Laughlin’s wave function, plasma analogies
and the fractional quantum Hall effect
on infinite cylinders
vorgelegt von
Diplom-Mathematikerin
Sabine Jansen
aus Paris
Von der Fakult¨at II - Mathematik und Naturwissenschaften
der Technischen Universit¨at Berlin
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften
- Dr. rer. nat. -
genehmigte Dissertation
Promotionsausschuss:
Vorsitzender: Prof. Dr. A. Bovier
Berichter: Prof. Dr. R. Seiler
Berichter: Prof. Dr. H. Schulz-Baldes
Zus¨atzlicher Gutachter: Prof. Dr. E. H. Lieb
Tag der wissenschaftlichen Aussprache: 22. Juni 2007
Berlin 2007
D 83
Zusammenfassung
Gegenstand dieser Dissertation ist die Untersuchung von Laughlins Wellenfunktion auf einem
Zylinder. Die Ergebnisse sind sowohl f¨ur die Theorie des fraktionalen Quanten-Hall-Effekts
als auch f¨ur die klassische statische Mechanik geladener Teilchen in einem neutralisierenden
Hintergrund von Interesse.
Wir zeigen, dass Laughlins Wellenfunktion als “Quanten-Polymer-System” dargestellt
werden kann. Die L2-Norm ist die Zustandssumme eines Polymersystems mit translations-
invarianter Aktivit¨at. Die Aktivit¨at kann zu einer stabilen Aktivit¨at reskaliert werden. Das
Polymersystem steht wiederum in engem Zusammenhang mit einem Erneuerungsprozess.
Unter der Voraussetzung einer endlichen mittleren Wartezeit in diesem Prozess zeigen wir,
dass Laughlins Zustand einen wohldefinierten thermodynamischen Limes besitzt. Dabei
lassen wir die L¨ange des Zylinders bei konstanter Dichte und konstantem Radius gegen un-
endlich gehen. Der Grenzzustand ist invariant und mischend bzgl. bestimmter Translationen
l¨angs der Zylinderachse. Bei F¨ullfaktor 1/p ist die Periode gerade das p-fache der Periode
des gef¨ullten Landaubandes. F¨ur hinreichend d¨unne Zylinder zeigen wir, dass die Wartezeit
des Erneuerungsprozesses tats¨achlich endlich ist und die soeben erw¨ahnte Periode auch die
kleinste Periode ist.
Weiter definieren wir modifizierte Zylinder- und Torusfunktionen, die ebenfalls in Ver-
bindung zu Polymersystemen stehen. Die Definition verwendet Funktionen mit kompaktem
Tr¨ager. Bei hinreichend kleinem Tr¨ager entspricht die Zylinderfunktion einem Monomer-
Dimer-System, w¨ahrend die Torusfunktion einem Monomer-Dimer-System auf einem Ring,
m¨oglicherweise noch mit einem zus¨atzlichen langen Polymer, entspricht. Dieses Monomer-
Dimer-System ist explizit l¨osbar; Torus- und Zylinder- Monomer-Dimer- Funktionen sind im
Limes langer Zylinder ¨aquivalent.
Wir gehen auf die Bedeutung unserer Ergebnisse im Sinne von Laughlin’s Plasmaanalogie
ein. Wir zeigen, dass im Limes d¨unner Zylinder unsere Ergebnisse bzgl. der Normierungskon-
stanten bzw. freier Energien und bzgl. der Einteilchendichten konsistent mit bekannten
Ergebnissen ¨uber eindimensionale Plasmasysteme sind. Ferner zeigen wir, dass unsere Schran-
ken ¨uber Normierungskonstanten gut zu Asymptotiken der freien Energien breiter halbperi-
odischer Streifen passen.
Laughlins Funktion bei F¨ullfaktor 1/3 ist der exakte Grundzustand eines geeigneten
Hamiltonoperators. Wir weisen nach, dass Grundzust¨ande dieses Hamiltonoperators mit un-
endlich vielen Teilchen bei F¨ullfaktor 1/3 notwendigerweise eine Symmetriebrechung aufweisen,
wenn es eine Spektrall¨ucke ¨uber dem Grundzustand gibt.
Schließlich betrachten wir ein einfaches Modell f¨ur Ladungstransport auf Zylindern und
untersuchen den Zusammenhang zwischen der Periodizit¨at der Einteilchendichte und frak-
tionalem Ladungstransport.
iii
Abstract
We investigate Laughlin’s wave function on a cylinder. The results are of interest for the
fractional quantum Hall effect and for the classical statistical mechanics of charged particles
moving in a neutralizing background.
We show that Laughlin’s cylinder function is related to a “quantum polymer” system.
The L2-norm squared is a polymer partition function with translationally invariant activity.
The activity can be rescaled to a stable activity. The polymer system is related to a renewal
process. We show that if the renewal process has finite mean, Laughlin’s state on the cylinder
has a well-defined thermodynamic limit as the cylinder gets infinitely long, the radius being
kept fixed. The limiting state is invariant and mixing with respect to shifts along the axis. At
filling factor 1/p, the period is ptimes the period of the filled Landau band. On sufficiently
thin cylinders, we show that the associated renewal process has indeed finite mean and the
period mentioned above is the smallest period.
We define a class of modified torus and cylinder Laughlin-type wave functions. These
are still associated with polymer systems. The definition uses functions of compact support.
If the support is small, the cylinder function is associated with a monomer-dimer system,
and the torus function with a monomer-dimer system on a ring with possibly one additional
long polymer. The monomer-dimer case is solvable, and monomer-dimer cylinder and torus
functions are equivalent in the limit of long cylinders.
We interpret our result in view of Laughlin’s plasma analogy. We show that in the limit
of thin cylinders, our results on normalization / free energies and the one-particle density
are consistent with existing results on one-dimensional systems. We show that our bounds
on normalization constants are consistent with large strip asymptotics of the free energy of
a jellium system on a semiperiodic strip.
Laughlin’s function at filling factor 1/3 is the ground state of a suitable Hamiltonian.
We show that, on the cylinder, gapped infinite volume ground states of this Hamiltonian
necessarily display a kind of translational symmetry breaking.
Finally, we look at a simple model of bulk charge transport on the cylinder and examine
the relationship between our symmetry breaking result and fractional charge transport.
v
Contents
Zusammenfassung iii
Abstract v
Introduction 1
1 Laughlin’s wave function 5
1.1 Particles in a magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Laughlin-type wave functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Filled Landau level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Truncated interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5 Incompressibility and gaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.6 Plasma analogies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2 Thermodynamic limits 29
2.1 Laughlin’s cylinder wave function . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.1.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.1.2 Associated polymer system . . . . . . . . . . . . . . . . . . . . . . . . 40
2.1.3 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.1.4 Correlation functions and clustering . . . . . . . . . . . . . . . . . . . 57
2.1.5 Symmetry breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.2 Solvable models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
2.2.1 Generalized Laughlin wave functions . . . . . . . . . . . . . . . . . . . 77
2.2.2 Cylinder wave functions . . . . . . . . . . . . . . . . . . . . . . . . . . 79
2.2.3 Torus wave functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.3 Jellium tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.3.1 Interpolation between one- and two-dimensional jellium . . . . . . . . 88
2.3.2 Minimal electrically neutral components . . . . . . . . . . . . . . . . . 91
2.4 A Lieb-Schultz-Mattis type argument . . . . . . . . . . . . . . . . . . . . . . . 94
3 Charge transport 103
3.1 Laughlin’s argument(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
3.2 Charge transport on a cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.2.1 Periodic boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 108
3.2.2 Spectral boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 113
Conclusion 121
Bibliography 123
vii
Introduction
The classical Hall effect, named after its discoverer E. Hall, is a well-known electrodynamic
phenomenon: when a current IHflows through a thin sheet of conducting material in the
form of a bar, subject to a magnetic field perpendicular to the sheet, a potential difference
VHbetween opposite sides of the bar appears. According to classical electrodynamics, the
Hall conductance σH=IH/VHis a linear function of the ratio n/B of the areal charge carrier
density and magnetic field strength.
Hundred years after the discovery of the Hall effect, K. von Klitzing and his collaborators
[KDP80] observed experimentally that at low temperatures, the Hall conductance is not a
linear function of n/B; instead, it has plateaus where the conductance equals an integer
multiple of a fundamental unit. This is the integer quantum Hall effect (IQHE). Later,
plateaus at fractional multiples were found [TSG82], giving rise to the fractional quantum
Hall effect (FQHE).
Since their discovery in the early 1980s, the integer and fractional quantum Hall effects
have spurred a wealth of research among experimental, theoretical and mathematical physi-
cists. A large body of knowledge is now general consensus and can be found in one of the
many books and reviews on the quantum Hall effect, see e.g. [PG87, Yos98, DDPR04]. A
number of mathematical results are available (a good review can be found in [BvESB94]).
Some questions are still controversial while others have well-accepted, but yet unproved an-
swers. Meanwhile, the theory of the quantum Hall effect has reverberated on other fields,
including quantum computation [Day05], models for Mott insulators [LL04], spin systems
[KL87] and rotating Bose condensates [WGS98].
The integer quantum Hall effect is in general explained in terms of independent elec-
trons (see however [SG04]). In contrast, electron interactions are of utmost importance for
the understanding of the fractional effect. Consequently, the FQHE led to an intense research
on the nature of the ground state of the electron gas in FQHE samples. Early numerical
computations confronted theoreticians with an interesting phenomenon: the degeneracy of
the ground state depends on the geometry of the sample. On rectangles with quasiperi-
odic boundary conditions - i.e., tori - a degeneracy, associated to a translational symmetry
breaking, was observed, whereas explicitly isotropic geometries like the disk and sphere had
non-degenerate ground states (see [Hal85] and the references therein). The degeneracy of
FQHE ground states on tori was initially dismissed as physically irrelevant [Hal85]. Later,
the very fact that degeneracy depends on geometry has been considered as an intrinsic fea-
ture of FQHE systems, in close connection with the notion of topological order now popular
among physicists [WN90]. From a more mathematical point of view however, it is not very
suprising that boundary conditions affect the finite volume degeneracy, and the interesting
question is whether infinite volume ground states are degenerate or not.
Today, it is widely accepted that electrons at FQHE fractions form an “incompressible
quantum fluid” whose bulk properties are well described by Laughlin’s wave function [Lau83].
1
2INTRODUCTION
The wave function has been initially proposed to describe electrons on a disk, but since then
has been adapted to the cylinder [Tho84], torus [HR85b] and sphere [Hal83]. Laughlin in-
voked a plasma analogy to conclude that his disk function describes a homogeneous electron
gas. While the existence of the thermodynamic limit of the free energy for Coulomb systems
has been proved a while ago [LL72], analytical results pertaining to correlation functions are
more sparse, see the review by Brydges and Martin [BM99]. The homogeneity of the plasma
related to Laughlin’s function follows from numerical results; proofs are to our knowledge
still missing.
The present work is devoted to the study of Laughlin’s wave function on a cylinder. This
choice is motivated by the use of the cylinder geometry in another famous contribution by
Laughlin [Lau81], which has now come to be known as Laughlin’s argument. Let us briefly
sketch the main mathematical problem underlying this thesis. Consider the function ΨNof
Ncomplex variables zj=xj+iyjdefined through
ΨN(z1, .., zN) := Y
1≤j<k≤N
(ezj/R −ezk/R)pe−1
2PN
j=1 x2
j,
where Ris a positive number and pan odd integer. The function ΨNhas the period 2πR in
each yj. Thus we may consider that it describes a gas of Nelectrons moving on a cylinder
of radius R. The position of each particle is specified by a complex number z=x+iy;xis
the coordinate along the cylinder axis, yis an angular variable. ΨNis Laughlin’s cylinder
wave function. The density of the gas at position zis
ρN(z) = N
||ΨN||2
L2Z(R×[0,2πR])N−1|ΨN(z, x2+iy2, .., xN+iyN)|2dx2dy2..dxNdyN.
One can show that the density is a function of the coordinate xalone and decays exponentially
outside [0, pN/R]. Thus Laughlin’s cylinder function describes a gas of electrons living on a
finite cylinder. The quantity ν= 1/p is related to the average density and is called the filling
factor. At filling factor 1/p = 1 (referred to as the filled Landau level), the density can be
computed explicitly and is a sum of equally weighted Gaussians centered in integer multiples
of 1/R. The following conjecture pertaining to p > 1 is at the heart of the present work:
In the limit N→ ∞, well in the middle of the finite cylinder, the electronic
density approaches a function of the coordinate along the cylinder axis that is
periodic with minimal period p/R.
We will be interested not only in the one-particle density but also in other correlation func-
tions. We will look at the problem both in the FQHE context and from the point of view of
classical Coulomb systems.
The question of non-trivial periodicity on the cylinder is motivated by the combination
of the contributions of Laughlin to the integer [Lau81] and the fractional [Lau83] quantum
Hall effect. In [Lau81], Laughlin uses a the cylinder geometry and gauge invariance argu-
ments to explain the IQHE. It has been suggested [TW84] that a ground state degeneracy is
required on cylinders in order to reconcile Laughlin’s argument with fractionally quantized
Hall conductances; the simplest picture is a p-fold degeneracy associated to a p-fold transla-
tional symmetry breaking, as conjectured for Laughlin’s cylinder state. This rather heuristic
connection between ground state degeneracy is rigorously established for systems on tori. In
the Chern number approach, fractional quantization of the Hall conductance implies ground
INTRODUCTION 3
state degeneracy [KS90].
Our results are backed by a number of existing works. Rezayi and Haldane [RH94] in-
vestigated the dependence of the cylinder density ρNon the cylinder radius. They showed
that if the numbers of particles is finite and fixed, there is a well pronounced periodicity as
the radius Rgets small. Similar results have been obtained numerically for the Laughlin-type
wave functions on tori [SFL+05, LL04]. Let us mention here that the thin torus picture has
also been used for investigations on more complicated filling fractions, see e.g. [BK06].
Laughlin connected his wave function to the theory of classical Coulomb systems by ob-
serving that the modulus squared of the disk function is the Boltzmann weight of a classical
one-component plasma. This plasma analogy can be transferred to the cylinder geometry.
The symmetry breaking phenomenon is well-known at coupling constant Γ = 2, correspond-
ing to filled Landau level functions (i.e., Laughlin functions with filling factor 1/p = 1)
[CFS83]. Translational symmetry breaking has been proved to occur for one-dimensional
systems [Kun74, BL75]. Based on the Γ = 2 and one-dimensional results, [ˇ
SWK04] have
investigated numerically jellium tubes at coupling constants Γ = 2p. Unlike in [RH94], they
vary the number of particles as well as the radius, and find that a periodicity appears in the
limit of long jellium strips; again, the amplitude of oscillations is large when the strip is thin.
The periodicity of the charge density in jellium tubes is conjectured also in [AGL01].
In view of the plasma analogy, results on the L2norm of Laughlin’s wave function give
information on free energies. The asymptotic behavior of the jellium strip free energy as the
strip gets large has been examined by Forrester [For91], who also studied the plasma analogy
on tori [For06], motivated by the question of universal finite-size corrections of Coulomb free
energies (see also [TT04]).
Laughlin’s wave function is essentially the power of a Vandermonde determinant. It can
be expanded in Slater determinants of lowest Landau level basis functions; the expansion
coefficients are closely related to the expansion coefficients of powers of Vandermonde deter-
minants, written as a sum over monomials. Di Francesco et al. [FGIL94] and Dunne [Dun93]
investigate Laughlin’s disk wave function from this point of view. The relation with the FQHE
has also spurred combinatorial work on the Vandermonde determinant [STW94, KTW01].
The main point behind the proof of our results is a product rule for expansion coefficients
of Vandermonde determinants established in [FGIL94]. We use this product rule to represent
Laughlin’s wave function as a quantum polymer system: we choose this denomination because
of the analogy to the quantum dimer model, proposed by Rokhsar and Kivelson [RK88] as
an idealization of resonating valence bond states.
The thesis is organized as follows:
Chapter 1 gives a synthetic presentation of the Laughlin-type wave functions and some back-
ground material on particles in magnetic fields and Laughlin’s plasma analogy.
In Chapter 2, we prove the main results of our thesis: For sufficiently small cylinder
radius, the thermodynamic limit of Laughlin’s cylinder state exists and defines a periodic,
mixing state with minimal period p/R. The key ingredient is the representation of Laughlin’s
wave function as a quantum polymer. The L2-norm is a one-dimensional polymer partition
function. It satisfies a recurrence relation known in stochastics as a renewal equation. If
the associated renewal process has finite mean interarrival time, the (non-quantum) polymer
system has a well-defined thermodynamic limit, closely related to the thermodynamic limit
4INTRODUCTION
of Laughlin’s state.
We define a class of solvable models by replacing Gaussians with functions of compact
support. The cylinder and torus functions can still be represented as polymer systems. When
the support of the compact functions is small, the system is a pure monomer system. As
the support gets larger, the system becomes a monomer-dimer system and then - without
intermediary steps - a system where polymers of arbitrary length have non-vanishing activity.
The monomer and monomer-dimer cases define solvable models. When the cylinder system
is a monomer-dimer system, the modified torus function is a monomer-dimer system on a
ring, with possibly one long additional polymer covering the whole ring, and we can prove
the equivalence of cylinder and torus wave functions in this setting.
We translate our results into the jellium picture. This allows an interpretation of results
on the asymptotics of normalization constants in terms of free energies, and also sheds some
light on the renewal equation.
In the last section, we reexamine the results in the light of the characterization of Laugh-
lin’s wave function at filling factor 1/3 as an exact ground state of truncated interactions
[Hal83, PT85, TK85]. We adapt an argument by Koma [Kom04] that shows that the sym-
metry breaking is to be expected if the truncated interaction has a gapped ground state.
This result should be understood as a consistency check of different assumptions on FQHE
systems: the FQHE ground state at simple filling fractions is expected to be gapped and
incompressible. If the truncated interaction with Laughlin’s wave function as ground state
reproduces this feature, symmetry breaking necessarily follows.
In Chapter 3, we examine the charge transport on cylinders (and tori). We recall Laugh-
lin’s argument and the Chern number approach. We present the Chern number approach in
a way that allows a simple comparison with the cylinder setting and prove that the adiabatic
curvatures in absence of background interactions can be expressed in terms of the one-particle
density. We examine a simple model of charge transport on cylinders, closely following the
approach to the IQHE in [Ric00], Chapter 1.
Acknowledgements At this point I would like to express my gratitude to Prof. R. Seiler
and Prof. E. H. Lieb for their continous support and their invaluable help. This thesis would
not have been possible without their contribution. I thank Prof. H. Schulz-Baldes for useful
discussions on quantum Hall systems and for the numerous references he pointed out to
me, and R. Siegmund-Schultze for his help on renewal processes. I would also like to thank
Prof. Y. Avron for bringing [Osh00] to my attention and Prof. D. Brydges for pointing
out [TT04]. Furthermore, I am deeply indebted to colleagues, friends and family for their
support, patience and encouragement.
Chapter 1
Laughlin’s wave function
The object of this thesis is the investigation of Laughlin’s wave function on a cylinder. The
Laughlin wave function is essentially a power of a Vandermonde polynomial; it is a very
simple function, and in principle we could write down the function and the integrals defin-
ing the reduced correlation matrices, and formulate in a few lines the problem we wish to
solve as sketched in the Introduction. We take a different approach and give some back-
ground information, in more detail than strictly necessary for the proofs of the main results
of this thesis. The connection to the fractional quantum Hall effect is explained in Chapter 3.
Laughlin’s wave function was introduced as an approximate ground state for electrons in a
magnetic field. We start by recalling some basic facts on particles in a magnetic field (Section
1.1). Then we give the definition of Laughlin-type wave function in different geometries (Sec-
tion 1.2). The Laughlin wave functions are indexed by odd integers p, related to the average
density of the electron gas they describe. The function pertaining to p= 1 corresponds to
the filled Landau level; on a cylinder, it describes an electron gas with a periodicity along
the cylinder axis (Section 1.3). The Laughlin functions can be characterized as exact ground
states of truncated interactions, which we present in detail in Section 1.4. The existence of
gaps, of truncated interactions as well as “real” Hamiltonians, is related to the notion of in-
compressibility. Since the connection between these notions is often left implicit, we explain
this precisely in Section 1.5. Finally, we present the plasma analogy relating the modulus
squared of the wave functions considered to Boltzmann weights of classical one-component
plasma (Section 1.6).
Most of the material presented in this chapter summarizes well-known facts. The presentation
given here differs from the usual presentation by the simultaneous treatment of different
geometries and the addition of some details and proofs generally omitted in the physics
literature.
1.1 Particles in a magnetic field
The Laughlin wave functions are considered as good approximations to the ground state
of a two-dimensional, spin-polarized interacting electron gas in a magnetic field. Before we
introduce the wave functions, let us summarize some facts about the quantum mechanics of
charged particles in a magnetic field.
5
6CHAPTER 1. LAUGHLIN’S WAVE FUNCTION
In the following, the same symbol is used for functions and their multiplication operators;
the notation will not distinguish between a formal differential operator and the closed or self-
adjoint operators arising with suitable domain of definition. We do not specify the precise
nature of commutators.
Consider an electron of mand charge −e < 0 moving in the x, y-plane. A magnetic field
B=Bzezis applied perpendicular to the plane. Let A= (Ax, Ay) : R2→R2be a
differentiable function such that ∂xAy−∂yAx=Bz.Ais the vector potential generated by
the magnetic field. The motion of the electron is described by the Landau Hamiltonian
H=1
2m(p+eA)2=1
2m(−i~∂x+eAx)2+ (−i~∂y+eAy)2.(1.1)
With a suitable domain of definition, Hdefines a self-adjoint operator with discrete spectrum
σ(H) = {(n+1
2)~ω|n∈N0}, ω := e|Bz|
m.(1.2)
Note that Hdoes not include any spin contribution: the electrons are assumed to have all
their spins aligned (spin-polarized gas). The quantity ωis called cyclotron frequency. Other
useful quantities are the flux quantum φ0and the magnetic length defined through
φ0:= h
e=2π~
e, l := s~
e|Bz|.
The magnetic flux through a disk of radius lequals the flux quantum: |Bz|2πl2=φ0. The
Hamiltonian commutes with the center operators
cx:= x+1
mω (py+eAy), cy:= y−1
mω (px+eAx)
and the magnetic translations
t(a) = ei(a1cy−a2cx)/l2, a = (a1, a2)∈R2.
The magnetic translations act like usual translations except for a phase factor: (t(a)ψ)(z) =
eiγ(z;a)ψ(z−a). Here and in the following, we identify Cand R2in the usual way: z=
x+iy ≡(x, y). The magnetic translations satisfy
t(a+b) = e−i(a∧b)/2l2t(a)t(b), a ∧b=a1b2−a2b1
whence the commutation relation t(a)t(b) = ei(a∧b)/l2t(b)t(a). As a consequence, two mag-
netic translations t(a) and t(b) commute if and only if the area of the parallelogram spanned
by aand bis an integer multiple of 2πl2. Let b:= 1
l√2(cx−icy).Then [b, b∗] = 1. The lowest
Landau level N(H−~ω/2) has a complete orthonormal set Ωm,m∈N0such that
(c2
x+c2
y)Ωm= (2b∗b+ 1)l2Ωm= (2m+ 1)l2Ωm.(1.3)
The center operators are closely related to the coordinates x, y: for example, for all φin the
lowest Landau level, the following identities hold:
(φ, xφ) = (φ, cxφ),(φ, (x2+y2)φ) = (φ, (c2
x+c2
y+l2)φ),
(φ, yφ) = (φ, cyφ) (φ, ei(q1x+q2y)φ) = (φ, ei(q1cx+q2cy)e−q2l2/4φ).
1.1. PARTICLES IN A MAGNETIC FIELD 7
There are different possible choices for the vector potential A, giving rise to different, but
unitarily equivalent Hamiltonians. Two gauges are particularly useful: the symmetric gauge
and the Landau gauge. We give some details on these gauges. In the following, we suppose
that the magnetic field points downwards: Bz<0. This leads to the expression of lowest
Landau level wave functions in terms of holomorphic functions. The opposite choice Bz>0
would lead to antiholomorphic functions.
Symmetric gauge. In the symmetric gauge A= (−Bzy/2, Bzx/2), the lowest Landau
level is
N(H−~ω/2) = {ψ∈L2(R2)|ψ(z) = f(z)e−|z|2/4l2for some holomorphic f}
and the set of functions (Ωm)m∈N0defined through
Ωm(z) := 1
p2πl2m!(2l2)mzme−|z|2/4l2.(1.4)
is a complete orthonormal set of the lowest Landau level fulfilling (1.3). The magnetic
translations are t(a)ψ(z) = ei
2l2a∧zψ(z−a).
Landau gauge. In the Landau gauge A= (0, Bzx),
N(H−~ω/2) = {ψ∈L2(R2)|ψ(z) = f(z)e−x2/2l2for some holomorphic f}.
The magnetic translations become
t(a)ψ(z) = ei(a1y−1
2a1a2)/l2ψ(z−a).
For φ∈ S(R), let
(V φ)(x, y) := 1
p2π√πZ∞
−∞
φ(k)eiky/le−(x−kl)2/2l2dk.
Note that (V φ)(x, ·) is related to the Fourier transform of k7→ φ(k)e−(x−kl)2/2l2.Vextends
to a unitary map from L2(R) onto the lowest Landau level. Moreover,
V kV −1=cx/l, V i d
dk V−1=cy/l
and we can set Ωm:= V Fmwhere Fmare the Hermite functions. The set of functions (Ωm)
then forms a complete orthonormal system of eigenfunctions of c2
x+c2
y.
The previous considerations apply to infinite samples. In practice, we wish to model finite
samples. The most popular geometries are the disk, cylinder or semiperiodic strip, and torus,
which we now describe.
Disk. In principle, the motion of an electron in a disk BR={z∈C≡R2| |z| ≤ R}should
be modeled by a Hamiltonian in L2(BR) with suitable boundary conditions. However, it is
more common to look at the Hamiltonian Hin L2(R2) restricted to a spectral subspace of
c2
x+c2
y. The corresponding lowest Landau level is simply span{Ωm|0≤m≤M}where
M∈Nis the largest integer such that (2M+ 1)l2≤R2, and Ωmare as in (1.3).
8CHAPTER 1. LAUGHLIN’S WAVE FUNCTION
Infinite cylinder. Let R > 0. The expression (1.1) with periodic boundary conditions
t(2πRey)ψ=ψdefines a self-adjoint operator in L2(R×[0,2πR]) with spectrum as in (1.2).
The semiperiodic strip can be seen as a cylinder of radius R, the y-coordinate plays the
role of an angular variable. In the Landau gauge, the lowest Landau level has the complete
orthonormal set ψk, k ∈Zwith
ψk(z) := 1
p2πRl√πeiky/Re−(x−kγl)2/2l2
where γ:= l/R. They are eigenfunctions of the center operator cx. The eigenvalue gives the
center of the Gaussian part of ψ:cxψk=kγlψk. The magnetic translations act as follows:
t(nγlex)ψk=einγcy/lψk=ψk+n, t(aey)ψk=e−iacx/l2ψk=e−ika/Rψk
for all n∈Zand a∈R. The Hamiltonian commutes with all translations in the y-direction,
but only with a discrete set of translation in the x-direction: t(aex) preserves the boundary
condition t(2πRey)ψ=ψif and only if a·2πR ∈2πl2Z.
Torus. Let L > 0. We wish to define a Hamiltonian Hin L2([0, L]×[0,2πR]) through
(1.1) and the boundary conditions
t(Lex)ψ=ψ, t(2πRey)ψ=ψ.
This is possible if t(Lex) and t(2πRey) commute, i.e., L2πR =Nf·2πl2for some Nf∈N.
In this case, the flux through the surface is an integer multiple of the flux quantum φ0:
|Bz| · 2πRL =Nfφ0, whence the denomination number of fluxes of Nf. The resulting
Hamiltonian has the same spectrum as in (1.2). Each eigenvalue is Nf-fold degenerate. In
the Landau gauge, the lowest Landau level consists of functions ψ(x, y) = f(z)e−x2/2l2where
fis holomorphic and satisfies
f(z+ 2πR) = f(z), f(z+L) = eNfz/R+L2/2l2f(z).(1.5)
The lowest Landau level has the orthonormal basis ˜
ψk,k∈Z/NfZ, where
˜
ψk=∞
X
r=−∞
ψk+rNf.
The Hamiltonian commutes with magnetic translations t(aex+bey) for vectors (a, b) in the
lattice γlZ×2πR
Nf
Z, and
t(γlex)˜
ψk=eiγcy/l ˜
ψk=˜
ψk+1, t(2πR
Nf
ey)˜
ψk=e−i2πγcx/Nfl˜
ψk=e−i2πk/Nf˜
ψk
Finite cylinder. Let L > 0, R > 0 and a, b ∈R,a < b. In the spirit of the disk approach
described above, the lowest Landau level on a cylinder is often modeled by a restriction to
the spectral subspace of cxwhere a≤cx≤b, i.e. the considered space is
span{ψk|a≤kγl ≤b} ⊂ L2(R×[0,2πR]).(1.6)
A more satisfactory approach uses the spectral or chiral boundary conditions introduced in
[AANS98]. These are described in more detail in Chapter 3. Let us only mention the follow-
ing: the expression (1.1) together with spectral boundary conditions defines a Hamiltonian
in L2([a, b]×[0,2πR]) with a spectrum of the form
σ(H) = {en(kl2/R)|n∈N0, k ∈Z}
1.2. LAUGHLIN-TYPE WAVE FUNCTIONS 9
where e0(ρ)< .. < en(ρ)< ... are the eigenvalues of a norm-continuous family of operators
h(ρ) in L2([a, b]). The “ground state curve” e0(·) satisfies e0(ρ)≥~ω/2 with equality if and
only if a≤ρ≤b. Thus the ground state energy of His ~ω/2. The ground state has the
orthogonal basis ψkwhere k∈Zis such that a≤kγl ≤band the functions ψkare considered
as elements of L2([a, b]×[0,2πR]). Thus we recover essentially the space (1.6).
1.2 Laughlin-type wave functions
The previous section describes particles moving in a two-dimensional world subject to mag-
netic field, but otherwise free. The one-particle Hamiltonians have a discrete spectrum and
explicitly known eigenfunctions. The next step is to ask for the ground state of a gas of elec-
trons that interact between themselves and with a neutralizing background. We are looking
for the ground state of a Hamiltonian of the type
HN=
N
X
j=1
1
2m(pj+eA(zj))2+
N
X
j=1
W(zj) + X
1≤j<k≤N
V(zj−zk).
Vis essentially a 1/r-Coulomb interaction, adapted to the geometry (torus, cylinder..). In
the limit of large magnetic fields |Bz| → ∞, the gap ~ω∝ |Bz|separating the lowest Landau
level from the excited states of the one-particle Hamiltonian becomes large. Thus as an
approximation, one can look for ground states of Π⊗NHNΠ⊗Nwhere Π is the projection
onto the lowest Landau level. In the lowest Landau level, the kinetic energy gives a constant
N~ω/2, and we are left with the Hamiltonian
Π⊗NN
X
j=1
W(zj) + X
1≤j<k≤N
V(zj−zk)Π⊗N.
Laughlin [Lau83] proposed a lowest Landau level wave function in the disk geometry that is
commonly accepted as a good approximation to the ground state, as far as bulk properties
are concerned. Later, Thouless [Tho84] adapted Laughlin’s wave function to the cylinder
geometry, and Haldane and Rezayi defined torus Laughlin-type wave functions [HR85b].
There is also an analogue for a spherical geometry, defined by Haldane [Hal83], but we shall
not use this wave function here. Let us mention however that the spherical geometry is
advantageous for numerical computations [HR85a, FOC86].
Definition 1.1. Let l, R, L > 0and N, p ∈Nand 2πRL =pN2πl2. Set γ:= l/R. Let
ΨD
N(z1, .., zN) := 1
√N!Y
N≥j>k≥1
(zj−zk)pe−PN
j=1 |zj|2/4l2
ΨC
N(z1, .., zN) := 1
√N!Y
N≥j>k≥1
(ezj/R −ezk/R)pe−PN
j=1 x2
j/2l2e−1
2p2γ2PN−1
j=0 j2
(2πRl√π)N/2
ΨT
N(z1, .., zN) := Fcm(
N
X
j=1
zj)Y
1≤j<k≤N
θ1(i(zj−zk)
2R|iL
2πR)pe−1
2l2PN
j=1 x2
j
where θ1is Jacobi’s odd elliptic theta function and Fis in the p-dimensional subspace of
holomorphic functions such that
F(Z+i2πR) = (−1)p(N−1)F(Z), F(Z+L) = (−1)p(N−1)epZ/RepL/(2R)F(Z).(1.7)
10 CHAPTER 1. LAUGHLIN’S WAVE FUNCTION
These functions will be called disk, cylinder and torus Laughlin wave functions.
For N= 1, the products Q1≤j<k≤Nare empty. We follow the convention that the product
over an empty set is 1. The multiplicative factor in the definition of ΨC
Nis chosen so that
the normalization constant CN=||ΨC
N||2has properties derived in the next chapter. We
refer the reader to [AS72] for a list of properties and to [FK01] for a modern account on
theta functions. Let us recall some facts: let τ∈Cwith ℑτ > 0. Jacobi’s odd elliptic theta
function is defined as
θ1(z|τ) = −i∞
X
n=−∞
(−1)neiπτ(n−1/2)2e2i(n−1/2)z.
As a function of z, it is holomorphic, doubly quasiperiodic (see (1.8) below) with periods 1
and τ, and
θ1(z|τ) = 0 ⇔ ∃m, n ∈Z:z=m+nτ.
The zeroes of θ1(· | τ) are simple. The fact that the holomorphic solutions of (1.7) form a
p-dimensional subspace follows from the theory of theta functions ([FK01], Chapter 7).
The wave functions of the previous definition are symmetric if pis even, and antisymmetric
if pis odd. They lie in the disk, cylinder and torus lowest Landau level introduced in the
previous section:
Lemma 1.2. 1. ΨD
N∈ ⊗Nspan{Ωm|0≤m≤p(N−1)}, the lowest Landau level on a
disk with radius ((2p(N−1) + 1))1/2l(symmetric gauge).
2. ΨC
N∈ ⊗Nspan{ψk|0≤k≤p(N−1)}, the spectral subspace 0≤cx≤p(N−1)γl in
the lowest Landau level on a cylinder (Landau gauge).
3. ΨN↾([a, b]×[0,2πR])Nis made up of wave functions in the ground state of the
Landau Hamiltonian in L2([a, b]×[0,2πR]) with spectral boundary conditions, provided
a≤0< p(N−1)γl ≤b(Landau gauge).
4. ΨT
N∈ ⊗Nspan{˜
ψk|k∈Z/pNZ}, the lowest Landau level on the torus R/pNγlZ×
R/2πRZ(Landau gauge).
Proof. 1. Let
VN(z1, .., zN) := detzk−1
j1≤j,k≤N=Y
N≥j>k≥1
(zj−zk)
be the Vandermonde determinant. Its p-th power can be expanded in monomials
VN(z1, .., zN)p=X
m1,..,mN
bN(m1, .., mN)zm1
1..zmN
N.
The Vandermonde determinant is a polynomial of maximal degree N−1 in each variable, thus
Vp
Nis a polynomial of maximal degree p(N−1) in each variable and only 0 ≤m1, .., mN≤
p(N−1) contributes to the sum above. The disk Laughlin function is Vp
Nmultiplied by
a Gaussian factor. The basis functions Ωmare monomials zmtimes the Gaussian factor
e−|z|2/4l2. Thus ΨD
Nis a sum of tensor products of functions Ωmwith 0 ≤m≤p(N−1).
1.3. FILLED LANDAU LEVEL 11
2. and 3. The cylinder function is VN(ez1/R, .., ezN/R)ptimes a Gaussian factor. Using
ψk(z) = (2πRl√π)1/2ekz/Re−k2γ2/2e−x2/2l2, we get
ΨC
N(z1, .., zN) = 1
√N!VN(ez1/R, .., ezM/R)pe−PN
j=1 x2
j/2l2e−p2γ2PN−1
j=0 j2
(2πRl√π)N/2
=1
√N!X
0≤m1,..,mN≤p(N−1)
aN(m1, .., mN)ψm1⊗.. ⊗ψmN
where
aN(m1, .., mN) := bN(m1, .., mN)eγ2PN
j=1(m2
j−p2(j−1)2)/2.
This proves 2.. The functions ψkare in the ground state of the Hamiltonian with spectral
boundary conditions in L2([a, b]×[0,2πR]) if a≤kγl ≤b, whence also 3.
4. The N-fold tensor product of the torus Landau level is the space of functions of the form
f(z1, .., zN) exp(−PN
j=1 x2
j/2l2) where fis holomorphic and satisfies the quasiperiodicity
condition (1.5) in each complex variable zj,Nf=pN. Theta functions are holomorphic,
thus it remains to prove that
f(z1, .., zN) := F(
N
X
j=1
zj)Y
1≤j<k≤N
θ1(i(zj−zk)
2R|iL
2πR)pe−1
2l2PN
j=1 x2
j
is quasiperiodic. This is checked using (1.7) and
θ1(z+π|τ) = −θ1(z|τ) = θ1(−z|τ), θ1(z+πτ |τ) = −e−2iz−iπτ θ1(z|τ).(1.8)
Filling factor. In view of the previous lemma, the Laughlin wave functions describe N-
particle states living on samples of size essentially pN ·2πl2. This leads to an average density
1/(p·2πl2) (see also Corollary 2.15). The one-particle lowest Landau level of e.g. the torus
space has dimension pN. Therefore the case p= 1 is called the filled Landau level. It
corresponds to an average density 1/(2πl2) (see also the next section). The ratio 1/p of the
average density and the filled Landau level density is called the filling factor.
1.3 Filled Landau level
The filled Landau level function (p= 1), e.g., on the cylinder, is
ΨC
N=ψ0∧ψ1∧.. ∧ψN−1.
The one-particle density and other correlation functions have simple expressions. The n-point
reduced density matrix associated to an N-particle wave function ΨN∈L2(ΩN) is
ρN(z′
1, .., z′
n;z1, .., zn) = N!
(N−n)! ZΩN−n
ΨN(z1, .., zn, wn+1, .., wN)
ΨN(z′
1, .., z′
n, wl+1, .., wN)dw/||ΨN||2.(1.9)
12 CHAPTER 1. LAUGHLIN’S WAVE FUNCTION
For ΨC
Nas above, it can be written as
ρN(z′
1, .., z′
n;z1, .., zn) = n!X
0≤m1<..<mn≤N−1
ψm1∧.. ∧ψml(z′
1, .., z′
n)ψm1∧.. ∧ψml(z1, .., zn).
If we shift the function on the cylinder so that the origin is in the middle of the cylinder, the
sum ranges from −Mto N−1−Mwhere Mis an integer close to N/2. Taking then the
limit N→ ∞, we get the expressions for the filled Landau level reduced matrices
ρ(z′
1, .., z′
n;z1, .., zn) = n!X
m1,..,mn∈Z:
m1<..<mn
ψm1∧.. ∧ψml(z′
1, .., z′
n)ψm1∧.. ∧ψml(z1, .., zn).
The one-particle density is
ρ(z) := ρ(z;z) = ∞
X
m=−∞|ψm|2(z) = 1
2πRl√πX
m∈Z
e−(x−ml2/R)2/l2.
Using Poisson’s summation formula, this can be rewritten as
ρ(z) = 1
2πl2(1 + 2 ∞
X
n=1
e−n2π2R2/l2cos(2πnxR/l2)).
In particular, the one-particle density is independent of the coordinate yaround the cylinder
and is periodic in the direction of the cylinder axis, of period l2/R =γl. The amplitude of the
oscillations is large when the radius of the cylinder is small. The average of the one-particle
density is 1/(2πl2), which is the value of the constant one-particle density of the filled Landau
level on the infinite plane:
ρplane(z) = ∞
X
n=0 |Ωn|2(z) = 1
2πl2
∞
X
n=1
|z|2n
n!(2l2)ne−|z|2/2l2=1
2πl2.
Note that as the radius Rgoes to infinity at fixed magnetic length l, we recover ρplane from
the cylinder density ρ.
The aim of the next chapter is to prove a similar statement for the cylinder function at
general filling factor 1/p, namely, that the one-particle density is a periodic function of the
coordinate along the axis. The period is just ptimes the filled Landau level period γl. The
amplitude of the oscillations is large when the radius of the cylinder is small. The average
density is 1/(p·2πl2).
1.4 Truncated interactions
Laughlin’s wave functions are widely accepted as good approximations to the ground state
of the “true” lowest Landau level projected Hamiltonian Π⊗NHNΠ⊗N. An additional inter-
esting feature is that Laughlin’s state is the exact ground state of a modified Hamiltonian,
as observed in [Hal83, PT85, TK85].
This observation is interesting for several reasons. First, it allows an indirect approach to
the question of how far away Laughlin’s state is from the exact ground state: in the spirit of
numerical backward error analysis, we can compare Hamiltonians instead of ground states. It
1.4. TRUNCATED INTERACTIONS 13
has been suggested to treat the difference between the lowest Landau level projected Coulomb
interaction and the truncated interaction as a small perturbation [HR85a].
Second, we can consider the truncated interaction as a toy Hamiltonian for the FQHE and ask
whether it reproduces features such as incompressibility and existence of gaps. The modified
Hamiltonian has been used also as a toy Hamiltonian for investigations on Mott insulators
[LL04]. Finally, the characterization of the Laughlin state as an exact ground state will be
used in section 2.4 to investigate the equivalence of the cylinder and torus functions.
There are two a priori distinct definitions of truncated interactions. Haldane (see his contri-
bution in [PG87]) defines the interaction in terms of projections onto eigenfunctions of the
relative angular momentum, where as [TK85, PT85] characterize Laughlin’s wave function
as a ground state of projected ∆jδ-interactions. Both approaches give the same result, as we
will see in this section.
The Laughlin wave functions at filling factor 1/p are characterized by the order of their ze-
roes: as two particles get close zj−zk→0, the wave function is O((zj−zk)p). This makes
Laughlin’s wave function an exact ground state of ∆jδ-interactions [TK85, PT85] projected
to the lowest Landau level. These lowest Landau level projected interactions define bounded
operators. In the lowest Landau level on infinite samples (or in the disk geometry), the
bounded operators are projections on eigenfunctions of the separation of the particles, mea-
sured through the center operators.
The definition of the operator in the disk geometry follows Haldane’s presentation in [PG87].
For the cylinder, the operators are presented in a slightly different way in [RH94]. The second
quantized expressions for the torus and cylinder that can be found in [SFL+05] and [LL04]
will be recovered from our definition in Corollary 1.7.
Let H ⊂ L2(R2) be the lowest Landau level in the symmetric gauge. The two-particle space
H⊗H has a complete orthonormal set
ψn,m(z1, z2) = cn,m(z1+z2)n(z1−z2)me−(|z1|2+|z2|2)/4l2, m, n ∈N0
(with cn,m suitable normalization constants). Let
M=1
4l2(c(1)
x−c(2)
x)2+ (c(1)
y−c(2)
y)2−1
2.
Here, we use the notation c(1)
x=cx⊗1,c(2)
x=1⊗cxand similarly for c(1)
y, c(2)
y. Then
Mψn,m =mψn,m. Thus σ(M) = N0. The operator Mis sometimes called relative angular
momentum.
Definition 1.3. Let ψn,˜
ψnbe the cylinder and torus lowest Landau level basis functions,
and γ=l/R. For m∈N0, let Fmbe the m-th Hermite function (eigenfunction of −∂2
k+k2)
and let Pm,PC
m,PT
mbe the following two-particle operators acting respectively in the lowest
Landau level on the plane, cylinder and torus:
•Pm=PD
mis the projection on the eigenspace N(M−m)of M.
•PC
mis the bounded operator determined through
(ψk1⊗ψk2, PC
mψn1⊗ψn2) = γ√2δk1+k2,n1+n2Fm((k1−k2)γ
√2)Fm((n1−n2)γ
√2).(1.10)
14 CHAPTER 1. LAUGHLIN’S WAVE FUNCTION
•PT
mis the unique bounded operator such that
(˜
ψk1⊗˜
ψk2, PT
m˜
ψn1⊗˜
ψn2)
=γ√2X
r∈Z
δk1+k2,n1+n2+rNfX
s,t∈Z:
s−t=rmod 2
Fm((k1−k2+sNf)γ
√2)Fm((n1−n2+tNf)γ
√2).
