University of Amsterdam - UvA · University of Amsterdam 1Utrecht University, Institute for Theoretical Physics 2Utrecht University, Mathematical Institute 3University of Amsterdam,

University of Amsterdam

MSc Physics

Track: Theoretical Physics

Master Thesis

The Quantum Hall Effects

Casimir Operators and Anyonic Quasi-Particle Excitations

by

I.A. Niesen

0312681

28 February 201460 ECTS

Research conducted between February 2013 and January 2014

Supervisors:Prof.dr. C. Morais Smith1

Dr. A.G. Henriques2

Official Supervisor:Prof.dr. C.J.M. Schoutens3

Institute for Theoretical PhysicsUniversity of Amsterdam

1Utrecht University, Institute for Theoretical Physics2Utrecht University, Mathematical Institute3University of Amsterdam, Institute for Theoretical Physics

Abstract

After an extensive mathematical treatment of the system comprised of a single charged particle in auniform perpendicular magnetic field, we provide an introduction to the integer and the fractionalquantum Hall effect. The quantization of kinetic energy in terms of Landau levels is discussed,followed by an explanation of the quantization of the Hall resistance at integer filling factor, Laugh-lin’s variational approach to the fractional quantum Hall effect, and a short discussion on compositefermion theory. We then focus on the fractional quantum Hall effect, where all low energy excita-tions lie within the fractionally filled Landau level. Because the kinetic energy is constant withineach Landau level, the effective dynamics are described by the density operators projected to thefractionally filled Landau level. Gaining insight in the projected density operators is therefore ofprime importance for understanding the fractional quantum Hall effect.

We derive the algebra that the projected density operators satisfy, and, in order to gain a betterunderstanding of this algebra of projected density operators, we compute several Casimir operatorsof the algebra: first on the infinite plane, which results in a Casimir that is rife with infinities due tothe infinite degeneracy of the Landau levels, and then on the torus, where we find explicit expressionsfor the Casimir operators of order two, three and four, generalizing Haldane’s expression [10] for thesecond order Casimir. Additionally, we recognize a pattern in the expressions found for the Casimiroperators that can be generalized to order NB , which is equal to the degeneracy of each Landaulevel.

Having computed the Casimir operators, we proceed to a related topic of interest: that ofthe low energy quasi-particle excitations of the fractional quantum Hall fluid. We show that thequasi-particles are anyonic in nature. Because the quasi-particles carry (fractional) charge, weexpect that they feel the external magnetic field that causes the quantum Hall effect. Motivated bythis realization, we construct an effective model that takes anyons as fundamental particles, ratherthan quasi-particle excitations, and we produce a method for generating exact N -particle anyonicenergy eigenfunctions for non-interacting charged anyons in a uniform perpendicular magnetic field.Comparing our method to other known methods for constructing exact anyonic wavefunctions, weconclude that, despite being less general, our method is more thorough and finds additional exactanyonic eigenfunctions that have not been constructed before.

Contents

1 Introduction 3

2 Free Electrons in a Magnetic Field 72.1 The Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Resistance and Resistivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.2 Cyclotron Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.3 The Classical Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.4 The Quantum Hall Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 A Quantum Mechanical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.1 The Lagrangian and Hamiltonian Formalisms . . . . . . . . . . . . . . . . . . 122.2.2 Gauge Fields in Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . . 132.2.3 Canonical Quantization and Gauge Invariance . . . . . . . . . . . . . . . . . 142.2.4 Construction of the One-Particle Hilbert Space . . . . . . . . . . . . . . . . . 172.2.5 The Vector Potential as a Connection 1-Form . . . . . . . . . . . . . . . . . . 19

3 The Quantum Hall Effects 243.1 The Integer Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1.1 Landau Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.1.2 Energy Level Degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.1.3 Mathematical Intermezzo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.1.4 The Hall Resistivity at Integer Filling Factor . . . . . . . . . . . . . . . . . . 323.1.5 Quantization of the Hall Resistance . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 The Fractional Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2.1 The Ground State Wavefunction at ν = 1 . . . . . . . . . . . . . . . . . . . . 383.2.2 Laughlin’s Plasma Analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.2.3 Laughlin’s Ground State Wavefunction . . . . . . . . . . . . . . . . . . . . . . 423.2.4 Elementary Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.2.5 Jain’s Composite Fermion Theory . . . . . . . . . . . . . . . . . . . . . . . . 46

4 The FQHE on the Infinite Plane 484.1 The Projected Density Operator Algebra . . . . . . . . . . . . . . . . . . . . . . . . 48

4.1.1 Second Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.1.2 Projection onto the Fractionally Filled Landau Level . . . . . . . . . . . . . . 504.1.3 Computation of the Form Factor . . . . . . . . . . . . . . . . . . . . . . . . . 524.1.4 Derivation of the Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.2 Casimir Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.2.1 Construction of the Casimir Operators . . . . . . . . . . . . . . . . . . . . . . 564.2.2 Computation of the Second Order Casimir . . . . . . . . . . . . . . . . . . . . 604.2.3 Physical Interpretation of the Second Order Casimir . . . . . . . . . . . . . . 62

1

CONTENTS 2

5 The FQHE on the Torus 655.1 Periodic Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.1.1 Construction of the One-Particle Hilbert Space . . . . . . . . . . . . . . . . . 655.1.2 The Projected Density Operators . . . . . . . . . . . . . . . . . . . . . . . . . 695.1.3 An NB-Dimensional Representation of the Magnetic Translation Algebra . . 725.1.4 The FQHE on the Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.2 Casimir Operators on the Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.2.1 Construction of the Rth Order Casimir . . . . . . . . . . . . . . . . . . . . . 765.2.2 Computation of the Rth Order Casimir . . . . . . . . . . . . . . . . . . . . . 785.2.3 Physical Interpretation of the Casimir Operators . . . . . . . . . . . . . . . . 81

6 Charged Anyons in a Magnetic Field 896.1 Introduction to Anyons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.1.1 Anyons in the Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . . 906.1.2 The Configuration Space of Identical Particles . . . . . . . . . . . . . . . . . 926.1.3 The Universal Cover of the Configuration Space . . . . . . . . . . . . . . . . 946.1.4 Anyon Wavefunctions and the Braid Group . . . . . . . . . . . . . . . . . . . 966.1.5 The Statistical Term in the Wavefunction . . . . . . . . . . . . . . . . . . . . 97

6.2 Construction of the Anyon Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . 996.2.1 The Wavefunction Ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.2.2 The Time Independent Schrodinger Equation . . . . . . . . . . . . . . . . . . 1026.2.3 A Polynomial Ansatz for the Wavefunction . . . . . . . . . . . . . . . . . . . 1036.2.4 Solutions to the Schrodinger Equation . . . . . . . . . . . . . . . . . . . . . . 1066.2.5 The Normalizability and Symmetry Conditions . . . . . . . . . . . . . . . . . 1096.2.6 Measuring Anyons: Angular Momentum . . . . . . . . . . . . . . . . . . . . . 1126.2.7 Explicit Formulas for Two-Particle Anyon Eigenfunctions . . . . . . . . . . . 115

7 Conclusion and Outlook 122

A The Quantum Hall Effects: Addendum 125A.1 Overview of the Single Particle Operators . . . . . . . . . . . . . . . . . . . . . . . . 125A.2 Fourier Transform of the Coulomb Potential in 2D . . . . . . . . . . . . . . . . . . . 127

B Anyon Wavefunctions: Addendum 128B.1 Solution to the Recurrence Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . 128B.2 Normalizability Condition for N=2 Particles . . . . . . . . . . . . . . . . . . . . . . . 142

Bibliography 147

Acknowledgements 150

Chapter 1

Introduction

Consider a gas of spinless electrons confined to the x-y plane, subject to a uniform perpendicularmagnetic field B = Bez. Contrary to ordinary electrical systems, a steady current I corresponds toa transverse Hall voltage difference VH . The ratio of VH and I is, by definition, the Hall resistance,

RH =VH

I=

B

−enel, (1.1)

where nel is the two-dimensional electron density, and −e is the electron charge. In particular, theHall resistance RH is perpendicular to the magnetic field strength B.

If there is no electric field present, the electrons move in circles, called cyclotron orbits. Thefrequency of such an orbit is given by the cyclotron frequency, ωC := eB/m, where m is the massof the electron. Because the frequency is constant, the velocity and the radius are proportional toeach other, and the latter is of the order of the magnetic length lB :=

√~/eB, which is the typical

length scale of the system.Due to the magnetic field, the kinetic energy of the electrons becomes quantized, forming Landau

levels: equidistant energy levels in steps of ~ωC . This quantization becomes apparent when thesystem is cooled to temperatures T for which the thermal energy kBT is much smaller than thekinetic energy ~ωC . Moreover, the Hall resistance, instead of being proportional to the magneticfield B as shown in equation (1.1), shows plateaus around values of the magnetic field for which theHall resistance equals

RH =h

e21ν,

where ν is the filling factor, which is defined as the number of electrons divided by the degeneracy of asingle Landau level (which depends on B). There are actually two underlying phenomena that causethe Hall resistance to form plateaus: the plateaus formed at integer filling factor are a consequenceof the integer quantum Hall effect (IQHE), and the plateaus occurring at fractional filling factor areexplained by the fractional quantum Hall effect (FQHE). The IQHE can be understood in terms ofnon-interacting particles, whereas the FQHE requires that electron-electron interactions be takeninto account.

There are several descriptions of the FQHE. The oldest is due to Laughlin [14], which explainsthe FQHE at so called Laughlin filling factors ν = 1

p with p an odd integer. Laughlin provided avariational wavefunction for the ground state based on symmetry properties, his knowledge of thenon-interacting ground state, and physical intuition; as well as an explanation of the fractionallycharged quasi-particle excitations that obey fractional statistics. Continuing in the same line ofreasoning, it is possible to view the FQHE states at filling factor p

2ps+1 , with p, s ∈ N, as integerquantum states of composite particles. These composite particles go by the name composite fermions(due to Jain [11]).

3

4

In addition to the construction of multi-particle ground state wavefunctions a la Laughlin, thereare also field theoretical approaches to the quantum Hall effects, such as the extended Hamiltonianformalism by Murthy and Shankar [19], or the Chern-Simons theory approach to composite fermionsby Lopez and Fradkin [18], which puts Jain’s composite fermion theory on a firm field-theoreticalfooting by describing the attachment of flux to electrons as a coupling between the electrons and aChern-Simons gauge field.

The discovery of the quantum Hall effects had a great impact on condensed matter physics. Forexample, the quantization of the Hall resistance is so precise, that nowadays it provides the mostaccurate way to measure the fine structure constant. Unrelated to the previous statement, but no lessinfluential, is the fact that quantum Hall effects provided the first example of a state of matter thatcannot be understood in terms of spontaneous symmetry breaking, because, contrary to ordinarymatter which condenses into crystal-like solids when the temperature is lowered, no symmetry islost when the temperature of the FQHE system is decreased. Instead of positional order that occursin crystals, the order in the FQHE fluid, known as topological order (see Wen [24]), refers to thesystematic way in which the electrons move around: the electrons are localized in the bulk, but netcharge transport does occur at the edges of the sample, where, irrespective of the geometry of thesample, each edge states contributes one quantum of conductance e2/h to the total conductance.The reason that the quantum Hall fluid does not condense into a crystal when the temperature islowered is that the mass of the constituents, the electrons, is too small, and quantum fluctuationsprevent the FQHE fluid from becoming a solid. Additionally, the quantum Hall effects are hometo a number of fascinating collective phenomena, such as fractionally charged particles, fractionalstatistics and even non-abelian statistics.

This thesis consists of two distinct but related parts:

• the first part focuses on Casimir operators of the algebra of projected density operators in theFQHE,

• the second part discusses the anyonic quasi-particle excitations of the quantum Hall fluid atfractional (Laughlin) filling factor.

In the low temperature limit kBT ~ωC , the Landau levels are filled from bottom to top.This means that, at fractional filling, the highest non-empty Landau level is fractionally filled.Because kBT ~ωC , all effective dynamics occur within this fractionally filled Landau level, wherethe kinetic energy is constant. Hence, the low energy excitations are governed by the electron-electron interactions, which can be completely described in terms of the density operators projectedto the fractionally filled Landau level. The so obtained project density operators form an algebra,first discovered by Girvin, MacDonald and Platzman [5], which is therefore also known as theGMP algebra. Understanding the algebra of projected density operators is of prime importance forunderstanding the FQHE fluid at low temperatures.

The aim of the first part of this thesis is to obtain a better understanding of the algebra ofprojected density operators, by computing several Casimir operators of the algebra. These arespecific operators1 that commute with all the elements of the algebra. Common examples of Casimiroperators are the total spin and total angular momentum operator, which are physically significantoperators, and our initial conjecture was that so too are the Casimir operators of the projecteddensity operator algebra.

Haldane [10] computed the second order Casimir of the projected density operator algebra. Wewill show how to generalize Haldane’s result up to fourth order and provide explicit expressionsfor the Casimir operators in terms of sums of products of permutation operators. The computedCasimir operators turn out to be constant on fixed-particle-number subspaces of Fock space, on

1In a general Lie algebra g with a countable basis Xi, i ∈ I, the second order Casimir ζ is the element of theuniversal enveloping algebra U(g) of g given by ζ =

Pi∈I XiX

i, where Xi is the dual vector of Xi with respect to afixed invariant bilinear form. There are additional Casimir elements for each order.

5

which each Casimir operator becomes a polynomial in the particle number. In these polynomialexpressions, a clear pattern appears that can be generalized up the the N th

B order Casimir, whereNB is the degeneracy of each Landau level.

After obtaining the expressions for the Casimirs, we will proceed with the second part of thisthesis, in which we attempt to give an exact description of the quasi-particle excitations of theFQHE fluid at the Laughlin fractions ν = 1

p , where p is an odd integer. It has been experimentallyverified (see Goldman and Su [9], and M. Heiblum et. al. [21]) that the quasi-particle excitationscarry fractional charge, and it is theoretically predicted that they obey anyonic statistics. Becausethe quasi-particles are charged, it is expected (see for example Khare [13], Stern [22], Johnson andCanright [12] or Lerda [17]) that the quasi-particles themselves feel the external magnetic field thatcauses the FQHE.

We will construct exact energy eigenfunctions for non-interacting charged anyons in a uniformperpendicular magnetic field. Our approach differs from Laughlin’s in that we take the anyon tobe a fundamental particle, instead of a collective excitation. This means that we are providing aneffective description of the system of quasi-particle excitation of the FQHE fluid.

Wilczek ([25] and [26]) introduced anyons as flux tube-charged particle composites. Wilczek’sdescription allows for two equivalent ways of describing anyons. One sees the anyon as a bosoniccharge to which a flux tube is attached by coupling the charge to a Chern-Simons gauge field (seee.g. Dunne [2]). The other involves a singular gauge transformation, which gauges away the vectorpotential due to the flux tube and transforms the flux tube-charged particle composites into non-interacing charged particles that obey anyonic exchange statistics. It is this second description thatis closest to the formalism implemented in this thesis: we will construct wavefunctions that obeyanyonic exchange statistics and are defined on the universal cover of the configuration space of Nidentical particles as introduced by Leinaas and Myrheim [16].

Motivated by the single-particle energy eigenstates for a charged particle in a uniform perpen-dicular magnetic field, we come up with a general ansatz, and develop a theory with which wecan construct exact N -particle energy eigenfunctions for non-interacting charged anyons in a uni-form perpendicular magnetic field. The energy spectrum of the found eigenstates is the same asthat of N non-interacting charged fermions in a uniform perpendicular magnetic field, given byEN,K = 1

2~ωC(N + 2K), where K ∈ Z≥0 labels the excitation energy that the system has on topof its ground state energy 1

2N~ωC . Observing that the energy eigenvalues do not explicitly dependon the statistical parameter α that determines the phase eiαπ picked up by the wavefunction underthe exchange of two identical anyons, we further diagonalize the energy eigenstates with respect tothe (z-component of the) total angular momentum operator, the spectrum of which does explicitlydepend on α. We conclude that the total angular momentum operator could in principle lead to amethod for directly measuring the anyonic statistical parameter α.

The main advantage of our approach is that it provides exact anyonic energy eigenfunctions.The disadvantage, as shown by comparing to Khare [13], Johnson and Canright [12] or Lerda [17],is that we are not able to generate a complete set of eigenstates2. The latter reference of the threeprovides the most extensive list of anyonic energy eigenstates, which were obtained by a methodthat is similar to ours, but with the use of a more general ansatz: Lerda considers what he callstype I and type II wavefunctions, the former of which coincide with the wavefunctions that we find.However, as discussed in the conclusion, we find additional wavefunctions of type I that Lerda doesnot produce. We are therefore led to conclude that, despite the fact that it is less general, ourapproach is more thorough than Lerda’s. This brings us to the further conclusion that it might beviable to repeat our approach for Lerda’s type II wavefunctions.

2As of today, no complete exact solution to the N non-interacting charged anyon problem in a uniform perpendicularmagnetic field is known, see Khare [13].

6

The outline of this thesis is as follows. Chapter 2 provides an introduction to the Hall effect, followedby a mathematically rigorous description of a single charged particle confined to two dimensions ina uniform perpendicular magnetic field. Chapter 3 gives an introduction to both the IQHE and theFQHE. In Chapters 4 and 5 the Casimir operators in the FQHE are discussed: first on the infiniteplane, and then on the torus. Chapter 6 is about anyons in the FQHE, and the construction ofexact anyonic energy eigenfunctions. Finally, in Chapter 7 we present our conclusions and providean outlook for possible further research.

Chapter 2

Free Electrons in a Magnetic Field

In order to get acquainted with the Quantum Hall Effects, it is necessary to obtain a firm under-standing of a simple toy model — that of a single electron moving freely in a constant perpendicularmagnetic field — which, despite its simplicity, exhibits most of the key features underlying thequantum Hall effects.

In the first part of this chapter, we will give a classical description of a free electron in a magneticfield. In the second part, we will set the stage for the remainder of this thesis by describing thesame system from a quantum mechanical perspective, and in doing so, we will keep a high level ofmathematical rigour.

2.1 The Hall Effect

Consider a gas of non-interacting electrons confined to an impurity free two-dimensional plane, whichwe take to be the x-y plane. Suppose we add a uniform magnetic field B = Bez that points in the+z direction. It turns out that this setup leads to very peculiar behavior of the electrons: especiallywhen a current is applied to the the 2D surface. Even though the electrons do not interact withanything, they still experience some kind of resistance; i.e. a tendency of the material (in this casemeaning the conduction plate together with the magnetic field) to stop the current from flowing.This resistance is called the Hall resistance. It is different from normal electrical resistance in thesense that the corresponding voltage difference is perpendicular to the direction of the current.

2.1.1 Resistance and Resistivity

In ordinary conducting materials, a given voltage difference V generates a current I, and both arerelated by Ohm’s law:

V = IR. (2.1)

The resistance R signifies to which extend the conducting material prevents the current from flowing.Ohm’s law is only applicable to conducting materials of a particular geometry wherein the current

is restricted to flow in one specific direction (e.g. wires), in which case I measures the amount ofcharge passing through a fixed area per unit of time. More generally, we can speak of a currentdensity J , which, in case of a conducting wire is the current per unit area I/A. The current densityalso makes sense in materials where the charge can flow in more than one direction.

When there are many electrons that contribute to the total charge flow, and we are looking atdistance scales on which the electron distribution can be approximated by a thee dimensional densityfunction nel(r, t), the current density J due to electrons moving around is given by

J = −e nel(r, t) v(r, t), (2.2)

7

2.1. THE HALL EFFECT 8

i.e. the product of the electron charge −e, the electron density nel(r, t), and the electron velocityvector field v(r, t), which are functions of the position r and the time t. In this more general setup,Ohm’s law takes the form:

E = ρJ . (2.3)

ρ is called the resistivity of the material. Contrary to resistance, resistivity is, generally speaking, alocal property of the material. In the case of a conducting wire of length l, cross sectional area Aand uniform resistivity ρ subject to a constant longitudinal electric field with field strength E, wehave V = El. Combining (2.1) and (2.3) yields,

R = ρl

A. (2.4)

The above formula intuitively makes sense: decreasing the cross-sectional area or increasing thelength of the wire should indeed increase the resistance.

2.1.2 Cyclotron Motion

Let us focus on the situation of a single electron confined to move in the x-y plane under the influenceof a perpendicular magnetic field B = Bez, without any external electric field being present. Theelectron experiences a magnetic Lorentz force Fm given by

Fm = −ev ×B. (2.5)

The force Fm also lies in the x-y plane, and it is perpendicular to v. It has a constant magnitude ofFm = e v B. As a consequence, any electron with a non-zero velocity will display a circular motion,called a cyclotron orbit. We will adopt the following coordinates for such an orbit: r denotes theposition of the electron, R the centre of the cyclotron orbit, and η is the vector that points from Rto r – see Figure 2.1.

Figure 2.1: Electron at position r making a cyclotron orbit. The magnetic fieldB points out of the page.

In any circular motion of radius η with constant speed v, the acceleration towards the centre isgiven by a = v2

η . By equating ma = Fm, we get

v =eB

mη = ωCη, (2.6)

whereωC :=

eB

m(2.7)


is called the cyclotron frequency. It is the angular frequency at which the electron preforms cyclotronorbits. Contrary to for example planetary orbits, the frequency of a cyclotron orbit is independent ofthe radius and the velocity of the orbiting particle: it only depends on the strength of the magneticfield.

From equation (2.6), we see that the radius and the speed are proportional to each other. Hence,there are two dynamical variables that describe the cyclotron orbit completely: the speed v at whichthe orbit is performed — or equivalently, the radius η — and the position R of the centre of thecyclotron orbit.

2.1.3 The Classical Hall Effect

In addition to the magnetic field B = Bez, let us turn on an external electric field E = Exex +Eyey

pointing in the x-y plane. The equation of motion for a single electron is

p = F = −eE − ev ×B. (2.8)

That is, (px

py

)= −e

(Ex

Ey

)+ ωc

(0 −11 0

)(px

py

), (2.9)

where we have left out the z-component because the electron is confined to the x-y plane. Takingthe time derivative of the above equation yields(

px

py

)= ωc

(0 −11 0

)(px

py

),

which is solved by (px

py

)= R

(cos(ωct)sin(ωct)

), (2.10)

where R is an integration constant. Plugging equation (2.10) into equation (2.9) and solving for pgives the general solution to equation (2.9):(

px

py

)=

R

ωc

(sin(ωct)−cos(ωct)

)+

e

ωc

(Ey

−Ex

)=: pc + pH .

From the solution above, we see that the motion of the electron is a superposition of a circularcyclotron motion (pc) and a linear motion (pH = mvH), where vH is the Hall drift velocity,

vH :=e

mωc

(Ey

−Ex

)=

E ×B

B2. (2.11)

The linear term pH is responsible for any net electrical current in the material: it specifies thedirection in which the charge carriers — in this case electrons — move.

Remark 2.1. There are two important things to note. First, contrary to ordinary electrical systems,the current flows in, or opposite to, the direction of vH , which is perpendicular to the direction of E.This phenomenon is called the Hall drift. It is the above mechanism that relates an electric currentto a perpendicular voltage difference, which in turn gives rise to the Hall resistance.

Second, the Hall drift velocity only depends on the electric and magnetic fields, and not on themass or charge of the charge carriers. In particular, this means that if the charge carriers wouldhave been positively charged, the current J = +enelvH would have flowed in the opposite direction.


Another way of thinking about the Hall resistance is the following. Suppose we have a currentI in the +x direction (meaning that the electrons, which have a negative charge, move in the −xdirection). The magnetic Lorentz force acting on the electrons points in the −y direction: it triesto stop the electrons from moving in a straight line. This is the tendency of the system to resistthe current. To ensure that the current is maintained, we need to apply an electric field in the −ydirection (perpendicular to I) that counters the magnetic Lorentz force.

In more mathematical language; we are looking for steady currents, i.e. currents for which theelectrons move at a constant velocity. That is to say that we are looking for stationary (p = 0)solutions to equation (2.8), i.e. 0 = −eE − ev×B. These are given by v = vH , or, in terms of thecurrent density (2.2), (

Jx

Jy

)= −enelvH =

(0 − e nel

Be nel

B 0

)(Ex

Ey

). (2.12)

Comparing to equation (2.3), we see that the matrix in equation (2.12) is the inverse of the resistivity,also called the conductivity. Its inverse ρ is given by

ρ = ρH

(0 1−1 0

), (2.13)

whereρH :=

B

enel(2.14)

is the Hall resistivity. The fact that the resistivity matrix is off-diagonal reflects the fact that thecurrents are induced by transverse electric fields. The ordinary resistivity terms will appear on thediagonal of ρ.

For example, if we alter equation (2.9) such that it incorporates interactions of the electrons withimpurities in the sample – e.g. by adding a term −p/τ which causes p to decrease over time,(

px

py

)= −e

(Ex

Ey

)+ ωc

(0 −11 0

)(px

py

)− 1τ

(px

py

), (2.15)

then the steady currents will be given by(Ex

Ey

)=(ρL ρH

−ρH ρL

)(Jx

Jy

),

where ρL = mnel e2 τ is the longitudinal resistivity. This is known as the Drude model.

The Hall effect was first measured by Edwin Herbert Hall in 1879. When passing a current Ithrough a thin metallic plate of thickness lz, he measured a transverse voltage difference given by

VH = −I ρH

lz, (2.16)

corresponding to a Hall resistance of

RH = −ρH

lz=

B

−e nel lz, (2.17)

where nel now refers to the average electron density. The minus sign is due to the fact that thatthe charge carriers are negatively charged. Indeed, if the charge carriers would have been positivelycharged, then, due to remark 2.1, vH would have been the same, but the sign in equation (2.12)would have changed. Hence, we conclude that for charge carriers of charge Q inside a conductor ofthickness lz, the measured Hall resistance is

RH =B

Qnel lz. (2.18)

As a matter of fact, one of the applications of the Hall effect is to determine the sign of the chargeof the charge carriers by determining the sign of the Hall resistance.


Remark 2.2. If nel is independent of the z direction, and the length lz is constant and small enoughfor us to be able to treat the system as being effectively two-dimensional, then nellz is equal to thetwo-dimensional electron density n2D

el , and the Hall resistance (2.17) becomes

RH =B

−e n2Del

,

which is in agreement with formula (1.1) in the Introduction.

The measured value of the Hall resistance found in (2.17) can be understood in terms of the Hallresistivity as follows. Let us assume that the conducting plate has dimensions length lx times widthly times thickness lz. Inject a uniform current I = Ixex = Jx ly lzex through the plate. Accordingto equation (2.12), this current comes with a transverse electric field E = Eyey, where

Ey = −ρHJx.

The Hall voltage difference VH is related to the electric field E = Eyey by VH = Ey ly. Pluggingthis into the equation above yields

VH = −IxρH

lz,

which is in agreement with equation (2.16).

Remark 2.3. The Hall resistance is independent of both the length lx and the width ly of the sample:only the thickness lz plays a role. This is a feature that holds for any kind of sample, not only forrectangular ones. As a matter of fact, it has been experimentally verified that the measured Hallresistance is independent of the shape of the sample as seen from the +z direction.

2.1.4 The Quantum Hall Effects

In case of the Classical Hall Effect, the Hall resistance (2.17) is proportional to the magnetic field,

RH ∝ B

enel. (2.19)

However, in 1980, Klaus von Klitzing measured a Hall resistance curve in a certain MOSFET(metal-oxide-semiconductor field-effect transistor) that is not linear in the magnetic field, but insteadhas a stair-case like shape as shown in Figure 2.2. This phenomenon, where the Hall resistancecurve remains constant for certain values of B, is commonly referred to as quantization of the Hallresistance.

It turns out that there are actually two different processes that cause the formation of plateausin the Hall resistance curve. For values of B for which the Hall resistivity equals

ρH =h

e21ν, (2.20)

where ν is a positive integer, the underlying phenomenon goes by the name integer quantum Halleffect, or IQHE for short. The discovery of the IQHE is attributed to Klaus von Klitzing, for whichhe was awarded the Nobel Prize in 1985. We will discuss the IQHE in Section 3.1.

There is also the fractional quantum Hall effect (FQHE), corresponding to plateaus in the Hallresistance at resistivities for which ν in (2.20) is fractional. Understanding the FQHE requires anunderstanding of the IQHE, so we will postpone any further discussion on the FQHE until we havetreated the IQHE. See Section 3.2 for the details.

However, before we can start discussing either the IQHE or the FQHE, we need to understand how2D electrons in a perpendicular magnetic field can be described within the framework of quantummechanics. This requires the setup of the appropriate mathematical formalism, which is the topicof the next section.

2.2. A QUANTUM MECHANICAL APPROACH 12

Figure 2.2: The quantum Hall effects: longitudinal (R) and Hall (RH) resistancein units of the von Klitzing constant h/e2 as a function of the magnetic fieldstrength in Tesla. The image shows that the Hall resistance goes up in steps,instead of linearly as predicted by the classical Hall effect, while the longitudinalresistance goes to zero at each step.

2.2 A Quantum Mechanical Approach

Consider a free electron with charge −e confined to move in the x-y plane, subject to a perpendic-ular magnetic field B = Bez. In order to describe this system quantum mechanically, we need aHamiltonian. We will take a small detour and set up the appropriate Lagrangian and Hamiltonianformalism for free electrons — in 3D — subject to a generic electric and a magnetic field, beforeturning to the 2D case.

2.2.1 The Lagrangian and Hamiltonian Formalisms

Suppose we have a charged particle with charge −e and mass m. The particle moves in configurationspace Q, which can be any submanifold of R3. Furthermore, we define the phase space TQ to bethe tangent bundle of Q. We will adopt local coordinates (r,v) on TQ, where vi is the i-componentof ∂

∂r . Physical observables are functions on the phase space; for example, we have the mechanicalmomentum of the particle, which is defined by

Π := mv. (2.21)

Let us denote the path the particle follows through TQ by (r(t), r(t)). The movement of the particlein question is dictated by Newton’s law:

F = mr, (2.22)

where, in the system that we are considering, the force F is the Lorentz force, given by,

FL = −e(E + r ×B). (2.23)

The above equation of motion can be derived from a action,

S[r(t)] =∫dtL(r(t), r(t)),


where L = L(r,v) is the Lagrangian, which we will specify below. Because (2.22) is a second orderdifferential equation, any point (r0,v0) in phase space together with a choice of a moment in timet0 determines a unique curve r(t) such that (r(t0), r(t0)) = (r0,v0), fixing the complete future andpast of the particle in question. Such a point (r0,v0) is called a classical state.

The action is a functional on the space of paths r(t) in Q. Fixing two endpoints r0 and r1, theaction principle says that the particle will go from r0 to r1 in such a way that its path extremizesthe action S. This is equivalent to saying that r(t) satisfies the Euler-Lagrange equations,

∂L

∂r

∣∣∣∣(r(t),r(t))

=d

dt

∂L

∂v

∣∣∣∣(r(t),r(t))

. (2.24)

Next, we deliberately choose a Lagrangian L such that equation (2.24) reduces to equation (2.22)with F = FL. A Lagrangian that fulfills this condition is

L(r,v) =12mv2 − eA(r) · v + eV (r),

where V and A are the electric and magnetic vector potential — which we will elaborate on in thenext section — respectively, and v2 = v ·v. The dot refers to the standard Euclidean metric on TQ.Because the system that we want to describe is of a quantum mechanical nature, we want to work inthe Hamiltonian formalism. The Hamiltonian can be obtained by preforming a fiberwise Legendretransformation to the Lagrangian in the variables v1, .., v3, making it a function on the cotangentbundle T ∗Q. Note that, for fixed r, L is quadratic (hence convex) in the velocity, so we can indeedperform a fiberwise Legendre transformation.

Having done the transformation, H is now a function of p instead of v, where p is given by:

p :=∂L

∂v= mv − eA(r). (2.25)

p is called the canonical momentum; it is not a physical quantity, because it depends on the gaugedependent vector potential A. The Hamiltonian is given by

H(r,p) =1

2m[p + eA(r)

]2 − eV (r). (2.26)

Before we proceed to canonical quantization, let us take some time to review the concept of gaugefields within the context of electromagnetism.

2.2.2 Gauge Fields in Electromagnetism

The electric and magnetic field are vector fields on R3. In the presence of a charge density ρ andcurrent density J , they satisfy Maxwell’s equations:

∇ ·E =ρ

ε0(2.27)

∇ ·B = 0 (2.28)

∇×E = −∂B

∂t(2.29)

∇×B = µ0J + µ0ε0∂E

∂t(2.30)

The first two equations are Gauss’s law for the electric and the magnetic field. The third equation isFaraday’s law of induction, and the last equation is Ampere’s circuital law with Maxwell’s correction.Loosely speaking, equations (2.27) and (2.30) relate the electric and magnetic field to their respective


sources: the charge density ρ for E and the current density J for B, while Equations (2.28) and(2.29) restrict the possible vector fields that we can take to be the electric or the magnetic field.

We now turn to the co-chain complex of differential forms on R3. Under the identification ofvector fields with 1 and 2-forms on R3, given by: ei 7→ dxi and ei 7→ 1

2εijkdxj ∧dxk, we can identify

the exterior derivative d with the gradient, rotation and divergence operators, as follows:

0 → Ω0(R3)d0=grad−−−−−→ Ω1(R3) d1=rot−−−−→ Ω2(R3) d2=div−−−−→ Ω3(R3) → 0

By de-Rham’s theorem, the k-th de-Rham cohomology group HkdR(R3) of R3 is isomorphic to the

k-th singular cohomology group Hk∆(R3) of R3;

HkdR(R3) ∼= Hk

∆(R3).

Also, by contractibility of R3, Hk∆(R3) can easily be computed to be equal to R when k = 0, and 0

otherwise. Thus, we conclude that

HkdR(R3) :=

Ker(dk)Im(dk−1)

=

R when k = 00 otherwise (2.31)

Now, equation (2.28) precisely states that B ∈ Ker(d2), which, by equation (2.31) equals Im(d1).Thus, there exists another vector field A ∈ Ω1(R3), such that

B = ∇×A. (2.32)

This vector field is called the magnetic vector potential, or vector potential for short. Additionally,from equation (2.29), we have E + ∂A/∂t ∈ Ker(d1), which, again by equation (2.31), is equal toIm(d0). Hence, there also exists a function V such that

E = −∇V − ∂A

∂t.

V is called the electric potential.Because the exterior derivatives d0 and d1 have a nonzero kernel, there is a choice in which A and

V to pick. Given two vector potentials A and A′ that satisfy ∇×A = ∇×A′ = B, the differenceA −A′ ∈ Ker(d1) = Im(d0), i.e. A′ = A + ∇χ for some function χ. Likewise, two different V ’ssatisfying ∇V = E + ∂A/∂t differ by a constant (since Ker(d0) = H0

dR(R3) = R).A particular choice of the electric and the magnetic vector potentials is called a gauge. The

fields A and V are called gauge fields. They are not physical fields in the sense that there is anarbitrariness to them, contrary to the electric and magnetic fields, which are completely determinedby their physical sources ρ and J . Thus, working with gauge fields is always a little bit tricky,because there are many gauge fields that correspond to the same physical situation, and we requirethat every measurable quantity is independent of the choice of the gauge.

Remark 2.4. When we restrict ourselves to two-dimensional systems, keep in mind that the electricand magnetic field should always be thought of as three-dimensional fields existing on all of R3

wherein the two-dimensional configuration space Q is embedded, because Maxwell’s equations do notmake sense in any dimension other than three.

2.2.3 Canonical Quantization and Gauge Invariance

When going from classical to quantum mechanics, the degrees of freedom r and p are promoted toself-adjoint operators on a — soon to be specified — Hilbert space H,

r r seen as a multiplication operatorp −i~∇


satisfying the commutation relations[ri, pj ] = i~δij

for i = 1, 2, 3, with all other commutators vanishing.Pure states are represented by vectors ψ ∈ H with unit norm. Their time dependence is governed

by the Schrodinger equation,

i~∂ψ

∂t= Hψ, (2.33)

where H is the Hamiltonian, which in our case is given by equation (2.26).For any operator O on H, the expectation value of O with respect to ψ is given by

〈ψ,Oψ〉,

where 〈, 〉 denotes the inner product on H. The expectation values are the measurable quantitiesthat can be obtained from a quantum mechanical system.

Back to the quantum Hall effect. Consider an electron with charge −e confined to a two-dimensional surface under the influence of a constant perpendicular magnetic field B = Bez. Fornow, we will set E = 0. Because of the gauge freedom in electrodynamics, a fixed physical statecan be described mathematically in many different ways, and we require that all these mathematicaldescriptions are equivalent, in the sense that the extractable physical information should be thesame for each description.

More concretely, after choosing a gauge, we obtain an isomorphism between the Hilbert space ofstates H and the space of square-integrable complex valued functions on R2;

H ∼= L2(R2,C), (2.34)

the mechanical momentum Π (2.21) becomes

Π = p + eA(r), (2.35)

and the Hamiltonian takes the form

H =Π2

2m=

12m

[p + eA(r)

]2. (2.36)

If we would have chosen a different gauge, then both the isomorphism in (2.34) and the explicitformulas for the mechanical momentum (2.35) and the Hamiltonian (2.36) would have been different.

Because we want different gauges to describe the same physics, we require that to each gaugetransformation

A 7→ A′ = A +∇χ

there corresponds a unitary operator U : H → H, such that

Π′ = p + eA(r) + e∇χ(r) = UΠU†, (2.37)

and, consequently

H ′ =1

2m

[p + eA(r) + e∇χ(r)

]2= UHU†. (2.38)

For, given such a unitary transformation U , we can go from the A gauge to the A′ gauge by applyingU to all the functions in L2(R2). Indeed, if ψ ∈ L2(R2) describes some fixed physical system in theA gauge, and ψ′ := Uψ, then, by unitarity of U ,

〈ψ,Πψ〉 = 〈ψ′,Π′ψ′〉.


As a matter of fact, for any physical operator O, which is always an analytic function of r and Π,we have O′ = UOU†, and therefore

〈ψ,Oψ〉 = 〈ψ′, O′ψ′〉.

Moreover, the time evolution of ψ and ψ′ is such that, even at later times, ψ(t) = exp(−itH/~)ψand ψ′(t) = exp(−itH ′/~)ψ′ are related by ψ′(t) = Uψ(t). We conclude that, for all times t,

〈ψ(t), Oψ(t)〉 = 〈ψ′(t), O′ψ′(t)〉,

which is another way of saying that O,ψ and O′, ψ′ describe the same physical system.

Lemma 2.1. The unitary transformation U that satisfies equations (2.37) and (2.38) is the multi-plication operator:

U(r) := exp(−ieχ(r)

~

)(2.39)

Proof. To see that (2.39) indeed solves equations (2.37) and (2.38), observe that we have the followingoperator equation, meaning that it holds whenever applied to an arbitrary wavefunction.

piU†(r) = −i~∂ie

ieχ(r)/~ = U†(r)[e(∂iχ)(r)− i~∂i

]= U†(r)

[e(∂iχ)(r) + pi

].

Therefore,

UΠU† = U

[p + eA(r)

]U† = e∇χ(r) + p + eA(r) = Π′, (2.40)

and

UHU† =1

2m

[U(p + eA(r))U†

]2=

12m

[p + eA(r) + e∇χ(r)

]2= H ′.

Example 2.1. Consider the two most frequently used gauges: the

symmetric gauge AS =B

2(−yex + xey) , (2.41)

and theLandau gauge AL = −Byex. (2.42)

The two gauges are related by the following equation,

AS = AL +∇χSL,

whereχSL(x, y) =

B

2xy.

According to equation (2.39), the unitary operator U : L2(R2,C) → L2(R2,C) that transformswavefunctions from the Landau gauge to the symmetric gauge is given by

USL(x, y) = exp

(−ieBxy

2~

). (2.43)

To summarize, we have found a unitary operator U : L2(R2,C) → L2(R2,C) that allows us togo from one gauge to the other. Because there are infinitely many vector potentials A that generatethe same magnetic field B = Bez = ∇×A, to a fixed physical state there corresponds an infiniteset of wavefunctions, each of which describes this particular physical state in a different gauge. Itseems more sensible to assign a single mathematical entity to this fixed physical state, instead of aninfinite set of wavefunctions. As it turns out, a physical state is naturally described by a section ofa complex line bundle L over R2, where a choice of the gauge corresponds to a global trivializationof the line bundle. The construction of this line bundle is the topic of the next section.


2.2.4 Construction of the One-Particle Hilbert Space

We will now construct the aforementioned line bundle L over R2. A physical one-particle statewill then be described by a square-integrable section of this line bundle. The line bundle will beequipped with a connection such that the connection 1-from associated to a particular trivializationcorresponds to the magnetic vector potential A = Axdx+ Aydy in a specific gauge. Consequently,the curvature 2-form associated to the connection will be equal to the magnetic field 2-form B = dA.The construction of L seems fairly abstract at first, but when most of the definitions are done theobject is actually quite manageable.

Definition 2.1. Let PM denote the space of piecewise smooth paths γ : [0, 1] → R2 that start atγ(0) = 0. We define the line bundle L as the set PM ×C modulo the following equivalence relation,

(γ1, z1) ∼ (γ2, z2) ⇔ γ1(1) = γ2(1) and z2 = exp[2πiκ

∫γ2,γ1

B

]z1, (2.44)

where κ is a constant that will be fixed later on. The equivalence class of the element (γ, z) will bedenoted by [γ, z].

Notation: in the above equivalence relation, the integral is over the domain D enclosed by thecurves γ1 and γ2, and oriented in such a way that the orientation on the boundary ∂D correspondsto first going along γ1, and then going back along γ2. Hence, by Stokes theorem, we could haveequally well written

(γ1, z1) ∼ (γ2, z2) ⇔ γ1(1) = γ2(1) and z2 = exp[2πiκ(

∫γ1

A−∫

γ2

A)]z1,

for any A with B = dA.Despite the use of the word line bundle, we have not yet shown that L actually is a line bundle

over R2.

Definition 2.2. Define the projection

π : L → R2 by π([γ, z]) = γ(1).

This makes the fibre Lr at the point r ∈ R2 the set of equivalence classes of curves ending at γ(1) = r.

Next, we will construct global trivializations of L, and in the process we will also show that each Lr

is actually a one-dimensional complex vector space. A trivialization of L will be given by a set ofpaths βr : [0, 1] → R2 for each point r ∈ R2, satisfying βr(0) = 0 and βr(1) = r, such that the pathsβr vary nicely with the endpoint r. Given such a set βr : r ∈ R2, for fixed r ∈ R2, we can thenconstruct a map λr : Lr → C, defined by

λr([γ, z]) := exp[2πiκ

∫βr,γ

B

]z. (2.45)

Lemma 2.2. For each r ∈ R2, the map λr : Lr → C given by equation (2.45) is well-defined. It isalso a bijection.

Proof. Suppose that (γ1, z1) ∼ (γ2, z2). Then,

λr([γ2, z2]) = exp[2πiκ

∫βr,γ2

B

]z2 = exp

[2πiκ

∫βr,γ2

B

]exp

[2πiκ

∫γ2,γ1

B

]z1

= exp[2πiκ

∫βr,γ1

B

]z1 = λr([γ1, z1]).

Also, for any fixed path γ, λr([γ, z] : z ∈ C) = C, which implies surjectivity. For injectivity,observe that exp(2πiκ

∫βr,γ1

B)z1 = exp(2πiκ∫

βr,γ2B)z2 precisely when (γ1, z1) ∼ (γ2, z2).


We now turn Lr into a C-vector space such that λr becomes a C-linear isomorphism.

Definition 2.3. Define addition on Lr by

[γ1, z1] + [γ2, z2] :=[γ1, z1 + z2 exp

(2πiκ

∫γ1,γ2

B

)],

and multiplication by a complex number c as

c[γ, z] := [γ, cz].

The last definition should not come as a surprise. The first one simply ensures additivity of λr. Atfirst sight, it might look like a non-abelian form of addition, but because λr maps bijectively intoC, which is abelian, so is the sum defined above.

Corollary 2.1. For each choice of paths βr : r ∈ R2, using the λr’s constructed above, we obtaina global trivialization of L over R2 through (r, [γ, z]) 7→ (r, λr([γ, z]), making L a line bundle overR2.

Definition 2.4. The space of sections of the line bundle L over R2 is denoted by Γ(R2,L).

Having constructed the line bundle, we can define the Hilbert space that we will be working with.

Definition 2.5. Define an inner product on Γ(R2,L) by

〈ψ, φ〉 :=∫

R2ψβ(r)φβ(r)dr, (2.46)

where ψβ , φβ : R2 → C are the trivialized expressions for ψ and φ through some trivialization βr.From (2.46), we obtain a norm in the usual way,

||ψ||2 := 〈ψ,ψ〉.

Note that the inner product defined above does not depend on the choice of trivialization, becausetwo trivialized expression ψβ and ψβ′ differ by a locally varying complex phase, which cancels out in(2.46) due to the bar on the first argument. However, the inner product might be infinite for somesections of L. Hence, to define the actual Hilbert space, we have to be a little bit more restrictive.

Definition 2.6. The one-particle Hilbert space H is defined as the space of measureable square-integrable sections of L modulo the equivalence relation that identifies two sections if the set onwhich they disagree has zero measure, i.e. H = L2(R2,L). For all practical purposes however, wecan simply think of H as the space of differentiable sections of L with finite norm.

Remark 2.5. The inner product (2.46) is well-defined on H due to Holders inequality. As a con-sequence, H is closed under addition: for ψ, φ ∈ H, ||ψ+φ||2 = ||ψ||2 + 〈ψ, φ〉+ 〈φ, ψ〉+ ||φ||2 <∞.Also, H is complete because L2(R2,C) is, making H a true Hilbert space.

As noted at the end of Section 2.2.3, we want to describe physical states as sections of the linebundle L. In order get a better idea of what the relation is between a section of L and a wavefunction,consider the following example.

Example 2.2. The symmetric gauge corresponds to the case where each of the paths βr = βSr is

a straight line from the origin to the point r = (x, y), and the Landau gauge corresponds to pathsβr = βL

r that go from 0 in a straight horizontal line to the point (x, 0), and then in a straight vertical


line to the point r = (x, y). Indeed, using (2.45), we see that λSr ([γ, z]) = exp[2πiκ

∫βS

r ,γ]z and

λLr ([γ, z]) = exp[2πiκ

∫βL

r ,γ]z are related by

λSr ([γ, z]) = λL

r ([γ, z]) exp

[2πiκ

∫βS

r ,βLr

B

]= λL

r ([γ, z]) exp[2πiκ

B

2xy

]= λL

r ([γ, z])USL(x, y), (2.47)

provided that we take κ to be minus one over the magnetic flux quantum:

κ := − eh.

Let eS(r) := (λSr )−1(1) and eL(r) := (λL

r )−1(1), where 1 denotes the complex number 1, be the globalframes corresponding to the symmetric and Landau trivializations respectively. A section ψ ∈ H canbe expressed in terms of both frames as follows:

ψ(r) = ψS(r) eS(r) = ψL(r) eL(r),

where ψS and ψL are what we used to call the wavefunction expressed in the symmetric or Landaugauge respectively. Applying formula (2.47) to the section [γ, z] = eS and then taking (λL

r )−1 onboth sides yields eL(r) = USL(r) eS(r). That is,

ψS(r) = USL(r)ψL(r),

which is in agreement with equation (2.43).

Remark 2.6. Taking κ := − e~ makes the unitary operator USL dimensionless, as it should be.

The example above shows that sections of L naturally transform appropriately under gaugetransformations. However, we still have to show that the trivializations βS and βL chosen aboveactually agree with the symmetric and the Landau gauges in the first place. In order to do so,we have to construct a specific connection on L, which we will do in the next section. It is theconnection that distinguishes the line bundle L from the trivial bundle R2 × C.

2.2.5 The Vector Potential as a Connection 1-Form

There exists a natural connection ∇ on the space of section of L: the one that is associated to thenotion of parallel transport coming from glueing paths together. After choosing a trivialization βr,the connection takes the trivialized form ∇ = d+A, where A is the connection 1-form that will turnout to be very closely related (see Remark 2.8) to the gauge field A, and the resulting trivializedexpression ψβ : R2 → C for ψ ∈ H will then agree with the original wavefunction corresponding tothe particular gauge A.

Remark 2.7. In order to distinguish the connection from the partial derivative operator, we use ∇for the connection, and reserve ∇ for the vector operator (∂x, ∂y).

