introduction to the quantum hall eﬀectsduine102/qfinqm08/...1 introduction to the quantum hall...

1

Introduction to the Quantum Hall Effects

Lecture notes, 2006

Pascal LEDERER Mark Oliver GOERBIG

Laboratoire de Physique des Solides, CNRS-UMR 8502Universite de Paris Sud, Bat. 510

F-91405 Orsay cedex

Contents

1 Introduction 71.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 History of the Quantum Hall Effect . . . . . . . . . . . . . . . 91.3 Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Charged particles in a magnetic field 192.1 Classical treatment . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1.1 Lagrangian approach . . . . . . . . . . . . . . . . . . . 202.1.2 Hamiltonian formalism . . . . . . . . . . . . . . . . . 22

2.2 Quantum treatment . . . . . . . . . . . . . . . . . . . . . . . . 232.2.1 Wave functions in the symmetric gauge . . . . . . . . . 252.2.2 Coherent states and semi-classical motion . . . . . . . 29

3 Transport properties– Integer Quantum Hall Effect (IQHE) 333.1 Resistance and resistivity in 2D . . . . . . . . . . . . . . . . . 333.2 Conductance of a completely filled Landau Level . . . . . . . . 343.3 Localisation in a strong magnetic field . . . . . . . . . . . . . 373.4 Transitions between plateaus – The percolation picture . . . . 42

4 The Fractional Quantum Hall Effect (FQHE)– From Laugh-lin’s theory to Composite Fermions. 454.1 Model for electron dynamics restricted to a single LL . . . . . 46

4.1.1 Matrix elements . . . . . . . . . . . . . . . . . . . . . . 484.1.2 Projected densities algebra . . . . . . . . . . . . . . . . 50

4.2 The Laughlin wave function . . . . . . . . . . . . . . . . . . . 514.2.1 The many-body wave function for ν = 1 . . . . . . . . 524.2.2 The many-body function for ν = 1/(2s+ 1) . . . . . . 554.2.3 Incompressible fluid . . . . . . . . . . . . . . . . . . . . 58

3

4 CONTENTS

4.2.4 Fractional charge quasi-particles . . . . . . . . . . . . 594.2.5 Ground state energy . . . . . . . . . . . . . . . . . . . 624.2.6 Neutral Collective Modes . . . . . . . . . . . . . . . . . 67

4.3 Jain’s generalisation – Composite Fermions . . . . . . . . . . 694.3.1 The effective potential . . . . . . . . . . . . . . . . . . 70

5 Chern-Simons Theories and Anyon Physics 755.1 Chern-Simons transformations . . . . . . . . . . . . . . . . . 755.2 Statistical Transmutation – Anyons in 2D . . . . . . . . . . . 78

5.2.1 Anyons and Chern-Simons theories . . . . . . . . . . . 805.2.2 Fractional charge and fractional statistics . . . . . . . 82

6 Hamiltonian theory of the Fractional Quantum Hall Effect 856.1 Miscroscopic theory . . . . . . . . . . . . . . . . . . . . . . . . 86

6.1.1 Fluctuations of ACS(r) . . . . . . . . . . . . . . . . . . 876.1.2 Decoupling transformation at small wave vector . . . . 91

6.2 Effective theory at all wave vectors . . . . . . . . . . . . . . . 956.2.1 Approximate treatment of the constraint . . . . . . . . 986.2.2 Energy gaps computation . . . . . . . . . . . . . . . . 1006.2.3 Self similarity in the effective model . . . . . . . . . . . 103

7 Spin and Quantum Hall Effect– Ferromagnetism at ν = 1 1097.1 Interactions are relevant at ν = 1 . . . . . . . . . . . . . . . . 109

7.1.1 Wave functions . . . . . . . . . . . . . . . . . . . . . . 1107.2 Algebraic structure of the model with spin . . . . . . . . . . . 1127.3 Effective model . . . . . . . . . . . . . . . . . . . . . . . . . . 115

7.3.1 Spin waves . . . . . . . . . . . . . . . . . . . . . . . . . 1177.3.2 Skyrmions . . . . . . . . . . . . . . . . . . . . . . . . . 1187.3.3 Spin-charge entanglement . . . . . . . . . . . . . . . . 1197.3.4 Effective model for the energy . . . . . . . . . . . . . . 121

7.4 Berry phase and adiabatic transport . . . . . . . . . . . . . . 1237.5 Applications to quantum Hall magnetism . . . . . . . . . . . . 127

7.5.1 Spin dynamics in a magnetic field . . . . . . . . . . . . 1277.6 Application to spin textures . . . . . . . . . . . . . . . . . . . 128

8 Quantum Hall Effect in bi-layers 1318.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1318.2 Pseudo-spin analogy . . . . . . . . . . . . . . . . . . . . . . . 133

CONTENTS 5

8.3 Differences with the ferromagnetic monolayer case . . . . . . 1348.4 Experimental facts . . . . . . . . . . . . . . . . . . . . . . . . 137