(1.11)
Remark. Pmis bounded because it is a projection. The boundedness of (1.11) follows from
the fact that the lowest Landau level on a torus is finite-dimensional. The decay properties
of the Hermite functions can be used to show that (1.10) defines a bounded operator of norm
||PC
m|| ≤ √2γX
k∈Z
Fm(kγ
√2)2;
see also (1.19) below.
The operators PR
m,R=D, C, T are associated to the projection of the delta function inter-
action 4πl2
j!e|z1−z2|2/4l2l2j∆j
z1−z2δ(z1−z2)
to the lowest Landau level. More precisely:
Lemma 1.4. Let ∆z1−z2:= ( ∂
∂z1−∂
∂z2)( ∂
∂¯z1−∂
∂¯z2)be the Laplacian with respect to z1−z2.
Let HDbe the lowest Landau level in the symmetric gauge, HCthe lowest Landau level on
the cylinder and torus. Let D=R2, C =R×[0,2πR], T = [0, L]×[0,2πR].Then for
R∈ {D, C, T}and f, g ∈ HR:
(f, P R
jg) = 4πl2
j!ZRl2j∆j
z1−z2(e|z1−z2|2/4l2¯
fg)(z, z)dxdy. (1.12)
Proof. Assume without loss of generality l= 1. For z1, z2∈Clet Z:= (z1+z2)/√2,
z:= (z1−z2)/√2.
We start with the symmetric gauge. First, remark that the right-hand side of (1.12) defines
a bounded sesquilinear form. To see this, consider first
φ(z) = f(z)e−|z|4/2, ψ(z) = g(z)e−|z|2/4
two functions in the lowest Landau level. Thus fand gare holomorphic. By a classical result
on Bergman spaces, for r > 0 and m∈N0there exists a constant cm>0 such that
|∂mf
∂zm(0)| ≤ cmZ|z|<r |f(z)|2dxdy ≤cmer2/4||φ||.
and similarly for g, ψ. Thus
|∆j¯
φψe|z|2/2(0)|=|(4 ∂2
∂z∂¯z)j¯
fg(0)| ≤ 4jer2/2c2
j||φ||||ψ||.
The boundedness of the right-hand side of (1.12) can be inferred from this. Since the left-
hand side is bounded too, it is enough to check (1.12) for functions f, g in a suitable complete
1.4. TRUNCATED INTERACTIONS 15
orthonormal system. Let Ωm,m∈N0be the functions defined in (1.4). Let M, N, m, n ∈N0
and
f(z1, z2) := ΩM(Z)Ωm(z), g(z1, z2) := ΩN(Z)Ωn(z).(1.13)
Then
∆j
z1−z2e|z1−z2|2/4(¯
fg)(w, w)
= (¯
ΩMΩN)(√2w)1
2π√2m+nm!n!∆j
z1−z2¯zmzn(0)
= (¯
ΩMΩN)(√2w)1
2π2m+n√m!n!)∆j
z1−z2(z1−z2)m(z1−z2)n(0)
= (¯
ΩMΩN)(√2w)1
2π2m+n√m!n!δm,jδn,jm!n!4j
=¯
ΩMΩN(√2w)j!
2πδm,jδn,j.
Since ZR2
¯
ΩMΩN(√2x, √2y)dxdy =1
2(ΩM,ΩN) = 1
2δM,m
we obtain Z∆j
z1−z2e|z1−z2|2/4¯
fg)(w, w) = j!
4π(f, P D
jg).
Functions of the form (1.13) form a complete orthonormal set in the lowest Landau level,
thus using the boundedness of PD
jwe get the desired result.
Now let us turn to a cylinder of radius R. Let γ=l/R = 1/R. Again, it is enough
to check (1.12) for functions in a complete orthonormal system. The lowest Landau level
basis functions are ψγ
n(z) = (γ/(2π√π))1/2exp(iny/R) exp(−(x−nγ)2/2), n∈Z. A simple
computation gives
ψn1(z1)ψn2(z2) = √2ψγ/√2
n1+n2(Z)ψγ/√2
n1−n2(z).
Hence for f:= ψm1⊗ψm2,g:= ψn1⊗ψn2
∆j
z1−z2e|z1−z2|2/4(¯
fg)(w, w)
= 2( ¯
ψγ/√2
m1+m2ψγ/√2
n1+n2)(√2w)1
2j∆je|z|2/2¯
ψγ/√2
m1−m2ψγ/√2
n1−n2(0) (1.14)
The Hermite polynomials Hn(x) satisfy the following generating function identity:
e2xt−t2=∞
X
n=0
Hn(x)tn
n!.
From this we deduce
¯
ψγ
k(z)ψγ
n(z)e|z|2/2=γ
2π√πe−γ2(k2+n2)/2eγk¯zeγnze−x2e(x2+y2)/2
=γ
2π√πe−γ2(k2+n2)/2eγk¯z−¯z2/4eγnz−z2/4
=γ
2π√πe−γ2(k2+n2)/2∞
X
r,s=0
Hr(kγ)Hs(nγ)
2r+sr!s!¯zrzs.
16 CHAPTER 1. LAUGHLIN’S WAVE FUNCTION
Since ∆ = 4∂¯
∂, we get
∆jψγ
k(z)ψγ
n(z)e|z|2/2(0) = γ
2π√πe−γ2(k2+n2)/2Hj(kγ)Hj(nγ) = 2jj!γ
2πFj(kγ)Fj(nγ)
(1.15)
where Fn(x) = (2nn!√π)−1/2Hn(x)e−x2/2is the n-th Hermite function. Combining (1.14)
and (1.15), we obtain
∆j
z1−z2e|z1−z2|2/4(¯
fg)(w, w) = j!γ
π√2(¯
ψγ/√2
m1+m2ψγ/√2
n1+n2)(√2w)Fj(kγ
√2)Fj(nγ
√2).
But now
Z2π/γ
0
dy Z∞
−∞
dx(¯
ψγ/√2
m1+m2ψγ/√2
n1+n2)(√2x, √2y)
=1
2Z2π√2/γ
0
dy Z∞
−∞
dx(¯
ψγ/√2
m1+m2ψγ/√2
n1+n2)(x, y) = 1
2δm1+m2,n1+n2.
Thus we get
Z∆j
z1−z2e|z1−z2|2/4(¯
fg)(w, w) = j!γ
2π√2δm1+m2,n1+n2Fj(kγ
√2)Fj(nγ
√2) = j!
4π(f, P C
jg).
Now let us turn to the torus. Let Tγ= [0, Nfγ]×[0,2π/γ], Nf∈N. The lowest Landau
level basis functions ˜
ψmare related to the cylinder functions through ˜
ψm=Pk∈Zψm+kNf.
If f, g are two smooth functions, let
qj(f, g) := ZT∆j
z1−z2e|z1−z2|2/4(¯
fg)(w, w)dxdy.
Mimicking the reasoning done for the cylinder case, we get
qj(ψk1⊗ψk2, ψn1⊗ψn2) = j!γ
2π√2δK,N 1
√πZ√2Nfγ
0
e−(x−Kγ/√2)2dxFj(kγ
√2)Fj(nγ
√2)
(1.16)
where we used capital letters for sums of indices and small letters for differences (K=
k1+k2, k =k1−k2). We are interested in
qj(˜
ψk1⊗˜
ψk2,˜
ψn1⊗˜
ψn2) = X
s1,s2,t1,t2∈Z
qj(ψk1+s1Nf⊗ψk2+s2Nf, ψn1+t1Nf⊗ψn2+t2Nf) (1.17)
In view of (1.16), the sum vanishes unless K+SNf=N+TNffor some S, T ∈Z. Suppose
that K−N=rNfwith r∈Z. Note that
X
s1,s2,t1,t2∈Z
δs1+s2−t1−t2,ra(s1+s2)b(s1−s2)b(t1−t2)
=X
S,s∈Z:
S=smod 2 X
T,t∈Z:
T=tmod 2
δS−T,ra(S)b(s)b(t)
=X
S∈Z
a(S)X
s=Smod 2
b(s)X
t=S+rmod 2
b(t)
=
1
X
α=0X
S∈Z
a(2S+α)X
s∈Z
b(2s+α)X
t∈Z
b(2t+r+α)
1.4. TRUNCATED INTERACTIONS 17
Combining this with (1.16) and (1.17), we get
qj(f, g) = j!γ
2π√2X
s,t∈Z:
s−t=rmod 2
Fj((k+sNf)γ
√2)Fj((n+tNf)γ
√2).
Lemma 1.5. The operators PR
jare bounded, self-adjoint and non-negative. Let m∈N0and
V0, V1, .., Vm>0. Let ψbe in the disk, cylinder, or torus lowest Landau level depending on
R∈ {D, C, T}. Then
ψ∈ N(
m
X
j=0
VjPR
j)⇔ ∀Z:ψ(Z+z, Z −z) =
z→0O(|z|m+1).(1.18)
Proof. The operators are bounded by definition (see also the remark following Definition
1.3), the self-adjointness is obvious for PC
m. For PC
mand PT
mthe symmetry follows from
Fj(−x) = (−1)jFj(x) or from (1.12). As a projection, PD
mis obviously non-negative. To see
that PC
mis positive, let
U:HC⊗HC→l2(Z2)≡l2(Z)⊗l2(Z2), ψn1⊗ψn27→ (δn1+n2,mδn1−n2,n)m,n∈Z.
Uis only a partial isometry since n1+n2=n1−n2mod 2. Then
UPC
j=1⊗(γ√2|Fγ
jihFγ
j|)U, Fγ
j:= (Fj(nγ
√2))n∈Z.(1.19)
This implies the positivity of PC
j. To treat the torus case, let ˆ
δn,s take the value 1 if n=s
mod 2Nfand 0 else, and let ˆen= (ˆ
δn,s)s∈Z. Define
˜
U:HT⊗HT→l2((Z/2NfZ)2)≡l2(Z/2NfZ)⊗2(1.20)
˜
ψn1⊗˜
ψn27→ 1
√2(ˆen1+n2⊗ˆen1−n2+ ˆen1+n2+Nf⊗ˆen1−n2+Nf).(1.21)
Again, ˜
Uis a partial isometry. Let
Gj:= (X
k∈Z
Fj((n+ 2kNf)γ
√2))n∈Z, Hj:= (Gj(n+Nf))n∈Z.
Then ˜
UPT
j=γ√21⊗(|GjihGj|+|HjihHj|)˜
U,
whence the positivity of PT
j. Thus the operators PR
jare positive and ψ∈ N(Pm
j=0 VjPj) if
and only if (ψ, PR
jψ) = 0 for j= 0,1.., m. Let ψ(z1, z2) be in the lowest Landau level of two
particles and let l= 1. Write ψ(z1, z2) = f(z1+z2
2, z1−z2)e−|z1−z2|2/8so that
(ψ, PR
jψ) = 4π
j!ZR
(∆j
z|f|2)(Z, 0)dXdY
where the Laplacian ∆zacts on the second variable of f(Z, z). Note that ψ(Z+z, Z −z) =
O(|z|m+1) if and only if f(Z, z) = O(|z|m+1). Since functions in the lowest Landau level are
holomorphic up to a factor of the type exp(−ax2−by2), we can write for sufficiently small z
f(Z, z) = X
m,n∈N0
am,n(Z)zm¯zn,|f(Z, z)|2=X
m,n∈N0
bm,n(Z)zm¯zn
18 CHAPTER 1. LAUGHLIN’S WAVE FUNCTION
and we get
(ψ, PR
jψ) = 4j+1j!πZR
bj,j(Z)dXdY.
We proceed by induction on m. Suppose first (ψ, P R
0ψ) = 0. Since b0,0(Z) = |f(Z, 0)|2we get
f(Z, 0) = 0 for all Zand thus f(Z, z) = O(|z|). Now suppose (ψ, PR
jψ) = 0 for j= 0, .., m+1
and by induction f(Z, z) = O(|z|m+1). Then
f(Z, z) =
m+1−k
X
k=0
ak,m+1−kzk¯zm+1−k+O(|z|m+2), bm+1,m+1(Z) =
m+1
X
k=0 |ak,m+1−k|2.
Thus (ψ, PR
m+1ψ) = 0 implies ak,m+1−k= 0 for k= 0, .., m + 1.
Now we can characterize the Laughlin-type wave functions as ground states of suitable Hamil-
tonians.
Proposition 1.6. Let p∈N,V0, .., Vp−1>0and N∈N,N≥2. Define the Hilbert spaces
HD
N:= ⊗Nspan{Ωj|0≤j≤p(N−1)}
HC
N:= ⊗Nspan{ψj|0≤j≤p(N−1)}
HT
N:= ⊗Nspan{˜
ψj|j∈Z/NfZ}.
Let ΠD
Nbe the projection onto HD
Nin ⊗Nspan{Ωn|n∈N0}, and ΠC
Nthe projection onto
HC
Nin ⊗Nspan{ψj|j∈Z}. Let ΠT
Nbe the identity in B(HT
N). For R∈ {D, C, T}let HR
N
be the N-particle Hamiltonian in B(HR
N)defined as
HR
N= ΠR
NX
1≤j<k≤N
(V0PR,jk
0+.. +Vp−1PR,jk
p−1) ΠR
N
where Pjk refers to Pacting on the j-th and k-th variables.Then HR
N,HC
Nhave unique ground
states ΨD
N,ΨC
N.HT
Nhas a p-fold degenerate ground state, the space of ΨT
Nfunctions.
Proof. By Lemma 1.5, HR
Nis positive and a function Ψ ∈ HR
Nis in the null space of HR
Nif
and only if for any j < k, Ψ(z1, .., zN) = O(|zj−zk|p) as |zj−zk|goes to zero, the sum
zj+zkand the variables zn, n 6=j, k being fixed. From this we see that the Laughlin-type
wave functions are indeed ground states, and it remains to show that these are the only ones.
The argument for the disk is simple (see Haldane’s article in [PG87]): let Ψ ∈ HD
N. Then
there is a a polynomial of degree ≤p(N−1) in each variable such that
Ψ(z1, .., zN) = P(z1, .., zN)e−Pj|zj|2/4l2.
If Ψ(z1, .., zN) = O(|zj−zk|p) as zj−zk→0, Pis a multiple of (zj−zk)p. Thus if
Ψ∈ N(HD
N), there is a polynomial Qso that
P(z1, .., zN) = Q(z1, .., zN)Y
N≥j>k≥1
(zj−zk)p.
Since Phas maximal degree p(N−1) in each variable, Qis a constant and Ψ is a multiple of
ΨD
N. The argument for the cylinder is similar: note that if Ψ is in HC
N, there is a polynomial
Pof degree ≤p(n−1) in each variable such that
Ψ(z1, .., zN) = PN(ez1/R, .., ezN/R)e−1
2l2PN
j=1 x2
j.
1.4. TRUNCATED INTERACTIONS 19
If Ψ ∈ HT
N, there is a holomorphic function fwith the quasiperiodicity
f(z1, .., zj+ 2πRi, .., zN) = f(z1, .., zN),
f(z1, .., zj+L, .., zN) = eNfzj/ReN2
fl2/2R2f(z1, .., zN)
in each variable zjsuch that Ψ(z) = f(z1, .., zN)e−1
2l2PN
j=1 x2
j.If Ψ ∈ N(HT
N), there is a
holomorphic function gsuch that
f(z1, .., zN) = g(z1, .., zN)Y
1≤j<k≤N
θ1(izj−zk
2R|iL
2πR)p.(1.22)
The function gsatisfies for each j∈ {1, .., N}
g(z1, ., zj+i2πR, .., zN) = (−1)p(N−1)g(z1, .., zN)
g(z1, ., zj+L, .., zN) = (−1)p(N−1)epPN
k=1 zk/RepL/2Rg(z1, .., zN).(1.23)
Define a function Gthrough g(z1, .., zN) = G(PN
j=1 zj, z2−z1, .., zN−z1); equivalently,
G(Z, z′
2, .., z′
N) = g(z1, z1+z′
2, .., z1+z′
N), z1:= 1
N(Z−
N
X
j=2
z′
j).
Gis holomorphic. It follows from (1.23) that for b1, .., bN∈Z,
g(z1+b1L, .., zN+bNL) = (−1)p(N−1)BepBZ/ReLB2/2Rg(z1, .., zN), B =
N
X
j=1
bj.
This can be used to show that Gis doubly periodic in z′
jwith periods iN2πR and NL in
each variable z′
j. Since it is holomorphic, this means that G(Z, z′
2, .., z′
N) = F(Z) for some
holomorphic function F. Thus g(z1, .., zN) = F(z1+.. +zN). Together with (1.22) and
(1.23), this proves that Ψ is a torus Laughlin wave function ΨT
Nas in Definition 1.1.
To summarize, we have shown that the lowest Landau level projections of ∆jδ-interactions
are the bounded operators defined in the beginning of this section, The Laughlin-type wave
functions can be characterized as the ground states of Hamiltonians defined with these trun-
cated interactions. This holds for even pas well as for odd pand requires no assumption
on the parity of the wave function. Some simplifications are possible when we do assume
something on the parity. We exemplify this for Laughlin’s 1/3-function and antisymmetric
functions.
Second quantized version. If we restrict to antisymmetric functions, we can express the
various Hamiltonians used in terms of fermionic creation and annihilation operators. Let us
recall their definition. For Ω ⊂R2and H ⊂ L2(Ω), let F=L∞
N=0 ∧NHbe the associated
fermionic Fock space. For r, s ∈N, the wedge product of f∈ ∧r(H)⊂L2(Ωr) and g∈ ∧sH
is the element of ∧r+sHdetermined through
(f∧g)(z1, .., zr+s) = r(r+s)!
r!s!Ar+s(f⊗g)(z1, .., zr+s)
=1
pr!s!(r+s)! X
π∈Sr+s
sgn(π)f(zπ(1), .., zπ(r))g(zπ(r+1), .., zπ(r+s)).
20 CHAPTER 1. LAUGHLIN’S WAVE FUNCTION
where Anis the antisymmetrisation operator in L2(Ωn)≡ H⊗n. The wedge product of
f∈ ∧0H ≡ Cwith g∈ ∧sHis f·g. The map thus defined extends uniquely to a bounded
bilinear map ∧:F ×F → F. For f∈ H, the creation operator c∗(f)∈ B(H) is the operator
ψ7→ f∧ψ, and the annihilation operator c(f) is its adjoint. The truncated interaction has
a simple expression [LL04, SFL+05]:
Corollary 1.7. Let p= 3,N∈N\{1}and R > 0. Let ψn,˜
ψnbe the cylinder and torus low-
est Landau level basis functions, and cn=c(ψn),˜cn=c(˜
ψn). Let f(n) = ne−n2γ2/4,˜
f(n) =
Pk∈Zf(n+ 6kN).The N-particle Laughlin states at filling factor 1/3on the cylinder and
torus can be characterized as the ground states of
X
0≤k1,k2,n1,n2≤3N−3
δk1+k2,n1+n2f(k1−k2)f(n1−n2)c∗
k2c∗
k1cn1cn2,(1.24)
and X
R,x,y∈Z/6NZ
R=x=ymod 2
˜
f(x)˜
f(y)˜c∗
R−x
2
˜c∗
R+x
2
˜cR+y
2˜cR−y
2(1.25)
acting in ∧Nspan{ψn|0≤n≤3N−3}resp. ∧N{˜
ψn|n∈Z/(3N)Z}.
Sketch of proof. (1.24) is obviously a second-quantized version of PC
1projected to ∧2span{ψk|
0≤k≤3N−3}, up to a positive multiplicative constant. The projection of PC
2and PC
0to
antisymmetric functions vanish. Applying Proposition 1.6 with p= 3, we see that ΨL
nis the
unique ground state of (1.24).
The argument for the torus is similar. The computation needed to check that (1.25) is the
second-quantized version of PT
1uses essentially the following: if (R, x)∈(Z/6NZ)2is such
that Rand xare both even or odd (this does not depend on the choice of representants
because 6Nis even), ((R+x)/2,(R−x)/2) is well-defined as an element of (Z/3NZ)2.
Furthermore if (k1, k2)∈(Z/3NZ)2,
(R+x
2=k1mod 3N
R−x
2=k2mod 3N⇔(k1+k2=Rmod 6N,
k1−k2=xmod 6Nor (k1+k2=R+ 3Nmod 6N,
k1−k2=x+ 3Nmod 6N.
Comparison of truncated interaction and projected Coulomb interactions. Let
V(|z1−z2|) be a rotationally invariant interaction and Π the projection on the lowest Landau
level Hof in the plane. Then Π⊗2V(|z1−z2|)Π⊗2can be expanded in terms of the projections
Pm:
Π⊗2V(|z1−z2|)Π⊗2=∞
X
m=0
VmPm.
The numbers Vm,m∈N0are called pseudopotential parameters (see Haldane’s contribution
in [PG87]). For the Coulomb interaction V(z) = 1/|z1−z2|, they can be computed as
Vm=Γ(m+1
2)
2m!.
The truncated interactions Pp−1
j=0 VmPmcan be viewed as the short-range part of ΠVΠ. This
raises the question how large the error done by cutting off the long-range part is. Here, one
should be aware that two questions show up:
1.5. INCOMPRESSIBILITY AND GAPS 21
1. Is Pp−1
j=1 VmPma good approximation to Π⊗2VΠ⊗2?
2. Is the truncated many-body Hamiltonian from Proposition 1.6 that has Laughlin’s
function as its ground state, a good approximation to the lowest Landau level projected
Hamiltonian
Π⊗NX
1≤j<k≤N
V(zj−zk)Π⊗N+ Π⊗N
N
X
j=1
W(zj)Π⊗N?
The comparison of truncated and “true” lowest-Landau level projected Hamiltonians has been
performed by [HR85a] in a spherical geometry. On the sphere, the background potential W
gives only a constant contribution to the total Hamiltonian and the distinction between the
two questions blurs. [HR85a] found that the truncated Hamiltonian is a good approximation
to the lowest Landau level projected Hamiltonian. In samples with boundaries, the situation
is quite different: the truncated interaction is in general not a good approximation to the
true interaction.
Consider for example two particles in a disk. Let Π3be the projection in ∧2L2(R2) onto
span{Ωm∧Ωn|0≤m, n ≤3}, the Hilbert space of two fermions in the lowest Landau
level with center c2
x+c2
y≤3. The ground state of the truncated interaction Π3P1Π3is
(z1−z2)3e−(|z1|2+|z2|2)/4l2. The ground state of Π3VΠ3, on the other hand, is Ω2∧Ω3, as
can be shown by a simple computation. This reflects the fact that excess charge in Coulomb
systems accumulates at the boundary; this feature of Coulomb interaction is due to its long
range part and thus not captured by a truncated interaction.
Thus in samples with boundaries, the answer to the first question is in general negative.
However, this does not exclude that truncated Hamiltonian and the lowest Landau level
Hamiltonian including the contribution from a neutralizing background have similar ground
states.
It is interesting to examine the way the physics literature treats samples with boundaries.
For example, Laughlin [Lau83] compares his wave function to various “true” ground states:
these are obtained as ground states of Hamiltonians without background potential, projected
to the space of lowest Landau level many-body functions with total angular momentum
pN(N−1)/2. The restriction of angular momentum is justified on grounds that this is the
angular momentum of Laughlin’s function at filling factor 1/p. A similar approach is used by
Dev and Jain [DJ91], and Girvin and Jach [GJ83] who even define the filling factor through
the total angular momentum (equation (9)). The conclusion of the numerical investigations
cited is that “true” ground states are close to the Laughlin state, but it seems to us that the
treatment or rather neglection of the background as well as the restriction to a given angular
momentum sector require more justification. Thus we feel that the numerical comparisons
of exact and Laughlin states in the spherical geometry [HR85a, FOC86] are much more
convincing.
1.5 Incompressibility and gaps
The aim of this section is to discuss the notion of a gap above the ground state and its relation
to incompressibility. We do this mainly because we will use gap conditions in Section 2.4 and
Chapter 3, and we wish to clarify the relation between the two in general different gaps that
22 CHAPTER 1. LAUGHLIN’S WAVE FUNCTION
are used. However, apart from making things (hopefully) a little bit clearer, the following
considerations are not necessary for the understanding of the remainder of the thesis.
In the ground state of a quantum Hall system at certain densities, the electrons form an
incompressible gas (see below). This is justified by numerical results on the ground state
energy per particle and by the existence of a gap above the ground state. The relation
between gaps and incompressibility is often only sketched (see however [Yos98]); we give some
details here. As soon as one refers to systems of interacting particles, there are several natural
notions of gaps; this is implicit when one talks of the gappedness of different excitations and
raises the question which gap leads to incompressibility of the system.
Incompressible fluid at zero temperature. Consider a system of Nparticles moving in
a domain of volume V, modeled by a Hamiltonian with ground state energy E(N, V ). Sup-
pose that the limit u(n) := limN→∞ E(N, V )/N taken along suitable sequences of domains
such that N→ ∞,N/V →n, exists. The limit uis the energy per particle at density nand
temperature 0. Numerical computations [YHL83] suggest that for quantum Hall systems, u
is a piecewise differentiable function of nwith a “cusp” at certain densities: for example,
du/dn is discontinuous at n∗= (3 ·2πl2)−1. The discontinuity of du/dn is the manifestation
of incompressibility. A system is usually called incompressible if its volume is independent
of the pressure. Let us briefly explain the connection to the cusp in u(n) and other charac-
terizations, e.g. discontinuity of the chemical potential (see also [Yos98]).
The bulk thermodynamics of a system with one species of particles is described by a set of
intensive parameters among which, at zero temperature, “only one is independent”: suppose
that in a neighborhood of a certain density n∗, the bulk thermodynamics is described by a
continuous curve J∋s7→ (u, n, p, µ)(s), where Jis an open interval and u, n, p, µ are the
energy per particle, density, pressure and chemical potential. We interpret the projections
of the curve to the (n, u) and (n, p) planes as the graphs of (possibly multi-valued) functions
u(n), p(n). Suppose that the energy per particle uis a continuously differentiable (single-
valued) function of the density nexcept at n=n∗where it has different left and right
derivatives. Then by the usual thermodynamic laws, for n6=n∗,
p(n) = n2du
dn
(this is just a different way of writing du =−pdv, with v= 1/n the volume per particle). In
particular, the pressure is a single-valued function of nexcept possibly at n=n∗. Let
p±:= n2
∗lim
n→n∗±
du
dn.
We require that the system has a non-negative compressibility, i.e., the density increases
when the pressure increases and vice versa. Then p−< p+and the pressure is a continuous,
increasing function of nexcept at n=n∗where it jumps from p−to p+. In view of the
continuity of s7→ (n(s), p(s)), it follows that at n=n∗, the pressure can take any value
between [p−, p+]. Equivalently, we may say that the density is constant along the pressure
interval [p−, p+]: the system is incompressible.
Now let us turn to the behavior of the chemical potential. At zero temperature, the chemical
potential is
µ=u+pv =u+p/n =d(nu)
dn (1.26)
1.5. INCOMPRESSIBILITY AND GAPS 23
where the last identity holds only in points of differentiability of u. Let µ±=u+p±/n. If
n=n∗, the chemical potential can take any value in [µ−, µ+]; by a slight abuse of language,
we may say that the chemical potential is a discontinuous function of the density. Equiva-
lently, the density is constant along the chemical potential interval [µ−, µ+].
To conclude, we have the following manifestations of incompressibility:
•du/dn is discontinous at n=n∗
•the volume per particle vis constant along a pressure interval [p−, p+]
•the chemical potential is discontinous at n=n∗
•the density is constant along a chemical potential interval [µ−, µ+].
Different energy gaps. The incompressibility of a system, characterized through the
discontinuity of the chemical potential, is related to the existence of gaps. For example,
Girvin [Gir05] writes the discontinuity of the chemical potential at filling factor 1/3 as
∆µ= 3(∆++ ∆−), where ∆±represent quasielectron / quasihole excitation energies (p.
158). On the other hand, the existence of a gap is also invoked to justify the application of
adiabatic theorems in the various quantum Hall effect arguments involving slow addition of a
flux quantum. This gap however is in general different from the gaps related to incompress-
ibility. For systems of interacting particles, two gaps can be considered: the gap at a fixed
number of particles, and the gap of a grand-canonical Hamiltonian H−µˆ
N. The disconti-
nuity of the chemical potential is related to these quantities, but in generals differs from them.
Let Hbe the Hilbert space for the motion of one particle moving. For simplicity, sup-
pose dim H=d < ∞. Let F:= Ld
N=0 ∧NHbe the corresponding fermionic Fock space,
HN∈ B(∧NH) the Hamiltonian for N-particle systems.
Gap at a fixed number of particles. The first natural notion of a gap is simply the gap at a
fixed number of particles: let E0(N)< E1(N)< .. be the (possibly degenerate) eigenvalues
of HN. We shall refer to E1(N)−E0(N) as the gap above the ground state of HN(we
assume that HNis not a multiple of the identity). This gap is essentially the gap used in
adiabatic theorems: typical quantum Hall effect gedanken experiments involve flux depen-
dent Hamiltonians HN(φ) and use a condition of the type E1(N, φ)−E0(N, φ)≥gmin >0
for some φ-independent gmin, at a given number of particles.
Discontinuity of the chemical potential. Since incompressibility can be characterized in terms
of the chemical potential, it is natural to turn to the grand-canonical ensemble. Let µ∈R
be a chemical potential, and
H−µˆ
N:=
d
M
N=0
(HN−µNPN)∈ B(F).
PNis the projection on ∧NH, considered as a subspace of F. At zero temperature, the
expected number of particles is N(µ) = trPµˆ
N/trPµ, where Pµis the projection on the
ground state of H−µˆ
N. For each number of particles N∈N, we define left and right
chemical potentials µ−(N) := E0(N)−E0(N−1) and µ+(N) := E0(N+ 1) −E0(N)1We
1The definition of the chemical potentials µ±(N) appears in the work [LW68] on Mott insulators; there,
the nonvanishing of µ+−µ−is related to Mott insulating behavior.
24 CHAPTER 1. LAUGHLIN’S WAVE FUNCTION
assume that
∀N∈N:µ−(N)< µ+(N),(1.27)
i.e., E0(N) is a strictly convex function of N. For µ−(N)< µ < µ+(N), Pµis the projection
on the ground state of HN, considered as a subspace of F, so that
µ−(N)< µ < µ+(N)⇒N(µ) = N. (1.28)
In order to apply one of the criteria of incompressibility, we have to take thermodynamic
limits. We do this only at a formal level and restore the dependence of the volume in the
notation by writing µ±(N, V ), E0(N, V ) instead of µ±(N), E0(N). For large N,E0(N, V )≈
Nu(N/V ) = Nu(n) with n=N/V . Then
E0(N+ 1, V )−E0(N, V )≈(N+ 1)u(n+1
V)−Nu(n)≈u(n) + nu′
+(n) =: µ+(n)
E0(N, V )−E0(N−1, V )≈Nu(n)−(N−1)u(n−1
V)≈u(n) + nu′
−(n) =: µ−(n)
where we assume that uhas left and right derivatives u′
−and u′
+. The previous relations are
consistent with (1.26). In the thermodynamic limit, (1.28) becomes
µ−(n)≤µ≤µ+(n)⇒n(µ) = n.
The system is incompressible at a density nif the limiting value µ+(n)−µ−(n) of the differ-
ence µ+(N, V )−µ−(N, V ) as N→ ∞,N/V →ndoes not vanish. Thus incompressibility
at zero temperature involves only the ground state energies E0(N, V ), and the existence of
gaps above the ground states of HNin ∧NHis irrelevant.
Gap of H−µˆ
N.The discontinuity of the chemical potential has nothing to do with the gap
of HNat fixed N. However, it is related to the gap of H−µˆ
Nin F. More precisely, suppose
µ−(N)< µ < µ+(N) so that the ground state of H−µˆ
Ncoincides with the ground state of
HN. Then the gap
g(H−µˆ
N) := min σ(H−µˆ
N)\{E0(N)−µN}−(E0(N)−µN)
above the ground state of H−µˆ
Nis bounded by the discontinuity of the chemical potential
and the gap E1(N)−E0(N) above the ground state of HNin HN:
g(H−µˆ
N)≤min(E1(N)−E0(N), µ+(N)−µ−(N)).(1.29)
Proof of (1.29). Note that
g(H−µˆ
N)≤min{E1(N)−E0(N)}∪{E0(N′)−E0(N)−µ(N′−N)|N′6=N}
= min{E1(N)−E0(N), µ+(N)−µ, µ −µ−(N)}
≤min{E1(N)−E0(N), µ+(N)−µ−(N)}.
The second line is obtained by using (1.27) and writing for N′≥N+ 1
E0(N′)−E0(N)−µ(N′−N) =
N′−N
X
k=1
(E0(N+k)−E0(N+k−1) −µ)
≥E0(N+ 1) −E0(N)−µ.
An analogous inequality holds for N′≤N−1.
1.6. PLASMA ANALOGIES 25
To summarize, we have the following relationship between the discontinuity of the chemical
potential and the two gaps: the discontinuity of the chemical potential has nothing to do
with the existence of gaps at a fixed number of particles, but involves only ground state en-
ergies E0(N). In contrast, if H−µˆ
Nhas a gapped ground state, then the chemical potential
is discontinous and there is a gap at fixed number of particles, and the gap of H−µˆ
Nis
bounded from above by these two quantities.
Finally, let us mention how the quantities are related for FQHE states: the FQHE state ad-
mits a curve of “collective excitations” depending on a wave vector k. The curve approaches
the discontinuity of the potential at small wavelengths (klarge), but has a minimum smaller
than ∆µ([HR85a, Gir05]). Thus the gap at a fixed number of particles is smaller than the
discontinuity of the chemical potential.
Remarks: 1. Independent particles. For non-interacting particles, the distinctions made
above are superfluous: the gaps and the discontinuity of the chemical potential equal the gap
of the one-particle Hamiltonian.
2. Different gaps for different excitations. In solid state physics, it is common to talk
of gaps for specific excitations. In this perspective, the gap at fixed number of particles
E1(N)−E0(N) is the gap for excitations that preserve the total number of particles. In
contrast, the differences E0(N′)−E0(N) show up when we allow excitations that do change
the number of particles. This aspect reappears in the C∗-algebraic definition of gaps for
systems with an infinite number of particles: two algebras of operators are introduced (in
Section 2.4, AU(1) and A), and there are two notions of ground states and gappedness.
3. Gapless edge excitations. At this point we stress that the incompressibility and gappedness
of the FQHE ground state refer to bulk properties. It is well-known that on samples with
boundaries, there are gapless excitations. This can be explained with the spectral boundary
conditions (see p. 8 and p. 113). The Hamiltonian Hwith spectral boundary conditions
determines a splitting of the Hilbert space into invariant subspaces Hband Hecorresponding
to bulk and edge states. The gap of Hrestricted to bulk states Hbis larger or equal to ~ω.
In contrast, the gap of Hon the whole space L2([a, b]×[0,2πR]) is bounded by
mine0(kl2/R)−~ω/2|k∈Z, kl2/R /∈[a, b],
where e0(ρ) is a continuous function with minimum ~ω/2. In the limit R→ ∞, the magnetic
length land the interval [a, b] being fixed, the gap vanishes. Thus in the limit of large samples,
the Hamiltonian has gapless excitations, associated to edge states.
1.6 Plasma analogies
Laughlin interpreted the modulus squared of his wave function as the Boltzmann weight of
a two-dimensional one-component plasma. This analogy can be made in different geometries
and provides useful intuition when it comes to the translational symmetry breaking. In
this section, we shall describe the relevant plasma models, and give the Laughlin - plasma
analogy in the various geometries used. In order to compare free energies, special care must
be devoted to additive constants in the definition of energy functions.
26 CHAPTER 1. LAUGHLIN’S WAVE FUNCTION
We use the following “Coulomb potentials”:
VD(z) = −log |z|/R0(z∈C\{0}),
VC(z) = −log |2R
R0
sinh z
2R|(z∈C\{2πRki |k∈Z}),
VT(z) = −log |2R
R0
θ1(iz
2R|iL
2πR )
θ′
1(0 |iL
2πR )|+x2
2RL,(z∈C\(LZ+ 2πRiZ)).
The index D, C, T refers to the geometry (disk, cylinder, torus). The cylinder has radius
R, the torus is R/LZ×R/2πRZ, and R0is a reference length scale. The notation θ′
1(z|τ)
refers to the derivative of θ1with respect to the complex variable z. The potentials fulfill
suitable Poisson equations, the cylinder and torus potentials have appropriate periodicity,
and all potentials behave as VDfor small |z|:
Properties: The functions VD, VC, VTdefine tempered distributions φ∈ S(R2)7→ RR2V φ.
Let δa:φ7→ φ(a). The following equations hold in the sense of distributions:
−∆VD= 2πδ0,−∆VC= 2πX
k∈Z
δ(0,2πRk)−∆VT= 2πX
m,k∈Z
δ(mL,k2πR)−1
2πRL.
Furthermore, for k, m ∈Z, we have
VC(z+ 2πRi) = VC(z), VT(z+mL +k2πRi) = VT(z).
and VC,T =VD(z) + o(1) as z→0.
Thus VDis a two-dimensional Coulomb potential. VDis the 2D potential adapted to a sys-
tem confined to a strip [0, L]×[0,2πR] with periodic boundary conditions in the y-direction
(i.e., a cylinder). Similarly, VTis a torus Coulomb potential, i.e., adapted to the rectangle
[0, L]×[0,2πR] with periodic boundary conditions. VTis the potential created by a charge
(the δ0-part in the Poisson equation), its images (the other Dirac distributions), and their
neutralizing background (the constant term).
Now consider a system of Nparticles of charge qand positions zj=xj+iyjmoving in a
neutralizing background of charge density −nq on a disk, a cylinder or a torus. For R, L > 0,
let
D:= {z∈C| |z|< R}, C := [−L/2, L/2] ×[0,2πR] =: T.
Now we can define the potential energy Uα, α ∈ {D, C, T}through
Uα(z1, .., zN) := X
1≤j<k≤N
q2Vα(zj−zk)−nq2
N
X
j=1 Zα
Vα(zj−z)dz
+1
2(nq)2Zα×α
Vα(z−z′)dzdz′(1.30)
where n=N/πR2if α=D(disk) and n=N/(2πRL) if α∈ {C, T }(cylinder or torus). The
1.6. PLASMA ANALOGIES 27
expression of the Boltzmann weights are:
e−βUD=Y
j<k |zj−zk
R0|Γe−Γπn
2PN
j=1 r2
jeΓN2
2(3
4−log |R
R0|)(1.31)
e−βUC=Y
j<k |2 sinh zj−zk
2R|Γe−Γπn PN
j=1 x2
je−λ2
12 N3,if R=R0(1.32)
e−βUT=Y
j<k |2R
R0
θ1(iz
2R|iL
2πR )
θ′
1(0 |iL
2πR )|Γe−Γπn Pj<k(xj−xk)2/N e−N3λ2/12e−N2Γ
2log R
R0
∞
Y
n=1
(1 −u2n)2.
(1.33)
where Γ = βq2,λ2= Γ/(4πnR2) and u:= e−2πR2/L. Details on the computation can be
found in [FGIL94] for the disk energy UD, [ˇ
SWK04] for the cylinder and [For06] for the torus
energy.
Remark: The reader may feel that we have done some overcounting in the definition of the
torus energy, since the interaction VTalready includes a background contribution. There is
however, a standard approach to the definition of energy functions for long range interactions
in periodic geometries (see [BST66] for the Coulomb interaction). In the following paragraph,
we briefly sketch this approach, which allows not only to check that our definition of the
torus energy UTcoincides with the more standard definition, but also to give a meaning to
a “periodic 1/r interaction” (used for quantum Hall samples on tori in [YHL83]).
To simplify matters, we replace first the logarithmic interaction with an interaction V∈
S(R2). The potential created by a uniform background, charges located in z1, .., zN∈[0, L1]×
[0, L2] and their images is
W(z;z1, .., zN) =
N
X
j=1 X
m∈Λ
V(z−zj−m)−N
L1L2ZR2
V(z−z′)dzdz′
where Λ = L1Z×L2Z. Let µ:= PN
j=1 δzj−N
L1L2be the charge density in the unit cell
[0, L1]×[0, L2]. Then
U(z1, .., zN) := 1
2(Z[0,L1]×[0,L2]
Wdµ −NV (0))
represents the energy of the unit cell minus the self-interaction of the particles. The energy
is in general reexpressed as [BST66]
U(z1, .., zN) = X
j<m
ψ(zj−zm) + N
2(ψ(0) −V(0)) (1.34)
where ψ(r) = 2π
L1L2Pk∈Λ′\{(0,0)}ˆ
V(k)eik·r,and Λ′denotes the lattice dual to Λ. But Ucan
also be expressed by the formula (1.30) with integrations carried out on [0, L1]×[0, L2] and
interaction potential
˜
V(z) = X
m∈Λ
V(z−m)−X
m∈Λ\{0}
V(m) = ψ(z)−ψ(0) + V(0)
so that ˜
V(0) = V(0). In view of VT(z) = VD(z) + o(1) as |z| → 0, this is the analogue of the
formula we used in order to define UT. In contrast, (1.34) gives the more familiar expression
U(r1, .., rN) = X
j<m X
k∈Z2\{(0,0)}
1
k2eik·(rj−rm)+ const.
28 CHAPTER 1. LAUGHLIN’S WAVE FUNCTION
A similar expression is used in [YHL83] to define the energy function used for numerical com-
putations for quantum Hall systems on a torus, except that 1/k2is replaced with const/|k|,
the two-dimensional Fourier transform of 1/|r|, and the resulting Hamiltonian is projected
to the lowest Landau level.
The plasma analogy relates the Boltzmann weights to the modulus squared of the Laughlin
type wave functions:
Proposition 1.8. Let ΨD
N,ΨC
N,ΨD
Nbe the Laughlin wave functions at filling factor 1/p
defined on p.9, and e−βUD, e−βUC, e−βUTthe Boltzmann weigths as in (1.31), (1.32) and
(1.33). Suppose that Γ = 2pand n= (p·2πl2)−1. Then
|ΨD
N|2=e−βUDe−ΓN2
2(3
4−log |R
R0|)(1.35)
|ΨC
N(z1, .., zN)|2=1
N!
1
(2πRl√π)NeNλ2
12 e−βUC({zj−p(N−1)l2/(2R)}) (1.36)
|ΨT
N(z1, .., zN)|2∝ |Fcm(
N
X
j=1
zj)|2e1
Nl2(PN
j=1 xj)2e−βUT(z1, .., zN).(1.37)
where the multiplicative constant in (1.37) can be determined from (1.33).
Sketch of proof. The first identity (1.35) is obvious. (1.36) uses the identities es−et=
2 exp s+t
2sinh s−t
2and
X
1≤j<k≤N
(xj+xk) = N−1
2
N
X
j=1
xj.
(1.37) is shown using
X
1≤j<k≤N
(xj−xk)2=N
N
X
j=1
x2
j−(
N
X
j=1
xj)2.
The plasma analogy was used by Laughlin [Lau81] in the disk geometry to transfer knowledge
on plasma systems to the electron gas described by Laughlin’s wave function. The existence
of the thermodynamic limit of the free energy of the two-dimensional jellium system has
been proved in various geometries [SM76]. There is a wealth of numerical results on two-
dimensional jellium systems (see the references in [FGIL94]). They suggest that for coupling
constants below Γ ∼140, the plasma is translationally invariant, whereas it becomes crys-
talline above.
Semi-periodic strips should be seen as quasi one-dimensional systems rather than two-di-
mensional. The formation of Wigner crystals in one-dimensional jellium has been proved
[Kun74, BL75]. Motivated by this and the exact results on plasmas at coupling Γ = 2 (corre-
sponding to the filled Landau level), [ˇ
SWK04] investigated numerically symmetry breaking
on jellium strips.
It is interesting to observe that the very analogy invoked by Laughlin to show that his function
describes a homogeneous gas suggests a non-trivial periodicity of the density on cylinders.
Chapter 2
Thermodynamic limits
The filled Landau level on a cylinder corresponds to an electron gas whose density is not
uniform, but has a periodicity in the direction along the cylinder axis, as we have seen in
Section 1.3. The main result of this thesis is that for Laughlin cylinder functions at lower
filling factor 1/p, a similar statement can be proved provided the cylinder radius is sufficiently
small. The period of the one-particle density is ptimes the period of the filled Landau level.
Furthermore, for thin cylinders the thermodynamic limit of all correlation functions exists.