Remark 2.8. There is a slight ambiguity about the magnetic vector potential. The physicists con-vention is that of equation (2.32), where A has units volt-second per meter. The mathematiciansconvention uses A for the connection 1-form – see equation (2.50) – where we require A to havethe same units as the derivative operator, i.e. 1/meter. To distinguish between the two, we keep Afor the ordinary physicists vector potential, and use A for the mathematicians vector potential. Thedifference between the two is a mere constant,

A =ie

~A.

The factor ie/~ comes from comparing iΠ/~ = (∇+ ieA/~) to the connection d+ A.


In this section, want to construct the aforementioned connection ∇. In order to do so, we willfirst define the corresponding notion of parallel transport. But first, a preliminary definition.

Definition 2.7. For two paths γ, γ′ : [0, 1] → R2 such that γ(1) = γ′(0), we define the compositionγ′ γ by gluing both paths together. We will parametrize γ′ γ such that it does γ on [0, 1

2 ] and γ′

on [ 12 , 1].

Note that, for our purposes, it doesn’t really matter how the new curve is parametrized as theequivalence relation in equation (2.44) considers all curves that only differ by a reparametrizationto be equivalent.

Having defined the composition of paths, we can make sense of the notion of parallel transport.

Definition 2.8. Given two points r, s ∈ R2 and a path η : [0, 1] → R2 such that η(0) = r andη(1) = s, we construct a linear isomorphism

P ηr,s : Lr → Ls,

defined byP η

r,s([γ, z]) := [η γ, z]. (2.48)

This map is trivially well-defined, and it is straightforward to check that it is an isomorphism.

Notation: For a vector w ∈ R2 and a point r ∈ R2, denote the path that connects r and r + w ina straight line by ηr,w : [0, 1] → R2. That is, for s ∈ [0, 1], ηr,w(s) = r + sw.

Definition 2.9. The connection ∇ is defined as follows. Given a vector field v ∈ Vec(R2), ∇ actson an arbitrary section ψ ∈ Γ(R2,L) by

∇v(ψ)∣∣r

= limt→0

(P

ηr,vt

r,r+vt

)−1 (ψr+vt)− ψr

t. (2.49)

Now, suppose we have a trivialization βr : r ∈ R2 of L. Each βr induces an isomorphismλr : Lr → C. Let er := (λr)−1(1) be the corresponding global frame. More concretely, er = [βr, 1].Using the definition of ∇ above, we will construct an explicit formula for the connection 1-form Acorresponding to the trivialization βr of L.

When acting on the frame e, we have,

∇v(e)∣∣r

= A(v)e∣∣r

(2.50)

for any point r ∈ R2 and any tangent vector v ∈ TrR2. Using (2.49), the left hand side of (2.50) canalso be obtained by taking a limit of parallel transported vectors,

∇v(e)∣∣r

= limt→0

(P

ηr,vt

r,r+vt)

)−1

(er+vt)− er

t. (2.51)

We can rewrite the RHS of (2.51) using the definition of parallel transport (2.48),

(P

ηr,vt

r,r+vt

)−1 (er+vt) = [η−1r,vt βr+vt, 1] =

[βr, exp

(− ie

~

∫D(ηr,vt)

B

)]= exp

(− ie

~

∫D(ηr,vt)

B

)er.

In the above expression, D(ηr,vt) refers to the domain enclosed by and oriented according to thecurve β−1

r η−1r,vt βr+vt — see Figure 2.3. Also, for any path γ, we have adopted the notation γ−1

for the path γ parametrized backwards. We conclude that parallel transporting the frame e resultsin an extra phase factor determined by the path over which we do the parallel transportation.


βr

βr+vt

r = ηr,vt(0)

r + vt = ηr,vt(1) v

O

D(ηr,vt)

Figure 2.3: Parallel transporting the vector er+vt over a straight line to the pointr yields an additional phase factor exp

(− ie

~∫

D(ηr,vt)B), where D(ηr,vt) is the

region enclosed by the curves as shown above.

Plugging the above expression into equation (2.51) yields

∇v(e)∣∣r

= limt→0

exp(− ie

~∫

D(ηr,vt)B)− 1

ter. (2.52)

Comparing equations (2.50) and (2.52) gives

A(v)∣∣r

= limt→0

exp(− ie

~∫

D(ηr,vt)B)− 1

t=

d

dtexp

(− ie

~

∫D(ηr,vt)

B

)∣∣∣∣t=0

. (2.53)

It might not be entirely obvious that formula (2.53) defines a 1-form, so let us verify this.

Lemma 2.3. Through formula (2.53), a trivialization βr : r ∈ R2 defines a 1-form A. (Recallthat D(ηr,vt) depends on βr.)

Proof. The smoothness of A comes from the fact that the paths βr vary smoothly with r. Hence, inorder to prove that (2.53) defines a 1-form, we have to show that it is linear in v. To see that thisis indeed the case, we introduce some more notation. For each r ∈ R2, define Er : R2 → R by

Er(w) = exp

(− ie

~

∫D(ηr,w)

B

).

Writing w = vt, equation (2.53) becomes

A(v)∣∣r

=d

dtEr(vt)

∣∣t=0

. (2.54)

Using the chain rule on the composition t 7→ w(t) 7→ Er(w(t)),

d

dtEr(w(t)) = w′(t) · E′r(w(t)) =

[(v1∂w1 + v2∂w2)

]Er(w)

∣∣w=vt

,

equation (2.54) readsA(v)

∣∣r

=[(v1∂w1 + v2∂w2)

]Er(w)

∣∣w=0

,

which is clearly linear in v.


In short, every trivialization βr defines a 1-form through equation (2.53). Let us perform onefinal consistency check.

Lemma 2.4. Suppose we have two 1-forms, A and A′, corresponding to trivializations βr andβ′r respectively. Denote the frames that correspond to βr and β′r by e and e′, and let u : R2 → Cbe the function that relates the two frames, i.e. e = ue′. Then, the 1-forms are related by

Ae = (u−1du+ A′)e.

Proof. We have

A(v)e∣∣r

= limt→0

exp(− ie

~∫

D(ηr,vt)B)− 1

ter

= limt→0

(P

ηr,vt

r,r+vt

)−1 (er+vt)− er

t

= limt→0

u(r + vt)(P

ηr,vt

r,r+vt

)−1 (e′r+vt)− er

t

= limt→0

u(r + vt) exp(− ie

~∫

D′(ηr,vt)B)e′r − er

t

= limt→0

u(r + vt)exp

(− ie

~∫

D′(ηr,vt)B)− 1

tu(r)−1er + lim

t→0

u(r + vt)− u(r)t

u(r)−1er

=[A′(v)e+ u−1du(v)e

]∣∣r,

as required.

We know how the connection ∇ acts on a global frame e corresponding to a trivialization βr,where, recall that the frame e is defined by er = [βr, 1]. What about arbitrary sections ψ ∈ H? Wecan express ψ in terms of e as follows: ψ = ψβe, where ψβ : R2 → C. Now, using equation (2.50)and the Leibniz rule,

∇(ψ) = ∇(ψβe) = dψβ ⊗ e+ ψβ∇(e) = (d+ A)ψβ ⊗ e.

That is, ∇ acts on the trivialized wavefunction ψβ as d+ A. In terms of x-y coordinates, we have

(d+ A)ψβ = (∂x + Ax)ψβx dx+ (∂y + Ay)ψβ

y dy = ∇xψβx dx+ ∇yψ

βy dy.

Remark 2.9. We see that the components Πx and Πy of the mechanical momentum operator are,up to the numerical factor i

~ mentioned in Remark 2.8, equal to the covariant derivative operators∇x = ∂x + Ax and ∇y = ∂y + Ay respectively.

We can also express the Hamiltonian in terms of the connection ∇. From equation (2.36), wehave

H =1

2m

([px + eAx]2 + [py + eAy]2

)= − ~2

2m

([∂x +

ie

~Ax

]2+[∂y +

ie

~Ay

]2)= − ~2

2m

([∂x + Ax

]2+[∂y + Ay

]2)= − ~2

2m

(∇2

x + ∇2y

),

which, in coordinate-free notation reads

H =~2

2m∆, (2.55)


where ∆ is the Bochner Laplacian, defined by

∆ := −tr∇2.

Finally, we will revisit Example 2.2 and show that equation (2.53) indeed gives the correctexpression for the magnetic vector potential in the Landau and symmetric gauge when using theaforementioned trivializations βL

r and βSr respectively.

Example 2.3. Let r = (x, y) ∈ R2 be arbitrary, and let v = (a, b) ∈ TrR2 be any tangent vector atr. For the Landau gauge, we have

∫D(ηr,vt)

B = B(yat+ 12abt

2). Therefore,

AL(v)|r =d

dt

∣∣∣∣t=0

exp[− ie

~B

(yat+

12abt2

)]= − ieB

~ya.

Because AL = ALxdx + AL

y dy, we have AL(v) = ALxa + AL

y b. Comparing with the above equationyields

AL =~ie

AL = −Bydx,

which agrees with the vector potential in the Landau gauge (2.42). Likewise, for the symmetric gaugewe have

∫D(ηr,vt)

B = 12B(ya− xb)t. Thus,

AS(v)|r =d

dt

∣∣∣∣t=0

exp[− ie

~B

2(ya− xb) t

]= − ieB

2~(ya− xb) .

Comparing to AS(v) = ASxa+ AS

y b gives

AS =~ie

AS =B

2(−ydx+ xdy) ,

which agrees with the vector potential in the symmetric gauge (2.41).

In conclusion, we have constructed a line bundle L over R2, such that its square-integrablesections correspond to the one-particle states available to a free electron in a constant perpendicularmagnetic field B. We have equipped L with a connection ∇, to which the Hamiltonian is relatedby (2.55). Globally trivializing L yields a connection 1-form A that is proportional to the magneticvector potential A. Moreover, the curvature 2-form Ω of ∇ is proportional to the magnetic field2-form: Ω = dA + A ∧ A = dA ∝ dA = B; that is, we can view the magnetic field as curving theline bundle L, making it a topologically trivial line bundle over R2, with no canonical trivialization.

In practice, we will always choose a gauge — i.e. a global trivialization of L — and work inthat particular gauge, allowing us to regard a section ψ ∈ H as a wavefunction ψ ∈ L2(R2,C). Forthe remainder of this thesis, we will use the words “section” and “wavefunction” interchangeably.However, it is good to keep in mind that a wavefunction ψ ∈ L2(R2,C) is only a particular realizationof a section of L, the latter of which represents the actual physical state.

Chapter 3

The Quantum Hall Effects

Now that we have set up the formalism with which we will be working, we can start to do someactual physics. In this chapter, we will discuss both the integer and the fractional quantum Halleffect. Section 3.1 deals with the quantization of kinetic energy in the form of Landau levels andhow this leads to the integer quantum Hall effect. In order to describe the fractional quantum Halleffect, we have to turn on the electron-electron interactions, which will be discussed in Section 3.2.

Throughout the rest of this thesis, we will assume that the system is in the low temperaturelimit. Concretely put, this meant that we require the temperature T to be such that thermal energyscale of the system kBT is much smaller than the kinetic energy scale ~ωC . This ensures that theLandau level structure is maintained in the presence of thermal fluctuations.

3.1 The Integer Quantum Hall Effect

As in the previous chapter, consider a 2D spinless free electron gas confined to the x-y plane in thepresence of a perpendicular magnetic field B = Bez. The magnetic field causes the kinetic energyof the electrons to become quantized. This is known as Landau quantization, which forms the basisof our understanding of the QHE’s. For now, we leave the the electric potential out of the equation;it will be turned on in Section 3.1.4.

3.1.1 Landau Quantization

Recall that the Hamiltonian for a single electron in a magnetic field is given by equation (2.36)

H =1

2m[p + eA(r)]2 .

where p is the canonical momentum, given by equation (2.25)

p = mv − eA(r) = Π− eA(r), (3.1)

and Π = mv is the mechanical momentum (2.21). Since we are working in the Hamiltonian picture,Π should be viewed as a function of r and p:

Π(r,p) = p + eA(r). (3.2)

When performing canonical quantization, the canonical momentum p gets replaced by the derivativeoperator p = −i~∇, and Π (up to a multiplicative constant, see Remark 2.8) becomes the covariantderivative operator ∇ that acts on sections of the one particle Hilbert space H. In particular, wehave the commutation relations

[ri, pj ] = i~δij .

24

3.1. THE INTEGER QUANTUM HALL EFFECT 25

In terms of the mechanical momentum (3.2), the Hamiltonian reads

H =Π2

2m=

12m

(Π2x + Π2

y). (3.3)

We compute,

[Πx,Πy] = e[px, Ay] + e[Ax, py] = −i~e(∂Ay

∂x− ∂Ax

∂y

)= −i~e(∇×A)z = −i~eB. (3.4)

Or, in terms of the magnetic length

lB =

√~eB

,

which is the typical length scale in our system, equation (3.4) becomes

[Πx,Πy] = −i~2

l2B. (3.5)

This is a remarkable result. We see that, because of the magnetic field, the x and y componentsof the mechanical momentum (which correspond to physical properties of the moving electron) nolonger commute. This means that we cannot know both Πx and Πy at the same time! Moreover, theHamiltonian in equation (3.3) mathematically resembles that of an ordinary quantum mechanicalharmonic oscillator.

Define the ladder operators a and a† by

a :=lB√2~

(Πx − iΠy) and a† :=lB√2~

(Πx + iΠy), (3.6)

where the normalization is chosen precisely such that

[a, a†] = 1.

Following exactly the same procedure as for the quantum mechanical harmonic oscillator, H can berewritten in terms of a and a†,

H =~2

2ml2B(aa† + a†a) = ~ωC

(a†a+

12

), (3.7)

where ωC = eBm is the cyclotron frequency that we encountered in the classical Hall effect. The

eigenvalues of the Hamiltonian are

En = ~ωC

(n+

12

), (3.8)

for n a non-negative integer.In conclusion, we see that our Hamiltonian is of the same form as the quantum mechanical

harmonic oscillator, only in this case, the energy levels correspond to kinetic energy of the electron.In the semi-classical picture, an electron moves in a cyclotron orbit, where the velocity, and thereforealso the radius, is quantized. Applying a† to an electron in a cyclotron orbit increases both its radiusand its velocity, hence its kinetic energy.

It seems that are done now, having diagonalized the Hamiltonian and found its spectrum. How-ever, we have missed a crucial feature of the system that we are describing. Recall that a cyclotronorbit is characterized by two degrees of freedom: the radius of the orbit, and the position of thecentre. This last degree of freedom we have completely neglected so far.


3.1.2 Energy Level Degeneracy

We want to introduce operators R = (X,Y ) for the coordinates of the centre – also called the guidingcentre – of the cyclotron motion. In order to express R in terms of the known operators x, y, px

and py, we refer to Figure 2.1. Clearly, R = r−η. Also, from equation (2.6), in magnitude η = vωC

.Because η is rotated clockwise over an angle of π

2 compared to v = Πm , we can express η in terms of

Π as follows,

η =ΠeB

× ez. (3.9)

Hence, the guiding centre operators X and Y are given by

R =(XY

)=(x− ηx

y − ηy

)=(x− Πy

eB

y + Πx

eB

)(3.10)

where Π is defined in terms of x, y, px and py by equation (3.2).Now that we have a proper expression for the operators corresponding to the guiding centre, let

us compute the commutator [X,Y ].

[X,Y ] =[x− Πy

eB, y +

Πx

eB

]=

1eB

[x, px]− 1eB

[py, y]−1

(eB)2[Πy,Πx]

=2i~eB

− 1(eB)2

i~2

l2B=

i~eB

= il2B . (3.11)

As with the mechanical momentum, also the x and y components of the guiding centre operator donot commute. This means that X and Y too, cannot be measured simultaneously.

We introduce a second set of ladder operators b and b†, defined by

b =1√2lB

(X + iY ) and b† =1√2lB

(X − iY ), (3.12)

where, as before, the prefactors are chosen such that

[b, b†] = 1.

Because X and Y commute with Πx and Πy – see appendix (A.3) and (A.4), the one-particle Hilbertspace can be written as a tensor product of two harmonic oscillators. The Hamiltonian is given by

H = ~ωC

(a†a+

12

)+ ~ω0

(b†b+

12

)(3.13)

where ω0 = 0, and the one-particle eigenfunctions of H are completely described by two parameters:n for the energy level, and m for the guiding centre,

|n,m〉 :=(a†)n

√n!

(b†)m

√m!

|0〉. (3.14)

Physically, we can think of the system comprised of a single electron in a magnetic field as asum of independent harmonic oscillators of frequency ωC located at different positions labelled bythe guiding center coordinate m. By saturating Heisenberg’s uncertainty relation, we can assign anaverage area to one such harmonic oscillator,

∆X∆Y = 2πl2B . (3.15)


Given that the (macroscopic) sample whereon the electron moves has total area A, we can fit

NB :=A

2πl2B= nBA, (3.16)

where nB is the flux density

nB :=1

2πl2B=

B

h/e=B

φ0, (3.17)

harmonic oscillators onto the sample. In the equation above, we have introduced the magnetic fluxquantum1 φ0 := h

e .In short, the situation is the following: we have NB positions to place a guiding centre, and at

each position an electron can be in any of the discrete harmonic oscillator energy levels En given byequation (3.8). Thus, we see that, in the case that we have more than one non-interacting spinless2

electron, a fixed energy eigenstate can be occupied by NB electrons. That is, each energy level isNB-fold degenerate.

Now, suppose that we have Nel = nel×A non-interacting spinless electrons in our system, whichwe take to be in the ground state. Because the electrons are fermions, the first NB electrons can allgo into the harmonic oscillator ground states, the second NB electrons must go into the first exitedstates, and so on. Hence, the multi-particle ground states are completely determined by the fillingfactor

ν :=Nel

NB=nel

nB. (3.18)

A final note: the degeneracy NB depends on the magnetic field. By increasing it we can decreasethe area of uncertainty 2πl2B in equation (3.15). Hence, we can change the filling factor either bychanging the electron density nel, or by changing the magnetic field strength B.

We now possess all the necessary machinery to understand the IQHE, which we will discuss inthe next section. But, before we do so, we will try to make the theory explained above a little bitmore precise from a mathematical viewpoint.

3.1.3 Mathematical Intermezzo

Let us try to put the theory or a more rigorous mathematical footing. As described in Section2.2.4, the one-particle Hilbert space H is the space L2(R2,L) of square integrable sections of L. TheHamiltonian is given by equation (2.55),

H =~2

2m∆,

Picking a gauge corresponds to globally trivializing the line bundle. This yields an isomorphismH ∼= L2(R2,C), and the Hamiltonian then takes the form

H =1

2m[p + eA(q)]2 =

12m

(Π2

x + Π2y

)= ~ωC

(a†a+

12

), (3.19)

which corresponds to equation (3.3). The ladder operators a and a† satisfy

[a, a†] = 1. (3.20)

Definition 3.1. Let A be the associative algebra over C with involution spanned by two elementsα and its conjugate α∗. Let I ⊂ A be the ideal I = C(αα∗ − α∗α− 1). Define

A := A /I ,

that is, A is the associative algebra spanned by α and α∗ such that [α, α∗] = αα∗ − α∗α = 1.1The total flux through a superconducting loop is always an integer multiple of the magnetic flux quantum.2We consider spinless electrons because we do not want the Zeeman effect to split each Landau level into two.


A choice of the gauge gives explicit expressions for the operators a and a† acting on L2(R2,C)satisfying (3.20). That is, it defines a representation

α 7→ a

of the algebra A on L2(R2,C). Different gauges correspond to unitarily equivalent representations ofA, because the explicit expressions for Π, and therefore also for a and a†, corresponding to differentgauges are related by a unitary transformation as shown in equation (2.40). Therefore, the operatorsa and a† are actually defined on sections of L2(R2,L).

In order to gain a better understanding in the system that we are investigating, we will nowidentify all relevant irreducible representations of A.

Definition 3.2. The standard representation of A is defined as follows. Let H0 be the Hilbert spacel2(N0), where N0 is the set of non-negative integers. That is,

H0 =∞⊕

n=0

Vn,

where each of the Vn’s is isomorphic to C. The inner product on H0 is defined as

〈x, y〉 :=∞∑

n=0

xnyn (3.21)

for x = (x1, x2, . . .) and y = (y1, y2, . . .) arbitrary elements of H0. We have an orthonormal basisen of H defined by (en)m = δnm, i.e. en = 1 ∈ Vn.

The standard representation of A is defined through the action of α and α∗ on the ONB en,

α(en) :=

√n en−1 if n ≥ 1

0 if n = 0

α∗(en) :=√n+ 1 en+1.

The most common example in physics of the standard representation of A is that of the quantumharmonic oscillator.

Theorem 3.1. Any irreducible representation of A on a separable Hilbert space H with non-emptyKer(α) ⊂ H is unitarily equivalent to the standard representation.

Remark 3.1. For the case of a single electron in a perpendicular magnetic field, Ker(a) is actuallynon-empty, so the above assumption holds. We will explicitly construct the Ker(a) in the symmetricgauge in Lemma 3.4.

Proof. Suppose we have such a representation A → End(H). Define V0 := Ker(α). BecauseKer(a) 6= ∅, there exists a 0 6= ψ ∈ V0. For each n ∈ N, define

Vn := C (α∗)nψ.

Claim 1: (α)nψ 6= 0 for all n ∈ N.The basic idea is the following: suppose α∗ψ = 0, then 0 = αα∗ψ = (α∗α + 1)ψ = ψ, whichcontradicts the fact that ψ 6= 0. Hence, we conclude that α∗ψ 6= 0. More formally,Base case: ψ 6= 0.Induction step: suppose that (α∗)nψ 6= 0, and assume that (α∗)n+1ψ = 0. Then, by exploiting thecommutation relation in (3.20) and the fact that αψ = 0, 0 = α(α∗)n+1ψ = (n + 1)(α∗)nψ 6= 0,


proving that the assumption (α∗)n+1ψ = 0 is false, which means that (α∗)n+1ψ 6= 0.Claim 2: for m 6= n, Vm ⊥ Vn.Proof: without loss of generality, assume that n > m. Now,

〈(α∗)nψ, (α∗)mψ〉 = 〈(α)m(α∗)nψ,ψ〉 ∝ 〈(α∗)n−mψ,ψ〉 = 〈(α∗)n−m−1ψ, αψ〉 = 0.

Hence, by claims 1 and 2 above, after properly normalizing each (α∗)nψ we see that H contains acopy of H0 as an A-invariant subspace. By irreducibility of H, we conclude that H ∼= H0.

Remark 3.2. We have swept a couple of non-trivial details under the rug. Theorem 3.1 has amathematically rigorously formulated counterpart, known as the Stone-von Neumann theorem. Thistheorem is more general than Theorem 3.1 because the extra assumption that Ker(α) 6= ∅ is notneeded. We will now briefly discuss the Stone-von Neumann theorem.

Instead of dealing with the ladder operators a and a†, the Stone-von Neumann theorem is for-mulated in terms of the self adjoint operators Q = 1√

2(a + a†) and P = i√

2(a† − a), which satisfy

[P,Q] = −i.Given a representation of A on a Hilbert space H, it follows (by taking the trace on both sides

of [P,Q] = −i) that H has to be infinite dimensional, and that at least one of the operators P andQ is represented by an unbounded operator. As a consequence, the relation PQ−QP = −i cannothold on all of H. However, if P and Q are self-adjoint, then both generate one-parameter subgroupsU(t) := e−itP , t ∈ R and V (t) := e−itQ, t ∈ R of the space of bounded linear operators B(H) onH. By PQ−QP = −i, the operators U(t) and V (s) satisfy

U(t)V (s) = eitsV (s)U(t). (3.22)

Because U(t) and V (s) are bounded, equation (3.22) could possibly hold on all of H.The most common representation of two operators P and Q acting on some Hilbert space H

and satisfying PQ − QP = −i is the example due to Schrodinger of the position and momentumoperators Q = x and P = −i∂x acting on L2(R). Indeed, both operators are unbounded, but theirexponentiated versions are not. For ψ ∈ L2(R2), we have

U(s)ψ(x) = ψ(x− s) and V (s)ψ(x) = e−isxψ(x).

Now, the Stone-von Neumann Theorem says that, if we have another pair of weakly continuousfamilies U(t), t ∈ R and V (t), t ∈ R of unitary operators acting irreducibly on a separable Hilbertspace H such that

U(t)U(s) = U(t+ s), V (t)V (s) = V (t+ s), (3.23)

U(t)V (s) = eitsV (s)U(t) (3.24)

holds, then there exists a (unitary) isomorphism W : H → L2(R2) such that

WU(t)W−1 = U(t) and WV (t)W−1 = V (t)

for all t ∈ R. In other words, any (U , V ,H) satisfying (3.23) – (3.24) is unitarily equivalent to(U, V, L2(R2)). A similar statement also holds for more than one set of conjugate operators.

We will not go any futher into the details of this beautiful but complicated theorem. The interestedreader is referred to [23].

Suppose we are given a general (not necessarily irreducible) representation of A on a separableHilbert space H with non-empty Ker(α), then H can be decomposed into a direct sum of irreduciblerepresentations of A. The following corollary shows that this is indeed the case.


Corollary 3.1. Given a representation of A on a separable Hilbert space H with Ker(α) 6= ∅.Then, H ∼= H0 ⊗Ker(α) as a representation of A, meaning that H is isomorphic to dim(Ker(α))A-invariant copies of H0.

Proof. Pick a basis ψ1, ψ2, . . . of Ker(α). Following the proof Theorem 3.1, each of the ψi gener-ates an A-invariant subspace isomorphic to H0, and the so obtained A-invariant subspaces do notintersect. For, suppose that we have linearly independent ψi, ψj ∈ Ker(a) and n > m ∈ N such that(a†)nψi = λ(a†)mψj for some 0 6= λ ∈ C. Applying am+1 to both sides yields (a†)n−m−1ψi = 0,which contradicts Claim 1 of Theorem 3.1. Hence, H ∼= H0 ⊗Ker(α).

Remark 3.3. Any representation of A on a separable Hilbert space H with Ker(α) 6= ∅ is completelycharacterized by the dimension of Ker(α).

We now return to the one-particle Hilbert space H = L2(R2,L) for a single charged particle ina uniform perpendicular magnetic field. Recall that A acts on H via α 7→ a with Ker(a) 6= ∅, andtherefore H ∼= H0 ⊗Ker(a) according to Corollary 3.1.

Definition 3.3. In the decomposition H ∼= H0 ⊗ Ker(a), for each non-negative integer n denotethe n-th subpace Vn ⊗Ker(a) of H by Ln. That is, H ∼=

⊕∞n=0 Ln, where Ln

∼= Ker(a).

Remark 3.4. Because the combination a†a acts by multiplication with n on Ln, by (3.19) each Ln

is an energy eigenspace with energy ~ωC(n+ 12 ). We call Ln the nth Landau level.

Definition 3.4. The subspace L0 = Ker(a) ⊂ H is called the degeneracy space, because it deter-mines the degeneracy of each energy level. In physics, it is known as the lowest Landau level.

In order to further understand the representation of A on the one-particle Hilbert space H, wewill now investigate the degeneracy space Ker(a) ⊂ H. Recall the guiding centre operators X andY defined in equation (3.10).

Lemma 3.1. For each n ≥ 0, The operators X and Y are endomorphisms of Ln.

Proof. According to (A.3) and (A.4), both X and Y commute with Πx and Πy, and therefore witha, a† and H. This means that X and Y leave each of the Ln’s invariant.

As explained in the previous section, because [X,Y ] = il2B , we can define another set of ladderoperators b and b† – see equation (3.12) – satisfying [b, b†] = 1, which are also endomorphisms ofeach Landau level Ln.

Corollary 3.2. The map α 7→ b defines a representation of A on each of the Ln’s. As a consequence,dim(Ln) is infinite for each n.

Using the guiding centre operators X and Y , we can construct an operator that plays a mathe-matically similar role to the Hamiltonian (3.19).

Definition 3.5. The guiding centre operator G is defined by

G := X2 + Y 2 = 2l2b

(b†b+

12

).

By working in the symmetric gauge, in Corollary 3.6 we will show explicitly that Ker(b) ∩ L0 isone-dimensional. This statement is actually gauge independent because the ladder operators a, a†,b and b† are well-defined on sections of L2(R2,L).

Corollary 3.3. For each n ≥ 0, Ker(b) ∩ Ln is one-dimensional


Proof. For n = 0, the above statement is exactly Corollary 3.6. For n > 0, observe that Ln is theimage of L0 under (a†)n. Thus, an arbitrary vector in Ln is of the form (a†)nψ, with ψ ∈ L0. Sincea† and b commute, Ker(b) ∩ Ln is the space of vectors a†ψ satisfying

0 = b(a†)nψ = (a†)nbψ.

By injectivity of a†, the above equation is equivalent to ψ ∈ Ker(b) ∩ L0, so Ker(b) ∩ Ln is one-dimensional.

Corollary 3.4. A acts irreducibly on each of the spaces Ln through α 7→ b. Moreover, for eachnon-negative integer n, Ln together with the ladder operators b and b† forms a quantum harmonicoscillator, where the guiding centre operator G assumes the role of the Hamiltonian.

The structure of the one-particle Hilbert space L(R2,L) is becoming more apparent. Let ussummarize the results of this section in the following Corollary.

Corollary 3.5. The one-particle Hilbert space H = L2(R2,L) is isomorphic to a tensor product oftwo harmonic oscillators:

H ∼= H0 ⊗Ker(a).

Since H0 is itself a direct sum of one-dimensional vector spaces, we can also decompose H into adirect sum

H =∞⊕

n=0

Ln

of energy eigenspaces Ln, where each subspace Ln∼= Ker(a) forms an harmonic oscillator with

corresponding ladder operators b and b†. As a consequence, we have a basis of H where each basisvector ψn,m is labeled by two non-negative integers n and m, where n labels the energy with respectto the Hamiltonian H, and m labels the ’energy’ with respect to the guiding centre operator G.

Remark 3.5. The above Corollary justifies the notation |n,m〉 = ψn,m in equation (3.14).

We conclude with a lemma on the nature of the guiding centre operators X and Y . The positionoperators x and y do not commute with the Hamiltonian H, However, we can make them commutewith H if we replace them by the operators x 7→

∑n pnxpn and y 7→

∑n pnypn, where

pn := H → Ln (3.25)

is the orthogonal projection onto the nth Landau level. It turns out that this procedure exactlygives us the guiding centre operators X and Y .

Lemma 3.2. The operators X and Y are given by

X =∞∑

n=0

pnxpn and Y =∞∑

n=0

pnypn.

Proof. We will only prove the lemma for X. The proof for Y is analogous. The following equalitiesare equivalent:

X =∞∑

n=0

pnxpn ⇔ 〈Xξ, η〉 =∑

n

〈pnxpnξ, η〉 for all ξ ∈ Ln1 and η ∈ Ln2 ,

where n1 and n2 are arbitrary. We will prove the second statement. For ξ ∈ Ln1 and η ∈ Ln2 wehave ∑

n

〈pnxpnξ, η〉 = 〈pn1xξ, η〉 = 〈xξ, pn1η〉 = δn1,n2〈xξ, η〉.


Thus, we have to prove that〈Xξ, η〉 = δn1,n2〈xξ, η〉 (3.26)

for all n1, n2 ∈ N0, with ξ ∈ Ln1 and η ∈ Ln2 .If n1 6= n2, then the left hand side is zero because X maps Ln1 into itself, so we have an equality.

If n1 = n2, then, from the definition of X in (3.10), 〈Xξ, η〉 = 〈xξ, η〉 − 1eB 〈Πyξ, η〉. However, recall

that Πy ∝ a− a†, and both a and a† change the energy level of ξ, so 〈Πyξ, η〉 = 0 by orthogonalityof the energy eigenspaces. Hence, we conclude that (3.26) is true, which proves the lemma.

We can also (trivially) express the Hamiltonian in terms of the projection operators (3.25).

Lemma 3.3. The Hamiltonian H is given by

H = ~ωC

∞∑n=0

(n+

12

)pn

Proof. Follows immediately from equation (3.19) and Remark 3.4.

Having our feet firmly planted on strong mathematical soil, we are now confident enough totackle the multi-particle case and discuss the integer quantum Hall effect.

3.1.4 The Hall Resistivity at Integer Filling Factor

The goal of this section is to explain the integer quantum Hall effect; that is, the peculiar shape ofthe Hall resistance curve shown in Figure 2.2. We keep the number of electrons in the system Nel

constant, while varying the magnetic field strength B, or equivalently, the filling factor ν (3.18), i.e.

ν =Nel

NB,

where NB = BA/φ0 is the degeneracy of each Landau level, and A is the sample area. Recall thatthe system is assumed to be in the ground state, and hence the Landau levels are filled up frombottom to top. For the remainder of this chapter, we will follow the reasoning of Goerbig [7].

Before explaining the actual shape of the Hall resistivity curve — which we will do in Section3.1.5 — we will first focus on the specific case where the magnetic field B is such that ν is exactlyinteger. For these values of B, it will turn out that the value of the Hall resistivity agrees with theclassical value, given by equation (2.14):

(ρH)classical =B

enel,

where nel is the electron density.Instead of computing the resistivity, we will compute the conductivity of the system. Because ν

is assumed to be integer, say ν = N ∈ N, each Landau level is either completely filled, or completelyempty. Only the completely filled Landau levels, of which we have N , contribute to the conductivity.Computing the total conductivity boils down to computing the conductivity of one completely filledLandau level (which will turn out to be the same for each Landau level), and then multiplying byN .

In order to compute the conductivity of a fully filled Landau level, we take a rectangular systemof length L and width W , such that LW . We choose our x− y coordinate system such that thecorners of the sample are at (0, 0), (0,W ), (L,W ) and (L, 0). Next, we apply a current I in thex-direction. We assume that the system is translationally invariant in the x-direction, meaning thatthe potential U = −eVH associated to the Hall voltage difference is independent of x: U = U(y),and we impose periodic boundary conditions in the x-direction. Also, we assume that the kinetic


energy ~ωC scale is much larger than the potential energy difference ∇U associated to the magneticlength lB , as to not spoil the Landau level structure. More precisely, promoting an electron to thenext Landau level means that its cyclotron radius increases by roughly lB . Thus, the potentialenergy gained is of the order of lB |∇U |. We want this increase in potential energy to be negligiblecompared to the gain in kinetic energy. That is, we require

~ωC lB |∇U | = lBeE. (3.27)

The most convenient gauge for this system is the Landau gauge (2.42), which respects thetranslational symmetry and assures that H is independent of the x-coordinate. When looking foreigenfunctions of H, given by

H =1

2m(Π2

x + Π2y), (3.28)

by separating the variables x and y, we see that functions of the form

ψk(x, y) = eikxξk(y) (3.29)

will do the trick. Using the ansatz from (3.29), we can effectively set px = ~k, where k = 2πmL for

m ∈ Z. This means that the Hamiltonian (3.28) of our system takes the following shape

H =(~k − eBy)2

2m+

p2y

2m+ U(y) =

12mω2

C(y − kl2b )2 +

p2y

2m+ U(y). (3.30)

Consider k to be fixed. We can expand the potential U around the point y = kl2B . Because thekinetic energy dominates the potential energy, we can view the additional term U(y) as a correctionto the first. Neglecting U(y) completely yields an harmonic oscillator centered at y = kl2B , which hasenergy eigenfunction that decay exponentially with decay constant lB . Because U varies very littleon length scales of the order of lB , we anticipate that the true energy eigenfunctions will resemblethe harmonic oscillator energy eigenfunctions, which are localized around kl2B . Therefore, in a firstorder approximation, we can linearize U(y) near y = kl2B :

U(y) ≈ U(kl2B) +∂U

∂y

∣∣∣∣kl2B

(y − kl2B) = U(kl2B) + eE(kl2B)(y − kl2B). (3.31)

Plugging equation (3.31) into equation (3.30) and completing the square yields

H ≈p2

y

2m+

12mω2

C

(y − kl2B +

eE(kl2B)mω2

C

)2

+ U(kl2B)− e2E2(kl2B)2

2mω2C

. (3.32)

Next, define

yk := kl2B +eE(kl2B)mω2

C

. (3.33)

Note that the second term in (3.33) is much smaller than the first due to equation (3.27). Hence, wecan replace U(kl2B) in (3.32) by U(yk) at the cost of an extra term −eE(kl2B)(yk−kl2B) = − e2E2(kl2B)2

mω2C

.

Both this term as well as the last term in (3.32) are proportional to mvD — where vD = E2/B2 isthe Hall drift velocity — which is much smaller than the sum of the kinetic and the potential energyand can therefore be neglected. This results in

H ≈p2

y

2m+

12mω2

C(y − yk)2 + U(yk), (3.34)

which, for each value of k, is an harmonic oscillator centered at yk shifted by a constant U(yk).


We conclude that the approximate eigenfunctions of the Hamiltonian (3.28) are of the formψn,k(x, y) = eikxξn,k(y), where ξn,k is an eigenfunction of (3.34). The wavefunctions ψn,k haveenergy eigenvalues

εn,k = ~ωC

(n+

12

)+ U(yk). (3.35)

Here, n ∈ N denotes the Landau level we are in, and yk signifies where the wavefunction is localized.Hence, m = Lk

2π is interpreted as the guiding centre eigenvalue. Because we want the wavefunctionto be localized somewhere on the sample, k must take values between certain kmin and kmax.When E = 0, yk = kl2B and k runs from kmin = 0 to kmax = W

l2Bsince the y-coordinate runs

from y = 0 to y = W . Hence, the guiding centre eigenvalue m takes values between mmin = 0,and mmax = LW

2πl2B= Area

2πl2B= NB , which agrees with the fact that each Landau level is NB fold

degenerate. In our case E 6= 0, but it is approximately constant, so the allowed values for yk will beshifted by the constant eE(kl2B)

mω2C

, but there are still NB values for k such that yk lies between 0 andW .

Having computed the approximate spectrum of the Hamiltonian, we can now proceed to computethe conductance of a single Landau level. We will work in the Heisenberg picture.

For n ∈ 1, . . . , N, the n-th Landau level contributes

In = −ekmax∑

k=kmin

〈n, k| xL|n, k〉 = − e

L

kmax∑k=kmin

vn,kx (3.36)

to the current I = Iex. In the above equation, we have defined vn,kx := 〈n, k|x|n, k〉 to be the

expectation value of the x-component of the velocity of the (n, k)th energy eigenstate ψn,k = |n, k〉.Now, the Heisenberg equation of motion for the operator x is

x =i

~[H,x] =

i

~∂H

∂px[px, x] =

∂H

∂px=

1~∂H

∂k.

Applying the above to the eigenfunction |n, k〉 and using that L 1 to approximate the derivativewith a fraction, gives

vn,kx =

1~∂εn,k

∂k≈ L

2π~∆εn,m

∆m=L

h(εn,m+1 − εn,m),

since ∆m = 1. Using equation (3.35), we see that the velocity expectation value vn,kx of the state

|n, k〉 is given by

vn,kx =

L

h(U(ym+1)− U(ym)),

where ym is short for yk(m).Performing the sum over k in (3.36), which is actually a sum over m, we see that all the inter-

mediate terms cancel, leaving us with

In = − eh

(εn,mmax− εn,mmin

) = − eh

(U(ymmax)− U(ymmin

)) =e2

hVH ,

where VH := − 1e (U(ymmax)−U(ymmin) is the Hall voltage difference between the lower (y = 0) and

the upper (y = W ) edges of the sample, which is perpendicular to the current flow in the x-direction.We conclude that the n-th Landau level contributes one

Gn =e2

h


quantum of conductance to the electronic transport, and this holds for each filled Landau level,independent of the value of n. Hence, the total conductivity G is given by

G =N−1∑n=0

Gn = Ne2

h.

The resistivity is the inverse of the conductivity. Thus, The Hall resistivity assumes the value

ρH =1G

=1N

h

e2

at integer filling factor ν = N . As noted earlier, the above value coincides with the classical Hallresistivity at integer filling. Indeed, if we set ν = N in equation (3.18), we have nel = NnB = NeB

h ,which implies that

(ρH)classical =B

enel=

h

Ne2= (ρH)quantum.

We conclude that a translationally invariant rectangular sample subject to an electric currentin the x-direction generates an electric Hall potential VH such that, at integer filling factor, thecorresponding quantum Hall resistivity ρH equals the classical Hall resistivity. This is in agreementwith Figure 2.2. The more intriguing part of the Hall resistance curve, however, is the part wherethe Hall curve deviates from the classical Hall curve, which happens when ν is not integer. We willdiscuss these deviations in the next section.

3.1.5 Quantization of the Hall Resistance

In order to truly understand the IQHE, we now consider non-interacting electrons moving in anactual physical sample. As before, we consider a sample of dimension L times W , and inject acurrent I in the x-direction. In order to prevent the electrons from leaving the sample at the topy = W and bottom y = 0 edges, we add a confining potential Uconf (y). Additionally, the electronsinteract with the sample itself, which we model by an impurity potential Uimp(x, y). Includingthe Hall potential from the previous section, which we now denote by UHall(y), the total potentialU(x, y) is given by

U(x, y) = UHall(y) + Uconf (y) + Uimp(x, y). (3.37)

In Section (3.1.2), the Hamiltonian was translation invariant, and, as a consequence, the guidingcentre coordinate R is a constant of motion. The appearance of U , which now also depends on x,breaks translational invariance, lifting the Landau level degeneracy. If the potential is too strong,as explained in the previous section, then the energy eigenstates get shifted by such amounts thatthe Landau level structure will disappear completely, thus preventing us from observing any kind ofquantum Hall effect. We therefore assume that ~ωC lB |∇U |.

First, let us do a consistency check. Recall that r = R+η. The Hamiltonian expressed in termsof η instead of Π — which are related by equation (3.9) — is given by

H =12mω2

C(η2x + η2

y) + U(R + η). (3.38)

Also, using equation (3.5), we have

[ηx, ηy] =1

(eB)2[Πy,−Πx] = −il2B . (3.39)

Now, recall Heisenberg’s equation of motion for any operator O,

O =i

~[H,O].


Thus, using equations (3.38) and (3.39), we get the following equations of motion for the the electroncoordinates η relative to the guiding centre R,

ηx =i

~mω2

Cηy[ηy, ηx] +i

~∂U

∂ηy[ηy, ηx] = −ωCηy −

l2B~∂U

∂ηy,

ηy =i

~mω2

Cηx[ηx, ηy] +i

~∂U

∂ηx[ηx, ηy] = ωCηx +

l2B~∂U

∂ηx.

Indeed, in the limit ~ωC lB |∇U |, the second term is negligible compared to the first one, and werecover the classical cyclotron equation of motion. This means that locally, the electrons behave asif they do not feel the impurity potential.

Having concluded that the electrons approximately preform a cyclotron motion, let us investigatethe dynamics of the guiding centres. Because we have assumed that the potential practically does notvary on length scales of the order of the cyclotron radius (∝ lB), we can approximate U(r) ≈ U(R).Hence, the equations of motion for R become:

X =i

~[U(R), X] =

i

~∂U

∂Y[Y,X] =

l2B~∂U

∂Y,

Y =i

~[U(R), Y ] =

i

~∂U

∂X[X,Y ] = − l

2B

~∂U

∂X.

Using l2B~ = 1

eB , we conclude that

R =1eB

∇U ×B

B=

E ×B

B2. (3.40)

The second equality sign follows from U = −eV and E = −∇V .Comparing to equation (2.11), we see that the guiding centre moves exactly as it does in the

classical Hall effect: along equipotential lines of U , in such a way that is goes counter-clockwisearound a positively charged potential hill. For this reason, the limit ~ωC lB |∇U | is referred to asthe semi-classical limit.

An example of the total potential U that the electrons feel is depicted in Figure 3.1(a). In thebulk of the sample, Uimp is the dominant term in (3.37), and the equipotential lines of U encirclethe impurities. Hence, electrons travelling along such equipotential lines are localized, and do notcontribute to net electronic transport. Near the edges on the other hand, the confining potentialUconf is dominant, which rises at the edges of the sample. Hence, electrons travelling at the edgescan travel from left to right and contribute to the macroscopic current I running through the sample.The corresponding states are called edge states. Moreover, due to (3.40), the edge currents are chiralbecause the impurity potential bends upwards near the edges. Thus, if an electron in an edge statehits an impurity, thereby forcing it to hop to a nearby available state, which is also a edge state,the electron keeps on moving in the same direction, because all the nearby available states “move”in the same direction. Hence, there is no backscattering. This explains why, at integer filling factor,the longitudinal resistance is zero.

We have now understood both the values of the Hall (RH = B−enellz

) and the longitudinal(RL = 0) resistance at integer filling factor ν = N , displayed in Figure 3.1(a). In order to understandwhy the quantum Hall resistivity curve (Figure 2.2) deviates from the classical Hall curve (which isa straight line), we will discuss what happens when the filling factor ν is perturbed around a somefixed integer N . We keep electron density nel fixed, and we adjust the filling factor by changing themagnetic field — see equation (3.18).

Suppose that the magnetic field B is such that ν = N . Reducing the magnetic field reduces thedegeneracy of each Landau level, which forces some electrons in the fully occupied Landau levels togo into the empty (N + 1)th Landau level. The first couple of electrons promoted to the (N + 1)th


Landau level will go into the potential valleys – see Figure 3.1(b): they will become localizedelectrons that do not contribute to the macroscopic current I. Thus, both the Hall resistivity aswell as the longitudinal resistivity remain unchanged. As we decrease the magnetic field further,the electron puddles grow larger, because the equipotential lines that the newly promoted electronsfollow encompass larger areas. Eventually, equipotential lines that connect the upper (y = W ) andthe lower (y = 0) end of the sample will become available, allowing conduction electrons at theedges to travel from the upper to the lower edge – see Figure 3.1(c). This allows for backscattering,and therefore simultaneously lowers the transverse Hall resistance and increases the longitudinalresistance, as shown the bottom graph of Figure 3.1(c).

Figure 3.1: A plot of the density of states (top), potential landscape as seenfrom above (middle) and the Hall and longitudinal resistance as a function ofthe magnetic field for constant electron density (bottom). In (a), the filling factorν is an integer n. When the magnetic field is lowered, electrons are forced topopulate the (n+1)th Landau level. The first excited electrons become localized(b), but when the magnetic field is decreased further (c), electrons are allowed tomove from top to bottom through the sample, both lowering the Hall resistanceand increasing the longitudinal resistance because the edge states at oppositeedges become connected.

If, instead of lowering the magnetic field, we would have increased it, the same phenomena occursin terms of electron holes instead of actual particles. This explains the plateaus in the quantum Hallresistances shown in Figure 2.2, which distinguish the quantum from the classical Hall effect.

The description above gives a very intuitive explanation of the peculiar shape of the Hall resis-tance curve. It is called the percolation picture. It explains the shape of the Hall resistivity curvearound the points for which the filling factor is integer. However, a closer look at the Hall resistiv-ity curve shown in Figure 2.2 shows that there are also plateaus at fractional filling factor. Theseplateaus go by the name fractional quantum Hall effect, which is the topic of the next section.

3.2. THE FRACTIONAL QUANTUM HALL EFFECT 38

3.2 The Fractional Quantum Hall Effect

In the preceding section we have seen how, at integer ν = N filling factor, a Hall resistivity plateauis formed due to the fact that there is an energy gap between the filled N th Landau level and theempty (N + 1)th Landau level. However, in 1982, H.L. Stormer and D.C. Tsui discovered a Hallplateau at fractional filling factor ν = 1

3 . The phenomena of the formation of Hall plateaus atfractional filling factor is called the fractional quantum Hall effect, or FQHE for short. From thetheory that we have developed so far, this newly discovered plateau does not make any sense. Whenslightly perturbing the filling factor around ν = 1

3 , there is no energy gap for electrons to cross:they can safely stay in the lowest Landau level, which is two-thirds empty. Hence, the reasoning wedid in the previous section for the IQHE cannot hold for fractional filling factors. Besides, in thenon-interacting limit the ground state at fractional filling factor is highly degenerate, because allavailable states in the fractionally filled Landau level have the same energy. Thus, there is no uniqueground state to do perturbation theory on. Nevertheless, R.B. Laughlin [14] succeeded in 1983 toreproduce a wavefunction that resembles the numerically computed ground state wavefunction upto high precision. We will motivate Laughlin’s educated guess in the section below.

The FQHE is only observed in samples with very low impurity. Hence, the impurity potentialis not expected to be the driving mechanism behind the FQHE. Throughout this section, we willassume

~ωC Uimp.

Also, as discussed above, the Landau level structure can by itself not be held accountable for theFQHE. There is one remaining mechanism that we have neglected when discussing the IQHE: thatof the electron-electron interaction. Hence, we additionally assume that the interaction energy is ofa order comparable to the kinetic energy,

~ωC ≈ Uint.