8.4.1 Phase Diagram . . . . . . . . . . . . . . . . . . . . . 1378.4.2 Excitation gap . . . . . . . . . . . . . . . . . . . . . . 1398.4.3 Effect of a parallel magnetic field . . . . . . . . . . . . 1398.4.4 The quasi-Josephson effect . . . . . . . . . . . . . . . . 1418.4.5 Antiparallel currents experiment . . . . . . . . . . . . . 142

8.5 Excitonic superfluidity . . . . . . . . . . . . . . . . . . . . . . 1458.5.1 Collective modes – Excitonic condensate dynamics . . 1488.5.2 Charged topological excitations . . . . . . . . . . . . . 1508.5.3 Kosterlitz-Thouless transition . . . . . . . . . . . . . . 1538.5.4 Effect of the inter layer tunneling term . . . . . . . . . 1548.5.5 Combined effects of a tunnel term and a parallel field B‖1558.5.6 Effect of an inter-layer voltage bias . . . . . . . . . . . 158

6 CONTENTS

Chapter 1

Introduction

1.1 Motivation

Almost thirty years after the discovery of the Integer Quantum Hall Effect(IQHE, 1980)[1], and the fractional one (FQHE, 1983)[2], two-dimensionalelectron systems submitted to a perpendicular magnetic field remain a veryactive field of research, be it experimentally or on the theory level [3]. Newsurprises arise year after year, exotic states of electronic matter, new ma-terials with fascinating quantum Hall properties keep triggering an intenseactivity in the field: see for example the number of papers on graphene whichappear on cond-mat since the discovery of the QHE in this material in 2005[4, 5] or the appearance of excitonic superfluidity in quantum Hall bilayers[6]. The continuous increase in sample quality over the years is a key fac-tor in the discovery of new electronic states of two dimensional matter. Inheterostructure interfaces such as the GaAs/AlGaAs system, one of the pro-totypical experimental systems, the electronic mobility µ has now reachedvalues up to µ ≃ 30× 106cm2/Vs, more than two orders of magnitude largerthan the values obtained at the time of the first QHE discoveries in the 80’s.

The discovery of the quantum Hall effects, in particular that of the FQHE,has taken a large part in a qualitative advance of condensed matter physicsregarding electronic fluids in conducting materials. In large band metallicsystems, the role of interactions was successfully taken into account until thesixties by the Landau liquid theory [7], which is a perturbation approach:interactions between electrons alter adiabatically the properties of the freeelectron model, so that the Drude-Sommerfeld model keeps its validity with

7

8 Introduction

renormalized coefficients. The theoretical tools corresponding to this physicshave involved sophisticated diagramatic techniques such as Feynman dia-grams, which are all based on the existence of a well controlled limit of zerointeraction Green’s function. It was realized in the fifties, with the theoryof BCS superconductivity [8] that attractive interactions cause a breakdownof the non interacting model, and a spontaneous symmetry breaking (in thesuperconductivity case a breakdown of gauge invariance), but even in thatcase Fermi liquid theory seemed an unescapable starting point for conductingsystems. The various other hints of the breakdown of perturbation theory,such as the local spin fluctuation problem, the Kondo problem, the Mott in-sulator problem, the physics of solid or superfluid 3He in the sixties/seventiesor even the Luttinger liquid problem in the eighties did little to suggest theintellectual revolutions which were in store with the discoveries of the frac-tional quantum Hall effects and of superconducting high Tc cuprates [9]. (Theheavy fermion physics is somewhat of a hybrid between the former and thelatter electronic systems.)

The fundamental novelty of both those phenomena, which involve elec-tron systems in two dimensional geometries, is that the largest term in theHamiltonian is the interaction term, so that perturbation theory is basicallyuseless. If one tries to do perturbation theory from a limiting case, suchas the incompletely filled Landau level in the Hall case, or the zero kineticenergy in the High Tc case, one is faced with the macroscopic degeneracy ofthe starting ground state. There is no way to evolve adiabatically from thisdegenerate ground state to the physical one. In both cases, interactions donot alter the quantitative properties of a pre-existing ground state. They areessential at determining the symmetry and properties of the ground statewhich results from the lifting of a formidable degeneracy. Thus the wholeaparatus of perturbation theory turned ou to be inadequate for a theoreticalunderstanding of the QHE, as well as of High Tc superconductivity. Newmethods have had to be devised, and new concepts emerged to account forunexpected exotic phenomena: new particles, new statistics, new groundstates,[11] etc.. The most intuitive method turned out to be very useful andfruitful. It amounts to guessing the many-body wave function for ≈ 1011

particles for the ground state. This is the incredibly original path chosen in1983 by Laughlin [12]. More or less the same method led P. W. Anderson topropose the RVB wave function as the basic new object to describe High Tc

superconductivity in 1987. That wave function is a resonating superpositionof singlet pair products involving all electrons. It looks like a BCS wave

1.2. HISTORY OF THE QUANTUM HALL EFFECT 9

function, where strong correlations prevent the simultaneous occupation ofany site by two electrons. This proposal has been at the center of activediscussions over the last twenty years.

