Guiding-center Hall viscosity and intrinsic dipole moment ......Zograf[2] showed there is a...
Transcript of Guiding-center Hall viscosity and intrinsic dipole moment ......Zograf[2] showed there is a...
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
3.4 1/3 Laughlin state occupation profile . . . . . . . . . . . . . . . . . . . . . . 49
3.5 1/4 Laughlin state occupation profile . . . . . . . . . . . . . . . . . . . . . . 50
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
3.10 1/3 Laughlin state ∆N(k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.11 1/4 Laughlin state ∆N(k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.12 2/2 Moore-Read state ∆N(k) : 2020. . . . . . . . . . . . . . . . . . . . . . . . 55
3.13 2/4 Moore-Read state ∆N(k) : 0110. . . . . . . . . . . . . . . . . . . . . . . . 56
3.14 2/4 Moore-Read state ∆N(k) : 0101. . . . . . . . . . . . . . . . . . . . . . . . 56
3.15 1/2 Laughlin state dipole moment . . . . . . . . . . . . . . . . . . . . . . . . 58
3.16 1/3 Laughlin state dipole moment . . . . . . . . . . . . . . . . . . . . . . . . 58
3.17 1/4 Laughlin state dipole moment . . . . . . . . . . . . . . . . . . . . . . . . 59
3.18 2/2 Moore-Read state dipole moment : 2020. . . . . . . . . . . . . . . . . . . 59
ix
3.19 2/4 Moore-Read state dipole moment : 0110. . . . . . . . . . . . . . . . . . . 60
3.20 2/4 Moore-Read state dipole moment : 0101. . . . . . . . . . . . . . . . . . . 60
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)
1/3 Laughlin state occupation numbers
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)
1/4 Laughlin state occupation numbers
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)
1/3 Laughlin state Luttinger sum
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)
1/4 Laughlin state Luttinger sum
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
1/3 Laughlin state dipole moment per unit length
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
1/4 Laughlin state dipole moment per unit length
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
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