In the remainder of this chapter we will discuss Laughlin’s variational ground state wavefunction,which only holds for filling fractions of the form ν = 1

2s+1 with s ∈ N; the anyonic single particleexcitations of the FQHE fluid; and the theory of composite fermions which works for the moregeneral fractions of the form ν = p

2sp+1 with s, p ∈ N. There are other fractions at which the FQHEis observed, like ν = 5

2 , which are not of the form ν = p2sp+1 . For a discussion on the non-composite

fermion fractions, the readers is urged to look elsewhere.

3.2.1 The Ground State Wavefunction at ν = 1

Before we discuss the Laughlin multi-particle wavefunction, let us briefly revisit the single-particlecase.

Remark 3.6. From now on, we will work in the symmetric gauge.

It is easier to work in complex coordinates. We define: z := x−iy and z = x+iy, with correspondingpartial derivative operators ∂ := (∂x + i∂y)/2 and ∂ := (∂x − i∂y)/2. Most mathematicians mightfeel uneasy with this convention, but it is the one used in the literature, so we will stick to it.3 Notethat, as a consequence, the word holomorphic will actually mean anti-holomorphic, and vice versa.

From equation (3.13), we see that the Hamiltonian is effectively a sum of two number operators:a†a and b†b, where the a and b operators are given by equations (3.6) and (3.12) respectively. Recallthat a† raises the energy of a state, and b† changes the position of the guiding centre. That is, a anda† are responsible for hopping between different Landau levels, while b and b† cause hopping within

3We are using this notation to ’compensate’ for the fact that the dynamical particles, in this case electrons, havea negative charge. Using our funny notation, the lowest Landau level will be described by holomorphic instead ofanti-holomorpic functions.


a fixed Landau level. Using equations (3.2) and (3.10) for Π and R = (X,Y ) respectively, we canexpress the ladder operators in terms of complex derivative and multiplication operators.

a = −i√

2(

z

4lB+ lB ∂

), a† = i

√2(

z

4lB− lB∂

), (3.41)

b = −i√

2(

z

4lB+ lB∂

), b† = i

√2(

z

4lB− lB ∂

). (3.42)

A particle is in the lowest Landau level precisely when its corresponding wavefunction ψ is in thekernel of a, that is aψ = 0. The following lemma gives a concrete description of Ker(a).

Lemma 3.4. Ker(a) =f(z) exp

(− |z|2

4l2B

): f is holomorphic

Proof. We will prove two inclusions. First, suppose that ψ ∈ Ker(a). Define f(z, z) := ψ(z, z) exp

(|z|24l2B

).

Since ψ is in the kernel of a, it is most certainly in the domain of a, which means that it is R-differentiable, and therefore so is f . Moreover, using aψ = 0, we get

∂f(z, z) =[∂ψ(z, z) +

z

4l2Bψ(z, z)

]exp

(|z|2

4l2B

)= ψ(z, z)

[− z

4l2B+

z

4l2B

]exp

(|z|2

4l2B

)= 0.

We conclude that f is both R-differentiable and satisfies the Cauchy-Riemann equations ∂f(z, z) = 0.Thus, f is holomorphic, and we write f(z, z) = f(z). The other inclusion follows immediately byapplying a to any element of the set on the RHS of the equation above, which will give zero.

In exactly the same manner, we have the following lemma.

Lemma 3.5. Ker(b) =g(z) exp

(− |z|2

4l2B

): g is anti-holomorphic

Corollary 3.6. Taking the intersection of Ker(a) and Ker(b) yields the one-dimensional subspaceof functions

λ · exp(− |z|2

4l2B), λ ∈ C

.

In particular, the n = 0, m = 0 wavefunction is the unique wavefunction in Ker(a) ∩Ker(b) withnorm 1,

ψ0,0(z) =1√2πl2B

exp(−|z|

2

4l2B

).

All other wavefunctions in Ker(a) are obtained by applying b† to ψ0,0. Each successive applicationof b† gives an extra factor of z. After normalization, we get

ψ0,m(z) =im√

2πl2Bm!

(z√2lB

)m

exp(−|z|

2

4l2B

), (3.43)

which is indeed of the form f(z) exp(− |z|2

4l2B

).

Next, we turn to the N -particle case. Instead of a function of one variable, the ground statewavefunction ψN is now an anti-symmetric wavefunction of N variables

ψN (zi) ∝ fN (zi) exp

(−

N∑i=1

|zi|2

4l2B

), (3.44)

where we again require fN (zi) to be holomorphic in all zi variables. In the above equation, zistands for z1, ..., zN . There is an additional normalization factor, not included in fN , that we willneglect for now.


Let us start by examining a case that we are already familiar with, the case of ν = 1. At ν = 1,the degeneracy of one Landau level NB equals the total number of particles, i.e. N = NB . Fromequation (3.43), we see that a factor of zmi

i in front of the exponential means that the i-th particlehas its guiding centre at position mi. Because the electrons are indistinguishable, The ground statewavefunction ψN (zi) for the case ν = 1 must be a linear combination of wavefunctions representingall possible ways of positioning the different electrons at each of the possible m = 0, ..., N−1 guidingcentre positions, in such a way that the total function is anti-symmetric with respect to the exchangeof individual electron coordinates. There is exactly one function fN that has the desired properties,

fN (zi) = l−N(N−1)

2B det

z01 . . . zN−1

1...

...z0N . . . zN−1

N

. (3.45)

The above N × N determinant is known as the Vandermonde determinant. Using a proof byinduction, it can be shown that the above expression is equal to

fN (zi) =∏i<j

(zi − zj

lB

).

We have found the ground state wavefunction without solving any kind of Schrodinger equation.Of course, for non-interacting particles, this is not surprising, because there is a standard methodfor solving the N -particle case once the one-particle case has been solved: using anti-symmetrizedsingle particle wavefunctions (in the case of fermions). However, Laughlin’s idea was to use thesame properties of holomorphicity and anti-symmetry, together with his physical intuition, on hisansatz ground state wavefunction for other filling factors, namely those of the form 1

2s+1 , to comeup with a trial wavefunction that turned out to describe the ground state extremely well. We willdiscuss Laughlin’s trial wavefunction in Section 3.2.3, but before we do so, let us pause of a secondand try to develop some physical intuition for the N -particle ground state wavefunction ψN givenby equation (3.44). Laughlin noticed that this wavefunction is related to the partition function of awell-known classical system.

3.2.2 Laughlin’s Plasma Analogy

There is a striking resemblance between ground state wavefunction (3.44) at ν = 1, and the partitionfunction of a classical 2D plasma. Let us define

φ(zi)) :=

∏i<j

(zi − zj

lB

) exp

(−

N∑i=1

|zi|2

4l2B

),

to be the unnormalized wavefunction (so that we can stop writing the “∝” symbol). Since |φ(zi)|2is proportional to the probability distribution |ψN (zi)|2, we can regard it as a classical Boltzmannweight

|φ(zi)|2 = exp(−βU) (3.46)

for a suitably chosen potential U . The corresponding partition function is

Z =∫ N∏

i=1

dzi|φ(zi)|2 =∫ N∏

i=1

dzi exp [−βU(zi)] .

Next, we choose β and U in such a way that we can interpret U as the potential for a 2D gas ofinteracting particles with charge q in a uniformly charged background. This requires that we set


β = 2q , and

U(zi) = −q2∑i<j

ln∣∣∣∣zi − zj

lB

∣∣∣∣+ q∑

k

|zk|2

4l2B, (3.47)

which indeed solves equation (3.46), assuming we take q = 1. The reason that we have introducedthe factor of q, is because it assumes the role of charge equation (3.47). At filling factor ν = 1, thecharge has to be 1 in order to make sure that (3.46) is satisfied, so we could have left q out of theequation. However, it will turn out that the plasma analogy also works for other filling factors, forwhich q 6= 1, so we will keep track of q in our calculations below.

Before we continue, let us check that the above potential indeed corresponds to a 2D gas ofinteracting charged particles in a uniformly charged background. In 2D, we have Gauss’ law inintegral form, ∮

E · dn = 2πq, (3.48)

where q is the enclosed charge (modulo a constant, which we have introduced to make the formula’scome out a little bit nicer). For a point charge q at the origin, by the rotational symmetry of theconfiguration, we deduce that E(r) = q

r r. Hence, the corresponding electric potential is given byV (r) = −qln(r/lB), where we have chosen lB as an integration constant. Thus, potential energy fortwo point charges of charge q separated by a distance r is given by

U(r) = −q2ln(r

lB

),

which explains the first term in (3.47).For the second term, which can be interpreted as N charges q each feeling a background potential

VB(z, z) = |z|24l2B

, we use Gauss’ law in differential form to relate the potential VB to some backgroundcharge density ρB . We have

∇2V (r) = −∇ ·E(r) = −2πρB(r).

Hence,

ρB = − 12π∇2

(|z|2

4l2B

)= − 1

2πl2B= − B

φ0.

Indeed, the second term in (3.47) corresponds to the interaction between the charges at positionsz1, . . . , zN and a uniform background charge with a charge density ρB that is equal to minus thedensity of magnetic flux quanta nB – see equation (3.17).

To conclude, equation (3.46) allows us to regard |φ(zi)|2, which is proportional to the proba-bility (distribution) that the N electrons are found at positions z1, . . . , zN , as a Boltzmann weightexp(−βU) of a 2D gas of particles of charge q in a uniformly charged background. Hence, the proba-bility of finding the electrons in the quantum Hall fluid at positions z1, . . . , zN is proportional to theprobability of finding N charge q particles in a 2D charged background at the exact same positions.Thus, we can now use our knowledge of the 2D plasma to gain insight in the quantum Hall fluid.

From electrostatics, it is known that charges in a uniformly charged background will arrangethemselves in such a way that the total charge is neutral, thereby minimizing U . This means thatthe charges at the positions z1, ..., zN will be distributed over the sample in such a way that, whenaveraged over length scales much larger than the average electron distance, the electron chargedensity nel satisfies

nelq + ρB = 0. (3.49)

Since q = 1 at ν = 1, we require nel = −ρB = 12πl2B

, which agrees with (3.18) for ν = 1. Moreover,we know that the wavefunction will be very close to zero whenever the nel does not satisfy the charge


neutrality condition in (3.49), because in that case the Boltzmann factor will be negligibly small.This means that the ν = 1 quantum Hall system is a strongly correlated system, where the electronsarrange themselves in such a way that the electron density nel is approximately constant 1

2πl2Bon

length scales larger than lB .Now that we have established Laughlin’s plasma analogy, we are ready to discuss Laughlin’s

ground state wavefunctions.

3.2.3 Laughlin’s Ground State Wavefunction

The charge neutrality condition (3.49) relates the imaginary charge q to the filling factor. Indeed,combining equations (3.18) and (3.49) yields

q = − ρB

nel=

1nel

12πl2B

=nB

nel=

1ν.

In the previous subsection, we had q = 1. If, instead, we keep q arbitrary, then equations (3.46) and(3.47) suggest that we take

φq(zi) :=

∏i<j

(zi − zj

lB

)q exp

(−

N∑i=1

|zi|2

4l2B

), (3.50)

as a trail unnormalized ground state wavefunction for a filling factor of ν = 1/q. This is Laughlin’strial wavefunction. Comparing to the non-interacting ground state wavefunction (3.44), we requirethat the prefactor is holomorphic, and locally integrable, which means that q has to be a positiveinteger. Moreover, φ has to be antisymmetric with respect to the the exchange of particle coordinates,implying that q has to be odd. This means that the wavefunction in (3.50) serves as a trail groundstate wavefunction for filling factors of the form

ν =1q

=1

2s+ 1,

with s ∈ N0 = 0, 1, 2, 3, . . .. Note that taking s = 0 corresponds to the familiar case of ν = 1.Now that we have obtained a trail ground state wavefunction by an educated guess, let us try to

show that this trial wavefunction indeed describes the ground state of the quantum Hall system atfilling factor ν = 1

2s+1 . In order to do so, we will artificially construct a different kind of interactionpotential of which φq is the exact ground state, and then we will argue that this artificial potential hasa 99% overlap with the Coulomb potential for as far as the ground state wavefunction is concerned.We will focus on the case s > 0, since the s = 0 case corresponds to ν = 1, for which we have alreadyshown that (3.50) describes the ground state.

At fractional filling factor ν = 1/(2s + 1) with s > 0, only the lowest Landau level is partiallyfilled, and the higher Landau levels are completely empty. Thus, the dynamics of the system occurwithin the lowest Landau level, meaning that all electrons have the same kinetic energy, which wecan therefore neglect. Put differently, the only energy relevant for this system is the interactionenergy due to the mutual Coulomb repulsion of the electrons. Before we look at the N -particle case,let us first focus on the case of N = 2 electrons.

Because the electrons have equal charge, the two-particle Coulomb interaction potential UC

should normally speaking not have any bound states: two electrons that feel each others repulsiveCoulomb force simply accelerate away from each other, exchanging potential for kinetic energy.However, since both electrons have fixed kinetic energy, they cannot exchange potential for kineticenergy, and instead rotate around each other, as they would do classically due to the externalperpendicular magnetic field. The wavefunction of two electrons orbiting each other with an angular


momentum ~m has a prefactor of the form (z1 − z2)m. The possible energy eigenstates of the two-electron system are the different bound states labeled by the exponent m. As a consequence, whenacting on the ground state, the potential UC can be expanded as follows

UC =∞∑

m=0

uCmPm,

where Pm is the operator that projects onto the mth bound state, and uCm is the corresponding

eigenvalue of UC , which, because the average distance between two electrons in the ground statewith relative angular momentum m is approximately4 lB

√2m, is equal to

uCm ≈ UC(z = lB

√2m). (3.51)

The above values uCm are called the Haldane’s pseudopotentials.

As with the two-particle case, the N -particle interaction potential U can be decomposed intoHaldane’s pseudopotentials,

U =∑i<j

∞∑m=0

umPm(i, j),

where now Pm(i, j) projects onto the (i, j)th two-particle state with relative angular momentum m.The Coulomb potential UC determines the values for um through formula (3.51). Likewise, anyarbitrary choice of um’s determines an interaction potential U .

We will now construct an artificial interaction potential UA by choosing specific values for theum’s such that φq is the exact ground state wavefunction of UA. We define

uAm :=

uC

m for m < q0 for m ≥ q

. (3.52)

With the above choice of um‘s, it easily follows that UAφq = 0. In order to see this, note that

for each pair (i, j), φq contains a prefactor (zi − zj)q, which means that each pair has a relativemomentum of at least q. Thus Pm(i, j)φq = 0 for m < q, and therefore UAφ

q = 0. Moreover, if φis any other wavefunction that describes the system in the same filling factor ν = 1/q as φq, thenφ can be expressed in terms of the standard exponential times a polynomial prefactor that can bedecomposed into powers of the form

∏i<j(zi − zj)mij . Now, if mij 6= q for any pair of (i, j), then,

in order to keep the filling factor constant, at least one other pair has to have an mi′j′ < q. Thisincreases the potential energy of the state φ by umi′j′ = uC

mi′j′> 0. Therefore, φq is indeed the

lowest energy state for the artificial interaction potential UA. Moreover, we conclude that there is astrictly positive energy gap between the ground state Laughlin wavefunction and the excited states.This means that we have at least one of the required ingredients for the formation of a plateau ofthe Hall resistance at fractional ν = 1/q filling5.

What are we to make of this? After all, we are interested in the Coulomb potential, not inthe artificial potential constructed above. It turns out that, for particle numbers where numericalapproximation of the ground state (using exact diagonalization) is viable, the Laughlin ground statewavefunction φq agrees with over 99% with the numerically computed ground state wavefunctionof the quantum Hall fluid at filling factor ν = 1/q. The reason for this is that, when computingHaldane’s pseudopotentials uC

m for the Coulomb potential, one obtains a series of numbers thatdecreases with m, because of equation (3.51) and the fact that the Coulomb potential decreases

4This value can be obtained by going to the centre of mass and relative coordinates, effectively reducing the two-particle problem to a one-particle problem, and then using the fact that the expectation value of the radius of theguiding centre for a single particle with guiding centre eigenvalue m is lB

√2m. See Goerbig [7] p.41 for details.

5Note that this energy gap is also present when using the Coulomb potential, because the numbers uCm decrease

discretely with m.


as 1/r. Combined with the fact that the actual ground state wavefunction contains but a smallcontribution of high relative angular momentum states, the interaction potential from (3.52) isa good approximation to the Coulomb potential for as far as the ground state wavefunction isconcerned.

3.2.4 Elementary Excitations

In the previous section we have seen that there is an energy gap between the Laughlin groundstate wavefunction and the excited states. In order to understand the occurrence of the QHE atfilling factors of the form ν = 1

2s+1 , we have to establish the nature of the excitations. We arelooking for effectively non-interacting charged single particle excitations, that, when created, due tothe presence of an energy gap become localized in the potential valleys of the impurity potential,causing the formation of a plateau in the Hall resistance exactly as in the IQHE – see Section 3.1.5.

It turns out that there are two kinds of excitations relative to the ground state: elementaryexcitations that involve changing the filling factor, and collective charge-density wave excitationsthat keep the overall filling factor fixed. We will only focus on the first kind of excitations, becausethese are specific to the FQHE. Regarding the charge-density wave excitations, we can mention thatthe dispersion relation has a minimum at some non-zero wavevector, suggesting that for the lowenergy excitations the electrons arrange themselves in a regular pattern such as a Wigner crystal;but we will not go into detail here.

The elementary excitations are created by changing the filling factor through the addition ofeither charge or flux to the system. For Laughlin’s wavefunction (3.50), the exponent q is theparameter that is related to the filling factor. However, we cannot perturb q around 2s+ 1, becauseq is required to be an odd integer. Instead, as an ansatz for a wavefunction that describes andelementary exited state, we manually add a prefactor of the form

∏j(zj − z0)/lB to the Laughlin

wavefunction

φqh(z0, zi) :=N∏

j=1

(zj − z0lb

)φq(zi). (3.53)

The subscript qh stands for quasi-hole, which we will elaborate on below.What influence does the extra prefactor have on the ground state? Since original prefactor of φq

is holomorphic in all its arguments, we can expand it around the point z0,

φq(zi) =∞∑

m1,...,mN=0

[cm1,...,mN

(z1 − z0lB

)m1

. . .

(zN − z0lB

)mN]

exp

− N∑j=1

|zj |2

4l2B

,

Thus, φqh becomes

φqh(z0, zi)∞∑

m1,...,mN=0

[cm1,...,mN

(z1 − z0lb

)m1+1

. . .

(zN − z0lB

)mN+1 ]exp

− N∑j=1

|zj |2

4l2B

.

(3.54)From equation (3.43) we see that the exponent of the factor (zi−z0) is related to the position of theguiding centre of zi relative to z0. Recall that there are at most NB = mmax possible positions for theguiding centre in the sample. Now, equation (3.54) says that adding the extra factor

∏j(zj− z0)/lB

to φqh increases all the powers mi by one. This means that relative to z0, all the particles have theirguiding centre increased by one, leaving a hole at the center z0. Therefore, we call excitations of theform φqh quasi-holes. Moreover, since all the mi are increased by one, so is mmax = NB . Hence,we conclude that the function φext in (3.53) corresponds to an exited state where the Landau leveldegeneracy NB , and therefore the total flux penetrating the system, has increased.

In order to determine the charge of a quasi-hole, we follow Murthy and Shankar [19] (see p.9) andobserve what happens when adiabatically adding a flux tube, i.e. a localized quantum of flux φ0 = h

e


— the flux associated to a quasi-hole — to the system. Let Σ be a disc of radius r centered at thepostion z0 of the quasi-hole, which we take to be origin for convenience. Increasing the magneticflux locally by an amount φ0 in a time interval [0, T ] generates an azimuthal electric field E dueto Faraday’s law of Induction: E(r, t) = Eθ(r, t)eθ. Because of the Hall effect, this electric fieldgenerates a radial current j(r, t) = jr(r, t)er with radial component jr = Eθ

ρH, where ρH is the Hall

resistivity. Using the fact that, at the Laughlin filling fractions, the quantum Hall resistivity equalsthe classical Hall resistivity (2.20), we have

ρH =B

enel=

B

enBν=

2πl2BBeν

=h

e2ν.

Hence, radial component jr of the current j is given by

jr(r, t) =e2ν

hEθ(r, t).

The total charge entering the disc Σ per unit time is given by Q = −j(r)2πr. Hence, the totalcharge accumulated in the time interval [0, T ] wherein the flux tube is inserted is equal to

Q = −∫ T

0

j(r, t)2πrdt = −e2ν

h

∫ T

0

Eθ(r, t)2πrdt. (3.55)

Using Faraday,

Eθ(r, t)2πr =∮

∂Σ

E(r, t) · dr = − d

dt

∫Σ

Bd2r = −dΦdt, (3.56)

where Φ(t) is the flux penetrating the disc at time t. Combining (3.55) and (3.56) gives that thetotal charge Q accumulated in the time interval [0, T ] is equal to

Q =e2ν

hΦ(T ) =

e2ν

hφ0 = eν.

Hence, by taking r → 0, adiabatically inserting the flux tube results in the accumulation of eν chargeat z0. We conclude that the quasi-hole carries a fractional charge of eν.

In addition to being fractionally charged, the quasi-holes also obey so called fractional statistics,meaning that the total wavefunction describing multiple quasi-holes picks up a complex phase eiαπ

when to quasi-holes are exchanged, where α is not necessarily an integer. Particles obeying fractionalstatistics are also called anyons. We will discuss anyons detail in Chapter 6. Specifically, the anyonicnature of the quasi-holes will be explained in Section 6.1.1.

In addition to quasi-holes, we can also create quasi-particles by multiplying the Laughlin wave-function with a prefactor

∏j(zj − z0)/lB . However, doing so means that the total prefactor is no

longer analytic, implying that we are no longer describing a state in the lowest Landau level. Toremedy this situation, we have to orthogonally project onto the lowest Landau level after adding theprefactor. That is,

φqp(z0, zi) := PLLL

N∏j=1

(zj − z0lB

)φq(zi),

where PLLL denotes the orthogonal projection operator that projects onto the lowest Landau level.To summarize, we have found wavefunctions that give excellent approximations to the ground

state wavefunctions at fractional filling factor ν = 12s+1 . We then identified the single particle ground

state excitations to be quasi-holes and quasi-particles, carrying fractional charge and one quantumof flux. When created, the quasi-holes and quasi-particles get pinned to the equipotential lines ofthe impurity potential, which, by the same reasoning as for the IQHE, cause the Hall resistance toform plateaus around filling factors of the form ν = 1

2s+1 .As mentioned at the beginning of this chapter, not all filling factors at which the FQHE is

observed are of the form ν = 12s+1 . In order to explain the FQHE for non-Laughlin fractions, we

turn to composite fermion theory.


3.2.5 Jain’s Composite Fermion Theory

The Laughling wavefunctions describe the FQHE ground states at filling factors of the form ν = 12s+1 ,

where s is a non-zero integer. However, plateaus in the Hall resistance have been observed at otherfilling factors. We will discuss those which can be described by composite fermion theory, a theoryproposed by Jain in 1989 [11] which concerns filling factors of the form

ν =p

2ps+ 1, (3.57)

where both p and s are non-negative integers. Experimentally observed factors not of the form(3.57), such as the 5/2 and 7/2 states, will not be discussed.

The theory of composite fermions is based on the idea of flux attachment, which we illustratewith the following example. Take the Laughlin ground state wavefunction (3.50), where q = 2s+ 1is an odd integer. We can split φ2s+1 into a product of two other functions,

φ2s+1(zi) =

∏i<j

(zi − zj

lB

)2s+1 exp

(−

N∑i=1

|zi|2

4l2B

)

=

∏i<j

(zi − zj

lB

)2s∏

i<j

(zi − zj

lB

) exp

(−

N∑i=1

|zi|2

4l2B

)

=

∏i<j

(zi − zj

lB

)2s ξ1(zi).

When going from the second to the third line, we have recognized the ν = 1 ground state wave-function (3.44), denoted by ξ1, as a part of the Laughlin ν = 1

2s+1 wavefunction. As explained inSection 3.2.4, the quasi-hole-like prefactor means that we have attached 2s quasi-holes, each carryingone flux quantum, to each of the electrons located at the positions z1, ..., zN . This means we caninterpret the ν = 1

2s+1 Laughlin ground state wavefunction as a ν∗ = 1 ground state wavefunctionfor composite particles, where each composite particle consists of one electron with 2s flux quantaattached to it. The star indicates that we are not talking about the actual filling factor, which isstill 1

2s+1 , but about the composite fermion filling factor. Indeed, after attaching 2s flux quanta toeach electron, the remaining flux quanta to composite fermion ratio is indeed 1 : 1, agreeing withν∗ = 1 found above.

Jain’s idea is to extend the splitting of the wavefunctions illustrated above to more general fillingfactors of the form p

2ps+1 , as follows. At ν = 12s+1 we showed above how to factorize the ground

state wavefunction into a quasi-hole factor and the ν∗ = 1 wavefunction ξ1. Similarly, we canfactorize a ν = p

2sp+1 ground state wavefunction into a quasi-hole part and a ν∗ = p ground statewavefunction, the latter of which we denote by ξp(zi, zi). The wavefunction ξp(zi, zi) consistsof the standard exponential multiplied by a prefactor of the form (3.45), where we now take thedeterminant of a pNB times pNB matrix, a fixed row of which contains all the possible polynomialprefactors associated to the pNB available single-particle eigenstates. The newly obtained compositefermion ground state wavefunction for ν∗ = p takes the form

φs,pJ (zi) = PLLL

∏i<j

(zi − zj

lB

)2s ξp(zi, zi),

where we have projected onto the lowest Landau level because the electron filling factor ν = p2ps+1 < 1,

and therefore φs,pJ has to be a lowest Landau level wavefunction, which it is not initially because ξp

contains anti-holomorphic terms.


The fact that the composite fermion filling factor ν∗ is equal to p can be seen as follows. Supposethat at ν = p

2sp+1 we have an electron density of nel and a flux quanta density of nB = 12πl2B

= Bφ0

.Recall that the filling factor is given by equation (3.18), i.e.

ν =nel

nB.

Once we attach 2s flux quanta carrying φ0 = eh flux each to every electron, the remaining magnetic

field B∗ that acts on the composite fermions is given by

B∗ = B − 2sφ0nel,

and the corresponding flux quanta density is equal to

n∗B = nB − 2snel.

We have as many composite fermions as electrons, Hence,

ν∗ =nel

n∗B=

nel

nB − 2snel=

ν

1− 2sν= p,

where we have used ν = p2sp+1 for the last equality sign.

In short, at filling factor ν = p2ps+1 , we can view our system of electrons as a system of composite

fermions at integer filling factor ν∗ = p. The formation of a plateau in the Hall resistance can beexplained in terms of the IQHE, but now applied to the composite fermions.

This concludes our general discussion on the quantum Hall effects. We will now proceed to thefirst topic of interest, which is that of Casimir operators in the FQHE.

Chapter 4

The FQHE on the Infinite Plane

From now on, we will restrict our attention to the FQHE. The goal of this chapter is to gain abetter understanding of the FQHE fluid in the low temperature limit, where the effective dynamicsoccur within the fractionally filled Landau level. The physics of this fractionally filled Landaulevel is completely captured by the projected density operators, which form an algebra, called theprojected density operator algebra. Understanding this algebra is therefore of prime importance tounderstanding the FQHE. In order to gain a better understanding of the algebra of projected densityoperators, we will try to find and compute certain Casimir operators, which are specific operatorsthat commute with all the elements of the algebra.

In Section 4.1, we will set up the formalism of second quantization and show that the projecteddensity operators satisfy the projected density operator algebra. Once the ground work has beenlaid out, in Section 4.2 we will turn our attention towards finding and computing Casimir operatorsof this same algebra.

4.1 The Projected Density Operator Algebra

We take our 2D system of strongly correlated spinless electrons in a uniform perpendicular magneticfield to be at fractional filling factor, and we assume that it is in the low temperature limit. Asmentioned at the beginning of Section 3.2, we neglect the impurities in the sample, but we do includeboth the kinetic and the interaction term in the Hamiltonian.

Let n0 denote the (fractionally filled) highest non-empty Landau level. The low energy excitationsof our system are intra-Landau level excitations within the nth

0 Landau level, because they do notinvolve enough energy to have an electron jump to the next unoccupied Landau level. This situationis best described with the language of second quantization.

4.1.1 Second Quantization

Because we will be working in the language of second quantization, we allow the number of particlesto fluctuate. That is, we introduce Fock space F as the direct sum of all n-particle states Fn,

F :=∞⊕

n=0

Fn,

where Fn is the anti-symmetrized (since the dynamical particles are electrons, which are fermions)tensor product of n copies of the single particle Hilbert space H defined in Section 2.2.4: Fn :=H∧H∧ . . .∧H (n-times), and F0 := C. Recall from Section 2.2.4 that the single particle space H is

48

4.1. THE PROJECTED DENSITY OPERATOR ALGEBRA 49

the space of square-integrable sections L2(R2,L) of the line bundle L over R2. We extend the innerproduct 〈−,−〉 on H to an inner product 〈−,−〉F on F as follows,

〈φ1 ∧ . . . ∧ φk, φ′1 ∧ . . . ∧ φ′l〉F :=

det(〈φi, φ

′j〉) if k = l

0 otherwise . (4.1)

As explained in Corollary 3.5, we have a basis of single particle energy eigensections φn,m ∈ H,where n refers to the Landau level andm to the guiding centre. Corresponding to these eigensections,we define the ladder operators cn,m and c†n,m, which are each other’s Hermitian conjugates. Bydefinition, c†n,m acts on an arbitrary state φ1 ∧ . . . ∧ φk ∈ Fk as follows,

c†n,m(φ1 ∧ . . . ∧ φk) := φn,m ∧ φ1 ∧ . . . ∧ φk. (4.2)

That is, c†n,m creates a particle in the eigenstate φn,m. Now, cn,m is defined to be the adjoint of c†n,m

wit respect to the inner product given by (4.1). Explicitly, for any basis state φn1,m1 ∧ . . .∧ φnk,mk,

cn,m(φn1,m1 ∧ . . . ∧ φnk,mk) :=

k∑i=1

(−1)i−1δ(n,m),(ni,mi) φn1,m1 ∧ . . . ∧ φni,mi ∧ . . . ∧ φnk,mk,

where the hat indicates that the term underneath it should be omitted. We see that cn,m destroysa particle in the state |n,m〉, if present. Hence, the ladder operators are commonly referred to ascreation (c†) and annihilation (c) operators.

Using the newly defined ladder operators cn,m and c†n,m, we can define the singular operatorsψ(r) and ψ†(r)

ψ(r) :=∑n,m

φn,m(r)cn,m (4.3)

ψ†(r) :=∑n,m

φ∗n,m(r)c†n,m, (4.4)

that create and destroy particles that are Dirac delta centered at the position r. Since the Diracdelta function is not an element of the one particle Hilbert space H, the operator ψ†(r) only makessense when integrated over. As before, ψ(r) is the adjoint of ψ†(r).

The operators defined above satisfy the following anti-commutation relations:

cn,m, c†n′,m′ = δn,n′δm,m′ , ψ(r), ψ†(r′) = δ(r − r′), (4.5)

cn,m, cn′,m′ = 0, ψ(r), ψ(r′) = 0, (4.6)

c†n,m, c†n′,m′ = 0, ψ†(r), ψ†(r′) = 0. (4.7)

The model that we are considering is that of interacting spinless electrons confined to a two-dimensional plane, subject to a perpendicular magnetic field B = Bez. The Hamiltonian H thatcorresponds to this system consists of a kinetic and an interaction term. In the language of secondquantization, H is given by the following formula,

H = Hkin +Hint

=∫

R2dr ψ†(r)

12m

(Π2x + Π2

y)ψ(r) +12

∫R4dr dr′ ψ†(r′)ψ†(r)U(r′ − r)ψ(r)ψ(r′). (4.8)

Here, Π is one-particle mechanical momentum operator defined by equation (3.2), which in termsof operators acting on the one-particle Hilbert space H reads

Π = −i~∇+ eA(r), (4.9)


and U is the Coulomb interaction potential,

U(r) =e2

4πε01|r|. (4.10)

We will massage the interaction term into a form that is more usefull for our purposes. Fromequations (4.5) – (4.7), we get

Hint =12

∫R4dr dr′ ψ†(r′)ψ†(r)U(r′ − r)ψ(r)ψ(r′)

= −12

∫R4dr dr′ ψ†(r)ψ†(r′)U(r′ − r)ψ(r)ψ(r′)

=12

∫R4dr dr′ ψ†(r)ψ(r)U(r′ − r)ψ†(r′)ψ(r′)− 1

2

∫R4dr dr′ ψ†(r)U(r′ − r)ψ(r′)δ(r − r′)

=12

∫R4dr dr′ ρ(r)U(r′ − r)ρ(r′)− U(0)

2

∫R2dr ρ(r).

where we have defined the density operator

ρ(r) := ψ†(r)ψ(r).

The extra term U(0)2

∫R2 dr ρ(r) that we have picked up through our manipulations, albeit infinite, is

merely a constant, since the integral∫

R2 dr ρ(r) is the operator that counts the number of electronsin the system, which is fixed for our purposes. Because it is the energy differences that determinethe dynamics of our system, we will ignore this infinite but constant term.

In short, the Hamiltonian that we will be working with is given by

H =∫

R2dr ψ†(r)

12m

(Π2x + Π2

y)ψ(r) +12

∫R4dr dr′ ρ(r)U(r − r′)ρ(r′). (4.11)

4.1.2 Projection onto the Fractionally Filled Landau Level

Since all the effective dynamics occur within the nth0 Landau level, we want to work with operators

that act only on the nth0 Landau level. The most obvious way to achieve this is to restrict ourselves

to using only ladder operators cn,m and c†n,m with n = n0. Hence, we define

ψn0(r) :=∑m

φn0,m(r)cn0,m,

ψ†n0(r) :=

∑m

φ∗n0,m(r)c†n0,m.

In the nth0 Landau level, the kinetic energy is constant ~ωC(n0 + 1/2), and can therefore be

ignored. Thus, the effective Hamiltonian consists only of the interaction term in equation (4.11),which, after projecting to the nth

0 Landau level, takes the form

Heff =12

∫dr dr′ψ†n0

(r)ψn0(r)U(r − r′)ψ†n0(r′)ψn0(r

′)

=12

∫drdr′ ρn0(r)U(r − r′)ρn0(r

′), (4.12)

where we have defined the projected density operator

ρn0(r) := ψ†n0(r)ψn0(r).


It is the projected density operators ρn0 that we are interested in, because, in the nth0 Landau

level, the only relevant type of energy is the electron-electron interaction energy, which depends onthe electron density distribution that is described by the projected density operators. Indeed, theeffective Hamiltonian (4.12) is completely described in terms of the projected density operators. Theremainder of Section 4.1 deals with the projected density operators, and the algebra that they form.

For computational convenience, we will describe our system in Fourier space. Fourier transform-ing the projected density operators yields for the effective Hamiltonian

Heff =12

∫drdr′

∫dq

(2π)2ρn0(q)eiq·r

∫dq′

(2π)2ρn0(q

′)eiq′·r′U(r − r′)

=12

∫dq

(2π)2dq′

(2π)2ρn0(q)ρn0(q

′)∫drdr′ ei(q·r+q′·r′)U(r − r′)

=12

∫dq

(2π)2dq′

(2π)2ρn0(q)ρn0(q

′)∫dsdt

2e

i2 (q·(s+t)+q′·(s−t))U(t)

=12

∫dq

(2π)2dq′

(2π)2ρn0(q)ρn0(q

′)(2π)2δ(q + q′)∫dt e

i2 t·(q−q′)U(t)

=12

∫dq

(2π)2ρn0(q)ρn0(−q)

∫dt eit·qU(t)

=12

∫dq

(2π)2U(q)ρn0(q)ρn0(−q), (4.13)

where U(q) = e2

2ε01|q| is the Fourier transform of the Coulomb potential (4.10) — see Appendix A.2

for details — and,

ρn0(q) :=∫dr e−iq·rρn0(r).

We adopt the convention that the factors of 12π always accompany the q-integrals.

Because we have projected to a single Landau level, we prefer to express our operators in termsof the energy eigenbasis |n,m〉 instead of the position basis |r〉. We have,

ρn0(q) =∫dr e−iq·rψ†n0

(r)ψn0(r) (4.14)

=∑

m,m′

∫dr φ∗n0,m(r)e−iq·rφn0,m′(r) c†n0,mcn0,m′

=∑

m,m′

〈n0,m|e−iq·r|n0,m′〉c†n0,mcn0,m′ , (4.15)

where, going from the second to the third line, we observed that∫dr φ∗n0,m(r)e−iq·rφn0,m′(r) is the

expectation value 〈n0,m|e−iq·r|n0,m′〉 of the operator e−iq·r.1

The next step is to simplify the matrix coefficients 〈n0,m|e−iq·r|n0,m′〉 of ρn0(q). In doing so,

we can make use of the fact that the one-particle Hilbert space H is a tensor product of two harmonicoscillator Hilbert spaces: H ∼= H0 ⊗Ker(a) — see Corollary 3.5 — and write the position operatorr as a sum of the guiding centre operator R and the cyclotron radius operator η: r = R+η. Recallthat η is related to the mechanical momentum Π by (3.9). Because Πx and Πy commute with Xand Y — see Appendix equations (A.3) and (A.4), so do ηx and ηy. Hence, we can split e−iq·r intotwo,

e−iq·r = e−iq·ηe−iq·R. (4.16)

1We will refrain from putting hats on r whenever it represents the position operator instead of the positioncoordinate, as it should be clear from the context which of the two it is meant to represent.


Now, using that η only acts on the H0 part and R only acts on the Ker(a) part of H, we get

〈n0,m|e−iq·r|n0,m′〉 = 〈n0|e−iq·η|n0〉〈m|e−iq·R|m′〉. (4.17)

The first term in the above expression is called the form factor :

Fn0(q) := 〈n0|e−iq·η|n0〉. (4.18)

With the above definition, equation (4.15) becomes

ρn0(q) = Fn0(q)ρn0(q), (4.19)

where, by definition,ρn0(q) :=

∑m,m′

〈m|e−iq·R|m′〉c†n0,mcn0,m′ , (4.20)

which we will also refer to as a projected density operator. Before we proceed to discuss the projecteddensity operator algebra, we will first compute the form factor in (4.18).

Remark 4.1. Instead of projecting to the nth0 Landau level by restricting the sums over n in (4.3)

and (4.4) to n = n0, we could have also first projected the first quantized single particle operatorσ(q) := e−iq·r to an intra-Landau level operator, and then proceded to second quantization. Indeed,using (4.16) - (4.18), projecting σ(q) to the nth

0 Landau level means

σ(q) = e−iq·r Pn0−−→ Fn0(q)e−iq·R = Fn0(q)σ(q),

where σ(q) := e−iq·R is a projected density operator that acts only within each single Landau level.This is the approach that we will take in the next chapter, when we consider a finite sample with

periodic boundary conditions.

4.1.3 Computation of the Form Factor

In order to compute the form factor, we will follow the reasoning of Lederer and Goerbig [15]. Tostart with, let us express the operator e−iq·η in terms of the single-particle ladder operators a anda†. In the computation below, we distinguish between the vector q = (qx, qy) ∈ R2 and the complexnumber q := qx + iqy. Now, from equation (A.7),

q · η = qxηx + qyηy =ilB√

2(−a† + a)qx −

lB√2(a† + a)qy

=lB√2

[(−iqx − qy)a† + (iqx − qy)a

]=

lB√2

[− i(qx − iqy)a† + i(qx + iqy)a

]=ilB√

2(−q∗a† + qa).

Hence, by the Baker-Campbell-Hausdorff formula

eA+B = eAeBe−12 [A,B], (4.21)

which holds whenever the commutator [A,B] commutes with both A and B, we have,

e−iq·η = elB√

2(−q∗a†+qa) = e

− lB√2q∗a†

elB√

2qae−

12

l2B2 [−q∗a†,qa] = e

− lB√2q∗a†

elB√

2qae−

l2B4 |q|

2.


Back to the form factor F (q). Recall that, because, in equation (4.17), we have split the state|n,m〉 into an |n〉 and an |m〉 part, we are only acting on the energy part of the Hilbert space,denoted by H0 (see Corollary 3.5). The idea is to insert the identity operator

∑k |k〉〈k| in the

middle and pull out the sum over k. Here, |k〉 refers to the energy eigenstates of the non-degenerateHarmonic oscillator H0. This yields

Fn0(q) = 〈n0|e−iq·η|n0〉 = e−l2B4 |q|

2〈n0|e−

lB√2q∗a†

elB√

2qa|n0〉

= e−l2B4 |q|

2∞∑

k=0

〈n0|e−lB√

2q∗a† |k〉〈k|e

lB√2qa|n0〉

= e−l2B4 |q|

2∞∑

k=0

〈n0|e−lB√

2q∗a† |k〉

(〈n0|e

lB√2q∗a† |k〉

)∗.

For fixed k, let us examine the term 〈n0|e−lB√

2q∗a† |k〉. Since the set of all |k〉 = 1√

k!(a†)k|0〉 is an

orthonormal basis of H0, at most one term in the exponential survives. That is, assuming k ≤ n0,we have,

〈n0|e−lB√

2q∗a† |k〉 = 〈n0|

1(n− k)!

(− lB√

2q∗a†

)n0−k

|k〉

=1

(n0 − k)!

√n0!k!

(− lBq

∗√

2

)n0−k

〈n0|n0〉

=1

(n0 − k)!

√n0!k!

(− lBq

∗√

2

)n0−k

, (4.22)

and for k > n0, we get,

〈n0|e−lB√

2q∗a† |k〉 = 0.

Finally, the form factor can be computed.

Fn0(q) = e−l2B4 |q|

2n0∑

k=0

1(n0 − k)!

√n0!k!

(− lBq

∗√

2

)n0−k 1(n0 − k)!

√n0!k!

(lBq√

2

)n0−k

= e−l2B4 |q|

2n0∑

k=0

n0!(n0 − k)!k!

1(n0 − k)!

(−l2B |q|2

2

)n0−k

= e−l2B4 |q|

2n0∑i=0

n0!i!(n0 − i)!

1i!

(−l2B |q|2

2

)i

Let us summarize the result of this section in the following Corollary.

Corollary 4.1. The form factor Fn0(q) = 〈n0|e−iq·η|n0〉 corresponding to the nth0 Landau level is

given by

Fn0(q) = e−l2B4 |q|

2Ln0

(l2B |q|2

2

), (4.23)

where Ln is the nth Laguerre polynomial, which is defined by

Ln(x) :=n∑

i=0

(n

i

)(−x)i

i!=

n∑i=0

n!i!(n− i)!

(−x)i

i!.

Having computed the form factor, we now turn to the projected density operators.


4.1.4 Derivation of the Algebra

For two projected density operators ρn0(q) and ρn0(s), given by equation (4.20),

ρn0(q) =∑

m,m′

〈m|e−iq·R|m′〉c†n0,mcn0,m′ ,

both ρn0(q) + ρn0(s) and ρn0(q)ρn0(s) make sense as operators acting on the nth0 Landau level.

Definition 4.1. Let Apd be the unital associative algebra over C generated by all the projecteddensity operators ρn0(q) with q ∈ R2. We can add a Lie algebra structure to Apd by defining theLie bracket of two elements ρn0(q), ρn0(s) ∈ Apd to be

[ρn0(q), ρn0(s)] := ρn0(q)ρn0(s)− ρn0(s)ρn0(q).

The topic of this section is to compute the Lie bracket on Apd.Before we focus on the Lie algebra structure of the (second quantized) projected density operators

ρn0(q), we will first look into the (first quantized) single particle projected density operators σ(q)introduced in Remark 4.1.

Lemma 4.1. The single particle projected density operators σ(q) = e−iq·R form a unital associativealgebra over C for which the product has the following simple expression,

σ(q)σ(s) = exp(− il

2B

2q × s

)σ(q + s). (4.24)

Notation: for vectors q, s ∈ R2, q × s means q1s2 − q2s1.

Proof. The fact that the operators form a unital associative algebra over C is obvious. Now, byequation (3.11),

[q ·R, s ·R] = il2Bq × s,

which commutes with both q ·R and s ·R. Hence, using Baker-Campbell-Hausdorff (4.21),

σ(q)σ(s) = e−iq·Re−is·R = e12 [−iq·R,−is·R]e−i(q+s)·R = exp

(− il

2B

2q × s

)σ(q + s).

Corollary 4.2. The single-particle projected density operators σ(q) = e−iq·R satisfy the followingLie bracket,

[σ(q), σ(s)] = −2i sin(l2B2

q × s

)σ(q + s). (4.25)

The associative algebra in (4.24) is called the magnetic translation algebra2 because the same algebrais satisfied by the magnetic translation operators, which we will discuss in Section 5.1.2.

With the help of Corollary 4.2, we can derive an explicit formula for the Lie bracket between twosecond quantized projected density operators ρn0(q). But, before we do, let us make a notationalremark.

2The magnetic translation algebra is usually written with a plus sign in the exponent. In our case, the minussign can be traced back to the negative charge of the electrons. Some authors perfer to work with a positive sign inthe algebra, and choose to have the magnetic field point in the −z direction to compensate for the negative electroncharge, but we will not do so.


Remark 4.2. To avoid cluttering, we will drop the n0 index in all formulas form now on. However,it should be kept in mind that everything that follows refers to the nth

0 Landau level. In particular,ψ(r) actually means ψn0(r), and, likewise, cm refers to cn0,m.

First, we compute the commutator [c†mcm′ , c†l cl′ ], for which we require the following two identities:

[AB,C] = A[B,C]± − [C,A]±B, (4.26)[A,BC] = [A,B]±C −B[C,A]±, (4.27)

which hold in any associative algebra. Using equations (4.26) and (4.27), we get

[c†mcm′ , c†l cl′ ] = c†m[cm′ , c†l cl′ ]− [c†l cl′ , c†m]cm′

= c†m

(cm′ , c†l cl′ − c†l · 0

)−(c†l cl′ , c

†m − 0 · cl′

)cm′

= c†mcl′δm′,l − c†l cm′δl′,m. (4.28)

Lemma 4.2. The Lie bracket between two projected density operators ρ(q) and ρ(s) is given by

[ρ(q), ρ(s)] = −2i sin[(q × s)l2B

2

]ρ(q + s). (4.29)

Proof. By equation (4.28),

[ρ(q), ρ(s)] =∑

m,m′,l,l′

〈m|e−iq·R|m′〉〈l|e−is·R|l′〉[c†mcm′ , c†l cl′ ]

=∑

m,m′,l′

〈m|e−iq·R|m′〉〈m′|e−is·R|l′〉c†mcl′ −∑

m′,l,l′

〈l|e−is·R|l′〉〈l′|e−iq·R|m′〉c†l cm′

=∑m,l′

〈m|e−iq·Re−is·R|l′〉c†mcl′ −∑m′,l

〈l|e−is·Re−iq·R|m′〉c†l cm′

=∑

m,m′

〈m|[e−iq·R, e−is·R]|m′〉c†mcm′ . (4.30)

Now use Corollary 4.2, which says that

[e−iq·R, e−is·R] = −2i sin[(q × s)l2B

2

]e−i(q+s)·R,

and then pull the sine factor out of the sum over m and m′ in (4.30).

The algebra of projected density operators (4.29) was first identified by Girvin, MacDonald andPlatzman [5], and is commonly referred to as the GMP algebra.

Remark 4.3. Despite the fact that the Fock space projected density operators ρ(q) inherit the Liealgebra structure (4.25) of the single particle projected density operators σ(q), the associative algebrastructure is completely different; that is, the ρ(q)’s do not satisfy (4.24). In the next section, wewill find out that this makes computing the Casimir operators, which are trivial in case of the singleparticle operators, a lot more difficult.

Let us summarize Section 4.1. The dynamics of the electrons in the partially filled nth0 Landau

level are determined by an algebra of projected density operators ρ(q) labelled by vectors q ∈ R2.The commutator of two projected density operators is given by

[ρ(q), ρ(s)] = −2i sin[(q × s)l2B

2

]ρ(q + s).

4.2. CASIMIR OPERATORS 56

The effective Hamiltonian that acts on the electrons in the nth0 Landau level can be expressed in

terms of the projected density operators: from equation (4.13),

Heff =12

∫dq

(2π)2U(q)ρ(q)ρ(−q),

where we have absorbed the form factors (4.19) into the Coulomb potential,

U(q) := U(q)Fn0(q)Fn0(−q) =e2

2ε0|q|e−

l2B2 |q|

2[Ln0

(l2B |q|2

2

)]2.