In the context of Quantum Hall Effects, new ideas such as Chern-Simonstheories, which investigate the formation of composite particles when elec-trons bind to flux tubes, have been very fruitful [15, 16, 17]. These theories ,based on their topological character, have given flesh to the notion of strangephenomena such as charge fractionalization, fractional statistics, and so forth,at work among the elementary excitations of the 2D electronic liquid undermagnetic field.

A new concept, also emerging over the last twenty years, is that of quan-tum phase transition and of quantum critical points [18]. Quantum phasetransitions occur at zero temperature. They are not controlled by tempera-ture, but by parameters such as pressure, magnetic field, or chemical doping[11, 18]. In heavy fermions for instance, the quantum critical point separatesa metallic paramagnetic phase from an insulating antiferromagnetic one. Atfinite temperature, the “quantum critical regime” involves a broader arrayof parameter values. Quantum phase transitions, as we shall see, are presentin a number of Quantum Hall Effects, as a function of magnetic field or ofelectron density.

1.2 History of the Quantum Hall Effect

The classical Hall effect

The classical Hall effect was discovered by Edwin Hall in 1879, as a minor cor-rection to Maxwell’s “Treatise on Electromagnetism ”, which was supposedto be a final and complete account of the physical properties of Nature. Hallnoted that, contrary to Maxwell’s opinion, if a current I is driven througha thin metallic slab in a perpendicular magnetic field B = Bez, an elec-tronic density gradient develops in the slab, in the direction orthogonal tothe current. This gradient is equivalent to a transverse voltage V , so thatthe resulting transverse resistance (the Hall resistance) is proportional to thefield, and inversely proportional to the electronic density: RH = −B/enel.Here nel is the density per unit surface, and −e is the electron charge.

Things are rather simple to understand with the Drude model, with theelectron equation of motion:

10 Introduction

résistance résistance de Halllongitudinale

gaz d’électrons 2DI

−I

Figure 1.1: Two dimensional electron system under perpendicular magnetic field. Thecurrent I is driven through the two black contacts. The longitudinal resistance is measuredbetween two contacts on the same edge, while the Hall resistance is measured across thesample on the two opposite edges.

dp

dt= −e

(

E +p

m× B

)

− p

τ,

where E is the electric field, m is the electron (band) mass , p its momentumτ the mean diffusion time due to impurities. The stationary solution for thisequation, i.e. that for dp/dt = 0, is

0 = −e(

Ex +py

mB)

− px

τ,

0 = −e(

Ey −px

mB)

− py

τ.

With the cyclotron frequency ωC ≡ eB/m and the Drude conductivity σ0 =nele

2τ/m, one gets

σ0Ex = −nelepx

m− nele

py

m(ωCτ),

σ0Ey = nelepx

m(ωCτ) − nele

py

m,

In terms of the current density j = −nelep/m, in matrix form

E = ρ j,

History of the Quantum Hall Effect 11

with the resistivity tensor

ρ =

(

σ−10

Benel

− Benel

σ−10

)

=1

σ0

(

1 ωCτ−ωCτ 1

)

. (1.1)

The conductivity follows by matrix inversion,

σ = ρ−1 =

(

σL −σH

σH σL

)

, (1.2)

with σL = σ0/(1 + ω2Cτ

2) et σH = σ0ωCτ/(1 + ω2Cτ

2). In the limit of a puremetal with infinite τ , ωCτ → ∞, one has

ρ =

(

0 Benel

− Benel

0

)

, σ =

(

0 − enel

Benel

B0

)

. (1.3)

Note that the diagonal (longitudinal ) conductivity is zero together with thelongitudinal resistivity.

The classical Hall effect, deemed by Hall of purely academic interest, andwith no foreseeable application whatsoever is nowadays of current industrialuse, and is still useful in condensed matter physics to measure the carrierdensity in conducting materials, as well as to determine their sign.

Landau quantization

Landau was the first to apply quantum mechanics, in 1930, in the study ofmetallic systems, to the quantum treatment of electronic motion in a staticuniform magnetic field. He found that problem to be quite analogous in2D to that of a harmonic oscillator, with an energy structure of equidistantdiscrete levels, with a distance hωC . Each level is highly degenerate. Thesurface density of states per Landau level, nB, is nb = B/φ0 per unit area,where φ0 = h/e is the flux quantum , so that nB is the density of flux quantathreading the surface in a perpendicular field B. Because of their fermioniccharacter, electrons added to the plane fill in successive Landau Levels (LL),so that it is natural and useful to define a filling factor

ν =nel

nB

. (1.4)

This quantum treatment will be reviewed in chapter 2.