The limiting state is mixing with respect to a suitable translation group. The proof of these
results is the object of Section 2.1. It makes crucial use of the representation of Laughlin’s
wave function as a polymer system and a relationship to discrete renewal processes. The
fundamental result is a statement on the asymptotics of normalization constants (Theorem
2.12) which allows to infer that on thin cylinders, the associated renewal process has finite
mean interarrival time.
Interestingly, the relationship with polymer systems subsists when we consider modified
Laughlin type wave functions. In Section 2.2, we define modified cylinder and torus wave
functions and show that, for a certain range of parameters, they are associated with monomer-
dimer systems, define solvable models, and monomer-dimer torus and cylinder wave functions
are equivalent in a suitable limit.
In Section 2.3, we look at the problem from the point of view of classical jellium systems.
We compare the jellium tube to one- and two-dimensional systems, and derive an inequality
from Newton’s electrostatic theorem. This inequality is related to a recurrence relation ful-
filled by the normalization constant of Laughlin’s wave function. The semiperiodic Coulomb
interaction is translationally invariant with respect to translations along the cylinder axis by
any real value. Therefore in the jellium tube picture, our results imply symmetry breaking
for all values of the radius. On thin strips, we know that the period is pl2/R, whereas on
large strips, the period might be l2/R, but in any case the one-particle density cannot be
constant.
In Section 2.4, we examine infinite volume ground states, among which we find the limits of
Laughlin’s wave function, of truncated interactions. The main interest lies in the relationship
between the existence of gaps and symmetry breaking.
29
30 CHAPTER 2. THERMODYNAMIC LIMITS
2.1 Laughlin’s cylinder wave function
In this section, we prove the main result of this thesis, namely that, at least for sufficiently
thin cylinders, the thermodynamic limit of Laughlin’s state at filling factor 1/p exists and
defines a state with a translational period that is ptimes the period of the filled Landau
level. The key ingredient is the representation of Laughlin’s cylinder function as a polymer
system. The norm squared CNof Laughlin’s function is the polymer partition function of
a system of hard rods on the one-dimensional lattice Zwith translationally invariant activ-
ity. It satisfies a recurrence relation known in stochastics as a renewal equation. When the
associated renewal process has finite mean, the thermodynamic limit of the polymer system
exists; Laughlin’s function essentially inherits this feature from the polymer system, as well
as clustering properties. The translational invariance of the limiting polymer state translates
into periodicity of Laughlin’s state. For sufficiently thin cylinders, the associated renewal
process has indeed finite mean, and pl2/R is the smallest period of the thermodynamic limit
of Laughlin’s state.
This section is organized as follows:
•In the first subsection, we give some basic properties of Laughlin’s wave function, and
in particular its expansion into Slater determinants of lowest Landau level basis func-
tions. The main result is that the expansion coefficients satisfy a product rule and that
Laughlin’s wave function can be represented as a polymer system. The product rule has
been shown to hold for expansion coefficients of powers of Vandermonde determinants
[FGIL94]. The novelty here is the observation that on cylinders, the product rule holds
also for the expansion coefficients of Laughlin’s function.
•In Subsection 2.1.2, we review well-known results on polymer systems and renewal
processes that will be used later on. The main message is that systems of hard rods
on a lattice with stable, translationally invariant activity are associated with discrete
renewal processes. When the associated renewal process has finite mean, the system
of rods has a nicely behaved thermodynamic limit, while processes with infinite mean
give rise to complications.
•As a consequence, in order to make the polymer representation of Laughlin’s wave
function useful, we have to answer two questions: Is the activity of the associated
system stable? Does the associated renewal process have a finite mean? These ques-
tions are answered in Subsection 2.1.3 by looking at the asymptotics of the L2norm
squared of Laughlin’s wave function. The answer to the first question is always positive.
For the second question, we have a positive answer only for thin cylinders; the proof
uses the polymer analogy and combines the well-known results presented in Subsection
2.1.2 with the fact that in the limit of large radii, all polymers except monomers have
vanishing activity. What happens for large radii is open.
•In Subsection 2.1.4, we prove that under the assumption of a finite mean, Laughlin’s
state has a well-defined thermodynamic limit. The limiting state is periodic in the
direction along the cylinder axis, and pl2/R is a period; furthermore, the limiting state
is mixing with respect to shifts by multiples of pl2/R. The results on one-dimensional
systems of hard rods presented in Subsection 2.1.2 serve as a guiding intuition, but our
results are quantum mechanical and thus go beyond Subsection 2.1.2.
•In the last subsection, we prove that on thin cylinders, the limiting state of Subsection
2.1.4 has pl2/R as its smallest period. Since Laughlin’s state can be characterized as
2.1. LAUGHLIN’S CYLINDER WAVE FUNCTION 31
the ground state of a Hamiltonian with period l2/R (see Section 2.4), this result shows
that there is symmetry breaking.
2.1.1 Basic properties
Laughlin’s wave function has a few number of basic, but important properties that we present
in this subsection. The simplest are two symmetry properties (Lemma 2.1). The second ob-
servation is that the one-particle density is a sum of Gaussians centered on the lattice lγZ,
and therefore, in the limit of infinite cylinders, the one-particle density cannot be constant:
at best, it can have a period that is a multiple of l2/R =γl (Lemma 2.2). Laughlin’s wave
function is a sum of products of lowest Landau level basis functions (see (2.1) below). The
most important result of this subsection is that the expansion coefficients satisfy a product
rule (Lemma 2.4) that allows the representation of Laughlin’s function as a polymer system
(Lemma 2.6). Later, we will give an estimate of the activities of the associated polymer
system based on an auxiliary result proved in this subsection (Lemma 2.5).
The starting point is the intimate relation between Laughlin’s wave function and powers of
a Vandermonde determinant. Recall from the proof of Lemma 1.2 that
ΨC
N(z1, .., zN) = 1
√N!X
0≤m1,..,mN≤p(N−1)
aN(m1, .., mN)ψm1⊗.. ⊗ψmN(2.1)
aN(m1, .., mN) := bN(m1, .., mN)eγ2PN
j=1(m2
j−p2(j−1)2)/2(2.2)
where ψm(z)∝eimy/Re−(x−mγl)2/2l2are the normalized cylinder lowest Landau level basis
functions and the coefficients bNare defined through the expansion of the p-th power of the
Vandermonde determinant:
VN(z1, .., zN)p=X
m1,..,mN
bN(m1, .., mN)zm1
1..zmN
N.
Remember that we give empty products the value 1, so that
V1(z)p=Y
1≤j<k≤1
(zk−zj)p= 1 = b1(0)
and
ΨC
1(z) = 1
p2πRl√πe−x2/2l2=a1(0)ψ0(z), a1(0) = 1.(2.3)
Let us explain here how the geometry affects the expansion. As observed by Dunne [Dun93],
the expansion coefficients of Laughlin’s function as a sum of normalized lowest Landau level
basis function is made up of two parts: in (2.2), we see that aN(m1, .., mN) is the product of
bN(m1, .., mN), coming from the expansion of the polynomial Vp
N, and an exponential factor
that accounts for the fact that z7→ ez/Re−x2/2l2is not normalized. A similar expansion holds
for the disk function. It suffices to replace the cylinder basis functions ψkin (2.1) with the
disk basis functions Ωk(z) = zke−|z|2/4l2(2πl2k!(2l2)k)−1/2and to set
aD
N(m1, .., mN) := bN(m1, .., mN)(2πl2)N/2N
Y
j=1
mj!(2l2)mj1/2.(2.4)
A comparison of this formula with (2.2) shows how the geometry enters the scene. The
difference in the contributions from normalization constants is responsible for the fact that
32 CHAPTER 2. THERMODYNAMIC LIMITS
cylinder expansions coefficients satisfy the product rule of Lemma 2.4 while disk expansion
coefficients do not.
The relationship of Laughlin’s function to powers of a Vandermonde determinant has been
exploited in [Dun93, FGIL94] and motivated combinatorial investigations [STW94, KTW01].
We use here two properties of the expansion coefficients bN({mj}) proved in [Dun93, FGIL94]:
the reversal invariance (2.10) and the product rule and admissibility conditions (Lemma 2.4
for the bN’s instead of aN’s).
Let us mention that although, to our knowledge, there is no closed expression giving bN({mj})
as a function of the mj’s, a number of formulas is available for specific mj’s: for example,
bN(0, p, .., pN −p) = 1,
bN(1, p, .., p(N−2), pN −p−1) = (−1)N−1p(p−1)N−2,(2.5)
bN(s(N−1), s(N−1) + 1, .., (s+ 1)(N−1)) = (−1)sN(N−1)
2(s+ 1)N!
N!(s+ 1)!N, p = 2s+ 1,
(2.6)
see [Dun93], Section 6 for the p= 3 case of the formulas and [FGIL94], Section 5, for a proof
of the third identity. The equality (2.5) is proved in a way similar to (2.75) in the proof of
Lemma 2.23 below. Similar formulas are verified for N≤8, podd, in [RH94]. Additional
formulas can be found also in [KTW01].
Laughlin’s cylinder function is an eigenvector of total y-momentum and of a rotation by 180◦,
which is defined as follows: let (s0ψ)(z) := ψ(−z) and for a∈R,
(saψ)(z) := (t(aex)s0t(−aex)ψ)(z) = ei2ay/l2ψ(2a−z).
The rotation sapreserves the boundary condition ψ(z+i2πR) = ψ(z) if and only if a∈γl
2Z.
Moreover,
∀r∈1
2Z,∀k∈Z:srγlψk=ψ2r−k.(2.7)
Lemma 2.1. Laughlin’s cylinder function is an eigenfunction of the total y-momentum and
of an overall 180◦rotation around the middle of the cylinder:
N
X
j=1
cx,jΨC
N=−il2
N
X
j=1
∂
∂yj
ΨC
N=pN(N−1)
2γlΨC
N(2.8)
s⊗N
p(N−1)γl/2ΨC
N= ΨC
N.(2.9)
Proof. The translational invariance in the y-direction (2.8) follows from the fact that Vp
Nis
a homogeneous polynomial of total degree p(0 + 1 + .. +N−1) = pN(N−1)/2 so that
aN(m1, .., mN) vanishes unless m1+.. +mN=pN(N−1)/2. In [Dun93] (Section 6) and
[FGIL94] (Section 4, Property 0) it is shown that
bN(p(N−1) −mN, ..., p(N−1) −m1) = bN(m1, .., mN).(2.10)
This follows from
N−1
Y
j=1
zp(N−1)
jVN(1
z1
, .., 1
zN
)p= (−1)pN(N−1)/2VN(z1, .., zN)p
2.1. LAUGHLIN’S CYLINDER WAVE FUNCTION 33
and the antisymmetry / symmetry bN(mπ(1), .., mπ(N)) = (sgn(π))pbN(m1, .., mN); the per-
mutation 1 7→ N, 27→ N−1, etc., has the sign (−1)⌊N/2⌋= (−1)N(N−1)/2where ⌊N/2⌋is
the integer part of N/2. Combining (2.10) and (2.2), we obtain
aN(p(N−1) −mN, ..., p(N−1) −m1) = aN(m1, .., mN).(2.11)
The rotational invariance (2.9) then follows from (2.7) and (2.1).
Remark. Following [Dun93], we will also say reversal invariance instead of rotational invari-
ance.
The symmetries have simple consequences for the reduced n-point matrices which we now
introduce. It is convenient to use the language of second quantization. We work out the
details only for odd p. Most of the statements however, hold for even palso; the most impor-
tant technical difference comes from the unboundedness of bosonic creation and annihilation
operators [BR79b].
Let F:= ∧H,H=L2(R×[0,2πR]) be the fermionic Fock space over the infinite cylinder
and Athe anticommutation algebra over H, i.e., the closed sub-algebra of B(F) generated
by 1B(F)and the creation and annihilation operators c∗(f), c(f), f∈ H. Laughlin’s wave
function defines a state on Athrough
haiΨC
N=hΨC
N, aΨC
Ni/||ΨC
N||2.
If h·i is a state on A, the reduced n-point matrix ([BR79b], Section 6.3.3), if it exists, is the
function ρ(n): (R×[0,2πR])n×(R×[0,2πR])n→Rsuch that for all f1, .., fn, g1, .., gn∈ H,
hc∗(f1)..c∗(fn)c(gn)..c(g1)i
=Z((R×[0,2πR])2n
g1(z′
1)..gn(z′
n)ρ(n)(z′
1, .., z′
n;z1, .., zn)f1(z1)..fn(zn)dz′
1..dzn.
Put differently, it is the integral kernel of the operator ρ(n)in ⊗nHwith expectation values
hg1⊗.. ⊗gn, ρ(n)f1⊗.. ⊗fni=hc∗(f1)..c∗(fn)c(gn)..c(g1)i.
Using (c(f)ΨN)(z2, .., zN) = √NRf(z1)ΨN(z1, .., zN)dz1, one sees that the reduced density
matrices of h·iΨC
Nare given by integrals of the type (1.9). Thus the definitions given in
the section on the filled Landau level are consistent with the definitions given here. Let
ck=c(ψk). The r-point reduced density matrix of ΨC
Ncan also be expressed as
ρ(r)
N(z′
1, .., z′
r;z1, .., zr) = r!X
0≤n1<..<nr≤pN−p
0≤m1<..<mr≤pN−p
n1+..+nr=m1+..+mr
hc∗
m1..c∗
mrcnr..cn1iΨC
N
ψm1∧.. ∧ψmr(z′
1, .., z′
r)ψn1∧.. ∧ψnr(z1, .., zr).
The restriction to m1+.. +mr=n1+.. +nrcomes from
ρ(r)
N(z′
1−ia, ..;.., z′
r−ia) = ρ(r)
N(z1, ..;..zr),
which is a consequence of the translational invariance (2.8). In particular, the one-particle
density
ρN(z) := ρ(1)
N(z;z) =
pN−p
X
k=0 hc∗
kckiΨC
N|ψk(z)|2=1
2πRl√π
pN−p
X
k=0 hc∗
kckiΨC
Ne−(x−kγl)2
34 CHAPTER 2. THERMODYNAMIC LIMITS
is a sum of Gaussians, independent of the coordinate yaround the cylinder. This will subsist
in the thermodynamic limit. Thus instead of looking at the one-particle density, we can look
at the sequence of occupation numbers hc∗
kcki. The following lemma ensures a one-to-one
relation between the periodicity of the one-particle density and of the sequence of occupation
numbers.
Lemma 2.2. Let (nk)k∈Zbe a sequence of numbers in [0,1] that is not identically zero and
ρ(x) := P∞
k=−∞ nk|ψk(x)|2.Then any period of ρ(·)is a multiple of l2/R =γl. Moreover,
for p∈N,ρis periodic with period pγl if and only if (nk)is p-periodic. In this case the
Fourier coefficients of ρcan be expressed as
1
pγl Zpγl
0
e−ikxρ(x)dx =1
p
1
2πl2e−π2k2
p2γ2
p−1
X
j=0
nje−i2πjk/p.(2.12)
Proof. ρ=f∗µis the convolution of a function fand a measure µ
f(x) := 1
2πRl√πe−x2/l2, µ := X
k∈Z
nkδkγl.
Taking Fourier transforms, we get
ˆρ=√2πˆ
f·ˆµ=1
2πRe−(lx)2/4ˆµ(2.13)
which should be read in the distributional sense, i.e., ∀φ∈ S(R) : R∞
−∞ ρˆ
φ=√2πR∞
−∞ c
ˆ
fφdµ.
Suppose µand νare tempered measures such that f∗µ=f∗ν. Then ˆ
f·ˆµ=ˆ
f·ˆν. Since ˆ
f
is a continuous, strictly positive function this implies ˆµ= ˆνand the injectivity of the Fourier
transform gives µ=ν. For h∈R, let τhµbe the shifted measure τhµ(B) = µ(B+h).
Suppose ρ=f∗µis periodic with period h, then
(f∗µ)(x) = ρ(x) = ρ(x−h) = Z∞
−∞
f(x−h−y)dµ(y) = (f∗τh)µ(x)
and thus τ−hµ=µ. Thus the support of µis invariant by a shift by h. Since the support
is non-empty by the assumption Pknk>0, and obviously suppµ⊂γlZ,hhas to be a
multiple of γl. Now let p∈N0. If ρis pγl periodic, so is µ, whence nk+p=nk. Conversely,
if nk+p=nk,ρis obviously pγl-periodic. Let
ak(ρ) := 1
pγl Zpγl
0
e−i2πkx/(pγl)ρ(x)dx,
ak(µ) = 1
pγl Z[0,pγl[
e−i2πkx/(pγl)dµ =1
pγl
p−1
X
j=0
nje−i2πkj/p.
Then
ˆρ=X
k∈Z
ak(ρ)√2πδ2πk/(pγl),ˆµ=X
k∈Z
ak(µ)√2πδ2πk/(pγl).
(See [Sch73], p. 224–229 for an account on periodic distributions and their Fourier series.)
Together with (2.13) we obtain (2.12).
2.1. LAUGHLIN’S CYLINDER WAVE FUNCTION 35
Remarks. 1. The previous lemma shows in particular that ρcannot be constant, since con-
stant functions admit periods that are not multiples of γl.
2. The mapping (nk)7→ ρis actually injective.
3. If we set p= 1, we recover the result of Poisson’s summation formula for the filled Landau
level (see Chapter 1).
4. There is an analogue of (2.12) for finitely many particles:
ρ(x) = X
k∈Z
nk|ψk(x)|2,X
k∈Z
nk=N
⇒1
Npγl Z∞
−∞
e−2πikx/(pγl)ρ(x)dx =1
p
1
2πl2e−π2k2
p2γ21
NX
j∈Z
nje−i2πjk/p.
5. For even p, Laughlin’s wave function describes bosons and the occupation numbers can
be greater than 1. We can relax the condition nk∈[0,1] to nk≥0 and the requirement that
(nk) is polynomially bounded in −∞ and +∞(i.e., it defines a tempered measure).
Di Francesco et al. proved a necessary condition for the non-vanishing of bN(m1, .., mN)
([FGIL94], Property 3) and a product rule for the expansion coefficients bN(Property 5).
These properties also hold for the expansion coefficients aN. Before we state them, we give
some auxiliary definitions.
Definition 2.3. Let N∈N,m= (m1, .., mN)∈ZN. We say
•mis N-admissible if 0≤m1, .., mN≤p(N−1) and
∀k∈ {1, .., N}:
k
X
j=1
mj≥
k
X
j=1
p(j−1)
with equality for k=N:PN
j=1 mj=pN(N−1)/2.
•kis a renewal point of mif k∈ {0, .., N}and Pk
j=1(mj−p(j−1)) = 0.R(m)denotes
the set of renewal points of m. If mis N-admissible, R(m)contains 0and N.
•mis reducible if it has a renewal point in {1, ....N −1}, and irreducible in the opposite
case.
The notion of admissibility is related to the majorization partial order ([HLP64], p.45): for
α, α′∈RN
+, one writes α≺α′if the finite sequences (αj) and (α′
j), rearranged in decreasing
order (ασ(1) ≥..ασ(N), α′
τ(1) ≥..α′
τ(N)), satisfy Pk
j=1 α′
τ(j)≤Pk
j=1 ασ(j)for all k∈ {1, .., N}
with equality for k=N. Then
(mπ(1), .., mπ(N))is N-admissible for all π∈ SN
⇔(m1, .., mN)≺(0, p, 2p, .., (N−1)p).(2.14)
Furthermore, by a classical theorem ([HLP64], p.49) a sequence is majorated by the reference
sequence (0, .., (N−1)p) if and only if there is a doubly stochastic matrix Psuch that
(m1, .., mN)T=P(0, p, 2p, .., (N−1)p)T.(2.15)
36 CHAPTER 2. THERMODYNAMIC LIMITS
Renewal points of mcorrespond to block-diagonal stochastic matrices: if Pis block-diagonal
with an upper left k×kblock, kis a renewal point of m. By the Birkhoff-von Neumann
theorem, the set of doubly stochastic matrices is the convex hull of permutation matrices.
We will see that if aN(m1, .., mN)6= 0, mis majorated by (0, ..., p(N−1)) and the stochastic
matrix in (2.15) can be chosen as a simple average of ppermutation matrices: P=1
p(Pπ1+
.. +Pπp).
Lemma 2.4. Let (m1, .., mN)∈ZN.
1. If aN(m1, .., mN)6= 0, then (m1, .., mN)is N-admissible.
2. Suppose mis reducible with a renewal point k∈ {1, .., N −1}. Then
aN(m1, .., mN) = ak(m1, .., mk)aN−k(mk+1 −pk, .., mN−pk).(2.16)
Moreover, if m1≤.. ≤mN, then (m1, .., mN)is N-admissible if and only if (m1, .., mk)
is k-admissible and (mk+1 −pk, .., mN−pk)is N−k-admissible.
Note that the product (2.16) is 0 if one of the sequences does not have the correct admissibility.
Before we turn to the proof of the lemma, let us stress that the product rule (2.16) really
uses the cylinder geometry: remember that the Laughlin function expansion coefficients
are made up of two contributions. The first, bN(m1, .., mN), coming from the power of
the Vandermonde polynomial, always satisfies the factorization rule [FGIL94], while the
normalization contribution (see (2.2) and (2.4)) does so only in the cylinder geometry.
Proof. The lemma is based on similar results holding for the coefficients bN(m1, .., mN),
proved in [FGIL94]. We give slightly different proofs. The proof of the equivalence of admis-
sibilities follows [KTW01]. For simplicity suppose p= 3. The reasoning for other values of p
is strictly analogous.
1. Suppose aN(m1, .., mN)6= 0. Then by (2.2), we must have bN(m1, .., mN)6= 0. Writing
VN(z1, .., zN)p= ( X
π∈SN
sgnπzπ(1)−1
1...zπ(N)−N
N)p(2.17)
=X
π,σ,τ∈SN
sgnπστ
N
Y
j=1
zπ(j)+σ(j)+τ(j)−3
j(2.18)
we obtain
bN(m1, ..., mN) = X
π,σ,τ∈SN:
∀j:mj=π(j)+σ(j)+τ(j)−3
sgnπστ. (2.19)
Thus bN(m1, .., mN)6= 0 implies that there exist permutations π, σ, τ ∈ SNsuch that mj=
π(j) + σ(j) + τ(j)−3 for all j∈ {1, .., N}. Since πis in SN, for all k∈ {1, .., N},
k
X
j=1
π(j)≥
k
X
j=1
j(2.20)
with equality if k=N. Similar inequalities hold for σand τ. Adding them up, we obtain
k
X
j=1
mj=
k
X
j=1
(π(j) + σ(j) + τ(j)−3) ≥
k
X
j=1
(3j−3),(2.21)
2.1. LAUGHLIN’S CYLINDER WAVE FUNCTION 37
with equality for k=N, whence 1.
2. Now suppose (m1, .., mN)∈ZNis such that m1+.. +mk= 3k(k−1)/2 for some kin
{1, .., N −1}. Let π, σ, τ ∈ SNbe such that mj=π(j) + σ(j) + τ(j)−3 for all j∈ {1, .., N}.
Then there is equality in (2.21) and in (2.20) for π, σ, τ. This implies that π, σ and τleave
{1, .., k}and {k+ 1, .., N}invariant, Let π′∈ Skbe the restriction of πto {1, .., k}and
π′′ ∈ SN−ksuch that π(k+j) = π′′(j) + kfor all j∈ {1, .., N −k}. Define σ′, τ′, σ′′, τ′′ in
a similar way. Then mk+j−pk =π′′(j) + σ′′(j) + τ′′(j) for all j∈ {1, .., N −k}. Therefore
(2.19) becomes
bN(m1, .., mN)
=X
π′,σ′,τ′∈Sk:
∀j:mj=π′(j)+σ′(j)+τ′(j)−3
sgnπ′σ′τ′ X
π′′,σ′′,τ′′∈SN−k:
∀j:mk+j−3k=π′′(j)+σ′′(j)+τ′′(j)−3
sgnπ′′σ′′τ′′
=bk(m1, .., mk)bN−k(mk+1 −3k, .., mN−3k).
Since
N
X
j=k+1
((mj−3k)2−32(j−k−1)2)
=
N
X
j=k+1
(m2
j−32(j−1)2)−6k
N
X
j=k+1
(mj−3(j−1)) =
N
X
j=k+1
(m2
j−32(j−1)2),
the exponential factors satisfy a product rule too, whence
aN(m1, .., mN) = ak(m1, .., mk)aN−k(mk+1 −3k, .., mN−3k).
Finally, let (m1, .., mN) be N-admissible and increasing 0 ≤m1, .., mN≤pN −p. We obtain
immediately the k-admissibility of (m1, ..., mk). It remains to see that (mk+1−pk, .., mN−pk)
is N−k-admissible. For r≥1, write
r
X
j=1
(mk+j−pk −p(j−1)) =
k+r
X
j=k+1
(mj−p(j−1))
=
k+r
X
j=1
(mj−p(j−1)) ≥0
with equality for r=N. If in addition m1≤.. ≤mN, the previous inequalities imply
0≤mk+1 −pk ≤.. ≤mN−pk ≤p(N−k−1), so that (mk+1 −pk, .., mN−pk) is
N−k-admissible. Conversely, if (m1, .., mk) is k-admissible, (mk+1 −pk, .., mN−pk) is
N−k-admissible, one can check that (m1, ., mN) is N-admissible.
Depending on the parity of p, the coefficients aN(m1, .., mN) are either symmetric or anti-
symmetric functions of the mj’s. Thus if aN(m1, .., mN)6= 0, any permutation of the mj’s
is N-admissible, i.e., (m1, .., mN) is majorized by the reference vector (0, p, .., p(N−1)) and
can be written as a doubly stochastic matrix times this vector. In the proof of the previous
lemma, for p= 3, the only sequences that occur are of the form
mj=π(j) + σ(j) + τ(j)−3, j = 1, .., N.
38 CHAPTER 2. THERMODYNAMIC LIMITS
Put differently,
(m1, .., mN)T=1
3(Pπ+Pσ+Pτ)(0,3, .., 3(N−1))T.
Thus we can choose the stochastic matrix as a simple average of permutation matrices. This
holds for all p∈Nwith the obvious modifications.
If aN(m1, .., mN)6= 0, the N-tupel (mj) is majorized by (p(j−1)). This means that the
mj’s are less spread out than 0, p, .., p(N−1). The following lemma quantifies this “less” for
irreducible sequences by comparing their variances. This amounts to a comparison of sums
of squares, since
1
N2X
j<k
(mj−mk)2=1
N
N
X
j=1
m2
j−(1
N
N
X
j=1
mj)2=1
N
N
X
j=1
m2
j−(1
N
N
X
j=1
p(j−1))2.
Lemma 2.5. Let m∈ZNbe N-admissible, irreducible and increasing: m1< .. < mN. Then
N
X
j=1m2
j−p2(j−1)2≤ −(p+ 1)(N−1).(2.22)
If m1≤.. ≤mN, the upper bound is −p(N−1).
Proof. Let 0 < m1< .. < mN≤p(N−1) be N-admissible. Then
∀k∈ {0, .., N}:nk:=
N
X
j=1mj−p(j−1)≥0,
n0=nN= 0, and mj=p(j−1) + nj−nj−1, j = 1, .., N. The monotonicity of (mj)
translates into
nj+1 −2nj+nj−1+p≥1, j = 1, .., N. (2.23)
A summation by parts gives
N
X
j=1
(m2
j−p2(j−1)2) = −
N
X
j=1
nj(p+nj+1 −2nj+nj−1)≤ −(p+ 1)
N−1
X
j=1
nj
The irreducibility of (m1, .., mN) implies nj≥1 for j∈1, .., N −1, whence (2.22). If
m1≤.. ≤mN, the lower bound in (2.23) is 0, and the summation by parts gives the upper
bound −p(N−1).
The previous lemma will be important later on. It allows to prove that long polymers in an
associated polymer system have small activity.
The factorization rule (2.16) leads to a representation of Laughlin’s wave function in terms
of “building blocks” that is reminiscent of polymer systems (see next section). Let Γ :=
{{j, .., j +n−1} | j∈Z, n ∈N}. Elements of Γ will be referred to as rods or polymers.
For X, Y ∈Γ, we write X < Y if x < y for all x∈X, y ∈Y. For N∈N, let PN
be the set of ordered partitions (X1, .., XD) of {0, .., N −1}into rods: X1< .. < XD,
{0, ..., N −1}=X1˙
∪.. ˙
∪XD. For j∈Z,N∈Nand odd p, define:
u{j,..,j+N−1}=X
0≤m1≤..≤mN≤pN−p
irreducible
aN(m1, .., mN)ψm1+pj ∧.. ∧ψmN+pj.
2.1. LAUGHLIN’S CYLINDER WAVE FUNCTION 39
(The definition for even pis similar: ⊗sinstead of ∧.) In view of (2.3), the monomer functions
are simply
u{j}=ψpj.(2.24)
Thus for X∈Γ with N(X)≥2,, uXis an antisymmetric function of N(X) = |X|complex
variables made up of Gaussians ψkwith k∈pX. Moreover, the magnetic translations act as
t(jpγlex)⊗N(X)uX=uj+X, j ∈Z.
Lemma 2.6. Let p∈Nodd, N∈Nand PNthe set of increasingly ordered partitions of
{0, .., N −1}as defined above. Let I⊂Rand Ω = I×[0,2πR]. Then
ΨN=X
DX
(X1,...,XD)∈PN
uX1∧.. ∧uXD(2.25)
||ΨC
N||2
L2(ΩN)=X
DX
(X1,..,XD)∈PN
Φ(X1)·.. ·Φ(XD) (2.26)
where Φ(X) := ||uX||2
L2(ΩN(X)).
If pis even, a formula similar to (2.25) holds with symmetric products ⊗sinstead of wedge
products; (2.26) holds unchanged.
Proof. We start with
ΨC
N=X
0≤m1<..<mN≤p(N−1)
aN(m1, .., mN)ψm1∧.. ∧ψmN.(2.27)
Only admissible sequences contribute to the sum. Let mbe an admissible sequence with
renewal points R(m) = {r0, .., rk}, 0 = r0< r1< .. < rk=N. Then
aN(m1, .., mN)ψm1∧.. ∧ψmN=ar1(m1, ..., mr)ψm1∧.. ∧ψmr
∧... ∧ark−rk−1(mrk+1 −prk, ..., mr−prk)ψmrk−1+1 ∧.. ∧ψmN,
and the subsequences (m1, ..., mr1),..,(mrk+1 −prk, ..., mr−prk) are irreducible. When R=
{r0, .., rk}is kept fixed and we sum over sequences mwith renewal points R(m) = R, we
obtain
ur1∧t(r1pγl)⊗(r2−r1)ur2−r1∧.. ∧t(rk−1pγl)⊗(N−rk−1)uN−rk−1.(2.28)
Each strictly increasing sequence 0 = r0< r1.. < rk=Ndefines a partition of {0, .., N −1}:
the points are interpreted as starting points of rods, we get the partition X1={0, .., r1−1},
X2={r2, .., r3−1},.., Xk={rk−1, .., N −1}. The function (2.28) is nothing else but
uX1∧.. ∧uXk. Thus if in (2.27) we sum first over sequences mwith a given set of renewal
points and then over all possible sets, identified with the corresponding partition, we obtain
(2.25). For the normalization constant, note that
||ΨC
N||2
L2(ΩN)=X
0≤m1<..<mN≤p(N−1) |aN(m1, .., mN)|2
N
Y
j=1 ||ψmj||2
L2(Ω)
||u{j,..,j+N−1}||2
L2(ΩN)=X
0≤m1<..<mN≤p(N−1)
irred.
|aN(m1, .., mN)|2
N
Y
j=1 ||ψmj||2
L2(Ω).
(2.29)
40 CHAPTER 2. THERMODYNAMIC LIMITS
This follows from the orthogonality of the functions with respect to y-integration:
{m1, .., mN} 6={n1, .., nN} ⇒
Z[0,2πR]N(ψm1∧.. ∧ψmN)ψn1∧.. ∧ψnN(x1+iy1, .., xN+iyN)dy1..dyN= 0.
Summing in two steps, as for the proof of (2.25), gives (2.26).
Example. It is instructive to work out by hand the polymer representation for two particles
and p= 3. We have
ΨC
2(z1, z2) = 1
√2(ez2/R −ez1/R)3e−(x2
1+x2
2)/2l2e−1
2γ232P2
j=1(j−1)2
2πRl√π
=1
√2(e3z2/R −e3z1/R)−3(e2z2/Rez1/R −ez2/Re2z1/R)e−9γ2/2
2πRl√π
=e1
2γ2(02+32−9)ψ0∧ψ3−3e1
2γ2(12+22−9)ψ1∧ψ2
=ψ0∧ψ3−3e−2γ2ψ1∧ψ2.
This can be rewritten as
ΨC
2=u{0}∧u{1}+u{0,1}, u{0}=ψ0, u{1}=ψ3, u{0,1}=−3e−2γ2ψ1∧ψ2.
The previous lemma gives the normalization constant as a polymer partition function. It
turns out that the correlation functions are closely related to those of the associated polymer
system, which we describe in the next section. Moreover, there is an analogy to the dimer
model introduced by Rokhsar and Kivelson [RK88] as an idealization of resonating valence
bond states (RVB). Consider a lattice of spin 1/2-particles. To each dimer covering of the
lattice, we can associate a state where lattice sites in a dimer form a singlet. The sum of states
corresponding to different dimer coverings is an RVB state. Different dimer coverings do not
give rise to orthogonal functions. Nonetheless, [RK88] propose a model of hard core dimers
on a lattice, where wave functions corresponding to different dimer coverings are defined as
orthogonal. The representation obtained in the previous lemma is similar: Laughlin’s wave
function is a sum of contributions from different polymer coverings, and the contributions
from different coverings are orthogonal. Thus we may call the Laughlin function a quantum
polymer system.
2.1.2 Associated polymer system
By the results of the previous subsection, the L2norm squared CNof Laughlin’s wave function
is a polymer partition function. More precisely, it is the grand-canonical partition function
for a system of hard rods on the one-dimensional lattice Z. The normalization CNsatisfies a
recurrence relation known in stochastics as a renewal equation. In this subsection, we give a
short overview of results on polymer systems, discrete renewal processes and thermodynamic
limits of systems of hard rods on Z. This will allow us to infer results on the asymptotics of
(CN) and will serve as a useful guiding intuition for the thermodynamic limits of correlation
functions in Laughlin’s state.
We begin with a brief overview on systems of polymers on a lattice. We proceed with a short
summary on discrete renewal processes, and then present results on the thermodynamic limits
2.1. LAUGHLIN’S CYLINDER WAVE FUNCTION 41
of systems of hard rods. Let us summarize the main aspect of the hard rod system. The
recurrence relation satisfied by the partition function is equivalent to the formal power series
identity
1 + ∞
X
n=1
Cntn=1
1−P∞
n=1 αntn.
The number αn≥0 is the activity of a rod of lenght n. We assume that the power series
have a non-vanishing radius of convergence (this amounts to a stability assumption on the
activity (αn)). Then one of the following conditions must hold:
1. There exists some r > 0 such that Pnαnrn= 1 and Pnnαnrn<∞. (The renewal
process with interarrival distribution (rnαn) is positive recurrent.)
2. There exists some r > 0 such that Pnαnrn= 1, but Pnnαnrn=∞. (The renewal
process with interarrival distribution (rnαn) is null recurrent.)
3. The radius of convergence rof Pnαntnsatisfies Pnrnαn<1. (The renewal process
with interarrival distribution (rnαn) is transient.)
In the first case, the polymer system has a well-behaved thermodynamic limit (see Propo-
sition 2.9 below). In the second and third cases, one needs in general additional knowledge
on the coefficients (αn). Typically, in those cases for large volumes, the polymer system will
tend to fill the whole volume with a few number of long rods ([Gia07]). For Laughlin’s cylin-
der function, we will see that we are actually in the first case for sufficiently thin cylinders.
What happens on thick cylinders is open.
The summary on lattice polymer systems follows [GK71]. An account on discrete renewal
processes can be found e.g. [Fel62] in the sections on recurrent events. The recurrence relation
for the one-dimensional system of hard rods is used in [IVZ06]. For monomer-dimer systems,
it boils down to the two-step recurrence relation of [HL72], Section III. Generating functions
and renewal theory have been applied to a large variety of biochemical models and the models
are still an active field of research in probability theory [Lif64, BD77, Fis84, CGZ06, Gia07].
These are frequently termed polymer models. Let us stress however that there is a slight
difference in the use of the word “polymer”: in this thesis, we use “polymer” essentially
as a synonym for “hard rod” and look at partitions of intervals into rods. Renewal points
correspond to starting points of rods. In the biochemical models evoked above, one typically
looks at the configuration of one single long polymer (a molecular chain, a double string of
DNA..), possibly in more than one dimension. Renewal points are points where “something
happens”, e.g., the polymer chain hits a wall.
In a lot of models, the coefficients (αn) are functions of some parameter. When the parameter
is varied, it is possible to switch between one of the three cases above. This gives rise to a
phase transition; depending on the model described, one finds among others, the denomina-
tions localization-delocalization transition, wetting transition, denaturation transitions. The
intermediate case 2. corresponds to a critical regime.
Lattice polymer systems
We start with a brief overview following [GK71]. A polymer is a set X⊂Zd. The cardinality
of Xis denoted N(X) and may be viewed as a number of sites in X.Monomers are polymers
with N(X) = 1, dimers correspond to N(X) = 2, n-mers to N(X) = n(n∈N). The activity
is a map Φ : Γ →[0,∞[. Allowed configurations of the system are subsets {X1, .., XD}of Γ
42 CHAPTER 2. THERMODYNAMIC LIMITS
that define a partition of Λ: Λ = X1˙
∪·.. ˙
∪XD. The activity Φ defines a probability measure
Pthrough
PΛ({{X1, .., XD}}; Φ) = Φ(X1)·.. ·Φ(XD)/QΛ[Φ]
with the polymer partition function
QΛ[Φ] := X′
{X1,..,XD}
Φ(X1)·.. ·Φ(XD).(2.30)
The sum ranges over all partitions of Λ into polymers. If the activity of monomers is Φ({j}) =
1, the monomers can be interpreted as empty sites and we can rewrite QΛas
QΛ[Φ] = X
X1,..,XD⊂Λ:
N(Xj)≥2
Φ(X1)·..Φ(XD)e−βU(X1,..,XD)
where U(X1, .., Xd) is 0 when Xi∩Xj= for all i6=j, and ∞else. Thus QΛis the grand-
canonical partition function for a hard core interaction. The pressure and correlation func-
tions are defined through
βpΛ:= 1
|Λ|log QΛ,
ρΛ(X1, .., Xp) := X
{Y1,..,Yn}:
X1˙
∪..XD˙
∪Y1˙
∪..Yn=Λ
Φ(X1)..Φ(Xp)Φ(Y1)·.. ·Φ(Yn)/QΛ.
The correlation functions give the probability of finding the polymers X1, .., Xp. The sum
rule
∀x∈Λ : X
X∋x
ρΛ(X) = 1 (2.31)
expresses the fact that every site of Zis covered by exactly one polymer. One can show that
QΛ˙
∪Λ′≥QΛQΛ′.(2.32)
For ξ > 0, ξNΦ denotes the rescaled activity
(ξNΦ)(X) := ξN(X)Φ(X).
The probability measure PΛand the correlation functions ρΛare invariant with respect to
such a rescaling, whereas
QΛ[ξNΦ] = ξ|Λ|QΛ[Φ].
The activity is called stable if X
X∋0
Φ(X)
N(X)<∞
and translationally invariant if Φ(a+X) = Φ(X) for all a∈Zd.
We will be interested in one-dimensional lattice systems where Φ(X) = 0 if Xis not of the
form {j, ..j +N(X)−1}for some j∈Z. Such systems are closely related to discrete renewal
processes, which we now briefly describe. Details can be found in the book by W. Feller
[Fel62] in the sections on recurrent events.
2.1. LAUGHLIN’S CYLINDER WAVE FUNCTION 43
Discrete renewal processes
A point process on Nis a (discrete) renewal process if the waiting times between consec-
utive events are independent identically distributed random variables. More precisely, let
(Ω,F, P) be a probability space, E:= P(N), and E0the standard σ-algebra on E1. Let
R: (Ω,F)→(E, E0) be a random variable. We call Rarenewal process with interarrival
distribution (pn)n∈N∪{∞} if there is a family (Tn)n∈Nof N∪{∞}-valued, independent iden-
tically distributed random variables on Ω such that P(Tj=n) = pnfor all n∈N∪ {∞}
and
R(ω) = {Sn(ω)|n∈N, Sn(ω)<∞}
for all ω∈Ω, where
Sn(ω) =
n
X
j=1
Tj(ω).
The elements of R(ω) are called renewal points, and the renewal probability is
un:= P(“nis a renewal point”) = P(n∈R) = P(∃k∈N0:Sk=n).
If Tjtakes the value ∞with a non-zero probability, the process is called transient.
Example: Bernoulli process. Let 0 ≤p, q ≤1 with p+q= 1, and let (Xn)n∈Nbe independent
random variables such that P(Xn= 0) = q,P(Xn= 1) = p. Define Ras
R(ω) := {n∈N|Xn(ω) = 1}.
Then Ris a renewal process with geometric interarrival distribution and constant renewal
probability:
∀n∈N:pn=pqn−1, un=p=1
Pkkpk
.
More generally, the renewal theorem says that if (pn) is aperiodic (i.e. the greatest common
divisor of {n∈N|pn>0}is 1)
lim
n→∞un= 1/µ, µ := X
n∈N∪{∞}
npn(2.33)
with the convention 1/∞= 0 and ∞ · 0 = 0. The proof makes heavy use of the renewal
equation
∀n≥1 : un=p1un−1+p2un−2+.. +pn−1u1+bn, bn:= P(S1=n).(2.34)
Here, bn=pn, but the equation holds also for renewal processes where T1=S1is allowed to
have a distribution different from (pj). These are called delayed renewal processes. This is
best seen by writing
un=P(∃k:Sk=n) = P(S1=n) +
n
X
k=2
n−1
X
j=1
P(Sk=nand Tk=j)
=P(T1=n) +
n−1
X
j=1
n
X
k=2
P(Sk−1=n−j)P(Tk=j)
=bn+
n−1
X
j=1
un−jpj.
1Subsets A⊂Nare identified with their characteristic functions χA∈ {0,1}N, and on {0,1}Nwe consider
the σ-algebra generated by the cylinder sets.
44 CHAPTER 2. THERMODYNAMIC LIMITS
This recurrence relation is equivalent to the formal power series identity
∞
X
n=1
untn=P∞
n=1 bntn
1−P∞
n=1 pntn
or, if bn=pnand with u0:= 1
∞
X
n=0
untn=1
1−P∞
n=1 pntn.
For a given interarrival distribution (pn)n∈N∪{∞}, the problem “find a probability distribu-
tion (bn)n∈Non Nso that the associated delayed renewal process has a constant renewal
probability” has a solution if and only if µ=Pn∈N∪{∞} npn<∞. In this case the unique
solution is bn=P∞
k=npk/µ and the constant renewal probability is 1/µ.
If µ < ∞, the delayed process with constant renewal probability can be extended to a
stationary process on Zas follows: Let S0: Ω → {0,−1,−2, ..},S1: Ω →Nbe two random
variables with joint distribution
m≤0,1≤n:P(S0=mand S1=n) = pn−m/µ.
S0and S1represent the renewal points that are closest to the origin 0. Note that
P(S1=n) = ∞
X
k=n
pk/µ =bn, P(S1−S0=n) = npn/µ.
Let (Tn)n∈Z\{1}be a family of independent identically distributed random variables with
values in N,P(Tk=n) = pn. For n≤ −1, let Sn:= S0−P|n|
k=1 T−k, and for n≥2,
Sn:= S1+Pn
k=2 Tk. We will call
R: Ω → P(Z), ω 7→ {Sn(ω)|n∈Z}
a stationary renewal process on Z. The renewal probability un=P(n∈R(ω)) is constant:
for all n∈Z,un= 1/µ. Moreover, the sequence (Sn)n≥1defines the delayed renewal process
with constant renewal probability described above. If we look at (Sn)n≥1and condition on
S0= 0, we recover the (non-delayed) renewal process on N.
A subset of Zcan also be described by the sequence of its points, arranged in increasing
order and suitably numbered. Thus the renewal process Ron Zis described by the sequence
(Sn)n∈Z, the numbering is chosen so that S0≤0< S1. Equivalently, we may describe the
process through its waiting intervals
Xn={Sn, Sn+ 1, ..., Sn+1 −1}, n ∈Z.
These define a partition of Z. If we interpret the waiting intervals as polymers, the dis-
tribution of Ris a probability distribution on infinite polymer configurations; it turns out
that this distribution arises as the thermodynamic limit of suitable finite volume polymer
distributions PΛ.