In the last equality sign above we have used U(q) = e2

2ε01|q| , where e is the electron charge and not

Euler’s number, and equation (4.23) for the form factor.As it stands, we have a complete description of the mathematical landscape that we are about

to explore, and in the next section, we set about doing exactly that.

4.2 Casimir Operators

In an attempt to get a better understanding of the algebra of projected density operators, we willtry to find Casimir operators of the algebra.

Casimir operators are specific operators that lie in the center of the algebra, meaning that theycommute with all the other elements. More precisely, in a general Lie algebra g with a countablebasis Xi, where i runs over some countable index set I, the second order Casimir ζ is the elementof the universal enveloping algebra U(g) of g given by ζ =

∑i∈I XiX

i, where Xi is the dual vectorof Xi with respect to a fixed invariant bilinear form. There are additional Casimir elements for eachorder, for which we will give explicit formulas in the section below.

Casimir operators are quite common in physics. By Dixmier’s lemma3, when acting on anirreducible vector space, Casimir operators are represented by scalar multiples of the identity. Thesescalars sometimes carry physical significance. For example, the total angular momentum and thetotal spin operator are Casimir operators, and the corresponding scalars that represent them arethe total angular momentum and total spin respectively. Keeping this in mind, we now endeavor tofind and compute Casimir operators of the algebra of projected density operators, in the hope thatthey too, carry physical significance.

4.2.1 Construction of the Casimir Operators

The first Casimir operator that we find is quite similar to the effective Hamiltonian itself: it differsonly in the fact that the Coulomb potential is absent.

Theorem 4.1. the operator

ζ :=∫

R2

dq

(2π)2ρ(q)ρ(−q) (4.31)

commutes with all elements of the associative algebra Apd generated by the ρ(q)’s.

Proof. We will prove the theorem by hand using the ladder operator expression of the projecteddensity operators (4.20). In order to prove the above lemma, we have to show that ζ commutes witheach of the ρ(q)’s. Explicitly, ζ is given by

ζ =∫

dq

(2π)2∑

m,m′,l,l′

〈m|e−iq·R|m′〉〈l|eiq·R|l′〉c†mcm′c†l cl′ . (4.32)

3An infinite dimensional generalization of Schur’s lemma for associative algebras over an uncountable and alge-braically closed field, which says that any central element of the algebra is represented by a scalar when the algebraacts irreducibly on some vector space.


Now, for fixed s ∈ R2, we have

[ζ, ρ(s)] =∫

dq

(2π)2∑

m,m′,l,l′

〈m|e−iq·R|m′〉〈l|eiq·R|l′〉∑p,p′

〈p|e−is·R|p′〉[c†mcm′c†l cl′ , c†pcp′ ],

where, using (4.26) – (4.28),

[c†mcm′c†l cl′ , c†pcp′ ] = c†mcm′ [c†l cl′ , c

†pcp′ ] + [c†mcm′ , c†pcp′ ]c

†l cl′

= c†mcm′(c†l cp′δl′,p − c†pcl′δp′,l) + (c†mcp′δm′,p − c†pcm′δp′,m)c†l cl′ .

Thus, by (4.25)

[ζ, ρ(s)] =∫

dq

(2π)2

[ ∑m,m′,l,p′

〈m|e−iq·R|m′〉〈l|eiq·Re−is·R|p′〉c†mcm′c†l cp′

−∑

m,m′,l′,p

〈m|e−iq·R|m′〉〈p|e−is·Reiq·R|l′〉c†mcm′c†pcl′

+∑

m,l,l′,p′

〈l|eiq·R|l′〉〈m|e−iq·Re−is·R|p′〉c†mcp′c†l cl′

−∑

m,l,l′,p

〈l|eiq·R|l′〉〈p|e−is·Re−iq·R|m′〉c†pcm′c†l cl′

]

=∫

dq

(2π)2

[ ∑m,m′,l,l′

〈m|e−iq·R|m′〉〈l|[eiq·R, e−is·R]|l′〉c†mcm′c†l cl′

+∑

m,m′,l,l′

〈l|eiq·R|l′〉〈m|[e−iq·R, e−is·R]|m′〉c†mcm′c†l cl′

]

=∫

dq

(2π)2

[2i sin

((q × s)l2B

2

)(ρ(q)ρ(−q + s)− ρ(q + s)ρ(−q)

)]

=∫

dq

(2π)2

[2i sin

((q × s)l2B

2

)(ρ(q + s)ρ(−q)− ρ(q + s)ρ(−q)

)]= 0.

Going to the last equality sign above, we relabeled q 7→ q+s in the first term. Note that (q+s)×s =q × s.

It turns out that the above computation can be shortened significantly by exploiting the com-mutation relation (4.29) directly. We have,

[ζ, ρ(s)] =∫

dq

(2π)2[ρ(q)ρ(−q), ρ(s)]

=∫

dq

(2π)2

(ρ(q)[ρ(−q), ρ(s)] + [ρ(q), ρ(s)]ρ(−q)

)=∫

dq

(2π)2

[2i sin

((q × s)l2B

2

)(ρ(q)ρ(−q + s)− ρ(q + s)ρ(−q)

)]= 0.


That is, the element ζ is a central element of the abstract algebra Apd itself, by which we mean thatwe forget about the action of Apd on Fock space.

The reasoning above can be generalized to Casimir operators of higher order. The expressionsfor the higher order Casimir operators are obtained from Fairly, Fletcher and Zachos [3], but theproof below is not.

Theorem 4.2. For each r ∈ N, we have a Casimir invariant, given by,

ζr+1 :=∫

dq1

(2π)2. . .

dqr

(2π)2

(∏i<j

e−i2 l2Bqi×qj

)ρ(q1) · ... · ρ(qr)ρ(−q1 − ...− qr). (4.33)

Remark 4.4. The index R := r + 1 refers to the amount of projected density operators that occurin the expression for the Casimir. It is called the order of the Casimir operator. Note that ζ2 = ζ,since the cross-product in the exponential vanishes.

Proof. We will prove that ζr+1 is central in Apd.Equation (4.26) for commutators can be generalized to an arbitrary product of n operators:

[A1 · . . . ·An, B] =n∑

i=1

A1 · . . . ·Ai−1 · [Ai, B] ·Ai+1 · . . . ·An. (4.34)

To verify the above equation, simply expand the RHS and note that all terms except the first andthe last cancel, or prove by induction using (4.26). Now, by (4.34),

[ζr+1, ρ(s)

]=∫

dq1

(2π)2. . .

dqr

(2π)2

(∏i<j

e−i2 l2Bqi×qj

)[ρ(q1) . . . ρ(qr)ρ(−q1 − . . .− qr), ρ(s)

]=∫

dq1

(2π)2. . .

dqr

(2π)2

(∏i<j

e−i2 l2Bqi×qj

)

×

r∑

l=1

ρ(q1) . . . ρ(ql−1)[ρ(ql), ρ(s)

]ρ(ql+1) . . . ρ(qr)ρ(−q1 − . . .− qr)

+ ρ(q1) . . . ρ(qr)[ρ(−q1 − . . .− qr), ρ(s)

]

=r∑

l=1

−∫

dq1

(2π)2. . .

dqr

(2π)2

(∏i<j

e−i2 l2Bqi×qj

)2i sin

(l2B ql × s

2

)

× ρ(q1) . . . ρ(ql−1)ρ(ql + s)ρ(q1+1) . . . ρ(qr)ρ(−q1 − . . .− qr)

+∫

dq1

(2π)2. . .

dqr

(2π)2

(∏i<j

e−i2 l2Bqi×qj

)2i sin

(l2B (q1 + . . .+ qr)× s

2

)× ρ(q1) . . . ρ(qr)ρ(−q1 − . . .− qr + s). (4.35)

Let us focus on the l-th term in the sum above. We will perform a change of variables to thenew variable ql := ql + s, and then remove the tilde afterwards, which amounts to replacing ql byql − s in the expression above. This change of variables leaves the measure unchanged, but we dopick up extra terms from the exponentials(∏

i<j

e−i2 l2Bqi×qj

)7→

(∏i<j

e−i2 l2Bqi×qj

)(∏i<l

ei2 l2Bqi×s

)(∏l<j

ei2 l2Bs×qj

),


while the sine term remains unaltered. Rewriting the sine term as a difference of two exponentials

−2i sin(l2B ql × s

2

)= exp

(− il2Bql × s

2

)− exp

(il2Bql × s

2

),

we have

−

(∏i<j

e−i2 l2Bqi×qj

)2i sin

(l2B ql × s

2

)

7→

(∏i<j

e−i2 l2Bqi×qj

)[(∏i<l

ei2 l2Bqi×s

)(∏l≤j

e−i2 l2Bqj×s

)−

(∏i≤l

ei2 l2Bqi×s

)(∏l<j

e−i2 l2Bqj×s

)]under ql 7→ ql − s.

Hence, performing the transformation ql 7→ ql − s in each term of the sum over l in (4.35)separately gives[ζr+1, ρ(s)

]=

r∑l=1

∫dq1

(2π)2. . .

dqr

(2π)2

(∏i<j

e−i2 l2Bqi×qj

)

×

[(∏i<l

ei2 l2Bqi×s

)(∏l≤j

e−i2 l2Bqj×s

)−

(∏i≤l

ei2 l2Bqi×s

)(∏l<j

e−i2 l2Bqj×s

)]

× ρ(q1) . . . ρ(qr)ρ(−q1 − . . .− qr + s)

+∫

dq1

(2π)2. . .

dqr

(2π)2

(∏i<j

e−i2 l2Bqi×qj

)2i sin

(l2B (q1 + . . .+ qr)× s

2

)× ρ(q1) . . . ρ(qr)ρ(−q1 − . . .− qr + s)

=∫

dq1

(2π)2. . .

dqr

(2π)2

(∏i<j

e−i2 l2Bqi×qj

)ρ(q1) . . . ρ(qr)ρ(−q1 − . . .− qr + s)

×

r∑

l=1

[(∏i<l

ei2 l2Bqi×s

)(∏l≤j

e−i2 l2Bqj×s

)−

(∏i≤l

ei2 l2Bqi×s

)(∏l<j

e−i2 l2Bqj×s

)]

+ 2i sin(l2B (q1 + . . .+ qr)× s

2

)= 0,

sincer∑

l=1

[(∏i<l

ei2 l2Bqi×s

)(∏l≤j

e−i2 l2Bqj×s

)−

(∏i≤l

ei2 l2Bqi×s

)(∏l<j

e−i2 l2Bqj×s

)]

=

(r∏

j=1

e−i2 l2Bqj×s

)−

(r∏

i=1

ei2 l2Bqi×s

)= −2i sin

(l2B (q1 + . . .+ qr)× s

2

).

Now that we have found a Casimir operator ζR of the algebra Apd for each order R ∈ 2, 3, 4, . . .,let us try to rewrite the abstract expression (4.33) into a concrete expressions for ζR in terms ofladder operators acting on Fock space, to see if there is any physically relevant information to beobtained from the Casimir operators.


4.2.2 Computation of the Second Order Casimir

We have the following explicit formula for the second order Casimir operator ζ = ζ2, given byequation (4.32), in terms of the ladder operators cm′ and c†m,

ζ =∑

m,m′,l,l′

∫dq

(2π)2〈m|e−iq·R|m′〉〈l|eiq·R|l′〉c†mcm′c†l cl′ . (4.36)

In this section, we will try to simplify the above expression by computing the prefactors 〈m|e−iq·R|m′〉.We will proceed in the same manner as we did in Section 4.1.3. From equation (A.5), we have

q ·R = qxX + qyY =lB√2

[qx(b† + b) + iqy(b† − b)

]=

lB√2(qb† + q∗b),

where, as before, we define q := qx + iqy ∈ C. Hence, by Baker-Campbell-Hausdorff (4.21),

e−iq·R = e− ilB√

2(qb†+q∗b) = e

− ilB√2

qb†e− ilB√

2q∗be

l2B4 |q|

2[b†,b] = e− ilB√

2qb†e− ilB√

2q∗be−

l2B4 |q|

2.

Following the same reasoning as in equation (4.22), we have

〈m|e−iq·R|m′〉 = e−l2B4 |q|

2∞∑

k=0

〈m|e−ilB√

2qb† |k〉〈k|e−

ilB√2

q∗b|m′〉

= e−l2B4 |q|

2M∑

k=0

√m!k!

1(m− k)!

(−ilBq√

2

)m−k√m′!k!

1(m′ − k)!

(−ilBq∗√

2

)m′−k

,

where M := minm,m′. Keep in mind that, although the mathematical manipulations are similarto those in Section 4.1.3, the labels m and k now refer to the guiding center instead of the energy.

Contrary to the computation in Section 4.1.3, m is in general not equal to m′. Hence, wedistinguish between the cases m ≥ m′ and m ≤ m′. In case m ≥ m′, we have M = m′, and therefore

〈m|e−iq·R|m′〉 = e−l2B4 |q|

2

√m′!m!

(−ilBq√

2

)m−m′ m′∑k=0

m!(m− k)!(m′ − k)!k!

(− l

2B |q|2

2

)m′−k

= e−l2B4 |q|

2

√m′!m!

(−ilBq√

2

)m−m′

L(m−m′)m′

(l2B |q|2

2

), (4.37)

where L(α)m is the nth generalized Laguerre polynomial,

L(α)n (x) :=

n∑i=0

(n+ α

n− i

)(−x)i

i!=

n∑i=0

(n+ α)!(n− i)!(α+ i)!

(−x)i

i!.

Similarly, in case m ≤ m′, we get

〈m|e−iq·R|m′〉 = e−l2B4 |q|

2

√m!m′!

(−ilBq∗√

2

)m′−m

L(m′−m)m

(l2B |q|2

2

). (4.38)

The next step towards finding a simplified expression for ζ is computing the matrix coefficients∫dq

(2π)2 〈m|e−iq·R|m′〉〈l|eiq·R|l′〉. From equations (4.37) and (4.38), we see that the differences m′−m

and l′ − l determine the outcome. For example, if m = m′ and l = l′, then, by orthogonality ofthe Laguerre polynomials (see Abramowitz and Stegun [1], Chapter 22 on Orthogonal Polynomials,p.775), ∫ ∞

0

dr rαe−rL(α)n (r)L(α)

m (r) =Γ(n+ α+ 1)

n!δn,m, (4.39)


which holds for α > −1, and Γ denotes the gamma function

Γ(z) =∫ ∞

0

dt tz−1et,

we have∫dq

(2π)2〈m|e−iq·R|m〉〈l|eiq·R|l〉 =

∫dq

(2π)2e−

l2B2 |q|

2Lm

(l2B |q|2

2

)Ll

(l2B |q|2

2

)=

1(2π)2

∫ 2π

0

dθ

∫ ∞

0

dr re−l2B2 r2

Lm

(l2Br

2

2

)Ll

(l2Br

2

2

)=

12πl2B

∫ ∞

0

e−sLm(s)Ll(s)

=1

2πl2Bδm,l.

When going from the second to the third line above, we substituted s = l2Br2

2 .In general, m and m′ are not equal, and neither are l and l′. Define a := m′−m and a+b := l− l′

to be their respective differences. Before we proceed to the most general case, let us examine thecase that b = 0 and a > 0, i.e. m′ = m + a and l = l′ + a. Why we have chosen a and b in such away that the order of the primes is reversed should become clear in the next few lines. Now,∫

dq

(2π)2〈m|e−iq·R|m+ a〉〈l′ + a|eiq·R|l′〉

=∫

dq

(2π)2e−

l2B2 |q|

2

√m!

(m+ a)!

√l′!

(l′ + a)!

(−ilBq∗√

2

)a(ilBq√

2

)a

L(a)m

(l2B |q|2

2

)L

(a)l′

(l2B |q|2

2

)

=

√m!

(m+ a)!

√l′!

(l′ + a)!1

2πl2B

∫ ∞

0

ds e−ssaL(a)m (s)L(a)

l′ (s)

=m!

(m+ a)!1

2πl2B

Γ(m+ a+ 1)m!

δm,l′ =1

2πl2Bδm,l′ ,

where we have used the same substitution as before followed by the orthogonality of the Laguerrepolynomials (4.39). The case a < 0 goes analogously. Note that the condition α > −1 mentionedabove is never violated, because, as can be seen from equations (4.37) and (4.38), it is always theabsolute value |a| of the difference m′ −m that occurs in expressions above.

Finally, we can tackle the remaining case, that of non-zero b. It turns out that this is actuallythe most trivial case of all. Note that, in the above computation, the fact that m′ −m = a = l − l′

guarantees that the exponents of q and q∗ are equal, meaning that any polar angle dependencecancels out. Whenever we take b 6= 0, there will be a θ dependence in the expression of the forme±ibθ which integrates to zero. For example, let us consider the case a, b > 0.∫

dq

(2π)2〈m|e−iq·R|m+ a〉〈l′ + a+ b|eiq·R|l′〉

=∫

dq

(2π)2e−

l2B2 |q|

2

√m!

(m+ a)!

√l′!

(l′ + a+ b)!

(−ilBq∗√

2

)a(ilBq√

2

)a+b

L(a)m

(l2B |q|2

2

)L

(a+b)l

(l2B |q|2

2

)

=

√m!

(m+ a)!

√l′!

(l′ + a+ b)!ib

(2π)2

∫ 2π

0

dθ eibθ

∫ ∞

0

dr re−l2B2 r2

(lBr√

2

)2a+b

L(a)m

(l2Br

2

2

)L

(a+b)l

(l2Br

2

2

)= 0,


because∫ 2π

0dθ eibθ = 0 and the fact that the r-integral is bounded due to the Gaussian factor. Any

other choice of a and b such that b 6= 0 gives zero in a similar fashion.Let us summarize the obtained results in the following corollary.

Corollary 4.3. For m, l′, a and b integers such that m,m+ a, l′ + a+ b, l′ > 0, we have∫dq

(2π)2〈m|e−iq·R|m+ a〉〈l′ + a+ b|eiq·R|l′〉 =

12πl2B

δm,l′δb,0.

In particular, the above expression is independent of a.

Equipped with the above corollary, we can simplify the expression (4.36) for ζ significantly.Originally, the sum in (4.36) goes over all non-negative integers. Rearranging the sum such that itgoes over m, l′, a and b in such a way that m,m+ a, l′ + a+ b, l′ > 0, we get

ζ =∑

m,l′,a,b

∫dq

(2π)2〈m|e−iq·R|m+ a〉〈l′ + a+ b|eiq·R|l′〉 c†mcm+ac

†l′+a+bcl′

=1

2πl2B

∑m,l′,a,b

δm,l′δβ,0c†mcm+ac

†l′+a+bcl′

=1

2πl2B

∞∑m=0

∞∑a=−m

c†mcm+ac†m+acm. (4.40)

Let us summarize our result.

Corollary 4.4. The second order Casimir operator ζ defined in (4.31) is given by the followingexplicit formula:

ζ =1

2πl2B

∞∑m,l=0

c†mclc†l cm. (4.41)

We have tried to apply a similar reasoning to the higher order Casimir operators, but the effortturned out to be fruitless. The difficulty lies in the fact that, for as far as our knowledge goes,formula (4.39) regarding the orthogonality of the Laguerre polynomials does not have a higher ordercounterpart.

With the above formula for ζ at our disposal, we can now try to find out what the physicalinterpretation of the second order Casimir ζ is. This is the topic of the next section.

4.2.3 Physical Interpretation of the Second Order Casimir

In order to give physical meaning to the Casimir operator ζ, we have to normal order the formulafor ζ in equation (4.41). Before we do so, let us briefly review the mechanism of the ladder operatorsand their normal ordering.

For starters, we write an arbitrary nth0 Landau level multi-particle basis state in terms of creation

operators,φm1 ∧ . . . ∧ φmk

= c†m1. . . c†mk

|0〉, (4.42)

where |0〉 ∈ F0 denotes the vacuum. Recall that we are suppressing the index n0 that labels theenergy level from the notation. As an illustrative example, using the fermionic anti-commutationrelations (4.5) – (4.7), we will compute the action of the operator c†mcm on the basis state (4.42). Thestrategy is always the same: (anti)-commute the lowering operators past all the creation operatorsin (4.42) until they annihilate the vacuum. This is the reason that we want our operators to benormal ordered, which means that the creation operators are always to the left of the annihilationoperators.


We have,

c†mcm c†m1. . . c†mk

|0〉 = c†m

k∑i=1

(−1)i−1δm,mi c†m1

. . . c†mi . . . c†mk|0〉

=k∑

i=1

δm,mic†m1

. . . c†m . . . c†mk|0〉

=

(k∑

i=1

δm,mi

)c†m1

. . . c†mk|0〉.

The hat above means that the operator underneath it does not occur in the sum. We conclude thatevery state of the form c†m1

. . . c†mk|0〉 is an eigenstate of c†mcm with eigenvalue

k∑i=1

δm,mi =

1 if m ∈ m1, . . . ,mk0 if m /∈ m1, . . . ,mk

.

That is, c†mcm checks whether φm occurs in (4.42) or not, i.e whether the state m is occupied by anelectron. We refer to c†mcm as the number operator4

Definition 4.2. We write nm for the number operator c†mcm.

In a similar fashion, we compute that every state of the form (4.42) is an eigenstate of c†mc†l clcm

with eigenvaluek∑

i,j=1

δm,miδl,mj =

1 if m, l ∈ m1, . . . ,mk0 otherwise .

That is, c†mc†l clcm counts whether the pair (m, l) is occupied by electrons or not.

Definition 4.3. We write nm,l := c†mc†l clcm.

Finally, we come to the second order Casimir ζ. We have,

ζ =1

2πl2B

∞∑m,l=0

c†mclc†l cm

=1

2πl2B

− ∞∑m,l=0

c†mc†l clcm +

∞∑m,l=0

c†mcm

=

12πl2B

− ∞∑m,l=0

nm,l +∞∞∑

m=0

nm

. (4.43)

The effect of normal ordering is that, for fixed m, we get a number operator for every l. Thus, weget infinitely many number operators5. This means the operator ζ returns the number ∞ wheneverthere is at least one particle present. We conclude that, despite the fact that the constituents in thefinal expression (4.43) for ζ have a physical meaning, ζ itself does not.

The most logical step to take in order to overcome the infinities arising in the final expression(4.43) for ζ is to truncate the sums overm and l at some arbitrary integerM , which would correspond

4The number operator nm counts the number of particles in the state m, hence the name.5Whenever m = l there is also a number operator hidden in nm,l, but this only gives one extra number operator

nm for each m, which cannot possibly compensate for the infinitely many number operators that the second term in(4.43) contains.


to restricting the number of available guiding center positions to be finite. This is exactly whathappens on a finite sample. However, in order to model a finite sample correctly, we have to includeboundary conditions, which come with difficulties of their own. How to deal with the boundary ofa finite sample is the subject of the next chapter.

Chapter 5

The FQHE on the Torus

The second order Casimir operator constructed in the previous chapter suffers from infinities arisingfrom the fact that each Landau level is infinitely degenerate, which is a consequence of the fact thatwe were working on an infinite plane. These infinities make it difficult to give physical meaning tothe second order Casimir operator. Moreover, the approach we took in computing the second orderCasimir does not seem to generalize to higher order Casimir operators.

To overcome these problems, we want to have finitely degenerate Landau levels. This can beachieved by working on a torus, i.e. a finite sample with periodic boundary conditions. In Section 5.1,we will construct the line bundle over the torus and set up the appropriate mathematical formalismto describe electrons in a uniform perpendicular magnetic field moving on a torus. Next, in Section5.2, we will redo the computation for the Casimir operators found in Theorem 4.2, and give physicalmeaning to them.

5.1 Periodic Boundary Conditions

In this section, we will construct the mathematical model that we use to describe the FQHE on thetorus.

Definition 5.1. Let T denote the torus of dimension a times b. That is, T := (R/aZ) × (R/bZ),where a and b are positive real numbers.

Moreover, we use p : R2 → T to denote the projection that maps r ∈ R2 to its correspondingequivalence class r ∈ T.

We apply a constant magnetic field B = Bez that is everywhere perpendicular to the surface of thetorus.

5.1.1 Construction of the One-Particle Hilbert Space

We will now construct the one-particle Hilbert space of sections over T. In doing so, we can copythe construction given in Section 2.2.4 word for word, keeping the following remark in mind.

Remark 5.1. In the part denoted by “notation” just below equation (2.44), it not clear what ismeant by the “domain enclosed by two given curves”, as we now have a choice between two possibledomains. We require that both domains give the same phase factor, which is to say that the integralover the difference between the two domains, i.e. the whole torus T, gives a total phase factor of 2π.In other words, we require

1 = exp(

2πiκ∫

TB

)= exp (2πiκBab)

65

5.1. PERIODIC BOUNDARY CONDITIONS 66

Recalling that κ = − eh = − 1

φ0, see Example 2.2, where φ0 is the flux quantum, the above equation

restricts the magnetic field B to take only specific values such that the total magnetic flux Φ = Babthrough the torus is given by

Φ = NBφ0.

That is, the total flux is an integer NB times the flux quantum φ0. Also, comparing to equation(3.16), we conclude that NB, being equal to

NB =Φφ0

=eBab

h=

Area(T)2πl2B

, (5.1)

is also equal to the degeneracy of each Landau level.

Repeating all the steps in Sections 2.2.4 and 2.2.5, we have now constructed a line bundle LTover T. However, working on a torus has its disadvantages. Notably, the singular homology, andtherefore also the de-Rham cohomology groups of the torus, given by

HkdR(T) :=

Ker(dk)Im(dk−1)

=

R for k = 0, 2R2 for k = 10 otherwise

,

are less trivial than those of the infinite plane. As a consequence, a given 2-form B ∈ Ω2(T) can ingeneral not be written as B = ∇×A on all of T (but it is possible to write B as the rotation of avector potential locally; not that this is of any use to us)1. In order to circumvent this drawback,we will work with the universal covering space of T, which is R2 itself. Of course, we do want tokeep the periodicity of the torus, so we equip R2 with a line bundle that is the pull-back of LT.

Definition 5.2. let L∗ := p∗(LT) be the pull-back bundle over R2, where p : R2 → T is the canonicalprojection from Definition 5.1.

For a fixed point r ∈ R2, the fibre of L∗ at the point r is given by L∗|r = LT|r. That is, we haveextended LT periodically over R2.

We can describe sections of L∗ as sections of the original line bundle L constructed in Section2.2.4, with an extra periodicity condition that we will specify below. That is, we can embed Γ(L∗)into Γ(L). As a consequence, we are now back to the more familiar situation of L over R2. Inparticular, the magnetic field B that causes the QHE takes the familiar form B = Bez. We canview this magnetic field as the curvature 2-form B = dA of a connection 1-form A, exactly asdescribed in Section 2.2.5.

In order to specify the periodicity condition mentioned above, we have to make sense of what itmeans to translate a section f ∈ Γ(L).

Definition 5.3. Let τu1 and τv

2 be the unique operators that, when acting on a section f ∈ Γ(L),(a) translate a distance u in the x-direction and a distance v in the y-direction respectively, and,(b) commute with the connection ∇ defined in Section 2.2.5.

To make the above definition more concrete, let us choose a gauge and compute the aforementionedtranslation operators in this particular gauge.

Remark 5.2. From now on, we will work in the symmetric gauge.

In order for the operators τu1 and τv

2 to commute with the connection, it is necessary that,in addition to translating, τu

1 and τv2 also multiply with a prefactor to compensate for the extra

term coming from the magnetic vector potential A, which plays the role of the connection 1-form.1By rotation, we mean the two-dimensional version ∇×A = ∂xAy − ∂yAx.


To illustrate this concept, let us try the following ansatz for τu1 and τv

2 . For an arbitrary sectionf ∈ Γ(L), write

τu1 (f)(x, y) = F1(x)G1(y)f(x− u, y),τv2 (f)(x, y) = F2(x)G2(y)f(x, y − v),

where F1(x)G1(y) and F2(x)G2(y) are still to be determined prefactors. We require that the trans-lation operators commute with ∇. Because Πx and Πy are (proportional to) the covariant derivativeoperators — see Remark 2.9 — this amounts to,

[Πx, τu1 ] = 0, [Πy, τ

u1 ] = 0, (5.2)

[Πx, τv2 ] = 0, [Πy, τ

v2 ] = 0. (5.3)

Now, the first equation of (5.2) applied to a section f ∈ Γ(L) reads(−i~∂x −

eB

2y

)F1(x)G1(y)f(x− u, y) = F1(x)G1(y)

(−i~∂x −

eB

2y

)f(x− u, y),

That is, F1(x) = 1. The second equation of (5.2) with F1(x) = 1 gives(−i~∂y +

eB

2x

)G1(y)f(x− u, y) = G1(y)

(−i~∂y +

eB

2(x− u)

)f(x− u, y),

which is solved by the less trivial G1(y) = exp(−ieBuy

2~

)= exp

(− iuy

2l2B

). Analogously, equation (5.3)

gives F2(x) = exp(

ivx2l2B

)and G2(y) = 1.

Corollary 5.1. The translation operators τu1 and τv

2 take the following form when expressed in thesymmetric gauge:

τu1 (f)(x, y) := exp

(− iuy

2l2B

)f(x− u, y), (5.4)

τv2 (f)(x, y) := exp

(ivx

2l2B

)f(x, y − v), (5.5)

where f ∈ Γ(L) is arbitrary.

Having defined the translation operators τu1 and τv

2 , let us summarize in the corollary belowwhich sections of L correspond to sections of L∗.

Corollary 5.2. The space of sections Γ(L∗) of the periodic line bundle L∗ over R2 is isomorphicto the subspace f ∈ Γ(L) : f = τa

1 (f) = τ b2(f) ⊂ Γ(L).

Note that the sections f ∈ Γ(L∗) correspond to actual periodic sections of L, despite the fact thatthe equation f = τa

1 (f) = τ b2(f) seems to imply otherwise due to the extra phase factors involved.

The additional phase factors in (5.4) and (5.5) appear because of the fact that we are working witha specific trivialization of the line bundle, which is done in such a way that two neighboring fibresare rotated with respect to one another.

Remark 5.3. At this point, one of the main differences between the infinite plane and the torusbecomes apparent. Because all square-integrable sections on the plane are rapidly-decreasing func-tions, no momentum or position operator (or function thereof) can map wavefunctions out of thespace of square-integrable sections. In the case of the torus, however, because of the extra periodicitycondition f = τa

1 (f) = τ b2(f) on sections f ∈ Γ(L∗), it is possible that some operators are no longer

well-defined on Γ(L∗) because they do not preserve periodicity.


Fortunately, the covariant derivative operators commute with the translation operators (by con-struction), so the mechanical momentum and therefore also the ladder operators a and a† are well-defined on sections of L∗. Hence, we can still use the ladder operators a and a† to split up theone particle Hilbert space — defined below — as we did on the infinite plane. The guiding centreoperators on the other hand, will turn out to be less fortunate.2

Definition 5.4. The one-particle Hilbert space, which we denote by3 H∗, is the space of measurablesquare-integrable sections of Γ(L∗) modulo the equivalence relation that identifies to sections when-ever the set on which they disagree has measure zero. Instead of integrating over all of R2, we nowonly require for each ψ ∈ H∗ that ∫

Tdr|ψ(r)|2 <∞. (5.6)

Note that we are back to the situation of a uniform perpendicular magnetic field on an infiniteplane: the only difference between H∗ and H is that we now have a weaker integrability condition,but an additional periodicity condition. Hence, we can re-use part of the theory constructed inChapter 4. In particular, the free Hamiltonian takes the same form at it did in Chapter 4, namelythat of equation (3.19),

Hkin =1

2m(Π2

x + Π2y

)= ~ωC

(a†a+

12

), (5.7)

where the ladder operators a and a† are defined by equation (3.6). As a consequence, we can useCorollary 3.1 to split the one-particle Hilbert space into a tensor product.

Corollary 5.3. The one-particle Hilbert space H∗ splits into a tensor product

H∗ ∼= H0 ⊗Ker(a)∗, (5.8)

where H0 is given by Definition 3.2. Equivalently, since H0 =⊕∞

n=0 Vn where each of the Vn’s is aone-dimensional complex vector space, we can also write H as a direct sum

H∗ =∞⊕

n=0

L∗n

where each of the Landau levels L∗n ∼= Ker(a)∗.

Remark 5.4. We put a star on Ker(a)∗ ⊂ H∗ to distinguish it from Ker(a) ⊂ H. Note that,Ker(a)∗ is not the same as all periodic sections of Ker(a) because the integrability condition (5.6)for H∗ is much weaker than the integrability condition for H.

Additionally, as noted just below equation (5.1), there is one crucial difference with the infiniteplane case discussed in Chapter 4.

Remark 5.5. Every Landau level L∗n has finite dimension dim(Ker(a)) = NB.

To summarize, we have constructed a line bundle LT over T, and pulled it back to a line bundleL∗ over R2 to return to the more familiar situation of sections of L over R2. We then defined theone-particle Hilbert space H∗ as the space of square-integrable sections of L∗, which, compared toH for the infinite plane has extra periodicity conditions and a weaker integrability condition. Themain differences between the infinite plane and the torus are that, for the case of the torus, (a)the magnetic field B is restricted to values such that the total flux Φ = Bab, where ab = Area(T),through the torus is an integer NB times the flux quantum φ0 = h

e , and (b) each Landau level hasa degeneracy of this same integer NB .

Our next task is to gain a better understanding of the separate Landau levels. This is what thenext section is about.

2The fact that the guiding centre operators are no longer well-defined is actually a good thing, because they arethe cause of the infinite degeneracy of each Landau level, see Corollary 3.2.

3To distinguish it from the Hilbert space H of not necessarily periodic sections — see Definition 2.6.


5.1.2 The Projected Density Operators

We want to construct and compute the Casimir operators defined in the previous Chapter for thecase of periodic sections of over R2. However, as mentioned in Remark 5.3, we do run in some seriousproblems if we try to blindly repeat the analysis of Chapter 4. For example, for f ∈ Γ(L∗), xf is notnecessarily in Γ(L∗). As a consequence, something as elementary as the position operators x andy, and therefore also the guiding centre coordinate operators X and Y , are no longer well-definedendomorphisms of the space of periodic sections Γ(L∗). Without the guiding centre operators, welose the eigenbasis |m〉 for each Landau level, and therefore the whole analysis of Chapter 4 fallsapart.

Fortunately, the situation is not as bad as it seems. For example, any T-periodic function of xand y is a well-defined operator on Γ(L∗). Moreover, it turns out that the single particle projecteddensity operators σ(q) = e−iq·R do conserve the periodicity of sections of L∗ for specific values ofq. But, before we make this claim more specific, let us first try to understand the meaning of theoperators σ(q) and their relation to the translation operators τu

1 and τv2 .

Recall that R = (X,Y ), which in the symmetric gauge is given by

X = x− Πy

eB= il2B∂y +

x

2= −il2B

(−∂y +

ix

2l2B

),

Y = y +Πx

eB= −il2B∂x +

y

2= il2B

(−∂x −

iy

2l2B

).

Comparing to equations (5.4) and (5.5), it looks like X and Y are closely related to the generatorsof the translation operators τu

1 and τv2 .

Lemma 5.1. The operators −iY/l2B and iX/l2B generate the translations in the x and y directionrespectively. That is, for u, v ∈ R,

τu1 = exp

(− iuYl2B

), (5.9)

τv2 = exp

(ivX

l2B

). (5.10)

Proof. A straightforward computation shows that for arbitrary f ∈ Γ(L),[exp

(− iuYl2B

)f

](x, y) =

[exp

(−u∂x −

iuy

2l2B

)f

](x, y) = exp

(− iuy

2l2B

)f(x− u, y) = τu

1 (f)(x, y),

and likewise for τv2 .

Remark 5.6. We see that the single particle projected density operators σ(q) are nothing but mag-netic translation operators that translate over a (rotated4) wavevector q. It is therefore not surprisingthat, when imposing periodic boundary conditions, only a subset of all σ(q)’s respect the periodicity.This also explains the fact that the Lie bracket of two single-particle projected density operators,given by (4.25),

[σ(q), σ(s)] = −2i sin(l2B2

q × s

)σ(q + s). (5.11)

is known as the magnetic translation algebra, as mentioned in Section 4.1.4. Using (5.9) and (5.10),we can write the translation operators τu

1 and τv2 as

τu1 = σ(0, u/l2B) and τv

2 = σ(−v/l2B , 0).

4because −Y corresponds to translations in the x direction, and X corresponds to translations in the y direction


Having established the relation between the magnetic translation operators and the single-particleprojected density operators, we can prove the following lemma.

Lemma 5.2. Write q = (q1, q2) = (2πk1/a, 2πk2/b). Now, we have the following equivalence:

k = (k1, k2) ∈ Z2 ⇐⇒ [σ(q), τa1 ] = [σ(q), τ b

2 ] = 0.

Proof. Using (5.11), for τa1 we have,

[σ(q), τa1 ] = [σ(q), σ(0, a/l2B)] (5.12)

= −2i sin(q1a

2

)σ

(q +

a

l2Bey

). (5.13)

Indeed, we see that the RHS equals zero if and only if q1 = 2πk1/a for some k1 ∈ Z. Likewise,[σ(q), τ b

2 ] = 0 if and only if q2 = 2πk2/b for some k2 ∈ Z.

Finally, we can make precise the statement made at the beginning of this section.

Lemma 5.3. As before, write q = (q1, q2) = (2πk1/a, 2πk2/b). The following two statements areequivalent:

k = (k1, k2) ∈ Z2 ⇐⇒ σ(q) ∈ End(Γ(L∗)).

In words, the above statement says that σ(q) maps periodic sections of L to periodic sections of L ifand only if q = (2πk1/a, 2πk2/b) is such that k = (k1, k2) ∈ Z2.

Proof. First, suppose that k = (k1, k2) ∈ Z2. We have to prove that σ(q) maps Γ(L∗) back intoitself. Pick f ∈ Γ(L) satisfying f = τa

1 (f) = τ b2(f). We have to show that σ(q)f ∈ Γ(L) also satisfies

σ(q)f = τa1 σ(q)f = τ b

2 σ(q)f . By Lemma 5.2, we have

σ(q)f = σ(q)τa1 (f) = τa

1 σ(q)f,

and, similarly, σ(q)f = τ b2 σ(q)f .

For the converse, suppose that σ(q) ∈ End(Γ(L∗)). Now, By equation (5.13), for every f ∈ Γ(L∗)we have,

σ(q)f = τa1 σ(q)f = σ(q)τa

1 f + 2i sin(q1a

2

)σ

(q +

a

l2Bey

)f

= σ(q)f + 2i sin(q1a

2

)σ

(q +

a

l2Bey

)f

Hence, we conclude that for every f ∈ Γ(L∗),

2i sin(q1a

2

)σ

(q +

a

l2Bey

)f = 0. (5.14)

Since σ(q + a

l2Bey

)is a combination of translations, it maps any nonzero f to a nonzero section.

Hence, the only possibility for (5.14) to hold is whenever k1 ∈ Z. The claim that k2 ∈ Z followsanalogously.

The above Lemma implies that the only operators relevant for our purposes are the σ(q)’s forwhich (q1, q2) = (2πk1/a, 2πk2/b). Hence the following definition.

Definition 5.5. From now on, we will use the notation σ(k) for σ(

2πk1a , 2πk2

b

).

For convenience, we mention the associative (4.24) and Lie algebra (4.25) formulas satisfied bythe single-particle projected density operators in terms of the integer vectors k, l ∈ Z2 below.


Lemma 5.4. The single particle projected density operators σ(k) satisfy

σ(k)σ(l) = exp(− iπk × l

NB

)σ(k + l), (5.15)

and also

[σ(k), σ(l)] = −2i sin(πk × l

NB

)σ(k + l). (5.16)

Proof. By equation (5.1),il2Bq × s

2=

2iπ2l2Bk × l

ab=iπk × l

NB

But, there is more.

Lemma 5.5. When acting on Γ(L∗), we have the following operator identity

σ(NBk) = eiπk1k2NB = ±1 (5.17)

for any k ∈ Z2.

Proof. Using Baker-Campbell-Hausdorff (4.21), equation (5.1) and [X,Y ] = il2B , we have

σ(NBk) = exp(−2πiNB

[k1X

a+k2Y

b

])= exp

(−ik1bX + k2aY

l2B

)= exp

(− ik1bX

l2B

)exp

(− ik2aY

l2B

)exp

(k1k2ab

2l4B[X,Y ]

)= (τ−b

2 )k1(τa1 )k2 exp (iπk1k2NB) .

Now, for any section f ∈ Γ(L∗), f = τa1 (f) = τ b

2(f). Hence,

σ(NBk)f = eiπk1k2NBf.

Corollary 5.4. For k, l ∈ Z2,

σ(k +Nbl) = e−iπk×leiπl1l2NB σ(k) = ±σ(k) (5.18)

when acting on Γ(L∗). As a consequence, we effectively only have N2B operators of the form σ(k)

acting on Γ(L∗). This means that any sum over q or k need only go over a Brillouin zone thatcontains N2

B adjacent values of q or k.

Proof. Using equation (5.15) and Lemma 5.5, we have

σ(k +NBl) = e−iπk×lσ(k)σ(NBl) = e−iπk×leiπl1l2NB σ(k)= ±σ(k).

Definition 5.6. The Brillouin zone BZ over which we will implicitly sum whenever there is a sumover k space is the set of integers 0, 1, 2, . . . , NB − 12.

We conclude that, despite the fact that something as fundamental as the position operator doesnot make sense when working with periodic boundary conditions, we do have a subalgebra of thesingle-particle projected density operators at our disposal, namely those σ(k)’s for which k ∈ BZ.Using these operators, we can construct the effective Hamiltonian and properly describe the FQHEon the torus. However, before we do, there is one more thing that we have to discuss.


5.1.3 An NB-Dimensional Representation of the Magnetic TranslationAlgebra

In the previous section, we constructed a subalgebra of single particle projected density operators,each of which is an endomorphism of every Landau level. We would like to know explicitly howa given single particle projected density operator acts on the fractionally filled nth

0 Landau level(or any other Landau level for that matter). What we will do is guess a finite dimensional matrixrepresentation of the algebra spanned by the σ(k)’s. We will then assume that there actually existsan orthonormal basis of Ln0 such that the operators σ(k) take the form of the matrices that we willconstruct, and then we will use these matrices to compute the Casimir operators ζr.

Recall from Lemma 5.4 that the single-particle projected density operators σ(k) satisfy

σ(k)σ(l) = exp(− iπ

NBk × l

)σ(k + l) (5.19)

in terms of the integer vectors k, l ∈ Z2. We are looking for NB ×NB matrices Mk that satisfy thesame relation.

Before we continue, anticipating the fact that we will later on use the relations derived in thissection extensively in the computation of the Casimir operators, we will switch notation. In theexpressions for the Casimir operators, the integer vector k will be summed over, and instead of oneor two, we will have r such summation indices k1, . . . ,kr. Because each ki also has an x and ay-component, which up to now we have labelled with a 1 and a 2 index, we want to come up witha different way of labelling the x and y-components than using the indices 1 and 2.

Notation: For a vector k ∈ Z2, from now on, we will use a prime to distinguish the componentsfrom each other. That is,

k = (k, k′) ∈ Z2.

So the x-component of k is k, and the y-component of k is k′.Now, (5.19) in terms of the to be constructed NB ×NB matrices Mk and Ml reads

MkMl = exp(− iπ

NBk × l

)Mk+l. (5.20)

Definition 5.7. The complex root of unity exp(− 2πi

NB

)will occur quite often in the remainder of

this chapter. That often, that we have decided to reserve the name5

α := exp(−2πiNB

)for it.

Definition 5.8. The orthonormal basis of Ker(a) in which we want to express the matrices Mk

will be labeled by the NB complex roots of unity

|α0〉, |α1〉, . . . , |αNB−1〉.

We choose this kind of labelling of the basis states over the more conventional |0〉, |1〉, . . . , |NB − 1〉because αm is periodic in m, which will be convenient when constructing the matrices Mk.

Next, we will define the matrices Mk by their action on the basis |αm〉.

5The complex root of unity α is completely unrelated to the statistical parameter α of the anyons in the nextchapter, which is a real number.


Educated guess:Quoting my supervisor Andre Henriques: “there are not too many things you can try” after a failedfirst attempt which we will not go into, let us try the following expression6 for Mk,

Mk|αm〉 := α−mk′ |αm+k〉,

where, we recall thatk = (k, k′).

In other words, the mth basis vector gets mapped (m + k)th basis vector (mod NB), and picks upan additional phase α−mk′ . The tilde is there because we anticipate that this matrix might notwork out perfectly, and possibly will require further adjustments. The above definition leads to thefollowing multiplication formula

MkMl|αm〉 = Mkα−ml′ |αm+l〉 = α−ml′Mk|αm+l〉

= α−ml′α−(m+l)k′ |αm+l+k〉

= α−k′lα−m(k′+l′)|αm+k+l〉

= α−k′lMk+l|αm〉. (5.21)

This looks similar to equation (5.20), apart from the prefactor α−k′l. Let us multiply the matricesMk with a yet to be determined factor c(k) in an attempt to get the correct prefactor in equation(5.21). That is, define

Mk := c(k)Mk.

By equation (5.21), the Mk’s satisfy the algebra

MkMl′= = α−k′l c(k)c(l)c(k + l)

Mk+l. (5.22)

For (5.22) to equal (5.20), we require

α−k′l c(k)c(l)c(k + l)

!= α12 (kl′−k′l).

That is,c(k)c(l) != c(k + l)α

12 (kl′+k′l).

The above equation is solved by c(k) = α−12 kk′ .

Let us summarize our result in the following corollary.

Corollary 5.5. There exists an NB-dimensional representation of the algebra of operators σ(k)given by

σ(k) 7→Mk, (5.23)

where the matrix Mk is defined by its action on the basis |α0〉, |α1〉, . . . , |αNB−1〉 through

Mk|αm〉 := α−12 kk′α−mk′ |αm+k〉. (5.24)

The representation (5.23) respects equation (5.18), as the next lemma shows.

6The suggested form of the matrices Mk should be credited entirely to A.G. Henriques. The author claims nooriginality on his part. Moreover, it was later pointed out by V. Gritsev that the matrices constructed in this sectionare also discussed in Fairly, Fletcher and Zachos [3].


Lemma 5.6. Just like the single-particle projected density operators σ(k), the matrices Mk areperiodic up to a sign over the Brillouin zone BZ.

Mk+NBl = e−iπk×leiπNBll′Mk = ±Mk. (5.25)

Proof. Indeed, from equation (5.24), we see that

MNBl|αm〉 = α−12 NBlNBl′α−mNBl′ |αm+NBl〉 = eiπNBll′ |αm〉.

Hence, MNBl = eiπNBll′1. Now follow the proof of Corollary 5.4.

As stated at the beginning of this section, we will assume that there exist an orthonormal basisof L∗n0

∼= Ker(a)∗, which we label by |αm〉 : m = 0, . . . , NB−1 such that the operators σ(k) act onthis basis via the identification (5.23). We now have an explicit formula for the matrices σ(k) thatwe will use to compute explicit expressions for the Casimir operators ζr in terms of ladder operatorsacting on Fock space.

5.1.4 The FQHE on the Torus

As we did in Chapter 4, we will work in the language of second quantization. That is, we defineFock space

F∗ :=∞⊕

n=0

F ∗n ,

where F ∗n is the n-particle Hilbert space H∗ ∧ . . .∧H∗ (n-times). Next, we choose a basis φn,m ofH∗ in accordance to the isomorphism in equation (5.8), where n ∈ N labels the Landau level, andm ∈ 0, . . . , NB − 1 refers to the basis vectors of Ker(a)∗ constructed in the previous section, anddefine corresponding creation and annihilation operators c†n,m and cn,m acting on F∗ by

c†n,m(φn1,m1 ∧ . . . ∧ φnk,mk) := φn,m ∧ φn1,m1 ∧ . . . ∧ φnk,mk

,

cn,m(φn1,m1 ∧ . . . ∧ φnk,mk) :=

k∑i=1

δ(n,m),(ni,mi)φn1,m1 ∧ . . . ∧ φni−1,mi−1 ∧ φni+1,mi+1 ∧ . . . ∧ φnk,mk.

Next, using the ladder operators cn,m and c†n,m, we define the periodic analogue of the projecteddensity operators ρn0(k) used in Chapter 4 — see equation (4.20) — by extending the single-particleprojected density operators σ(k) to act on all of Fock space,

ρn0(k) =∑

m,m′

〈m|σ(k)|m′〉c†n0,mcn0,m′ . (5.26)

In terms of the basis |αm〉 described in Section 5.1.3, σ(k) = Mk, and the matrix coefficients〈m|σ(k)|m′〉 can be readily computed.