12 Introduction

The Quantum Hall Effect : a macroscopic quantum phe-nomenon

The IQHE, discovered by von Klitzing in 1980 [1] is, at first sight, a directconsequence of Landau quantization, and disorder. In fact, as we shall see,impurity disorder is also a necessary feature: in a tanslationaly invariantsystem, the Hall resistivity would have the classical value. In fact Hall quan-tization appears because of the sample impurity potential, not in spite of it.The IQHE appears at low temperature, when kBT ≪ hωC , and is defined bythe formation of plateaus in the Hall resistance, which become quantized, forcertain ranges of values of B, as RH = (h/e2)1/n, where n is an integer, theinteger part of the filling factor: n = [ν]. Each plateau in the Hall resistancecoincides with a zero (exponentially small value in fact) of the longitudinalresistance (Fig. 1.2). A remarkable fact about the resistance quantization isthat its value is independent of the sample geometry, of its quality (densityand/or distribution of impurities, etc.). The Hall resistance is given entirelyin terms of fundamental constants, e and h. The accuracy of the determi-nation of the n = 1 plateau value reaches 1 part in 109, so that it is nowused in metrology as a universal resistance standard, the v. Klitzing constantRK−90 = 25812, 807Ω.

Another surprise followed shortly after the discovery of the IQHE. In1983, D. Tsui, H. Stormer and A. Gossard found the Fractional QuantumHall Effect (FQHE) [2]. This occurs for ”magical” values of the filling factor,especially within the lowest LL. The first observed fractional plateaus wereat ν = 1/3 and ν = 2/3. Since then, a whole series of plateaux have beendetected. The remarkable aspect is that for fractional ν values, there is ahuge degeneracy of the N body states. Since, apart from impurities, theonly relevant energy is the Coulomb repulsion between particles, one is fac-ing a strongly correlated electron system. Our understanding of the FQHEis still to-day essentially based on a revolutionnary theory put forward byLaughlin in 1983: he proposed, by a series of educated guesses, a wave func-tion for N ≈ 1011 particles, written in the first quantization language, whichdescribes an incompressible electronic liquid state, i.e. one such that ele-mentary and collective excitations are separated from the ground state by agap[12]. Following the discovery of other families of fractional QH plateauswhich are not described by the initial Laughlin wave functions, various gener-alizations have been proposed. B. Halperin generalised in 1983 the Laughlinwave function to the case of an additional discrete degree of freedom, such as

History of the Quantum Hall Effect 13

8 12 160 4Magnetic Field B (T)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

ρ xy

(h/e

)2

0

0.5

1.0

1.5

2.0

ρΩ

xx(k

)

2/3 3/5

5/9

6/11

7/15

2/53/74/9

5/11

6/13

7/13

8/15

1 2/3 2/

5/7

4/5

3 4/

Vx

VyIx

4/7

5/34/3

8/57/5

123456

champ magnétique B[T]

Figure 1.2: Experimental signature of the quantum Hall effect. Each plateau coincideswith a zero longitudinal resistance. The classical Hall resistance curve is the dotted line.Numbers label the filling factor ν = n for the IQHE, and ν = p/q (p and q integers for theFQHE.

14 Introduction

spin [19]. In 1989, Jain generalised the theory to account for observed frac-tional states with ν = p/(2sp+ 1) with s and p integers. He introduced thenotion of “Composite Fermions” (CF). The CF theory allows to understandthe FQHE as an IQHE of CF. This will be dealt with in chapter 4.

1.3 Samples

The discovery of the IQHE and of the FQHE is intimately connected tothe evolution of semiconducting sample preparation to produce 2D electrongases. The order of magnitude of electron densities in thin metallic filmswas not appropriate for the QHE discovery. The electronic surface density ofmetallic thin films is of order nel = 1018m−2 = 1014cm−2. As we shall see, theQHE become observable when the electronic surface density is of the orderof the magnetic flux density, i. e. nel ∼ nB = eB/h. This would amountto a magnetic field of order ≈ 1000 T, quite out of reach in the laboratorynowadays, when the largest available fields in a dc regime amount to lessthan 50 T, and less than 80 T for pulsed magnetic fields. More intensefields are available in destructive experiments or nuclear blasts. A usefulquantity which sets a length scale for the QH physics is the magnetic length,

lB =√

h/eB = 25, 7nm/√

B[T]. The magnetic length is such that the flux

which threads a surface equal to 2πl2B is the flux quantum φ0 = h/eLower electronic densities, typically nel ∼ 1011cm−2 are reached in semi-

conducting structures. The samples used at the time of the IQHE discoverywere MOSFETs, shown schematically in the figure 1.3. In such a device, ametallic film is separated from a semiconductor, which is doped with accep-tors, by an oxyde insulating layer. The metal chemical potential is controlledwith a voltage bias VG. When VG = 0, the Fermi level EF lies in the gapbetween the valence band and the conduction band, below the acceptor levels[Fig. 1.3(a)]. Upon lowering the chemical potential in the metal with VG > 0,one introduces holes, which attract electrons from the semiconductor towardthe interface with the insulating layer. This results in a downward bend-ing of the semiconductor band close to the interface. Electrons attractedto the interface first fill in acceptor levels, which are below the Fermi level[Fig. 1.3(b)]. By further lowering of the metal chemical potential, the semiconductor conduction band can be bent below the Fermi level close to theinsulating layer, so that electrons which occupy states in that part of the