2.1. LAUGHLIN’S CYLINDER WAVE FUNCTION 45
Rods on a one-dimensional lattice
Let us go back to the one-dimensional lattice polymer system. Suppose the activity is trans-
lationally invariant and supported by “rods”:
Φ(X) = (αN(X),if X={j, j + 1, .., j +N(X)−1}for some j∈Z,
0,else
for some sequence (αn) of non-negative numbers with Pαn>0. For N∈N, let
CN=Q{0,..,N−1}[Φ].
Then (2.30) can be rewritten as
CN=X
n1+..+nD=N
αn1·.. ·αnD.
Together with C0:= 1, this implies the formal power series identity
∞
X
n=0
Cntn= (1 −∞
X
n=1
αntn)−1(2.35)
and the recurrence relation
Cn=α1Cn−1+α2Cn−2+.. +αn−1C1+αn(n≥1).(2.36)
The thermodynamic limit of the pressure exists and can be expressed in terms of (αn).
Lemma 2.7. Let r:= max{t≥0|P∞
n=1 αntn≤1}. Then with the convention −log 0 := ∞
we have
βp := lim
N→∞
1
Nlog CN= sup 1
Nlog CN=−log r.
Proof. The supermultiplicativity (2.32) together with the translational invariance of the ac-
tivity implies CN+M≥CNCM. This shows that lim 1
Nlog CN= sup 1
Nlog CN. The limit is
−log rC, with rCthe radius of convergence rCof PCntn. It remains to show that r=rC.
Note first that (2.36) implies Cn≥αn, thus rC≤rα, where rαis the radius of convergence
of Pαntn. We continue by looking more closely at r. It follows from Pnαntn=∞if
t > rαthat r≤rα. If rα= 0, obviously r= 0. If rα>0, there are two possibilities: either
r < rαand α(r) = 1, r > 0, or r=rα, in which case Abel’s theorem on power series with
nonnegative coefficients implies
α(r) = α(rα) = lim
tրrα
α(t)≤1.
Several cases can be distinguished:
1. rα=r= 0. Then by rC≤rαwe also have rC= 0.
2. rα> r > 0. Then α(r) = 1, (2.35) is an identity of functions on [0, r[, and rC≥r.
Moreover ∞
X
n=0
rnCn= lim
tրr
1
1−α(r)=∞
whence rC=r.
46 CHAPTER 2. THERMODYNAMIC LIMITS
3. rα=r > 0. Again, (2.35) is an identity of functions on [0, r[, rC≥r=rα. Since
rC≤rαbecause of Cn≥αn, we have rC=rα=r.
Remarks: 1. The quantity rcan be characterized in a slightly more eloquent way as follows:
the equation P∞
n=1 αntn= 1, t∈[0,∞[ has either no solution or a unique solution. In the
first case, requals the radius of convergence Rof Pnαntn. In the second case, ris the
unique solution of the equation and lies necessarily in ]0, R].
2. Generalized ensemble. Lifson [Lif64] points out a relation between the “generating func-
tions approach” and “Guggenheim’s divergent generalized partition functions”. The generat-
ing function PCntncan be given the meaning of a partition function. The partition function
CVis the grand-canonical partition function for a system of rods on the lattice {0, .., V −1}
with a hard core interaction. Let p > 0 be a pressure, βan inverse temperature. Then the
generating function
C(e−βp) = ∞
X
V=0
e−βpV CV
can be thought of as a partition function too. It corresponds to the so-called generalized
ensemble. More generally, let Q(β, {Nj}, V ) be the canonical partition function for a system
of particles of different species k= 1, ., K, each present with Nkparticles, at temperature
β−1, in a volume V; then the generalized partition function would be
Γ(β, pe,{µj}) = X
{Nj}
∞
X
V=0
e−β(pV +PjµjNj)Q(β, {Nj}, V ) = ∞
X
V=0
e−βpV Ξ(β, V, {µj}).
This partition functions differs from other partition functions (e.g. canonical, grand-canonical,
isothermal-isobaric..): it is a function of intensive parameters only. There is no free exten-
sive parameter. Thus the usual procedure of letting extensive parameters go to infinity for
taking thermodynamic limits does not work. In the grand-canonical ensemble, the pressure
is obtained through
βp = lim
V→∞
1
Vlog Ξ(β, V, {µj})
if the limit exists. For this value of p, the generalized partition function Γ diverges. Thus
the procedure that replaces thermodynamic limits in the generalized ensemble is to look for
the point where Γ diverges, whence the name divergent partition function. This is just what
we do: in Lemma 2.7, we express the pressure in terms of the radius of convergence of C(t),
i.e., we look for the point where C(t) starts to diverge.
The recurrence relation (2.36) is similar to the renewal equation (2.34). Let us make the
connection to renewal processes more explicit. In the following, we suppose that the power
series Pαntnhas a non-vanishing radius of convergence; equivalently, r > 0 or βp < ∞.
This is the case if and only if ξNΦ is stable for some ξ > 0, since
X
X∋0
ξN(X)Φ(X)
N(X)=∞
X
N=1
ξNαN
NX
j∈Z:
0∈{j,..,j+N−1}
1 = ∞
X
N=1
αNξN.
Let rbe defined as in Lemma 2.7 and pn:= rnαn,p∞:= 1−Ppn. This defines a probability
distribution on N∪{∞}. Let R0be a renewal process on Nwith interarrival distribution (pn)
and renewal probabilities (un)n∈N. Combining pn=rnαn,C0=u0= 1 and the recurrence
relations (2.34), (2.36), we see that un=rnCnfor all n∈N0. The renewal theorem (2.33)
gives us a more refined version of Lemma 2.7.
2.1. LAUGHLIN’S CYLINDER WAVE FUNCTION 47
Lemma 2.8. Let rbe defined as in Lemma 2.7. If {n∈N|αn6= 0}has greatest common
divisor 1,
lim
n→∞rnCn=((P∞
n=1 nαnrn)−1,if P∞
n=1 αnrn= 1,
0,if P∞
n=1 αnrn<1.
Remark: If Pnαnrn= 1 and Pnnrnαn=∞,rnCn→0. If Pnαnrn<1, (2.35) implies
the stronger statement
∞
X
n=1
rnCn= (1 −∞
X
n=1
rnαn)−1.
The renewal process associated to the polymer system is not only useful to derive something
about the asymptotics of the partition functions, but also serves a better understanding of
correlation functions and their thermodynamic limit.
The finite volume correlation functions can be described in terms of conditional probabilities
of the renewal process. Let 0 < S1< .. < Sn< .. be the renewal points, and let
I0:= {{0, .., S1−1},{S1, .., S2−1}, .., {Sn−1, .., Sn−1}, ..}
be the set of waiting intervals. Then if Xi6=Xjfor i6=j,
ρ{0,..,N−1}(X1, .., Xk) = P(X1, .., Xk∈ I0|N∈R0).
If the associated renewal process has finite mean, we can define a stationary renewal process
Ron Z. Let Ibe the collection of its waiting intervals. The previous relation becomes
ρ{a,..,b−1}(X1, .., Xk) = P(X1, .., Xk∈ I | a, b ∈R).
The thermodynamic limit a→ −∞,b→ ∞ is given by the probability distribution of R,
without any conditions.
Proposition 2.9. Suppose Pnαnrn= 1,µ:= Pnnαnrn<∞, and gcd{n|αn6= 0}= 1.
Let Rbe the associated stationary renewal process and Pthe distribution of R. Let Ibe the set
of waiting intervals (Iis a random variable). Then for all rods X1, .., Xk, the thermodynamic
limit of the correlation functions for the polymer system with activities Φ(X) = αN(X)exists
and equals
ρ(X1, .., Xk) := lim
a→−∞
b→∞
ρ{a,..,b−1}(X1, .., Xk) = (P({X1, .., Xk} ⊂ I),if Xi6=Xjfor i6=j,
0,else.
The correlation functions are translationally invariant and mixing: for all sets of rods X1, .., Xm,
Y1, .., Yn, we have
ρ(X1, .., Xm, Y1+k, .., Yn+k) =
k→∞ ρ(X1, .., Xm)ρ(Y1, .., Yn) + O(|rkCk−µ−1|).(2.37)
Proof. Let X1, .., Xkbe some rods. If Xi∩Xj6=∅for some i6=j, the probability of finding
these rods as well as the probability that they are waiting intervals for the renewal process is
0. Thus we can suppose that they are non-overlapping and without loss of generality assume
X1< ... < Xk, that is they are numbered from left to right. Write Xk={sk, sk, .., sk+
48 CHAPTER 2. THERMODYNAMIC LIMITS
N(Xk)−1}={sk, .., ek−1}and take |a|, b large enough so that X1, .., Xkare contained in
{a, .., b −1}. Then, the correlation function can be written as
ρ{a,..,b−1}(X1, .., Xk) = X
Y1,..,YD:
Y1˙
∪.. ˙
∪Xk={a,..,b−1}
αN(Y1)...αN(YD)αN(X1)...αN(Xk)/Q{a,..,b−1}
=αN(X1).αN(Xk).Q{a,..,b=1}\(X1·∪..·∪XD}/Q{a,..,b−1}
=rb−aαN(X1).αN(Xk)Cs1−aCs2−e1..Cb−ek/(rb−aCb−a)
=us1−apN(X1)..pN(Xk)ub−ek/ub−a
−→
b−a→∞ µ−1pN(X1)us2−e1...ub−ekµ−1/µ−1
=µ−1pN(X1)us2−e1...pN(Xk)(2.38)
=P(X1, .., Xk∈ I).
The expression (2.38) shows that ρis translationally invariant. For the clustering properties,
let X1, .., Xm,Y1, ., Ynand denote as before their starting and endpoints with s1, .., sm, e1, .., em,
s′
1, .., e′
n. Then, for sufficiently large k,
ρ(X1, .., Xm, Y1+k, .., Yn+k) = µ−1pN(X1)us2−e1...pN(Xm)us′
1+k−empN(X1)us′
2−e′
1...pN(Yn)
=ρ(X1, .., Xm)µus′
1+k−emρ(Y1, .., Yn)
whence (2.37).
Remarks: 1. Semi-infinite line. The same reasoning shows that the limits of correlation
functions ρ{0,..,N−1},N→ ∞, exist and can be described in terms of the renewal process on
N.
2. Probability distribution on the length of rods. The probability for finding a given rod
X={j, .., j +N(X)−1}is ρ(X) = qrN(X)αN(X). Let k∈Z. The probability that kis in a
rod, or waiting interval, Xof length N(X) = nis, in loose notation
P(k∈X, N(X) = n) = X
X:k∈X,
N(X)=n
ρ(X) = X
j∈Z:
j≤k≤j+n−1
qrnαn=qnrnαn=µ−1npn.
The sum over nis 1: this is the sum rule PX∋0ρ(X) = 1. Thus if µ=q−1is finite, there
is a natural probability distribution (qnrnαn) on the length of polymers. In the renewal
theory picture, this distribution is called the size-biased distribution. The expected length
of the polymer containing a given point (in the renewal setting, the expected lifetime or age
at death) is ∞
X
n=1
qn2rnαn=∞
X
n=1
µ−1n2pn.
The lifetime expectation can be infinite although the process has finite mean µ. This is
related to the so-called inspection paradox.
3. Rate of clustering. In the previous proposition, the rate of clustering is determined by the
speed of convergence of un→µ−1, where un=rnCnis the renewal probability. We have
the following: ∞
X
n=1 |un−µ−1|<∞ ⇔ ∞
X
n=1
n2pn<∞.
2.1. LAUGHLIN’S CYLINDER WAVE FUNCTION 49
The implication ”⇒” can be shown by taking the limit tր1 in
∞
X
n=0
(un−µ−1)tn=1
1−P∞
n=1 pntn−1
P∞
n=1 npn(1 −t)=1
1−p(t)−1
p′(1)(1 −t).
The converse uses results on rates of convergence in renewal theory, see [Sto65]. A closer
look at the generating functions also shows that if the radius Rof convergence of Pnrnαn
is strictly larger than r,ungoes to µ−1exponentially fast:
rnCn−q=un−µ−1=O((r/R)N).
µ=∞as a close-packing limit. The previous proposition is of no help when the renewal
process has infinite mean µ. The case µ=∞should be thought of as a close-packing limit
(in [Gia07], Section 1.2 it is shown that in this regime the average density of renewal points,
i.e. the “contact density” in the picture used in the book, is typically 0). We illustrate this
by an example. Let ν∈]0,1[, c > 0 and
αn:= δ
S
1
n1+ν, S := ∞
X
n=1
n−1−ν.
Then α(t) = P∞
n=1 αntnhas radius of convergence 1. We look at sequences of volumes ΛN=
{−N/2, .., N/2−1},Neven, and ask for the limiting behavior of the polymer correlation
function
ρN(X) = Ca+N/2αb−aCN/2−b
CN
, X ={a, .., b −1}.
Depending on the value of the parameter c, we are in one of the three cases sketched in the
beginning of this subsection:
1. When δ > 1, the equation α(t) = 1 has a solution r < 1 and we are in the case where the
associated renewal process has finite mean, and the thermodynamic limit is described
by Proposition 2.9. In particular, limN→∞ ρN(X) exists and is strictly positive.
2. When δ= 1, the solution to the equation α(t) = 1 is r= 1, and Pnnαnrn=
Pnn−νδ/S =∞. Then it is known ([Don97], Theorem B, [Gia07], Theorem A.7)
that there exists a cν>0 such that
Cn∼
n→∞
cν
n1−ν.
As a consequence
Ca+n/2Cn/2−b
Cn∼
n→∞
c2
ν
n1−ν(1
2+a
n)(1
2−b
n)ν−1
and limN→∞ ρN(X) = 0 for every fixed X.
3. When δ < 1, the equation α(t) = 1 has no solution. It is known [Gia07], Theorem A.4,
that
un∼
n→∞
αn
(1 −δ)2=δ
S(1 −δ)2
1
n1+ν
and we have again limNρN(X) = 0 for each fixed rod X.
50 CHAPTER 2. THERMODYNAMIC LIMITS
Thus for this example, if the associated renewal process has infinite mean, each fixed rod X
has a probability that vanishes in the limit N→ ∞. Since on the other hand the sum rule
X
Y:k∈Y
ρN(Y) = 1
(see (2.31)) must hold for all N∈2Nand k∈ {−N/2, .., N/2−1}, we conclude that long
intervals tend to be filled with long polymers.
Later, we will apply the results of this subsection to Laughlin’s wave function. We can think
of this function as a polymer system where each polymer has internal degrees of freedom. We
prove in the next subsection that on thin cylinders the associated renewal process has finite
mean. What happens for large radius is open. If it happens that the associated renewal
process has infinite mean, the previous considerations tell us that the system is governed
by long polymers, and any statement about the thermodynamic limit would require a closer
look at the internal structure of long polymers.
2.1.3 Normalization
By Lemma 2.6, Laughlin’s cylinder wave function is closely related to a system of hard rods
on a one-dimensional lattice with activity
Φ(X) = ||uX||2
L2(I×[0,2πR])N(X).
The polymer representation is valid for any choice of interval I⊂R. Here, we will choose
I=R. This choice leads to a translationally invariant activity. Indeed, we have seen that
translation of a rod Xresults in magnetic translation of the function:
ua+X=t(a·pγlex)⊗N(X)uX.
Hence if we choose the infinite cylinder as domain of integration, I=R, we obtain a trans-
lationally invariant activity and we can write
Φ(X) = ||uX||2
L2((R×[0,2πR])N(X))=αN(X)
αN=X
0≤m1≤..≤mN≤p(N−1)
irred.
|aN(m1, .., mN)|2.(2.39)
Let us recall that the one-particle density of Laughlin’s N-particle function is a sum of
Gaussians with centers inside [0, pNγl]. Thus the one-particle density decays exponentially
outside the finite cylinder [0, pNγl]×[0,2πR], even though we integrate on infinite cylinders
R×[0,2πR]. By a slight abuse of language, we shall refer to the limit N→ ∞ as the limit
of infinite cylinders.
In view of the results summarized in the previous section, two questions on the associated
polymer system naturally arise. The first question is whether the activity is stable or, more
generally, whether it can be rescaled to a stable activity. On p.46, we have observed that the
activity (αn) can be rescaled to a stable activity if and only if the radius of convergence of
Pnαntnis strictly positive. From Lemma 2.6 we know that the L2norm squared of Laugh-
lin’s function, CN=||Ψc
N||2is the polymer partition function associated with the activity
(αn). It follows from the relation (2.35) that Pnαntnhas a positive radius of convergence if
2.1. LAUGHLIN’S CYLINDER WAVE FUNCTION 51
and only if PnCntnhas. Thus the question of the stability of the associated activity boils
down to the question whether the radius of convergence rof PnCntnis strictly positive.
Lemma 2.10 below shows that r > 0 for all (finite) cylinder radii R.
The second question is whether the associated renewal process has a finite mean µ. By
Lemma 2.8, the normalization constant (Cn) obeys the asymptotics rnCn→q(the aperiod-
icity assumption of Lemma 2.8) is fulfilled because in the system associated with Laughlin’s
function, monomers have activity α1= 1). The limit qis the inverse of the expected inter-
arrival time of the renewal process, q=µ−1. Thus the question whether µ < ∞reduces
to the question whether limnrnCnis strictly positive or not. This question is addressed in
Theorem 2.12, where we prove that q > 0 on sufficiently thin cylinders. The key ingredient
is an estimate on the radius dependence of the activity (Lemma 2.11).
The following lemma gives the asymptotics of the normalization constant. In particular, we
prove that the radius of convergence of PnCnrnis strictly positive, for all values of the
radius. Apart from the information r > 0, we do not use the bounds (2.40) below in the
quantum Hall context. The bounds are however of interest in the plasma analogy, which is
why we prove them.
Lemma 2.10. For all p∈Nand γ=l/R > 0, there exist r > 0and q≥0such that
−log r= lim
N→∞
1
Nlog CN= sup
N
1
Nlog CN, q = lim
n→∞Cnrn.
Moreover, r=rp(γ)satisfies the bound
epX
n1,..,np∈Z:
n1+..+np=0
e−π2
pγ2(n2
1+..+n2
p)−1≤rp(γ)p1−p
2(eγ
√π)1−p≤1.(2.40)
Before we turn to the proof, let us briefly look at p= 1 (filled Landau level). Laughlin’s
cylinder function at filling factor 1 is simply
ΨC
N=ψ0∧.. ∧ψN−1.
It follows that CN= 1 for all N∈N, thus r= 1. This should be compared to the bound
(2.40) which for p= 1 reads e−1≤r≤1. Thus for the filled Landau level, the lower bound
is actually too small whereas the upper bound is exact.
Proof of Lemma 2.10. By Lemma 2.6 and the supermultiplicativity (2.32), we have CN+M≥
CNCM. In particular, limN→∞ 1
Nlog CN= supN1
Nlog CNand ris well-defined (but possibly
0). Now let
φn(z) := 1
(2πRl√π)1/2peiny/Re−1
2pl2(x−pnγl)2.
Then
ΨN(z1, .., zN) = 1
√N!(det(φk−1(zj))1≤j,k≤N)p=: 1
√N!(det A(z1, .., zN))p
and |ΨN|2= (det A∗A)p/N!. Lower and upper bounds on CNare now obtained with
Hadamard’s and H¨older’s inequalities. Using Hadamard’s inequality
|det A| ≤
N
Y
j=1
(
N
X
k=1 |Ajk|2)1/2
52 CHAPTER 2. THERMODYNAMIC LIMITS
we see
CN≤1
N!ZR×[0,2πR]N−1
X
k=0 |φk(z)|2pdxdyN
=1
N!Z∞
−∞
1
l√πN−1
X
k=0
e−1
pl2(x−pkγl)2pdxN
=1
N!Z∞
−∞
1
√πN−1
X
k=0
e−1
p(s−pkγ)2pdsN
By Poisson’s summation formula,
∞
X
k=−∞
e−1
p(x−kpγ)2=√π
γ√p
∞
X
n=−∞
e−π2n2/(pγ2)ei2πnx/(pγ)=: f(x).
Thus
Z(N−1/2)pγ
−pγ/2
1
√πN−1
X
k=0
e−1
p(x−pkγ)2pdx ≤1
√πNZpγ
0
f(x)pdx
=Npγ
√π√π
γ√ppX
n1+..+np=0
e−π2
pγ2(n2
1+..+n2
p)
=Np1−p
2(√π
γ)p−1X
n1+..+np=0
e−π2
pγ2(n2
1+..+n2
p)
=: Nb(γ).
The integral from −∞ to −pγ/2 can be bounded from above by
Z−pγ/2
−∞
(∞
X
k=0
e−(x−pkγ)2/p)pdx =: c
which is an N-independent, finite number. For symmetry reasons this is also a bound to the
integral from (N−1/2)pγ to ∞. Thus we obtain
CN≤1
N!(Nb(γ) + 2c)N.
Using Stirling’s formula, we obtain the upper bound r≥1/(eb(γ)) >0.Now we turn to a
lower bound for CN. Let IN:= [−pγl/2,(N−1/2)pγl]×[0,2πR]. With H¨older’s inequality
|RΩf| ≤ |Ω|1−1/p(Rfp)1/p , we get
CN≥1
N!ZIN
N|det A|2p≥1
N!
1
|IN
N|p−1ZIN
N|det A|2p
=N!p−1
(Npγl2πR)p−1||φ0∧.. ∧φN−1||2p
L2(IN
N)
=N!p−1
(Npγl2πR)N(p−1) (2πR)N(p−1)
N−1
Y
k=0 Z(N−k−1/2)pγ
−(k+1
2)pγ
e−s2/pds
=N!√πN
(Npγ)Np−1√ppN
N−1
Y
k=0 1−1
2erfc((N−k−1
2)√pγ)−1
2erfc((k+1
2)√pγ)p
2.1. LAUGHLIN’S CYLINDER WAVE FUNCTION 53
where erfcx:= 2
√πR∞
xe−t2dt is the complementary error function. The decay properties of
erfc can be used to show that the product QN−1
k=0 stays bounded away from zero. Making use
again of Stirling’s formula, we get
r≤(pγ)p−1ep−1
√πp−1√pp= ( γe
√π)p−1pp
2−1.
The asymptotics rnCn→q≥0 follows from Lemma 2.8. 1/q is the expected interarrival time
of the associated renewal process. Note that the monomer activity of the system associated
to Laughlin’s function is α1= 1, thus the aperiodicity condition gcd{n∈N|αn>0}= 1
needed to apply Lemma 2.8 is fulfilled.
It follows from the previous lemma that r > 0, and therefore Pnαntnmust have a non-zero
radius of convergence. Put differently, the activity can be rescaled to a stable activity. It is
still open whether the associated renewal process has finite mean µ < ∞, or equivalently, if
q=µ−1>0.
Remarks: 1. The bounds on rare interesting for small γ. They can be related to a conjecture
on the relation between strip free energies and bulk free energies of Coulomb systems (see
Section 2.3). We now turn to the large γbehavior.
2. For γ= 0, the activity cannot be rescaled to a stable activity. Let us first define the
activity for γ= 0. Note that for general γ, the activity and the normalization constants are
polynomials of e−γ2with integer coefficients:
CN=X
0≤m1≤..≤mN≤pN−p|bN(m1, .., mN)|2(e−γ2)PN
j=1(p2(j−1)2−m2
j)
αN=X
0≤m1≤..≤mN≤pN−p
irred
|bN(m1, .., mN)|2(e−γ2)PN
j=1(p2(j−1)2−m2
j)(2.41)
where bN(m1, .., mN) are the expansion coefficients of the p-th power of the Vandermonde
determinant. The bN’s are integers since they are sums of signs of permutations, see p.37.
These expressions are obtained by combining (2.29) and (2.2). Note that by Lemma 2.5,
the sums in (2.41) contain only non-negative powers of e−γ2. For γ= 0, corresponding to
the limit of infinite radii, we may define CNand αNthrough (2.41). are well-defined for
γ= 0, corresponding to the limit of infinite radii. The value of CNfor γ= 0 is known: it is
CN= (pN)!/(N!p!N) see e.g. [FGIL94], Section 5, Property 6. It follows that for p≥2 and
γ= 0, the series PnCntnhas a vanishing radius of convergence: r= 0.
Lemma 2.10 shows that for all (finite) values of the radius R, the activity of the polymer
system associated with Laughlin’s function can be rescaled to a stable activity. Now it
remains to see whether the associated renewal process has finite mean. Before we turn to
this question, let us have a closer look at the activity (αn).
Lemma 2.11. The monomer activity is α1= 1. Furthermore, there exist integers bN,m ∈N0
such that
∀N≥1 : αN=∞
X
m=p(N−1)
bN,me−mγ2.
In particular, as γ→ ∞,N-mers have activity O(e−p(N−1)γ2).
54 CHAPTER 2. THERMODYNAMIC LIMITS
Proof. Recall from (2.24) that the monomer functions are u{j}=ψpj. Thus the monomer
activity is ||u{j}||2= 1. By Lemma 2.5, PN
j=1(p2(j−1)2−m2
j)≥p(N−1) for any increas-
ing, N-admissible irreducible (m1, .., mN). The second statement of the lemma now follows
immediately from the formula (2.41).
Remark. Rezayi and Haldane [RH94] have observed that for systems with a fixed, finite
number of particles, Laughlin’s wave function approaches the Tao-Thouless state 2in the
limit of vanishing cylinder radius:
ΨC
N≃ψ0∧ψp∧... ∧ψp(N−1) as γ→ ∞.
Using the polymer representation, this can be rephrased as: in the limit of thin cylinders,
the system is a pure monomer system. We will see that a similar statement holds for systems
with infinitely many particles. This behavior is consistent with the fact that in the limit
of thin cylinders, all polymers except monomers have vanishing activity, as shown in the
previous lemma.
Now let us come back to the question whether q= limnrnCn>0, or equivalently, whether
the associated renewal process has finite mean. Recall that this is the case if and only if
P∞
n=1 αnrn= 1 and P∞
n=1 nαnrn<∞. The crucial observation is that these two conditions
are automatically fulfilled if Pnαntnhas a radius of convergence Rstrictly larger than the
radius of convergence rof PnCntn. Thus we are going to compare domains of convergence.
Furthermore, observe that the monomer case is rather trivial: for a monomer system,
∞
X
n=1
αntn=t, ∞
X
n=0
Cntn=1
1−t
whence r= 1 and the “series” Pntnαn=thas an infinite radius of convergence.
Lemma 2.11 tells us that on thin cylinders, the system looks almost like a monomer system
and the associated renewal process should have finite mean. This heuristic argument gives
the main intuition behind the next theorem.
We wish to apply Lemma 2.11. Thus it is useful to keep track of the γ-dependence of the
various quantities involved. By (2.41), the activity and the normalization constants are
polynomials of e−γ2with coefficients in N(see also the previous lemma). We make the
γ-dependence more explicit in the notation by writing αn(e−γ2), Cn(e−γ2). The power series
C(t, e−γ2) := 1 + ∞
X
n=1
Cn(e−γ2)tn, A(t, e−γ2) := t+∞
X
n=2
αn(e−γ2)tn
are actually power series of two variables, tand e−γ2. They have non-negative integer
coefficients. By the results of Section 2.1.2, Cand Aare related through
C(t, e−γ2) = 1
1−A(t, e−γ2).
2At filling factor 1/p, the Tao-Thouless state is a simple Slater determinant where every p-th Landau
orbital is occupied. It was proposed as a candidate FQHE ground state in [TT83] but later abandoned
[Tho85].
2.1. LAUGHLIN’S CYLINDER WAVE FUNCTION 55
The quantities r=r(e−γ2) and R=R(e−γ2) are γ-dependent too. They define curves that
may be characterized through the boundaries of the domains of convergence of C(t, u) and
A(t, u) in R2
+, see Figure 2.1. We know that r(u)≤R(u) for all u. The case u= 0 corre-
sponds to the infinitely thin cylinder. By Lemma 2.11, in this limit we have a pure monomer
system. Consequently, r(0) = 1, R(0) = ∞and r(0) < R(0).
The strategy in the proof of the next theorem is to show the strict inequality r(u)< R(u)
subsists for sufficiently small u. This will imply that on thin cylinders the associated re-
newal process has finite mean, or equivalently, rnCn→q > 0. Furthermore, we show some
analyticity results.
II
u=e−γ2
t
1
1e−γ2
p
t=R(u)
III
t=r(u)
I
I: Aand Cconverge
II: A < ∞,C=∞
III: Aand Cdiverge
0
Figure 2.1: Domains of convergence of A(t, u) and C(t, u) for p≥2. The curve r(u) delimits the
domain of convergence of A,R(u) the domain of convergence of C. Both series diverge when u≥1.
We know that r(u) = 1 + O(u) and R(u)≥const ·u−pas u→0. When u < e−γ2,r(u)< R(u). It
is an open question whether the curves rand Rtouch for some up=e−γ2
pstrictly below 1.
Theorem 2.12. Let p≥2be fixed. Define the functions C(t, u), A(t, u), r(u), R(u)as above.
Let r=rp(γ),q=qp(γ)as in Lemma 2.10. The following holds:
1. There exists a γp>0such that for u < e−γ2
p,r(u)< R(u).
2. The functions ]γp,∞[∋γ7→ rp(γ), qp(γ)are analytic and strictly positive. As γ→ ∞,
rp(γ) = 1 + O(e−γ2), qp(γ) = 1 + O(e−γ2).
Proof. The main idea is to use Lemma 2.11 and look at what happens when e−γ2→0.
Observe that A(t, 0) = t. It follows that R(0) = ∞,C(t, 0) = 1/(1 −t) and r(0) = 1 < R(0).
56 CHAPTER 2. THERMODYNAMIC LIMITS
Let
µ(u) := ∞
X
n=1
nαn(u)(r(u))n=r(u)∂A
∂t (r(u), u).(2.42)
We have µ(0) = 1. The crucial obervation is that
lim
u→0R(u) = ∞,∀u∈[0,1[: 0 < r(u)≤1.
To see this, note first that R(u)≥r(u) and r(u)>0 for u=e−γ2∈]0,1[ by Lemma 2.10.
Next, let 0 < u < v < 1; by Lemma 2.5 , there exist nonnegative numbers bmn so that
αn(u) = X
m≥p(n−1)
bmnum≤(u
v)p(n−1) X
m≥p(n−1)
bmnvm= (u
v)p(n−1)αn(v)
Since R(u) = 1/(lim supn(αn(u))1/n), this shows
upR(u)≥vpR(v).
For v < 1, R(v)>0, thus letting u→0 in the previous inequality, we obtain limu→0R(u) =
∞. On the other hand, since A(1, u)≥1≥A(r(u), u) for all u≥0, r(u)≤1 for all u≥0.
Thus for sufficiently small u,r(u)≤1< R(u).
Let γp>0 such that r(u)< R(u) if u < e−γ2
p=: up. On [0, up[, the curve r(u) is in the
interior of the domain of convergence of A(t, u), thus µ(u)<∞. Furthermore r(u) is the
solution of A(r(u), u) = 1. The power series A(t, u) defines a holomorphic function of two
variables, defined in
DA={(z, w)∈C2| |z|< R(|w|)}.
We know A(1,0) = 1 and ∂zA(1,0) = 1 6= 0. Thus by an implicit function theorem, there
is a neighborhood U⊂C2of (1,0) such that for (z, w)∈U, the equation A(z, w) = 1 has
a unique solution z(w); furthermore, z(w) is holomorphic. But for w=u∈R,z(u) = r(u),
thus r(u) is analytic in a neighborhood of 0. Let u∈[0, e−γ2
p[. We know that (r(u), u) is in
the interior of DA, and (∂zA)(r(u), u)≥r(u)>0. Thus the same reasoning as before gives
the analyticity of r(·) in a neighborhood of u. It follows that r(u) is analytic on [0, e−γ2[.
Around u= 0, it has a power series expansion
r(e−γ2) = 1 + X
n
βn(e−γ2)n
whence r(e−γ2) = 1 + O(e−γ2). The quantity µ(u) from (2.42) inherits the analyticity, and
from µ(0) = 1 we get µ(e−γ2) = 1 + O(e−γ2). Since q=µ−1, this concludes the proof.
Remark. The characterization of r(u) and R(u) as boundaries of domains of convergence
gives access to convexity properties: for example, the set
{(log t, log u)|t, u > 0, C(t, u)<∞}
is convex (see e.g. the book [H¨or90]). It follows that the functions log u7→ log r(u) and
]0,∞[∋γ27→ −log rp(γ)
are convex.
2.1. LAUGHLIN’S CYLINDER WAVE FUNCTION 57
2.1.4 Correlation functions and clustering
The normalization of Laughlin’s wave function was expressed as a polymer partition function.
For sufficiently thin cylinder (i.e. sufficiently large γ=l/R), the associated renewal process
has finite mean. In Section 2.1.2, we showed that this implies the existence of the thermo-
dynamic limit of correlation functions for the associated polymer system, the resulting state
is translationally invariant and mixing.
Laughlin’s wave function inherits the properties of the associated polymer system. The
thermodynamic limit of all correlation functions exist (Theorem 2.13). Since uj+X=t(j·
pγl)⊗N(X)uX3, the translational invariance of the polymer system becomes invariance of
Laughlin’s state with respect to magnetic translations by multiples of pγl; moreover Laugh-
lin’s state is reversal invariant (Theorem 2.14). We prove also that Laughlin’s state is mixing
with respect to the group of translations t(n·pγl) (n∈Z) (Theorem 2.16). The question
whether pγl is the smallest period of the state is deferred to the next section.
In the following, the meaning of Adepends on the parity of p∈N: when pis odd, Ais the
canonical anticommutation algebra over H=L2(R×[0,2πR], i.e. the C∗-algebra generated
by 1and the creation and annihilation operators c∗(f), c(g), f, g ∈L2(R×[0,2πR]). When
pis even, it is the canonical commutation relation algebra over H, i.e. the algebra gener-
ated by the Weyl operators W(f) = exp(iΦ(f)) where Φ(f) is the closure of the operator
(c(f)+c∗(f))/√2 defined in terms of the bosonic creation and annihilation operators, on the
bosonic Fock space (see [BR79b], Section 5.2).
Details of proofs will be worked out only for odd p.
Theorem 2.13. Suppose rnCn→q > 0. Then there is a state h·i on Asuch that for all
sequences (aN)Nof integers with aN→ −∞ and N+aN→ ∞, the state h·iNassociated to
t(aNpγl)⊗NΨNconverges weakly to h·i: for all a∈ A,
haiN=ht(aNpγ)⊗NΨN, at(aNpγ)⊗NΨNi/||ΨN||2→
N→∞ hai.(2.43)
Proof. It is enough to prove the convergence (2.43) for operators a=c∗(f1)..c∗(fn)c(g1)..c(gm)
where f1, .., gmare elements of a complete orthonormal system of L2(R×[0,2πR]). The ex-
pectation haiNvanishes if m6=n. Since c(f)ΨN= 0 if fis orthogonal to the lowest Landau
level, it is enough to consider functions in the lowest Landau level. Thus we are left with the
case a=c∗
ℓ′
1..c∗
ℓrcℓr..cℓ1where again for k∈Z.ck=c(ψk). These operators anticommute
and we can take ℓ1< .. < ℓrand similarly ℓ′
1< .. < ℓ′
r.
Strategy. For L={ℓ1< .. < ℓr} ⊂ Z, let cL:= cℓ1..cℓrand c∗
L:= (cL)∗. Let bN:= N+aN.
We will see that hc∗
L′cLiNcan be written as
hc∗
L′cLiN=
N
X
n=1
bN−n
X
j=aN
Cj−aNCbN−j−n
CN
fn(L′−pj, L −pj).(2.44)
for a suitable family of functions (fn)n∈N. We use the notation L−pj ={ℓ1, .., ℓr}−pj for
the set {ℓ1−pj, .., ℓr−pj}. Let us anticipate and mention some of the properties of (fn)n∈N:
• ∀n∈N∀L⊂Z:fn(L, L)≥0.
3For a∈R,t(a) is a shorthand for a translation in the x-direction: t(a) = t(aex).
58 CHAPTER 2. THERMODYNAMIC LIMITS
•The functions have compact support: fn(L′, L)6= 0 implies L∪L′⊂ {0, .., pn −p}.
For fixed jand n,Cj−aNCbN−j−n/CN→qrndue to Cnrn→q > 0. This suggests the
following definition:
hc∗
L′cLi:= ∞
X
n=1
qrn∞
X
j=−∞
fn(L′−pj, L −pj).(2.45)
The theorem is proved in three steps:
1. Define fnand prove (2.44).
2. Show hc∗
LcLi ≤ 1 and hc∗
LcLiN→ hc∗
LcLi.
3. Prove Pj,n qrn|fn(L′−pj, L −pj)| ≤ hc∗
LcLi1/2hc∗
L′cL′i1/2and hc∗
L′cLiN→ hc∗
L′cLi.
1. Let L′, L ⊂Zwith |L′|=|L|=r. We start with the representation
hΨN, c∗
L′cLΨNi=X
m′,m
aN(m′)aN(m)hψm, c∗
L′cLψmi.(2.46)
The sum ranges over increasing N-admissible sequences m= (m1, ., mN), m′= (m′
1, ., m′
N),
and aN(m), ψmare short-hands for aN(m1, ., mN), ψm1∧.. ∧ψmN. Suppose mand m′have
common renewal points s, t such that L∪L′⊂ {ps, .., pt −p}. Then
hψm, c∗
L′cLψmi=Y
j∈{1,..,s}∪{t,..,N}
δmj,m′
j|as(m1, .., ms)|2|aN−t(mt+1 −pt, .., mN)|2
·at−s(m′
s+1 −ps, .., m′
t−ps)at−s(ms+1 −ps, .., mt−ps)
·hψm′
s+1 ∧.. ∧ψm′
t, c∗
L′cLψms+1 ∧.. ∧ψmti.(2.47)
This motivates the following definitions: we call m, m′L∪L′-irreducible if mand m′have
no common renewal points except 0 and Nthat are below or above L∪L′:
k∈R(m)∩R(m′)∩{1, .., N −1}
⇒(L∪L′)∩{0, .., pk −p} 6=∅and (L∪L′)∩{pk, .., pN −p} 6=∅.
Similarly, we call a single sequence m L ∪L′-irreducible if it has no renewal point except
possibly 0 and Nbelow or above L∪L′. The function fN(L′, L) is defined by (2.46) except
that the sum runs over increasing N-admissible pairs of sequences that are L∪L′-irreducible.
With this definition, combining (2.46) and (2.47) we obtain (2.44).
2. For L=L′, the definition of fngives
fN(L, L) = X′
m|aN(m)|2χL⊂{m1,..,mN}
where the sum ranges over increasing N-admissible sequences that are L-irreducible. In par-
ticular, fN(L, L)≥0. Moreover, if fN(L, L)6= 0, there exists an N-admissible sequence m
such that L⊂ {m1, .., mN}, whence L⊂ {0, .., pN −p}.
2.1. LAUGHLIN’S CYLINDER WAVE FUNCTION 59
Let d∈N. From (2.44) and fn(L, L)≥0 we get
d
X
n=1
bN−j−n
X
j=aN
Cj−aNCbN−j−n
CN
fn(L−pj, L −pj)≤ hc∗
LcLiN≤1
If fn(L−pj, L −pj)6= 0 we must have L−pj ⊂ {0, .., pn −p}, thus only a finite number of
j’s contribute to the sum and we can take the limit N→ ∞, which gives
d
X
n=1
∞
X
j=−∞
qrnfn(L−pj, L −pj)≤1.
Letting d→ ∞, we obtain the convergence of the sum defining hc∗
LcLi. The proof of
hc∗
LcLiN→ hc∗
LcLiis completed by an ǫ/3 argument. For ǫ > 0, there exists an m∈N
such that
∞
X
n=d+1
∞
X
j=−∞
qrnfn(L−pj, L −pj)≤ǫ/3.(2.48)
By Lemma 2.10, 0 < Cnrn≤1 and Cnrn→q. By the assumption q > 0, we get
minnrnCn=: c > 0. Therefore
CjCN−j−n
CN≤CN−n
CN≤r−(N−n)
cr−N=1
crn.
It follows that
∞
X
n=d+1
bN−n
X
j=aN
Cj−aNCbN−j−n
CN
fn(L−pj, L −pj)≤ǫ
3qc (2.49)
Finally, there is an M∈Nsuch that for N≥M,
|
d
X
n=1bN−n
X
j=aN
Cj−aNCbN−j−n
CN
fn(L−pj, L −pj)−∞
X
j=−∞
qrnfn(L−pj, L −pj)| ≤ ǫ/3.(2.50)
Putting together (2.48), (2.49) and (2.50) we see that for N≥M,
|hc∗
LcLiN−hc∗
LcLi| ≤ 2 + (qc)−1
3ǫ
and we obtain the desired convergence.
3. Let d∈N. As before, we have
d
X
n=1
qrn∞
X
j=−∞|fn(l′−pj, l−pj)|= lim
N→∞
m
X
n=1
bN−j−n
X
j=aN
CjCbN−n−j
CN|fn(l′−pj, l−pj)|
60 CHAPTER 2. THERMODYNAMIC LIMITS
since due to n≤donly finitely many terms contribute to the sum. But now
d
X
n=1
bN−n
X
j=aN
Cj−aNCbN−n−j
CN|fn(L′−pj, L −pj)|
≤
N
X
n=1
bN−n
X
j=aN
Cj−aNCbN−n−j
CN|fn(L′−pj, L −pj)|
≤X
m,m′|aN(m′)aN(m)hψm′, c∗
L′cLψmi|/CN
=X
k∈ZN−r|aN(ℓ′∪k)aN(ℓ∪k)|/CN
≤X
k∈ZN−r|aN(ℓ′∪k)|21/2X
k∈ZN−r|aN(ℓ∪k)|21/2/CN
=hc∗
L′cL′i1/2
Nhc∗
LcLi1/2
N≤1.
Here r=|L|=|L′|and ℓ, ℓ′∈Zrare such that L={ℓ1, ., ℓr},L′={ℓ′
1, .., ℓ′
r}. The
notation k∪ℓrefers to the increasing sequence obtained by rearranging k1, .., kN−r, ℓ1, .., ℓr.
Letting first dand then Ngo to infinity, we get the absolute convergence of the sum defining
hc∗
L′cLiand the bound |hc∗
L′cLi| ≤ hc∗
L′cL′i1/2hc∗
LcLi1/2(Cauchy-Schwarz for the state h·i).
An argument strictly analogous to 2. then shows hc∗
L′cLiN→ hc∗
L′cLi.
Now let us turn to the symmetries of Laughlin’s state. A unitary map Uin B(H), H=
L2(R×[0,2πR]) induces the unitary map Γ(U) := LN∈N0U⊗Nin Fand a map
τU:A → A, τU(a) := Γ(U)aΓ(U)−1.
The map τUis the unique C∗-automorphism such that τU(c(f)) = c(Uf) for all f∈ H
(the Bogoliubov automorphism induced by U). This construction can be applied to the
180◦rotation (or reversal)s0, introduced at the beginning of the chapter, to the magnetic
translation t(γlex) in the x-direction and to the translations t(aey) (a∈R) in the y-direction;
note that in the Landau gauge, magnetic translations in the ydirection coincide with the
usual translation ψ7→ ψ(·−ia). We will call the resulting morphisms of Aτs,τxand τa
y.
Theorem 2.14. Suppose rnCn→q > 0and let ω(·) = h·i be the limiting state of the previous
theorem. Then ωis invariant with respect to reversals, translations in the xdirection by
multiples of pγl, and arbitrary translations in the ydirection:
∀n∈Z,∀a∈R:ω=ω◦τs=ω◦τnp
x=ω◦τa
y.
Proof. The invariance with respect to y-translations and reversal follow from Lemma 2.1:
the y-translational invariance is a direct consequence of (2.8). For the reversal invariance,
suppose Nis odd and let aN:= −(N−1)/2. By (2.9),
s0t(aNpγlex)⊗NΨN=t(aNpγlex)⊗Nsp(N−1)γl/2ΨN=t(aNpγlex)⊗NΨN,
whence the reversal invariance of h·iN. Taking limits along sequences of odd integers N=
(2M+ 1) → ∞, we obtain the reversal invariance of h·i. The invariance with respect to τp
x
2.1. LAUGHLIN’S CYLINDER WAVE FUNCTION 61
follows from (2.45):
hτp
x(c∗
L′cL)i=hc∗
L′+pcL+pi=∞
X
n=1
qrn∞
X
j=−∞
fn(L′+p−pj, L +p−pj)
=∞
X
n=1
qrn∞
X
j=−∞
fn(L′−pj, L −pj)
=hc∗
L′cLi.
