS expansion is used for the model fractional quantum Hall

states. For a complete treatment, we refer readers to other works.[69, 10] The (unnormalized)

model wavefunctions Ψν({zi}) at ν = 1/q can be written as a holomorphic correlator of a

product of vertex operators V (zi) [43]

Ψν({zi}) = 〈N√q|V (ZN) . . . V (z1)|0〉, (B.1)

where |0〉 and 〈N√q| are asymptotic states which carry U(1) charges, zero and N√q re-

spectively. The many-particle wavefunction Ψν({zi}) which is a polynomial in {zi} can be

expanded in terms of monomials mλ for bosons (or in terms of Slater determinants slλ for

fermions):

Ψν({zi}) =∑λ

aλmλ({zi}). (B.2)

Here, each monomial mλ is labeled by a partition λ whose N parts {λi} are non-negative

integers less than NΦ, the total number of single-particle states. aλ are actually the coef-

98

ficients that appear in (3.3) up to overall constant. We can extract the coefficients aλ by

application of the residue theorem,

aλ =N∏i=1

1

2πi

∮dzi

zλi+1i

Ψν({zi}). (B.3)

In order to have a matrix product representation, we can insert an identity operator I =∑α |α〉〈α| between each pair of vertex operators in (B.1). Here, the states |α〉 form the basis

of the underlying chiral conformal field theory. Then, we obtain

aλ =∏{αi}

N∏i=1

1

2πi

∮dzi

zλi+1i

〈αi|V (zi)|αi−1〉, (B.4)

where |α0〉 = |0〉 and 〈αN | = 〈N√q|. Thus, the problem of expanding Ψν({zi}) in terms of

mλ reduces to calculating the 3-point function,

〈α|V (z)|α′〉 (B.5)

for all possible CFT states |α〉 and |α′〉.

Now, we can translate the partition λ into a set of occupation numbers {nm = 0, 1 : m =

0, 1, . . . , NΦ − 1}: if m ∈ {λi : i = 1, 2, . . . , N}, then nm = 1, and nm = 0 otherwise. Then,

we can write

aλ = (Bn0 [0]Bn1 [1] . . . BnNΦ−1 [NΦ − 1])Ne√q,0, (B.6)

where the matrices Bnm [m] are defined as

B0[m]αα′ = δα,α′ (B.7a)

B1[m]αα′ =1

2πi

∮dz

zm+1〈α|V (z)|α′〉. (B.7b)

Because the Hilbert space spanned by the states |α〉 is infinite, the exact representation

(B.6) involves infinite dimensional matrices. We can use an approximation by “truncating”

99

the MPS.[10] The model wavefunction Ψν is constructed from a CFT involving only a finite

number of primary fields ϕa, i.e. from a minimal model. Then, given a positive integer Pmax,

which is called the “truncation level,” we can consider the finite number of states, |ϕa〉 and

their descendants up to Pmax-th Virasoro level. For these states, we can construct the B

matrix which is an approximation to the exact B matrix.

100

Appendix C

Relation between structure factors

Note that the usual structure factor S(k) is

S(k) = N−1e 〈Ψ|ρ0(−k)ρ0(k)|Ψ〉. (C.1)

Its relationship with S(q; g) (guiding-center structure factor) is (See Haldane in [48])

S(q) = 1− fn(q)2 + ν−1fn(q)2S(q; g). (C.2)

where ν = Ne/NΦ. The “regularized” guiding-center structure factor1 is

S(q; g)c := S(q; g)− 1

NΦ

〈Ψ(g)|ρ(q)|Ψ(g)〉〈Ψ(g)|ρ(−q)|Ψ(g)〉G

= S(q; g)− δq,0 ν2NΦ, (C.3)

with the assumption that the ground state is uniform: 〈Ψ(g)|ρ(q)|Ψ(g)〉G = δq,0 νNΦ.

Thus, we may write the usual structure factor as

S(q) = 1− fn(q)2 + δq,0 νNΦ + ν−1fn(q)2S(q; g)c. (C.4)

1It can also be called the “connected” guiding-center structure factor.

101

Bibliography

[1] Daniel Arovas, J. R. Schrieffer, and Frank Wilczek. Fractional statistics and the quan-tum hall effect. Phys. Rev. Lett., 53:722–723, Aug 1984.