Samples 15

métal oxyde

E

z

F

bande deconduction

bande de valence

niveauxd’accepteurs

métal oxyde semiconducteur

E

z

F

bande deconduction

bande de valence


métal semiconducteur

E

z

F

bande deconduction

bande de valence


(a)oxyde

(isolant)

(isolant) (isolant)

(b) (c)

VVG

G

II

I

métaloxyde

semiconducteurV

G

z

z

E

E

E

1

0

électrons 2D

Figure 1.3: Metal-Oxyde Field Effect Transistor (MOSFET). The inset I is a schematicview of a MOSFET. (a) Energy level structure. In the metallic part, the band states areoccupied up to the Fermi level EF . The oxyde is an insulating film. The Fermi level inthe semiconductor falls in the gap between the valence band and the conduction band.There are acceptor states doped close to the valence band, but above the Fermi level EF

(b)The chemical potential in the metal is controlled by a gate bias VG. The introductionof holes results in a band bending in the semiconducting part and (c) when the gate biasexceeds a certain value, the conduction band is filled close to the insulating interface, anda 2D electron gas is formed. The confining potential has a triangular profile with electricsubbands which are represented in the inset II.

16 Introduction

dopants(récepteurs)

AlGaAs

z

EF

GaAs

dopants(récepteurs)

AlGaAs

z

EF

GaAs(a) (b)

électrons 2D

Figure 1.4: Semiconducting (GaAs/AlGaAs) heterostructure. (a) A layer of (receptor)dopants lies on the AlGaAs side, at a certain distance from the interface. The Fermienergy is locked to the dopant levels. The bottom of the GaAs conduction band lies lowerthan those levels so electrons close to the interface migrate to the GaAs conduction band.(b) This polarisation leads to a band bending close to the interface, and a 2D electron gasforms, on the GaAs side.

conduction band form a 2D gas. Electron motion , in spite of a finite extentof the wave function in the z direction is purely 2D if confinement is suchthat the energy separation between electronic sub-bands E0 (partially filled)and E1 (empty) is significantly larger than kBT (inset II in Fig. 1.3).

The problem with MOSFETS is the small distance between the 2D elec-tron gas and the dopants. The latter also act as scattering centers, so thatthe mean free path is relatively small, and the electron mobility relativelylow. This problem is dealt with by forming a 2D electron gas at the inter-face of a semiconducting heterostructure, such as for example in the III-Vheterostructure GaAs/AlGaAs. The two semi-conductors have different gapsbetween their valence bands and their conduction bands. When the side withthe largest gap, AlxGa1−xAs, is doped, the receptor dopant levels are occu-pied by electrons, and the Fermi level is tied to receptor levels, which mayhave a higher energy than the bottom of the conduction band in GaAs. Theelectrons close to the interface migrate in this conduction band [Fig. 1.4(a)].This polarisation produces a band bending, now on the GaAs side, whichis not disordered by the dopants. This spatial separation between the 2Delectron gas and the impurities allows to reach larger mobility values thanin MOSFETS. Technological progress in the fabrication of semi-conductingheterostructures along the last twenty years has allowed to increase mobilities

Samples 17

V =15V

Density of states

B=9T

T=30mK

T=1.6K

∼ ν

∼ 1/ν

Graphene IQHE:

R = h/e

at = 2(2n+1)

at = 2n

ν

ν

H ν2

(no Zeeman)

Usual IQHE:

g

Figure 1.5: Quantum Hall Effect as observed in graphene by Zhang et al (Nature 438,197 (2005)), and Novoselov et al. (Nature 438, 201 (2005))

by two orders of magnitude: The FQHE was discovered in 1983 in a samplewith mobility µ ≃ 0, 1× 106cm2/Vs [2] while samples of the same type reachnowadays a mobility of µ ≃ 30 × 106cm2/Vs.

The discovery of the QHE in graphene in 2005 opens up a new avenueto experiments and theory in the QHE, because graphene is a qualitativelynew 2D material, with original electronic structure. See figure 1.5

18 Introduction

Chapter 2

Charged particles in a magneticfield

Our understanding of integer or fractional quantum Hall effects relies mostlyon the quantum mechanics of electrons in a 2D plane, or thin slab, whensubmitted to a perpendicular magnetic field. There is a notable exception,that of the Integer Quantum Hall Effect (IQHE) observed in anisotropic 3Dorganic salts such as Bechgaard salts. The IQHE may arise in 3D systemsunder magnetic field provided the electronic structure of the material undermagnetic field exhibits the suitable gap structure. However, in the presentlectures, I will adress the main stream of quantum Hall effects physics, thatof electrons the dynamics of which is restricted to a plane. The topic of thischapter is the single electron quantum mechanics in a plane under magneticfield. I start with a discussion of the classical mechanics, as a limiting caseof the quantum mechanical case.