We were initially interested in the thermodynamic limit of the one-particle density. Using
the representation
ρN(z) =
bN
X
k=aNhc∗
kckiN|ψk(z)|2=1
2πRl√π
bN
X
k=aNhc∗
kckiNe−(x−kγl)2/l2
we obtain immediately that, under the assumptions of Theorem 2.13, ρN(z)→ρ(z), where
ρ(z) = 1
2πRl√π
∞
X
k=−∞hc∗
kckie−(x−kγl)2/l2.(2.51)
Remark: Occupation numbers with even p.Strictly speaking, the previous expression holds
true only when pis odd. When pis even, the occupation number operator ˆnkis unbounded
and thus ˆnk/∈ A. The eigenprojections of ˆnkare in A, thus we may write ˆnk=PknkPkand
set hˆnki=PknkhPki. It is implicit in the proofs of the following theorems that the sums
involved stay finite. Another procedure to give a meaning to hˆnkiis to check that the state
ω=h·i is regular ([BR79b] p.24), in which case there is a natural definition of an operator
ˆnk,ω in the cyclic representation (Hω, πω,Ωω) and one can set hˆnki:= hΩω,ˆnk,ωΩωi. In the
following, we will keep on writing hˆnkiwithout explicitly requiring podd.
The one-particle density ρ(z) is pγl-periodic in the direction along the cylinder axis and has
the correct average value:
Corollary 2.15. Suppose rnCn→q > 0and let h·i be the state of Theorem 2.13. The
one-particle density ρassociated to h·i is independent of the coordinate yaround the axis and
is periodic with period pγl in the direction along the axis. Moreover, it has average density
(p·2πl2)−1:
1
pγl Zpγl
0
ρ(x)dx =1
2πl2
1
p
p−1
X
k=0hc∗
kcki=1
p
1
2πl2.
Proof. The one-particle density ρ(z) is given by (2.51) and obviously independent of y=ℑz.
The periodicity of ρfollows from the periodicity of h·i proved in the previous theorem. By
Lemma 2.2,
1
pγl Zpγl
0
ρ(x)dx =1
2πl2
1
p
p−1
X
k=0hc∗
kcki.
Let nk:= hc∗
kcki. It remains to prove n0+n1+.. +np−1= 1. A close look at the definition
of fnin the proof of Theorem 2.13 shows that fn({k},{k}) can be expressed in terms of the
polymer functions uXfrom Lemma 2.6:
fn({k}−pj, {k}−pj) = hu{j,..,j+n−1},ˆnku{j,..,j+n−1}i.
62 CHAPTER 2. THERMODYNAMIC LIMITS
The formula (2.45) can be rewritten in terms of the polymer correlation functions ρP(X) =
qrN(X)αN(X)from Proposition 2.9 as
nk=hˆnki=X
X
ρP(X)huX,ˆnkuXi
||uX||2.(2.52)
This formula is of interest in itself: in a probabilistic language, the probability for finding
a particle in the lattice site kgiven that there is a given polymer Xcovering the site is
huX,ˆnkuXi/||uX||2. For our purpose it is more convenient to write it as
nk=∞
X
n=1
qrn∞
X
j=−∞hun,ˆnk−pjuni, un=u{0,..,n−1}.
Using P∞
k=−∞hun,ˆnkuni=nαn(unis an n-particle wave function with norm ||un||2=αn),
it follows that
p−1
X
k=0
nk=∞
X
n=1
qrn
p−1
X
k=0
∞
X
j=−∞hun,ˆnk−pjuni=∞
X
n=1
qrn∞
X
k=−∞hun,ˆnkuni
=∞
X
n=1
qrnnαn= 1
(recall q= 1/(Pnnrnαn)).
The formula (2.52) can be generalized: any expectation value of a product of occupation
numbers ˆnkcan be expressed in terms of the polymer correlation functions ρPfrom Proposi-
tion 2.9 and the functions uXof Lemma 2.6. For example, the diagonal two-point correlation
is
hˆnkˆnℓi=X
X:
pX⊃{k,ℓ}
ρP(X)huX,ˆnkˆnℓuXi
||uX||2+X
X,Y :
pX∋k, pY ∋ℓ
ρP(X, Y )huX,ˆnkuXi
||uX||2huY,ˆnℓuYi
||uY||2.
(2.53)
This suggests that the clustering properties of ρPtransfer to the state h·i: if kand ℓare far
apart, the probability that they are in the same polymer is small, thus in (2.53) the second
summand dominates; but then recall ρP(X, Y )−ρP(X)ρP(Y) = O(|udist (X,Y )−µ−1|), where
un=rnCn→µ−1=q > 0. One can show that
hˆnkˆnℓi−hˆnkihˆnℓi −→
|k−ℓ|→∞ 0.
The speed of convergence is essentially determined by Pn≥|k−ℓ|nrnαn(this gives a bound to
the probability that kand ℓare in the same polymer) and |u|k−ℓ|−µ−1|. Similar statements
hold for general correlation functions.
Theorem 2.16. Suppose rnCn→q > 0. Then the state ω(·) = h·i of Theorem 2.13 is
mixing with respect to the shifts τnp
x, n ∈Z:
∀a, b ∈ A : lim
n→∞ω(aτpn
x(b)) = ω(a)ω(b).(2.54)
Proof. Again, it is enough to check (2.54) for operators a=c∗
L′cL,b=c∗
K′cK, where
L, L′, K, K′⊂Zand we agree c∅=c∗
∅=1. The main idea is to write a formula simi-
lar to (2.53). The correlation function can be written as a sum of two parts: one where
2.1. LAUGHLIN’S CYLINDER WAVE FUNCTION 63
L∪L′and K∪K′are in a same compound block (this will be the sum Png(..) defined
below) this part will be bounded by the probability amplitude that a long polymer occurs.
Another part of the sum consists of contributions where L∪L′and K∪K′are in differ-
ent blocks; this converges to the product of the correlations as L∪L′and K∪K′are far away.
We start by treating some special cases. Recall that hc∗
L′cLi 6= 0 implies |L|=|L′|and
Pk∈L′k=Pk∈L=k(this comes from the translational invariance ω◦τa
y=ω,τa
y(ck) =
e−ika/Rck).
1. |L|+|K| 6=|L′|+|K′|. Then ω(aτnp
x(b)) = 0 and we must have |K| 6=|K′|or |L| 6=|L′|,
whence ω(a)ω(b) = 0.
2. |L|+|K|=|L′|+|K′|but |K′| 6=|K|. Then ω(a)ω(b) = 0. Suppose for example
|K′|>|K|. Then
X
k∈L′∪(pn+K′)
k−X
k∈L∪(pn+K)
k=X
k∈L′∪K
k−X
k∈L∪K
k+pn(|K′|−|K|)>0
for sufficiently large n, whence ω(aτpn
x(b)) = 0 for sufficiently large n.
3. |L|=|L′|,|K|=|K′|, but Pk∈L∪Kk6=Pk∈L′∪K′k. Then
X
k∈L′∪(pn+K′)
k−X
k∈L∪(pn+K)
k=X
k∈L′∪K′
k−X
k∈L∪K
k6= 0
thus ω(aτnp
x(b))0. Moreover Pk∈Kk6=Pk∈K′kor Pk∈Lk6=Pk∈Lk, therefore ω(a)ω(b) =
0.
We are left with the case |L′|=|L|,|K′|=|K|,Pk∈K∪Lk=PK′∪L′k. We will show that
ω(ab)−ω(a)ω(b) is small when d:= min K∪K′−max L∪L′is large.
Suppose m,m′are increasing N-admissible and have a common renewal point between L∪L′
and K∪K′, i.e.:
s∈R(m)∩R(m′) : L∪L′⊂ {0, .., ps −p}and K∪K′⊂ {ps, .., pN −p}.
Then
aN(m′)aN(m)hψm′, abψmi
=as({m′
j}1≤j≤s)aN−s({mj}1≤j≤s)hψm′
1∧.. ∧ψm′
s, aψm1∧.. ∧ψmsi
·aN−s({m′
j−ps}j≥s+1)aN−s({mj−ps}j≥s+1)hψm′
s+1 ∧.. ∧ψm′
N, bψms+1 ∧.. ∧ψmNi
(2.55)
Thus if we define
gn(L′, L;K′, K) := X′aN(m′)aN(m)hψm′, c∗
L′cLc∗
K′cKψmi
with a sum over L∪L′∪K∪K′- irreducible pairs of increasing N-admissible sequences
m, m′with no renewal point between L∪L′and K∪K′and the functions fnas in the proof
of Theorem 2.13, we obtain
fN(L′∪K′, L ∪K) = X
0<i≤j<N
fi(L′, L)Cj−ifN−j(K′−pj, K −pj) + gN(L′, L;K′, K).
64 CHAPTER 2. THERMODYNAMIC LIMITS
For X={j, .., j +N−1}, it is convenient to define
fX(L′, L) = fN(L′−pj, L −pj), gX(L′, L;K′, K) = gN(L′−pj, L −pj;K′−pj, K −pj).
Using the representation (2.45), we get habi=F+Gwhere
F=X
X<Y
qrN(X)+N(Y)fX(L′, L)(Cd(X,Y )rd(X,Y )−q)fY(K′, K)
G=X
X
qN(X)gX(L′, L;K′, K)
Here, if X={sX, .., sX+N(X)−1}and Y={sY, .., sY+N(Y)−1},d(X, Y ) := sY−(sX+
N(X)). The previous expressions should be compared with
haihbi=X
X,Y
q2rN(X)+N(Y)fX(L′;L)fY(K′;K).
We start by estimating F−haihbi. Write X={sX, .., eX−1},Y={sY, .., eY−1}. The
difference of Fand haihbiis a sum over X, Y which we split into two parts. Let M∈N. The
first part is a sum over X, Y with sY−eX≥M. It is bounded by
sup
n≥M|1−q−1rnCn|X
X
qrN(X)|fX(L′, L)|X
Y
qrN(Y)|fY(K′, K)|
≤sup
n≥M|1−q−1rnCn|(hc∗
L′cL′ihc∗
LcLihc∗
K′cK′ihc∗
KcKi)1/2≤sup
n≥M|1−q−1rnCn|.
The second part is a sum over X, Y with sY−eX≤M−1. Writing d= (min(K∪K′)−
psY) + p(sY−eX) + (peX−max(L∪L′)), we obtain
min(K∪K′)−psY≥(d−pM)/2 or peX−max(L∪L′)≥(d−pM)/2.
Using Cnrn≤q, we can bound give a bound on the second part of the sum as
2X
X:
peX−max(L∪L′)≥(d−pM)/2
qrN(X)|fX(L′, L)|+ 2 X
Y:
min(K∪K′)−psY≥(d−pM)/2
qrN(Y)|fY(K′, K)|.
If we go back to the definition of fXas a sum over N-admissible, L∪L′irreducible sequences
m′, m, we see that only msequences with no renewal point between L∪L′and eXcontribute.
By a procedure similar to step 3. of the proof of Theorem 2.13, one can show that
X
X:
peX−max(L∪L′)≥(d−pM)/2
qrN(X)|fX(L′, L)| ≤ X
n≥(d−pM)/2p
nrnαn
the right-hand side representing the probability in the polymer ensemble of Proposition 2.9
that a given point is in a polymer of length greater or equal to (d−pM)/2p. The sum over
Ycan be estimated in a similar way. Thus we get
|F−haihbi| ≤ sup
n≥M|1−q−1rnCn|+ 4 X
n≥(d−pM)/2p
nrnαn.
It remains to give a bound on G. We start by looking at the non-vanishing contributions to
gn(L′, L;K′, K). Without loss of generality, we may assume Pk∈L′k≥Pk∈Lk(otherwise,
2.1. LAUGHLIN’S CYLINDER WAVE FUNCTION 65
use ha∗i=haito interchange L′and L).
Claim. Suppose m′, m are increasing N-admissible sequences and Pk∈L′k≥Pk∈Lk. and
hψm, c∗
L′cLc∗
K′cKψm′i 6= 0. Then if sis a renewal point of m′such that L∪L′⊂ {0, .., ps−p},
K∪K′⊂ {ps, .., pN −p},sis also a renewal point of m.
Proof of the claim. Let m′, m, K, L, K′, L′, s be as described above, and M={m1, .., mN},
similarly for M′. Then we must have M′\(K′∪L′) = M\(K∪L). Intersecting with {0, .., ps−
p}we get M′\L′∩ {0, .., ps −p}=M\L∩ {0, .., ps −p}. In particular Ps
j=1(m′
j−mj) =
Pk∈L′k−Pk∈Lk≥0. Thus
0 =
s
X
j=1
(m′
j−p(j−1)) ≥
s
X
j=1
(mj−p(j−1)) ≥0.
The inequality on the right-hand side must be an equality, and sis a renewal point of m.
As a consequence, if a pair m′, m contributes to gn(L′, L;K′, K) and X′
1, .., X′
Dis the partition
of {0, .., N −1}determined by the renewal points of m′, there is a rod X′
jsuch that pX′
j∩
(L′∪L)6= 0 and pX′
j∩(K′∪K)6= 0. In particular, N(X′
j)≥min(K′∪K)−max(L∪L′) = d.
Thus in the spirit of step 3. in the proof of Theorem 2.13, we have
|
N
X
n=1
N−n
X
j=0
CjCN−n−j
CN
gn(L′−pj, L −pj;K′−pj, K −pj)|
≤X′
m,m′
aN(m)aN(m′)hψm′, abψmi|/CN
≤ hc∗
KcKc∗
LcLi1/2
ΨNX′
m′|aN(m′)|2hψm′, c∗
L′cL′c∗
K′cK′ψm′i1/2/CN.
where P′is the sum over sequences m′having no renewal point between L∪L′and K′∪K.
Taking the limit N→ ∞ after shifting the origin to the middle of the cylinder, we obtain a
bound of Gin terms of the square root of the probability that max L∪L′/p is in a polymer
of length greater or equal to p. Finally, we obtain the bound
|habi−haihbi| ≤ sup
n≥M|1−q−1rnCn|+ 4 X
n≥(d−pM)/2p
nrnαn+X
n≥d/p
nrnαn)1/2(2.56)
for any M∈N. Choosing Mof the order of d/(2p), we see that |habi−haihbi| goes to zero
as d→ ∞.
Remark: Rate of convergence. Suppose Pnαntnhas radius of convergence R > r. By
Theorem 2.12, this is the case for sufficiently large γ. For Λ ⊂Z,|Λ|<∞, let AΛbe the
algebra generated by 1, ck, c∗
m, k, m ∈Λ. If a∈ AΛ,b∈ AΛ′, then habi−haihbigoes to zero
like (r/R)dist (Λ,Λ′)/4p: there is exponential clustering.
2.1.5 Symmetry breaking
In Theorem 2.14, we proved that the state ω=h·i of Theorem 2.13 is invariant with respect
to translations τnp
xby multiples of pγl. Thus pγl is a translational period. This leaves the
question open whether pγl is the smallest period. We will prove that for sufficiently thin
66 CHAPTER 2. THERMODYNAMIC LIMITS
cylinders, pγl is indeed the smallest period of ω. Since ωis an infinite volume ground state
of a Hamiltonian invariant with respect to translations by multiples of γl (see Section 2.4),
this means that there is symmetry breaking.
In principle, it is possible that the state has pγl as its smallest period, but a γl-periodic
one-particle density. We will prove that for sufficiently thin cylinders, this does not happen
and pγl is the smallest period of the one-particle density.
The proof of the non-trivial periodicity of the one-particle density uses the analyticity results
of Theorem 2.10. They are consistent with numerical results by [ˇ
SWK04] and related to
numerical results for tori [SFL+05]. This work looks at torus wave functions at filling factor
1/3 and gives numerical estimates of the order parameter
O=1
N
3N
X
j=1
ei2πj/3hc∗(˜
ψj)c(˜
ψj)i
(˜
ψk, k ∈Z/3NZare the torus lowest Landau level basis functions). The authors find that
the curve e−γ27→ O looks like e−c2/γ2for some constant c > 0. In particular, for large radius
R=l/γ,Ois very small but non-zero.
As a complement, we prove that the states ω,ω◦τx,..,ω◦τp−1
xare orthogonal if
∞
X
n=1
n2rnαn<∞.(2.57)
We call the states ω1, .., ωpon Aorthogonal if for all positive functionals ˜ωon A, and all
k∈ {1, .., p},˜ω≤ωkand ˜ω≤X
j6=k
ωj⇒˜ω= 0
or equivalently, if in the cyclic representation (πω,Hω,Ωω) associated to ω:= ω1+... +ωp,
there exist mutually orthogonal projections P1, .., Pp∈πω(A)′such that
ωj(a) = hPjΩω, πω(a)Ωωi(j∈ {1, .., p}), a ∈ A
(see [BR79a] Lemma 4.1.19, p.333). The analogous notion for probability measures is dis-
jointness of supports.
For the proof of the orthogonality, it is enough to restrict to a commutative subalgebra and
we can use a probabilistic picture. The proof then follows ideas of [AM80], who look at
thermodynamic limits of a classical one-dimensional jellium systems. They notice a simple
condition on the range of the electric field. This condition subsists in the thermodynamic
limit and shows that the distribution of the electric field cannot be translationally invariant.
[AM80] then use an ergodic theorem to prove that the electric field is a function of particle
positions and conclude that the distribution of particle positions cannot be translationally
invariant (see also p.71 for more details). Something similar can be done here. The second
moment condition (2.57) ensures the integrability of an auxiliary random variable, needed
for the ergodic theorem. A related approach is also used in [AGL01], who conjectured that
their results could be applied to jellium tubes.
2.1. LAUGHLIN’S CYLINDER WAVE FUNCTION 67
Theorem 2.17. Suppose Cnrn→q > 0and let ω=h·i be the state of Theorem 2.13.
1. Suppose that the second moment Pnn2rnαnis finite. Then the states ω◦τk
x,k=
0, ..., p −1are orthogonal. In particular, ω◦τk
x=ωif and only if kis an integer
multiple of p.
2. Let γpbe as in Theorem 2.12. Then we have:
(a) The second moment condition and the conclusions of 1. hold for all γ > γp.
(b) Let ρ,nk=ω(c∗
kck)be the one-particle density and occupation numbers associated
to ω. Then the order parameter
hOpi:= 1
pγl Zpγl
0
ρ(x)e−i2πx/(pγl)dx =1
p
1
2πl2e−π2
p2γ2
p−1
X
j=0
nje−i2πj/p
is an analytic function of γ∈]γp,∞[, and hOpi= 1 + O(e−γ2)as γ→ ∞. Hence
on ]γp,∞[hOpicannot vanish except possibly on a discrete set, and for sufficiently
large γ,hOpi>0.
Remark: For p= 3, the reversal symmetry implies n1=n−1=n2and the order parameter
equals
hO3i=1
3γl Z3γl
0
ρ(x)e−i2πx/(3γl)dx =1
3
1
2πl2e−π2k2
9γ2(n0−n1).
Thus hO3iis really a measure of the amplitude of oscillations.
Again, we give details of the proof only for odd p. When pis even, the occupation numbers
ˆnkare unbounded operators (thus ˆnk/∈ A). We start with the simpler part of the proof,
namely with the proof of 2.
Proof of 2. Recall from Theorem 2.12 that for γ > γp, the series Pnαntnhas a radius of
convergence Rstrictly larger than the radius of convergence rof PnCntn. It follows that for
γ > γp, the second moment condition (2.57) holds. Now let us turn to the order parameter.
The analyticity is essentially a consequence of the analyticity of rp(γ) and qp(γ) (see Theorem
2.12). For the asymptotics of the order parameter as γ→ ∞, remember that the natural
variable is u=e−γ2, and u= 0 describes a pure monomer system. The asymptotics follows
from an expansion around the point u= 0. We give more details now.
Let ˆnk=c∗
kck. Remember
nk=∞
X
n=1
qrnαnX
j∈Zhun,ˆnk−pjuni/||un||2.
For n∈N, we have
p−1
X
k=0
e−i2πk/p ∞
X
j=−∞hun,ˆnk−pjuni=pX
k∈Z
e−i2πk/phun,ˆnkuni=p
pn−p
X
k=0
e−i2πk/phun,ˆnkuni
Let hOpi′:= 1
pPp−1
k=0hˆnkie−i2πk/p. Then
hOpi′=∞
X
n=1
qnrnαnhOpi′
n,hOpi′
n:= 1
pnhun,pn−p
X
k=0
e−i2πk/pˆnkuni/||un||2.
68 CHAPTER 2. THERMODYNAMIC LIMITS
Recall that (qnrnαn) represents a probability distribution on the lengths of polymers. The
previous formula expresses the fact that each polymer contributes a part to hOpi, depending
only on its length. Both ||un||2=αnand hunˆnkuniare polynomials of e−γ2, for example
hunˆnkuni=X
0≤m1≤..≤mN≤p(N−1)
irred.
χ{m1,..,mN}(k)|bN(m1, .., mN)|2eγ2PN
j=1(m2
j−p2(j−1)2)
where bN(m1, .., mN) are integers. By Theorem (2.12), r(e−γ2) and qare analytic functions
of e−γ2∈[0, e−γ2
p[. Since |hOpi′
n| ≤ 1, and αn(e−γ2)≤αn(e−γ′2) for γ≥γ′, the sum
PnnrnαnhOpi′
nis a uniformly convergent sum of analytic functions, hence it is analytic
itself. In particular, it has a convergent power series expansion in e−γ2= 0 around u= 0.
Since αn(e−γ2= 0) = 0 for n≥1, r(0) = 1 and q(0) = 1, we obtain hOpi′
n= 1+O(e−γ2).
Proof of 1. The main ingredients of the proof of 1. are a lemma that we will state and prove
below and the observation that orthogonality of states needs only be checked on a subalge-
bra. In the problem at hand, there is a natural choice of subalgebra: it is the algebra A0
generated by the occupation numbers ˆnk. The algebra A0is abelian and thus can be treated
using a probabilistic language. The associated probability space Ω describes point particles
on a lattice. The state ωon Ais associated with a probability measure Pon Ω, and shifted
states ω◦τk
xcorrespond to shifted measures Pk. In order to prove orthogonality of the states,
we need to prove that the shifted measures are mutually singular. The singularity of the
measures follows from the existence of a random variable Qwith the following property:
there is a set of pairwise different values q0, .., qp−1such that for each k∈ {0, .., p −1}, the
random variable Qtakes the value qkPk-almost surely. This means that the measures Pk
have their supports in the disjoint preimages under Qof q0, .., qp−1.
The existence of such a random variable Qis the object of Lemma 2.18 below. The main
work to be done is the proof of this lemma, which we postpone to p.71 where the reader will
find also a description of the physical intuition behind this lemma.
Let A0be the closed subalgebra of Agenerated by the occupation number operators ˆnk=
c∗
kck, k ∈Zand the identity 1. We start by showing that it is enough to check orthogonality
on A0. Let ωn:= ω◦τpn
x, and let ˜ωbe a positive functional on Awith ˜ω(a)≤ω0(a) and
˜ω(a)≤(ω1(a) + ... +ωp−1(a)) for all a∈ A. In particular, the inequalities hold for a∈ A0.
If the states ωj|A0are orthogonal, it follows that ˜ω|A0= 0. Since 1∈ A0, we have ˜ω(1) = 0.
By the positivity of ˜ω, for all a∈ A,|˜ω(a)| ≤ ˜ω(1)||a|| = 0, whence ˜ω= 0.
Thus we need only show that the restrictions of ω0, ., ωp−1to A0are orthogonal. But A0
is a commutative algebra and we can use a probabilistic language. Let us briefly recall
the construction of the probability space. The occupation number operator ˆnkgives the
number of particles at lattice site k. The probability space Ω we will consider describes
particles on a lattice. Each particle configuration X∈Ω is characterized uniquely by the
sequence of occupation numbers (nk(X))k∈Z, where nk(X) gives the number of particles at
the lattice site k. For simplicity, suppose podd. Then Laughlin’s function is antisymmetric
and thus describes fermions. There is at most one particle per lattice site and we may take
Ω = P(Z)≡ {0,1}Z, where we identify sets with their characteristic functions. We give {0,1}
the discrete topology and {0,1}Zthe product topology, and let Ebe the Borel-σ-algebra. The
mapping ˆnk7→ nk(·) extends uniquely to a homomorphism φfrom A0to C(Ω), and there is
a unique probability measure Pon (Ω,E) such that for all a∈ A0,ω(a) = Rφ(a)dP .
2.1. LAUGHLIN’S CYLINDER WAVE FUNCTION 69
The probability of cylinder sets can be specified as follows: for K, A ⊂Z,
P({X|X∩K=A}) = ωY
k∈A
ˆnkY
k∈K\A
(1 −ˆnk).(2.58)
Equivalently, for all K={k1, .., kr} ⊂ Z,|K|=r < ∞, and all (ǫ1, .., ǫr)∈ {0,1}r,
P({X|nk1(X) = ǫ1, .., nkr(X) = ǫr}) = ωY
k∈K
ǫk=1
ˆnkY
k∈K
ǫk=0
(1 −ˆnk).
Shifted states area associated with shifted measures: let τ: Ω →Ω, X7→ X+ 1. Then the
measures Pjassociated to the states ωj=ω◦τj
xare given by Pj=P◦τj. To conclude the
proof, we need the following lemma:
Lemma 2.18. There is a map Q: Ω →RZ,X7→ (QX(x))x∈Zsuch that:
•for all X∈Ω, for all x, s ∈Z:QX+s(x) = QX(x−s), i.e., Qis “covariant”;
•for P-almost all X∈Ω, for all x∈Z:QX(x)∈ −x/p +Z.
The lemma will be proved a little later. For a given set of particles, QX(x) should be thought
of as the total charge accumulated in ]−∞, x[, where the particles account for a charge 1 and
a uniform background of density −1/p is added. By Lemma 2.18, there exists a measurable
subset Mof Ω such that P(M) = 1 and for all X∈ M and x∈Z:QX(x)∈ −x/p +Z. We
have
P(M) = 1 and for all X∈ M :QX(0) ∈Z.
It follows that for j∈Z, the shifted measures Pj=P◦τjsatisfy
Pj(M−j) = 1 and for all X∈ M+j:QX(0) ∈ −j/p +Z.(2.59)
To see this, let X∈ M−j; we may write X=Y−jwith y∈ M and
QX(0) = QY−j(0) = QY(j)∈ −j/p +Z.
By (2.59), the measures P=P0, P1, .., Pp−1are mutually singular.
As a consequence, the states ωjon A0are orthogonal. Indeed, let ˜ωbe a positive functional on
A0with ˜ω≤ω0and ˜ω≤(ω1+..+ωp−1). Let µbe the measure (not necessarily a probability
measure) on (Ω,F) associated to ˜ωthrough a relation similar to (2.58). Then, for all E∈ F,
µ(E)≤P0(E) and µ(E)≤(P1(E) + .. +Pp−1(E)). Applying this to Ej=M+j, we get
µ(Ej) = 0 for all j∈ {0, .., p −1}, hence µ= 0 and ˜ω= 0.
The proof of Lemma 2.18 follows ideas of [AM80, AGL01]. We start with a summary of the
ideas of these works that we shall use. In [AM80], Section 4, Aizenman and Martin consider
a one-dimensional classical jellium system, consisting of Nparticles of negative charge −e
moving in a neutralizing background of charge density ρe on a line segment [−L/2, L/2] ⊂R.
For simplicity, we will assume e= 1. The one-dimensional Coulomb potential is −|x|. Each
particle configuration is characterized by a set of particle positions x1, .., xN. The electric
field at xis a function of the particle positions: for x∈[−L/2, L/2],
E(x;{xj}) = −
N
X
j=1
sgn(x−xj) + 2ρx. (2.60)
70 CHAPTER 2. THERMODYNAMIC LIMITS
(Actually, the field is not well-defined at the location of a particle, x=xj. We shall not
deal with this difficulty here.) Due to the overall neutrality of the system, the electric field
vanishes outside the volume containing the charges:
E(−L/2; {xj}) = 0 = E(L/2; {xj}).(2.61)
Furthermore, for x≤y,
E(y;{xj})−E(x;{xj}) = 2ρ(y−x)−2X
xj∈[x,y]
1.(2.62)
It follows that
E(x;{xj}) = E(x;{xj})−E(−L/2; {xj}) = 2ρx +N−2X
xj∈[−L/2;x]
1
Suppose we take Neven; then
E(x;{xj})∈2ρx + 2Z.(2.63)
This constraint will subsist in the limit N→ ∞, the density ρbeing kept fixed, and shows
that the limiting distribution of the field Ecannot be translationally invariant. It can at
best be periodic with period 1/ρ. We would like to know whether the particle positions have
a translationally invariant distribution. In the limit of infinite volume, the particle positions
can be recovered from the electric field as the locations of its discontinuities. But a function
of a periodic quantity may very well be translationally invariant, so this is of no help. In
order to conclude that the particle positions have a non translationally invariant distribution,
we need to know that the field is a function of the particle positions. This is of course the
case for finite volumes, but for systems with infinitely many particles the formula (2.60) no
longer makes sense. Nevertheless, the field can be recovered from the particle positions. This
is shown in [AM80] using an ergodic theorem.
An abstract version of the argument previously described is considered in [AGL01], who look
at point processes and at one-dimensional charge distributions. Let us sketch some aspects
of [AGL01] for the special case of one species of particles moving on a lattice. Let Pbe
a translationally invariant probability distribution on the space of particle configurations
P(Z)≡ {0,1}Z. Let ρbe the average number of particles at a given site. We might think of
the particles as point charges with charge 1 moving in a neutralizing background of charge
density −ρ. For a given configuration X⊂Zand a, b ∈Z(a < b), define
FX(]a, b]) := |X∩]a, b]|−ρ(b−a).(2.64)
FX(]a, b]) is the difference between the number of particles in ]a, b] in the configuration X
and the average number of particles in ]a, b]. It can also be considered as the total “charge”
in ]a, b]. [AGL01] prove that if the random variables X7→ FX(]a, b]), a, b ∈Z,a < b have
variances uniformly bounded in a, b (actually, [AGL01] give weaker conditions), there exists
acovariant antiderivative, i.e., there exists a family of random variables X7→ EX(a), a∈Z,
such that for P-almost all X⊂Zand all a, b ∈Z,a < b:
EX+1(a) = EX(a−1), FX(]a, b]) = EX(b)−EX(a).(2.65)
Combining the second identity with (2.64) gives an expression for EX(b)−EX(a) analogous
to (2.62). Thus EX(a) can be thought of as the “field” in a, or simply as the total charge
2.1. LAUGHLIN’S CYLINDER WAVE FUNCTION 71
accumulated in ] − ∞, a]. Now the probability space can be partitioned into level sets of
X7→ EX(0) (modulo 1). For θ∈R/Z, let
Mθ:= {X∈ P(Z)| EX(0) = θmod 1}.
Using (2.65), one easily sees that the shift Tk:X7→ X+kmaps Mθonto Mθ+kρ. If ρis
not an integer, there are several sets Mθ, and [AGL01] prove that the probability measure
Pdecomposes into a mixture of mutually singular measures Pθ, supported in Mθ. In the
particular case that the density is rational and not integer: ρ=p/q,p, q coprime and q≥2,
we obtain a decomposition of the probability space into qsubsets invariant under shifts by
multiples of q. In particular, the probability measure Pcannot be ergodic with respect to
shifts by multiples of q: in this sense, there is symmetry breaking.
Now let us come back to Lemma 2.18. The idea of proof is a combination of ideas found in
[AM80] and [AGL01]. From [AGL01] we take the idea to construct a generalized “electric
field” or “total charge” random variable. This is the variable QX(x) of Lemma 2.18. For the
construction of the random variable, we follow the idea [AM80] of looking at finite volumes
first and then making a detour involving a larger probability space and an ergodic theorem.
As we have seen in the proof of Theorem 2.17, the level sets of the field play an important
role for the symmetry breaking, just as in [AGL01] and [AM80].
Proof of Lemma 2.18. The starting point is the following: the state ωis associated with a
probability measure Pon Ω = P(Z), where a particle configuration is characterized through
the set X⊂Zof occupied lattice points. The measure Pis invariant with respect to shifts
(on Z) by multiples of p. By Corollary 2.15, the average density is
1
p
p−1
X
j=0hˆnji=1
p.
We think of the particles as point charges of charge 1 moving in a neutralizing background
of charge density −1/p. Each configuration X⊂Zdefines a “charge distribution” Fon Z
through
FX(A) := |X∩A|− 1
p|A|, A ⊂Z.
The previous considerations suggest to search for a covariant antiderivative of FX, i.e. to
search for a map X7→ (QX(x))x∈Zsuch that
QX+s(x) = QX(x−s), FX([x, y[) = QX(y)−QX(x)
and to show that the probability measure Pis supported in the set where QX(x) = −x/p
mod 1. We will proceed as follows:
1. We start by looking at finite volumes. For systems with finitely many particles N,
we can simply define QN
X(x) as the total charge accumulated in ] − ∞, x[ (with a
suitable background distribution). The field thus constructed is not covariant, but
essentially satisfies the condition QN
X(x) = −x/p mod 1. We will see that finite volume
configurations may be characterized through an auxiliary space (Ω′
N,F′
N, P′
N); this
serves as a guide for the infinite volume case.
2. As a second step, we construct an auxiliary probability space (Ω′,F′, P′) space from
which the particle positions may be recovered, i.e., there is a random variable M: Ω′→
72 CHAPTER 2. THERMODYNAMIC LIMITS
Ω = P(Z) such that the measure Pis the image of P′under M:P=P′◦M. The
approach in [AM80] suggests to take as larger probability space Ω′the configurations
of the field, but we will see that in our setting, a different construction is more natural;
the auxiliary space arises roughly as infinite volume limit of the space Ω′
Nof 1.
3. We proceed by showing that there exists a field Ω′∋ν7→ (Qν(x))x∈Z, defined on the
auxiliary space, that is an antiderivative of the particle positions in the sense that
FM(ν)([x, y[) = Qν(y)−Qν(x),
where M: Ω′→Ω reconstructs the particle locations from the auxiliary variable ν∈Ω′.
We show that Qhas a covariance property, that Qν(x) = −x/p mod 1 P′-almost surely
and that ν7→ Qν(x) has finite expectation value; this is the place where the second
moment condition (2.57) enters.
4. Just as in [AM80], we use an ergodic theorem for Qν(x) to show that the field Qνcan
be recovered from the particle locations, i.e., there is a map Q: Ω ∋X7→ (QX(x))x∈Z
such that
∀x∈Z:Qν(x) = QM(ν)(x) for P′−almost all ν∈Ω′.
The map Q, suitably defined, is the map of Lemma 2.18.
1. Finite volumes. For N∈N, let h·iNbe the state defined by ΨNand PNthe corresponding
probability measure on Ω = P(Z). Note that X⊂ {0, .., pN −p}and |X|=N PN-almost
surely. More precisely, let aN(m1, .., mN) be the expansion coefficients of Laughlin’s N-
particle state, CNits L2-norm squared, then the probability PNis defined as
PN({m1, .., mN}) = |aN(m1, .., mN)|2/CN
where for X⊂Zwe write P(X) instead of P({X}). Think of PNas describing Nparticles
located in {0, .., pN −1}, with a uniform neutralizing background so that the total charge
accumulated in ] −∞, x[ vanishes if x≤0 or x≥pN. Let
QN
X(x) := (|X∩[0, x[|− 1
px, if x∈ {0, .., pN},
0,else.
For X⊂ {0, .., pN −1},|X|=N, we have
QN
X(0) = QN
X(pN) = 0.
This equation is the analogue of (2.61). If x, y are in {0, .., pN}and x≤y, we have obviously
QN
X(y)−QN
X(x) = |X∩[0, x[|− 1
px;
in this sense, QN
Xis an antiderivative. Furthermore, it is obvious from the definition of QN
that
∀X⊂Z,∀x∈ {0, .., pN}:QN
X(x) = −x/p mod 1.
This is the analogue of the constraint (2.63) on the electric field for one-dimensional jellium.
Now let us use the additional structure present here. First, note that if X⊂Zhas a
non-zero probability, PN(X)>0, there exists an increasing, N-admissible finite sequence
m= (m1, .., mN) such that X={m1, .., mN}. Now, we claim that:
2.1. LAUGHLIN’S CYLINDER WAVE FUNCTION 73
Let m= (m1, .., mN) be increasing m1≤.. ≤mNand N-admissible. Let X=
{m1, .., mN}with PN(X)>0. Suppose kis a renewal point of m. Then QN
X(pk) =
0 and for pk ≤x≤pN,
QN
X(x) = |X∩[pk, x[|−(x−pk)/p.
Thus renewal points give zeroes of the field, and the field at xdepends only on
the particle configuration up to the next renewal point.
To see why this holds, let m,X,k.PN(X)>0 implies that (m1, .., mk) and (mk+1 −
pk, .., mN−pk) are k(resp. N−k)-admissible. In particular, 0 ≤m1, .., mk≤pk −p, and
pk ≤mk+1, .., mN≤pN −p, therefore if x≥pk
QN
X(x) = |X∩[0, pk[|−k+|X∩[pk, x[|−(x−pk)/p
=|{m1, ., mk}|−k+|X∩[pk, x[−(x−pk)/p
=|X∩[pk, x[−(x−pk)/p,
whence the claim.
Definition of an auxiliary probability space: finite volumes. We would like to use the obser-
vation on renewal points to define QXfor infinite systems. However, defining renewal points
for infinite sets X⊂Zseems even harder than defining a total charge, thus we have to make
a little detour. Recall from the proof of Lemma 2.5 that N-admissible sequences can be
characterized in terms of sequences ν0, .., νNwith
mj=p(j−1) + νj−νj−1, ν0=νN= 0
νj≥0, νj+1 −2νj+νj−1+p≥1.
Furthermore νk= 0 if and only if kis a renewal point of m. The probability measure PN
defines a measure P′
Non the space Ω′
Nof (νj)-sequences through
P′
N({(0, ν1, .., νN,0)}) = PN({(m1, .., mN)}) = |aN(m1, .., mN)|2/CN.
The factorization rule from Lemma 2.4 leads to
P′
N({0, ν1, .., νk−1,0, νk+2, .., νN−1,0)}) =
CkP′
k({(0, ν1, .., νk−1,0)})CN−kP′
N−k({(0, νk+1, .., νN−1,0)})/CN,
moreover P′
N(νk= 0) = CkCN−k/CNfor all k∈ {0, .., N}. Thus for k→ ∞,N−k→ ∞,
P′
N(νk= 0) →µ−1=q.
2. Auxiliary probability space: infinite volume. More generally, there is a natural probability
measure on the space of infinite sequences. Let Ω′⊂NZ
0be the set of sequences (νj)j∈Zsuch
that
∀j∈Z:νj+1 −2νj+νj−1+p≥1
and the set {j|νj= 0}is bounded neither from below nor above. Define a probability
measure P′on Ω′by requiring that the zeroes of (νj) are distributed like a stationary renewal
process with interarrival distribution (pn) = (rnαn), constant renewal probability
P′(νj= 0) = q=µ−1= 1/(X
n
nrnαn),
74 CHAPTER 2. THERMODYNAMIC LIMITS
and for all a∈Z,N∈N, and ǫa+1, .., ǫa+N−1in N0,
P′({(νj)|νa= 0, νa+1 =ǫa+1, ..., νa+N−1=ǫa+N−1, νa+N= 0})
=µ−1CNrNP′
N({(0, ǫa+1, .., ǫa+N−1,0)}).
The measure P′is invariant and mixing with respect to shifts (νj)7→ (νj−1). The probability
measure Pon Ω is recovered with
M: Ω′→Ω,(νj)7→ {p(j−1) + νj−νj−1|j∈Z}
as P=P′◦M. Note that M((νj−1)) = p+M((νj)).
3. Definition of an auxiliary map Q.For ν∈Ω′and x∈Z, let j∈Zbe such that νj= 0
and pj ≤x. Define Qν(x) := FM(ν)([pj, x[). This definition is motivated by the observation
made in 1. that a renewal point arising through νj= 0 give zeroes of the field Qν(pj) = 0.
Note that the definition does not depend on the choice of j: suppose pj′< pj ≤xand
νj′=νj= 0. Let mk:= p(k−1) + νk+1 −νk. By the convexity property of (νk)k, (mk) is
increasing, furthermore νk≥0 implies
mj′≤p(j′−1), pj′≤mj′+1 ≤.. ≤mj≤p(j−1), pj ≤mj+1..
whence M(ν)∩[pj′, pj[= {mj′+1, .., mj}and
FM(ν)([pj′, x[) −FM(ν)([pj, x[) = FM(ν)([pj′, pj[) = 0.
Qνhas a series of properties:
1. Qνsatisfies a constraint on its range: Qν(x) = −x/p mod 1 for all ν∈Ω
2. Qνis an antiderivative of the charge density F: if x < y,Qν(y)−Qν(x) = FM(ν)([x, y[)
3. Qνis covariant: Qτν (x) = Qν(x−p), where (τν)j=νj−1
4. The field at xhas finite expectation value: RΩ′|Qν(x)|dP′(ν)<∞for all x∈Z.
Indeed, let x∈Z, j ∈Zwith νj= 0, pj ≤x, then
Qν(x) = FM(ν)([pj, x[) = |M(ν)∩[pj, x[|−(x−pj)/p =−x/p mod 1.
If x < y,
Qν(y)−Qν(x) = FM(ν)([pj, y[) −FM(ν)([pj, x[) = FM(ν)([x, y[).
Choose jsmall enough so that p(j+ 1) ≤xand let ν′
j:= νj−1. Then ν′
j+1 = 0, and
Qτν(x) = Qν′(x) = FM(τν)([p(j+ 1), x[)
=Fp+M(ν)([pj +p, x[) = FM(ν)([pj, x −p[) = Qν(x−p).
Finally, suppose pj ≤x≤p(j+n−1) and νj=νj+n= 0, j∈Z,n∈N. Then M(ν)∩
[pj, p(j+n)−p] = {mj+1, .., mj+n}and |Qν(x)| ≤ n+(x−pj)/p ≤2n. For ν∈Ω′, let nν(x)
be the smallest possible n. Then
Z|Qν(x)|dP′(ν)≤2ZΩ′
nν(x)dP′(ν) = 2 ∞
X
n=1
n·nrnαn<∞,
2.1. LAUGHLIN’S CYLINDER WAVE FUNCTION 75
where we used that P(nν(x) = n) = nrnαn.
4. Definition of QX(x).We wish to define QX(x) such that Qν(x) = QM(ν)(x). Note that
by P=P′◦M, for P-almost all X, there is a ν∈Ω′such that X=M(ν). We need to check
however, that Qν(x) depends only on M(ν), i.e. M(ν) = M(ν′) implies Qν=Qν′. This is
done by means of the ergodic theorem (P′is mixing with respect to shifts). Let
¯
Q:= 1
pZ(Qν(x) + Qν(x+ 1) + .. +Qν(x+p−1))dP′(ν).
Then for P′-almost all ν∈Ω′,
¯
Q= lim
k→∞
1
pk
k−1
X
r=0Qτ−rν(x) + .. +Qτ−rν(x+p−1)
= lim
k→∞
1
pk
k−1
X
r=0Qν(x+pr) + .. +Qν(x+pr +p−1)
= lim
k→∞
1
pk
x+pk−1
X
y=x
Qν(y) = lim
k→∞
1
pk
x+pk−1
X
y=x
(Qν(x) + FM(ν)([x, y[))
It follows that for P′-almost all ν∈Ω,
Qν(x) = ¯
Q−lim
k→∞
1
pk
x+pk−1
X
y=x
FM(ν)([x, y[).
The right-hand side depends on M(ν) only, hence Qν(x) = QM(ν)(x) for P′-almost all νwith
QX(x) := (¯
Q−limk→∞ 1
pk Px+pk−1
y=xFX([x, y[),if the limit exists
0,else.
If the limit QX(x) exists for some x∈Z, it exists for all y∈Z. Let M′be the set of X∈Ω
such that the limit exists. Then
∀X∈ M′:QX(y)−QX(x) = FX([x, y[).
For s, x ∈Z,
−1
pk
x+pk−1
X
y=x
FX+s([x, y[) = −1
pk
x+pk−1
X
y=x
FX+s([x, y[) = −1
pk
x+pk−1
X
y=x
FX([x−s, y −s[)
=−1
pk
x+pk−1−s
X
y′=x−s
FX([x−s, y′[)
hence if X∈ M′, then also X+s∈ M′and QX+s(x) = QX(x−s). This relation is also
true if X /∈ M′, in which case it reads 0 = 0. Thus
∀X∈Ω,∀x, s ∈Z:QX+s(x) = QX(x−s).