[2] J. E. Avron, R. Seiler, and P. G. Zograf. Viscosity of quantum hall fluids. Phys. Rev.Lett., 75:697–700, Jul 1995.

[3] Alexander A Belavin, Alexander M Polyakov, and Alexander B Zamolodchikov. Infi-nite conformal symmetry in two-dimensional quantum field theory. Nuclear Physics B,241(2):333–380, 1984.

[4] B. Andrei Bernevig and F. D. M. Haldane. Model fractional quantum hall states andjack polynomials. Phys. Rev. Lett., 100:246802, Jun 2008.

[5] B. Andrei Bernevig and N. Regnault. Anatomy of abelian and non-abelian fractionalquantum hall states. Phys. Rev. Lett., 103:206801, Nov 2009.

[6] Barry Bradlyn, Moshe Goldstein, and N. Read. Kubo formulas for viscosity: Hallviscosity, ward identities, and the relation with conductivity. Phys. Rev. B, 86:245309,Dec 2012.

[7] H. C. Choi, W. Kang, S. Das Sarma, L. N. Pfeiffer, and K. W. West. Activation gapsof fractional quantum hall effect in the second landau level. Phys. Rev. B, 77:081301,Feb 2008.

[8] J. Dubail, N. Read, and E. H. Rezayi. Real-space entanglement spectrum of quantumhall systems. Phys. Rev. B, 85:115321, Mar 2012.

[9] L. W. Engel, S. W. Hwang, T. Sajoto, D. C. Tsui, and M. Shayegan. Fractional quantumhall effect at ν=2/3 and 3/5 in tilted magnetic fields. Phys. Rev. B, 45:3418–3425, Feb1992.

[10] B. Estienne, Z. Papic, N. Regnault, and B. A. Bernevig. Matrix product states for trialquantum hall states. Phys. Rev. B, 87:161112, Apr 2013.

[11] Benoit Estienne, B Andrei Bernevig, and Raoul Santachiara. Electron-quasihole dualityand second-order differential equation for read-rezayi and jack wave functions. PhysicalReview B, 82(20):205307, 2010.

102

[12] Richard P Feynman. Atomic theory of the two-fluid model of liquid helium. PhysicalReview, 94(2):262, 1954.

[13] Richard P Feynman. Statistical mechanics: a set of lectures, 1998.

[14] RP Feynman and Michael Cohen. Energy spectrum of the excitations in liquid helium.Physical Review, 102(5):1189, 1956.

[15] PD Francesco, P Mathieu, and D Senechal. Conformal field theory, graduate texts incontemporary physics, 1999.

[16] S. M. Girvin, A. H. MacDonald, and P. M. Platzman. Collective-excitation gap in thefractional quantum hall effect. Phys. Rev. Lett., 54:581–583, Feb 1985.

[17] SM Girvin, AH MacDonald, and PM Platzman. Magneto-roton theory of collectiveexcitations in the fractional quantum hall effect. Physical Review B, 33(4):2481, 1986.

[18] F. D. M. Haldane. Fractional quantization of the hall effect: A hierarchy of incompress-ible quantum fluid states. Phys. Rev. Lett., 51:605–608, Aug 1983.

[19] F D M Haldane. Proceedings of the International School of Physics ”Enrico Fermi”,Course CXXI ”Perspectives in Many-Particle Physics” : Luttinger’s theorem andBosonization of the Fermi surface. north-holland, Amsterdam, 1993.

[20] F D M Haldane. “hall viscosity” and intrinsic metric of incompressible fractional hallfluids. 2009.

[21] F D M Haldane. Geometrical description of the fractional quantum hall effect. Phys.Rev. Lett., 107(11):116801, 2011.

[22] F D M Haldane and YeJe Park. unpublished.

[23] F. D. M. Haldane and E. H. Rezayi. Periodic laughlin-jastrow wave functions for thefractional quantized hall effect. Phys. Rev. B, 31:2529–2531, Feb 1985.

[24] FDM Haldane. Self-duality and long-wavelength behavior of the landau-level guiding-center structure function, and the shear modulus of fractional quantum hall fluids. arXivpreprint arXiv:1112.0990, 2011.

[25] FDM Haldane and EH Rezayi. Finite-size studies of the incompressible state of thefractionally quantized hall effect and its excitations. Physical review letters, 54(3):237,1985.

[26] B. I. Halperin. Quantized hall conductance, current-carrying edge states, and the ex-istence of extended states in a two-dimensional disordered potential. Phys. Rev. B,25:2185–2190, Feb 1982.