2.1 Classical treatment

The equation of motion of a particle (with charge −e and mass m in amagnetic field B = Bez is as follows:

x = −ωC y, y = ωC x, (2.1)

This follows from the Lorentz force F = −er×B – By definition, the cyclotronfrequency is ωC = eB/m. The equation is solved as:

x = −ωC(y − Y ), y = ωC(x−X), (2.2)

19

20 Charged particle in a static uniform magnetic field

η

B

r

R

Figure 2.1: Cyclotron motion of an electron in a magnetic field, around the guidingcenter R.

where R = (X,Y ) is a constant of motion. With η = (ηx, ηy) = r − R, onehas

ηx = −ω2Cηx, ηy = −ω2

Cηy, (2.3)

and the solution is

x(t) = X + r sin(ωCt+ φ), y(t) = Y + r cos(ωCt+ φ), (2.4)

where r is the cyclotron motion radius, and φ is an arbitrary angle (constantof motion). The physical meaning of the constant of motion R is transparent:it is the “guiding center”, around which the electron moves on a circle ofradius r (Fig 2.1).

2.1.1 Lagrangian approach

Lagrangian mechanics starts from the energy function L and the minimumaction principle, which reproduce the equations of motion of the classicalsystem. This function is defined in configuration space (positions qµ andvelocities qµ). The minimum action principle results in the Euler-Lagrangeequations

d

dt

∂L

∂qµ− ∂L

∂qµ= 0, (2.5)

Classical approach 21

valid for any index µ. The appropriate function in our case is

L(x, y; x, y) =1

2m(

x2 + y2)

− e [Ax(x, y)x+ Ay(x, y)y] , (2.6)

where A = (Ax, Ay) is a vector potential which is independent of time.This represents the minimal coupling theory for a charged particle and anelectromagnetic field, written in a covariant form, with Einstein’s convention,

Lrel =1

2mxµxµ − exµAµ.

The conjugate (or ”canonical”) momenta, which will be needed in the Hamil-tonian formulation of clasical or quantum mechanics are

px ≡ ∂L

∂x= mx− eAx, py ≡ ∂L

∂y= my − eAy. (2.7)

The Euler-Lagrange equations yield the equations of motion [Eq. (2.1)]

mx = −ey(∂xAy − ∂yAx), my = ex(∂xAy − ∂yAx), (2.8)

where ∂x ≡ ∂/∂x, ∂y ≡ ∂/∂y, and (∂xAy − ∂yAx) = (∇× A)z = B is the zcomponent of the magnetic field.

Gauge invariance

A gauge transformation of the vector potential is defined as A′ = A + ∇χ,where χ is an arbitrary function. The magnetic field is independent of thegauge (it is “gauge invariant”) since ∇ × ∇χ = 0. A usual gauge in nonrelativistic physics is the Coulomb gauge, ∇ ·A = 0.1 The gauge (the gaugefunction) is not completely determined by the Coulomb gauge condition,which demands only ∆χ = 0, where ∆ = ∇2 is the Laplacian. Gaugetransformations in 2D are thus defined as harmonic functions. Two gaugechoices are especially useful in the quantum treatment of our problem: theLandau gauge (e.g. for problems defined on a rectangular sample)

AL = B(−y, 0, 0) (2.9)

1Relativistic mechanics use rather the Lorentz gauge, ∂µAµ = 0.

22 Charged particle in a static uniform magnetic field

and the symmetric gauge(e.g. for problems defined on a disc)

AS =B

2(−y, x, 0), (2.10)

the function which transforms from one of these two gauges to the other isχ = (B/2)xy.

Since velocities x and y, are also gauge invariant, it is clear that conjugate(or ”canonical”) momenta in equation (2.7) are not. The gauge invariantmomenta (or ”mechanical momenta”) are

Πx = mx = px+eAx = −mωCηy, Πy = my = py+eAy = mωCηx, (2.11)

where we used Eq. (2.2).

2.1.2 Hamiltonian formalism

For the quantum treatment of a one particle system, it is often preferred touse the Hamiltonian formalism of classical mechanics. The Hamiltonian isderived from the Lagrangian by a Legendre transformation,

H(x, y; px, py) = xpx + ypy − L.

It is an energy function defined in phase space (positions/conjugate mo-menta). One must express velocities in terms of conjugate momenta, usingequations (2.7), and one finds for the Hamiltonian

H =1

2m

[

(px + eAx)2 + (py + eAy)

2]

. (2.12)

The Hamiltonian may also be written in a concise fashion, using the ”relative”variables, (ηx, ηy) (using 2.11),

H =1

2mω2

C(η2x + η2

y), (2.13)

where the ”new” variables are nevertheless defined by the variables in phasespace, i.e. (x, y, px, py).