Finally, QXinherits from Qνthe condition on its range:
QX(x) = −x/p mod 1 for P-almost all X.
76 CHAPTER 2. THERMODYNAMIC LIMITS
Theorem 2.17 answers the question whether there is symmetry breaking in Laughlin’s state
only for thin cylinders. It leaves open what happens on large strips. In view of the numerical
results mentioned at the beginning of this section, we believe that, at least for small p, the
result applies to broad cylinders, but this still awaits a proof.
2.2 Solvable models
In this section, we define generalized Laughlin-type cylinder and torus wave functions. The
starting point for the cylinder function is the representation as a power of a Slater deter-
minant of Gaussians. The generalized function is obtained by allowing functions fdifferent
from Gaussians. These functions are interesting mainly for two reasons.
The first reason is that for particular choices of f, the modified cylinder function defines a
solvable model: if we choose fas a function with sufficiently small, compact support, the
modified cylinder function is associated with a monomer-system on Z. The partition func-
tion of a monomer-dimer system on a linear chain satisfies a two-step recurrence relation (see
e.g. [HL72], p.196). Consequently the L2-norm squared of the modified function satisfies a
two-step recurrence relation, which is of course explicitly solvable. The correlation functions
can be computed explicitly as well. We look only at the one-particle density.
More generally, the modified cylinder functions retain a lot of the structure present for
Laughlin’s cylinder function, and most of the results of Section 2.1 have an analogue. In
particular, the modified wave functions are associated with polymer systems. An important
parameter in the description of Laughlin’s wave function is the ratio γ=l/R of magnetic
length and cylinder radius. We have seen that as γ→ ∞, the system approaches a monomer
system. A similar phenomenon appears here: the definition of modified functions involves
not only the function fbut also its translates f(·−pnγl), n ∈Z. We choose fas a function
of compact support. Then we observe the following phenomenon:
1. When γis very large, f(·) and its neighbors f(·−nγl) have non-overlapping supports,
and the associated polymer system is a monomer system.
2. When γis such that only nearest neighbors f(·) and f(·± γl) overlap, we obtain the
monomer-dimer system mentioned above.
3. As soon as γis small enough so that next nearest neighbors f(·) and f(·−2nγl) over-
lap, the associated polymer system has polymers of non-vanishing activity of arbitrary
length.
The second motivation is the comparison of cylinder and torus functions. Up to now, this
chapter has only been concerned with cylinder functions, and our goal was to show that they
display a kind of symmetry breaking. This kind of symmetry breaking is well-known for
torus functions [HR85b, Hal85]. A torus can be seen as a cylinder with periodic boundary
conditions. It is widely accepted that for small p(p≤7), Laughlin’s function describes an
incompressible liquid. Thus boundary conditions should not affect the bulk behavior, and we
expect that the cylinder and torus wave functions are equivalent in the limit of long cylinder
/ tori. We check this equivalence for the solvable monomer-dimer model: when the cylinder
modified function is associated with a monomer-dimer system on a linear chain, a suitably
defined modified torus function is associated with a monomer-dimer system on a ring, with
possibly one additional long polymer covering the whole ring.
2.2. SOLVABLE MODELS 77
2.2.1 Generalized Laughlin wave functions
We start by giving the definition of generalized cylinder and torus functions ΦC
N, ΦT
N. In
Proposition 2.20, we verify that the modified functions have periodicity properties that jus-
tify the denominations “cylinder” and “torus” functions. The definition of the modified
functions involves a function f:R→R. In Proposition 2.21, we check that we recover the
Laughlin wave functions of Definition 1.1 if we choose fas a suitable Gaussian.
Let f:R→Cbe a continuous even function (f(x) = f(−x)) such that R∞
−∞ |f|2p= 1. For
n∈Nand N∈N, define
φn(z) = 1
2p
√2πReiny/Rf(x−pnγl),˜
φn(z) = ∞
X
k=−∞
φn+kN .(2.66)
Definition 2.19. Let f, φn,˜
φnbe as above. The generalized Laughlin cylinder and torus
wave functions for Nelectrons at filling factor 1/p are the functions
ΦC
N(z1, .., zN) = 1
√N!det(φn−1(zj))1≤n,j≤Np.
ΦT
N(z1, .., zN) = 1
√N!
p
Y
s=1
det(eiXsyj/(pNl2)˜
φn−1(zj−Zs
N))1≤n,j≤N
where Z1, .., Zp∈Csatisfy Pp
s=1 Xs/(pNγl)∈Z,Pp
s=1 Ys/(2πR)∈Z.
The conditions on Z1, .., Zpare similar to conditions imposed on the zeroes of theta functions,
see [FK01], Chapter 7. Let us remark that there is a small difference between the definitions
of ΦC
Nand ΦT
N: the cylinder function is, up to a prefactor, the p-th power of a determinant,
whereas the torus function is in general only a product of pdeterminants.
Recall the expression of the magnetic translation in the x-direction:
(t(aex)ψ)(z) = eiay/l2ψ(z−a).
If Ψ is an N-particle wave function, we denote by tj(a), j∈ {1, .., N}the magnetic translation
in the j-th variable. For example, (t1(aey)Ψ)(z1, .., zN) = Ψ(z1−ia, z2, .., zN).
Proposition 2.20. The functions ΦC
Nand ΦT
Nhave the following periodicity:
tj(2πRey)ΦT
N= ΦT
N, tj(2πRey)ΦC
N= ΦC
N,
tj(pNγlex)ΦT
N= ΦT
N
for all j∈ {1, .., N}.
Proposition 2.21. Let
f(x) = 1
2p
pl√πe−1
2pl2x2.
Then the modified cylinder functions equal Laughlin’s function ΦC
N= ΨC
N, and the functions
ΦT
Nare torus Laughlin wave functions. Conversely, any torus Laughlin wave function ΨT
Nis
of the form ΨT
N=λΦT
Nwith λ∈Cand ΦT
Nas in Definition 2.19.
78 CHAPTER 2. THERMODYNAMIC LIMITS
Proof of Proposition 2.20. The periodicity in the direction around the cylinder follows from
φn(z+ 2πRi) = φn(z) and
eiPp
s=1 Xs2πR/(pNl2)=ei2πPp
s=1 Xs/(pNγl)= 1.
For the quasiperiodicity of the torus wave function, we compute
t1(pNγl)ΦT
N(z1, .., zN) = eipNy1/RΦT
N(z1−pNγl, .., zN).
Using
eiNy1/R ˜
φ0(z1−pNγl −Zs
N) = eiYs/R ˜
φ0(z1−Zs
N)
and Pp
s=1 Ys∈2πRZ, one can check t1(pNγl)ΦT
N= ΦT
N. The periodicity in the variables
z2, .., zNis shown in an analogous way.
Proof of Proposition 2.21: cylinder function. The cylinder Laughlin wave function is
ΦC
N(z1, .., zN) = 1
√N!Y
N≥j>k≥1
(ezj/R −ezk/R)pe−PN
j=1 x2
j/2l2e−1
2p2γ2PN−1
j=0 j2
(2πRl√π)N/2
=1
√N!
1
(2πRl√π)N/2det(e(n−1)zj/R)1≤n,j≤Ne−1
2pl2PN
j=1 x2
je−1
2pγ2PN
n=1(n−1)2p
=1
√N!
1
(2πRl√π)N/2det(ei(n−1)yj/Re−1
2pl2(xj−p(n−1)γl)2)1≤j,n≤Np
This is of the form given in Definition 2.19 provided we choose f(x) as a Gaussian with the
correct multiplication constant.
Torus functions. We start by showing that any ΦT
Nfunction is a torus Laughlin wave function.
Note first that
ΦT
N(z1, .., zN) = f(z1, .., zN)e−1
2l2PN
j=1 x2
j
for some holomorphic function f. This is checked using
eiXsy/R ˜
φn(z−Zs
N)∝eXsz/(pNl2)∞
X
k=−∞
e(n+kN)z/R−pγ2(n+kN)2/2e−x2/(2pl2)e−X2
s/(2pl2N2).
Together with the quasiperiodicity of Proposition 2.20, this shows that ΦT
Nis in the torus
lowest Landau level. On the other hand, ΦT
Nhas a zero of order at least pas two particles
get close (zj−zk)→0. Thus by the results of Section 1.4 it is a torus Laughlin wave function.
The functions ˜
φncan be considered as the basis functions of the lowest Landau level on a
torus [0, pNγl]×[0,2πR] with a modified magnetic length l′=√pl. The corresponding filled
Landau level wave functions are
G(z1, .., zN) = F(
N
X
j=1
zj)Y
1≤j<k≤N
θ1(i(zj−zk)
2R|iL
2πR)e−1
2pl2PN
j=1 x2
j
where L=pNγl and Fis a holomorphic function satisfying
F(Z+i2πR) = (−1)N−1F(Z), F (Z+L) = (−1)N−1eZ/ReL/(2R)F(Z).
2.2. SOLVABLE MODELS 79
But the space of filled Landau level wave functions is one-dimensional, hence there exists a
c∈Rsuch that √N!˜
φ0∧.. ∧˜
φN−1=cG. (2.67)
This gives
ΦT
N(z1, .., zN) = 1
√N!cpei(Pp
s=1 Xs)(PN
j=1 yj)/(pNl2)
p
Y
s=1
G(z1−Zs
N, .., zN−Zs
N)
=1
√N!cpFcm(
N
X
j=1
zj)Y
1≤j<k≤N
θ1(i(zj−zk)
2R|iL
2πR)pe−1
2l2PN
j=1 x2
j(2.68)
Fcm(Z) = F(Z−Z1)F(Z−Z2)..F(Z−Zp).(2.69)
Conversely, any torus wave function is of the form (2.68) for some holomorphic function Fcm
satisfying the conditions (1.7). We know from the theory of theta functions ([FK01], Chapter
7) that any such function can be written, up to a multiplicative constant, as a product of
the form (2.69), where Z1, .., Zpare related to the zeroes of Fcm and fulfill PXs∈LZ,
PYs∈2πRZ, and Fis an odd or even theta function depending on the parity of N. As
a consequence, every torus Laughlin wave function can be written in the form of Definition
2.19.
Remark. A more direct way of getting (2.67) uses a generalized Vandermonde identity in-
volving theta functions, see [For06], Proposition 3.1.
Now that we have defined the generalized functions, we can turn to their thermodynamic
limits. We start with the cylinder functions.
2.2.2 Cylinder wave functions
In this subsection we give some properties of the modified Laughlin function and investigate
the solvable monomer-dimer model. In Lemma 2.22, we prove that the modified cylinder
function is associated with a polymer system; this holds true for general f. In Proposition
2.23, we choose fas a function of compact support and show how the size of the support
compared to the “unit cell” [−pγl/2, pγl/2] affects the corresponding polymer system. When
the support of fis contained in [−pγl, pγl], the associated polymer system is a monomer-
dimer system. This system is solvable. In Proposition 2.24, we take p= 3 and determine the
normalization constant and the one-particle density for the solvable model (finite volumes),
from which we deduce the corresponding expressions in the infinite volume limit (Corollary
2.25).
Let N∈Nand π∈ S{0,..,N−1}. We define the set of renewal points of the permutation πas:
R(π) := {k∈ {0, .., N −2} | π({0, .., k})⊂ {0, .., k}}.
For n∈N,j∈Zand podd, define
u{0,..,n−1}:= √n!AX
σ2,..,σp∈Sn:
R(σ2)∩..∩R(σp)=∅
sgn(σ2..σp)
φ0φσ2(1)..φσp(1) ⊗.. ⊗φN−1φσ2(N−1)..φσp(N−1)(2.70)
u{j,..,j+n−1}:= t(jpγlex)⊗nu{0,..,n−1}.(2.71)
80 CHAPTER 2. THERMODYNAMIC LIMITS
For even p, the antisymmetrization Ais replaced with the symmetrization S. In the following
||·||refers to the L2((R×[0,2πR])N) for suitable N. The one-particle density of ΦC
Nis denoted
ρN(z).
Lemma 2.22. Let p∈Nodd, N∈N, then
ΦC
N=X
X1,..,XD
uX1∧.. ∧uXDρN(z) = X
X
ρP
N(X)vX(z)
CN:= ||ΦC
N||2=X
n1+..+nD=N
αn1..αnD
where the first sum ranges over ordered partitions X1< .. < XDof {0, .., N −1}and
αN(X)=||uX||2, vX(z) := N(X)
||uX||2Z|uX|2dx2..dyN(X).
Similar formulas hold for even p, provided we replace the wedge product with the symmetric
product and, in the definition of uX, the antisymmetrization Awith the symmetrization
operator S.
Proof. To fix ideas, we take p= 3. ΨC
Ncan be expanded as
ΨN=√N! AX
σ,τ
sgn(στ)φ0φτ(0)φσ(0) ⊗.. ⊗φN−1φτ(N−1)φσ(N−1),(2.72)
where the sum ranges over permutations of {0,1, .., N −1}. The sum can be written as a sum
over (possibly empty) sets K⊂ {0, .., N −2}and permutations σ, τ with R(σ)∩R(τ) = K.
The empty set K=∅gives (N!)−1/2u{0,..,N−1}. If K={k1, .., kD}with k1< .. < kD,
the summation over corresponding σ, τ gives rise to Q(N(Xj))−1/2uX1⊗.. ⊗uXDwith
Xj={kj, .., kj+1 −1},k0= 0, kD+1 =N. Using A(Af ⊗Ag) = A(f⊗g), we obtain
ΨC
N=X
X1,..,XDN!
N(X1)!...N(XD)!1/2A(uX1⊗.. ⊗uXD) = X
X1,..,XD
uX1∧.. ∧uXD.
Let Xj={kj, .., kj+1 −1}, 0 = k0< k1< .. < kD+1 =N. Then
uX1∧.. ∧uXD=A(X′
m1<..<mN
ei(m1y1+..+mNyN)/Rfm(x1, ..., xN))
for suitable functions fm, and the sum goes only over N-admissible partitions with renewal
points R(m) = {k1, .., kD+1}. This can be used to show
{X1, .., XD} 6={Y1, .., YD′} ⇒ ZuX1∧.. ∧uXDuY1∧.. ∧uYD′dy2..dyN= 0 (2.73)
from which the representations of the norm CNand the one-particle density can be deduced.
The previous lemma is of particular interest when the Gaussian is replaced with a function
fof compact support suppf⊂[−s, s]. The parameter scontrols the size of the support and
the overlap between the function and its translates f(·−kpγl), k∈Z. It takes the role of γ
as a parameter controlling the activities.
2.2. SOLVABLE MODELS 81
Proposition 2.23. For large γ, the associated polymer system is a monomer system. As γ
is decreased, it becomes first a monomer-dimer system and then a system with polymers of
arbitrary length:
1. If pγ ≥2s,φkφn= 0 unless k=l, and uXvanishes unless N(X) = 1.
2. If pγ ≥s,φkφn= 0 when |k−l| ≥ 2, and uXvanishes unless N(X)∈ {1,2}.
3. If φkφk+2 6= 0,uX6= 0 for all rods X.
Proof. 1. Suppose pγ ≥2sso that suppf⊂[−pγ/2, pγ/2] and φkφl= 0 unless k=l. To fix
ideas, consider the case p= 3. The only permutations contributing to the sum in (2.72) are
σ=τ= id, so that
ΦC
N=φ3
0∧.. ∧φ3
N−1=u{0}∧.. ∧u{N−1}, u{n}=φ3
n.
2. Suppose pγ ≥s. Again, chose p= 3. If σ, τ contribute to the sum in (2.72), they must
fulfill
∀k∈ {0, .., N −1}:|σ(k)−k| ≤ 1,|τ(k)−k| ≤ 1,|σ(k)−τ(k)| ≤ 1.(2.74)
This condition can also be expressed in terms of the permutation matrices Pid =1,Pσ,Pτ:
the matrices Pσ,Pτand PσP−1
τmust be band-diagonal (only matrix elements pj,j and pj,j±1
are allowed to be non-zero). The first inequality in (2.74) implies that the cycle decompo-
sition of σconsists of nearest neighbor transpositions (r r + 1). If (r r + 1) is in the cycle
decomposition of σ, the condition ∀k:|σ(k)−τ(k)| ≤ 1 prevents (r−1r) and (r+ 1 r+ 2)
from appearing in the cycle decomposition of τ. Thus there is a set of disjoint nearest neigh-
bor transpositions (i1i1+ 1), .., (ikik+ 1) consisting of the transpositions appearing in the
decomposition of σor τ. The supports of the transpositions {i1, i1+ 1}, .., {ik, ik+ 1}can be
completed by singletons {n}to give a partition of {0, .., N −1}into monomers and dimers.
Following the arguments in the proof of Lemma 2.22, one sees that ΦC
Ncan be written as a
sum over monomer-dimer partitions.
3. Suppose φkφk+2 6= 0. Let σ1σ2, .., σp∈ S{0,..,n−1}and (m0, .., mN−1) := (σ1(0) + σ2(0) +
.. +σp(0), .., σ1(N−1) + .. +σp(N−1)). The N-tuple (m1, .., mN) can be considered as a
set of angular momenta, since
(φσ1(0)..φσp(0))⊗.. ⊗(φσ1(N−1)..φσp(N−1)) = eiPN
j=1 mjyj/RFσ,τ (x1, .., xN)
for some function F. Suppose that (m0, m1, .., mN−1) = (1, p, 2p, .., (N−2)p, (N−1)p−1).
We wish to determine the corresponding permutations σ1, .., σp. To do this, we start by
looking at the partition of m0= 1 obtained by arranging σ1(0), .., σp(0) in descending order,
then at the partition of m1, etc. The only way of writing 1 as a sum of pnon-negative integers
arranged in decreasing order is 1 = 1 + 0 + .. + 0, hence the partition of m0may be written
as (1 0p−1). Thus the sum m1=p=σ1(1) + .. +σp(1) has pnon-negative numbers, among
which at least one is different from 1 and at most one is equal to 0. This leaves (2 1p−20)
as the only possible partition. Hence the partition associated to 2pconsists of pintegers
≥1 among which at least one is different from 2 and at most one is equal to 1. The only
possibility is (3 2p−11). Proceeding in this way, we get the sequence of partitions
(1 0p−1),(2 1p−20),(3 2p−21), ..., (N−1 (N−2)p−2N−3),((N−1)p−1N−2).
The corresponding product of functions
φ1φp−1
0⊗φ2φp−2
1⊗.. ⊗φN−1φp−2
N−2φN−3⊗φp−1
N−1φN−2
82 CHAPTER 2. THERMODYNAMIC LIMITS
does not vanish since we assume that nearest neighbor functions overlap. Now we have to
figure out the corresponding permutations. Up to a numbering of the permutations, the only
way to get the first partition is by choosing σ1(0) = 1, σ2(0) = .. =σp(0) = 0. Therefore
σ2(1), ..σp(1) must be different from 0. Thus in order to obtain the partition (2,1p−2,0) we
must have, up to a renumbering of σ2, .., σp,σ1(1) = 0, σ2(1) = 2, σ3(1) = .. =σp(1) = 1. It
follows that σ2(2) = 3 and one of the permutations σ0, σ3, ., σptakes the value 1. Continuing
in this way and restoring the degree of freedom in the numbering of the permutations, we
see that the only cycles in the decompositions of σ1, .., σpare transpositions (r r + 1), and
each such transposition occurs in the decomposition of exactly one permutation. There are p
choices for the permutation that takes (0 1), p−1 possibilities for the one taking (1 3), etc.,
in total p(p−1)N−2choices. For each of these, the product of signs is sgn(σ1..σp) = (−1)N−1.
In total we get
u{0,..,n−1}
=p(p−1)N−2(−1)N−1φ1φp−1
0⊗φ2φp−2
1⊗.. ⊗φN−1φp−2
N−2φN−3⊗φp−1
N−1φN−2+rn(2.75)
where the remainder rnhas no contribution of the momentum type (1, p, .., p(N−2), pN −
p−1) and in particular is orthogonal to the first part of the sum. It follows that
u{0,..,n−1}6= 0
and the associated polymer system has polymers of arbitrary length and non-vanishing ac-
tivity Φ({0, .., n −1}) = ||u({0, .., n −1})||2.
Remark: Formula (2.75) should be compared to the exact expression (2.5) of bN(1, p, .., pN −
2p, pN −p−1).
When the associated polymer system is a monomer-dimer system, the normalization con-
stants and correlation functions can be determined explicitly. The following proposition
gives the one-particle density at filling factor 1/p = 1/3.
Proposition 2.24. Let p= 3 and suppose that φkφm= 0 if |k−m| ≥ 2. Let α:= ||φ2
0φ1||2=
||φ0φ2
1||2be a measure of the overlap between nearest neighbor functions 4. The one-particle
density of ΦC
Nis
ρN(x) =
N−1
X
j=0
CjCN−j−1
CN|φ3
j|2(x) +
N−2
X
j=0
CjCN−j−2
CN
9α|φ2
jφj+1|2(x) + |φjφ2
j+1|2(x)
where
CN=λN+1
+−λN+1
−
λ+−λ−
, λ±=1
2±r1
4+ 9α2.
Proof. We start by computing the monomer and dimer functions u{n},u{n,n+1}. This can
be done directly from their definition (2.70), but it is more instructive to determine them
from Lemma 2.22. Remark that ΦC
1=u{0}by that lemma, and ΦC
1=φ3
0by definition, thus
u{0}=φ3
0and we get from (2.71) u{n}=φ3
n. Next, looking at the 2-particle wave function
4The equality ||φ2
0φ1||2=||φ0φ2
1||2is checked using that fis even.
2.2. SOLVABLE MODELS 83
ΦC
2, we get
ΦC
2=1
√2(φ0⊗φ1−φ1⊗φ0)3
=1
√2(φ3
0⊗φ3
1−3φ2
0φ1⊗φ0φ2
1+ 3φ0φ2
1⊗φ2
0φ1−φ3
1⊗φ3
0)
=φ3
0∧φ3
1−3φ2
0φ1∧φ0φ2
1
!
=u{0}∧u{1}+u{0,1}
which gives
u{0,1}=−3φ2
0φ1∧φ0φ2
1, u{n,n+1}=−3φ2
nφn+1 ∧φnφ2
n+1.
We know from the previous propositions that CNis the polymer partition function of a
monomer-dimer system. The recurrence relation (2.36) becomes
C0= 1, C1= 1, CN+2 =CN+1 + 9α2CN
where we used that α1=||φ3
0||2= 1, ||u{0,1}||2= 9α2. The normalization constant is then
expressed in terms of the roots λ±of the characteristic equation λ2=λ+ 9α2as
CN=λN+1
+−λN+1
−
λ+−λ−
.
By Lemma 2.22, the one-particle density is a sum of monomer and dimer contributions
ρN(x) =
N−1
X
j=0
CjCN−j−1
CN
v{j}(x) +
N−1
X
j=0
Cjα2CN−j−2
CN
v{j,j+1}(x),
v{j}(x) = |φ3
j|2(x),
v{j,j+1}(x) = 2
||v{j,j+1}||2ZR×[0,2πR]|v{j,j+1}|2(x, x2+iy2)dx2dy2.
=9α
α2|φ2
jφj+1|2(x) + |φjφ2
j+1|2(x).
The previous proposition leads immediately to the existence of the thermodynamic limit of
the one-particle density, away from the boundaries of the finite cylinder [0, pNγl]×[0,2πR].
Corollary 2.25. Under the assumptions of the previous proposition, the one-particle density
has a limit away form the boundaries of the support of ΦC
N: for x∈[0, pNγl],|ρN(x)−ρ(x)| ≤
f(|x|+|pNγl −x|)with f(m)→m→∞ 0and
ρ(x) = 1
λ+−λ−X
j∈Z|φ3
0|2+1
λ+|φ2
0φ1|2+|φ0φ2
1|2(x−3jγl).
In particular, we see that the one-particle density is periodic with period 3γl. We have
observed in Proposition 2.23 that the system becomes a pure monomer system as γgets
large. Our results allow us to check that, loosely speaking, the transition is smooth: indeed,
recall the roots λ±are defined are simple functions of the overlap parameter α:= ||φ2
0φ1||2
(see Proposition 2.24). The transition from the monomer-dimer to the monomer system
corresponds to the transition from α > 0 to α= 0, and we see that λ±is a smooth function
of the overlap parameter α.
84 CHAPTER 2. THERMODYNAMIC LIMITS
2.2.3 Torus wave functions
The Laughlin-type torus wave functions proposed in [HR85b] form a p-dimensional space;
they are eigenfunctions of suitable center-of-mass translations. This invariance entails the
periodicity of the one-particle density, which is thus much simpler as in the cylinder case.
In Lemma 2.26, we check that this feature is reproduced by the generalized torus functions.
This holds for general functions f.
Next, we take fas a function of compact support such that the associated cylinder polymer
model is a monomer-dimer system and show that for a suitable choice of Z1, .., Zp(the de-
grees of freedom in the definition of torus functions), the torus function is associated with
a monomer-dimer system on a ring, with one additional long polymer covering the whole
ring (Proposition 2.27). As a corollary, we obtain that the one-particle density of the corre-
sponding torus function equals the one-particle density of the cylinder function in the limit
N→ ∞ (Corollary 2.28).
Let us start by looking at the translational invariance of the torus functions. We have
seen that the torus functions are quasiperiodic in each complex variable z1, .., zNof periods
L=pNγl and i2πR. The space of such functions is invariant with respect to center-of-mass
translations by multiples of γlexand 2πR/(pN)ey(the lattice γlZ×(2πR/pN)Zalready
appeared on p.8; see also p.94). The modified torus functions are invariant with respect to
overall shifts by ptimes the lattice vectors γlexand (2πR/pN)ey:
Lemma 2.26. For a∈R2, let T(a) := t(a)⊗Nbe the magnetic translation of all the parti-
cles. Then the torus wave functions are invariant with respect to overall shifts by pγlexand
2πR/Ney:
T(pγlex)ΦT
N= (−1)p(N−1)ΦT
N, T(2πR
Ney)ΦT
N= (−1)p(N−1)ΦT
N.
The periodicity is enhanced for particular choices of Z1, .., Zp. For example, if Zj=ij2πR/N,
j= 1, .., p:
T(2πR
pN ey)ΦT
N= (−1)N−1ΦT
N.
Proof. Using eiy/R ˜
φk(z−pγ) = ˜
φk+1(z) and ˜
φ0=˜
φN, we see that
˜
φ1∧.. ∧˜
φN= (−1)N−1˜
φ0∧.. ∧˜
φN−1
and
(t(pγl))⊗NΦT
N(z1, .., zN) = 1
√N!eip Pjyj/R
p
Y
s=1det(eiXsyj/(pNl2)˜
φk−1(zj−Xs
N−pγ))j,k
=1
√N!
p
Y
s=1det(eiXsyj/(pNl2)˜
φk(zj−Xs
N))j,k)
=1
√N!(−1)p(N−1)ΦT
N(z1, .., zN).
2.2. SOLVABLE MODELS 85
Similarly, ˜
φn(z−i2πR/N) = e−i2πn/N ˜
φn, hence
T(i2πR/N)ΦT
N=1
√N!
p
Y
s=1det(eiXs(yj−i2πR/N)/(pNl2)e−i2π(k−1)/N ˜
φk−1(zj−Xs
N))j,k)
=e−i(PN
s=1 Xs)2πR/(pNl2)e−i2πp PN
k=1(k−1)/N ΦT
N
= (−1)p(N−1)ΦT
N.
Suppose in addition that Zj=ij2πR/(pN), then
T(i2πR/pN)ΦT
N=1
√N!
p
Y
s=1det(˜
φk−1(zj−ij 2πR
pN −i2πR
pN ))1≤j,k≤N)
=1
√N!(−1)N−1
p
Y
s=1det(˜
φk−1(zj−ij 2πR
pN ))1≤j,k≤N)
= (−1)N−1ΦT
N.
The previous lemma implies that the one-particle density of the torus wave functions is
doubly periodic with the period pγl in the x-direction and 2πR/N in the y-direction. The
periodicity of the one-particle density of the cylinder function would follow if we knew that
both functions are equivalent in the thermodynamic limit. We prove the equivalence only for
the special case of nearest neighbor overlapping.
Note that there is only one Laughlin function, but a p-dimensional space of torus functions.
One must thus be careful as to which torus function one compares the cylinder function.
The most natural candidates are the torus functions ΦT
N,1and ΦT
N,2obtained by choosing
Z1=.. =Zp= 0 for ΦT
N,1and Zj=ij2πR/(pN) for ΦT
N,2. The first function has in common
with ΦC
Nthe representation as the power of a determinant, while ΦT
N,2and ΦC
Nare both
eigenvectors of T(i2πR/(pN)) for the eigenvalue (−1)N−1.
For X={j, .., j +n−1},N∈N, let uXbe defined as in the previous section and
ˆuX:= X
k∈Z
t(kpNγlex)⊗N(X)uX
its periodification. The torus wave functions for Z1=.. =Zs= 0 have a representation of
the form
ΦT
N=X
X1,..,XD±ˆuX1∧.. ∧ˆuXD+ ˜uN
where X1, .., XDare partitions of the ring Z/NZ. However, in general ˜uNis not orthogonal
anymore to PX1,..,XD. For this reason we restrict to nearest neighbor overlapping. For
simplicity, we take p= 3.
Proposition 2.27. Let N≥3. Suppose φkφl= 0 if |k−l| ≥ 2, take p= 3 and let ˆuXbe
defined as above. Let ˆu{0,N−1}:= ˆu{−1,0}and
˜uN= 3(−1)N−1˜
φ2
0˜
φ1∧.. ∧˜
φ2
N−1˜
φN+ 3˜
φ0˜
φ2
1∧.. ∧˜
φN−1˜
φ2
N.
86 CHAPTER 2. THERMODYNAMIC LIMITS
Then the torus function ΦT
Nfor Z1=.. =Zp= 0 is represented as a polymer system on a
ring, made up of monomers, dimers, and one additional long chain:
ΦT
N=X′
X1,...,XD
ǫ{Xj}ˆuX1∧.. ∧ˆuXD+ ˆuN(2.76)
The sum ranges over two types of partitions:
1. partitions X1< .. < XDof {0, .., N −1}into monomers {j}and dimers {j, j +1}. For
these partitions, ǫ{Xj}= +1;
2. X1={0, N −1},X2< .. < XDpartition of {1, .., N −2}into monomers and dimers.
For this type of partition, ǫ{Xj}= (−1)N−1.
The different contributions to (2.76) are orthogonal and the normalization constants in
L2(([0,3Nγl]×[0,2πR])N)as well as the one-particle density have representations similar
to Lemma 2.22, for the correct associated polymer system.
Sketch of proof. We start with
ΦT
N=1
√N!A(X
σ,τ∈S{0,..,N−1}
sgn(στ)˜
φ0˜
φσ(0) ˜
φτ(0) ⊗.. ⊗˜
φN−1˜
φσ(N−1) ˜
φτ(N−1)).
To derive (2.76), we use a reasoning similar to the proof of 2. in Proposition 2.23. If σ, τ
give a non-vanishing contribution to ΦT
N, (k, σ(k)) must be nearest neighbors on the ring
Z/NZ, and similarly for (k, τ(k)), (τ(k), σ(k)), for all k∈Z. This leaves two possibilities.
First, σand τcan be products of nearest neighbor transpositions (including (0 N−1)) with
disjoint support. Second, we can have σ, τ ∈ {1, c}or σ, τ ∈ {1, c−1}where cis the cycle
c= (0 1 2 ..N −1). The transpositions give the sum over PX1,..,XDin (2.76). The sign
ǫ{Xj}is (−1)N−1if the transposition (0 N−1) occurs: in this case we have to permute in the
wedge product. The cycles give ˜uN(note that ˜
φN=˜
φ0). For N= 2, the cycle cis a nearest
neighbor transposition; this is why we assume N= 3. In order to get representations of the
normalization constant and the one-particle density, one has to check that an orthogonality
relation similar to (2.73) holds. Again, this is done by looking at the y-momenta. The only
difference is that y-momenta are only defined modulo 3N.
Remark. Similar representations can be derived for torus functions with X1=.. =Xp= 0 but
Y1, ., Yppossibly non-zero. The shift in yby Ysthen leads to phase factors in front of the poly-
mer functions ˆuX. If the zeroes are chosen in a way that T(i2πR/(pN)ey)ΦT
N= (−1)N−1ΦT
N,
we obtain a formula with ˜uN= 0, there is no ”ring contribution”.
The previous proposition allows to check the equivalence of modified cylinder and torus
functions in the nearest neighbor overlapping case.
Corollary 2.28. Under the assumptions of the previous proposition, and with the overlap
parameter α=||φ2
0φ1||2
L2(R×[0,2πR]), and λ±as in Proposition 2.24, the torus normalization
constant satisfies
CT
N∼
N→∞ λN
+
and the one-particle density of the torus function has as thermodynamic limit the function ρ
given in Corollary 2.25.
2.3. JELLIUM TUBES 87
Sketch of proof. Let ˆ
Ω = [0, pNγl]×[0,2πR] and Ω = R×[0,2πR]. Note that ||uX||L2(ΩN(X))=
||ˆuX||L2(ˆ
ΩN(X))and ||˜un||2
L2(ˆ
Ωn)= 18αn.For N≥3, write ||ΦT
N||2
L2(ˆ
ΩN)=DN+ 18αn.DNis
a sum over monomers and dimers and can be written in terms of the cylinder normalization
constant as
DN=CN−1+ 2α2CN−2, α2=||u{0,1}||2= 9α2
(see also [HL72], Section III, equation (3.8)). The first summand corresponds to partitions
where 0 is covered by a monomer, while 2CN−2comes from partitions where 0 is in the dimer
{0,1}or {N−1,0}. Thus we obtain, using λ+λ−=−9α2and λ++λ−= 1:
CT
N∼
N→∞
1
λ+−λ−
(1 + 2 ·9α21
λ+
)λN
+=λN
+
We get for the finite volume probabilities of finding a monomer or a dimer
ρP
N,T ({0}) = CN−1
CT
N∼λN
+
λ+−λ−
1
λN
+
=1
λ+−λ−
ρP
N,T ({0,1}) = 9α2CN−2
CT
N∼λN−1
+
λ+−λ−
1
λN
+
=9α2
λ+
1
λ+−λ−
.
The limits are the same as for the polymer system with non-periodic boundary conditions; the
long chain ˜uNdoes not contribute in the limit N→ ∞. We conclude with a representation
of the type
ρN,T (z) = X
X
ρP
N,T (X)ˆvX(z).
Thus for the solvable monomer-dimer case, the modified cylinder and suitable torus functions
give the same one-particle density.
2.3 Jellium tubes
In the beginning of the thesis, we have presented the plasma analogy: the modulus squared
of Laughlin’s wave function is essentially the Boltzmann weight of a classical one-component
plasma, with a logarithmic Coulomb interaction. The results on the normalization constants
and correlation functions of Laughlin’s wave function lead to results on the free energy and
correlation functions of the plasma system.
The plasma system related to Laughlin’s cylinder wave function lives on a cylinder, or
semiperiodic strip, and following [AGL01] we will refer to it as a jellium tube. When the
cylinder is very thin, the system is close to a one-dimensional jellium system, while for large
cylinder radius, the jellium tube approaches a fully two-dimensional one-component plasma.
For one-dimensional jellium systems, the symmetry breaking has been proved a while ago
[Kun74, BL75]. When Nelectrons are placed on a line of length Land interact between
themselves and with a neutralizing background through a one-dimensional Coulomb interac-
tion, they tend to minimize their Coulomb interaction by taking positions on a lattice. The
spacing between two consecutive positions is L/N. If we view the jellium tube as a quasi
one-dimensional system rather than a two-dimensional system, we may expect the same peri-
odicity. The tubes we consider have length pNl2/R. In the jellium context, the period pl2/R
is thus quite natural.
88 CHAPTER 2. THERMODYNAMIC LIMITS
Forrester [For91] has examined the asymptotics of jellium strip free energies as the strip width
gets large; he has also looked at torus Coulomb free energy finite size corrections, motivated
by the question of universal finite size corrections and a relation to Sine-Gordon fields (see
[For06] and the references therein).
In the first subsection, we look at the jellium tube in both limits: large and thin strips. We
give an expression for the jellium tube free energy. We compare the free energy and the
correlation functions of thin cylinders to the corresponding quantities for one-dimensional
systems and find that they are in good agreement with existing results. We also compare the
free energy of broad jellium strips to the bulk free energy; this leads to a conjecture consis-
tent with results by [For91] and with the bounds on the normalization constant of Laughlin’s
wave function derived in Lemma 2.10.
The plasma analogy also sheds some light on the recurrence relation satisfied by the nor-
malization constants. In the second subsection, we recall Lenard’s combinatorial treatment
of the one-dimensional two-component plasma. Lenard derived a renewal equality for the
isothermal-isobaric partition functions and interpreted it in terms of minimal electrically
neutral components. Then we derive a recurrence inequality for jellium tubes with the help
of Newton’s electrostatic theorem; the inequality looks similar to a renewal equation. Again,
there is a relation to minimal neutral blocks.
2.3.1 Interpolation between one- and two-dimensional jellium
The plasma analogy relates Laughlin’s cylinder wave function to a one-component plasma on
a semiperiodic strip of width 2πR, with a “Coulomb” interaction VC(z) = −log |2R
R0sinh z
2R|,
where R0>0 is a reference length scale. When the width of the strip goes to infinity, we
recover the full two-dimensional logarithmic interaction: if z6= 0 and R0>0 are kept fixed,
lim
R→∞VC(z) = −log |z/R0|=VD(z).
In the opposite limit of thin strips, for all z∈Cwith x=ℜz6= 0,
VC(z) = −log R
R0−|x|
2R+o(1) as R→0.
The right-hand side represents a one-dimensional Coulomb interaction. Thus the semiperi-
odic strip interpolates in a natural way between the two-dimensional and the one-dimensional
systems, and it is instructive to compare our results to existing results on these systems. We
start by giving an expression for the cylinder free energy.
Let C:= [−L/2, L/2] ×[0,2πR]. The free energy of the jellium tube is
βF := −log1
N!h2NZ(R2)N×CN
e−β(1
2mPN
j=1 |pj|2+UC)=−log1
N!λ2N
βZCN
e−βUC
(2.77)
where λβ:= pβh2/(2πm) is the thermal wavelength and hPlanck’s constant. We will be
interested in the free energy per particle in the thermodynamic limit N→ ∞ with a fixed
density n=N/|Ω|. For the cylinder and torus systems, we keep the radius Rfixed when
taking the limit. The corresponding free energies (per particle) are
βf =βf0−lim
N→∞
1
Nlog1
(2πRL)NZCN
e−βUC, βf0= log nλ2
β−1.(2.78)
2.3. JELLIUM TUBES 89
f0is the free energy of a system of noninteracting particles: UC= 0 would give f=f0. In
view of the plasma analogy (1.36), if Γ = βq2= 2pand n= (p·2πl2)−1for p∈N, the free
energy can be written as
βf =βf0+λ2
12 −lim
N→∞
1
Nlog N!lN√πN
LNZ(I×[0,2πR])N|ΨC
N|2
where I:= [−L
2+p(N−1)γl
2,L
2+p(N−1)γl
2]. In the following we assume that the result is
unchanged if integrations are carried out on the infinite cylinder instead of I×[0,2πR]
(remember that ΨC
Ndecays exponentially outside (I×[0,2πR])N. Under this assumption,
using Stirling’s formula, L=Npγl and
λ2=Γ
4πnR2=2p
4π(p·2πl2)−1R2=p2γ2,(2.79)
we get
βf =βf0+λ2
12 + log eλ
√π+ log rp(γ) (2.80)
where rp(γ) comes from Lemma 2.10.
Comparison with a one-dimensional Coulomb gas. Let V(x) = −|x|/(2R) and for
L > 0, N∈N,n0:= N/L
U(x1, .., xN) := q2X
1≤j<k≤N
V(xj−xk)−n0q2
N
X
j=1 ZL
0
V(xj−x)dx+
1
2n2
0q2ZL
0ZL
0
V(x−x′)dxdx′
be the energy of a one-dimensional Coulomb gas of Nparticles moving in a neutralizing
background. A useful quantity is Γ1:= βq2
2πRn0, the plasma parameter or coupling constant; it
is the ratio of a typical potential and kinetic energy. Large coupling constants correspond to
kinetic energies that are small compared to typical potential energies. In the thermodynamic
limit N, L → ∞ at fixed density n, the free energy per particle including kinetic energy can
be expressed as
βf(1) =βf(1)
0+Γ1
12 −log z0(√Γ1)
e, βf(1)
0= log n0λβ−1,(2.81)
where z0(λ) is an analytical function such that limλ→∞ z0(λ)λ/√π= 1 see [Kun74], and the
one-particle density ρ(x) is
ρ(x) = n0
∞
X
k=−∞
e−Γ1(n0x−k)2FΓ1(n0x−k),lim
Γ1→∞FΓ1(x) = χ[−1,1](x),(2.82)
see [Kun74, BL75]. If we look at a one-dimensional system with density n0= 2πRn, the
one-dimensional plasma parameter is nothing else but the quantity λ2of (2.79):
Γ1=βq2
2R·2πRn =λ2.
90 CHAPTER 2. THERMODYNAMIC LIMITS
Comparing (2.80) and (2.81), we see that the one- and two-dimensional free energies f−f0
and f(1) −f(1)
0get close as the strip gets thin, i.e., R→0 or λ=pγ =pl/R → ∞ at
fixed two-dimensional density n. (Recall that limγ→∞ rp(γ) = 1.) The one-particle density
of Laughlin’s wave function on thin strips is
ρ(x) = 1
p·2πl2X
k∈Z
nke−(x−kγl)2/l2with nk=(1 + O(e−γ2),if k∈pZ,
O(e−γ2),else.
This compares well to the one-dimensional density, since
Γ1(n0x−k)2= (pγ 2πR
p·2πl2x−pkγ)2=1
l2(x−pγl)2.
Comparison with a two-dimensional Coulomb gas. The free energy βF of a two-
dimensional jellium system (particles of charge q+ neutralizing background of charge density
−nq) with logarithmic Coulomb interaction VD(z) = −log |z/R0|is defined by a formula
similar to (2.77). The thermodynamic limit of the free energy per particle exists and is given
by an expression of the type
βf(2) =βf0+g(Γ) −Γ
4log nR2
0, βf0= log nλ2
β−1
for a suitable function gand Γ = βq2, see [SM76]. The value of the function at coupling
constant Γ = 2 is known and equals
g(2) = −1
2log(2π) + log e
√π.
We expect that the strip free energy (2.79) approaches the two-dimensional energy as R→ ∞
if the length scale R0is chosen consistently; we will choose R=R0. Forrester [For91] derives
more refined asymptotics for the strip free energies. He shows that if the thermodynamic
limit of correlation functions in the bulk (i.e., in the fully two-dimensional geometry) and in
the strip geometry exist, if the strip correlation functions converge to the bulk correlation
functions as the strip width goes to infinity, and if in addition the bulk correlation functions
satisfy sum rules typical for conducting Coulomb systems, then the following asymptotics
hold:
β(f−f(2)) = π
6n(2πR)2+O(1
R4) as R→ ∞.(2.83)
Using (2.80), the expressions of βf(2) and g(2) given above, and the identifications Γ = βq2=
2p,n= (p2πl2)−1, one finds that (2.83) is equivalent to
log Ap(γ) = (p−p2)γ2
12 +g(2p)−pg(2) + O(γ4) as γ→0 (2.84)
where the auxiliary quantity
Ap(γ) = rp(γ)( eγ
√π)1−pp1−p
2
is exactly the quantity on which we have given bounds in Lemma 2.10. As observed by
[For91], for Γ = 2, the relation (2.83) can be checked explicitly: the normalization constant
for the filled Landau level (p= 1) is CN= 1 whence r1(γ) = 1, A1(γ) = 1 and (2.84) holds.
For p≥2, we do not have a proof of (2.83) (note that Forrester’s assumptions are quite
strong); however, (2.84) is consistent with Lemma 2.10 if
p(g(2) −1) ≤g(2p)≤pg(2).