We shall write(Mk)m,m′ := 〈αm|Mk|αm′

〉

for the (m,m′)th matrix element of Mk in the |αm〉 basis.

Definition 5.9. Let δm,m′ denote the delta function that takes its arguments mod NB. That is

δm,m′ := 1 if m = m′ mod NB

0 otherwise .

Using equation (5.24), we can compute the matrix elements of Mk in the |αm〉 basis without toomuch difficulty.


Lemma 5.7. The matrix elements of Mk are given by

(Mk)m,m′ = α−12 kk′α−m′k′ δm,m′+k. (5.27)

Proof. By equation (5.24), we have

(Mk)m,m′ = 〈αm|Mk|αm′〉 = α−

12 kk′α−m′k′〈αm|αm′+k〉,

where, because of the orthonormality of the basis |αm〉,

〈αm|αm′+k〉 = δαm,αm′+k = δm,m′+k.

Corollary 5.6. We can rewrite equation (5.26) as

ρn0(k) = α−12 kk′

∑m,m′

α−m′k′ δm,m′+k c†n0,mcn0,m′ . (5.28)

Lemma 5.8. The projected density operators ρn0(k) acting on F∗ satisfy the same Lie algebra asthe projected density operators from Chapter 4 that act on F :

[ρn0(k), ρn0(l)] = −2i sin(πk × l

NB

)ρn0(k + l). (5.29)

Proof. Using Lemma 5.4, we can copy the proof of Lemma 4.2.

Remark 5.7. As for the FQHE on the infinite plane, the projected density operators ρn0(k) onlyinherit the Lie algebra structure (5.16) of the single particle projected density operators σ(k), butnot the associative algebra structure (5.15).

Remark 5.8. The projected density operators ρn0(k) are also periodic up to a sign in k.

ρn0(k +Nbl) = e−iπk×leiπNBll′ ρn0(k) = ±ρn0(k). (5.30)

This follows directly from equation (5.25).

Finally, we can discuss the FQHE on the torus. Going to the low temperature limit, where all theeffective dynamics occur within the highest non-empty (fractionally filled) Landau level labelled byn0, we proceed as in Section 4.1.2, neglecting the kinetic energy term which is an irrelevant constant.The effective Hamiltonian is now given by the periodic analogue of equation (4.13),

Heff =1

2ab

∑k∈BZ

U(k)ρn0(k)ρn0(−k), (5.31)

where U(k) is the Fourier transform of U(r).7

We have now set up all the necessary mathematical formalism to describe the FQHE on thetorus. In the next section, we will attempt to compute Casimir operators of the Lie algebra (5.29)satisfied by the projected density operators.

7There is a caveat in the above expression, which does not really matter for our present purpose: the interactionpotential U(r) cannot be equal to the Coulomb potential, because it has to be T-periodic. We will not discuss thespecific form of U(r). All that matters is that U is periodic, has the correct short distance behavior, and is Fouriertransformable. For more details, see Haldane [10].

5.2. CASIMIR OPERATORS ON THE TORUS 76

5.2 Casimir Operators on the Torus

In this section we will redo the computations of Section 4.2 for the case of the torus. The computa-tions will turn out to be less complicated because, when working with periodic boundary conditions,the partially filled nth

0 Landau level is merely a finite dimensional vector space of dimension NB .

Notation: as we did in Chapter 4, from now on, we will suppress the index n0 that refers to thefractionally filled Landau level in the notation.

5.2.1 Construction of the Rth Order Casimir

According to Theorem 4.2, we have Casimir operator ζr+1 for each r ∈ N, given by

ζr+1 :=1

(ab)r

∑q1,...,qr

∏i<j

exp(− il

2Bqi × qj

2

) ρ(q1) · ... · ρ(qr)ρ(−q1 − ...− qr)

=1

(ab)r

∑k1,...,kr

∏i<j

exp(− iπki × kj

NB

) ρ(k1) · ... · ρ(kr)ρ(−k1 − ...− kr).

It turns out that the proof of Theorem 4.2 requires some modifications in order to work on thetorus. The reason is, that in the proof of Theorem 4.2 we use a shift in the integration variableq, which, as equation (5.30) shows, introduces extra minus signs possibly leading to not all termscancelling out properly. Hence, we will carefully repeat the proof of Theorem 4.2 for the case of thetorus while keeping (5.30) in mind.

Theorem 5.1. For each r ∈ N, the operator ζr+1, given by

ζr+1 =1

(ab)r

∑k1,...,kr

(∏i<j

e− iπ

NBki×kj

)ρ(k1) · ... · ρ(kr)ρ(−k1 − ...− kr), (5.32)

is a central operator of the projected density operator algebra.

Proof. Following the reasoning of (4.35), we obtain for arbitrary l ∈ Z2,

[ζr+1, ρ(l)

]=

r∑m=1

−

∑k1,...,kr

(∏i<j

e− iπ

NBki×kj

)2i sin

(πkm × l

NB

)

× ρ(k1) . . . ρ(km−1)ρ(km + l)ρ(km+1) . . . ρ(kr)ρ(−k1 − ...− kr)

+∑

k1,...,kr

(∏i<j

e− iπ

NBki×kj

)2i sin

(π(k1 + . . .+ kr)× l

NB

)× ρ(k1) . . . ρ(kr)ρ(−k1 − . . .− kr + l). (5.33)

As in the proof of Theorem 4.2, we now focus on the mth term in the sum above. Keep in mindthat all of the ki’s above run over the Brillouin zone BZ := 0, . . . , NB − 12. We do a change ofsummation variables to the new variable km := km + l, and then remove the tilde afterwards, whicheffectively amounts to two things: (1) in the expression above we replace km by km − l, and (2)the new variable km now runs over the shifted Brillouin zone BZ + l. We will discuss (2) below.Plugging (1) into (5.33) means that we substitute(∏

i<j

e− iπ

NBki×kj

)7→

(∏i<j

e− iπ

NBki×kj

)( ∏i<m

eiπ

NBki×l

)( ∏m<j

e− iπ

NBkj×l

).


On to (2). Having done the change of variables describe above, the mth term in the sum in (5.33)becomes

−∑m

k1,...,kr

(∏i<j

e− iπ

NBki×kj

)( ∏i<m

eiπ

NBki×l

)( ∏m<j

e− iπ

NBkj×l

)2i sin

(πkm × l

NB

)× ρ(k1) . . . ρ(kr)ρ(−k1 − . . .− kr + l), (5.34)

where the tilde on the sum means that the mth variable km runs over the shifted Brillouin zoneBZ + l, while the other variables run over the ordinary Brillouin zone BZ. We would like to shiftthe summation over km back to a summation over BZ. However, such a shift might give rise toextra signs due to (5.30), which could spoil the argument we used in proving Theorem 4.2. Let usinvestigate this further.

Each km that does not lie in BZ we write as km + zNB , with km ∈ BZ, for a unique integervalued vector z ∈ Z2. As before, we drop the tilde, which means that, effectively, we have to replacekm by km + zNB for some terms in (5.34). This is what happens if we do so for a single term in(5.34). Using (5.30), we get(∏

i<j

e− iπ

NBki×kj

)7→

(∏i<j

e− iπ

NBki×kj

)( ∏i<m

e−iπki×z

)( ∏m<j

e−iπz×kj

)

2i sin

(πkm × l

NB

)7→ 2i sin

(πkm × l

NB

)eiπz×l

ρ(km) 7→ e−iπkm×zeiπNBzz′ ρ(km)

ρ(−k1 − . . .− kr + l) 7→ e−iπ(−k1−...−kr+l)×(−z)eiπNBzz′ ρ(−k1 − . . .− kr + l)

That is, replacing km by km + zNB yields a total extra factor of( ∏i<m

e−iπki×z

)( ∏m<j

e−iπz×kj

)eiπz×le−iπkm×zeiπNBzz′eiπ(−k1−...−kr+l)×zeiπNBzz′

=

( ∏i<m

e−2iπki×z

)e−2iπkm×ze2iπNBzz′ = 1.

Hence, we can safely change the domain of summation for km in (5.34) to BZ without any con-sequences. Note that, what we have actually shown is that (5.34) without the summation sign infront, is perfectly periodic in each of the variables ki, despite the fact that ρ(ki) is not. We concludethat, after changing the variable km 7→ km − l, (5.34) becomes a sum over the unshifted Brillouinzone for all summation variables:

−∑

k1,...,kr

(∏i<j

e− iπ

NBki×kj

)( ∏i<m

eiπ

NBki×l

)( ∏m<j

e− iπ

NBkj×l

)2i sin

(πkm × l

NB

)× ρ(k1) . . . ρ(kr)ρ(−k1 − . . .− kr + l). (5.35)

We perform the change of variables km 7→ km − l for all m ∈ 1, . . . , r, and then we proceedanalogously to the proof of Theorem 4.2: we split the sine above into a sum of exponentials and


then plug (5.35) into (5.33) to get

[ζr+1, ρ(l)

]=

∑k1,...,kr

(∏i<j

e− iπ

NBki×kj

)ρ(k1) . . . ρ(kr)ρ(−k1 − . . .− kr + l)

×

r∑

m=1

[( ∏i<m

eiπ

NBki×l

)( ∏m≤j

e− iπ

NBkj×l

)−

( ∏i≤m

eiπ

NBki×l

)( ∏m<j

e− iπ

NBkj×l

)]

+ 2i sin(π(k1 + . . .+ kr)× l

NB

)= 0.

5.2.2 Computation of the Rth Order Casimir

Contrary to the case of the infinite plane, we can now compute all the Casimir operators ζr+1 forr ∈ N in one go.

Theorem 5.2. For r ∈ N, the (r + 1)th order Casimir operator (5.32) takes the following form

ζr+1 =1

(2πl2B)r

∑m1,m′

1,...,mr+1,m′r+1

[δm′

1,mr+1

r∏i=1

δm′i+1,mi

]c†m1

cm′1c†m2

cm′2· · · c†mr+1

cm′r+1

(5.36)

when acting on the Fock space F∗ through the representation given by equation (5.28).

Proof. We start with the expression given by equation (5.32), and simply plug in equation (5.28).

ζr+1 =1

(ab)r

∑k1,...,kr

[∏i<j

exp(− iπki × kj

NB

)]ρ(k1) · ... · ρ(kr)ρ(−k1 − ...− kr)

=1

(2πl2BNB)r

∑k1,...,kr

[∏i<j

α12 ki×kj

][r∏

i=1

α−12 kik

′i

]α−

12

Pi,j kik

′j

×∑

m1,m′1,...,mr+1,m′

r+1

[(r∏

i=1

α−m′ik′i

)αm′

r+1P

i k′i

×

(r∏

i=1

δmi,m′i+ki

)δmr+1,m′

r+1−k1−...−kr

× c†m1cm′

1c†m2

cm′2· · · c†mr+1

cm′r+1

]

=1

(2πl2BNB)r

∑m1,m′

1,...,mr+1,m′r+1

∑k1,...,kr

α12

Pi<j(kik

′j−k′ikj)− 1

2

Pi kik

′i− 1

2

Pi,j kik

′j

× α−P

i m′ik′i+m′

r+1P

i k′i

(r∏

i=1

δmi,m′i+ki

)δmr+1,m′

r+1−k1−...−kr

× c†m1

cm′1c†m2

cm′2· · · c†mr+1

cm′r+1

(5.37)


Let us focus on the part in curly brackets in the two lines above (5.37) for now, keeping all them’s fixed. We will denote this part by the symbol (*).

(*) :=∑

k1,...,kr

α12

Pi<j(kik

′j−k′ikj)− 1

2

Pi kik

′i− 1

2

Pi,j kik

′j α−

Pi m′

ik′i+m′

r+1P

i k′i

×

(r∏

i=1

δmi,m′i+ki

)δmr+1,m′

r+1−k1−...−kr(5.38)

We will rewrite the first exponent in (5.38) as

α12 [

Pi<j(kik

′j−k′ikj)−

Pi kik

′i−

Pi,j kik

′j ] = α

12

Pi k′i(−

Pi<j kj+

Pj<i kj−ki−

Pj kj)

= α12

Pi k′i(−2

Pi<j kj−2ki)

= α−P

i k′i(ki+P

i<j kj) =: (A)

Next, we will perform the sums over k1, . . . , kr in (*). Each Dirac delta δmi,m′i+ki

forces ki takeany value satisfying mi = m′

i + ki(mod NB). Because each ki runs over the set 0, . . . , NB − 1,there is exactly one value for ki that solves mi = m′

i + ki(mod NB), say ki = mi −m′i + ziNB for

some integer zi ∈ Z. Plugging this equation for ki back into the exponents yields

(A) = α−P

i k′i(mi−m′i+ziNB+

Pi<j [mj−m′

j+zjNB ])

= α−P

i k′i(mi−m′i+

Pi<j [mj−m′

j ])α−NB

Pi k′i(zi+

Pi<j zj)

= α−P

i k′i(mi−m′i+

Pi<j [mj−m′

j ]) =: (B)

because αNB = 1. Plugging (B) back into (*) gives8

(*) =∑

k′1,...,k′r

α−P

i k′i(mi−m′r+1+

Pi<j [mj−m′

j ])δmr+1,m′r+1−

Pj [mj−m′

j ]

=r∏

i=1

∑k′i

α−k′i(mi−m′r+1+

Pi<j [mj−m′

j ])

δmr+1,m′r+1−

Pj [mj−m′

j ]. (5.39)

Now, observe that for any integer n ∈ Z,

NB−1∑k′=0

α−k′n =

NB for n = 0 (mod NB)1−(α−n)NB

1−α−n = 0 for n 6= 0 (mod NB)

= NB δn,0. (5.40)

Applying (5.40) to the term within brackets in (5.39) yields

(∗) = (NB)r

(r∏

i=1

δmi−m′r+1+

Pi<j [mj−m′

j ],0

)δmr+1,m′

r+1−P

j [mj−m′j ].

We conclude that ζr+1 is given by

ζr+1 =1

(2πl2B)r

∑m1,m′

1,...,mr+1,m′r+1

(r∏

i=1

δmi−m′r+1+

Pi<j [mj−m′

j ],0

)δmr+1,m′

r+1−P

j [mj−m′j ]

× c†m1cm′

1c†m2

cm′2· · · c†mr+1

cm′r+1

. (5.41)

8Note that there is a second exponent α−P

i m′ik′i+m′

r+1P

i k′i in (*) that we have multiply with (B).


In order to get to equation (5.36), we will have to simplify the product of Dirac delta’s. EachDirac delta represents an equation, so the product represents a system of equations. Hence, in orderto simplify the product of Dirac delta’s, all we have to do is simplify the corresponding system ofequations. We have (everything mod NB),

m′r+1 = m1 +

r∑1<j

(mj −m′j)

m′r+1 = m2 +

r∑2<j

(mj −m′j)

...m′

r+1 = mr−1 + (mr −m′r)

m′r+1 = mr

m′r+1 = mr+1 +

r∑j=1

(mj −m′j)

Next, we plug the last equation into all the other ones, and then subtract∑r

j=1(mj −m′j) on both

sides.

mr+1 = m1 − (m1 −m′1)

mr+1 = m2 − (m1 −m′1)− (m2 −m′

2)...

mr+1 = mr−1 − (m1 −m′1)− (m2 −m′

2)− . . .− (mr−1 −m′r−1)

mr+1 = mr − (m1 −m′1)− (m2 −m′

2)− . . .− (mr −m′r)

m′r+1 = mr+1 +

r∑j=1

(mj −m′j)

Now, we simplify each equation separately.

mr+1 = m′1 (1)

mr+1 = m′2 − (m1 −m′

1) (2)...

mr+1 = m′r−1 − (m1 −m′

1)− (m2 −m′2)− . . .− (mr−2 −m′

r−2) (r− 1)mr+1 = m′

r − (m1 −m′1)− (m2 −m′

2)− . . .− (mr−1 −m′r−1) (r)

mr+1 = m′r+1 − (m1 −m′

1)− (m2 −m′2)− . . .− (mr −m′

r) (r + 1)

Next, for i = 1, . . . r+1, we take equation number (i) and subtract equations (1)+ (2)+ . . .+(i− 1)


from it. This yields the following set of equations:

m′1 = mr+1 (1)

m′2 = m1 (2)− (1)

...m′

r−1 = mr−2 (r− 1)− (1)− (2)− . . .− (r− 2)m′

r = mr−1 (r)− (1)− (2)− . . . (r− 1)m′

r+1 = mr (r + 1)− (1)− (2)− . . .− (r)

Finally, we notice that because all of the mi’s and m′i’s run over the set 0, . . . , NB − 1, the only

possible way for the above equations to be satisfied mod NB , is if they are actually satisfied asequations in Z. This shows that(

r∏i=1

δmi−m′r+1+

Pi<j [mj−m′

j ],0

)δmr+1,m′

r+1−P

j [mj−m′j ]

=

[δm′

1,mr+1

r∏i=1

δm′i+1,mi

], (5.42)

where the Dirac delta’s on the RHS are ordinary Dirac delta’s. Now, plugging (5.42) into (5.41)proves the theorem.

Having computed the (r + 1)th order Casimir ζr+1 for each r ∈ N, the next logical step is to tryto give physical meaning to the operators computed. This is the topic of the next section.

5.2.3 Physical Interpretation of the Casimir Operators

In the previous section we have computed the (r+1)th order Casimir operator ζr+1 for each r ∈ N inthe representation given by (5.28). We now address the important question of whether it is possibleto assign any physical meaning to the operators ζr+1. When working in the formalism of secondquantization, as explained in Section 4.2.3, all operators with a clear physical interpretation arenormal ordered, meaning that all raising operators are to the left of the lowering operators. Hence,our approach will be to normal order the expression for ζr+1 in equation (5.36), and then try tointerpret the results.

We will start with the simplest Casimir, the second order operator ζ2, which, according to (5.36),is given by,

ζ2 =1

2πl2B

∑m1,m′

1,m2,m′2

δm′1,m2δm′

2,m1c†m1cm′

1c†m2

cm′2.

The ladder operators can be normal ordered as follows,

c†m1cm′

1c†m2

cm′2

= c†m1(−c†m2

cm′1+ δm2,m′

1)cm′

2

= −c†m1c†m2

cm′1cm′

2+ δm2,m′

1c†m1

cm′2.

Hence,

ζ2 =1

2πl2B

NB

∑m

c†mcm −∑

m,m′

c†mc†m′cm′cm

.

The two operators occurring in the above expression have a clear physical meaning. We followthe convention of Definitions 4.2 and 4.3.


Definition 5.10. For k ∈ N, let

nm1,...,mk:= c†m1

· · · c†mkcmk

· · · cm1 .

Recall that the operator nm = c†mcm counts the number of electrons occupying state |αm〉. That is,its action on a multi-particle basis state φm1 ∧ . . . ∧ φmk

is given by

nm φm1 ∧ . . . ∧ φmk=

φm1 ∧ . . . ∧ φmk

if m ∈ m1, . . . ,mk0 otherwise

Thus, the term∑

m nm counts the total number of electrons. Likewise, the operator nm,m′ =c†mc

†m′cm′cm acts on a basis state φm1 ∧ . . . ∧ φmk

as

nm,m′ φm1 ∧ . . . ∧ φmk=

φm1 ∧ . . . ∧ φmk

if m,m′ ∈ m1, . . . ,mk0 otherwise

Hence, the operator∑

m,m′ nm,m′ counts the number of pairs with multiplicity, meaning that eachpair is counted twice. We conclude that the second order Casimir only counts particles and pairs,it does not care about how the particles are distributed among the available states.

Let us summarize the result in the following corollary.

Corollary 5.7. The second order Casimir ζ2 takes the form

ζ2 =1

2πl2B

NB

∑m

nm −∑

m,m′

nm,m′

(5.43)

when acting on Fock space F∗. Moreover, ζ2 does not distinguish between two different basis stateswith the same amount of electrons: when restricted to the Nel-particle subspace F∗Nel

of Fock spaceF∗ =

⊕∞N=0 F∗N , ζ2 is equal to the multiplication operator

ζ2∣∣F∗

Nel

=Nel

2πl2B(1 +NB −Nel). (5.44)

Proof. When acting on φm1 ∧ . . . ∧ φmNel,∑

m nm gives a factor of Nel, and∑

m,m′ nm,m′ gives afactor of Nel(Nel− 1). Hence, whenever ζ2 acts on a basis state φm1 ∧ . . .∧φmNel

with Nel electronsin it, it gives

ζ2 φm1 ∧ . . . ∧ φmNel=

12πl2B

[NBNel −Nel(Nel − 1)] φm1 ∧ . . . ∧ φmNel

=Nel

2πl2B(1 +NB −Nel)φm1 ∧ . . . ∧ φmNel

.

Before we proceed to compute the higher order Casimir operators, we will perform a consistencycheck. Haldane [10] also computed the second order Casimir operator, which he calls C2.

C2 :=1

2NB

∑q 6=0

ρ(q)ρ(−q),

where ρ(q) is the sum

ρ(q) :=Nel∑i=1

ρi(q)


of first quantized one-particle operators ρi(q) := e−iq·Ri , and the index i is the particle label.Haldane’s definition of the second order Casimir differs from ours in that Haldane excludes ρ(0)from the sum over the Brillouin zone. Note that this does not affect the centrality of C2. Indeed,ρ(0) =

∑Nel

i=1 ρi(0) = Nel is a constant, hence a central operator, and the difference between twocentral operators is again a central operator, therefore, so is C2. Haldane computes

C2 =Nel(N2

B −Nel)2NB

+∑

m<m′

Pm,m′ , (5.45)

where Pm,m′ is the operator that exchanges the particles at guiding center positions m and m′.If we want to compare C2 to ζ2, we will have to add ρ(0) to the sum in C2. Let us call the

adjusted operator C ′2. By (5.45),

C ′2 :=1

2NB

∑q

ρ(q)ρ(−q) = C2 +(ρ(0))2

2NB=NelNB

2+∑

m<m′

Pm,m′ .

Next, we want to write the permutation operator Pm,m′ in terms of ladder operators. This isstraightforward.

Pm,m′ = c†m′c†mcm′cm.

Notice that, compared to nm,m′ = c†mc†m′cm′cm the order of the raising operators c†m′ and c†m is

reversed. That is, Pm,m′ annihilates particles at positions m and m′, and then creates them inreversed order, which is the same thing as exchanging the particles with guiding centre coordinatesm and m′. Using fermionic exchange statistics,

Pm,m′ = −nm,m′ .

Hence, we see that (5.43) restricted to the Nel-particle subspace F∗Nelreduces to

ζ2∣∣F∗

Nel

=1

2πl2B

NB

∑m

nm +∑

m,m′

Pm,m′

=1πl2B

(NBNel

2+∑

m<m′

Pm,m′

).

Corollary 5.8. We conclude that both Casimir operators agree

ζ2 =1πl2B

C2,

up to a numerically irrelevant constant.

Note that the overall prefactor of a Casimir operator does not necessarily carry any physical signif-icance, because for any Casimir C, λC is also a Casimir for every λ ∈ C.

Let us move on to the third order Casimir ζ3.

ζ3 =1

(2πl2B)2∑

m1,m′1,...,m3,m′

3

δm′1,m3δm′

2,m1δm′3,m2c

†m1cm′

1c†m2

cm′2c†m3

cm′3

=1

(2πl2B)2∑

m1,m2,m3

c†m1cm3c

†m2cm1c

†m3cm2 . (5.46)


Normal ordering the ladder operators gives

c†m1cm3c

†m2cm1c

†m3cm2 = c†m1

(δm2,m3 − c†m2cm3)(δm1,m3 − c†m3

cm1)cm2

= δm2,m3δm1,m3c†m1cm2 − δm2,m3c

†m1c†m3

cm1cm2

− δm1,m3c†m1c†m2

cm3cm2 + c†m1c†m2

cm3c†m3cm1cm2

= δm2,m3δm1,m3c†m1cm2 − δm2,m3c

†m1c†m3

cm1cm2


cm3cm2 + c†m1c†m2

cm1cm2

− c†m1c†m2

c†m3cm3cm1cm2

= δm2,m3δm1,m3nm1 + δm2,m3nm1,m2

+ δm1,m3nm1,m2 − nm1,m2

+ nm1,m2,m3 . (5.47)

Finally, performing the sums yields the following corollary.

Corollary 5.9. The third order Casimir ζ3 takes the form

ζ3 =1

(2πl2B)2

∑m

nm + (2−NB)∑

m,m′

nm,m′ +∑

m,m′,m′′

nm,m,′,m′′

when acting on Fock space F∗. When restricted to the Nel-particle subspace F∗Nel

of Fock space F∗,ζ3 becomes the multiplication operator

ζ3∣∣F∗

Nel

=Nel

(2πl2B)2[1 +NB −Nel(1 +NB) +N2

el

]. (5.48)

Proof. Substituting Nel, Nel(Nel − 1) and Nel(Nel − 1)(Nel − 2) for∑

m nm,∑

m,m′ nm,m′ and∑m,m′,m′′ nm,m,′,m′′ respectively in the above expression for ζ3 yields

ζ3∣∣F∗

Nel

=1

(2πl2B)2[Nel + (2−NB)Nel(Nel − 1) +Nel(Nel − 1)(Nel − 2)]

=Nel

(2πl2B)2[1 +NB −Nel(1 +NB) +N2

el

].

Remark 5.9. The higher order Casimir’s ζr+1 all have expressions similar to the formulas in (5.44)for ζ2 and (5.48) for ζ3: for each r ∈ 1, 2, 3, . . . , NB − 1, ζr+1 is constant on subspaces F∗Nel

ofFock space, and on each such subspace it acts by multiplication with a polynomial in Nel of degreer+1. For r ≥ NB, ζr+1 acts on each subspace F∗Nel

of Fock space by multiplication with a polynomialin Nel of degree NB, because we can have at most NB particles in the nth

0 Landau level.

Besides the above remark, there is no obvious relation between r and the polynomial ζr+1

∣∣F∗

Nel

.

Let us compute one more Casimir operator to see if some kind of pattern emerges.The fourth order Casimir is given by

ζ4 =1

(2πl2B)3∑

m1,m′1,...,m4,m′

4

δm′1,m4δm′

2,m1δm′3,m2δm′

4,m3c†m1cm′

1c†m2

cm′2c†m3

cm′3c†m4

cm′4

=1

(2πl2B)3∑

m1,...,m4

c†m1cm4c

†m2cm1c

†m3cm2c

†m4cm3 .


As before, we normal order the ladder operators,

c†m1cm4c

†m2cm1c

†m3cm2c

†m4cm3

= c†m1(δm2,m4 − c†m2

cm4)(δm1,m3 − c†m3cm1)(δm2,m4 − c†m4

cm2)cm3

= δ2m2,m4δm1,m3c

†m1cm3 (A)

−δm2,m4δm1,m3c†m1c†m4

cm2cm3

−δ2m2,m4c†m1

c†m3cm1cm3

−δm1,m3δm2,m4c†m1c†m2

cm4cm3

(B)

+δm2,m4c†m1c†m3

cm1c†m4cm2cm3


cm4c†m4cm2cm3


cm4c†m3cm1cm3

(C)

− c†m1c†m2

cm4c†m3cm1c

†m4cm2cm3 . (D)

The terms (A) and (B) are already normal ordered. (C) need one more adjustment,

(C) = δm2,m4c†m1c†m3

(δm1,m4 − c†m4cm1)cm2cm3

+ δm1,m3c†m1c†m2

(1− c†m4cm4)cm2cm3


(δm3,m4 − c†m3cm4)cm1cm3

= δm2,m4δm1,m4c†m1c†m3

cm2cm3


cm2cm3

+ δm2,m4δm3,m4c†m1c†m2

cm1cm3


c†m4cm1cm2cm3


c†m4cm4cm2cm3


c†m3cm4cm1cm3 .

Lastly, (D) requires a bit more work,

(D) = −c†m1c†m2

(δm3,m4 − c†m3cm4)(δm1,m4 − c†m4

cm1)cm2cm3

= −δm3,m4δm1,m4c†m1c†m2

cm2cm3


c†m4cm1cm2cm3


c†m3cm4cm2cm3

− c†m1c†m2

c†m3(1− c†m4

cm4)cm1cm2cm3 .

Performing the summations over m1, . . . ,m4 gives

(A) = NB

∑m

nm

(B) = (NB − 2)∑

m,m′

nm,m′

(C) = (NB − 2)∑

m,m′

nm,m′ − 3∑

m,m′,m′′

nm,m′,m′′

(D) = −∑

m,m′

nm,m′ + (NB − 2)∑

m,m′,m′′

nm,m′,m′′ −∑

m,m′,m′′,m′′′

nm,m′,m′′,m′′′ .

Finally, adding (A) – (D) yields the following corollary.


Corollary 5.10. The fourth order Casimir ζ4 takes the form

ζ4 =1

(2πl2B)3

[NB

∑m

nm + (2NB − 5)∑

m,m′

nm,m′ + (NB − 5)∑

m,m′,m′′

nm,m′,m′′

−∑

m,m′,m′′,m′′′

nm,m′,m′′,m′′′

]

when acting on Fock space F∗. When restricted to the Nel-particle subspace F∗Nelof Fock space F∗,

ζ4 becomes the multiplication operator

ζ4∣∣F∗

Nel

=Nel

(2πl2B)3[(1 +NB)−Nel(1 +NB) +N2

el(1 +NB)−N3el

]. (5.49)

Proof. Substituting Nel, Nel(Nel − 1), Nel(Nel − 1)(Nel − 2) and Nel(Nel − 1)(Nel − 2)(Nel − 3) for∑m nm,

∑m,m′ nm,m′ ,

∑m,m′,m′′ nm,m′,m′′ and

∑m,m′,m′′,m′′′ nm,m′,m′′,m′′′ respectively, gives

ζ4∣∣F∗

Nel

=Nel

(2πl2B)3[NB + (2NB − 5)(Nel − 1) + (NB − 5)(Nel − 1)(Nel − 2)− (Nel − 1)(Nel − 2)(Nel − 3)]

=Nel

(2πl2B)3[(1 +NB)−Nel(1 +NB) +N2

el(1 +NB)−N3el

].

Looking at the obtained expressions for ζ2, ζ3 and ζ4, we see the following pattern emerging.

Conjecture 5.1. When acting on the Nel-particle sector of Fock space, for r ∈ 1, 2, . . . , NB − 1the (r + 1)-th order Casimir ζr+1 takes the following form

ζr+1

∣∣F∗

Nel

=Nel

(2πl2B)r

[(1 +NB)

r−1∑i=0

(−Nel)i + (−Nel)r

]. (5.50)

Moreover, using∑r−1

i=0 xi = 1−xr

1−x , formula (5.50) reduces to

ζr+1

∣∣F∗

Nel

=Nel

(2πl2B)r

[(1 +NB)

1− (−Nel)r

1 +Nel+ (−Nel)r

]. (5.51)

Remark 5.10. We cannot generalize the obtained formulas for ζr+1 to orders for which r ≥ NB,since for r ≥ NB, the highest order terms will drop out because there are only NB different creationoperators, and (c†m)2 = 0 for any m. It is not obvious how exactly this will affect the obtainedformulas for the Casimir operators of order R = r + 1 > NB.

Lastly, inspired by Haldane’s expression (5.45) for the second order Casimir, we mention thatthere is a way to express the obtained Casimir operators ζ2, ζ3, and ζ4 in terms of permutationoperators. In the case of ζ2, there is only one permutation operator that comes to mind: the ex-change operator. However, when going to higher order Casimir operators, there are more candidatesavailable, because there are more than 2 elements in the permutation group SN for N > 2. When werewrite the expressions for the Casimirs in terms of permutation operators, we will not discriminateand use all possible permutation operators.


To start with, rewrite ∑m,m′

nm,m′ =∑σ∈S2

∑m>m′

sgn(σ)Pσm,m′ (5.52)

∑m,m′,m′′

nm,m′,m′′ =∑σ∈S3

∑m>m′>m′′

sgn(σ)Pσm,m′.m′′ (5.53)

∑m,m′.m′′,m′′′

nm,m′.m′′.m′′′ =∑σ∈S4

∑m>m′>m′′>m′′′

sgn(σ)Pσm,m′,m′′,m′′′ (5.54)

where, for example, Pσm,m′ applies σ ∈ S2 to the electrons with guiding center positions m and

m′, and likewise for the higher order permutation operators. To see that the above equations hold,observe that for (5.52),∑

m,m′

nm,m′ =∑

m>m′

2c†mc†m′cm′cm =

∑m>m′

(c†mc†m′cm′cm − c†m′c

†mcm′cm)

=∑

m>m′

(P

(Id)m,m′ − P

(12)m,m′

)=∑σ∈S2

∑m>m′

sgn(σ)Pσm,m′ .

Equations (5.53) and (5.54) follow analogously.Using equations (5.52) – (5.54), we obtain the following expressions for the Casimir operators

ζ2, ζ3 and ζ4:

ζ2 =1

2πl2B

[NBNel −

∑σ∈S2

∑m>m′

sgn(σ)Pσm,m′

]

ζ3 =1

(2πl2B)2

[Nel + (2−NB)

∑σ∈S2

∑m>m′

sgn(σ)Pσm,m′ +

∑σ∈S3

∑m>m′>m′′

sgn(σ)Pσm,m′.m′′

]

ζ4 =1

(2πl2B)3

[NBNel + (2NB − 5)

∑σ∈S2

∑m>m′

sgn(σ)Pσm,m′

+ (NB − 5)∑σ∈S3

∑m>m′>m′′

sgn(σ)Pσm,m′.m′′ −

∑σ∈S4

∑m>m′>m′′>m′′′

sgn(σ)Pσm,m′,m′′,m′′′

].

The advantage of the expressions for ζ2, ζ3 and ζ4 as a polynomial in Nel over the expressionsabove, is that the former expressions generalize to arbitrary order (R = r + 1 = NB), whereas thelatter do not. However, to the physicist, the latter expressions might capture the imagination more,which is why we have listed both.

ConclusionWe have computed the Casimir operators ζ2, ζ3 and ζ4, generalizing Haldane’s result for ζ2. Addi-tionally, we were able to extrapolate the expressions found for the lower order Casimirs up to orderR = NB . The original idea was that, because Haldane’s Casimir contains a permutation operator,perhaps the Casimir operators would be able to give information about the exchange statistics ofthe particles, even if the particles are not necessarily fermions. For example, for bosons, equation(4.28) holds9, and therefore the projected density operator algebra remains intact. However, howthe rest of the construction of Chapter 5 is affected by a change of statistics is not immediatelyclear. Besides bosonic and fermionic statistics, there exist even more general anyonic statistics,which we will discuss in the next chapter — but we will not be able to compute Casimirs in case of

9To prove this, use the bosonic versions of (4.26) – (4.27).


anyons, unfortunately, because there is no straightforward generalization of the formalism of secondquantization to anyonic systems (see the introduction of Section 6.1).

The results of this chapter also apply to composite fermions, described in Section 3.2.5, becausethe projected composite fermion density operators ¯ρ(q) — see equation (5) of Goerbig, Lederer andMorais Smith [6] — satisfy the same Lie algebra as the ordinary projected density operators (5.29):

[ ¯ρ(k), ¯ρ(l)] = 2i sin(πk × l

N∗B

)¯ρ(k + l), (5.55)

where B∗ = B−2sφ0nel, as explained in Section 3.2.5. Hence, all the expressions for the Casimirs ob-tained in this chapter also hold for composite fermions,10 only with B replaced by B∗. In particular,(5.51) for composite fermions reads

ζr+1

∣∣F∗

Nel

=Nel

(2π(l∗B)2)r

[(1 +N∗

B)1− (−Nel)r

1 +Nel+ (−Nel)r

].

Having computed the Casimir operators, we can conclude that, despite the fact that we foundnice formulas, ultimately, from a physicists perspective, the Casimir operators are perhaps not asinteresting as they could have been. Instead of pursuing the line of reasoning followed thus far anyfurther, we will change course and proceed to a topic that is intimately related to the FQHE, butnot directly related to the Casimir operators we computed in Chapters 4 and 5.

10The absence of the minus sign in (5.55) does not affect the form of the Casimir or the computation done in thischapter – as the author found out the hard way.

Chapter 6

Charged Anyons in a MagneticField

The fundamental particles of nature fall into two categories; depending on their exchange statistics,a fundamental particle can either be a bosons or a fermion. The type of the particles in ques-tion determines the symmetry properties of the wavefunction: N -particle wavefunctions describingidentical bosons (fermions) pick up a plus (minus) sign whenever two particles are interchanged.

In turns out that, in two dimensions, there is a possibility for particles to exist that are neitherfermions nor bosons. These particles are called anyons, a term invented by Wilczek [26], because thestatistical phase an anyonic N -particle wavefunction picks up under the exchange of two identicalanyons is — in principle — arbitrary. Since all fundamental particles in nature are either fermionsor bosons, it is expected that, if anyons exist at all, they exist in the form of quasi-particles.

The only physical system known to date in which anyons are expected to appear is that of theFQHE, where the anyons are effective quasi-particle excitations of the FQHE fluid in the groundstate at a Laughlin filling factor ν = 1

2s+1 with s ∈ N, as described in Section 3.2.4. We will discussthe anyonic nature of these quasi-particle excitations in Section 6.1.1. As described in Section 3.2.4,the quasi-particles carry charge. Therefore, we expect that they also feel the magnetic field thatcauses the QHE. Motivated by this conjecture, the goal of this chapter is to find an exact descriptionof charged non-interacting anyons in a uniform perpendicular magnetic field.

The approach that we will take is less conventional than the one normally found in literature,where, as will be explained in Section 6.1.1, an anyon is seen as a charged particle-flux tube compos-ite. Instead, we will take the anyon to be our fundamental particle, meaning that we are workingwith an effective description of the system of quasi-particle excitations, and we will attempt toconstruct exact anyonic energy eigenfunctions of the Hamiltonian that describes non-interactingcharged particles in a uniform perpendicular magnetic field. The difficulty in constructing anyonwavefunctions lies in the fact that even non-interacing anyons feel each other presence, by which wemean that their exchange statistics give rise to a statistical term in the wavefunction that relatesanyons at different positions to each other, making non-interacting anyons equally complicated asinteracting bosons or fermions.

In Section 6.1, we will introduce anyons in the FQHE, and we will set up the mathematicalformalism that describes anyons as fundamental particles. Next, in Section 6.2, we will constructenergy eigenfunctions of the Hamiltonian for non-interacting charged anyons in a uniform perpen-dicular magnetic field.

89

6.1. INTRODUCTION TO ANYONS 90

6.1 Introduction to Anyons

For identical bosons and fermions, the multi-particle wavefunction ψ picks up a phase eiαπ, where αis an even integer for bosons (i.e. eiαπ = 1), and an odd integer for fermions (i.e. eiαπ = −1), underthe exchange of two particles. For anyons, we allow α to be any real number. At first sight, it seemsas if anyons are not very different from bosons or fermions. Therefore, one might guess that muchof the existing theory for bosons and fermions can readily be applied to anyons. This assumptionturns out to false.

For example, one of the first things a physics student might try when constructing a mathematicalformalism that describes anyons, is to use the formalism of second quantization, and impose thefollowing commutation relations on the creation and annihilation operators for anyons,

ψ(r)ψ(r′)− eiαπψ(r′)ψ(r) = 0. (6.1)

However, closer inspection shows that this naive approach is doomed to fail. For, combining equation(6.1) with the same equation with r and r′ interchanged:

ψ(r′)ψ(r)− eiαπψ(r)ψ(r′) = 0,

leads to(1− e2iαπ)ψ(r′)ψ(r) = 0.

Hence, whenever α /∈ Z, we must have ψ(r′)ψ(r) = 0, which, after taking the Hermitian conjugate,means that we can create at most one anyon! This is undesirable for many reasons, one of which isthe fact that anyons statistics only show up when there are at least two anyons present, which wecannot create in this naive approach. The simple argument provided above shows that any secondquantized approach to anyons is likely to fail1.

Now that we have seen how not to describe anyons, we will proceed with the construction of acorrect mathematical formalism for describing anyons. There are multiple ways to do this. First,we will discuss the anyon as it is usually described within the context of the QHE: as a chargedparticle-flux tube composite. Next, we will switch the the mathematically more elegant descriptionthat takes anyons as fundamental particles. We will also elaborate on why anyons are only expectedto occur in two dimensions.

6.1.1 Anyons in the Quantum Hall Effect

As explained in Section 3.2.4, the quasi-particle excitations of the FQHE fluid at a Laughlin fillingfactor ν = 1

2s+1 are quasi-holes and quasi-particles, which consist of a magnetic flux tube Φ and anelectric charge e∗ = eν both localized at the same position. A flux tube, refers to an infinitely thinsolenoid pierced by a magnetic flux Φ, such that the magnetic flux outside of the solenoid is zero.The combination of a charged particle bound to a flux tube is commonly referred to as a chargedparticle flux-tube composite. Its anyonic nature was first noted by Wilczek [25] and [26]. In thissection, we will follow Wilczek and describe how a charged particle-flux tube composite gives rise toanyonic statistics.

Suppose that we have two charged particle-flux tube composites of charge −e∗, mass m andflux Φ, with corresponding position coordinates r1 and r2. We will assume that the electrostaticinteraction is negligible, and that the only interaction between the two particles is due to the fact

1That is, the author is not aware of any description of anyons that uses creation and annihilation operators foranyons. Note that the creation and annihilation operators used in Chern-Simons theory related approaches onlycreate and destroy the charged particle part of the anyon, but they leave the flux-tube part of the anyon untouched.See Dunne [2] for the details.


that the charge of one particle feels the magnetic flux attached to the other. Hence, the Hamiltonianfor the two particles is given by

H =1

2m(p1 + e∗a1) +

12m

(p2 + e∗a2),

where a1 is the magnetic vector potential that particle 1 feels due to the flux attached to particle 2,and likewise for a2. That is,

a1(r1, r2) =Φ2π

ez × (r1 − r2)|r1 − r2|2

,

a2(r1, r2) =Φ2π

ez × (r2 − r1)|r1 − r2|2

.

Note that the above vector potentials mimic the properties of a flux tube. Indeed, the vectorpotential

a(r) =Φ2π

ez × r

|r|2(6.2)

is irrotational outside the origin, so the corresponding magnetic field b(r) = ∇×a(r) = 0 for r 6= 0,but the total flux penetrating any disc D centered at the origin is∫

D

b · ezdr =∮

∂D

A · dr = Φ.

Additionally, we take the charged particles to be bosonic2.Going to the center of mass R := 1

2 (r1 + r2) and relative r := r1− r2 position coordinates, withcorresponding momenta P := MR = 2m r1+r2

2 = p1 + p2 and p := m2 r = p1−p2

2 , the Hamiltonianbecomes

H =P 2

4m+

[p + e∗a(r)]2

m,

where a given by equation (6.2). Thus, the two-charged particle-flux tube system is equivalent toa tensor product ψCM ⊗ ψrel of a free center of mass particle with mass M = 2m together withanother particle of charge −e∗ and mass m

2 that feels the vector potential a due to a magnetic fluxtube Φ at the origin.

We are interested in the exchange process of two anyons, which only involves the relative coordi-nate r. Exchanging the anyons amounts to adiabatically — as to not disturb the system and keepit in the same energy state — moving r to −r. In terms of polar coordinates, r = (r, θ), the anyonexchange process is θ 7→ θ+π. Note that, because the charged particles are bosons, the wavefunctionψrel describing the relative motion of the particles satisfies the boundary condition

ψrel(r, θ + π) = ψrel(r, θ). (6.3)

Remark 6.1. The polar angle θ is multi-valued on the configuration space. However, as will beexplained in the Section 6.1.3, we are actually describing wavefunctions defined on the universalcover of the configuration space, where θ ∈ R is globally defined and single-valued.

The remark above justifies the following gauge transformation:

a 7→ a′ = a−∇χ,

withχ(r, θ) =

Φ2πθ,

2We could have taken them to be fermionic, but for the sake of the model it does not really matter, because wecan decide on the total anyon exchange statistics later on by fixing the flux Φ.


and hence3

a′ = 0.

Therefore, in the primed gauge, also known as the anyon gauge, the Hamiltonian reduces to the freeparticle Hamiltonian

H ′ =P 2

4m+

p′2

m.

However, the wavefunction ψrel in the primed gauge changes as well. As explained in Section 2.2.3,ψrel gets multiplied by a factor U(r) = exp

(−ie∗χ(r)

~

)from equation (2.39). That is,

ψ′rel(r, θ) = exp(−ie∗Φh

θ

)ψrel(r, θ).

We conclude that, because of (6.3), the system comprised of two-charged particle-flux tubecomposites is equivalent to the system of two free particles with anyonic exchange statistics, meaningthat, under particle exchange θ 7→ θ + π,

ψ′rel(r, θ + π) = exp(−ie∗Φπ

h

)ψ′rel(r, θ).

That is, the total wavefunction picks up a phase eiαπ under the exchange of two particles, wherethe statistical parameter α is given by

α =−e∗Φh

.

Remark 6.2. Note that, for anyons, it matters whether the exchange is clockwise θ 7→ θ − π oranti-clockwise θ 7→ θ + π, because the statistical phase that the wavefunction picks up differs for theclockwise (e−iαπ) and anti-clockwise (eiαπ) exchanges.

In the case of quasi-holes, which carry a charge of e∗ = eν and a flux Φ = φ0 = he , the statistical

parameter is given byαqh = −ν

for quasi-holes. The statistical parameter for quasi-particles in the FQHE has the same value.To summarize, we have explained how charged particle-flux tube composites show anyonic behav-

ior. In particular, the quasi-hole excitations of the FQHE fluid at fractional filling factor ν = 12s+1 ,

which consist of a charged particle of charge −e∗ = −eν bound to a flux tube with flux φ0, areanyons with statistical parameter αqh = −ν. Moreover, we now have an idea of how anyonic statis-tics can arise in nature. However, when working with anyons, it seems cumbersome to think of theanyon as a composite of two separate objects. Instead, we prefer to describe an anyon as a singlemathematical entity.

The remainder of Section 6.1 will be devoted to constructing the mathematical framework thatwe will use to describe anyons.

6.1.2 The Configuration Space of Identical Particles

In order to give a rigorous description of anyons, we will have go all the way back to the fundamentalsof identical particles in quantum mechanics.

Since the discovery of the Gibbs paradox4, it has been known that it is necessary to differ-entiate between multi-particle systems comprised of identical particles and multi-particle systems

3Recall that the ∂∂θ

part of the gradient in polar coordinates is 1r

∂∂θ

.4The problem that the entropy of a gas of identical particles does not scale with the number of particles when two

identical containers of the same type of gas are joined together. This problem is solved by dividing the multiplicityby N !, which accounts for the fact that the particles are indistinguishable.


consisting of non-identical particles. The usual way to deal with identical particles, is to im-pose an (anti)symmetry condition on the multi-particle wavefunction. Other than the fact that(anti)symmetric wavefunctions give rise to experimentally observed distributions in statistical me-chanics, the origin of this (anti)symmetrization condition is not immediately clear.

Instead of imposing an (anti)symmetry condition, we will encode the indistinguishability of theidentical particles in the configuration space. We assume that each particle can move through allof Rd, with d an arbitrary positive integer, but that no two particles can be at exactly the sameposition, because we don’t know what it means for two particles to occupy the same space.

Definition 6.1. The set of points of coincidence C ⊂ (Rd)N is set C := (r1, . . . , rN ) ∈ (Rd)N :ri = rj for some 1 ≤ i < j ≤ N of configurations for which two or more particle coordinatescoincide.

As mentioned above, we want C to not be a part of the configuration space. Taking the indistin-guishability into account, the configuration space is defined as follows.

Definition 6.2. The configuration space of N identical particles CN,d is defined as the space

CN,d :=(Rd)N − C

SN,

where C ⊂ (Rd)N is set of elements of coincidence, and the division by SN means that we considerany two elements in (Rd)N − C related by a permutation of particle positions to be equal.

For example, in the case of N = 2 particles, the elements (a, b) ∈ (Rd)2 and (b, a) ∈ (Rd)2

represent the same element in C2,d, which reflects the fact that “two identical particles at positionsa and b” is the same statement as “two identical particles at positions b and a”.

having defined the configuration space, we can now discuss the fundamental difference betweend = 2 and d > 2, and why d = 2 allows for more general exchange statistics than d = 3. In order todo so, we only have to consider the two particles that we want to exchange. We follow the well-knownpublication by Leinaas and Myrheim [16].