2.2. QUANTUM TREATMENT 23

2.2 Quantum treatment

The Hamiltonian formalism allows to introduce the canonical quantization,where one imposes the non commutativity of position with its conjugatemomenta, in terms of Planck’s constant h,

[x, px] = [y, py] = ih, [x, y] = [px, py] = [x, py] = [y, px] = 0.

Since [x, y] = 0, one sees immediately that [ηx, ηy] = −[X,Y ]. The fact thatthe guiding center components are constants of motion is expressed by [seealso Eq. (2.13)]

[X,H] = [Y,H] = 0. (2.14)

To compute the commutator between components ηx and ηy, one may usethe formula

[A, f(B)] =∂f

∂B[A,B]. (2.15)

That formula is valid for two arbitrary operators which commute with theircommutator, [A, [A,B]] = [B, [A,B]] = 0. One gets

[ηx, ηy] =e

m2ω2C

([px, Ay] − [py, Ax])

=1

eB2(∂xAy[px, x] − ∂yAx[py, y])

=−iheB

or, in terms of magnetic length lB ≡√

h/eB,

[ηx, ηy] = −il2B, [X,Y ] = il2B. (2.16)

The result is of course gauge invariant. A remarkable point is that the dynam-ics of a charged particle in a magnetic field is perhaps the simplest exampleof a non commutative geometry. Notice that, without any knowledge on theenergy level structure, the latter has to be degenerate. In any level chosen atrandom, each state must occupy a minimal surface given by the Heisenberguncertainty principle,

σ = ∆X∆Y = 2πl2B.

24 Charged particle in a magnetic field

In that sense, the real 2D space looks like the phase space of a 1D particle,where each state occupies a ”surface” 2πh. The level degeneracy may thusbe written directly in terms of this minimal surface: the number of statesper level and per unit surface being nB = 1/σ = B/φ0 – i.e. the flux densityin units of the flux quantum φ0 = h/e. Since electrons follow fermionicstatistics, each quantum state is occupied at most by one particle, because ofthe Pauli principle. When there are many electrons in the system, the fillingν of energy levels is thus described by the ratio between the electron surfacedensity nel and the flux density nB, ν = nel/nB. This ratio is also called thefilling factor.

The Hamiltonian form (2.13), along with the commutation relations (2.16),is that of a harmonic oscillator – ηx and ηy may be interpreted as conjugatevariables. To exhibit explicitly the harmonic oscillator structure, we intro-duce two sets of ladder operators, (a, a†) with

a =1√2lB

(ηx − iηy), a† =1√2lB

(ηx + iηy)

ηx =lB√2(a† + a), ηy =

lB√2i

(a† − a), (2.17)

and (b, b†) with

b =1√2lB

(X + iY ), b† =1√2lB

(X − iY )

X =lB√2(b† + b), Y =

ilB√2(b† − b), (2.18)

with [a, a†] = [b, b†] = 1 et [a, b(†)] = 0. In terms of ladder operators, theHamiltonian writes

H = hωC

(

a†a+1

2

)

. (2.19)

The energy spectrum is thus given by En = hωC(n + 1/2), where n is theeigenvalue of operator a†a. In the context of an electron in a magnetic fieldthe equidistant levels of the oscillator are called ”Landau Levels” (LL, seeFig. 2.2). Formally, in fact, the system may be viewed as a system of twoharmonic oscillators,

H = hωC

(

a†a+1

2

)

+ hω′(

b†b+1

2

)

,

Quantum treatment 25

m

1

3

2

4

n = 0

nive

aux

de L

anda

u

Figure 2.2: Landau Levels. The quantum number n labels the levels, and m , which isassociated to the guiding center, describes the level degeneracy.

where the frequency of the second oscillator vanishes, ω′ = 0. The secondquantum number m is the eigenvalue b†b.

The eigenstates are thus determined by the two integer quantum numbers,n and m, associated with the two species of ladder operators,

a†|n,m〉 =√n+ 1|n+ 1,m〉, a|n,m〉 =

√n|n− 1,m〉 (pour n >0);

b†|n,m〉 =√m+ 1|n,m+ 1〉, b|n,m〉 =

√m|n,m− 1〉 (pour m >0).

When n = 0 ou m = 0, one finds

a|0,m〉 = 0, b|n, 0〉 = 0, (2.20)

and negative numbers are prohibited. An arbitrary state may thus be con-structed with the help of ladder operators, starting from the state |0, 0〉,

|n,m〉 =(a†)n

√n!

(b†)m

√m!

|0, 0〉. (2.21)

The wave functions, which are the state representation in real space, dependon the gauge chosen for the vector potential.