[27] Bertrand I Halperin. Theory of the quantized hall conductance. Helv. Phys. Acta,56(1-3):75, 1983.

103

[28] Carlos Hoyos and Dam Thanh Son. Hall viscosity and electromagnetic response. Phys.Rev. Lett., 108:066805, Feb 2012.

[29] J. K. Jain. Composite-fermion approach for the fractional quantum hall effect. Phys.Rev. Lett., 63:199–202, Jul 1989.

[30] C. Kallin and B. I. Halperin. Excitations from a filled landau level in the two-dimensionalelectron gas. Phys. Rev. B, 30:5655–5668, Nov 1984.

[31] Moonsoo Kang, A. Pinczuk, B. S. Dennis, L. N. Pfeiffer, and K. W. West. Observation ofmultiple magnetorotons in the fractional quantum hall effect. Phys. Rev. Lett., 86:2637–2640, Mar 2001.

[32] K. v. Klitzing, G. Dorda, and M. Pepper. New method for high-accuracy determinationof the fine-structure constant based on quantized hall resistance. Phys. Rev. Lett.,45:494–497, Aug 1980.

[33] A. Kumar, G. A. Csathy, M. J. Manfra, L. N. Pfeiffer, and K. W. West. Nonconventionalodd-denominator fractional quantum hall states in the second landau level. Phys. Rev.Lett., 105:246808, Dec 2010.

[34] LD Landau. The theory of superfluidity of helium ii. J. Phys, 5(1):71–90, 1941.

[35] LD Landau and EM Lifshitz. Theory of Elasticity, 3rd. Pergamon Press, Oxford, UK,1986.

[36] Lev Davidovich Landau, EM Lifshitz, JB Sykes, JS Bell, and ME Rose. Quantummechanics, non-relativistic theory: Vol. 3 of course of theoretical physics. Physics Today,11(12):56–59, 2009.

[37] R. B. Laughlin. Anomalous quantum hall effect: An incompressible quantum fluid withfractionally charged excitations. Phys. Rev. Lett., 50:1395–1398, May 1983.

[38] Sung-Sik Lee, Shinsei Ryu, Chetan Nayak, and Matthew PA Fisher. Particle-holesymmetry and the ν= 5 2 quantum hall state. Physical review letters, 99(23):236807,2007.

[39] Michael Levin, Bertrand I Halperin, and Bernd Rosenow. Particle-hole symmetry andthe pfaffian state. Physical review letters, 99(23):236806, 2007.

[40] Hui Li and F. D. M. Haldane. Entanglement spectrum as a generalization of entangle-ment entropy: Identification of topological order in non-abelian fractional quantum halleffect states. Phys. Rev. Lett., 101:010504, Jul 2008.

[41] J. M. Luttinger. Fermi surface and some simple equilibrium properties of a system ofinteracting fermions. Phys. Rev., 119:1153–1163, Aug 1960.

[42] J. Maciejko, B. Hsu, S. A. Kivelson, YeJe Park, and S. L. Sondhi. Field theory of thequantum hall nematic transition. Phys. Rev. B, 88:125137, Sep 2013.

104

[43] G Moore and N Read. Nonabelions in the fractional quantum hall effect. Nucl. Phys.B, 360(2):362–396, 1991.

[44] Michael Mulligan, Chetan Nayak, and Shamit Kachru. Isotropic to anisotropic transi-tion in a fractional quantum hall state. Phys. Rev. B, 82:085102, Aug 2010.

[45] W. Pan, J. S. Xia, H. L. Stormer, D. C. Tsui, C. Vicente, E. D. Adams, N. S. Sullivan,L. N. Pfeiffer, K. W. Baldwin, and K. W. West. Experimental studies of the fractionalquantum hall effect in the first excited landau level. Phys. Rev. B, 77:075307, Feb 2008.

[46] YeJe Park and F. D. M. Haldane. Guiding-center hall viscosity and intrinsic dipolemoment along edges of incompressible fractional quantum hall fluids. Phys. Rev. B,90:045123, Jul 2014.

[47] Yeje Park, Bo Yang, and FDM Haldane. Geometrical description of fractional quantumhall quasiparticles. In APS Meeting Abstracts, volume 1, page 24006, 2012.