Suppose we have two particles in Rd. Using center of mass R := 12 (r1 + r2) and relative

r := r1 − r2 position coordinates, the configuration space C2,d is isomorphic to

C2,d∼= Rd ⊗

(Rd − 0

Z2

),

since the center of mass R coordinate can be anywhere in Rd, but the relative position coordinater has to be non-zero, and it has to satisfy r ∼ −r, because the particles are identical. Now,(

Rd − 0Z2

)∼= (0,∞)× RP d−1,

where RP d−1 is the real projective space of dimension d−1, which, by definition, is equal to Rd/ ∼,with x ∼ y ⇔ x = λy for some λ ∈ R.

Next, we quantize the theory. This means that we construct a complex line bundle L over C2,d,such that its square-integrable sections correspond to two-particle wavefunctions. The line bundle isequipped with a connection which has zero curvature. In quantum mechanical language, the processof adiabatically exchanging particles becomes that of parallel transporting localized sections of Lover a closed loop. Performing a parallel transportation over a closed curve γ : [0, 1] → C2,d yields aunitary map

P γx : Lx → Lx,

where x = γ(0) = γ(1). Note that Lx∼= C, so P γ

x = eiαπ for some α ∈ R, which is the phase thatthe wavefunction picks up after the two particles have been exchanged via the loop γ. Because the


curvature is zero, P γx = P γ′

x for any γ′ that can be continuously deformed into γ. In particular,this means that we can restrict our attention to loops for which the center of mass coordinate R isconstant, because any closed loop for which R is not constant can be continuously deformed intoone for which R is constant. Hence, we will forget about the center of mass coordinate, and onlyconsider closed loops in the relative coordinate configuration space (0,∞)× RP d−1.

Now comes the point where we distinguish between d = 2 and d > 2. For d > 2, the fundamentalgroup of RP d−1 is given by

π1(RP d−1) ∼= Z2.

In particular, any closed loop in (0,∞)×RP d−1 traversed twice can be continuously deformed backto the trivial loop. Hence, for every x ∈ (0,∞)×RP d−1 and for every γ such that x = γ(0) = γ(1),

(P γx )2 = 1,

This means that any kind of exchange process in dimension d > 2 can result in the wavefunctionpicking up a phase eiαπ = ±1. However, in d = 2 dimensions,

π1(RP 1) ∼= Z,

and there is no restriction on P γx = eiαπ, which means that the wavefunction can pick up any

arbitrary phase under the exchange of two particles.Now that we have explained why anyons can possibly occur in d = 2 dimensions, let us focus on

the case d = 2, and discuss the anyon configuration space and its universal cover, which we need toproperly define anyonic wavefunctions.

6.1.3 The Universal Cover of the Configuration Space

From now on, all particles are restricted to move in d = 2 dimensions.

Notation: because d = 2 is fixed, we will simply write CN instead of CN,2. Additionally, an arbi-trary element of CN,d will be denoted as a set r1, . . . , rN. We use curly brackets to emphasize thefact that the order of the elements is irrelevant, i.e. b, a = a, b.

Next, we discuss an important issue that we have swept under the rug in the previous section.

Remark 6.3. An anyonic wavefunction is a section of a complex line bundle L over the configurationspace. However, given a fixed quantum mechanical state, there is some ambiguity in which sectionψ ∈ Γ(CN ,L) of the line bundle to assign to this particular state. For example, suppose we have asection ψ that has the same expectation values as the fixed quantum state. Now, we can do a particleexchange, after which ψ picks up a global phase eiαπ, meaning that ψ(x) 7→ ψ′(x) = eiαπψ(x) foreach x ∈ CN . The transformed section ψ′ describes precisely the same state as ψ does; in particular,all expectation values of ψ and ψ′ are the same. Instead of having different descriptions of the samestate, we would rather have one mathematical entity that contains, in a well-defined way, both ψand ψ′.

In more mathematical language, the section ψ ∈ Γ(CN ,L) is multi-valued. This means that,strictly speaking, it is not a function at all! In order to remedy this issue, we have to go tothe universal cover of the configuration space: a space that itself contains many copies5 of theconfiguration space CN , each copy corresponding to different labellings6 of the anyons.

5Modulo a measure zero set.6See next page.


Definition 6.3. The universal cover of CN is the simply connected covering space of CN , which isunique up to homeomorphism. It is denoted by CN . The corresponding covering map from CN to CN

will be denoted byP : CN → CN .

Without proof, we mention that there is a nice way to visualize the space CN . To each pointr1, . . . , rN ∈ CN we associate a collection of N non-intersecting lines γ1, . . . , γN extending toy → −∞ such that each line begins at precisely one of the points in the set r1, . . . , rN. Weconsider two elements r1, . . . , rN, γ1, . . . , γN and r′1, . . . , r′N, γ′1, . . . , γ′N to be equiv-alent if r1, . . . , rN = r′1, . . . , r′N as sets, and, γ1, . . . , γN can be continuously deformed intoγ′1, . . . , γ′N without the any of the lines intersecting each other, while keeping the endpoints r1, . . . , rNfixed throughout the deformation process. Now, CN is defined as the collection of equivalence classesof sets r1, . . . , rN, γ1, . . . , γN. We will use square brackets to denote equivalence classes:[r1, . . . , rN, γ1, . . . , γN].

Having defined the configuration space CN and its universal cover CN , we can discuss the processof labelling identical anyons. With bosons or fermions, a labelling is an assignment that assigns aposition variable to each fermion or boson. For anyons, the situation is a little bit more complicated.

Given a set of non-intersecting lines γ′1, . . . , γ′N starting at points r1, . . . , rN and extendingto y → −∞, we can always find another set of lines γ1, . . . , γN that are homotopy-equivalent toγ′1, . . . , γ′N, such that the lines γ1, . . . , γN after some point y ≤ c are straight and parallel. Wethen label the lines γ1, . . . , γN with numbers 1, 2 . . . , N from left to right.

Definition 6.4. Given a set of points r1, . . . , rN ∈ CN , a labelling is a choice of an element inthe preimage P−1(r1, . . . , rN) ∈ CN . The anyons are then labelled according to the labelling of thelines — as described above — that connect to the anyon positions.

1 2 1 2 1 2

Figure 6.1: The two dots represent two fixed anyon position coordinates in C2.The figures show three different ways of connecting these two dots with non-intersecting lines going to y → −∞. We then label the dots with either a 1 ora 2 depending on whether the dot in question is connected to the blue line (1)or the red line (2), where the labelling of the lines is always from left to right inincreasing order.

Having defined what it means to label indistinguishable anyons, we can now define the conceptof exchanging two indistinguishable anyons.

Definition 6.5. For a given configuration r1, . . . , rN, the process of exchanging two indistin-guishable anyons is that of physically moving two anyons from one labelling in P−1(r1, . . . , rN)to another labelling in P−1(r1, . . . , rN). In the exchange process, the lines γ1, . . . , γN initiallyconnected to the points in r1, . . . , rN remain connected to their respective (moving) endpoints, anddo not intersect during the exchange process.


Let us try to visualize the process of anyon exchange. In Figure 6.1, focus on the middle picture,where the first anyon is to the left of the second anyon. Now, we rotate both anyons clockwisearound their center of mass over an angle of π, while keeping the red and blue strands attachedto the moving endpoints. Doing so bring us to the left picture in Figure 6.1. This completes theexchange process, as we are now back at the starting point, in the sense that we have two anyons atexactly the same positions as we had before; the only difference is the anyons are labelled differently.

We could have also exchanged the anyons the other way around: using an anti-clockwise exchangeprocess. This would have taken us form the middle picture to the right picture in Figure 6.1. Asnoted in Remark 6.2, clockwise and anti-clockwise exchanges produce different overall phase factorsthat multiply the wavefunction. This feature distinguishes anyons from bosons or fermions, becausefor the latter two it does not matter how the particles are exchanged.

In the next section, we will explain how the difference between clockwise and anti-clockwiseexchanges will be encoded in the anyon multi-particle wavefunction.

6.1.4 Anyon Wavefunctions and the Braid Group

We now turn to the wavefunction ψ describing N indistinguishable anyons. Keeping Remark 6.3in mind, we want ψ to assign a complex number not only to each configuration, but also to eachlabelling of a fixed configuration. In other words, ψ is a section of a complex line bundle L over theuniversal cover of the configuration space: ψ ∈ Γ(CN ,L). For our present purposes, it is sufficientto consider only globally trivializable line bundles, so we will restrict ourselves to wavefunctions ofthe form

ψ : CN → C.

Recall that, in contrast to fermions and bosons, there are now two ways to exchange identicalanyons: clockwise and anti-clockwise. Since clockwise and anti-clockwise exchanges undo each other,the phases related to clockwise and anti-clockwise exchanges must be each others multiplicativeinverse. This hints at the presence of the braid group BN .

Before we continue, let us recall the correspondence between a covering space and the fundamentalgroup of the corresponding base space.

Remark 6.4. For a general topological space X covered by a (not necessarily universal) coveringspace p : X ′ → X, where p is the covering map, there is a well-defined action of the fundamentalgroup π1(X) on the covering space X ′, constructed as follows. For any point x′ ∈ X ′, observe theimage x = p(x′) ∈ X, and consider a closed loop γ : [0, 1] → X with basepoint γ(0) = γ(1) = x.The loop γ can be lifted to a unique loop γ : [0, 1] → X ′ such that γ(0) = x′. Now, γ acts on x′ bymapping x′ to γ(1) ∈ X ′. It can be shown that this action is independent of the homotopy class ofγ. Additionally, the action respects the multiplication of loops by composition, which means that theaction of closed loops on X ′ transcends to a group action of the fundamental group π1(X) on X ′.

Obviously, the universal cover CN covers the configuration space CN . Moreover, we have thefollowing theorem, for which we omit the proof.

Theorem 6.1. The fundamental group of CN is isomorphic to the braid group on N strands:

π1(CN ) ∼= BN .

Due to Remark 6.4, we see that the braid group BN naturally acts on CN . Pictorally, the elementsb12 and b−1

12 of BN map the middle picture in Figure 6.1 to the left and right pictures respectively.We conclude that the process of exchanging two anyons can be seen as an action of a particularelement of the braid group on CN .


Notation: we will adopt the shorthand notation ξ for r1, . . . , rN ∈ CN , and ξ for an element[r1, . . . , rN, γ1, . . . , γN] ∈ CN that projects to ξ. Moreover, for any element ξ ∈ CN and anyb ∈ BN , we denote the action of b on ξ by a dot, i.e. ξ 7→ b · ξ.

The requirement that the N -particle anyonic wavefunction ψ : CN → C picks up a phase e±iαπ

when two anyons are interchanged can finally be made precise.

Requirement. For any b ∈ BN∼= π1(CN ) and ξ ∈ CN , we require that

ψ(b · ξ) = ρ(b)ψ(ξ), (6.4)

whereρ : BN → C (6.5)

is the one-dimensional representation of the braid group that maps each left-over-right generatorbj,j+1 ∈ BN to ρ

(bj,j+1

)= e−iαπ ∈ C, and each right-over-left generator b−1

j,j+1 to ρ(bj,j+1

)−1 =eiαπ ∈ C.

Remark 6.5. It is the representation ρ : BN → C of the braid group that determines the type ofparticle that we are modelling. Setting α = 0 will give back the bosonic wavefunctions, as equation(6.4) then reduces to the statement that ψ takes the same value on each sheet of CN embedded in CN ,meaning that ψ is symmetric with respect to the exchange of two particles. Likewise, when settingα = 1, equation (6.4) reduces to the familiar anti-symmetry condition for fermionic N -particlewavefunctions. Only when we take α /∈ Z do we get to see the fact that we are actually dealing witha representation of the braid group, as only in this case we can distinguish between clockwise andanti-clockwise exchanges of particles, since then eiαπ 6= e−iαπ.

Now that we have made explicit the condition (6.4) for the wavefunction to satisfy under particleexchange, let us try to construct a function on the universal cover with the required exchangeproperties.

6.1.5 The Statistical Term in the Wavefunction

We will now construct functions on the universal cover CN of the configuration space that behavecorrectly under particle exchange. Because, later on, we want the obtained wavefunctions to beeigenfunctions of the Hamiltonian, we put the obvious additional restriction on the to be constructedwavefunctions that they are differentiable.

Remark 6.6. A possible approach to constructing multi-particle anyon wavefunctions is to try togeneralize the construction of multi-particle bosonic (fermionic) wavefunctions by (anti-)symmetrizingthe single-particle wavefunctions. Goldin and Majid [8] came up with such a generalization, by mak-ing use of what they call the braided tensor product. The problem with their approach, however, isthat the multi-particle wavefunctions obtained with the use of the braided tensor product are discon-tinuous, and can therefore not be eigenfunctions of a differential operator such as the Hamiltonian.

Notation: We will use complex coordinates z = x − iy on R2. An element of CN will now bedenoted by complex numbers z1, . . . , zN.

The complex coordinates introduced above allow for a local7 identification of CN and CN with CN ,thereby making CN and CN complex manifolds of complex dimension N . Hence, we know what itmeans for a function ψ : CN → C to be holomorphic.

Because we require that ψ picks up a phase when two anyons are exchanged, the dependence ofψ on the paths γ1, . . . , γN should appear only as a phase factor. We also require that ψ depends

7Dividing by a discrete group (like SN ) does not change the local properties of a space.


differentiably on its argument. A way to achieve this is by using the paths γ1, . . . , γN to defineangles θjk ∈ R for each pair of particles8, as shown in Figure 6.2, and then taking these angles asthe argument of some periodic function.

Figure 6.2: Forget about all anyons but the jth and the kth anyon, labelled by theblue and red dots respectively, and then define θjk as the angle swiped out overthe grey area as shown above. The two pictures give an example of a negative(left) and positive (right) angle θjk.

Using the angles θjk introduced above, we are now able to construct holomorphic functions onCN by considering products of the form |zj − zk|αeiαθjk , where j, k ∈ 1, . . . , N, which we formallywrite as

(zj − zk)α := |zj − zk|αeiαθjk . (6.6)

The notation above makes sense because we can locally express the right hand side of (6.6) in termsof a holomorphic branch of the complex logarithm,

(zj − zk)α = exp [α(log |zj − zk|+ iθjk)] = exp [α logC(zj − zk)] , (6.7)

where the choice of the branch of the complex logarithm is determined by the value of θjk. Thisway of writing (zj − zk)α shows that (zj − zk)α is actually holomorphic on CN .

Remark 6.7. We will use the notation (zi − zj)α for |zj − zk|αeiαθjk throughout the rest of thischapter, but it must be kept in mind that (zi − zj)α is a function that not only depends on zj andzk, but also on the angle θjk.

Remark 6.8. The fact that (6.6) behaves as required under the exchange of the jth and the kth

particle can be verified directly: e.g. for (j, k) = (1, 2), applying either b12 or b−112 increases θ12 by

−π or π respectively, which results in an extra factor of e−iαπ or eiαπ in (6.6). Note that the factor|z1 − z2|α does not give any extra signs because of the absolute value.

Differentiating (zi − zj)α with respect to zi yields

∂

∂zi(zi − zj)α =

∂

∂ziexp [α(logC(zj − zk)] =

α

zi − zjexp [α(logC(zj − zk)]

=α

(zi − zj)(zi − zj)α. (6.8)

To summarize, we have constructed the configuration space for identical particles, and we haveseen that, due to difference in the topology9 between the two and the more-dimensional configurationspaces, the phase picked up by the multi-particle wavefunction under the exchange of two particlescan be any complex phase in two dimensions, whereas it has to be ±1 in three or more dimensions.

8More formally, we are actually constructing functions on the abelian cover of CN .9More precisely, the fundamental group.

6.2. CONSTRUCTION OF THE ANYON EIGENFUNCTIONS 99

We then focused on the situation of two dimensions, and we argued that, in order to properlydescribe multi-particle wavefunctions — which are in general of anyonic nature — we have to considerfunctions defined on the universal cover of the configuration space. We then constructed statisticalterms of the form (zi − zj)α, which are holomorphic and have the required exchange properties.However, in order to construct a physically meaningful wavefunction, we will have to involve theN -particle Hamiltonian.

This concludes the mathematical setup for anyons. In the next section, we will proceed with thephysically relevant model of non-interacting charged anyons in a uniform perpendicular magneticfield, and we will attempt to construct eigenfunctions of the corresponding Hamiltonian.

6.2 Construction of the Anyon Eigenfunctions

In this section, we will constructN -particle anyonic wavefunctions describing non-interacting chargedanyons in a uniform perpendicular magnetic field. Our approach is based on a general ansatz in-spired by our physical knowledge of the one-particle case, the wavefunctions of which we discussedin Section 3.2.1.

Before we start with the construction of the anyon eigenfunctions, there is one important issuethat want to touch upon briefly. Anyons arise in the FQHE as effective quasi-particles due to thestrong magnetic field that the actual particles — i.e. the electrons — feel. It is not completely agreedupon that the anyons, being quasi-particle excitations of the FQHE fluid, feel the external magneticfield that causes the FQHE themselves, because, in principle, collective excitations can behavecompletely differently from their constituent particles. However, in the case of the quasi-particleexcitations of the FQHE fluid, it is known that the excitations carry charge, which should interactwith the external magnetic field. Moreover, the charged particle-flux tube composite description ofanyons is based on the interaction between a (localized) magnetic field on one anyon and the chargeof another. This suggests that the FQHE anyons should indeed also feel the external magnetic fieldthat causes the FQHE.

To further support our claim that anyons do feel the external magnetic field, here are three papersor books that claim the same: page 58 of Khare [13], page 17 of Stern [22] and the introduction ofJohnson and Canright [12]. Additionally, Lerda (page 60 of [17]) claims that the quasi-particles andquasi-holes in the FQHE feel the underlying electrons exactly as a charged particle would feel anexternal uniform magnetic field.

The author is of the opinion that the arguments presented above give enough motivation to studythe system of non-interacting charged anyons in a uniform perpendicular magnetic field. Besides,irrespective of whether anyons feel a magnetic field or not, the Hamiltonian that we will consider isalso applicable to free anyons that rotate around the z-axis with constant angular frequency Ω, aswill be explained in Remark 6.9.

6.2.1 The Wavefunction Ansatz

We now turn to an actual physical system of non-interacting charged anyons in a perpendicularmagnetic field B = Bez. We take our anyons to have mass m, and charge −e∗. In the case of theFQHE at filling factor ν,, we know that e∗ = eν. However, our description of anyons is in principlenot restricted to the FQHE, so we will leave the charge −e∗ as a free parameter.

We want to find exact anyonic eigenfunctions ψ of the N -particle Hamiltonian

HN =N∑

i=1

12m

(Π2

xi+ Π2

yi

), (6.9)

where ri = (xi, yi) ∈ R2 denotes the position of the i-th anyon, and

Πi = −i~∇i + e∗Ai.


As usual, we will be working in the symmetric gauge (2.41),

A =B

2(−yex + xey).

Remark 6.9. The Hamiltonian HN in (6.9) in the symmetric gauge is not only applicable tocharged anyons in a magnetic field; it is more general, because it also describes free anyons thatare set spinning with angular frequency Ω around the z-axis. Indeed, the coordinate transformationfrom a stationary to a rotating reference frame amounts to replacing the momentum p by p+r×Ω,where Ω := Ωez, which is mathematically equivalent to the minimal substitution p 7→ p + eA in thesymmetric gauge.

Going to complex coordinates z = x− iy, z = x+ iy, ∂z = (∂x + i∂y)/2 and ∂z = (∂x − i∂y)/2,the Hamiltonian (6.9) becomes

HN =~ω∗C

2

N∑i=i

[1

4(l∗B)2zizi − zi∂zi

+ zi∂zi− 4(l∗B)2∂zi∂zi

]. (6.10)

Recall that the magnetic length l∗B and the cyclotron frequency ω∗C are defined by l∗B :=√

~/e∗Band ω∗C := e∗B/m.10

Before we tackle the problem at hand, let us change to dimensionless variables to get rid of themagnetic length in the Hamiltonian.

Definition 6.6. A more computational friendly set of variables to work with is given by

w :=z

2l∗B, w :=

z

2l∗B, (6.11)

∂ := ∂w = 2l∗B∂z, ∂ := ∂w = 2l∗B∂z.

The θjk variables remain invariant under the rescaling given by (6.11).

In terms of the newly defined variables, the Hamiltonian (6.10) becomes

HN =~ω∗C

2

N∑i=i

[wiwi − wi∂i + wi∂i − ∂i∂i

], (6.12)

The approach that we take is the following. In order to construct eigenfunctions of the Hamilto-nian, we will use an smart ansatz based on the following ingredients: (a) the one-particle wavefunc-tions for charged particles in a magnetic field11, and (b) the statistical terms introduced in Section6.1.5.

Let us focus on (b) first. We require that ψ contains a statistical term that accounts for thecorrect exchange properties. The function that does the job is the statistical prefactor, given interms of the w-coordinates by

S(w1, . . . , wN , θ12, θ13, . . . , θN−1,N ) :=N∏

k>j=1

(wj − wk)α. (6.13)

This is the only term in the wavefunction that will depend on the angles θjk, the remaining termswill depend on the position coordinates alone. Before we proceed, let us check that this prefactorindeed has the correct exchange properties.

10The star in l∗B and ω∗C reminds us of the fact that the charge of the anyons is −e∗. It is not related to star weuse for composite fermions, which refers to a weakened magnetic field B∗.

11Note that the one-particle anyon wavefunctions are the same as the one-particle electron wavefunctions, becausethere are no exchange statistics in play when there is only one particle present.


Lemma 6.1. For any element B ∈ BN , the statistical prefactor S satisfies equation (6.4): that is,for any ξ ∈ CN ,

S(b · ξ) = ρ(b)S(ξ).

Proof. Because ρ : BN → C is a homomorphism, we can restrict ourselves to the case that b ∈ BN

is a generator b−1j,j+1 of the braid group. We will focus on the case of an anti-clockwise exchange, so

ρ(b−1j,j+1) = eiαπ. Since the endpoints remain invariant under an exchange process, we only have to

focus on the angular part of (6.13), given by,

N∏k>j=1

exp (iαθjk) = exp

iα N∑k>j=1

θjk

.

Applying b−1j,j+1 to all the different angles separately gives the following transformations,

b−1j,j+1θj,j+1 = θj,j+1 + π

b−1j,j+1θk,j = θk,j+1

b−1j,j+1θk,j+1 = θk,j

b−1j,j+1θj+1,l = θj,l

b−1j,j+1θj,l = θj+1,l

b−1j,j+1θk,l = θk,l

where k < l such that k, l /∈ j, j + 1. Hence, we see that the net effect of b−1j,j+1 on the sum∑N

k>j=1 θjk is that we get an extra factor of π, resulting in an extra overall phase factor of eiαπ.For clockwise exchanges bj,j+1, we get a −π added to the sum instead of a +π, which results in anextra overall phase factor of e−iαπ.

Next, we take (a) into account. In addition to the statistical prefactor, we will include theubiquitous exponential factor that occurs in all fermionic and bosonic N -particle wavefunctions forcharged particles in a uniform perpendicular magnetic field,

E(z1, . . . , zN , z1, . . . , zN ) := exp

[−

N∑i=1

|zi|2

4(l∗B)2

],

which in terms of the w-coordinates takes the form

E(w1, . . . , wN , w1, . . . , wN ) := exp

[−

N∑i=1

wiwi

],

into our ansatz. Including this extra factor will have the added benefit of making all obtained wave-functions rapidly decreasing at infinity.

Notation: We will abbreviate the vectors (w1, . . . , wN ), (w1, . . . , wN ) and (θ12, θ13, . . . , θN−1,N )with boldface characters w, w and θ respectively.

Ansatz. Taking everything together, we arrive at the following ansatz for the N -particle anyoniceigenfunction of HN .

ψ(w, w,θ) = S(w,θ)E(w, w)F (w, w), (6.14)

where F is a yet undetermined function of w and w.


Remark 6.10. We have constructed S such that it transforms as an anyon wavefunction shouldunder particle exchange. Hence, to ensure that ψ transforms correctly, we require that the remainingterm FE is symmetric. By symmetry of E, this means that F has to be symmetric with respect topairwise exchange of variables (wi, wi) ↔ (wj , wj).

In short, based on physical intuition and the statistical properties that we want our wavefunctionto satisfy, we have come up with an ansatz of the form (6.14) for a general N -particle anyonic energyeigenfunction. To see if our ansatz makes any sense at all, in the next section, we will try to solvethe time-independent Schrodinger equation with it.

6.2.2 The Time Independent Schrodinger Equation

Finally, we come to the time-independent Schrodinger equation. Motivated by our physical intuitionthat anyons in a magnetic field will form Landau levels just like ordinary charged particles do,we expect that (at least some of) the N -particle energy eigenvalues will be of the form EN,K =12~ω∗C(N + 2K), where K ∈ N is a natural number that signifies the amount of energy packets ~ω∗Cthe system has on top of its ground state energy 1

2N~ω∗C . Henceforth, we will refer to K as theexcitation energy.

The time-independent Schrodinger equation for ψ(w, w,θ) with energy eigenvalue E = EN,K

readsHN ψ(w, w,θ) =

12

~ω∗C(N + 2K)ψ(w, w,θ), (6.15)

where HN is given by (6.12).

Lemma 6.2. Plugging the ansatz (6.14) into Schrodinger equation (6.15) yields the following equa-tion for F :

N∑i=1

wi −12

(∂i +

N∑j=1,j 6=i

α

wi − wj

) ∂iF (w, w) = K F (w, w). (6.16)

Proof. We expand the left hand side of (6.15) by applying H in (6.12) to the ansatz (6.14).

HN ψ(w, w,θ) =~ω∗C

2

N∑i=i


]S(w,θ)E(w, w)F (w, w)

=~ω∗C

2

N∑i=i

S(w,θ)wiwiE(w, w)F (w, w)− wi∂i

[S(w,θ)

]E(w, w)F (w, w)

− S(w,θ)wi∂i

[E(w, w)

]F (w, w)− S(w,θ)E(w, w)wi∂i

[F (w, w)

]+ S(w,θ) wi∂i

[E(w, w)

]F (w, w) + S(w,θ)E(w, w) wi∂i

[F (w, w)

]− ∂i

[S(w,θ)

]∂i

[E(w, w)

]F (w, w)− ∂i

[S(w,θ)

]E(w, w) ∂i

[F (w, w)

]− S(w,θ) ∂i∂i

[E(w, w)

]F (w, w)− S(w,θ) ∂i

[E(w, w)

]∂i

[F (w, w)

]− S(w,θ) ∂i

[E(w, w)

]∂i

[F (w, w)

]− S(w,θ)E(w, w) ∂i∂i

[F (w, w)

].

The above expression can be simplified significantly. First of all, by taking the first terms of the first,second, third and fifth line together, we recognize the term S(w,θ)F (w, w)HN E(w, w). BecauseE(w, w) is the unnormalized wavefunction describing N particles in the lowest Landau level, it hasan energy eigenvalue of 1

2N~ω∗C . Hence, this term cancels with 12N~ω∗Cψ(w, w,θ) on the right hand

side of (6.15).


Furthermore, using ∂iE(w, w) = −wiE(w, w), we see that the second terms on the first andsecond line cancel with the first terms on the fourth and sixth line. Thus, equation (6.15) becomes

~ω∗C2

N∑i=i

S(w,θ)E(w, w) wi∂i

[F (w, w)

]− ∂i

[S(w,θ)

]E(w, w) ∂i

[F (w, w)

]− S(w,θ) ∂i

[E(w, w)

]∂i

[F (w, w)

]− S(w,θ)E(w, w) ∂i∂i

[F (w, w)

]= ~ω∗CKS(w,θ)E(w, w)F (w, w). (6.17)

Now, observe that ∂iE(w, w) = −wiE(w, w). Also, by equation (6.8),

∂iS(w,θ) =

N∑j=1,j 6=i

α

wi − wj

S(w,θ).

Hence, (6.17) can be rewritten as

~ω∗C2S(w,θ)E(w, w)

N∑i=i

2wi −

N∑j=1,j 6=i

α

wi − wj− ∂i

∂iF (w, w)

= ~ω∗CKS(w,θ)E(w, w)F (w, w),

which is indeed equivalent to (6.16).

In short, we have been able to rewrite the time-independent Schrodinger equation (6.15) appliedto the ansatz (6.14) to equation (6.16) for the unknown function F (w, w). In the next section wewill solve equation (6.16) for F .

6.2.3 A Polynomial Ansatz for the Wavefunction

We will now attempt to solve equation (6.16) obtained in the previous section for the unknownfunction F (w, w). Before we do so, let us mention two things. First, observe that there is a hiddenstructure to be found in equation (6.16).

Definition 6.7. Define the operators τi by

τi := −12

(∂i +

N∑j=1,j 6=i

α

wi − wj

). (6.18)

Lemma 6.3. The operators τi commute

Proof. Follows straight from the definition.

Corollary 6.1. Using the newly defined operators τi, equation (6.16) simplifies to:

N∑i=1

[wi + τi] ∂iF (w, w) = K F (w, w). (6.19)

Remark 6.11. The strength in the above simplification lies not only in the notation. Because theoperators τi commute with each other, and also with wi and ∂i, we can regard the τi’s as numberswhen working with equation (6.19) as an equation in w.


Second, using our physical intuition, we will restrict the form of F . The one-particle wave-functions for anyons should coincide with the one-particle wavefunctions for electrons, because thedifference between anyons and fermions only becomes apparent when there is more than one particlepresent. Recalling equation (3.14) and the formulas for the ladder operators a, a†, b and b† in thesymmetric gauge given in Section 3.2.1, the one-particle wavefunctions for electrons in the symmetricgauge are of the form

ψelectron(z, z) = P (z, z) exp(−|z|

2

4l2B

),

where P (z, z) is a polynomial in z and z such that:(1) the highest power of z in P (z, z) corresponds to the guiding centre coordinate of the

electron in state ψelectron, and,(2) the highest power of z in P (z, z) corresponds to the energy of the electron in state ψelectron.

Hence, we expect that F in the ansatz (6.14) is also polynomial in w1, . . . , wN and w1, . . . , wN . More-over, recall that we are trying to solve the time-independent Schrodinger equation (6.15)

HN ψ(w, w,θ) =12

~ωC(N + 2K)ψ(w, w,θ),

where K is the number of energy packets that separates the anyonic N -particle state ψ from theground state. Now, because of comment (2) above, the highest power of wi in F should correspondto the energy of the ith anyon. Therefore, we expect that the sum of the exponents of the variablesw1, . . . , wN in every term of F is at most K.

The above arguments naturally lead to the following formula for F :

F (w, w) =∑

0≤i1+...+iN≤K

ci1,...,iN(w)wi1

1 · · · wiN

N , (6.20)

where the ci1,...,iN(w) are unknown polynomials in w1, . . . , wN .

Remark 6.12. As noted by Ouvry [20], taking F (w, w) = F (w) to be a holomorphic polynomialyields anyonic N -particle eigenfunctions of the Hamiltonian HN with eigenvalue 1

2N~ω∗C , that is,N -particle lowest Landau level anyon eigenfunctions. This can easily be checked by applying HN

to ψ = S E F with F = F (w). However, we will see in the next section that the challenge lies infinding wavefunctions for the higher Landau levels.

It turns out that, when we restrict ourselves to a general ansatz of the form ψ = S E F , whereF is a generic polynomial in w and w, equation (6.20) is the only formula for F that will yield anysolutions to the Schrodinger equation (6.19). Before we show this, here is a definition that will comein handy.

Definition 6.8. For each coefficient ci1,...,iN(w), we define the order of ci1,...,iN

(w) to be i1+. . .+iN .Additionally, we introduce the function Ord, defined by

Ord[ci1,...,iN(w)] := (i1 + . . .+ iN )ci1,...,iN

(w).

Lemma 6.4. The only solutions to the Schrodinger equation (6.19) for an arbitrary polynomialfunction F in the variables w and w are those given by formula (6.20), where the only non-zerocoefficients ci1,...,iN

are those for which i1 + . . .+ iN ≤ K.

Proof. Assume that F is an arbitrary polynomial in w and w. Reorder F such that it is a polynomialin w with coefficients that depend on w — as in (6.20) — and let D be the degree of F whenconsidered as a polynomial in w. We will prove that D = K.


First, observe what happens when we apply (wk+τk)∂k to a arbitrary term ci1,...,iN(w)wi1

1 · · · wiN

N .

(wk + τk)∂k ci1,...,iN(w)wi1

1 · · · wiN

N = ikci1,...,iN(w)wi1

1 · · · wiN

N

+ ikτk[ci1,...,iN(w)]wi1

1 · · · wik−1k · · · wiN

N .

Hence, the operator∑N

k=1 [wk + τk] ∂k contains parts wk∂k that preserves the order, and parts τk∂k

that lower the order.Next, we focus on the highest order terms. Because of the above, the only terms on the

LHS of the Schrodinger equation (6.19) that contain terms of order D are the terms of the formwk∂kci1,...,iN

(w)wi11 · · · w

iN

N with i1 + . . .+ iN = D. In particular, for each fixed highest order termci1,...,iN

(w)wi11 · · · w

iN

N , we have,

N∑k=1

[wk + τk] ∂kci1,...,iN(w)wi1

1 · · · wiN

N =N∑

k=1

ikci1,...,iN(w)wi1

1 · · · wiN

N + lower order terms

= Dci1,...,iN(w)wi1

1 · · · wiN

N + lower order terms.

Comparing to the RHS of the Schrodinger equation, where each term is multiplied by K, we concludethat, the only way for the LHS to equal the RHS is when K = D.

Corollary 6.2. For as far as eigenfunctions of the form ψ = S E F , with F a polynomial in w andw, are concerned, the only possible energy eigenvalues 1

2~ω∗C(N + 2K) that we can hope to find arethose for which K is a non-negative integer.

Let us summarize where we stand. We wrote the ansatz eigenfunction ψ of HN as a product ofthree functions ψ = S E F , where F is given by formula (6.20), with yet to be determined coefficientsci1,...,iN

(w).

Lemma 6.5. Plugging (6.20) for F into the Schrodinger equation (6.19) yields the following recur-rence relation for the coefficients ci1,...,iN

(w) of F ,

(K−Ord) ci1,...,iN(w) =

N∑k=1

(ik+1)τk ci1,...ik−1,ik+1,ik+1,...,iN(w) for 0 ≤ i1+. . .+iN ≤ K. (6.21)

That is, the coefficient ci1,...,iNon the LHS is equal to a weighted sum over all coefficients obtained

by increasing one of the indices of ci1,...,iN.

Remark 6.13. Before we prove this lemma, let us make the following important remark: the aboverecurrence relation has plenty of solutions! As a matter of fact, the highest order coefficients, i.e.the ci1,...,iN

’s with order K, can be chosen freely, and then (6.21) gives the expressions for the lowerorder coefficients in terms of the highest order coefficients.

Proof. We have

N∑k=1

[wk + τk] ∂kF (w, w) =N∑

k=1

[wk + τk] ∂k

∑0≤i1+...+iN≤K

ci1,...,iN(w)wi1

1 · · · wiN

N

=

N∑k=1

∑0≤i1+...+iN≤K

ik ci1,...,iN(w)wi1

1 · · · wiN

N (A)

+N∑

k=1

∑0≤i1+...+iN≤K

ik τk ci1,...,iN(w)wi1

1 · · · wik−1k · · · wiN

N (B)


For (A), we have, after interchanging the sums,

(A) =∑

0≤i1+...+iN≤K

Ord[ci1,...,iN(w)]wi1

1 · · · wiN

N .

Regarding (B), for fixed index k, we have,∑0≤i1+...+iN≤K


1 · · · wik−1k · · · wiN

N

=K∑

ik=1

∑0≤i1+...+ik−1+ik+1+...+iN≤K−ik


1 · · · wik−1k · · · wiN

N

=K−1∑ik=0

∑0≤i1+...+ik−1+ik+1+...+iN≤K−1−ik

(ik + 1) τk ci1,...,ik+1,...,iN(w)wi1

1 · · · wiN

N

=∑

0≤i1+...+iN≤K−1

(ik + 1) τk ci1,...,ik+1,...,iN(w)wi1

1 · · · wiN

N

where∑

means that we are summing over all indices but ik. Summing over all k yields

(B) =∑

0≤i1+...+iN≤K−1

[N∑

k=1

(ik + 1) τk ci1,...,ik+1,...,iN(w)

]wi1

1 · · · wiN

N .

Taking everything together gives

N∑k=1

[wk + τk] ∂kF (w, w)

=∑

i1+...+iN=K

K ci1,...,iN(w)wi1

1 · · · wiN

N

+∑

0≤i1+...+iN≤K−1

[Ord[ci1,...,iN

(w)] +N∑

k=1

(ik + 1) τk ci1,...,ik+1,...,iN(w)

]wi1

1 · · · wiN

N

For the Schrodinger equation (6.19) to hold, the above has to be equal to

KF (w, w) =∑

0≤i1+...+iN≤K

K ci1,...,iN(w)wi1

1 · · · wiN

N .

By comparing the coefficients of wi11 · · · w

iN

N on both side of the Schrodinger equation, we see that thehighest order coefficients are free to choose, and the lower order coefficients must satisfy (6.21).

To summarize, we plugged the ansatz ψ = S E F into the Schrodinger equation, and we got arecurrence formula (6.21) for the coefficients of F seen as a polynomial in w. In the next section,we will try to find solutions to the recurrence relation (6.21).

6.2.4 Solutions to the Schrodinger Equation

We are looking for anyonic eigenfunctions of the Hamiltonian for N non-interacting charged anyonsin a uniform perpendicular magnetic field, which, in terms of the dimensionless complex coordinatesw and w, is given by (6.12), that is

HN =~ω∗C

2

N∑i=i


].


Using the ansatz ψ = S E F , where F is a polynomial in w and w, we have come tot the conclusionthat, for this particular ansatz, the eigenvalues of HN are given by EN,K = 1

2~ω∗C(N + 2K), whereK is a non-negative integer, and the to-be-determined polynomial F is of the form (6.20), i.e.

F (w, w) =∑

0≤i1+...+iN≤K

ci1,...,iN(w)wi1

1 · · · wiN

N .

As shown in Lemma 6.5, the Schrodinger equation for ψ,

HNψ =12

~ω∗C(N + 2K)ψ,

becomes a recurrence relation for the unknown coefficients ci1,...,iN(w) of F ,

(K−Ord) ci1,...,iN(w) =

N∑k=1

(ik+1)τk ci1,...ik−1,ik+1,ik+1,...,iN(w) for 0 ≤ i1+. . .+iN ≤ K. (6.22)

Additionally, by Remark 6.10, we require that F is symmetric with respect to pairwise exchange(wi, wi) ↔ (wj , wj) of anyon position coordinates. We will discuss the symmetry conditions on F inthe next section.

In this section, we will provide solutions to the above recurrence relation. To solve (6.22) directlyseems quite difficult. Instead, we have opted for a more brute force approach. For different particlenumbers N and excitation energies K, using (6.22), we computed the lower order coefficients interms of the highest order coefficients, and looked for a pattern. The details of these computationscan be found in Appendix B.1. The pattern that we found is the following.

Solution to the recurrence relation (6.22).

(1) Highest order coefficients ci1,...,iNwith i1 + . . .+ iN = K free to choose.

(2) Lower order coefficients ci′1,...,i′Nwith 0 ≤ i′1 + . . .+ i′N < K are given by

ci′1,...,i′N=

∑i1+...+iN=K

(i1i′1

)· · ·(iNi′N

)τ

i1−i′11 · · · τ iN−i′N

N ci1,...,iN. (6.23)

In equation (6.23), we use the convention that(

ii′

)= 0 whenever i < i′.

Remark 6.14. The above solution might seem a little intimidating. However, taking a closer lookat equation (6.23) reveals that it is actually quite an elegant equation. As a matter of fact, (6.23)is equivalent12 to the following set of equations that express the lower order coefficients ci′1,...,i′N

interms of the highest order coefficients ci1,...,iN

,

c0,0,...,0 =∑

i1+...+iN=K

τ i11 · · · τ iN

N ci1,...,iN, (6.24)

ci′1,...,i′N=

1i′1! · · · i′N !

[(∂

∂τ1

)i′1

· · ·(

∂

∂τN

)i′N]c0,0,...,0 for 0 < i′1 + . . .+ i′N < K, (6.25)

where the derivatives in equation (6.25) are formal derivatives with respect to the operators τi.

Let us go over one example to illustrate the workings of the above equations.12When looking for a pattern in the solutions to (6.22), it was actually (6.24) and (6.25) that were found first.


Example 6.1. We will keep it simple, and look at the situation of N = 2 particles with excitationenergy K = 3. We will express all the lower order coefficients in terms of the highest order coefficientsc30, c21, c12 and c03, which are free to choose.

From equation (6.24), the lowest order coefficient c00 is given in terms of the highest ordercoefficients by

c00 = τ31 c30 + τ2

1 τ2c21 + τ1τ22 c12 + τ3

2 c03. (6.26)

The first and second order coefficients are now given by differentiating the expression for c00 (6.26)with respect to the operators τi and then dividing by the corresponding factorials as shown in equation(6.25). This procedure yields

c10 = 3τ21 c30 + 2τ1τ2c21 + τ2

2 c12

c01 = τ21 c21 + 2τ1τ2c12 + 3τ2

2 c03

for the first order coefficients, and,

c20 = 3τ1c30 + τ2c21

c11 = 2τ1c21 + 2τ2c12c02 = τ1c12 + 3τ2c03

for the second order coefficients. Combining all of the above formulas, the full (N,K) = (2, 3)solution ψ2,3 to the Schrodinger equation is given by

ψ2,3 = S E F

= (w1 − w2)αe−w1w1−w2w2

[c00 + c10w1 + c01w2 + c20w

21 + c11w1w2 + c02w

22

+ c30w31 + c21w

21w2 + c12w1w

22 + c03w

32

]= (w1 − w2)αe−w1w1−w2w2

[τ31 c30 + τ2

1 τ2c21 + τ1τ22 c12 + τ3

2 c03

+(3τ2

1 c30 + 2τ1τ2c21 + τ22 c12

)w1 +

(τ21 c21 + 2τ1τ2c12 + 3τ2

2 c03)w2

+ (3τ1c30 + τ2c21) w21 + (2τ1c21 + 2τ2) w1w2 + (τ1c12 + 3τ2c03) w2

2

+ c30w31 + c21w

21w2 + c12w1w

22 + c03w

32

]where the highest order coefficients are all arbitrary polynomials of w1 and w2 obeying the symmetrycondition — see next section, and the operators τi are given by (6.18), that is

τi := −12

(∂i +

N∑j=1,j 6=i

α

wi − wj

). (6.27)

It seems that our work is done: we have found a recipe for producing eigenfunctions of theN -particle Hamiltonian for arbitrary particle number N and excitation energy K. However, thereis one problem that we have swept under the rug. The operators τi introduce local singularities bymultiplying with factors of the form 1

wi−wj, which are not locally square-integrable. This means

that certain wavefunctions obtained through the above procedure are not normalizable. How to dealwith this problem is the subject of the next section.


6.2.5 The Normalizability and Symmetry Conditions

There are two more conditions that we have to take into account, the symmetry and the normaliza-tion condition. We shall start with the easiest of the two: the symmetry condition.

Recall that we wrote F in equation (6.20) as,

F (w, w) =∑

0≤i1+...+iN≤K

ci1,...,iN(w)wi1

1 · · · wiN

N . (6.28)

We now focus on the coefficients ci1,...,iN(w), which are polynomials in the coordinates w1, . . . , wN .

Further expanding ci1,...,iN(w) yields,

ci1,...,iN(w) =

∑0≤j1+...+jN≤d

γj1,...,jN

i1,...,iNwj1

1 · · ·wjN

N . (6.29)

where d is the maximum degree of all coefficients together, and the γj1,...,jN

i1,...,iN’s are complex numbers.

Definition 6.9. We define an action of the permutation group SN on the coefficients γj1,...,jN

i1,...,iNby

σ(γj1,...,jN

i1,...,iN

):= γ

jσ(1),...,jσ(N)iσ(1),...,iσ(N)

(6.30)

for each σ ∈ SN .

The above definition is chosen such that a permutation σ applied to the particle coordinates exactlycorresponds to applying σ to the coefficients γj1,...,jN

i1,...,iNof the corresponding N -particle wavefunction.

With the above definition, the symmetry condition on F can be made explicit.

Symmetry condition: we require that F is symmetric under pairwise exchange (wi, wi) ↔(wj , wj). In terms of the coefficients γj1,...,jN

i1,...,iN, the symmetry condition reads

σ(γj1,...,jN

i1,...,iN

)= γj1,...,jN

i1,...,iNfor all σ ∈ SN and for all indices i1, . . . , iN , j1, . . . , jN . (6.31)

Remark 6.15. Because expression (6.23) — after inserting equation (6.29) — for the lower orderterms ci′1,...,i′N

of order less than K is symmetric with respect to pairwise exchange of the indices,we only have to impose the symmetry condition on the highest order terms ci1,...,iN

of order K, andthe lower order terms will inherit them automatically.

Next, is the more difficult normalizability condition. The problem is that the operators τi createadditional local singularities, which are not square-integrable. We can get rid of these singularitiesby putting extra constraints on the coefficients γj1,...,jN

i1,...,iN. The key idea is captured by the following

lemma.

Lemma 6.6. A homogeneous polynomial P (x, y) =∑

i+j=d ai,jxiyj in two variables of degree d is

divisible by the polynomial (x− y)l, for a fixed l ∈ N, l ≤ d, if and only if∑i+j=d

ikai,j = 0 for all k ∈ 0, 1, . . . , l − 1. (6.32)

Note that our unconventional choice of index placement for the coefficients ai,j is because the coef-ficients γj1,...,jN

i1,...,iNalso have indices at the top.


Proof. In order to prove the lemma, we use a basic algebra theorem on the divisibility of polynomialsin one variable.13 If f is a polynomial over the field of complex numbers (or any domain for thatmatter), then the following two statements are equivalent:

f has a zero of order l at y ∈ C ⇐⇒ f(y) = f ′(y) = f ′′(y) = . . . = f (l−1)(y) = 0, (6.33)

Next, we apply the statement in (6.33) to the polynomial f(x) = P (x, y) considered only as apolynomial in x. That is,

f(x) =d∑

i=0

bixi,

where bi is a function of y,bi(y) = ai,d−iyd−i.

Now, the LHS of (6.33) taken to be true for all y ∈ C is precisely the statement that thetwo-variable polynomial (x− y)l divides the polynomial P (x, y). The RHS of (6.33) reads

0 = f(y) =d∑

i=0

ai,d−iyd

0 = f ′(y) =d∑

i=0

iai,d−iyd−1

0 = f ′′(y) =d∑

i=0

i(i− 1)ai,d−iyd−2

...

0 = f (l−1)(y) =d∑

i=0

i(i− 1) · · · (i− (l − 2))ai,d−iyd−(l−1).

Note that, in each line, the exponent of y is independent of i, so we can divide out all the factors ofy when y 6= 0. We conclude that the RHS of (6.33), taken to be true for all y ∈ C, is equivalent to

0 =d∑

i=0

ai,d−i

0 =d∑

i=0

iai,d−i

0 =d∑

i=0

i(i− 1)ai,d−i

...

0 =d∑

i=0

i(i− 1) · · · (i− (l − 2))ai,d−i.

13See for example the (Dutch) lecture notes by G. van der Geer [4], Proposition 4.12, for the case l = 2. Thegeneralization of the theorem to arbitrary order is straightforward. If f has an lth order zero at y, then writef(x) = fl(x)(x − y)l and differentiate with respect to x up to l − 1 times at the point x = y. For the converse, byf(y) = 0, we have f(x) = f1(x)(x − y) for some polynomial f1. Differentiate with respect to x at x = y and usef ′(y) = 0 to obtain f1(y) = 0, hence f1(x) = f2(x)(x−y) for some polynomial f2, and therefore f(x) = f2(x)(x−y)2.Proceed by induction.


To complete the proof, note that the above set of equations is equivalent to (6.32) by taking linearcombinations.

Using the Lemma 6.6, we can find out which constraints we have to put on the γj1,...,jN

i1,...,iN’s for the

singularities to disappear. We shall illustrate how in the example below. Because Lemma 6.6 onlyputs a restriction on a fixed degree part, we take our highest order coefficients ci1,...,iN

to consist ofa degree d part only.