2.3. JELLIUM TUBES 91
Is there a phase transition? The previous paragraphs picture the jellium tube as an
interpolation between one- and two-dimensional jellium systems. It can be characterized in
terms of the two parameters Γ = βq2,λ2=βq2/(4πnR2) that are the coupling constants
of the related two- and one-dimensional plasmas. An interesting feature is that the one-
dimensional system exhibits no phase transition (Kunz [Kun74] proved that the free energy
is an analytic function of the coupling constant). On the other hand, there is numerical
evidence of a phase transition for jellium systems of dimensions 2 or 3 (see e.g. [BST66] for
three dimensions, [CLWH82] for two dimensions). It is believed that above Γ ≈140, the
one-component plasma is crystalline.
We proved that for all even plasma parameters Γ = 2p, there is a value γpsuch that for
γ=ℓ/R ≥γpthe free energy is an analytic function of γ, or equivalently of λ2= (pγ)2.
One might speculate that for small γ, the nature of the limiting two-dimensional system and
hence the magnitude of Γ = 2pshould play a role for large strips.
2.3.2 Minimal electrically neutral components
The normalization constants of Laughlin’s wave function satisfy a recurrence relation, the
renewal equation described in Section 2.1. This relation was used extensively to prove the
existence of thermodynamic limits. However, we did not give any physical meaning to it. The
aim of this subsection is to show that the plasma analogy gives some intuition that makes
the relation less surprising.
One-dimensional two-component plasma: Lenard’s combinatorial treatment. It
is interesting to observe that the renewal equality already appears in Lenard’s combinatorial
treatment of the two-component, one-dimensional plasma [Len61]. Let us briefly summarize
the model. Consider 2Nparticles of charges σ1, .., σN∈ {±σ}moving on a semi-infinite line
[0,∞[. The potential energy of the system is
U(q1, σ1;..;qN, σN) = −2πX
1≤i<j≤N
σiσj|qi−qj|.
The system is assumed to be neutral, Piσi= 0. Let β > 0 be an inverse temperature and
P > 0 an external pressure. It is convenient to work with a different set of parameters: write
σj=ǫjσ,ǫj=±1, xj= 2πβσ2qj, ¯γ=P/2πσ2. Lenard considers the isothermal-isobaric
partition function
e−βG := 1
N!2X
σ1,..,σN=±σZR2N×R2N
+
e−βP2N
j=1 p2
j/2+U(q1,σ1;..;qN,σN)+Pmaxjqjdpdq (2.85)
=r2π
β
2N1
(2πβσ2)2NQN(¯γ),
QN(¯γ) := X
ǫ1,..,ǫN=±1Z∞
0
dx2NZx2N
0
dx2N−1.. Zx2
0
dx1eP1≤i<j≤2Nǫiǫj|xi−xj|−¯γx2N).(2.86)
The sequence (ǫ1, .., ǫN) can be characterized in terms of the charges νi=Pi
k=1 ǫkaccu-
mulated to the left of charge number i. Lenard calls a configuration (ǫ1, .., ǫN)irreducible
if νi6= 0 unless i= 0 or i= 2N. Let ¯
QN(¯γ) be defined through (2.86) except that the
summation is over irreducible ǫ-sequences only. Then
QN=
N
X
k=1 X
n1+..+nk=N
¯
Qn1.. ¯
Qnk.
92 CHAPTER 2. THERMODYNAMIC LIMITS
This can be expressed in terms of a renewal equation and as a formal power series identity.
Q(z) = 1
1−¯
Q(z), Q(z) = 1 + ∞
X
n=1
Qnzn,¯
Q(z) = ∞
X
n=1
¯
Qnzn.
Lenard then proves (p. 688) that Q(z) is meromorphic in a disk of radius Rwith a unique
pole rinside that disk; ris real and
QNrN=1
r¯
Q′(r)+O((r/R)N), r ¯
Q′(r)<∞.
Later (p. 691), Lenard comes back to the significance of the irreducible configurations and
characterizes them as “the minimal electrically neutral parts of a general configuration”.
Let Pn:= rn¯
Qn. The numbers (Pn) define a probability distribution on N, they give the
probability that an irreducible block of a given length appears. The first moment
∞
X
n=1
nPn=r¯
Q′(r)
is finite for all ¯γ > 0 and represents “a measure of the efficacy of electrostatic shielding”.
However, in the limit ¯γ→ ∞, the moment gets infinite: “the number of particles within a
shielding distance tends to infinity”.
A renewal inequality for jellium tubes. Newton’s electrostatic theorem plays a crucial
role in the investigation of thermodynamic limits of Coulomb systems [LL72]. In one di-
mension, the Coulomb potential is essentially −|x|and disjoint neutral subsystems on a line
do not interact: this may be viewed as a one-dimensional version of Newton’s electrostatic
theorem. The y-average of the semi-periodic version of the logarithmic interaction gives the
one-dimensional Coulomb interaction. These observations and a clever partitioning of the
configuration state into neutral blocks allow the derivation of a renewal inequality.
Consider Nparticles of charge qmoving on a cylinder [0, NT ]×[0,2πR] in a neutralizing
background of charge density −nq =−q/(2πRT). The Coulomb interaction is taken as
VC(z) = −log |2 sinh z
2R|,z=x+iy ≡(x, y)∈R2. Let
ZN:= 1
N!Z([0,NT ]×[0,2πR])N
e−βUN
be the associated partition function. We will be interested in a splitting of the system into
neutral subsystem and define the total charge accumulated to the left of x∈[0, NT] as
Q(x;x1, .., xN) = q|{j∈ {1, .., N} | 0≤xj≤x}|−nq ·2πRx.
If the subsystem to the left of xis neutral, the background charge qn2πRx =x/T com-
pensates an integer number of charged particles, so that xmust be a multiple of T. It
is convenient to label particles from left to right and partition the configuration space ac-
cording to the set of zeroes of Q(·;x1, .., xN): for k∈ {1, .., N}, let RN,k be the set of
((x1, y1), .., (xN, yN)) ∈([0, NT]×[0,2πR])Nsuch that
x1≤.. ≤xNand min{x > 0|Q(x;x1, .., xN) = 0}=kT.
The set RN,k may be seen as a minimal electrically neutral component, in the spirit of
Lenard’s treatment.
Now we are ready to state the renewal inequality:
2.3. JELLIUM TUBES 93
Proposition 2.29. For N∈N, let αN:= RRN,N e−βUN.Then Z1=α1and for N≥2,
ZN≥α1ZN−1+α2ZN−2+... +αN−1Z1+αN.
Proof. Z1=α1is obvious; for N≥2, note that ZN=PN
k=1 RRN,k e−βUNso that the
proposition follows from RRN,k e−βUN≥αkZN−k. We prove the inequality for k= 1, the
case k≥2 is analogous. If Q(T;x1, .., xN) = 0, the system splits into two neutral subsystems,
consisting of the charges and the background located to the left /right of x=T; there is only
one charge to the left of x=T. The potential energy can be written as
UN(z1, .., zN) = U11(z1) + U12(z1, .., zN) + U22(z2, .., zN)
where U11,U22 represent the energies of the separate subsystems and UN,12 their interaction.
Using Jensen’s inequality, we get
ZRN,1
e−βUN≥e−βRRN,1U12e−β(U11+U22 )ZRN,1
e−β(U11+U22).(2.87)
The neutrality of the subsystems located to the left and right of x=Timplies
Z[0,2πR]N
U12(z1, .., zN)dy1dy2..dyN= 0.
Indeed,
−1
2πR Z2πR
0
log |2 sinh x+iy
2R|dy =−|x|
2R,
hence the yaverage of U12 reduces to a one-dimensional interaction which we may write
hU12iy(x1, ., xN) = −Z[0,T ]×[T,NT ]
|x′
1−x′
2|
2Rdρ1(x′
1)dρ2(x′
2) = 0
where ρ1:= δx1−ndx′
1and ρ2=PN
j=2 δxj−ndx′
2represent the charge densities of the two
subsystems, and ρ1([0, T]) = 0 = ρ2([T, NT ]) due to neutrality. Since e−β(U11+U22)and RN,1
are invariant with respect to translations (y1, y2, .., yN)7→ (y1−a, .., yN−a), it follows that
ZRN,1
U12e−β(U11+U22)= 0.(2.88)
Due to translational invariance,
ZRN,1
e−βU11 e−βU22 =α1ZN−1.
Combining this with (2.87) and (2.88), we obtain RRN,1e−βUN≥α1ZN−2.
The previous proposition is an analogue of Lenard’s formula for the two-component plasma
partition function. The result is not completely satisfying and we would prefer an equality
instead of an inequality. However, we believe that it connects the renewal inequality to phys-
ical intuition, at least in the jellium picture. It would be interesting to have an interpretation
of the renewal equality in the quantum-mechanical setting of Laughlin’s wave function.
94 CHAPTER 2. THERMODYNAMIC LIMITS
2.4 A Lieb-Schultz-Mattis type argument
Up to now, we have not used the characterization of Laughlin’s wave function as the ground
state of a truncated interaction. Laughlin described his state as incompressible; in order to
apply the characterizations of incompressibility of Section 1.5, we need a Hamiltonian that
has Laughlin’s state as its ground state. The truncated Hamiltonians of Section 1.4 serve
this purpose. In this section, we show that if the truncated Hamiltonian reproduces the in-
compressibility of the ground state, then there must symmetry breaking. This can be shown
with a Lieb-Schultz-Mattis method as given by Koma [Kom04]. A drawback is that it has
not been proved that the truncated Hamiltonian give an incompressible ground state, see
the discussion by Haldane in [PG87], p.319 and the end of this section. Moreover, the argu-
ment in this form applies to short-range, translationally invariant interactions, and cannot
be applied to the lowest Landau level projected Coulomb Hamiltonian. Note that not only is
the Coulomb interaction long-ranged, but the neutralizing background on the cylinder gives
a potential attracting the electrons to the center of the cylinder: the background creates a
potential that is not translationally invariant.
Koma [Kom04] considers infinite volume ground states of electrons on a cylinder, confined to
finitely many Landau levels and interacting via a short range potential. Using a Lieb-Schultz-
Mattis type argument, he establishes a relation between the existence of a gap, spatial periods
and average density of a ground state. This section gives a simplified version of his argu-
ments, adapted to the truncated Hamiltonian that has Laughlin’s p= 3 wave function as
its ground state. Before we turn to the proof, we give a glimpse of the Lieb-Schultz-Mattis
argument for spin chains, its extension to electrons on a lattice by [Tas04, YOA97] (see also
[Osh00]), and point out a connection to the usual argument for degeneracy of ground states
on a torus.
In [LSM61], Appendix B, Lieb, Schultz and Mattis prove the nondegeneracy of the ground
state and absence of an energy gap for a Heisenberg spin chain with periodic boundary
conditions. The non-degeneracy is proved first; the existence of low-energy excitations is
then inferred from a nice interplay of two transformations ([LSM61] p. 457): an overall shift
Txand a “twist” Ok= exp(ik PN
n=1 nSz
n), where Nis the length of the chain. Using the
commutation rule
TxOkT−1
x=OkeikNSz
1e−ik PN
n=1 Sz
n,(2.89)
it is shown that for suitable kthe candidate excited state OkΨ0is orthogonal to the ground
state Ψ0. A related argument is given in a setting with infinite spin chains in [AL86], where
the global twist was replaced with a local twist exp(iπ
ℓPℓ
j=−ℓSz
j(j+ℓ)).
Later, the argument has been adapted to fermions on a one-dimensional lattice, for the finite-
volume [YOA97] and the infinite volume [Tas04] case. The twist operator that is used is of
the type exp(ik Pjjˆnj), where ˆnjis the occupation number operator of lattice site j. This
is applied in [Kom04] to electrons in a magnetic field on a lattice.
For electrons in a magnetic field on a torus, there is a simple argument establishing a relation
between the density and the degeneracy. It rests on the non-trivial commutation rules of
magnetic translations, reminiscent of (2.89). Let us briefly recall the argument: consider a
Hamiltonian of the type
HN=1
2m
N
X
j=1
(pj+eA(xj, yj))2+X
1≤j<k≤N
V(zj−zk)
2.4. A LIEB-SCHULTZ-MATTIS TYPE ARGUMENT 95
in L2([0, L]×[0,2πR]) with quasiperiodic boundary conditions as discussed on p.8. Define
the center of mass translations
Tx:= t(L
Nf
ex)⊗N, Ty:= t(2πR
Nf
ey)⊗N.
Then t(L/Nfex)t(2πR/Nfey) = ei2π/Nft(2πR/Nfey)t(L/Nfex) and
[HN, Tx] = 0,[HN, Ty] = 0, TxTy=ei2πN/NfTyTx.
Suppose that Hhas an eigenspace of finite dimension d. Consider Tx,Tyas unitary operators
in this space, then
det Txdet Ty= det TxTy= det ei2πN/NfTyTx=ei2πdN/Nfdet Tydet Tx
whence ei2πdN/Nf= 1 and dN/Nf∈N. Write N/Nf=p/q with p, q coprime and suppose
ν=p/q /∈N. Then qdivides d, and every eigenspace of HNis at least q-fold degenerate.
Moreover, if Ψ0is a translationally invariant state, i.e. TxΨ0=eiαΨ0for some α∈R, it
breaks the y-invariance:
TxTyΨ0=ei2πp/qTyTxΨ0=ei2πp/qeiαTyΨ0,
Ψ0and TyΨ0belong to different eigenspaces of Txand thus hΨ0, TyΨ0i= 0.
This can be nicely related to the Lieb-Schultz-Mattis type argument for electrons on a lattice,
as given in [YOA97], if we restrict to the lowest Landau level. The lowest Landau level has
a basis ˜
ψk,k∈Z/NfZ. Each basis function can be thought of as a lattice site. The effect
of the magnetic translations is t(L/Nfex)˜
ψk=˜
ψk+1,t(2πR/Nfey)˜
ψk=e−i2πk/Nf˜
ψk. Thus
the shift in the y-direction may be rewritten as
Ty= exp(−i
Nf
X
j=1
jˆnj)
and we recognize the twist operator of [YOA97]. From this point of view, Koma’s approach
looks like an infinite volume version of the torus degeneracy argument (let us mention how-
ever, that Koma allows a one-particle potential V(y)). The overall y-shift is replaced by a
local twist, and instead of orthogonality of Ψ0and TyΨ0we obtain orthogonality only in the
limit where the local twist becomes global (see Lemma 2.34 below).
In the following we restrict to the lowest Landau level. Let ALL be the canonical anti-
commutation algebra over the lowest Landau level, i.e., the sub-algebra of Agenerated by
1and ck, c∗
m, k, m ∈Z. For Λ ⊂Z, let AΛbe the sub-algebra of ALL generated by 1
and ck, c∗
m, k, m ∈Λ, and let Aloc := ∪Λ⊂Z,|Λ|<∞AΛbe the set of local observables. Let
ˆnk:= c∗
kckand consider the formal expressions
H=X
k,k′,n,n′∈Z
δk+k′,n+n′f(k−k′)f(n−n′)c∗
k′c∗
kcncn′,ˆ
N=X
k∈Z
ˆnk
where f(n) := nexp(−γ2
4n2) and ˆnk=c∗
kck. The Hamiltonian His the second quantized
form of the truncated interaction of Definition 1.3, see Corollary 1.7. For A∈ Aloc, the sums
[H, A] := X
k,k′,n,n′∈Z
f(n−n′)f(k−k′)[c∗
k′c∗
kcncn′, A] [ ˆ
N, A] := X
k∈Z
[ˆnk, A].
96 CHAPTER 2. THERMODYNAMIC LIMITS
are absolutely convergent: suppose A∈ AΛ, Λ ⊂Z,|Λ|<∞. The commutator [ˆnk, A]
vanishes unless k∈Λ, thus [ ˆ
N, A] is a finite sum. Similarly, [c∗
k′c∗
kcncn′, A] vanishes unless
one of the indices k, k′, n, n′is in Λ. Let C, c > 0 such that |f(n)| ≤ Cexp(−cn2). Then
X
k∈ΛX
k′,n,n′∈Z
δk+k′,n+n′||f(n−n′)f(k−k′)[c∗
k′c∗
kcncn′, A]||
≤2C||A||X
k∈ΛX
k′,n,n′∈Z
δk+k′,n+n′|f(n−n′)f(k−k′)|
≤2C||A|| X
k∈Λ,n∈ZX
k′,n′∈Z:
n′−k′=k−n
e−c
2((n′+k′−n−k)2+4(k−n)2)
≤2C||A|| X
k∈Λ,n∈Z
e−2c(k−n)2X
t∈Z
e−c
2t2<∞.
For µ∈R,A∈ Aloc, let δµ(A) := i[H−µˆ
N, A] = i[H, A]−iµ[ˆ
N, A]. Then δµis a symmetric
derivation on Aloc. Its closure is the generator of a strongly continuous group (αµ
t)t∈Rof
C∗-automorphisms of ALL:
∀t∈R,∀A∈ Aloc :d
dtαµ
t(A) = iαµ
t([H−µˆ
N, A])
see [Mat96], Theorems 1.1 and 2.9. The algebra AU(1) is the subalgebra of operators of ALL
that preserve the total particle number. More precisely, let γtbe the ∗-automorphism of A
such that for all k∈Z,γt(ck) = e−itckand AU(1) the set of A∈ ALL such that γt(A) = Afor
all t∈R. Note that ∂tγt(ck) = +iγt([ ˆ
N, ck]), γtis associated to the derivation a7→ i[ˆ
N, a].
A local observable is in AU(1) if and only if [ ˆ
N, A] = 0.
Definition 2.30. Let µ∈Rand ωbe a state on ALL.
•ωis a ground state of H−µˆ
Nif
∀a∈ Aloc :ω(a∗[H−µˆ
N, a]) ≥0.(2.90)
•ωis a ground state at fixed density, or ground state for AU(1), if it is gauge-invariant
(ω◦γt=ωfor all t∈R) and (2.90) holds for all a∈ AU(1) ∩Aloc.
For 2., the precise value of the chemical potential is irrelevant. This is due to [H−µˆ
N, a] =
[H, a] for all a∈ AU(1) ∩ Aloc. It follows from 1. that ω◦αµ
t=ω(see Lemma 5.3.16 in
[BR79b]), and there exists a positive Hamiltonian Hωsuch that
HωΩω= 0, πω(αµ
t(a)) = eitHωπω(a)e−itHω,
where (πω, Hω,Ωω) is the cyclic representation associated to ω. We will say that the state ω
is gapped if Hωhas a gap above its (possibly degenerate) ground state. An equivalent, more
intrinsic definition is the following (see also [AL86], p.65):
Definition 2.31.
1. Let µ∈R, and ωbe a ground state of H−µˆ
N. Then ωhas a gap w0>0if for all
a, b ∈ ALL, the function ρab(t) = ω(aαµ
t(b)) has a distributional Fourier transform ˆρab with
suppˆρab∩]0, w0[= ∅.
2. Let ωbe a ground state at fixed density. Then ωis gapped with respect to AU(1), i.e.,
excitations preserving the number of particles, with gap ≥w0, if the previous condition on
ρab holds for all a, b ∈ AU(1).
2.4. A LIEB-SCHULTZ-MATTIS TYPE ARGUMENT 97
The thermodynamic limits of cylinder Laughlin wave functions are ground states at fixed
density, but also of H−µˆ
Nat chemical potential µ= 0. This follows from the positivity of
the truncated Hamiltonians. To see this, let (an)n∈Nbe such that an→ −∞,n+an→ ∞
and let ωnbe the state on ALL associated to t(an3γlex)⊗NΨC
N. The weak-* limit points of
ωare ground states of H. Recall that the states form a weak*-compact set.
Proposition 2.32. Suppose that for a subsequence (ω)nkof (ωn)n,ωnk∗
⇀ ω. Then ωis a
ground state of H.
Proof. For simplicity suppose ωn∗
⇀ ω. Let Λn:= {3an,3an+ 1, .., 3(an+n−1)}and let
HΛn:= P3an≤k,k′,n,n′≤3an+3n−3δk+k′,n+n′f(k−k′)f(n−n′)c∗
k′c∗
kcncn′be the finite volume
truncated Hamiltonian. The shifted Laughlin state is a ground state of HΛnconsidered
as an operator in ∧nspan{ψk|k∈Λn}, of energy 0. But HΛnis a positive operator in
∧span{ψk|k∈Λn}, and the shifted Laughlin state can also be considered as a ground state
in this larger space (i.e., changing the number of particles does not lower the energy, and we
can take µ= 0). Thus
∀a∈ AΛn:ωn(a∗[Hn, a]) ≥0.
Let a∈ Aloc. By the absolute convergence of the sum [H, a], [Hn, a] converges to [H, a].
Thus
|ω(a∗[H, a]) −ωn(a∗[Hn, a])| ≤ |ω(a∗[H, a]) −ωn(a∗[H, a])|+||a∗[H−Hn, a]|| → 0
thus ω(a∗[H, a]) = limnωn(a∗[Hn, a]) ≥0.
Remarks: 1. Equivalence of torus and cylinder functions. In Section 2.2.3, we compared
modified cylinder and torus wave functions and proved their equivalence in special cases.
The description in terms of truncated Hamiltonians allows an indirect approach to a similar
question. The torus wave functions can be used to define lowest Landau level states (we map
˜
ψkto ψk); these are ground states of Hon finite chains, with periodic boundary conditions,
and their limit points should be ground states of Has well. Thus in the thermodynamic limit,
torus and cylinder states are ground states of the same Hamiltonian. If they are different,
the infinite volume Hamiltonian has a degenerate ground state.
2. Equivalence of ensembles. It is often easily checked that the limits of N-particle finite
volume ground states are ground states at fixed density. To prove that they are also ground
states of H−µˆ
Nfor suitable µ(equivalence of canonical and grand-canonical ensembles)
may require more effort and can be done by using general theorems (see [Mat96]).
3. Value of the chemical potential. The limit of Laughlin’s wave function at p= 1/3 is a
ground state of H−0·ˆ
N, but not the only one: the limits of Laughlin’s wave functions at
lower filling fractions 1/p′give ground states as well, with a lower density. However, in view
of the incompressibility, we expect that there exists a positive chemical potential µ > 0 such
that the limits of the 1/3 functions are ground states of H−µˆ
N, and the 1/p wave functions,
p > 3, are not (µ > 0 favors higher density). In Theorem 2.33 below, we have in mind such
a positive chemical potential µ > 0.
The remainder of the section aims at the proof of the following theorem on ground states of
H−µˆ
N(among which we find Laughlin’s states):
98 CHAPTER 2. THERMODYNAMIC LIMITS
Theorem 2.33. Let µ∈Rbe a fixed chemical potential.
1. Suppose H−µˆ
Nhas a unique ground state ωand ωis gapped. Then ωmust be
translationally invariant and have an integer filling factor ν.
2. Suppose all ground states of H−µˆ
Nhave a spatial period. Let ωbe a pure, gapped
ground state with filling factor ν. Write ν=p/q,p, q ∈Nrelatively prime. Then any
period of ωis a multiple of q, and the ground state of αµ
tis at least q-fold degenerate.
The theorem says essentially that at non-integer filling factor, the existence of a gap implies
symmetry breaking. If there is a gap, there are either ground states with no translational
period at all, or all ground states have a period related to the average density. If one tries
to write this down as a mathematical statement, one has to assign an average density to a
state that may have no translational period. The theorem is formulated slightly differently
to avoid that trouble.
In view of the discussion on gaps in the first chapter, let us remark that the low-lying energy
excitations are constructed by means of local twists Un∈ AU(1) ∩ Aloc. Thus we actually
only use the gappedness of ωwith respect to excitations that do not change the number
of particles. Nevertheless, we chose to formulate the theorem in terms of H−µˆ
Nbecause
ground states for AU(1) are less likely to fulfill the assumptions on the degeneracy of the
ground states (e.g. 1. will never hold, since there are different ground states for different
densities).
The strategy of the proof is to take a gapped ground state ωwith period qand construct
candidate low energy states using a local twist. One proves that under suitable assumptions
among which an incommensurability of the period qand the filling factor, these candidate
states are orthogonal in the limit of global twists (Lemma 2.34) and have energies converging
to the ground state energy (Lemma 2.35). Thus if ωis gapped, the constructed low energy
states are close to a ground state, but orthogonal to ω. This gives a ground state degeneracy
and 1. follows. Lemma 2.36 below then allows to show that the candidate states are close to
ground states with no translational period at all, which is used to prove 2.
Lemma 2.34. (Orthogonality.) Let ωbe a state on ALL with a period q∈N, i.e., ω◦τq
x=ω.
Let ν:= 1
qPq
j=1 ω(ˆnj)be the filling factor. For n∈N, let
Un:= exp(i2π
n
n−1
X
j=0
jˆnj).
Suppose qν /∈Nand limn→∞(1
n2Pn−1
i,j=0 ˆniˆnj) = ν2(this is the case if ωis ergodic with
respect to τq
x). Then limn→∞ ω(Un) = 0.
Proof. Let ℓ∈N. By a direct computation,
τq
x(Uℓ) = Uℓe−i2πq
ℓPℓ+q−1
j=qˆnjRℓ, Rℓ:= ei2π
ℓPq−1
j=0 j(ˆnj+l−ˆnj).
Thus
ω(Uℓ) = ω(τq
x(Uℓ)) = ω(Uℓe−i2πq
ℓPℓ+q−1
j=qˆnjRℓ)
(1 −e−i2πqν )ω(Uℓ) = ω(RℓUℓ(e−i2π
ℓPℓ+q−1
j=qˆnj−e−i2πqν 1)) + e−i2πqνω(Uℓ(Rℓ−1)).
2.4. A LIEB-SCHULTZ-MATTIS TYPE ARGUMENT 99
The second member of the sum goes to zero since ||Rℓ−1|| =O(1/ℓ). Note that for A∈ A,
α∈C,
(eiA −eiα)∗(eiA −eiα) = 4|sin A−α
2|2≤(A−α)∗(A−α).
RℓUℓis unitary. Using Cauchy-Schwarz, we see
|ω(RℓUℓ(e−i2π
ℓPℓ+q−1
j=qˆnj−e−i2πqν 1))|2≤4π2q2ω((1
ℓ
ℓ+q−1
X
j=q
ˆnj−ν)2).
The upper bound goes to zero by the assumptions of the Lemma. Thus
(1 −e−i2πqν )ω(Uℓ)→0 (ℓ→ ∞).
If qν /∈N, this implies ω(Uℓ)→0.
Lemma 2.35. (Low energy states.) Let ωbe a state on A,µ∈R. Then
lim
ℓ→∞(ω(U∗
ℓ[Hµ, Uℓ]) + ω(Uℓ[Hµ, U∗
ℓ])) = 0.
If ωis a ground state at fixed density, the two parts of the sum go to zero individually.
Proof. Let ℓ∈N. First note Uℓ∈ Aloc, and Uℓcommutes with every number operator ˆnj.
Thus [H−µˆ
N, Uℓ] is well-defined and equals [H, Uℓ]. Similarly, [H−µˆ
N, U∗
ℓ] = [H, U∗
ℓ].
Note also Uℓ, U∗
ℓ∈ AU(1)
loc . Thus if ωis a ground state for AU(1),ω(Uℓ[H, U∗
ℓ]) ≥0 and
ω(U∗
ℓ[H, Uℓ]) ≥0. In this case the sum converges to zero if and only if each summand
goes to zero. Let g(k) := kχ[1,ℓ−1](k).so that UℓckU∗= exp(i2π
ℓg(k)ˆnk)ck. Then, using
cos x−1 = −2 sin2(x/2), we get
ω(U∗
ℓ[Hµ, Uℓ]) + ω(Uℓ[Hµ, U∗
ℓ]) = −4X
k,k′,n,n′∈Z
δk+k′,n+n′f(n−n′)f(k−k′)
sin2(2π
ℓ(g(n) + g(n′)−g(k)−g(k′)))c∗
k′c∗
kcncn′
hence
|ω(U∗
ℓ[Hµ, Uℓ]) + ω(Uℓ[Hµ, U∗
ℓ])|
≤4C′X
k,k′,n,n′∈Z
k+k′=n+n′
e−c((n−n′)2+(k−k′)2)sin2(2π
ℓ(g(n) + g(n′)−g(k)−g(k′)))
where c, C′are chosen so that |f(n)| ≤ C′e−cn2. The sum on the right-hand side can be split
into two parts. Let δ∈]0,1/3[. Let Σ1denote the sum over indices such that k−n=n′−k′
has modulus smaller or equal to ℓδ/2 for some δ∈]0,1/3[, and Σ2the sum over indices where
k−n=n′−k′has modules greater than ℓδ/2. We claim that there is a constant C > 0 so
that for sufficiently large ℓ,
|Σ1| ≤ Cℓ3δ
ℓ,|Σ2| ≤ Cℓ X
k≥ℓδ/2
e−ck2..
Thus Σ1and Σ2go to zero as ℓ→ ∞.
100 CHAPTER 2. THERMODYNAMIC LIMITS
Estimate of Σ2.The only contributions to the sum come from indices k, k′, n, n′where at
least one of them is in {1, .., ℓ −1}. By a reasoning similar to the one used to prove the
convergence of [H, A] for A∈ Aloc, we get
X
k∈{1,..,ℓ−1}X
k,k′,n′,n∈Z
k−n=n′−k′,|k−n|>ℓδ/2
e−c((n−n′)2+(k−k′)2)
≤X
k∈{1,..,ℓ−1}X
n∈Z:|k−n|>ℓδ/2
e−2c(k−n)2X
t∈Z
e−ct2/2
The parts of the sum where k′, n′or nlie in {1, .., ℓ −1}are treated in a similar way. This
gives the claimed estimate on Σ2.
Estimate of Σ1.Let k, n ∈Zsuch that |k−n| ≤ ℓδ/2. We claim that dist (g(k)−
g(n),{0,−ℓ, ℓ})≤ℓδ. If k, n ∈ {1, .., ℓ −1}.|g(k)−g(n)|=|k−n| ≤ ℓδ/2. If n≥ℓ > k ≥1,
g(k)−g(n) = k≥ℓ−ℓδ, thus |g(k)−g(n)−ℓ| ≤ ℓδ/2. The other cases are treated in a
similar fashion. Using sin(x+π) = −sin xand |sin x| ≤ |x|, we see
X
k∈{1,..,ℓ−1}X
k′,n,n′∈Z
k+k′=n+n′,|k−n|≤ℓδ/2
e−c((n−n′)2+(k−k′)2)sin2(2π
ℓ(g(n) + g(n′)−g(k)−g(k′)))
≤4π2ℓ2δ
ℓX
k∈{1,..,ℓ−1}X
n∈Z:
|n−k|≤ℓδ/2
(X
n′∈Z
e−c(n−n′)2)(X
k′∈Z
e−c(k−k′)2)
≤4π2(X
t∈Z
e−ct2)2ℓ3δ
ℓ.
Again, the cases k′, n′or n∈ {1, .., ℓ −1}can be treated in a similar way, and we get the
bound on |Σ2|.
Lemma 2.36. Let ωbe a fixed state on Awith a period q,l∈N. Let ωℓ(A) := ω(U∗
ℓAUℓ).
ωℓdefines a state on A. Suppose that ωℓis close to a periodic state, i.e., there exist an ǫ > 0
and a periodic state ω′with ||ω′−ωℓ|| < ǫ. Then ωℓis close to ω:
||ωℓ−ω|| ≤ 2ǫ.
Proof. Let ω′be a periodic state on A, i.e. ω′◦τT
x=ω′for some T∈N. Without loss of
generality we may assume that Tis a multiple of q. Let A∈ Aloc. Then for n∈Z
||ω′−ω||||A|| ≥ |ω′(τnT
x(A)) −ωℓ(τnT
x(A))|| =|ω′(A)−ω(U∗
ℓτnT
x(A)Uℓ)|
But for sufficiently large n,U∗
ℓτnT
x(A)Uℓ=τnT
x(A) and ω(U∗
ℓτnT
x(A)Uℓ) = ω(τnT
x(A)) =
ω(A), and we get
|ω′(A)−ω(A)| ≤ ||ω′−ω||||A||
for all local elements. Since Aloc is norm-dense in A, this shows ||ω′−ω|| ≤ ||ω′−ωℓ||,
whence
||ω−ωℓ|| ≤ ||ω′−ωℓ||+||ω′−ω|| ≤ 2||ω′−ωℓ|| ≤ 2ǫ.
2.4. A LIEB-SCHULTZ-MATTIS TYPE ARGUMENT 101
Proof of Theorem 2.33. 1. Let ωbe the ground state of H−µˆ
N. Since αµ
tis translationally
invariant, ω◦τxis a ground state too and by the unicity of the ground state ω◦τx=ω. Let
ν:= ω(ˆn1) be the filling factor. Suppose ν /∈N. Since ωis the unique ground state and any
extremal ground state is pure, ωis a pure state. In particular, it is ergodic with respect to
τx. By the previous lemmata,
lim
ℓ→∞ω(Uℓ) = 0,lim
ℓ→∞ω(U∗
ℓ[H, Uℓ]) = 0.(2.91)
Let (Hω, πω,Ωω) be the cyclic representation associated to ωand Hωthe unique self-adjoint
operator in Hωsuch that HωΩω= 0 and πω(αµ
t(A)) = eitHωπω(A)e−itHω. Let ψℓ:=
πω(Uℓ)Ωω.Note ||ψℓ|| = 1. (2.91) can be rewritten as
lim
ℓ→∞hΩω, ψℓi= 0,lim
ℓ→∞hψℓ, Hωψℓi= 0.(2.92)
Write ψℓ=˜
ψℓ+hΩω, ψℓiΩω. Then
||˜
ψℓ|| → 1,h˜
ψℓ, Hω˜
ψℓi → 0,h˜
ψℓ,Ωωi= 0,
contradicting the uniqueness of the ground state and the existence of a gap above the ground
state.
2. Let ωbe a gapped pure ground state of αµ
twith spatial period Tand filling factor ν=p/q,
p, q coprime. Suppose Tν /∈N. Let Hω, πω,Ωω, Hω, ψℓbe as above. Let Gbe the projection
on N(Hω) in Hω. Then (2.92) stays valid, thus hψℓ, Hωψℓi → 0. The existence of a gap
above the ground state implies limℓ→∞(1−G)ψℓ= 0.Thus for ǫ > 0 there exists l∈Nand
a normalized ψ′∈ N(Hω) such that ||ψ′−ψℓ|| < ǫ and |(Ωω, ψℓ)| ≤ ǫ. Let ω′be the state
defined through ω′(A) = hψ′, πω(A)ψ′i.ω′is a ground state of H−µˆ
N, therefore by our
assumptions it has a period. Moreover, ||ω′−ωℓ|| ≤ 2ǫ. Thus by Lemma 2.36, ||ωℓ−ω|| ≤ 4ǫ,
i.e.,
∀A∈πω(ALL) : |hψℓ, Aψℓi−hΩω, AΩωi| ≤ 4ǫ||A||.(2.93)
Since the bicommutant πω(ALL)′′ ⊂ B(Hω) is the closure of πω(ALL) with respect to the
seminorms |hξ, ·ηi|, (2.93) extends to πω(ALL)′′. But since ωis pure, Hωis irreducible with
respect to πω(ALL) and (πω(A))′′ = (C1)′=B(Hω). Thus we may take A=|ΩωihΩω|, the
projection on CΩωin (2.93) and obtain
|hψℓ,Ωωi|2−1≤4ǫ, |hψℓ,Ωωi| ≤ ǫ.
This is contradictory for sufficiently small ǫ. Thus qT ∈N, i.e., Tis a multiple of q.
Finally, recall that a state ωis called pure if for any positive functional ω′,ω′≤ωimplies
ω′=λω for some λ∈[0,1]. From this we see that ω◦τj
x(j∈Z) inherits the purity of ω. Let
Tbe the minimal period of ω. Then ω, ω ◦τx, .., ω ◦τT−1
xmust be distinct. The translational
invariance of αµ
timplies that they are all ground states. Thus we have Tdistinct pure ground
states, and the affine dimension of the ground state is greater or equal to T. Since Tis a
multiple of q, this concludes the proof.
Is the truncated Hamiltonian gapped? Theorem 2.33 shows that the existence of a
gap above the ground state, at filling factor 1/3, implies some kind of symmetry breaking.
However, the question whether the the truncated interaction has a gap above its ground state
is open. Numerical computations [SFL+05] suggest that the truncated Hamiltonian has a
102 CHAPTER 2. THERMODYNAMIC LIMITS
non-vanishing gap above its ground state, for all values of the radius, but to our knowledge
this is yet unproven.
In the previous section on jellium tubes, we mentioned that the jellium tube interpolates
between a system with a phase transition (the one-dimensional jellium) and a system without
(the two-dimensional system). A similar aspect appears in relation to the question of gaps of
truncated interactions on cylinders. We start by observing that in the limit of thin cylinders,
the truncated interaction at filling factor 1/3 may formally be written as
H=HT T +O(e−5γ2), HT T := X
k∈Z
(e−γ2ˆnkˆnk+1 + 4e−2γ2ˆnkˆnk+2) (2.94)
see also [SFL+05], equation (9). The limiting Hamiltonian HT T has obviously the ground
state energy 0. Ground states consist of configurations where electrons sit in lattice sites
spaced apart at least by 3 sites. At density 1/3, the unique ground state having a particle in
0 is the state which we loosely write
... ∧ψ−3∧ψ0∧ψ3∧ψ6...
(the Tao-Thouless state). This state is exactly the limit of Laughlin’s state as γ→ ∞, on
finite samples (see the remark on p.54) as well as on infinite samples (this is implicit in the
proof of the second part of Theorem 2.17). Furthermore, the Hamiltonian HT T has obviously
a gap of size 4e−2γ2above the ground state at density 1/3.
By the results of Section 1.4, the cylinder Hamiltonian may be considered as a periodified
version of the plane interaction (see also the expressions of the Hamiltonians in [RH94] and
the corresponding expressions of the disk interaction in [PG87], p.307). Thus we expect that
in the limit γ→0, the truncated interaction Hon the cylinder has a gap above its ground
state if the disk truncated interaction has.
More generally, we can consider truncated interactions for different filling fractions 1/p,
p∈Nand ask whether these have a gap. In the limit γ→ ∞, we will again obtain a gapped
Hamiltonian without hopping terms as in (2.94) and thus expect that the Hamiltonian on
thin cylinders is gapped. In the limit γ→0, the Hamiltonian approaches the disk truncated
interactions. In [PG87], p.319, Haldane argues that the disk Hamiltonian should be gapped
for p < 70 but gapless for p > 70; this is related to the expected phase transition of the
two-dimensional one-component plasma at coupling constant Γ ∼140.
Chapter 3
Charge transport
Laughlin’s work [Lau81, Lau83] is important for the understanding of both the integer and the
fractional quantum Hall effect. Laughlin’s argument refers to the subtle use of a clever geom-
etry - the cylinder - and gauge periodicity in an explanation of the integer effect. Laughlin’s
wave function is considered as a good approximation of fractional Hall effect ground states
at simple filling fractions. It has been suggested [TW84] that a degeneracy in the ground
states of electrons on cylinders is required in order to reconcile Laughlin’s cylinder argument
for the integer quantum Hall effect with fractional Hall conductances. If this is the case, and
if Laughlin’s function is indeed a good approximation to the ground state, it should reflect
this degeneracy. This is a heuristic argument; nevertheless, it is interesting to observe that it
is consistent with the symmetry breaking (on thin cylinders) proved in the previous chapter.
For quantum Hall systems on tori, the Hall conductance can be characterized in terms of a
Chern number, and fractional quantization of the Hall conductance requires indeed a ground
state degeneracy. This is a rigorous result. It is less clear what the significance, beyond
heuristics, of our symmetry breaking results is for the FQHE. This chapter is devoted to that
question. Let us anticipate and say that we do not give a full answer. The Hall conductance
for systems without impurities is related to the electronic density, and our results do imply
that the charge transport in Laughlin’s state has the correct quantization. However, as we
will see below, this uses only the fact that the density is periodic and has average value
(p·2πl2)−1, but not that the minimal period is pγl. Another drawback is that the results
only pertain to samples without impurities. However, we think that this chapter is useful in
replacing our results in their FQHE context.
We start with a brief summary of the gedanken experiments involving slow addition of a flux
quantum, as described in [Lau81, Lau83]. Then we turn to a simple model of bulk charge
transport on a cylinder. First we consider cylinders with quasiperiodic boundary conditions
in the direction along the cylinder axis, i.e., we look at a system on a torus. We show that
in the absence of a background potential, the adiabatic curvature appearing in the Chern
number approach can be expressed in terms of the one-particle density. We pursue with
spectral boundary conditions, more suitable for the cylinder Laughlin wave functions, and
give an account on the number of approximations involved to recover the widely accepted
fact that Laughlin’s function describes a fractionally quantized charge transport.
103
104 CHAPTER 3. CHARGE TRANSPORT
3.1 Laughlin’s argument(s)
Integer quantum Hall effect: cylinder geometry
In his paper on the integer quantum Hall effect [Lau81], Laughlin considers a gas of electrons
(charge −e) on a finite cylinder of radius R. A magnetic field is perpendicular to the surface,
an electric field of strength Exis present on the cylinder in the direction along the cylinder
axis, and a flux φthreads the cylinder. This system is represented by the Hamiltonian
H(φ) = 1
2m
N
X
j=1(−i~∂xj)2+ (−i~∂yj+e(Bz−φ
2πR)2+
N
X
j=1
eExxj
+
N
X
j=1
W(xj, yj) + X
1≤j<k≤N
V(xj−xk, yj−yk)
Vis an interaction potential and Wa background potential. For the moment, we leave open
the modelization of the cylinder edges. At fixed flux φ, the electric and magnetic fields lead
to a Hall current around the cylinder axis. The corresponding current operator is
Iy=
N
X
j=1
1
2πR −e
m(−i∂yj+e(Bz−φ
2πR)) = ∂H
∂φ .
Now imagine that we are given a projection P(0) onto a many particle wave function at flux
φ= 0. If we consider the electrons as non-interacting, we may fill the one-particle states up
to a given Fermi energy and take P(0) to be the projection onto the corresponding many-
particle state. We wish to evaluate the current hIy(0)i:= trP(0)Iy(0). Suppose that P(0)
can be extended to a smooth family of projections P(φ) commuting with the Hamiltonian
H(φ): [H(φ), P(φ)] = 0. Then
hIy(0)i= trP(0)∂H
∂φ (0) = ∂trPH
∂φ (0).(3.1)
If slow variation of the flux φresults in an adiabatic time evolution UA(φ), we may take
P(φ) = UA(φ)P(0)UA(φ)∗. Therefore the derivatives in (3.1) are referred to as adiabatic
derivatives. Now, the Hamiltonian is gauge periodic:
H(φ+φ0) = eiPN
j=1 yj/RH(φ)e−iPN
j=1 yj/R
for all φ. The period is the flux quantum φ0=h/e. This suggests the following approxima-
tion:
hIy(0)i ≃ trP(φ0)H(φ0)−trP(0)H(0)
φ0
,(3.2)
i.e., the derivative is approximated by a difference quotient with a flux difference ∆φ:= φ0.
Now comes the key sentence of [Lau81]:
“Since, by gauge invariance, adding ∆φmaps the system back into itself , the
energy increase due to it results form the net transfer of nelectrons (..) from one
edge to the other. The current is thus
I=cneV
∆φ=ne2V
h.”
3.1. LAUGHLIN’S ARGUMENT(S) 105
(Here, Vis the difference in electrochemical potential between the left and right edge.) Several
difficulties are encountered when one tries to turn Laughlin’s argument into a mathematical
statement, giving rise to different versions of the argument, compare e.g. [FGW00, ASY87].
We do not give a detailed discussion of the various difficulties encountered.
Suppose first H(φ) has a non-degenerate ground state, for each φ, and let P0(φ) be the pro-
jection onto the ground state. If the ground state energy stays separated from the rest of the
spectrum by a gap g(φ)> gmin >0 (see hypothesis H2 below), adiabatic time evolution fol-
lows the ground state: P0(φ) = P(φ). In this case, P(φ) is gauge periodic and the difference
quotient in (3.2) is 0. This is obviously not a good approximation to hIy(0)i.
To gain a better understanding, we have to take into account edges. We will consider only
independent particles (V= 0). The one-particle Hilbert space can be written as direct sum
of three orthogonal subspaces representing left edge, bulk and right edge states. Let P0(φ)
represent a state where bulk states are filled up to a given Fermi energy and edge states up
to local chemical potentials. The one-particle Hamiltonian restricted to the bulk state has
discrete spectrum with large gaps, whereas the restriction to the edge states has a spectrum
that becomes continuous in the limit of large radii. Thus intuitively, adiabatic time evolution
follows the ground state in the bulk, but not in the edges. In the words of [Lau81]:
“isothermal differentiation with respect to φ(..) is equivalent to adiabatic differ-
entiation in the sample interior.”