[48] Richard E Prange and Steven M Girvin. The quantum Hall effect, volume 2. Springer-Verlag New York, 1987.

[49] R-Z Qiu, FDM Haldane, Xin Wan, Kun Yang, and Su Yi. Model anisotropic quantumhall states. Physical Review B, 85(11):115308, 2012.

[50] N. Read. Non-abelian adiabatic statistics and hall viscosity in quantum hall states andpaired superfluids. Phys. Rev. B, 79:045308, Jan 2009.

[51] N Read and E Rezayi. Beyond paired quantum hall states: parafermions and incom-pressible states in the first excited landau level. Physical Review B, 59(12):8084, 1999.

[52] N. Read and E. H. Rezayi. Hall viscosity, orbital spin, and geometry: Paired superfluidsand quantum hall systems. Phys. Rev. B, 84:085316, Aug 2011.

[53] N Regnault, B Andrei Bernevig, and FDM Haldane. Topological entanglement andclustering of jain hierarchy states. Physical review letters, 103(1):016801, 2009.

[54] EH Rezayi and FDM Haldane. Laughlin state on stretched and squeezed cylinders andedge excitations in the quantum hall effect. Phys. Rev. B, 50(23):17199, 1994.

[55] Steven H Simon, EH Rezayi, NR Cooper, and I Berdnikov. Construction of a pairedwave function for spinless electrons at filling fraction ν= 2/ 5. Physical Review B,75(7):075317, 2007.

[56] R. Tafelmayer. Topological vortex solitons in effective field theories for the fractionalquantum hall effect. Nuclear Physics B, 396(23):386 – 410, 1993.

[57] D. C. Tsui, H. L. Stormer, and A. C. Gossard. Two-dimensional magnetotransport inthe extreme quantum limit. Phys. Rev. Lett., 48:1559–1562, May 1982.

[58] Hong-Hao Tu, Yi Zhang, and Xiao-Liang Qi. Momentum polarization: An entanglementmeasure of topological spin and chiral central charge. Phys. Rev. B, 88:195412, Nov 2013.

105

[59] Daniel Varjas, Michael P. Zaletel, and Joel E. Moore. Chiral luttinger liquids and ageneralized luttinger theorem in fractional quantum hall edges via finite-entanglementscaling. Phys. Rev. B, 88:155314, Oct 2013.

[60] Susanne Viefers. Quantum hall physics in rotating bose–einstein condensates. Journalof Physics: Condensed Matter, 20(12):123202, 2008.

[61] X. G. Wen. Gapless boundary excitations in the quantum hall states and in the chiralspin states. Phys. Rev. B, 43:11025–11036, May 1991.

[62] X. G. Wen and A. Zee. Shift and spin vector: New topological quantum numbers forthe hall fluids. Phys. Rev. Lett., 69:953–956, Aug 1992.

[63] Xiao-gang Wen and Anthony Zee. Classification of abelian quantum hall states andmatrix formulation of topological fluids. Physical Review B, 46(4):2290, 1992.

[64] Wikipedia. Dominance order — wikipedia, the free encyclopedia, 2014. [Online; accessed12-July-2014].

[65] Tai Tsun Wu and Chen Ning Yang. Dirac monopole without strings: monopole har-monics. Nuclear Physics B, 107(3):365–380, 1976.

[66] Tai Tsun Wu and Chen Ning Yang. Some properties of monopole harmonics. PhysicalReview D, 16(4):1018, 1977.

[67] Bo Yang, Zi-Xiang Hu, Z Papic, and FDM Haldane. Model wave functions for thecollective modes and the magnetoroton theory of the fractional quantum hall effect.Physical review letters, 108(25):256807, 2012.

[68] Bo Yang, Z Papic, EH Rezayi, RN Bhatt, and FDM Haldane. Band mass anisotropyand the intrinsic metric of fractional quantum hall systems. Physical Review B,85(16):165318, 2012.

[69] Michael P. Zaletel, Roger S. K. Mong, and Frank Pollmann. Topological characterizationof fractional quantum hall ground states from microscopic hamiltonians. Phys. Rev.Lett., 110:236801, Jun 2013.

[70] S. C. Zhang, T. H. Hansson, and S. Kivelson. Effective-field-theory model for thefractional quantum hall effect. Phys. Rev. Lett., 62:82–85, Jan 1989.

106

guiding-center hall viscosity and intrinsic dipole moment ......zograf[2] showed there is a...

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