Laughlin's wave function, plasma analogies and the fractional quantum Hall effect on infinite

136 Pages·2007·1.06 MB·English

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
GegenstanddieserDissertationistdieUntersuchungvonLaughlinsWellenfunktionaufeinem
Zylinder. Die Ergebnisse sind sowohl fu¨r die Theorie des fraktionalen Quanten-Hall-Effekts
als auch fu¨r 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-
endlichgehen. DerGrenzzustandistinvariantundmischendbzgl. bestimmterTranslationen
l¨angs der Zylinderachse. Bei Fu¨llfaktor 1/p ist die Periode gerade das p-fache der Periode
des gefu¨llten Landaubandes. Fu¨r hinreichend du¨nne 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.
WirgehenaufdieBedeutungunsererErgebnisseimSinnevonLaughlin’sPlasmaanalogie
ein. Wirzeigen,dassimLimesdu¨nnerZylinderunsereErgebnissebzgl. derNormierungskon-
stanten bzw. freier Energien und bzgl. der Einteilchendichten konsistent mit bekannten
Ergebnissenu¨bereindimensionalePlasmasystemesind. Fernerzeigenwir,dassunsereSchran-
ken u¨ber Normierungskonstanten gut zu Asymptotiken der freien Energien breiter halbperi-
odischer Streifen passen.
Laughlins Funktion bei Fu¨llfaktor 1/3 ist der exakte Grundzustand eines geeigneten
Hamiltonoperators. Wirweisennach, dassGrundzusta¨nde diesesHamiltonoperatorsmitun-
endlichvielenTeilchenbeiFu¨llfaktor1/3notwendigerweiseeineSymmetriebrechungaufweisen,
wenn es eine Spektrallu¨cke u¨ber dem Grundzustand gibt.
Schließlich betrachten wir ein einfaches Modell fu¨r 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. Weshowthatiftherenewalprocesshasfinitemean,Laughlin’sstateonthecylinder
has a well-defined thermodynamic limit as the cylinder gets infinitely long, the radius being
keptfixed. Thelimitingstateisinvariantandmixingwithrespecttoshiftsalongtheaxis. At
filling factor 1/p, the period is p times 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 I flows through a thin sheet of conducting material in the
H
form of a bar, subject to a magnetic field perpendicular to the sheet, a potential difference
V between opposite sides of the bar appears. According to classical electrodynamics, the
H
Hallconductanceσ =I /V isalinearfunctionoftheration/B ofthearealchargecarrier
H H H
density and magnetic field strength.
HundredyearsafterthediscoveryoftheHalleffect,K.vonKlitzingandhiscollaborators
[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
theunderstandingofthefractionaleffect. Consequently,theFQHEledtoanintenseresearch
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
quantumfluid”whosebulkpropertiesarewelldescribedbyLaughlin’swavefunction[Lau83].
1
2 INTRODUCTION
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 forCoulomb systems
has been proved awhile ago [LL72], analytical results pertaining tocorrelation 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.
ThepresentworkisdevotedtothestudyofLaughlin’swavefunctiononacylinder. 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 Ψ of
N
N complex variables z =x +iy defined through
j j j
ΨN(z1,..,zN):= (ezj/R ezk/R)pe−21PNj=1x2j,
−
1 j<k N
≤Y≤
where R is a positive number and p an odd integer. The function Ψ has the period 2πR in
N
each y . Thus we may consider that it describes a gas of N electrons moving on a cylinder
j
of radius R. The position of each particle is specified by a complex number z =x+iy; x is
the coordinate along the cylinder axis, y is an angular variable. Ψ is Laughlin’s cylinder
N
wave function. The density of the gas at position z is
N
ρ (z)= Ψ (z,x +iy ,..,x +iy )2dx dy ..dx dy .
N ||ΨN||2L2 Z(R×[0,2πR])N−1| N 2 2 N N | 2 2 N N
Onecanshowthatthedensityisafunctionofthecoordinatexaloneanddecaysexponentially
outside [0,pN/R]. Thus Laughlin’s cylinder function describes a gas of electrons living on a
finitecylinder. Thequantityν =1/pisrelatedtotheaveragedensityandiscalledthefilling
factor. At filling factor 1/p = 1 (referred to as the filled Landau level), the density can be
computedexplicitlyandisasumofequallyweightedGaussians centeredinintegermultiples
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-
tionalsymmetry breaking, as conjecturedforLaughlin’s cylinderstate. This ratherheuristic
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