Therefore the adiabatically evolved bulk state should be gauge periodic, but changes may
occur in the edge states, the “net transfer of nelectrons”.
However, it is not clear how to make sense of the separate application to edge and bulk
of adiabatic time evolution. The decomposition into edge and bulk Hilbert spaces is flux
dependent. During the flux increase left edge states typically flow into the bulk and bulk
states into the right edge. Another problem is related to the integrality of the charge transfer.
Fractional quantum Hall effect: infinite plane with flux tube
It is instructive to compare the previously described argument with the effect of slow addition
of a flux quantum as performed in [Lau83]. Consider a gas of electrons moving in a plane
with a perpendicular magnetic field. A solenoid carrying a flux φpierces the plane at the
origin. A one-particle Hamiltonian H1(φ) can be defined whose spectrum consists again of
the eigenvalues ~ω(n+ 1/2), n∈N0, and the lowest Landau level N(H1(φ)−~ω/2) has a
complete orthonormal set
ψn,φ(z) = cn,φ|z|φ/φ0zne−|z|2/4l2, n ∈Z:n≥ −φ/φ0,
see [Eˇ
SV02], Section III. Now consider the N-particle Hamiltonian
H(φ) =
N
X
j=1
H1,j(φ) + W(zj) + X
1≤j<k≤N
V(zj−zk)
describing Nparticles. Imagine that we start in a many-body ground state ΨNat flux φ= 0
and slowly increase the flux from 0 to the flux quantum φ0. If a suitable adiabatic theorem
can be applied, the final state in the adiabatic limit will be an eigenstate of H(φ0). But
H(φ0) is gauge periodic, thus applying a unitary Uto the adiabatically evolved state we
106 CHAPTER 3. CHARGE TRANSPORT
obtain an eigenvector of H(0). We wish to apply this procedure to Laughlin’s state as a
starting vector. (We suppose that it is an eigenvector for some interaction.) In [Lau83], the
adiabatically evolved state is given as
U−1UAΨL
N(z1, .., zN)≃CN
N
Y
j=1
zjΨL
N(z1, .., zN),(3.3)
CN∈C. In [PG87], p. 260, Laughlin explains that (3.3) should be considered as a good
approximation to the adiabatically evolved state. This is motivated through the help of
several approximations. The background potential is chosen to be a lowest Landau level
projected Gaussian: W(zj) = Πα|zj|2Π. With this choice, Wψn,0= 2α(n+ 1)ψn,0. It is
stated that adiabatic time evolution away from the solenoid is not affected by interactions,
so that we can set V= 0. The one-particle states ψn,0evolve, up to a phase, to ψn,φ0, which
is unitarily related to ψn+1,0. The n-dependence of the normalization constants is neglected:
ψn+1,0(z)≃zψn,0(z). Thus if ΨL
N=Pa(m1, .., mN)ψm1,0∧.. ∧ψmN,0, the adiabatically
evolved state will be, up to the unitary U
U−1UAΨL
N(z1, .., zN)≃X
m1,..,mN
a(m1, .., mN)ψm1+1,0∧.. ∧ψm1+1,0
≃X
m1,..,mN
a(m1, .., mN)z1..zNψm1,0∧.. ∧ψm1,0
=z1..zNΨL
N(z1, .., zN).
A plasma analogy gives access to the one-particle density ρh(z) of the obtained excited state:
ρh(z) should converge very fast to (3 ·2πl2)−1away from the origin (|z| → ∞), and
ZR2
(ρh(x, y)−1
3·2πl2)dx dy =−1
3.(3.4)
The excited state is therefore called a quasihole state: in the limit of infinitely many particles,
it represents a uniform gas of charge density −e(3 ·2πl2)−1except that there is a hole at
the origin which leads to an overall missing charge −e/3 (fractional charge of the quasihole).
Hence during the adiabatic flux increase, a charge −e/3 has been transferred from the origin
to infinity. This leads to a fractional Hall conductance.
It is interesting to observe that there is a double use of the adiabatic flux increase: it is
invoked to show that the Hall conductance is fractionally quantized, but also to show the
existence of exited states describing a fractionally charged hole.
Hall conductance as a relative index. The (integer) Hall conductance for non-interac-
ting electrons can be characterized as a Chern number when there is a doubly periodic
background potential on the plane, or when the electrons are placed on a torus. When the
background potential is not periodic, but macroscopically homogeneous, it can be expressed
as a non-commutative Chern number (see [BvESB94]), which in turn can be expressed as
the relative index of a pair of projections. In [ASS94] the Hall conductance is related to
an index without non-commutative geometry. Let H1(φ) be a one-particle Hamiltonian
representing a particle in an infinite plane with a flux tube in the origin and a magnetic
field perpendicular to the surface. Let P1(φ) be the projection onto the ground state of
H1(φ). Due to gauge periodicity, P1(φ0) = UP(0)U−1. The relative index gives a precise
meaning to trP1(φ0)−trP1(0) in case the traces are infinite-dimensional. The difference may
3.2. CHARGE TRANSPORT ON A CYLINDER 107
be interpreted as a charge deficiency, i.e., the difference in the number of electrons in the
ground states at the two values of the flux. If we regard P1(φ) as the one-particle reduced
matrix of a many-particle state, trP1(φ0)−trP1(0) is an analogue of (3.4). Note however
that subtleties show up: due to gauge invariance, the one-particle densities of P1(φ0) and
P1(0) are identical, so that a strict analogue of (3.4) would give zero.
3.2 Charge transport on a cylinder
Based on Laughlin’s cylinder argument, Tao and Wu [TW84] have suggested that a fractional
transport on the cylinder requires ground state degeneracy. They present the picture that,
at filling factor 1/3, there are three ground states; adding one flux quantum shifts the system
from one ground state to the other. Only after addition of three units of flux does the system
return to itself, so that Laughlin’s argument would guarantee an integer transferred charge
only after addition of three flux units, leading to a fractional charge.
For Hall systems on tori, the Chern number approach requires indeed a ground state de-
generacy in order to explain fractional quantization of the Hall conductance. For cylinder
systems, the significance of ground state degeneracy is not quite clear. Since the connection
to charge transport has motivated the investigations of this thesis, let us go into some detail.
In the following, units are chosen so that ~= 1, eBz=−1, l = 1, m = 1. We adopt a
simple model for the bulk charge transport, induced by a variable flux threading a cylinder
[0, L]×R/(2πRZ). Consider the N-particle Hamiltonian
H(φ) = X
j
1
2(v2
xj+vyj(φ)2) + X
j
W(zj) + X
j<k
V(zj−zk),(3.5)
where the velocity operators are
vxj=−i∂xj, vyj(φ) = −i∂yj−(x−φ
2πR).
We do not yet specify boundary conditions. Let χbe a switch function, i.e., a function with
derivative χ′∈C∞
0(]0, L[) such that χ′≥0 and χ(0) = 0, χ(L) = 1. We suppose that χ′
has its support in the middle of the cylinder, away from boundaries. According to Faraday’s
law, a time-dependent flux φ(t) will generate an electric field of strength ˙
φ/(2πR) around
the cylinder axis which together with the magnetic field gives rise to a Hall current along the
cylinder axis. The χ′-averaged current in the x-direction is
Iχ=−1
2
N
X
j=1
(χ′(xj)vxj+vxjχ′(xj)).
The flux φis slowly increased from 0 to the flux quantum φ0= 2π: let f∈C1([0,1]; R) be a
monotone increasing function with f(0) = 0, f(1) = 2πand let φ(t) := f(t/τ) for some time
scale τ > 0. The associated propagator in the rescaled time s=t/τ satisfies
i∂sUτ(s) = τH(f(s))Uτ(s), Uτ(0) = 1.(3.6)
Let the system be initially in a state described by a projection Pof trace q, and let Pτ(s) :=
Uτ(s)P(0)Uτ(s)∗be the time evolved projection. The current and total charge transport are
108 CHAPTER 3. CHARGE TRANSPORT
given by
Iτ(s) := 1
qtrPτ(s)Iχ, Qτ(s) := τZ1
0
Iτ(s)ds. (3.7)
We are interested in expectation values of this current when the flux φis adiabatically
changed, i.e. in limits of the quantities above in the limit τ→ ∞. In view of Ey=
˙
φ(t)/(2πR) = −f′(t/τ)/(τ2πR) this corresponds to the limit of small electric fields Ey. The
time average over the period of the flux increase of the field is φ0/2πR. A fractionally
quantized Hall conductance σH=1
3e2/h arises when the charge transported during the flux
increase by one flux quantum φ0=h/e is fractional: Q=e/p, or when the charge trans-
ported during an increase by three flux units is Q=e.
We will look at two different choices of boundary conditions: quasiperiodic boundary condi-
tions lead to the characterization of Hall conductance as a Chern number, whereas spectral
boundary conditions are the natural setting for the cylinder Laughlin wave function.
It is commonly stated that the quantization of charge transport in Laughlin’s state follows
from the correct value of the average density. This is based on the picture that, in the
absence of impurities, the (bulk) conductance equals the electronic charge density. This can
be checked for the torus geometry (see the next subsection), where the adiabatic increase of
the flux results in an overall translation of the system. As a result, one gets, for W= 0, a
charge transport in terms of the one-particle density ρ(x, y)
QA(φ)∝1
2πR Zφ
0Z[0,L]×[0,2πR]
χ′(x)ρ(x−φ′
2πR, y)dxdydφ′.(3.8)
For a cylinder with boundaries, such a formula holds at best in the limit of infinitely long
cylinders. If the one-particle density has period 3/R in the direction of the cylinder axis,
and we extend the x-integration in (3.8) to R, the average charge transport after addition of
three flux units is proportional to
1
2πR Z3·2π
0ZR×[0,2πR]
χ′(x)ρ(x−φ
2πR, y)dxdydφ
= (ZR
χ′(x)dx)(Z[0,3/R]×[0,2πR]
ρ(u, y)dudy) = 3 ·2π¯ρ
where ¯ρis the average density. Thus if the average density is ¯ρ= (3 ·2π)−1, we obtain an
integer transport after addition of three flux units and a correct quantization of the Hall
conductance. The question whether 3/R is the smallest period of the one-particle density
does not play a role. However, the symmetry breaking is consistent with the heuristic that
“it takes three flux units before the system returns to itself”.
The following two subsections are devoted to the proof of formulas of the type (3.8) on
tori (i.e., cylinders with periodic boundary conditions) and cylinders with spectral boundary
conditions. The main interest on tori is to compare the formula to the interpretation of the
Hall conductance in terms of Chern numbers.
3.2.1 Periodic boundary conditions
This subsection presents the classical Chern number approach (see [KS90, ASS94]) in a way
slightly different from the usual way. Our presentation simplifies the comparison with charge
3.2. CHARGE TRANSPORT ON A CYLINDER 109
transport on cylinders as defined in the next section. This section also aims at clarifying
the relationship between the characterization of the Hall conductance as a Chern number
and the common saying “the correct quantization of Hall conductance in Laughlin’s state is
obvious provided it has the right one-particle density”.
Quasiperiodic boundary conditions in the direction along the cylinder axis are defined in
terms of the magnetic translations t(L)ψ(z) = eiLyψ(z−L). It proves useful to consider the
family of boundary conditions t(L)ψ=eiβψ. Thus we should consider the Hamiltonian of
(3.5) with these boundary conditions; call it ˆ
H(β, φ). In order to make the β-dependence
more explicit and get a nice representation of the current Iχ, we will work with the unitarily
related Hamiltonian
H(β, φ) =
N
X
j=1
1
2(vxj(β)2+vyj(φ)2) + X
j
W(zj) + X
j<k
V(zj−zk),(3.9)
vxj(β) = −i∂xj−βχ′(xj), vyj(φ) = −i∂yj−(x−φ
2πR) (3.10)
with the β-independent boundary conditions t(L)ψ=ψ. The current can be represented as
a derivative:
Iχ(β, φ) = ∂H
∂β (β, φ).
We will need the following hypotheses:
H1: We say a function ψ: [0, L]×[0,2πR] fulfills the condition (QP) if it can be extended
to a function ˜
ψ∈C∞(R2) satisfying the quasiperiodicity condition
˜
ψ(x+L, y) = e−iLy ˜
ψ(x, y),˜
ψ(x, y + 2πR) = ˜
ψ(x, y).(3.11)
Let D0be the set of N-particle functions ψ∈L2(([0, L]×[0,2πR])N) that satisfy (QP) in
each variable. There is a family of self-adjoint operators H(β, φ) with common domain D
and common core D0, given on D0by the expression (3.9).
H2: The family of Hamiltonians is smooth and has a band bordered by gaps:
(i) Consider (β, φ)7→ H(β, φ) as a map from R2to B(D, HN), the space of bounded
operators from Dequipped with the graph norm of H(0,0) to HN. Then H(·,·) is
twice continuously differentiable.
(ii) There are two real-valued, continous functions g−≤g+and ǫ > 0 such that for all β, φ:
dist g+(β, φ), g−(β, φ)}, σ(H(β, φ))> ǫ
and the spectral projection P(β, φ) of of H(β, φ) associated with [g−, g+](β, φ) is finite-
dimensional, of (constant) dimension q > 0.
Remarks: 1. The family of Hamiltonians is gauge-periodic:
H(β+ 2π, φ) = ei2πPN
j=1 χ(xj)H(β, φ)e−i2πPN
j=1 χ(xj)
H(β, φ + 2π) = eiPN
j=1 yj/RH(β, φ)e−iPN
j=1 yj/R.
110 CHAPTER 3. CHARGE TRANSPORT
2. The Hamiltonian H(β, φ) is unitarily related to a more standard operator, used in [ASS94]:
let ˜
H(β, φ) be defined exactly as H(β, φ) except that the velocity operator in the y-direction
is vy(β) = −i∂y−β/L. Then
H(β, φ) = eiβ(χ(xj)−xj/L)˜
H(β, φ)e−iβ PN
j=1(χ(xj)−xj/L).
H1 and H2(i) ensure the existence and uniqueness of the propagator Uτ(β, s) (i∂sUτ(β, s) =
τH(β, f(s))Uτ(β, s)). Let Pτ(β, s) := Uτ(β, s)P(β, 0)Uτ(β, s)∗be the time evolved projec-
tion. The adiabatic limits (τ→ ∞) of the current and charge transport are characterized by
the following two theorems:
Theorem 3.1. ([ASY87, KS90]) Suppose H1 and H2 hold and define the current Iτand
charge transport Qτas in (3.7). Then the current is a 2π-periodic function of βwhose
average in the large τlimit is
1
2πZ2π
0
Iτ(β, s)dβ =f′(s)
τ
1
2πq Z2π
0
trP[(∂βP),(∂φP)](β, f(s))dβ +O(1/τ2) (3.12)
uniformly in s∈[0,1]. As a consequence, the average charge transport is
1
2πZ2π
0
Qτ(β)dβ =1
qc(P) + O(1/τ)
where c(P) = i
2πR[0,2π]×[0,2π]trP[(∂φP),(∂βP)](β, φ)dβdφ is an integer, the Chern number
associated with the gauge periodic family of projections P(β, φ).
Theorem 3.2. Suppose in addition to H1 and H2 that there is no background potential:
W= 0 in (3.6). Let ρ(x, y;β, φ)be the one-particle density associated to the density operator
q−1P(β, φ). Then
Iτ(β, s) = −f′(s)
τ
1
2πR Z[0,L]×[0,2πR]
ρ(x, y;β, f(s))χ′(x)dxdy +O(1/τ2) (3.13)
ρ(x, y;β, φ) = ρ(x−φ
2πR, y +β
L; 0,0).(3.14)
(3.13) holds uniformly in (β, s)∈R×[0,1]. The current and the charge transport are
independent of β.
Remark. The one-particle density is the function ρ: [0, L]×[0,2πR]→Rsuch that for all
continuous f: [0, L]×[0,2πR]→C,
1
qtrP
N
X
j=1
f(xj, yj) = Z[0,L]×[0,2πR]
ρ(x, y)f(x, y)dxdy.
In (3.14) we consider ρas a doubly periodic function from R2to R.
Proof of Theorem 3.1. Details of the proof can be found in [KS90]. We recall the key idea
behind (3.12): the current can be rewritten in terms of the persistent current formula
Iτ(β, s) = ∂Eτ
∂β (β, s) + i
qτ trPτ[(∂sPτ),(∂βPτ)](β, s).(3.15)
3.2. CHARGE TRANSPORT ON A CYLINDER 111
where Eτ(β, s) := q−1trP(β, s)H(β, f(s)) is a 2π- periodic function of β. The adiabatic
theorem allows to prove
trPτ[(∂sPτ),(∂βPτ)](β, s) = f′(s)trP[(∂φP),(∂βP)](β, f(s)) + O(1/τ)
uniformly in β, s. Inserting this into (3.15) and taking the average over β, we get (3.12).
Proof of Theorem 3.2. The key observation is that the Hamiltonian is not only gauge-periodic,
but gauge-constant. For ψ∈L2([0, L]×[0,2πR]), let
u(β, φ)ψ(x, y) := eiβχ(x)ψ(x−φ
2πR, y +β
L)
where ψis identified with its quasiperiodic extension as in (3.11). The unitary upreserves
the set of smooth quasiperiodic functions. To see this, consider χas a function in C∞(R)
with the quasiperiodicity χ(x+L) = χ(x) + 1 (this is compatible with χ′∈C∞
0(]0, L[),
χ(0) = 0 and χ(L) = 1). For ψ∈D0, let ˜
ψ:= u(β, φ)˜
ψ. Then ˜
ψ∈C∞(R2) and
˜
ψ(x+L, y) = eiβχ(x+L)ψ(x+L, y +β
L)
=eiβ(χ(x)+1)e−iL(y+β/L)ψ(x, y +β
L)
=eiβχ(x)e−iLyψ(x, y +β
L) = e−iLy ˜
ψ(x, y).
The y-periodicity of ˜
ψis obtained in a similar way. Thus u(β, φ) satisfies (QP). Let V(β, φ) :=
u(β, φ)⊗N. Then V(β, φ) leaves the domain of definition of H(β, φ) invariant, and
H(β, φ)V(β, φ) = V(β, φ)H(0,0)V(β, φ)−1, H(β, φ) = V(β, 0)H(0, φ)V(β, 0)−1(3.16)
for all β, φ ∈R, and (3.14) follows from P(β, φ) = V(β, φ)P(0,0)V(β, φ)−1. Moreover, the
propagators are related through Uτ(β, s) = V(β, 0)Uτ(0, s) so that the time evolved projec-
tions Pτ(β, s) and Pτ(0, s) are unitarily related. As a consequence, the energy Eτ(β, s) =
Eτ(0, s) does not depend on β, the persistent current ∂βEτin (3.15) vanishes and
Iτ(β, s) = if′(s)
qτ trP[(∂φP),(∂βP)](β, f(s)) + O(1/τ2) (3.17)
uniformly in β, s. In order to evaluate trP[(∂φP),(∂βP)], we use the Chern-Simons formula
(see [KS90], Lemma 3.2) which gives
trP[∂βP, ∂φP](β, φ) = ∂βtrP(0, φ)V(β, 0)−1(∂φ(V(β, 0))) −∂φtrP(0, φ)V(β, 0)−1(∂β(V(β, 0))
=−∂φtrP(0, φ)V(β, 0)−1(∂βV)(β, 0).
By an explicit computation, i∂βu(β, 0)ψ= ( i
L∂y−χ(x))u(β, 0)ψ, whence
trP[∂βP, ∂φP] = −∂φtrP(β, φ)
N
X
j=1
(i
L∂yj−χ(xj)).(3.18)
112 CHAPTER 3. CHARGE TRANSPORT
Now let ψ∈L2([0, L]×[0,2πR]) fulfill (QP). For β, φ ∈R, let ˜
ψ:= u(β, φ)ψ. Again, consider
χas a quasiperiodic function (χ(x+L) = χ(x) + 1).
hu(β, φ)ψ, (i
L∂y−χ(x))u(β, φ)ψi
=ZL+φ/2πR
φ/2πR Z2πR
0
(ψ(x, y)i
L(∂yψ)(x, y)−χ(x+φ
2πR)|ψ(x, y)|2)dxdy
=ZL
φ/2πR Z2πR
0
(ψ(x, y)i
L(∂yψ)(x, y)−χ(x+φ
2πR)|ψ(x, y)|2)dxdy
+Zφ/2πR
0Z2πR
0
(ψ(x+L, y)i
L(∂yψ)(x+L, y)−χ(x+φ
2πR +L)|ψ(x+L, y)|2)dxdy
=ZL
0Z2πR
0
(ψ(x, y)i
L(∂yψ)(x, y)−χ(x+φ
2πR)|ψ(x, y)|2)dxdy.
In passing to the last line, we used the relation
ψ(x+L, y)∂yψ(x+L, y) = ψ(x, y)∂yψ(x, y)−iL|ψ(x, y)|2
deduced from (3.11). Thus we get
hu(β, φ′)ψ, (i
L∂y−χ(x))u(β, φ′)ψi|φ′=φ
φ′=0 =ZL
0Z2πR
0
(χ(x)−χ(x+φ
2πR))|ψ(x, y)|2dxdy.
A similar computation gives
trP(β, φ′)
N
X
j=1
(i
L∂yj−χ(xj))|φ′=φ
φ′=0 =ZL
0Z2πR
0
(χ(x)−χ(x+φ
2πR)ρ(x, y)dxdy.
If we differentiate this equation, change variables (note that χ′and ρare periodic) and use
(3.18), we get
trP[∂βP, ∂φP](β, φ) = 1
2πR ZL
0Z2πR
0
χ′(x)ρ(x−φ
2πR, y)dxdy
which gives, together with (3.17) and (3.14), the desired identity (3.13).
Thus we see how the representation of the Hall conductance through an adiabatic curvature
and the saying “Hall conductance = one-particle density” (in the absence of background
potential) fit together.
Theorem 3.2 is just one among several arguments relating the Hall conductance to the one-
particle density in the absence of a background potential. This way of proving it is interesting
insofar as it allows a direct comparison with the Chern number approach and facilitates the
comparison with the spectral boundary conditions setting of the next subsection.
Furthermore, this theorem generalizes a result by Tao and Haldane. In [TH86], they compute
the Chern number associated to the family of projections onto flux-dependent Laughlin-type
wave functions on a torus.
3.2. CHARGE TRANSPORT ON A CYLINDER 113
3.2.2 Spectral boundary conditions
Instead of (quasi)periodic boundary conditions, we can use the spectral or chiral boundary
conditions introduced in [AANS98]. The treatment of charge transport in this subsection
follows closely the PhD thesis by Richter [Ric00]. Before we turn to the charge transport, let
us recall some facts about spectral boundary conditions.
Spectral boundary conditions: one particle. Let [a, b]⊂R,R > 0 and φ∈R.
Consider the formal operator in L2([a, b]×[0,2πR]) given by
H(φ) = 1
2(−i∂x)2+1
2(−i∂y−(x−φ
2πR))2.
We will consider H(φ) with periodic boundary conditions in the y-direction, thus it is useful
to work with Fourier series: let Ube the unitary
U:M
k∈Z
L2([a, b]) →L2([a, b]×[0,2πR]),(fk)k∈Z7→ X
k∈Z
eiky/Rfk(x).
Then formally,
U−1H(φ)U=M
k∈Z
h((k+φ
2π)1
R), h(ρ) = 1
2(−∂2
x+ (x−ρ)2).(3.19)
For ρ∈R, let Bρbe the set of functions f: [a, b]→Cthat are C∞and satisfy the ρ-
dependent boundary conditions
ρ < a :f′(a) = 0, f′(b) = (ρ−b)f(b)
a≤ρ≤b:f′(a) = (ρ−a)f(a), f′(b) = (ρ−b)f(b)
ρ > b :f′(a) = (ρ−a)f(a), f′(b) = 0.
Then Bρ→L2([0, L]), f 7→ 1
2(−∂2
x+ (x−ρ)2)fis essentially self-adjoint. We take h(ρ)
as its self-adjoint closure and define H(φ) through (3.19). The spectrum of h(ρ) consists of
simple eigenvalues e0(ρ)< e1(ρ)< ... where the smallest eigenvalue e0(ρ) is always ≥1/2 and
e0(ρ) = 1/2 if and only if ρ∈[a, b]. The resolvents ρ7→ (h(ρ) + i)−1are norm-continuous,
but not norm-differentiable: the smallest eigenvalue has a non-zero left derivative in a. From
this we get the following properties of H(φ):
1. H(φ) is norm-resolvent continuous in φ.
2. For n∈Z,φ∈R, let ψn(x, y;φ) = exp(iny/R) exp(−(x−n+φ/2π
R)2/2). The Hamilto-
nian has the finite dimensional ground state
N(H(φ)−1
2) = span{ψn|n∈Z:a≤n+φ/2π
R≤b}.(3.20)
Loosely speaking, one may say that the spectral boundary condition leaves the form
of the lowest Landau level basis functions unchanged; the only effect of the boundary
conditions is to pick out those Gaussians that are centered inside [a, b].
3. H(φ) is gauge periodic: H(φ+ 2π) = eiy/RH(φ)e−iy/R.
114 CHAPTER 3. CHARGE TRANSPORT
In view of the definition of charge transport, it is important to notice that H(φ) does not
satisfy H1, since the domain of definition is φ-dependent. Moreover, the ground state does
not satisfy H2. Indeed, the gap above the ground state
g(φ) := min
E∈σ(H(φ)\{1/2}(E−1
2)≤min{e0(n+φ/2π
R)−1
2|n+φ/2π
R/∈[a, b]}.
is strictly positive for all values of the flux φ, but discontinuous for flux values where
n+φ/2π∈ {Ra, Rb}for some integer n∈Zand approaches zero in those points. These
values also correspond to discontinuities of the projection onto the ground state (3.20) of
H(φ).
The fact that H(φ) does not satisfy H1 and H2 leads to technical complications. For example,
the existence of time evolution becomes more complicated.
Proposition 3.3. Let τ > 0and φ(·)∈C∞([0,1]) such that ˙
φ > 0. Then there exists
a unique map of unitaries Uτ(s), s ∈[0,1] that is strongly differentiable, and such that
DH(φ(0)) ⊂ D(H(φ(s)) for all s∈[0,1] and
i∂sUτ(s) = τH(φ(s))Uτ(s), Uτ(0) = 1.
Proof. We give only a sketch of proof, following [Ric00], Lemma 1.4.1 but correcting a small
mistake done there. The main idea is to note that the Hamiltonian is equivalent to a (non
self-adjoint) operator with constant boundary conditions. First remark that it is enough to
prove the existence of the solution to the time-dependent Schr¨odinger equation
i∂su(s) = τh(ρ(s))u(s) (3.21)
in L2([a, b]). Define the ρ-dependent function
fρ(s) =
1
2
b−ρ
b−a(x−a)2, ρ < a,
1
2(x−ρ)2, a ≤ρ≤b.
1
2
a−ρ
a−b(x−b)2, ρ > b.
A smooth function gsatisfies the boundary conditions (Bρ) if and only if ˜g=efρgsatisfies the
von Neumann boundary conditions ˜g′(a) = ˜g′(b) = 0. Moreover, if i∂sg(s) = τh(ρ(s))g(s),
then
i∂s˜g(s) = (τefρ(s)h(ρ(s))e−fρ(s)+i∂fρ
∂ρ
dρ
ds)˜g(s) =: A(s)˜g(s)
The right-hand side is well-defined for ρ(s)/∈ {a, b}. On those intervals the operator A(s),
although not self-adjoint, is better behaved than h(ρ(s)) and a classical result (cf. [RS75])
ensures the solvability of the equation. We can patch together a continous, piecewise dif-
ferentiable solution and then apply e−fρto obtain a continuous, piecewise differentiable
solution g(s) of the time-dependent Schr¨odinger equation (in particular, g(s)∈ D(h(ρ(s))
for all s∈[0,1]). Then (3.21) and the continuity of h(ρ) imply that g(·) is actually strongly
differentiable.
Multiparticle Hamiltonian. The previous considerations lead us to a different set of
assumptions on H(φ). Let Wbe a background potential and Van interaction potential
between the particles.
3.2. CHARGE TRANSPORT ON A CYLINDER 115
H’1: For each φ∈R, the operator
H(φ) =
N
X
j=1
Hj(φ) +
N
X
j=1
W(zj) + X
1≤j<k≤N
V(zj−zk)
is selfadjoint, and its domain is the same as for W=V= 0; Hj(φ) is the spectral boundary
Hamiltonian for the j-th particle.
H’2: The time evolution exists and has an adiabatic limit:
(i) Let φ(·)∈C∞([0,1]) with ˙
φ > 0, and τ > 0. There is a strongly differentable unitary
Uτ(·) such that Uτ(0) = 1,Uτ(s)DH(φ(0))⊂ DH(φ(s))for all s∈[0,1], and
i∂sUτ(s) = τH(φ(s))Uτ(s).
(ii) There is a continuous, piecewise differentiable family of finite-dimensional projections
P(φ) leaving the domain of H(φ) invariant and commuting with H(φ), such that
lim
τ→∞Uτ(s)P(φ(0))Uτ(s)∗=P(φ(s))
for all s∈[0,1].
Lemma 3.4. Assume H’1 and H’2. Let the charge transport be defined as in (3.7). Let
q:= trP(φ)and ρ(x, y;φ)be the one-particle density associated to q−1P(φ). Then the charge
transport in the adiabatic limit is
lim
τ→∞Qτ=−q−1trP(φ(s))
N
X
j=1
χ(xj)|s=1
s=0
=−Zb
aZ2πR
0
χ(x)(ρ(x, y;φ(1)) −ρ(x, y;φ(0))dxdy.
Sketch of proof. We observe that i[H(φ),−PN
j=1 χ(xj)] = Iχin the sense of quadratic forms,
i.e. for all functions ψin the domain of H(φ), we have
ihH(φ)ψ, −
N
X
j=1
χ(xj)ψi−ih−
N
X
j=1
χ(xj)ψ, H(φ)ψi=hψ, Iχψi.
This is shown by an integration by parts. The spectral boundary conditions on ψare needed
to ensure that the boundary terms arising in the integration by parts vanish. (With quasiperi-
odic boundary conditions, the previous commutator relation does not hold in the sense of
quadratic forms.) It follows that for ψ∈ D(H(φ)),
∂shUτψ, −
N
X
j=1
Uτψi=hUτψ, IχUτψi.
This implies that
Qτ=−1
qtrUτ(s)P(0)Uτ(s)∗
N
X
j=1
χ(xj)|s=1
s=0.
Passing to the limit τ→ ∞, we obtain the desired result.
116 CHAPTER 3. CHARGE TRANSPORT
Lemma 3.4 motivates the following definition:
Definition 3.5. Let P(φ), φ ∈Rbe a norm-continuous family of projections in L2(([a, b]×
[0,2πR])N)and χ∈C∞([a, b]) a switch function of finite dimension q > 0. Then
QA(φ) := −q−1tr(P(φ)−P(0))
N
X
j=1
χ(xj)
is called the adiabatic charge transport through suppχ′when the flux is varied from 0to φ.
Charge transport as adiabatic curvature. The map ψ∈L2([a, b]×[0,2πR]) 7→ eiβχ(x)ψ
leaves the spectral boundary conditions unchanged. Define H(β, φ) through (3.5) with spec-
tral boundary conditions. Then H(β, φ) is unitarily related to H(φ) through
H(β, φ) = eiβ PN
j=1 χ(xj)H(φ)e−iPN
j=1 βχ(xj).
(This is to be compared with (3.16).) If P(φ) is a projection as in the previous lemma, let
P(β, φ) := eiβ PN
j=1 χ(xj)P(φ)e−iβ PN
j=1 χ(xj).Then the adiabatic charge transport QA(φ) can
be expressed as an integral of the adiabatic curvature P[∂βP, ∂φP] and is independent of β:
QA(φ) = −q−1Zφ
0
trP[∂βP, ∂φP](β, φ′)dφ′.
This is to be compared with Theorem 3.12. Thus the adiabatic charge transport on a finite
cylinder (spectral boundary conditions) is an integral of an adiabatic curvature, just as it is
for the torus. The projections P(β, φ) that arise typically are gauge constant in β, but not
gauge periodic in φ. Hence contrary to the torus case, there is no Chern number associated
to the projection.
Flux dependent Laughlin functions. The advantage of Lemma 3.4 is that it allows
to associate a “charge transport” to a family of projections without making reference to a
Hamiltonian or a time evolution. If we wish to apply this to Laughlin’s state, we need to
define a flux-dependent wave function that coincides with Laughlin’s wave function at flux
φ= 0. There are two natural candidates:
Ψ(1)
N,φ(z1, .., zN) := Y
1≤j<k≤N
(ezj/R −ezk/R)3e−PN
j=1(xj−φ
2πR )2/2(3.22)
Ψ(2)
N,φ(z1, ..., zN) := X
0≤m1<..<mN≤3N−3
aN(m1, .., mN)ψm1,φ ∧.. ∧ψmN,φ (3.23)
In (3.23), aN(m1, .., mN) are the expansion coefficients of Chapter 2, and ψm,φ are normalized
ground state functions of H(φ) (spectral boundary conditions) restricted to the y-momentum
sector i∂yψ=m
Rψ: let fρ∈L2([a, b]) be a normalized ground state function of h(ρ), then
ψm,φ(z) = 1
√2πReimy/Rf(m+φ/2π)/R(x).
Note that
a≤m+φ/2π
R≤b⇒ψm,φ(z)∝eimy/Re−(x−(m+φ/2π)/R)2.
Both choices have their advantages and inconvenients. The function (3.22) shares the multi-
plicity of zeroes of Laughlin’s wave function: as two particles get close zj−zk→0, it vanishes
3.2. CHARGE TRANSPORT ON A CYLINDER 117
as O((zj−zk)3). It is the cylinder version of the function given in [TG91] for electrons on an
annulus. However, it is made of wedge products of eigenfunctions of the one-particle Hamil-
tonian with spectral boundary conditions in L2([a, b]×[0,2πR]) only as long as a≤φ/2πR,
(3N−3 + φ/2π)/R ≤b.
The function (3.23) is built using adiabatic time evolution in the absence of interactions and
background potential. It is always a sum of Slater determinants of eigenfunctions of H1(φ).
But it does not have zeroes of order 3 as two particles approach.
Of course, we expect that both functions have the same bulk properties as the cylinder gets
infinite. Intuitively, in the bulk, for both functions, increasing the flux from 0 to φjust shifts
the function by an amount of φ/2πR along the cylinder axis, and we expect that the charge
transport is essentially given by a formula of the type (3.8). We check this for Ψ(2)
Nand time
evolutions that leave the wave function in the bulk.
Proposition 3.6. Let φ∈R, and P(φ)be the projection in L2(([a, b]×[0,2πR])N)on Ψ(2)
N,φ,
d > 0, and χa switch function with suppχ′⊂]a+d, b−d[. Let QA(φ)be the charge transport
associated with P(φ)through Definition 3.5. Suppose Ψ(2)
N,0and Ψ(2)
N,φ are bulk states, i.e.,
a≤0≤(3N−3)/R ≤band a−φ/2π≤0≤(3N−3)/R ≤b−φ/2π. Then
QA(φ) = Z[a,b]×[0,2πR]
(χ(x+φ/2πR)−χ(x))ρ(x, y)dxdy +o(1) (3.24)
as dist (suppχ′,{a+|φ|/2πR, b − |φ|/2πR})→ ∞, uniformly in the number of particles N.
Here, ρ(x, y)is the one-particle density of Ψ(2)
N,φ.
Proof. Let ρ(x;φ) := R2πR
0ρ(x, y;φ)dy be the y-integrated density of Ψ(2)
N,φ. For φ∈R, let
φ′:= φ/2π. We consider only φ > 0 and suppose d≥φ′. Then
ρN(x;φ) =
3N−3
X
k=0
nkdk+φ′e−(x−(k+φ′)/R)2
where 0 ≤nk≤1 are φ-independent, and dρ= (Rb
ae−(x−ρ/R)2dx)−1. Let 0 ≤m < M ≤
3N−3 and set
ρL(x;φ) =
m−1
X
k=0
nkdk+φ′e−(x−(k+φ′)/R)2ρB(x;φ) =
M
X
k=m
nkdk+φ′e−(x−(k+φ′)/R)2
ρR(x;φ) =
3N−3
X
k=M+1
nkdk+φ′e−(x−(k+φ′)/R)2
(roughly, left edge, bulk and right edge contributions). We have to estimate the difference
Zb
a
χ(x)(ρ(x;φ)−ρ(x; 0))dx −Zb
a
(χ(x+φ′/R)−χ(x))ρ(x; 0)dx =R+B+L
where R, B, L are obtained from the left-hand side by replacing ρwith ρR, ρB, ρL. The three
contributions are estimated separately.
118 CHAPTER 3. CHARGE TRANSPORT
Bulk. The bulk remainder can estimated as
|B|=|Zb
a+φ′
χ(x)(ρB(x;φ)−ρB(x−φ′/R; 0))dx +Zb
b−φ′
ρB(x; 0)dx|
≤
M
X
k=m
nkZR|dk+φ′−dk|e−(x−(k+φ′)/R)2dx +
M
X
k=m
nkdkZ∞
b−φ′
e−(x−k/R)2dx
≤√π
M
X
k=m|dk+φ′−dk|+√π
2( max
m≤k≤Mdk)(
M
X
k=m
erfc(b−(k+φ′)/R)).
where erfc(u) := 2
√πR∞
ue−x2dx is the complementary error function. Let c≥maxρ/R∈[a,b]dρ
(cis essentially 2/√πwhen b−ais large). Then one can show that
|dk+φ′−dk| ≤ c2√π
2(erfc( k
R−a) + erfc(b−k+φ′
R)).
By the good decay properties of erfc, we get a bound of the type
|B| ≤ F(m
R−a) + F(b−M+φ′
R),
where Fis independent of N, m, M, φ and goes to zero at infinity.
Left edge. Suppose (m+φ′)/R ≤a+d. Then
|L| ≤ Zb
a+d−φ′/R
ρ(x;φ)dx +Zb
a+d
ρ(x; 0)dx
≤√π
m
X
k=0
erfc(a+d−k+φ′
R)≤√π
m
X
k=0
erfc(a+d−k+φ′
R)
≤F(a+d−m+φ′
R)
for some function Fgoing to zero at infinity.
Right edge. Using Zb
a
ρR(x;φ)dx =
3N−3
X
k=M+1
nk=Zb
a
ρR(x; 0)dx,
one rewrite Ras an integral involving χc:= 1 −χ. As for the left edge, we obtain a bound
|R| ≤ F(M+φ′
R−(b−d))
if (M+φ′)/R ≥b−d.
Choosing m/R well between aand a+d−φ′/R, (M+φ′)/R well in between b−dand band
putting together the estimates of B, L, R one sees that the sum B+L+Ris small when d
is large.
Differentiating (3.24) with respect to φand passing to the limit of infinite cylinders, we obtain
a formula for the charge transport of the type of (3.8). The discussion of the beginning of
3.2. CHARGE TRANSPORT ON A CYLINDER 119
the section applies, and we obtain the “correct” charge transport if the one-particle density
has the “correct” value.
To summarize, we went into some lengths to show that if Laughlin’s wave function has
the right one-particle density, it is associated to a fractionally quantized charge transport.
The picture behind this is that in the absence of impurities, slow addition of a flux acts
like translation. This can be proved rigorously on cylinders with periodic boundary con-
ditions, i.e., tori. In contrast, cylinders with spectral boundary conditions model samples
with boundaries. We expect that in the bulk, adiabatic addition of a flux quantum still
acts like a translation. A proof of this would require a simultaneous on an adiabatic and a
thermodynamic limit. Especially for systems of interacting particles, this involves some work.
Additional difficulties come from the flux-dependence of the boundary conditions and the
fact that eigenvalues cross. The first problem can be handled, on finite cylinders, by relating
the Hamiltonian to an operator with flux independent boundary conditions, at the price of
obtaining an operator that is not self-adjoint. On an infinite plane pierced by a flux tube, the
situation is more delicate, see [AHˇ
S05]. The second problem can be addressed by restricting
to an invariant subspace (e.g., a given y-momentum) where the crossing does not occur, or
by the use of an adiabatic theorem without a gap condition [AE99].
The violation of the gap condition reflects the fact that adiabatic time evolution does not
follow the ground state. This is clearly behind Laughlin’s use of the adiabatic flux in-
crease to obtain the quasihole excitation. On the cylinder, the situation is as described in
[TG91, GT93]: as the flux increases, a bulk ground state moves into the right edge. This is
accompanied by an energy increase. Meanwhile, a many-body state partially living on the
edge moves into the bulk, thereby decreasing its energy. In the process, energy levels may
cross.
Conclusion
We have shown that on sufficiently thin cylinders, the thermodynamic limit at filling factor
1/p of Laughlin’s state exists. The limiting state is translationally invariant and mixing with
respect to shifts by multiples of pl2/R along the cylinder axis. The state is not l2/R-periodic.
Since Laughlin’s state is the ground state of a l2/R-periodic Hamiltonian, this means that
there is symmetry breaking.
The key ingredient to our proof is the use of a product rule of expansion coefficients
in the expansion of Laughlin’s wave function into lowest Landau level basis functions. The
product rule allows the representation of Laughlin’s cylinder function as a quantum polymer.
The normalization constant is the partition function of a polymer system with translation-
ally invariant activity. Using bounds on the normalization constant, we have shown that the
activity can be rescaled to a stable activity. The polymer system is associated with a one-
dimensional renewal process. Our proof of symmetry breaking relies on a condition on the
interarrival distribution of the renewal process. Using information on the radius dependence
of the activities, we have shown that for thin cylinders, the associated renewal process has
finite mean. What happens for broad cylinders remains open.
A torus can be considered as a cylinder with periodic boundary conditions. In this sense,
cylinder and torus wave functions differ merely by boundary conditions. It is commonly ac-
cepted that Laughlin’s wave function, for small values of p, describes an incompressible liquid.
Thus intuitively, boundary conditions should not affect the bulk behavior and we expect that
the torus and cylinder functions are equivalent in the limit of long cylinders. We have defined
a class of solvable models, using functions of compact support. Under certain conditions on
the supports of the functions (next nearest neghbors overlap), the torus and cylinder func-
tions are related to monomer-dimer systems and we have shown their equivalence in the limit
of long cylinders. We believe that the solvable model gives a hint on the mechanism behind
the equivalence of the cylinder and Laughlin-type wave functions and that the method of
proof may be extended to the general case.
In view of the plasma analogy, our results can be interpreted in terms of jellium tubes.
Semiperiodic Coulomb interactions are invariant with respect to any shift. But the limiting
state of jellium tubes cannot be completely translationally invariant, since any period must
be a multiple of l2/R. Thus there is symmetry breaking on jellium tubes, for all radii. If
we view jellium tubes as quasi one-dimensional systems and invoke known results on the
one-dimensional one-component plasma, we expect a periodicity L/N =pl2/R. From the
plasma point of view, the smaller period l2/R is surprising, and it would be interesting to
have a physical interpretation.
In the limit of thin strips, we recover the expressions of the excess free energies and one-
particle densities for one-dimensional jellium systems. The free energy at large radii should
approach the fully two-dimensional free energy. This leads to a conjecture consistent with
121
122 CONCLUSION
bounds on the normalization constants that we have derived.
Thus the tube interpolates nicely between one- and fully two-dimensional systems. By
what is known on those systems, it follows that it interpolates between a system with no
phase transition and a system with a phase transition. This observation might be relevant
for large strips; in particular, the magnitude of pshould play a role.
It is widely accepted that the ground state of FQHE system is a gapped, incompressible
liquid at FQHE fractions. It is believed that the truncated interactions reproduce this fea-
ture, at least for small p. We have shown that if the truncated interaction have a gapped
ground state, then at fractional filling there must be some kind of symmetry breaking. This
leaves open the question whether they do indeed have a gapped ground state, and of course
says nothing about “real” FQHE systems.
Finally, our results are consistent with heuristic arguments adapting Laughlin’s IQHE
argument on the cylinder to the fractional effect. In systems without background interac-
tion, the Hall conductance is directly related to the one-particle density; we have given one
derivation of this common wisdom using the Chern number approach. On cylinders with
spectral boundary conditions, we have obtained a similar result. The periodicity and the
correct average value of the density imply fractional quantization of the Hall conductance.
However, we have used a number of assumptions that are rather unsatisfying, although they
are commonly used. It would be worthwhile to obtain better results on the significance of
cylinder ground state degeneracy for the FQHE. We hope that our detailed presentation will
be useful for future investigations in this direction.
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