Example 6.2. We shall investigate the (N,K) = (2, 1) case. In this case, the free to choose highestorder coefficients are c10 and c01, and the lowest order coefficient c00 can be expressed in terms ofthe highest order coefficients through (6.24). Hence, the polynomial F takes the form

F (w, w) = c00 + c10w1 + c01w2

= τ1c10 + τ2c01 + c10w1 + c01w2,

Using the definition of the τ -operators (6.27), the above expression for F becomes

F (w, w) = −12

(∂1c10 + ∂2c01)−12

α

w1 − w2(c10 − c01) + c10w1 + c01w2. (6.34)

We see that, in the case of (N,K) = (2, 1), there is only one local singularity. We expand

c10(w) =∑

j1+j2=d

γj1,j21,0 wj1

1 wj22 and c01(w) =

∑j1+j2=d

γj1,j20,1 wj1

1 wj22 .

Now, the condition for the local singularity to disappear, is that we want c10 − c01 to be divisible by(w1 − w2). By Lemma 6.6, this equivalent to the condition that∑

j1+j2=d

(γj1,j21,0 − γj1,j2

0,1

)= 0. (6.35)

Equation (6.35) is the only equation that has to be satisfied for the (N,K) = (2, 1) wavefunction tobe normalizable.

Remark 6.16. From Example 6.2, we see that the normalizability condition(s) and the symmetrycondition(s) are related. As a matter of fact, in the (N,K) = (2, 1) case, the normalizability followsdirectly from the symmetry requirements. Also for higher excitation energies, we will find that thesymmetry condition reduces the number of normalizability conditions, but for K > 1 symmetry alonewill no longer be sufficient.

Having explained the mechanism behind the normalizability condition, let us now turn to thegeneral case. The approach we take here is the same approach that we took for solving the recurrenceequation for F : we list the normalizability conditions for several cases separately, and then look fora pattern. Unfortunately, the normalizability conditions become quite numerous very quickly, so wehave only been able to find a general normalizability condition for N = 2 particles (and arbitraryexcitation energy K). See Appendix B.2 for the details. The result is summarized below.

Normalizability condition for N = 2 particles: for each degree d and K separately, we requirethat for any l ∈ 1, 2, . . . ,K and for all k ∈ 0, 1, 2, . . . , l − 1,∑

j1+j2=d

jK−l1

∑i1+i2=K

(−1)i1(i1)kγj1,j2i1,i2

= 0. (6.36)

The above equations do not take the symmetry conditions into account: they have to be imposedseparately.


We have developed a method to construct N -particle anyonic energy eigenfunctions for non-interacting charged anyons in a uniform perpendicular magnetic field. However, we have only beenable to find the normalizability conditions for N = 2 particles. This does not mean that the restof the theory is useless: quite the contrary. However, when constructing anyonic wavefunctions forthree or more particles, the normalizability conditions have to be imposed manually, which doesrequire a bit of labor, because, for every choice of N and K there are different — and at first sightunrelated — conditions that have to be met. Note that, as in the N = 2 case, the normalizabilityconditions for N > 2 will involve only fixed degree d parts of the polynomials ci1,...,iN

(w).

Corollary 6.3. Because the normalizability conditions have to be imposed for each particle numberN and degrees d and K separately, we obtain different energy eigenfunctions

ψN,d,K(w, w,θ) = SN (w,θ)EN (w, w)FN,d,K(w, w)

for every choice of N , d and K, with energy 12~ω∗C(N+2K). Note that the normalizability conditions

relate different highest order coefficients ci1,...,iNto each other. In particular, all highest order

coefficients of the energy eigenfunction ψN,d,K have the same degree d in w.

Remark 6.17. We have not shown that the labels N , d and K uniquely determine an energyeigenstate. All that we know is that all the energy eigenstates that we have constructed carry labelsN , d and K. As we will see in Section 6.2.7, for fixed N , d and K there can be more than one energyeigenfunction ψN,d,K , which means that the labels d and K actually refer to specific eigenspaces ofthe N -particle Hamiltonian HN .

To summarize, we have found exact eigenfunctions ψ = S E F of the Hamiltonian HN . Withinthe set of eigenfunctions that we found, certain eigenfunctions turned out to be not normalizable. Inorder to filter out the normalizable eigenfunctions, we have to impose (6.36). Additionally, we alsohave to impose the symmetry condition on F mentioned in Remark 6.10, which can be translatedto equation (6.31) in terms of the coefficients γj1,...,jN

i1,...,iNof F .

This completes our construction of the exact anyonic energy eigenfunctions for non-interactingcharged anyons in a uniform perpendicular magnetic field. Of course, it would be a shame to stophere; having constructed all these energy eigenfunctions, we are now able to extract actual physicalinformation from them, and this is exactly what we will do in the next section.

6.2.6 Measuring Anyons: Angular Momentum

As of today, there is no experiment that determines the statistical parameter α of the anyons directly.Now that we have a plethora of anyonic energy eigenfunctions that describe non-interacting chargedanyons in a uniform perpendicular magnetic field, our hope is that there exist operators with anα-dependent spectrum.

The only physical operator that we have considered so far, is the Hamiltonian HN . Because wehave only found eigenfunctions corresponding to the eigenvalues

EN,K =12

~ω∗C(N + 2K),

the energy spectrum is independent of α for as far as the eigenfunctions we have constructed areconcerned. Actually, this statement is not entirely true, because in the FQHE — which, up to now,is the only physical system wherein anyons are expected to exist — the charge e∗ of the quasi-holesis given by e∗ = eν, where ν is the filling factor, which coincidentally is also equal to the statisticalangle α. Thus, from the energy spectrum EN,K we can read-off the statistical angle14 α = ν from

14Thanks to L. Fritz for pointing this out. Note that, previously, the author used e to denote the anyon charge,which obscured the fact that in the case of the FQHE, the anyon charge is related to the filling factor, and hence tothe statistical angle.


the spacing between the Landau levels, which is determined by the cyclotron frequency

ω∗C =e∗B

m.

However, note that we could have also measured the filling factor directly to determine the sta-tistical angle, and for that all the theory developed in this chapter is not even necessary. Instead,what we mean by an α-dependent spectrum, is empirical evidence that the quasi-particle excita-tions in the FQHE show anyonic behavior in the first place, and for that purpose, we can use thetheory developed in this chapter, because the wavefunctions we have found apply to any kind ofnon-interacting charged anyons in a uniform perpendicular magnetic field, irrespective of what theunderlying phenomenon is that causes the anyons to appear.

Anyonic statistics show up when two anyons are exchanged, which we do by adiabatically trans-porting the anyons around each other. This suggests that the angular momentum operator mightbe dependent on the statistical parameter α.

Definition 6.10. LetLi := −i~(xi∂yi − yi∂xi)

be the operator associated to the z-component of the angular momentum of the ith anyon around theorigin. Furthermore, we denote the (z component of) the total angular momentum operator relativeto the origin by

LN :=N∑

i=1

Li.

In terms of the dimensionless complex coordinates w and w, the total angular momentum operatorreads

LN = ~N∑

i=1

(−wi∂i + wi∂i

).

As can be checked by hand, LN commutes with the N -particle Hamiltonian HN , so we can hope tosimultaneously diagonalize both operators. We will apply LN to an arbitrary energy eigenfunctionof the form ψN = SN EN FN . Since LN is a differential operator, we can apply LN to each of thethree factors separately, and then add up the results.

First, the statistical term SN (w,θ) =∏N

j>i=1(wi−wj)α. For a fixed factor (wi−wj)α, we have

LN (wi − wj)α = ~ (−wi∂i − wj∂j) (wi − wj)α = −~α(wi − wj)α. (6.37)

Hence, using equation (6.37) multiple times, we get

LNSN (w) = LN

N∏j>i=1

(wi − wj)α

= −α~2N(N − 1)

N∏j>i=1

(wi − wj)α

= −α~2N(N − 1)SN (w),

where 12N(N − 1) is the number of pairs that we can make out of N anyons. As expected, the

parameter α shows up in the spectrum of LN .Next is the exponential term. But first, a useful lemma.

Lemma 6.7. When acting with LN on polynomials in (w, w) of fixed degree (d,K), by which wemean that in each term of the polynomial the orders of the coordinates w1, . . . , wN add up to d, andthe orders of the coordinates w1, . . . , wN add up to K, we have the following operator identity

LN = ~(−d+K)

where the operator on the RHS is a multiplication operator.


Proof. When acting on a term wj11 · · ·w

jN

N wi11 · · · w

iN

N ,

wk∂k

(wj1

1 · · ·wjN

N wi11 · · · w

iN

N

)= jkw

j11 · · ·w

jN

N wi11 · · · w

iN

N

wk∂k

(wj1

1 · · ·wjN

N wi11 · · · w

iN

N

)= ikw

j11 · · ·w

jN

N wi11 · · · w

iN

N .

Now use that i1 + . . .+ iN = K and j1 + . . .+ jN = d.

Thus, by Lemma 6.7,

LNEN (w, w) = LN exp

(−

N∑i=1

wiwi

)= 0,

since∑N

i=1 wiwi has d = K = 1.Finally, we get to the polynomial part FN (w, w). Recall that, by equation (6.28), FN is a

polynomial of the form

FN (w, w) =∑

0≤i1+...+iN≤K

ci1,...,iN(w)wi1

1 · · · wiN

N , (6.38)

where all the lower order coefficients ci′1,...,i′Ncan be expressed in terms of the highest order coeffi-

cients ci1,...,iNof order K through equation (6.23),

ci′1,...,i′N=

∑i1+...+iN=K

(i1i′1

)· · ·(iNi′N

)τ

i1−i′11 · · · τ iN−i′N

N ci1,...,iN. (6.39)

Furthermore, to satisfy the normalizability conditions (6.36) we restrict ourselves to the case whereall highest order coefficients have the same degree d in w,

ci1,...,iN(w) =

∑j1+...+jN=d

γj1,...,jN

i1,...,iNwj1

1 · · ·wjN

N .

This means that, for as far as the highest order coefficients ci1,...,iN(w) are concerned, by Lemma

6.7,LN ci1,...,iN

(w)wi11 · · · w

iN

N = ~(−d+K) ci1,...,iN(w)wi1

1 · · · wiN

N .

But what about the lower order coefficients?By equation (6.39), each time we go down one step in the order of the coefficients ci′1,...,i′N

, weget an additional factor of one of the τi’s in each term in the sum. Now, the operator τi lowers theorder of w by one, because each term in τi, see (6.18),

τi := −12

(∂i +

N∑j=1,j 6=i

α

wi − wj

),

differentiates or divides by the coordinates w1, . . . , wN . This means that, each time we go down oneorder in w by decreasing i1 + . . .+ iN by one, we simultaneously decrease the order j1 + . . .+ jN ofw by one. Hence, the difference i1 + . . .+ iN − (j1 + . . .+ jN ) remains constant. We conclude that

LNFN,d,K(w, w) = ~(−d+K)FN,d,K(w, w),

where FN,d,K is a polynomial corresponding to particle number N , excitation energy K and highestorder coefficients of degree d in w.

Let us summarize the obtained results in the following corollary.


Corollary 6.4. If we label the obtained energy eigenfunctions with particle number N , excitationenergy K and w degree d, by which we mean that all highest order coefficients

ci1,...,iN(w) =

∑j1+...+jN=d

γj1,...,jN

i1,...,iNwj1

1 · · ·wjN

N ,

have degree j1 + . . . + jN = d in w, then the energy eigenfunctions ψN,d,K = SN EN FN,d,K aresimultaneously eigenfunctions of the total angular momentum operator LN with eigenvalue

LN ψN,d,K(w, w) = ~[−α

2N(N − 1)− d+K

]ψN,d,K(w, w).

Corollary 6.5. From the above corollary, we see that the spectrum of the total angular momentumoperator depends on the statistical parameter α. This means that the statistical parameter α for non-interacting charged anyons in a uniform perpendicular magnetic field can theoretically be measureddirectly by looking at the spectrum of the (z component of the) total angular momentum operatorLN .

Having concluded that the total angular momentum operator could possibly lead to a directmethod for detecting anyons by measuring their statistical phase, in the next and final section,we will finish our discussion on anyon wavefunctions by constructing some explicit formulas fortwo-particle anyonic wavefunctions using the method constructed in this chapter.

6.2.7 Explicit Formulas for Two-Particle Anyon Eigenfunctions

In this section, we want to give the reader an idea of what the anyon eigenfunctions look like. Wewill restrict ourselves to case of N = 2 particle eigenfunctions. The particle number N = 2 will besuppressed in the notation.

Recall that the complete eigenfunction ψd,K = ψN=2,d,K is of the form

ψd,K(w1, w1, w2, w2) = (w1 − w2)αe−w1w1−w2w2Fd,K(w1, w1, w2, w2),

where F is a polynomial of degree d in w and K in w, i.e. by (6.38):

Fd,K(w1, w1, w2, w2) =∑

0≤i1+i2≤K

ci1,i2(w)wi11 w

i22 .

We shall consider different values of K and d. In trying to find a balance between being completebut not too exhaustive, we will to discuss the values (0, 0) – (3, 0), (0, 1) – (3, 1) and (0, 2) – (2, 2)for (d,K).

Remark 6.18. We will leave the ubiquitous prefactor (w1 − w2)αe−w1w1−w2w2 , which contributes12N~ω∗C = ~ω∗C to the energy15 and −α

2 ~N(N − 1) = −α~ to the angular momentum out of thepicture, and only focus on the polynomial part Fd,K of the solution. Do keep in mind that, whenconsidering eigenvalues, we will include ~ω∗C and −α~ in the energy and angular momentum eigen-values respectively.

K=0, E = ~ω∗C .For K = 0, Fd,K is just a symmetric polynomial in w1 and w2,

Fd,0(w, w) = c00(w).

Below is a table of all linearly independent symmetric polynomials up to degree d = 3. The columnlabelled by Ld,0 gives the angular momentum eigenvalue ~(−α− d) of the total wavefunction ψd,0.

15In the sense that K signifies the energy that the system has on top of the ground state energy ~ω∗C .


d F Ld,0

0 F0,0(w) = 1 −~α

1 F0,0(w) = w1 + w2 ~(−α− 1)

2 F 1,00,0 (w) = w2

1 + w22 ~(−α− 2)

2 F 0,10,0 (w) = w1w2 ~(−α− 2)

3 F 1,00,0 (w) = w3

1 + w32 ~(−α− 3)

3 F 0,10,0 (w) = w1w2(w1 + w2) ~(−α− 3)

The upper indices label the different basis vectors of the corresponding (d,K)-eigenspace, which, forthe cases of d = 2 and d = 3, is two-dimensional.

K=1, E = 2~ω∗C .For K = 1, Fd,K is equal to

Fd,1(w, w) = c00(w) + c10(w)w1 + c01(w)w2, (6.40)

where the lowest order coefficient c00 is given by

c00(w) = τ1c10(w) + τ2c01(w)

= −12

[∂1c10(w) + ∂2c01(w) +

α

w1 − w2

(c10(w)− c01(w)

)](6.41)

in terms of the highest order coefficients. The highest order coefficients have to satisfy the symmetrycondition (6.31)

γj1,j21,0 = γj2,j1

0,1

and the normalizability condition, the latter of which, by Remark 6.16, follows from the former.

Remark 6.19. We shall use upper case characters A, B, C, etc. to denote the coefficients γj1,j2i1,i2

because the characters are easier on the eyes.

d = 0.For d = 0, c10 and c01 are constants, which, by the symmetry condition, are related by

c10 = c01 = A (6.42)

for some constant A ∈ C. Hence, the (d,K) = (0, 1) eigenfunctions form a one-dimensional subspace.Setting A = 1 and plugging (6.42) into (6.41) and then into (6.40) yields the polynomial F0,1 in thissubspace, given by

F0,1(w, w) = w1 + w2.

d = 1.For d = 1, using the symmetry condition, the coefficients c10 and c01 are given by

c10(w) = Aw1 +Bw2 (6.43)c01(w) = Bw1 +Aw2 (6.44)

for arbitrary constants A and B. Hence, the (d,K) = (1, 1) subspace of eigenfunctions is two-dimensional. Setting A = 1, B = 0 and A = 0, B = 1 respectively and inserting (6.43) and (6.44)into (6.41) and then into (6.40) yields the polynomials FA,B

1,1 , given by

F 1,01,1 (w, w) = −1− α

2+ w1w1 + w2w2,

F 0,11,1 (w, w) =

α

2+ w2w1 + w1w2.


The two polynomials above form a basis for the (d,K) = (1, 1) subspace of eigenfunctions.In what follows, we shall always use the symmetry condition to obtain the expressions for c10

and c01, which we will then insert into (6.41) and then into (6.40) to obtain the solution.

d = 2.For d = 2, the coefficients c10 and c01 are given by

c10(w) = Aw21 +Bw1w2 + Cw2

2

c01(w) = Cw21 +Bw1w2 +Aw2

2

for arbitrary constants A, B and C. Setting A = 1, B = C = 0; A = C = 0, B = 1; and A = B = 0,C = 1 respectively, yields the following basis for the three-dimensional (d,K) = (2, 1) subspace ofeigenfunctions

F 1,0,02,1 (w, w) = −2 + α

2(w1 + w2) + w2

1w1 + w22w2,

F 0,1,02,1 (w, w) = −1

2(w1 + w2) + w1w2(w1 + w2),

F 0,0,12,1 (w, w) =

α

2(w1 + w2) + w2

2w1 + w21w2.

d = 3.For d = 3, the coefficients c10 and c01 are given by

c10(w) = Aw31 +Bw2

1w2 + Cw1w22 +Dw3

2

c01(w) = Dw31 + Cw2

1w2 +Bw1w22 +Aw3

2

for arbitrary constants A, B, C and D. Setting A = 1, B = C = D = 0; A = C = D = 0, B = 1;A = B = D = 0, C = 1; and A = B = C = 0, D = 1 respectively, yields the following basis for thefour-dimensional (d,K) = (3, 1) subspace of eigenfunctions

F 1,0,0,03,1 (w, w) = −3

2(w2

1 + w22)−

α

2(w2

1 + w1w2 + w22) + w3

1w1 + w32w2,

F 0,1,0,03,1 (w, w) = −1

2(α+ 4)w1w2 + w1w2(w1w1 + w2w2),

F 0,0,1,03,1 (w, w) = −1

2(w2

1 + w22) +

α

2w1w2 + w1w2(w2w1 + w1w2),

F 0,0,0,13,1 (w, w) =

α

2(w2

1 + w1w2 + w22) + w3

2w1 + w31w2.

The obtained polynomials for K = 1 together with the total angular momentum eigenvalues~(−α− d+ 1) of the corresponding wavefunctions ψd,1 are shown in the table below.


d F Ld,1

0 F0,1(w, w) = w1 + w2 ~(−α+ 1)

1 F 1,01,1 (w, w) = −1− α

2 + w1w1 + w2w2, −~α

1 F 0,11,1 (w, w) = α

2 + w2w1 + w1w2 −~α

2 F 1,0,02,1 (w, w) = − 2+α

2 (w1 + w2) + w21w1 + w2

2w2 ~(−α− 1)

2 F 0,1,02,1 (w, w) = − 1

2 (w1 + w2) + w1w2(w1 + w2) ~(−α− 1)

2 F 0,0,12,1 (w, w) = α

2 (w1 + w2) + w22w1 + w2

1w2 ~(−α− 1)

3 F 1,0,0,03,1 (w, w) = − 3

2 (w21 + w2

2)− α2 (w2

1 + w1w2 + w22) + w3

1w1 + w32w2 ~(−α− 2)

3 F 0,1,0,03,1 (w, w) = − 1

2 (α+ 4)w1w2 + w1w2(w1w1 + w2w2) ~(−α− 2)

3 F 0,0,1,03,1 (w, w) = − 1

2 (w21 + w2

2) + α2w1w2 + w1w2(w2w1 + w1w2) ~(−α− 2)

3 F 0,0,0,13,1 (w, w) = α

2 (w21 + w1w2 + w2

2) + w32w1 + w3

1w2 ~(−α− 2)

K=2, E = 3~ω∗C .For K = 2, Fd,K is equal to

Fd,2(w, w) = c00(w) + c10(w)w1 + c01(w)w2 + c20(w)w21 + c11(w)w1w2 + c02(w)w2

2, (6.45)

where the lower order coefficients c00, c10 and c01 are given by

c00(w) = τ21 c20(w) + τ1τ2c11(w) + τ2

2 c02(w)

=14

∂21c20(w) + ∂1∂2c11(w) + ∂2

2c02(w)

+α

w1 − w2

[∂1

(2c20(w)− c11(w)

)+ ∂2

(c11(w)− 2c02(w)

)]

+α(α− 1)

(w1 − w2)2

[c20(w)− c11(w) + c02(w)

], (6.46)

c10(w) = 2τ1c20(w) + τ2c11(w)

= −12

2∂1c20(w) + ∂2c11(w) +

α

w1 − w2

(2c20(w)− c11(w)

), (6.47)

c01(w) = τ1c11(w) + 2τ2c02(w)

= −12

∂1c11(w) + 2∂2c02(w) +

α

w1 − w2

(c11(w)− 2c02(w)

). (6.48)

In the case of K = 2, the symmetry condition

γj1,j21,0 = γj2,j1

0,1


does not imply the normalizability conditions16, which read∑j1+j2=d

j1

(γj1,j22,0 − γj1,j2

1,1 + γj1,j20,2

)= 0,

∑j1+j2=d

γj1,j22,0 − γj1,j2

1,1 + γj1,j20,2 = 0.

See equations (B.8) - (B.10). Note that, by the symmetry condition, (B.9) and (B.10) are equivalent,so we can neglect (B.10).

We proceed as before: for each separate value of d we will list the general form of the highestorder coefficients c20, c11 and c02 satisfying the symmetry and normalizability conditions, and thenwe plug these into equations (6.45) - (6.48) to obtain a basis for the corresponding (d,K) eigenspaces.

d = 0For d = 0, the coefficients c20, c11 and c02 are given by

c20 = c02 =12c11 = A

for arbitrary A. Taking A = 1 yields

F0,2(w) = (w1 + w2)2.

It might seem suprising that (w1−w2)2 is not also a (d,K) = (0, 2) eigenfunction of the Hamiltonian.However, as can be checked by hand, (w1 − w2)2 does not solve the Schrodinger equation (6.19) forthe polynomial F .


c20(w) = Aw1 +Bw2

c11(w) = (A+B)(w1 + w2)c02(w) = Bw1 +Aw2

for arbitrary constants A and B. Taking A = 1, B = 0 and A = 0, B = 1 respectively yields thefollowing basis for the two-dimensional (d,K) = (1, 2) subspace of eigenfunctions

F 1,01,2 (w, w) = −1

2(3 + α)(w1 + w2) + w1w

21 + (w1 + w2)w1w2 + w2w

22,

F 0,11,2 (w, w) = −1

2(1− α)(w1 + w2) + w2w

21 + (w1 + w2)w1w2 + w1w

22.


c20(w) = Aw21 +Bw1w2 + Cw2

2

c11(w) = Dw21 + 2(A+B + C −D)w1w2 +Dw2

2

c02(w) = Cw21 +Bw1w2 +Aw2

2

for arbitrary constants A, B, C, and D. Setting A = 1, B = C = D = 0; A = C = D = 0, B = 1;A = B = D = 0, C = 1; and A = B = C = 0, D = 1 respectively, yields the following basis for the

16But it does reduce the number of normalizability conditions to two.


four-dimensional (d,K) = (2, 2) subspace of eigenfunctions

F 1,0,0,02,2 (w, w) =

14(6 + 5α+ α2)− (3 + α)(w1w1 + w2w2) + (w1w1 + w2w2)2,

F 0,1,0,02,2 (w, w) =

12− (w1 + w2)(w1 + w2) + w1w2(w1 + w2)2,

F 0,0,1,02,2 (w, w) =

14(2 + α+ α2)− w1w1 − w2w2 + α(w2w1 + w1w2) + (w2w1 + w1w2)2,

F 0,0,0,12,2 (w, w) = −1

4(2 + 5α+ α2) +

12(2 + α)(w1 − w2)(w1 + w2) + (w1 − w2)2w1w2.

The table below shows all the K = 2 eigenfunctions constructed above, together with the angularmomentum eigenvalues ~(−α− d+ 2) of the corresponding wavefunctions ψd,2.

d F Ld,2

0 F0,2(w, w) = (w1 + w2)2 ~(−α+ 2)

1 F 1,01,2 (w, w) = − 1

2 (3 + α)(w1 + w2) + w1w21 + (w1 + w2)w1w2 + w2w

22 ~(−α+ 1)

1 F 0,11,2 (w, w) = − 1

2 (1− α)(w1 + w2) + w2w21 + (w1 + w2)w1w2 + w1w

22 ~(−α+ 1)

2 F 1,0,0,02,2 (w, w) = 1

4 (6 + 5α+ α2)− (3 + α)(w1w1 + w2w2) + (w1w1 + w2w2)2 −~α

2 F 0,1,0,02,2 (w, w) = 1

2 − (w1 + w2)(w1 + w2) + w1w2(w1 + w2)2 −~α

2 F 0,0,1,02,2 (w, w) = 1

4 (2 + α+ α2)− w1w1 − w2w2 + α(w2w1 + w1w2) + (w2w1 + w1w2)2 −~α

2 F 0,0,0,12,2 (w, w) = − 1

4 (2 + 5α+ α2) + 12 (2 + α)(w1 − w2)(w1 + w2) + (w1 − w2)2w1w2 −~α

This concludes our discussion on N -particle anyon eigenfunctions. Below is the conclusion ofthis part of the thesis — i.e. the part on anyonic wavefunctions — and an overview of the obtainedresults. In the final chapter, we shall compare our results to those obtained by other physicists, andpresent an outlook for possible further research.

ConclusionUsing a smart ansatz we have found energy eigenstates ψN,d,K for energies 1

2~ω∗C(N + 2K) for eachparticle number N , excitation energy K and w degree d. We realized that not all constructedeigenfunctions are normalizable, so we had to impose extra normalizability conditions, which wewere able to formulate explicitly for the case of N = 2 particles.

Moreover, we showed that the energy eigenstates we found are simultaneously angular momentumeigenstates with eigenvalue

LψN,d,K = ~(−α

2N(N − 1)− d+K

)ψN,d,K .

Comparing to the one-particle angular momentum operator, which, in terms of the one-particleladder operators a, a†, b and b† — see Appendix A.1 — can be computed to be equal to

L = ~(a†a− b†b) = ~(n−m),

where n labels the energy and m the guiding center of a single charged particle in a uniform per-pendicular magnetic field, it seems plausible that the degree d is related to some kind of position17

of the N -anyon state ψN,d,K .

17Perhaps the center of mass position?


Additionally, through the angular momentum operator we have obtained a theoretical way tomeasure the statistical angle α directly for any of the N -particle anyon states that reside in thelinear span of the ψN,d,K energy eigenstates.

As can been seen from the tables in Section 6.2.7, the parameters N , d and K do not uniquelyspecify a state, so the labelling ψN,d,K that we use for any state with N particles, excitation energyK and w degree d refers to a subspace of states, rather than a single state. A possible direction forfurther research would be to find additional operators that commute with the Hamiltonian, such thattheir eigenvalues do constitute a complete set, in the sense that specifying the eigenvalues completelyclassifies a state. However, before going into this question, a more important question should beaddressed first: we do not know whether we have found all energy eigenstates. This question willbe addressed in the next chapter.

Chapter 7

Conclusion and Outlook

In this thesis we have been discussing two related topics: Casimir operators in the fractional quantumHall effect (FQHE), and multi-particle wavefunctions for non-interacting charged anyons in a uniformperpendicular magnetic field.

The first topic we studied to gain insight in the FQHE fluid in the low temperature limit, where alldynamics occur within a single Landau level, and are therefore described by the algebra of projecteddensity operators. In order to obtain a better understanding of this algebra, we sought to find andcompute Casimir operators, which are specific operators that commute with all the elements of thealgebra.

In a first attempt, we computed the second order Casimir operator for the FQHE on the infiniteplane. The obtained expression is rife with infinities, which led us to consider Casimir operatorsfor the FQHE on the torus instead. The torus turned out to be computationally friendlier than itsplanar counterpart, enabling us to compute a Casimir operator ζr+1 for each r ∈ 1, 2, . . . , NB − 1,

ζr+1

∣∣F∗

Nel

=Nel

(2πl2B)r

[(1 +NB)

1− (−Nel)r

1 +Nel+ (−Nel)r

]. (7.1)

The obtained expressions for the Casimir operators are in agreement with Dixmier’s lemma1, whichsays that any central element of an associative algebra over C is represented by a scalar whenthe algebra acts irreducibly on a vector space with a countable basis. Indeed, in the FQHE, thealgebra of projected density operators acts on Fock space, which is a direct sum of countably infinite-dimensional fixed-particle-number subspaces, on each of which the algebra acts irreducibly. Equation(7.1) gives the scalar that represents the (r+1)th order Casimir ζr+1 when acting on the irreduciblesubspace F∗Nel

⊂ F∗ of constant particle number Nel.Many quantum numbers in physics, such as total spin and total angular momentum, arise as a

scalar that represents a Casimir operator acting on an irreducible vector space. It was our hope that,similar to total angular momentum and total spin, the scalars on the RHS of equation (7.1) wouldalso be physically significant. However, thus far, the physical meaning of the scalars representingthe Casimir operators remains unclear.

We then proceeded on a tangent, and focused on the elementary quasi-particle excitations of theFQHE fluid, which are of anyonic nature. Because these quasi-particle excitations are fractionallycharged, we expected that they feel the external magnetic field that causes the FQHE to occur.This led us to an effective description of non-interacting charged anyons in a uniform perpendicularmagnetic field. By effective, we mean a description that takes the anyon as a fundamental particle,instead of a collective excitation. It is at this particular point that our approach deviates from

1An infinite dimensional generalization of Schur’s lemma for associative algebras over an uncountable and alge-braically closed field.

122

123

the standard approach to anyons: instead of thinking of anyons as interacting charged boson-fluxtube composites, we treat anyons as non-interacting charged particles with actual anyonic exchangestatistics2.

Using a general ansatz inspired by the single particle wavefunctions for a charged particle ina uniform perpendicular magnetic field, we constructed a plethora of exact energy eigenfunctions,which we simultaneously diagonalized with respect to the (z component of) the total angular momen-tum operator relative to the origin. In contrast to the energy eigenvalues, the angular momentumeigenvalues that we found turned out to be explicitly dependent on the statistical angle α. Thisrealization brought us to the conclusion that the angular momentum operator could possibly be usedto design a method for measuring the anyonic nature of the quasi-particle excitations directly.

The most relevant question that we have not yet discussed, is whether we have found all energyeigenfunctions of the system of N non-interacting charged anyons in a uniform perpendicular mag-netic field. The answer is given by several sources, such as Khare [13], which discusses mostly thetwo-anyon system in a perpendicular magnetic field, and, Johnson and Canright [12], which pro-duces multi-particle anyon wavefunctions using the single particle ladder operators; but, the mostextensive discussion on N anyons in a perpendicular magnetic field is given by Lerda [17], which wenow discuss in some detail.

Lerda uses an approach that is comparable to ours. However, the ansatz he considers is moregeneral, in the sense that his statistical prefactor is also allowed to contain an anti-holomorpicpart. This leads to what Lerda calls type I and type II wavefunctions, corresponding to statisticalprefactors of the form

SI(w,θ) =N∏

j>i=1

(wi − wj)α

and

SII(w,θ) =N∏

j>i=1

(wi − wj)2−α

respectively. The statistical prefactor used in this thesis is exactly SI . As a consequence, all wave-functions with statistical prefactor SII do not appear in this thesis. We can therefore immediatelyconclude that we have not found all energy eigenfunctions. Also, on a sidenote, the statistical phaseα that is absent in the energy eigenvalues of type I states, does appear in the energy eigenvalues oftype II states, which implies that the statistical angle should in principle be measurable from theenergy spectrum alone.

Does this mean that our approach completely falls short with respect to Lerda’s? Let us compareLerda’s results for type I wavefunctions to ours. Lerda constructs explicit formulas for anyonicwavefunctions in Section 6.3 of his book [17]. The eigenfunctions he obtains are of the form

ψI(w, w,θ) = SI(w,θ)E(w, w)P d+(w)Ld++N−1+ α2 N(N−1)

n

(−2

N∑i=1

wiwi

),

where E(w, w) = exp(−∑N

i=1 wiwi

)is the ubiquitous exponential occurring in all wavefunctions

constructed in this thesis, P d+(w) is an arbitrary symmetric polynomial of degree d+, and Lan is the

nth associated Laguerre polynomial. Comparing to our energy eigenfunctions, we see that n is equalto the excitation energy K, and the sum n+d+ is equal to the degree d, because the Laguerre poly-nomials also contain factors of w. It turns out that, all solutions obtained by Lerda that fall withinthe range of the N = 2 particle wavefunctions that we computed explicitly in Section 6.2.7, can bewritten as linear combinations of the solutions that we have found. Specifically, the following table

2In other words, we are using Wilczek’s singular gauge transformation to gauge away the vector potentials due tothe flux tubes — see Wilczek [25] and [26] for details.

124

shows how, for different choices of n and d+, P d+(w)Ld++1+αn (−2w1w1 − 2w2w2) can be written as

a linear combination of Fd,K(w) = Fn+d+,n(w)’s. Note that, for n = 0, Lerda’s solution consists ofsymmetric polynomials in w, which agrees with our K = 0 solutions.

Comparison of Lerda’s wavefunctions to the wavefunctions constructed in Section 6.2.7

n d+ P d+Ld++1+αn lin. comb. of Fn+d+,n’s

1 0 2 + α− 2(w1w1 + w2w2) −2F 1,01,1

1 1 (w1 + w2)[3 + α− 2(w1w1 + w2w2)

]−2(F 1,0,0

2,1 + F 0,1,02,1

)1 2 (w1 + w2)2

[4 + α− 2(w1w1 + w2w2)

]−2(F 1,0,0,0

3,1 + 2F 0,1,0,03,1 + F 0,0,1,0

3,1

)1 2 (w1 − w2)2

[4 + α− 2(w1w1 + w2w2)

]−2(F 1,0,0,0

3,1 − 2F 0,1,0,03,1 + F 0,0,1,0

3,1

)2 0 (α+2)(α+3)

2 − 2(α+ 3)(w1w1 + w2w2)

+2(w1w1 + w2w2)2 2F 1,0,0,02,2

Hence, we predict that all the wavefunctions found by Lerda are also obtained by following ourconstruction. Moreover, our approach is more thorough, because it leads to additional wavefunctionsthat Lerda does not find. We conclude that the method for constructing anyonic eigenfunctionsdiscussed in this thesis is of added value compared to the existing methods for generating anyoniceigenfunctions.

One might be tempted to conjecture that, if our approach were to be applied also to Lerda’stype II ansatz, all energy eigenfunctions could be found. However, as Lerda explains, even hisansatz is not general enough, as there exist three-particle statistical prefactors not of the form SI

or SII . Instead, is seems that there are more promising approaches than those based on tryingdifferent types of statistical prefactors. For example, one could try to look for the complete algebraof operators3 that is responsible for the degeneracy of the energy spectrum, which could possibly beused to generate solutions to the Schrodinger equation.

In the long term, there is plenty of research to be done in the field of charged anyons in a uniformperpendicular magnetic field. For example, we could include anyon interactions in the theory, or, wecould try to repeat the setup of Chapter 6 for non-abelian anyons, in which case the N -particle anyonwavefunctions must be considered as functions on the full universal cover of the anyon configurationspace, instead of the abelian cover. On a more hypothetical level, by comparing the Laughlin groundstate wavefunction to the d = 0 anyon ground state wavefunction for quasi-holes, where α is equalto the filling factor ν,

ψAnyon(w, w,θ) =

N∏j>i=1

(wi − wj)ν

exp

(−

N∑i=1

wiwi

)

ψLaughlin(w, w) =

N∏j>i=1

(wi − wj)1ν

exp

(−

N∑i=1

wiwi

)

it seems as if there could be some kind of duality between the two, where ν ↔ 1ν . But, before

embarking on any of the above topics, it is noteworthy to emphasize that even the case of non-interacting charged anyons in a uniform perpendicular magnetic field is far from completely solved.Looking at the wavefunctions constructed in this thesis, it seems that there is a lot of hidden structureready to be found, which should provide more than enough challenge for future anyon-specialists tosink their teeth into.

3Analogous to the guiding centre operators for a charged particle in a uniform perpendicular magnetic field.

Appendix A

The Quantum Hall Effects:Addendum

A.1 Overview of the Single Particle Operators

DefinitionsPresented here is an overview of the one-particle operators for a particle of charge −e and massm confined to two dimensions in a uniform perpendicular magnetic field. All operators are definedin terms of the fundamental position and momentum operators acting on the one-particle Hilbertspace L2(R2,C). In the position representation, they are given by

x = x, px = −i~ ∂

∂x,

y = y, py = −i~ ∂

∂y,

where x = x means that the operator x acts on ψ(x) by multiplying with the variable x, and likewisefor y. The magnetic vector potential A(x, y) is an analytic function in the variables x and y – itis actually polynomial in the Landau and Symmetric gauge, so A := A(x, y) makes sense as anoperator. We will drop the hats distinguishing the operator from the corresponding variable fromnow on.

Using the operators r = (x, y), p = (px, py) and A(r) defined above we define the mechanicalmomentum operator Π as

Π := p + eA.

Classically, an electron of mass m and charge −e confined to the x-y plane subject to a perpendicularmagnetic field B = Bez moves in (anti-clockwise – when viewed from the +z direction) circles withangular frequency

ωC :=eB

m,

called the cyclotron frequency. Since the radius η and the velocity v of the electron’s orbit are relatedby v = Π

m = ωCη, we have

η =ΠeB

× ez.

We use the above equation to define the operator η, which, in the classical picture, corresponds tothe radial vector pointing from the centre of the cyclotron orbit R to the position of the electron r,i.e. r = R + η. Either η or Π can be used to measure the electrons kinetic energy.

125

A.1. OVERVIEW OF THE SINGLE PARTICLE OPERATORS 126

Having defined the operator η, we can make sense of the operator R corresponding to the classicalguiding centre of the cyclotron orbit,

R := r − η.

Commutation relationsUsing the basic commutation relations

[x, px] = i~ = [y, py], (A.1)

with all remaining commutators of r and p vanishing, the other commutators will be computedbelow.

As computed in sections 3.1.1 and 3.1.2, we have

[X,Y ] = il2B and [Πx,Πy] = −i~2

l2B. (A.2)

Now, for r, and Π, we have, for example, [x,Πx] = [x, px + eAx] = [x, px] = i~. In general,

[x,Πx] = i~, [x,Πy] = 0,[y,Πx] = 0, [y,Πy] = i~.

Likewise, for R, and Π, we have, for example, [X,Πx] = [x − ηx,Πx] = [x,Πx] − 1eB [Πy,Πx] =

i~− i 1eB

~2

l2B= 0. More generally,

[X,Πx] = 0, [X,Πy] = 0, (A.3)[Y,Πx] = 0, [Y,Πy] = 0, (A.4)

which shows that indeed X and Y commute with the free particle Hamiltonian

H =1

2m(Π2

x + Π2y).

Finally, for r, and R, we have, for example, [x, Y ] = [x, y − ηy] = + 1eB [x,Πx] = i ~

eB = il2B . Ingeneral,

[x,X] = 0, [x, Y ] = il2B ,

[y,X] = −il2B , [y, Y ] = 0.

Ladder operatorsFrom equation (A.2), we see that (X,Y ) and (Πx,Πy) are canonically conjugate observables. Thissuggests introducing corresponding ladder operators a, a†, b and b†,

a :=lB√2~

(Πx − iΠy), a† :=lB√2~

(Πx + iΠy)

Πx =~√2lB

(a† + a), Πy =−i~√2lB

(a† − a),

and

b :=1√2lB

(X + iY ), b† :=1√2lB

(X − iY )

X =lB√2(b† + b), Y =

ilB√2(b† − b), (A.5)

We can also express η in terms of a and a†,

a =i√2lB

(−ηx + iηy), a† =i√2lB

(ηx + iηy), (A.6)

ηx =ilB√

2(−a† + a), ηy =

−lB√2

(a† + a). (A.7)

A.2. FOURIER TRANSFORM OF THE COULOMB POTENTIAL IN 2D 127

A.2 Fourier Transform of the Coulomb Potential in 2D

The Coulomb interaction potential U(r) gives the potential energy two electrons have when separatedby a distance r = |r|. It is defined by the following formula,

U(r) :=e2

4πε01r.

In the usual case, where U is seen as a function on R3, U is not in any Lp(R3) space with 1 ≤ p ≤ 2, sothe Coulomb potential can technically not be Fourier transformed. The way physicists circumventthis problem is by Fourier transforming the Yukawa potential, which is the Coulomb potentialmultiplied by a damping term e−ur, and then taking the limit u→ 0. In two dimensions, however,no such trickery is needed as the Coulomb potential can be Fourier transformed directly.

For a fixed wavevector q, we will adopt polar coordinates such that q = qex.

U(q) =∫

R2dr

e2

4πε01re−iq·r =

e2

4πε0

∫ ∞

0

dr

∫ 2π

0

dθ e−iqr cos θ

=e2

4πε0

∫ ∞

0

dr 2πJ0(qr) =e2

2ε01q

∫ ∞

0

ds J0(s)

=e2

2ε01|q|,

where we have used that the zeroth order Bessel function of the first kind J0 is normalized to 1.

Appendix B

Anyon Wavefunctions: Addendum

B.1 Solution to the Recurrence Relation

In this part of the appendix, we will elaborate on the solution (6.23) to the recurrence relation (6.21),given by

(K−Ord) ci1,...,iN(w) =

N∑k=1

(ik +1)τk ci1,...ik−1,ik+1,ik+1,...,iN(w) for 0 ≤ i1+ . . .+iN ≤ K. (B.1)

As noted in Remark 6.13, the highest order coefficients ci1,...,iNwith i1 + . . . + iN = K are free to

choose, and, using (B.1), the lower order coefficients can be expressed in terms of the highest ordercoefficients. Indeed, if we rewrite (B.1) as

ci1,...,iN=

N∑k=1

ik + 1K − (i1 + . . .+ iN )

τk ci1,...ik−1,ik+1,ik+1,...,iN(w) for 0 ≤ i1 + . . .+ iN ≤ K − 1,

(B.2)then it becomes more obvious that we can express a fixed coefficient of order R in terms of all theneighboring coefficients of order R + 1. Hence, starting from the highest order coefficients, andthen working our way down using (B.2) repeatedly, we can express every coefficient in terms of thehighest order coefficients.

In the next couple of pages, using Mathematica, we have expressed all the lower order coefficientsin terms of the highest order coefficients for the cases of N = 2, K = 1, 2, 3, 4, 5, 6, 7; N = 3,K = 1, 2, 3, 4, 5; and N = 4, K = 1, 2, 3. As can be verified directly for any of the expressionscomputed in the enclosed mathematica file, all cases are in agreement with formula (6.23), whereinthe lower order coefficient ci′1,...,i′N

is expressed in terms of the highest order coefficients ci1,...,iNas

ci′1,...,i′N=

∑i1+...+iN=K

(i1i′1

)· · ·(iNi′N

)τ

i1−i′11 · · · τ iN−i′N

N ci1,...,iN, (B.3)

where we use the convention that(

ii′

)= 0 whenever i < i′. The author is of the opinion that this is

convincing evidence that (B.3) is indeed the solution to the recurrence relation (B.1).The mathematica files for the cases N = 2, K = 1, 2, 3, 4, 5, 6, 7; N = 3, K = 1, 2, 3, 4, 5; and

N = 4, K = 1, 2, 3 can be found below.

Notation: instead of writing τ1, τ2, τ3 and τ4, we have opted for A, B, C, and D respectively,because it adds to the readability of the Mathematica file, and also because mathematica orders thelatter characters more naturally.

128

H* Recurrence relation for two particles *Lc@K_, i_, j_D := Hi + 1LHK - i - jL A c@i + 1, jD + Hj + 1LHK - i - jL B c@i, j + 1D

H* K=1 *L

c@0, 0D = Simplify@c@1, 0, 0DDB c@0, 1D + A c@1, 0D

H* K=2 *L

c@1, 0D = Simplify@c@2, 1, 0DDc@0, 1D = Simplify@c@2, 0, 1DDB c@1, 1D + 2 A c@2, 0D

2 B c@0, 2D + A c@1, 1D

c@0, 0D = Simplify@c@2, 0, 0DD

B2 c@0, 2D + A B c@1, 1D + A2 c@2, 0D

H* K=3 *L

c@2, 0D = Simplify@c@3, 2, 0DDc@1, 1D = Simplify@c@3, 1, 1DDc@0, 2D = Simplify@c@3, 0, 2DDB c@2, 1D + 3 A c@3, 0D

2 HB c@1, 2D + A c@2, 1DL

3 B c@0, 3D + A c@1, 2D

c@1, 0D = Simplify@c@3, 1, 0DDc@0, 1D = Simplify@c@3, 0, 1DDB2 c@1, 2D + 2 A B c@2, 1D + 3 A2 c@3, 0D

3 B2 c@0, 3D + 2 A B c@1, 2D + A2 c@2, 1D


B3 c@0, 3D + A B2 c@1, 2D + A2 B c@2, 1D + A3 c@3, 0D

H* K=4 *L

c@3, 0D = c@4, 3, 0Dc@2, 1D = c@4, 2, 1Dc@1, 2D = c@4, 1, 2Dc@0, 3D = c@4, 0, 3DB c@3, 1D + 4 A c@4, 0D

2 B c@2, 2D + 3 A c@3, 1D

3 B c@1, 3D + 2 A c@2, 2D

4 B c@0, 4D + A c@1, 3D

B.1. SOLUTION TO THE RECURRENCE RELATION 129

c@2, 0D = Simplify@c@4, 2, 0DDc@1, 1D = Simplify@c@4, 1, 1DDc@0, 2D = Simplify@c@4, 0, 2DDB2 c@2, 2D + 3 A B c@3, 1D + 6 A2 c@4, 0D

3 B2 c@1, 3D + 4 A B c@2, 2D + 3 A2 c@3, 1D

6 B2 c@0, 4D + 3 A B c@1, 3D + A2 c@2, 2D

c@0, 1D = Simplify@c@4, 0, 1DDc@1, 0D = Simplify@c@4, 1, 0DD4 B3 c@0, 4D + 3 A B2 c@1, 3D + 2 A2 B c@2, 2D + A3 c@3, 1D

B3 c@1, 3D + 2 A B2 c@2, 2D + 3 A2 B c@3, 1D + 4 A3 c@4, 0D


B4 c@0, 4D + A B3 c@1, 3D + A2 B2 c@2, 2D + A3 B c@3, 1D + A4 c@4, 0D

H* K=5 *L

c@4, 0D = c@5, 4, 0Dc@3, 1D = c@5, 3, 1Dc@2, 2D = c@5, 2, 2Dc@1, 3D = c@5, 1, 3Dc@0, 4D = c@5, 0, 4DB c@4, 1D + 5 A c@5, 0D

2 B c@3, 2D + 4 A c@4, 1D

3 B c@2, 3D + 3 A c@3, 2D

4 B c@1, 4D + 2 A c@2, 3D

5 B c@0, 5D + A c@1, 4D

c@3, 0D = Simplify@c@5, 3, 0DDc@2, 1D = Simplify@c@5, 2, 1DDc@1, 2D = Simplify@c@5, 1, 2DDc@0, 3D = Simplify@c@5, 0, 3DDB2 c@3, 2D + 4 A B c@4, 1D + 10 A2 c@5, 0D

3 B2 c@2, 3D + 6 A B c@3, 2D + 6 A2 c@4, 1D

6 B2 c@1, 4D + 6 A B c@2, 3D + 3 A2 c@3, 2D

10 B2 c@0, 5D + 4 A B c@1, 4D + A2 c@2, 3D

c@2, 0D = Simplify@c@5, 2, 0DDc@1, 1D = Simplify@c@5, 1, 1DDc@0, 2D = Simplify@c@5, 0, 2DDB3 c@2, 3D + 3 A B2 c@3, 2D + 6 A2 B c@4, 1D + 10 A3 c@5, 0D

4 B3 c@1, 4D + 6 A B2 c@2, 3D + 6 A2 B c@3, 2D + 4 A3 c@4, 1D

10 B3 c@0, 5D + 6 A B2 c@1, 4D + 3 A2 B c@2, 3D + A3 c@3, 2D

c@1, 0D = Simplify@c@5, 1, 0DDc@0, 1D = Simplify@c@5, 0, 1DDB4 c@1, 4D + 2 A B3 c@2, 3D + 3 A2 B2 c@3, 2D + 4 A3 B c@4, 1D + 5 A4 c@5, 0D

5 B4 c@0, 5D + 4 A B3 c@1, 4D + 3 A2 B2 c@2, 3D + 2 A3 B c@3, 2D + A4 c@4, 1D

2 Recurrence Relation N=2.nb



B5 c@0, 5D + A B4 c@1, 4D + A2 B3 c@2, 3D + A3 B2 c@3, 2D + A4 B c@4, 1D + A5 c@5, 0D

H* K=6 *L

c@5, 0D = c@6, 5, 0Dc@4, 1D = c@6, 4, 1Dc@3, 2D = c@6, 3, 2Dc@2, 3D = c@6, 2, 3Dc@1, 4D = c@6, 1, 4Dc@0, 5D = c@6, 0, 5DB c@5, 1D + 6 A c@6, 0D

2 B c@4, 2D + 5 A c@5, 1D

3 B c@3, 3D + 4 A c@4, 2D

4 B c@2, 4D + 3 A c@3, 3D

5 B c@1, 5D + 2 A c@2, 4D

6 B c@0, 6D + A c@1, 5D

c@4, 0D = Simplify@c@6, 4, 0DDc@3, 1D = Simplify@c@6, 3, 1DDc@2, 2D = Simplify@c@6, 2, 2DDc@1, 3D = Simplify@c@6, 1, 3DDc@0, 4D = Simplify@c@6, 0, 4DDB2 c@4, 2D + 5 A B c@5, 1D + 15 A2 c@6, 0D

3 B2 c@3, 3D + 8 A B c@4, 2D + 10 A2 c@5, 1D

6 B2 c@2, 4D + 9 A B c@3, 3D + 6 A2 c@4, 2D

10 B2 c@1, 5D + 8 A B c@2, 4D + 3 A2 c@3, 3D

15 B2 c@0, 6D + 5 A B c@1, 5D + A2 c@2, 4D

c@3, 0D = Simplify@c@6, 3, 0DDc@2, 1D = Simplify@c@6, 2, 1DDc@1, 2D = Simplify@c@6, 1, 2DDc@0, 3D = Simplify@c@6, 0, 3DDB3 c@3, 3D + 4 A B2 c@4, 2D + 10 A2 B c@5, 1D + 20 A3 c@6, 0D

4 B3 c@2, 4D + 9 A B2 c@3, 3D + 12 A2 B c@4, 2D + 10 A3 c@5, 1D

10 B3 c@1, 5D + 12 A B2 c@2, 4D + 9 A2 B c@3, 3D + 4 A3 c@4, 2D

20 B3 c@0, 6D + 10 A B2 c@1, 5D + 4 A2 B c@2, 4D + A3 c@3, 3D

c@2, 0D = Simplify@c@6, 2, 0DDc@1, 1D = Simplify@c@6, 1, 1DDc@0, 2D = Simplify@c@6, 0, 2DDB4 c@2, 4D + 3 A B3 c@3, 3D + 6 A2 B2 c@4, 2D + 10 A3 B c@5, 1D + 15 A4 c@6, 0D

5 B4 c@1, 5D + 8 A B3 c@2, 4D + 9 A2 B2 c@3, 3D + 8 A3 B c@4, 2D + 5 A4 c@5, 1D


Recurrence Relation N=2.nb 3


c@1, 0D = Simplify@c@6, 1, 0DDc@0, 1D = Simplify@c@6, 0, 1DDB5 c@1, 5D + 2 A B4 c@2, 4D + 3 A2 B3 c@3, 3D + 4 A3 B2 c@4, 2D + 5 A4 B c@5, 1D + 6 A5 c@6, 0D

6 B5 c@0, 6D + 5 A B4 c@1, 5D + 4 A2 B3 c@2, 4D + 3 A3 B2 c@3, 3D + 2 A4 B c@4, 2D + A5 c@5, 1D


B6 c@0, 6D + A B5 c@1, 5D + A2 B4 c@2, 4D + A3 B3 c@3, 3D + A4 B2 c@4, 2D + A5 B c@5, 1D + A6 c@6, 0D

H* K=7 *L

c@6, 0D = c@7, 6, 0Dc@5, 1D = c@7, 5, 1Dc@4, 2D = c@7, 4, 2Dc@3, 3D = c@7, 3, 3Dc@2, 4D = c@7, 2, 4Dc@1, 5D = c@7, 1, 5Dc@0, 6D = c@7, 0, 6DB c@6, 1D + 7 A c@7, 0D

2 B c@5, 2D + 6 A c@6, 1D

3 B c@4, 3D + 5 A c@5, 2D

4 B c@3, 4D + 4 A c@4, 3D

5 B c@2, 5D + 3 A c@3, 4D

6 B c@1, 6D + 2 A c@2, 5D

7 B c@0, 7D + A c@1, 6D

c@5, 0D = Simplify@c@7, 5, 0DDc@4, 1D = Simplify@c@7, 4, 1DDc@3, 2D = Simplify@c@7, 3, 2DDc@2, 3D = Simplify@c@7, 2, 3DDc@1, 4D = Simplify@c@7, 1, 4DDc@0, 5D = Simplify@c@7, 0, 5DDB2 c@5, 2D + 6 A B c@6, 1D + 21 A2 c@7, 0D

3 B2 c@4, 3D + 10 A B c@5, 2D + 15 A2 c@6, 1D

2 H3 B2 c@3, 4D + 6 A B c@4, 3D + 5 A2 c@5, 2DL

2 H5 B2 c@2, 5D + 6 A B c@3, 4D + 3 A2 c@4, 3DL

15 B2 c@1, 6D + 10 A B c@2, 5D + 3 A2 c@3, 4D

21 B2 c@0, 7D + 6 A B c@1, 6D + A2 c@2, 5D



c@4, 0D = Simplify@c@7, 4, 0DDc@3, 1D = Simplify@c@7, 3, 1DDc@2, 2D = Simplify@c@7, 2, 2DDc@1, 3D = Simplify@c@7, 1, 3DDc@0, 4D = Simplify@c@7, 0, 4DDB3 c@4, 3D + 5 A B2 c@5, 2D + 15 A2 B c@6, 1D + 35 A3 c@7, 0D

4 HB3 c@3, 4D + 3 A B2 c@4, 3D + 5 A2 B c@5, 2D + 5 A3 c@6, 1DL

2 H5 B3 c@2, 5D + 9 A B2 c@3, 4D + 9 A2 B c@4, 3D + 5 A3 c@5, 2DL

4 H5 B3 c@1, 6D + 5 A B2 c@2, 5D + 3 A2 B c@3, 4D + A3 c@4, 3DL

35 B3 c@0, 7D + 15 A B2 c@1, 6D + 5 A2 B c@2, 5D + A3 c@3, 4D

c@3, 0D = Simplify@c@7, 3, 0DDc@2, 1D = Simplify@c@7, 2, 1DDc@1, 2D = Simplify@c@7, 1, 2DDc@0, 3D = Simplify@c@7, 0, 3DDB4 c@3, 4D + 4 A B3 c@4, 3D + 10 A2 B2 c@5, 2D + 20 A3 B c@6, 1D + 35 A4 c@7, 0D




c@2, 0D = Simplify@c@7, 2, 0DDc@1, 1D = Simplify@c@7, 1, 1DDc@0, 2D = Simplify@c@7, 0, 2DDB5 c@2, 5D + 3 A B4 c@3, 4D + 6 A2 B3 c@4, 3D + 10 A3 B2 c@5, 2D + 15 A4 B c@6, 1D + 21 A5 c@7, 0D

2 H3 B5 c@1, 6D + 5 A B4 c@2, 5D + 6 A2 B3 c@3, 4D + 6 A3 B2 c@4, 3D + 5 A4 B c@5, 2D + 3 A5 c@6, 1DL

21 B5 c@0, 7D + 15 A B4 c@1, 6D + 10 A2 B3 c@2, 5D + 6 A3 B2 c@3, 4D + 3 A4 B c@4, 3D + A5 c@5, 2D

c@1, 0D = Simplify@c@7, 1, 0DDc@0, 1D = Simplify@c@7, 0, 1DDB6 c@1, 6D + 2 A B5 c@2, 5D + 3 A2 B4 c@3, 4D + 4 A3 B3 c@4, 3D + 5 A4 B2 c@5, 2D + 6 A5 B c@6, 1D + 7 A6 c@7, 0D

7 B6 c@0, 7D + 6 A B5 c@1, 6D + 5 A2 B4 c@2, 5D + 4 A3 B3 c@3, 4D + 3 A4 B2 c@4, 3D + 2 A5 B c@5, 2D + A6 c@6, 1D


B7 c@0, 7D + A B6 c@1, 6D + A2 B5 c@2, 5D +

A3 B4 c@3, 4D + A4 B3 c@4, 3D + A5 B2 c@5, 2D + A6 B c@6, 1D + A7 c@7, 0D



H* Recurrence relation for three particles *Lc@K_, i_, j_, k_D := Hi + 1LHK - i - j - kL A c@i + 1, j, kD +

Hj + 1LHK - i - j - kL B c@i, j + 1, kD + Hk + 1LHK - i - j - kL C c@i, j, k + 1D

H* K=1 *L

c@0, 0, 0D = Simplify@c@1, 0, 0, 0DDC c@0, 0, 1D + B c@0, 1, 0D + A c@1, 0, 0D

H* K=2 *L

c@1, 0, 0D = c@2, 1, 0, 0Dc@0, 1, 0D = c@2, 0, 1, 0Dc@0, 0, 1D = c@2, 0, 0, 1DC c@1, 0, 1D + B c@1, 1, 0D + 2 A c@2, 0, 0D

C c@0, 1, 1D + 2 B c@0, 2, 0D + A c@1, 1, 0D

2 C c@0, 0, 2D + B c@0, 1, 1D + A c@1, 0, 1D

c@0, 0, 0D = Simplify@c@2, 0, 0, 0DD

C2 c@0, 0, 2D + B C c@0, 1, 1D + B2 c@0, 2, 0D + A C c@1, 0, 1D + A B c@1, 1, 0D + A2 c@2, 0, 0D

H* K=3 *L

c@2, 0, 0D = c@3, 2, 0, 0Dc@1, 1, 0D = c@3, 1, 1, 0Dc@1, 0, 1D = c@3, 1, 0, 1Dc@0, 2, 0D = c@3, 0, 2, 0Dc@0, 1, 1D = c@3, 0, 1, 1Dc@0, 0, 2D = c@3, 0, 0, 2DC c@2, 0, 1D + B c@2, 1, 0D + 3 A c@3, 0, 0D

C c@1, 1, 1D + 2 B c@1, 2, 0D + 2 A c@2, 1, 0D

2 C c@1, 0, 2D + B c@1, 1, 1D + 2 A c@2, 0, 1D

C c@0, 2, 1D + 3 B c@0, 3, 0D + A c@1, 2, 0D

2 C c@0, 1, 2D + 2 B c@0, 2, 1D + A c@1, 1, 1D

3 C c@0, 0, 3D + B c@0, 1, 2D + A c@1, 0, 2D

c@1, 0, 0D = Simplify@c@3, 1, 0, 0DDc@0, 1, 0D = Simplify@c@3, 0, 1, 0DDc@0, 0, 1D = Simplify@c@3, 0, 0, 1DDC2 c@1, 0, 2D + B C c@1, 1, 1D + B2 c@1, 2, 0D + 2 A C c@2, 0, 1D + 2 A B c@2, 1, 0D + 3 A2 c@3, 0, 0D

C2 c@0, 1, 2D + 2 B C c@0, 2, 1D + 3 B2 c@0, 3, 0D + A C c@1, 1, 1D + 2 A B c@1, 2, 0D + A2 c@2, 1, 0D

3 C2 c@0, 0, 3D + B2 c@0, 2, 1D + 2 C HB c@0, 1, 2D + A c@1, 0, 2DL + A B c@1, 1, 1D + A2 c@2, 0, 1D


c@0, 0, 0D = Simplify@c@3, 0, 0, 0DD

C3 c@0, 0, 3D + B3 c@0, 3, 0D + C2 HB c@0, 1, 2D + A c@1, 0, 2DL + A B2 c@1, 2, 0D +

C HB2 c@0, 2, 1D + A B c@1, 1, 1D + A2 c@2, 0, 1DL + A2 B c@2, 1, 0D + A3 c@3, 0, 0D

H* K=4 *L

c@3, 0, 0D = c@4, 3, 0, 0Dc@2, 1, 0D = c@4, 2, 1, 0Dc@2, 0, 1D = c@4, 2, 0, 1Dc@1, 2, 0D = c@4, 1, 2, 0Dc@1, 1, 1D = c@4, 1, 1, 1Dc@1, 0, 2D = c@4, 1, 0, 2Dc@0, 3, 0D = c@4, 0, 3, 0Dc@0, 2, 1D = c@4, 0, 2, 1Dc@0, 1, 2D = c@4, 0, 1, 2Dc@0, 0, 3D = c@4, 0, 0, 3DC c@3, 0, 1D + B c@3, 1, 0D + 4 A c@4, 0, 0D

C c@2, 1, 1D + 2 B c@2, 2, 0D + 3 A c@3, 1, 0D

2 C c@2, 0, 2D + B c@2, 1, 1D + 3 A c@3, 0, 1D

C c@1, 2, 1D + 3 B c@1, 3, 0D + 2 A c@2, 2, 0D

2 C c@1, 1, 2D + 2 B c@1, 2, 1D + 2 A c@2, 1, 1D

3 C c@1, 0, 3D + B c@1, 1, 2D + 2 A c@2, 0, 2D

C c@0, 3, 1D + 4 B c@0, 4, 0D + A c@1, 3, 0D

2 C c@0, 2, 2D + 3 B c@0, 3, 1D + A c@1, 2, 1D

3 C c@0, 1, 3D + 2 B c@0, 2, 2D + A c@1, 1, 2D

4 C c@0, 0, 4D + B c@0, 1, 3D + A c@1, 0, 3D

c@2, 0, 0D = Simplify@c@4, 2, 0, 0DDc@1, 1, 0D = Simplify@c@4, 1, 1, 0DDc@1, 0, 1D = Simplify@c@4, 1, 0, 1DDc@0, 2, 0D = Simplify@c@4, 0, 2, 0DDc@0, 1, 1D = Simplify@c@4, 0, 1, 1DDc@0, 0, 2D = Simplify@c@4, 0, 0, 2DDC2 c@2, 0, 2D + B C c@2, 1, 1D + B2 c@2, 2, 0D + 3 A C c@3, 0, 1D + 3 A B c@3, 1, 0D + 6 A2 c@4, 0, 0D

C2 c@1, 1, 2D + 3 B2 c@1, 3, 0D + 2 C HB c@1, 2, 1D + A c@2, 1, 1DL + 4 A B c@2, 2, 0D + 3 A2 c@3, 1, 0D

3 C2 c@1, 0, 3D + 2 B C c@1, 1, 2D + B2 c@1, 2, 1D + 4 A C c@2, 0, 2D + 2 A B c@2, 1, 1D + 3 A2 c@3, 0, 1D


3 C2 c@0, 1, 3D + 4 B C c@0, 2, 2D + 3 B2 c@0, 3, 1D + 2 A C c@1, 1, 2D + 2 A B c@1, 2, 1D + A2 c@2, 1, 1D




c@1, 0, 0D = Simplify@c@4, 1, 0, 0DDc@0, 1, 0D = Simplify@c@4, 0, 1, 0DDc@0, 0, 1D = Simplify@c@4, 0, 0, 1DDC3 c@1, 0, 3D + B3 c@1, 3, 0D + C2 HB c@1, 1, 2D + 2 A c@2, 0, 2DL + 2 A B2 c@2, 2, 0D +

C HB2 c@1, 2, 1D + 2 A B c@2, 1, 1D + 3 A2 c@3, 0, 1DL + 3 A2 B c@3, 1, 0D + 4 A3 c@4, 0, 0DC3 c@0, 1, 3D + 4 B3 c@0, 4, 0D + C2 H2 B c@0, 2, 2D + A c@1, 1, 2DL + 3 A B2 c@1, 3, 0D +

C H3 B2 c@0, 3, 1D + 2 A B c@1, 2, 1D + A2 c@2, 1, 1DL + 2 A2 B c@2, 2, 0D + A3 c@3, 1, 0D4 C3 c@0, 0, 4D + B3 c@0, 3, 1D + 3 C2 HB c@0, 1, 3D + A c@1, 0, 3DL + A B2 c@1, 2, 1D +

2 C HB2 c@0, 2, 2D + A B c@1, 1, 2D + A2 c@2, 0, 2DL + A2 B c@2, 1, 1D + A3 c@3, 0, 1D

c@0, 0, 0D = Simplify@c@4, 0, 0, 0DD

C4 c@0, 0, 4D + B4 c@0, 4, 0D + C3 HB c@0, 1, 3D + A c@1, 0, 3DL +

A B3 c@1, 3, 0D + C2 HB2 c@0, 2, 2D + A B c@1, 1, 2D + A2 c@2, 0, 2DL + A2 B2 c@2, 2, 0D +

C HB3 c@0, 3, 1D + A B2 c@1, 2, 1D + A2 B c@2, 1, 1D + A3 c@3, 0, 1DL + A3 B c@3, 1, 0D + A4 c@4, 0, 0D



H* K=5 *L

c@4, 0, 0D = c@5, 4, 0, 0Dc@3, 1, 0D = c@5, 3, 1, 0Dc@3, 0, 1D = c@5, 3, 0, 1Dc@2, 2, 0D = c@5, 2, 2, 0Dc@2, 1, 1D = c@5, 2, 1, 1Dc@2, 0, 2D = c@5, 2, 0, 2Dc@1, 3, 0D = c@5, 1, 3, 0Dc@1, 2, 1D = c@5, 1, 2, 1Dc@1, 1, 2D = c@5, 1, 1, 2Dc@1, 0, 3D = c@5, 1, 0, 3Dc@0, 4, 0D = c@5, 0, 4, 0Dc@0, 3, 1D = c@5, 0, 3, 1Dc@0, 2, 2D = c@5, 0, 2, 2Dc@0, 1, 3D = c@5, 0, 1, 3Dc@0, 0, 4D = c@5, 0, 0, 4DC c@4, 0, 1D + B c@4, 1, 0D + 5 A c@5, 0, 0D

C c@3, 1, 1D + 2 B c@3, 2, 0D + 4 A c@4, 1, 0D

2 C c@3, 0, 2D + B c@3, 1, 1D + 4 A c@4, 0, 1D

C c@2, 2, 1D + 3 B c@2, 3, 0D + 3 A c@3, 2, 0D

2 C c@2, 1, 2D + 2 B c@2, 2, 1D + 3 A c@3, 1, 1D

3 C c@2, 0, 3D + B c@2, 1, 2D + 3 A c@3, 0, 2D

C c@1, 3, 1D + 4 B c@1, 4, 0D + 2 A c@2, 3, 0D

2 C c@1, 2, 2D + 3 B c@1, 3, 1D + 2 A c@2, 2, 1D

3 C c@1, 1, 3D + 2 B c@1, 2, 2D + 2 A c@2, 1, 2D

4 C c@1, 0, 4D + B c@1, 1, 3D + 2 A c@2, 0, 3D

C c@0, 4, 1D + 5 B c@0, 5, 0D + A c@1, 4, 0D

2 C c@0, 3, 2D + 4 B c@0, 4, 1D + A c@1, 3, 1D

3 C c@0, 2, 3D + 3 B c@0, 3, 2D + A c@1, 2, 2D

4 C c@0, 1, 4D + 2 B c@0, 2, 3D + A c@1, 1, 3D

5 C c@0, 0, 5D + B c@0, 1, 4D + A c@1, 0, 4D



c@3, 0, 0D = Simplify@c@5, 3, 0, 0DDc@2, 1, 0D = Simplify@c@5, 2, 1, 0DDc@2, 0, 1D = Simplify@c@5, 2, 0, 1DDc@1, 2, 0D = Simplify@c@5, 1, 2, 0DDc@1, 1, 1D = Simplify@c@5, 1, 1, 1DDc@1, 0, 2D = Simplify@c@5, 1, 0, 2DDc@0, 3, 0D = Simplify@c@5, 0, 3, 0DDc@0, 2, 1D = Simplify@c@5, 0, 2, 1DDc@0, 1, 2D = Simplify@c@5, 0, 1, 2DDc@0, 0, 3D = Simplify@c@5, 0, 0, 3DDC2 c@3, 0, 2D + B C c@3, 1, 1D + B2 c@3, 2, 0D + 4 A C c@4, 0, 1D + 4 A B c@4, 1, 0D + 10 A2 c@5, 0, 0D

C2 c@2, 1, 2D + 2 B C c@2, 2, 1D + 3 B2 c@2, 3, 0D + 3 A C c@3, 1, 1D + 6 A B c@3, 2, 0D + 6 A2 c@4, 1, 0D


C2 c@1, 2, 2D + 3 B C c@1, 3, 1D + 6 B2 c@1, 4, 0D + 2 A C c@2, 2, 1D + 6 A B c@2, 3, 0D + 3 A2 c@3, 2, 0D

3 C2 c@1, 1, 3D + 3 B2 c@1, 3, 1D + 4 C HB c@1, 2, 2D + A c@2, 1, 2DL + 4 A B c@2, 2, 1D + 3 A2 c@3, 1, 1D






c@2, 0, 0D = Simplify@c@5, 2, 0, 0DDc@1, 1, 0D = Simplify@c@5, 1, 1, 0DDc@1, 0, 1D = Simplify@c@5, 1, 0, 1DDc@0, 2, 0D = Simplify@c@5, 0, 2, 0DDc@0, 1, 1D = Simplify@c@5, 0, 1, 1DDc@0, 0, 2D = Simplify@c@5, 0, 0, 2DDC3 c@2, 0, 3D + B3 c@2, 3, 0D + C2 HB c@2, 1, 2D + 3 A c@3, 0, 2DL + 3 A B2 c@3, 2, 0D +

C HB2 c@2, 2, 1D + 3 A B c@3, 1, 1D + 6 A2 c@4, 0, 1DL + 6 A2 B c@4, 1, 0D + 10 A3 c@5, 0, 0DC3 c@1, 1, 3D + 4 B3 c@1, 4, 0D + 2 C2 HB c@1, 2, 2D + A c@2, 1, 2DL + 6 A B2 c@2, 3, 0D +

C H3 B2 c@1, 3, 1D + 4 A B c@2, 2, 1D + 3 A2 c@3, 1, 1DL + 6 A2 B c@3, 2, 0D + 4 A3 c@4, 1, 0D4 C3 c@1, 0, 4D + B3 c@1, 3, 1D + 3 C2 HB c@1, 1, 3D + 2 A c@2, 0, 3DL + 2 A B2 c@2, 2, 1D +

2 C HB2 c@1, 2, 2D + 2 A B c@2, 1, 2D + 3 A2 c@3, 0, 2DL + 3 A2 B c@3, 1, 1D + 4 A3 c@4, 0, 1DC3 c@0, 2, 3D + 10 B3 c@0, 5, 0D + C2 H3 B c@0, 3, 2D + A c@1, 2, 2DL + 6 A B2 c@1, 4, 0D +

C H6 B2 c@0, 4, 1D + 3 A B c@1, 3, 1D + A2 c@2, 2, 1DL + 3 A2 B c@2, 3, 0D + A3 c@3, 2, 0D4 C3 c@0, 1, 4D + 4 B3 c@0, 4, 1D + 3 C2 H2 B c@0, 2, 3D + A c@1, 1, 3DL + 3 A B2 c@1, 3, 1D +

2 C H3 B2 c@0, 3, 2D + 2 A B c@1, 2, 2D + A2 c@2, 1, 2DL + 2 A2 B c@2, 2, 1D + A3 c@3, 1, 1D10 C3 c@0, 0, 5D + B3 c@0, 3, 2D + 6 C2 HB c@0, 1, 4D + A c@1, 0, 4DL + A B2 c@1, 2, 2D +

3 C HB2 c@0, 2, 3D + A B c@1, 1, 3D + A2 c@2, 0, 3DL + A2 B c@2, 1, 2D + A3 c@3, 0, 2D



c@1, 0, 0D = Simplify@c@5, 1, 0, 0DDc@0, 1, 0D = Simplify@c@5, 0, 1, 0DDc@0, 0, 1D = Simplify@c@5, 0, 0, 1DDC4 c@1, 0, 4D + B4 c@1, 4, 0D + C3 HB c@1, 1, 3D + 2 A c@2, 0, 3DL + 2 A B3 c@2, 3, 0D +

C2 HB2 c@1, 2, 2D + 2 A B c@2, 1, 2D + 3 A2 c@3, 0, 2DL + 3 A2 B2 c@3, 2, 0D +

C HB3 c@1, 3, 1D + 2 A B2 c@2, 2, 1D + 3 A2 B c@3, 1, 1D + 4 A3 c@4, 0, 1DL + 4 A3 B c@4, 1, 0D + 5 A4 c@5, 0, 0DC4 c@0, 1, 4D + 5 B4 c@0, 5, 0D + C3 H2 B c@0, 2, 3D + A c@1, 1, 3DL +

4 A B3 c@1, 4, 0D + C2 H3 B2 c@0, 3, 2D + 2 A B c@1, 2, 2D + A2 c@2, 1, 2DL + 3 A2 B2 c@2, 3, 0D +

C H4 B3 c@0, 4, 1D + 3 A B2 c@1, 3, 1D + 2 A2 B c@2, 2, 1D + A3 c@3, 1, 1DL + 2 A3 B c@3, 2, 0D + A4 c@4, 1, 0D5 C4 c@0, 0, 5D + B4 c@0, 4, 1D + 4 C3 HB c@0, 1, 4D + A c@1, 0, 4DL +

A B3 c@1, 3, 1D + 3 C2 HB2 c@0, 2, 3D + A B c@1, 1, 3D + A2 c@2, 0, 3DL + A2 B2 c@2, 2, 1D +

2 C HB3 c@0, 3, 2D + A B2 c@1, 2, 2D + A2 B c@2, 1, 2D + A3 c@3, 0, 2DL + A3 B c@3, 1, 1D + A4 c@4, 0, 1D

c@0, 0, 0D = Simplify@c@5, 0, 0, 0DD

C5 c@0, 0, 5D + B5 c@0, 5, 0D + C4 HB c@0, 1, 4D + A c@1, 0, 4DL +

A B4 c@1, 4, 0D + C3 HB2 c@0, 2, 3D + A B c@1, 1, 3D + A2 c@2, 0, 3DL + A2 B3 c@2, 3, 0D +

C2 HB3 c@0, 3, 2D + A B2 c@1, 2, 2D + A2 B c@2, 1, 2D + A3 c@3, 0, 2DL + A3 B2 c@3, 2, 0D +

C HB4 c@0, 4, 1D + A B3 c@1, 3, 1D + A2 B2 c@2, 2, 1D + A3 B c@3, 1, 1D + A4 c@4, 0, 1DL +

A4 B c@4, 1, 0D + A5 c@5, 0, 0D



H* Recurrence relation for four particles *Lc@K_, i_, j_, k_, l_D :=

Hi + 1L HK - i - j - k - lL A c@i + 1, j, k, lD + Hj + 1L HK - i - j - k - lL B c@i, j + 1, k, lD +

Hk + 1L HK - i - j - k - lL C c@i, j, k + 1, lD + Hl + 1L HK - i - j - k - lL D c@i, j, k, l + 1D

H* K=1 *L

c@0, 0, 0, 0D = c@1, 0, 0, 0, 0DD c@0, 0, 0, 1D + C c@0, 0, 1, 0D + B c@0, 1, 0, 0D + A c@1, 0, 0, 0D

H* K=2 *L

c@1, 0, 0, 0D = c@2, 1, 0, 0, 0Dc@0, 1, 0, 0D = c@2, 0, 1, 0, 0Dc@0, 0, 1, 0D = c@2, 0, 0, 1, 0Dc@0, 0, 0, 1D = c@2, 0, 0, 0, 1DD c@1, 0, 0, 1D + C c@1, 0, 1, 0D + B c@1, 1, 0, 0D + 2 A c@2, 0, 0, 0D

D c@0, 1, 0, 1D + C c@0, 1, 1, 0D + 2 B c@0, 2, 0, 0D + A c@1, 1, 0, 0D

D c@0, 0, 1, 1D + 2 C c@0, 0, 2, 0D + B c@0, 1, 1, 0D + A c@1, 0, 1, 0D

2 D c@0, 0, 0, 2D + C c@0, 0, 1, 1D + B c@0, 1, 0, 1D + A c@1, 0, 0, 1D

c@0, 0, 0, 0D = Simplify@c@2, 0, 0, 0, 0DD

D2 c@0, 0, 0, 2D + C2 c@0, 0, 2, 0D + B C c@0, 1, 1, 0D +

B2 c@0, 2, 0, 0D + D HC c@0, 0, 1, 1D + B c@0, 1, 0, 1D + A c@1, 0, 0, 1DL +

A C c@1, 0, 1, 0D + A B c@1, 1, 0, 0D + A2 c@2, 0, 0, 0D


H* K=3 *L

c@2, 0, 0, 0D = c@3, 2, 0, 0, 0Dc@1, 1, 0, 0D = c@3, 1, 1, 0, 0Dc@1, 0, 1, 0D = c@3, 1, 0, 1, 0Dc@1, 0, 0, 1D = c@3, 1, 0, 0, 1Dc@0, 2, 0, 0D = c@3, 0, 2, 0, 0Dc@0, 1, 1, 0D = c@3, 0, 1, 1, 0Dc@0, 1, 0, 1D = c@3, 0, 1, 0, 1Dc@0, 0, 2, 0D = c@3, 0, 0, 2, 0Dc@0, 0, 1, 1D = c@3, 0, 0, 1, 1Dc@0, 0, 0, 2D = c@3, 0, 0, 0, 2DD c@2, 0, 0, 1D + C c@2, 0, 1, 0D + B c@2, 1, 0, 0D + 3 A c@3, 0, 0, 0D

D c@1, 1, 0, 1D + C c@1, 1, 1, 0D + 2 B c@1, 2, 0, 0D + 2 A c@2, 1, 0, 0D

D c@1, 0, 1, 1D + 2 C c@1, 0, 2, 0D + B c@1, 1, 1, 0D + 2 A c@2, 0, 1, 0D

2 D c@1, 0, 0, 2D + C c@1, 0, 1, 1D + B c@1, 1, 0, 1D + 2 A c@2, 0, 0, 1D

D c@0, 2, 0, 1D + C c@0, 2, 1, 0D + 3 B c@0, 3, 0, 0D + A c@1, 2, 0, 0D

D c@0, 1, 1, 1D + 2 C c@0, 1, 2, 0D + 2 B c@0, 2, 1, 0D + A c@1, 1, 1, 0D

2 D c@0, 1, 0, 2D + C c@0, 1, 1, 1D + 2 B c@0, 2, 0, 1D + A c@1, 1, 0, 1D

D c@0, 0, 2, 1D + 3 C c@0, 0, 3, 0D + B c@0, 1, 2, 0D + A c@1, 0, 2, 0D

2 D c@0, 0, 1, 2D + 2 C c@0, 0, 2, 1D + B c@0, 1, 1, 1D + A c@1, 0, 1, 1D

3 D c@0, 0, 0, 3D + C c@0, 0, 1, 2D + B c@0, 1, 0, 2D + A c@1, 0, 0, 2D

c@1, 0, 0, 0D = Simplify@c@3, 1, 0, 0, 0DDc@0, 1, 0, 0D = Simplify@c@3, 0, 1, 0, 0DDc@0, 0, 1, 0D = Simplify@c@3, 0, 0, 1, 0DDc@0, 0, 0, 1D = Simplify@c@3, 0, 0, 0, 1DDD2 c@1, 0, 0, 2D + C2 c@1, 0, 2, 0D + B C c@1, 1, 1, 0D +

B2 c@1, 2, 0, 0D + D HC c@1, 0, 1, 1D + B c@1, 1, 0, 1D + 2 A c@2, 0, 0, 1DL +

2 A C c@2, 0, 1, 0D + 2 A B c@2, 1, 0, 0D + 3 A2 c@3, 0, 0, 0DD2 c@0, 1, 0, 2D + C2 c@0, 1, 2, 0D + 2 B C c@0, 2, 1, 0D +

3 B2 c@0, 3, 0, 0D + D HC c@0, 1, 1, 1D + 2 B c@0, 2, 0, 1D + A c@1, 1, 0, 1DL +

A C c@1, 1, 1, 0D + 2 A B c@1, 2, 0, 0D + A2 c@2, 1, 0, 0DD2 c@0, 0, 1, 2D + 3 C2 c@0, 0, 3, 0D + B2 c@0, 2, 1, 0D +

D H2 C c@0, 0, 2, 1D + B c@0, 1, 1, 1D + A c@1, 0, 1, 1DL +

2 C HB c@0, 1, 2, 0D + A c@1, 0, 2, 0DL + A B c@1, 1, 1, 0D + A2 c@2, 0, 1, 0D3 D2 c@0, 0, 0, 3D + C2 c@0, 0, 2, 1D + B C c@0, 1, 1, 1D +

B2 c@0, 2, 0, 1D + 2 D HC c@0, 0, 1, 2D + B c@0, 1, 0, 2D + A c@1, 0, 0, 2DL +

A C c@1, 0, 1, 1D + A B c@1, 1, 0, 1D + A2 c@2, 0, 0, 1D

c@0, 0, 0, 0D = Simplify@c@3, 0, 0, 0, 0DD

D3 c@0, 0, 0, 3D + C3 c@0, 0, 3, 0D + B3 c@0, 3, 0, 0D +

D2 HC c@0, 0, 1, 2D + B c@0, 1, 0, 2D + A c@1, 0, 0, 2DL + C2 HB c@0, 1, 2, 0D + A c@1, 0, 2, 0DL +

A B2 c@1, 2, 0, 0D + D HC2 c@0, 0, 2, 1D + B C c@0, 1, 1, 1D + B2 c@0, 2, 0, 1D +

A C c@1, 0, 1, 1D + A B c@1, 1, 0, 1D + A2 c@2, 0, 0, 1DL +

C HB2 c@0, 2, 1, 0D + A B c@1, 1, 1, 0D + A2 c@2, 0, 1, 0DL + A2 B c@2, 1, 0, 0D + A3 c@3, 0, 0, 0D



B.2. NORMALIZABILITY CONDITION FOR N=2 PARTICLES 142

B.2 Normalizability Condition for N=2 Particles

In this section of the appendix, we will elaborate on the normalizability condition given in equation(6.36), which reads: For every excitation energy K and degree d separately, we require that for alll ∈ 1, 2, . . . ,K and for all k ∈ 0, 1, 2, . . . , l − 1,∑

j1+j2=d

jK−l1

∑i1+i2=K

(−1)i1(i1)kγj1,j2i1,i2

= 0. (B.4)

The normalizability condition guarantees that we get rid of the local singularities introduced by thedifferential operators (6.27),

τi := −12

(∂i +

N∑j=1,j 6=i

α

wi − wj

). (B.5)

We will not take the symmetry condition (6.31) into account here: it has to be imposed in additionto the normalizability condition.

The approach that we take is the following: we will explicitly list the normalizability conditionsfor N = 2, K = 1, 2, 3, 4, and then we will look for a pattern that generalizes to arbitrary K.

The normalizability conditions are obtained by substituting (B.5) into the expression for F ,

F (w, w) =∑

0≤i1+...+iN≤K

ci1,...,iN(w)wi1

1 · · · wiN

N ,

where, from (6.23), the lower order coefficients ci′1,...,i′Nare given by

ci′1,...,i′N=

∑i1+...+iN=K

(i1i′1

)· · ·(iNi′N

)τ

i1−i′11 · · · τ iN−i′N

N ci1,...,iN

in terms of the highest order coefficients ci1,...,iNwith i1 + . . .+ iN = K, and then sorting all terms

together that get multiplied by the same singular prefactor (w1−w2)−k for different values of k ∈ N.Doing so yields quite a large number of equations that have to be satisfied. We will then get rid ofthe redundant equations, and from what remains we will try to find a pattern that can be generalizedto arbitrary K.

K=1Recall the easiest case, that of K = 1, which we have already discussed in Example 6.2. As followsfrom equation (6.34), the only condition for the wavefunction to be normalizable is,

w1 − w2 | c10 − c01, (B.6)

where the long bar means “divides”. We write ci1,i2(w1, w2) =∑

j1+j2=d γj1,j2i1,i2

wj11 w

j22 for some fixed

but arbitrary degree d. Now, by Lemma 6.6, (B.6) is equivalent to∑j1+j2=d

γj1,j21,0 − γj1,j2

0,1 = 0

which indeed agrees with (B.4) for K = 1, and therefore l = 1 and k = 0.

K=2For K = 2, F is given by

F (w, w) = c00 + c10w1 + c01w2 + c20w21 + c11w1w2 + c02w

22,


where,

c00 = τ21 c20 + τ1τ2c11 + τ2

2 c02,

c10 = 2τ1c20 + τ2c11,

c01 = τ1c11 + 2τ2c02.

Recall that each of the coefficients ci1,i2 is a polynomial in w1 and w2.Next, we insert (B.5), and collect all terms that are accompanied by the same singular term.

This yields,

c00 =α

4(α− 1)

1(w1 − w2)2

(c20 − c11 + c02

)+α

41

w1 − w2

(2∂1c20 − ∂1c11 + ∂2c11 − 2∂2c02

)+ regular terms,

c10 = −α2

1w1 − w2

(2c20 − c11

)+ regular terms,

c01 = −α2

1w1 − w2

(c11 − 2c02

)+ regular terms.

Hence, the normalizability conditions for K = 2 are:

(w1 − w2)2 | c20 − c11 + c02 (A)(w1 − w2) | ∂1(2c20 − c11) + ∂2(c11 − 2c02) (B)

for c00, and

(w1 − w2) | 2c20 − c11 (C)(w1 − w2) | c11 − 2c02 (D)

for c10 and c01.As before, we write ci1,i2(w1, w2) =

∑j1+j2=d γj1,j2

i1,i2wj1

1 wj22 for some fixed but arbitrary degree

d, and use Lemma 6.6 to turn equations (A) - (D) into equations on the coefficients γj1,j2i1,i2

. Thus,for (A), we have two equations,∑

j1+j2=d

γj1,j22,0 − γj1,j2

1,1 + γj1,j20,2 = 0, (A1)

∑j1+j2=d

j1

(γj1,j22,0 − γj1,j2

1,1 + γj1,j20,2

)= 0 (A2)

Likewise, for (C) and (D) we have∑j1+j2=d

2γj1,j22,0 − γj1,j2

1,1 = 0 (C)

∑j1+j2=d

γj1,j21,1 − 2γj1,j2

0,2 = 0 (D)

Notice that there is already one redundancy here, as (C)− 2 (A1) = (D) so we can safely drop (D)form our list of conditions. Continuing in the same fashion, for (B) we have

∂1[2c20(w)− c11(w)] + ∂2[c11(w)− 2c02(w)]

=∑

j1+j2=d

j1

(2γj1,j2

2,0 − γj1,j21,1

)wj1−1

1 wj22 +

∑j1+j2=d

j2

(γj1,j21,1 − 2γj1,j2

0,2

)wj1

1 wj2−12

=∑

j1+j2=d

j1

(2γj1,j2

2,0 − γj1,j21,1

)wj1−1

1 wj22 +

∑j1+j2=d

(d− j1)(γj1,j21,1 − 2γj1,j2

0,2

)wj1

1 wj2−12


The polynomial on the RHS is a polynomial of constand degree d−1, so by Lemma 6.6, it is divisibleby (w1 − w2) if and only if the sum of all the coefficients is zero, that is, if and only if

0 =∑

j1+j2=d

j1

(2γj1,j2

2,0 − γj1,j21,1

)+

∑j1+j2=d

(d− j1)(γj1,j21,1 − 2γj1,j2

0,2

)= 2

∑j1+j2=d

j1

(γj1,j22,0 − γj1,j2

1,1 + γj1,j20,2

)+ d

∑j1+j2=d

(γj1,j21,1 − 2γj1,j2

0,2

)(B.7)

However, the above equation is implied by (A2) and (D), so (B) can be dropped as well.What remains the following set of equations:∑

j1+j2=d

j1

(γj1,j22,0 − γj1,j2

1,1 + γj1,j20,2

)= 0, (B.8)

∑j1+j2=d

γj1,j22,0 − γj1,j2

1,1 + γj1,j20,2 = 0, (B.9)

∑j1+j2=d

2γj1,j22,0 − γj1,j2

1,1 = 0. (B.10)

which agrees with (B.4) for K = 2, where l = 1, k = 0 corresponds to (B.8), l = 2, k = 0 correspondsto (B.9), and l = 2, k = 1 corresponds to (B.10).

For K = 3 and K = 4, the reasoning is analogous to the arguments used above. We will thereforenot go into all the details, but we shall list the set of initial equation that have to be satisfied aswell as the reduced set of equations after all redundancies have been taken care of.

K=3The normalizability conditions for K = 3 are:

(w1 − w2)3 | c30 − c21 + c12 − c03

(w1 − w2)2 | ∂1(3c30 − 2c21 + c12) + ∂2(c21 − 2c12 + 3c03)

(w1 − w2) | ∂21(3c30 − c21) + ∂1∂2(2c21 − 2c12) + ∂2

2(c12 − 3c03)

for c00,

(w1 − w2)2 | 3c30 − 2c21 + c12

(w1 − w2) | ∂1(3c30 − c21) + ∂2(c21 − c12)

(w1 − w2)2 | c21 − 2c12 + 3c03(w1 − w2) | ∂1(c21 − c12) + ∂2(c12 − 3c03)

for c10 and c01, and

(w1 − w2) | 3c30 − c21 (B.11)(w1 − w2) | c21 − c12 (B.12)(w1 − w2) | c12 − 3c03 (B.13)

for c20, c11 and c02 respectively.As in the case of K = 2, there is a lot of redundancy to be found in the above equations. For

example, (B.13) and (w1 − w2)2|(c21 − 2c12 + 3c03) imply (w1 − w2)|[∂1(c21 − c12) + ∂2(c12 − 3c03)]using a similar argument as in (B.7).


After all the redundant equations have been removed, the following equations remain∑j1+j2=d

j21

(γj1,j23,0 − γj1,j2

2,1 + γj1,j21,2 − γj1,j2

0,3

)= 0

∑j1+j2=d

j1

(γj1,j23,0 − γj1,j2

2,1 + γj1,j21,2 − γj1,j2

0,3

)= 0

∑j1+j2=d

j1

(3γj1,j2

3,0 − 2γj1,j22,1 + γj1,j2

1,2

)= 0

∑j1+j2=d

(γj1,j23,0 − γj1,j2

2,1 + γj1,j21,2 − γj1,j2

0,3

)= 0

∑j1+j2=d

(3γj1,j2

3,0 − 2γj1,j22,1 + γj1,j2

1,2

)= 0

∑j1+j2=d

(9γj1,j2

3,0 − 4γj1,j22,1 + γj1,j2

1,2

)= 0

which is in agreement with (B.4). Note that last three equations are equivalent to (B.11) - (B.13).

K=4The normalizability conditions for K = 4 are:

(w1 − w2)4 | c40 − c31 + c22 − c13 + c04

(w1 − w2)3 | ∂1(4c40 − 3c31 + 2c22 − c13) + ∂2(c31 − 2c22 + 3c13 − 4c04)

(w1 − w2)2 | ∂21(6c40 − 3c31 + c22) + ∂1∂2(3c31 − 4c22 + 3c13) + ∂2

2(c22 − 3c13 + 6c04)

(w1 − w2) | ∂31(4c40 − c31) + ∂2

1∂2(3c31 − 2c22) + ∂1∂22(2c22 − 3c13) + ∂3

2(c13 − 4c04)

for c00,

(w1 − w2)3 | 4c40 − 3c31 + 2c22 − c13

(w1 − w2)2 | ∂1(12c40 − 6c31 + 2c22) + ∂2(3c31 − 4c22 + 3c13)

(w1 − w2) | ∂21(12c40 − 3c31) + ∂1∂2(6c31 − 4c22) + ∂2

2(2c22 − 3c13)

(w1 − w2)3 | c31 − 2c22 + 3c13 − 4c04(w1 − w2)2 | ∂1(3c31 − 4c22 + 3c13) + ∂2(2c22 − 6c13 + 12c04)

(w1 − w2) | ∂21(3c31 − 2c22) + ∂1∂2(4c22 − 6c13) + ∂2

2(3c13 − 12c04)

for c10 and c01,

(w1 − w2)2 | 6c40 − 3c31 + c22

(w1 − w2) | ∂1(12c40 − 3c31) + ∂2(3c13 − 2c22)

(w1 − w2)2 | 3c31 − 4c22 + 3c13(w1 − w2) | ∂1(6c31 − 4c22) + ∂2(4c22 − 6c13)

(w1 − w2)2 | c22 − 3c13 + 6c04(w1 − w2) | ∂1(2c22 − 3c13) + ∂2(3c13 − 12c04)


for c20, c11 and c02, and,

(w1 − w2) | 4c40 − c31

(w1 − w2) | 3c31 − 2c22(w1 − w2) | 2c22 − 3c13(w1 − w2) | c13 − 4c04

for c30, c21, c12 and c03 respectively.Removing all redundant equations results in,∑

j1+j2=d

j31

(γj1,j24,0 − γj1,j2

3,1 + γj1,j22,2 − γj1,j2

1,3 + γj1,j20,4

)= 0

∑j1+j2=d

j21

(γj1,j24,0 − γj1,j2

3,1 + γj1,j22,2 − γj1,j2

1,3 + γj1,j20,4

)= 0

∑j1+j2=d

j21

(4γj1,j2

4,0 − 3γj1,j23,1 + 2γj1,j2

2,2 − γj1,j21,3

)= 0

∑j1+j2=d

j1

(γj1,j24,0 − γj1,j2

3,1 + γj1,j22,2 − γj1,j2

1,3 + γj1,j20,4

)= 0

∑j1+j2=d

j1

(4γj1,j2

4,0 − 3γj1,j23,1 + 2γj1,j2

2,2 − γj1,j21,3

)= 0

∑j1+j2=d

j1

(42γj1,j2

4,0 − 32γj1,j23,1 + 22γj1,j2

2,2 − γj1,j21,3

)= 0

∑j1+j2=d

(γj1,j24,0 − γj1,j2

3,1 + γj1,j22,2 − γj1,j2

1,3 + γj1,j20,4

)= 0

∑j1+j2=d

(4γj1,j2

4,0 − 3γj1,j23,1 + 2γj1,j2

2,2 − γj1,j21,3

)= 0

∑j1+j2=d

(42γj1,j2

4,0 − 32γj1,j23,1 + 22γj1,j2

2,2 − γj1,j21,3

)= 0

∑j1+j2=d

(43γj1,j2

4,0 − 33γj1,j23,1 + 23γj1,j2

2,2 − γj1,j21,3

)= 0

which is in agreement with (B.4).Since (B.4) holds up to K = 4, and there is no reason to assume that the apparent structure

in the normalizability conditions will change for higher excitation energies, we will stop here withchecking the validity of (B.4). Besides, for larger K, the work gets too tedious to do by hand.1

1It would have been possible to write a program that checks whether the normalizability conditions also hold forhigher excitation energies or not, if there were more time available, which there is not.

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Acknowledgements

I would not have been able to write this thesis without the help of a number of people, to whom Iam deeply grateful.

First of all, I would like to thank my supervisors Cristiane Morais Smith and Andre Henriquesfor their excellent supervision. They were both very involved in my project, which resulted in myundertaking the journey from Amsterdam to Utrecht at least once a week. In addition to keepingtrack of the long term goal, both Cristiane and Andre were also prepared to delve deeply into thematerial I was working on and to give input on detailed matters. Despite the fact that Cristiane,being a physicist, and Andre, being a mathematician, both have their own interests and priorities,it was always possible to find common ground. As a matter of fact, it is exactly the distinct kinds ofexpertise of Andre and Cristiane that allowed me to view the same phenomenon from two differentperspectives. I do not believe I could have learned as much as I have if it were not for Cristiane’sand Andre’s guidance, knowledge, and involvement.

Besides my two supervisors, I have benefitted greatly from the support and input of EmilioCobanera from Leiden University. Occupying a position in between Cristiane the physicist and Andrethe mathematician, Emilio served as a great interpreter and mediator when required. Moreover, itwas Emilio who came up with the first research topic of my thesis.

Additionally, my official supervisor at the University of Amsterdam, Kareljan Schoutens, turnedout to be very helpful when finishing my thesis, especially in discussing my results and providingadditional literature that was of great benefit to my conclusion.

Finally, I want to thank all my fellow students for making all the time spent at the Science Parkvery enjoyable, my parents and my not-so-little brother for their love and support throughout allthese years, and, last but definitely not least, my girl Eva for being there for me whenever I neededher.

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