2.2.1 Wave functions in the symmetric gauge

To find the real space representation of eigenstates, φn,m(x, y) = 〈x, y|n,m〉,wemust choose a gauge. Here we discuss the symmetric gauge [Eq. (2.10)],A = (B/2)(−y, x, 0); we must translate equations (2.20) and (2.21) in dif-ferential equations, using px = −ih∂x and py = −ih∂y. With the help of


equations (2.11) and (2.17), one finds the representation of ladder operatorsin the symmetric gauge

a =√

2(

z

4lB+ lB∂

)

, a† =√

2

(

z∗

4l2B− lB∂

)

b =√

2

(

z∗

4l2B+ lB∂

)

, b† =√

2(

z

4lB− lB∂

)

(2.22)

where z = x− iy is the electron position in the complex plane2, z∗ = x+ iyits complex conjugate, ∂ = (∂x − i∂y)/2 et ∂ = (∂x + i∂y)/2. A state in theLowest LL (LLL) is thus determined by the differential equation

(

z + 4l2B∂)

φn=0(z, z∗) = 0. (2.23)

The solution of equation (2.23) is a gaussian multiplied by an arbitrary an-alytic function f(z), with ∂f(z) = 0,

φn=0(z, z∗) = f(z)e−|z|2/4l2B , (2.24)

Similarly one finds for the state with m = 0

(

z∗ + 4l2B∂)

φm=0(z, z∗) = 0, (2.25)

the solution of which is

φm=0(z, z∗) = g(z∗)e−|z|2/4l2B , (2.26)

where the function g(z∗) is anti-analytic, ∂g(z∗) = 0. The state |n = 0,m =0〉 must thus be represented by a function which is both analytic and anti-analytic. The only function which satisfies both requirements is a constant.With the normalisation, one gets

φn=0,m=0(z, z∗) = 〈z, z∗|n = 0,m = 0〉 =

1√

2πl2Be−|z|2/4l2B , (2.27)

2The sign we chose for the imaginary part is unusual, but is convenient for electrons.For positively charged particles, we would chose the opposite sign, corresponding to theopposite chirality.


A state corresponding to the quantum number m in the LLL is found fromequations (2.20) and (2.22),

φn=0,m(z, z∗) =

√2

m

√

2πl2Bm!

(

z

4lB− lB∂

)m

e−|z|2/4l2B

=1

√

2πl2Bm!

(

z√2lB

)m

e−|z|2/4l2B , (2.28)

and

φn,m=0(z, z∗) =

√2

n

√

2πl2Bn!

(

z∗

4l2B− lB∂

)n

e−|z|2/4l2B

=1

√

2πl2Bn!

(

z∗√2lB

)n

e−|z|2/4l2B , (2.29)

for a state centered at the origin m = 0 in LL n. An arbitrary state writes

φn,m(z, z∗) =

√2

m

√

2πl2Bm!n!

(

z

4lB− lB∂

)m(

z∗√2lB

)n

e−|z|2/4l2B (2.30)

which generates the associated Laguerre polynomials [21].It is remarkable that, even if functions (2.28) and (2.29) have the same

probability density ,3

|φn=0,m=j(z, z∗)|2 = |φn=j,m=0(z, z

∗)|2 ∼(

|z|22

)je−|z|2/2l2B

j!,

with a probability maximum at radius r0 =√

2jlB (Fig. 2.3), they do notrepresent equal energy states.

To conclude the discussion of states |n = 0,m〉 represented in the sym-metric gauge, we compute the average value of the guiding center operator.With the help of equations (2.18), one finds that

〈R〉 ≡ 〈n = 0,m|R|n = 0,m〉 = 0,

but

〈|R|〉 =⟨√

X2 + Y 2⟩

= lB

⟨√

2b†b+ 1⟩

= lB√

2m+ 1. (2.31)

3It is a Poissonian distribution.


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 1 2 3 4 5 6

|φn,

m=

0(z,

z* )|2

r/lB=|z|/lB

(a) n=1n=3n=5

y/lB

y/lB y/lB

y/lB

x/lB x/lB

x/lB x/lB

-4 -2 0 2 4

-4

-2

0

2

4

-4 -2 0 2 4

-4

-2

0

2

4

-4 -2 0 2 4

-4

-2

0

2

4

-4 -2 0 2 4

-4

-2

0

2

4

(b)n=0 n=1

n=3 n=5

Figure 2.3: Probability density for a state |n,m = 0〉 for various values of n. (a) Thedensity depends only on the radius |z| = r and is maximum at r0 =

√2jlB . (b) When

plotted on the plane, the wave function for n ≥ 1 have a ring shape.


x x

x

p

x

p

y y(b)

(a)

x

p

y

x

0

0

0

0

<x,y|n=0,m=0>

<x,y|x ,y >0 0

Figure 2.4: Coherent states .

This means that both the particle and its guiding center are located onthe circle of radius lB

√2m+ 1, but the phase in undetermined. We may

use this to count states, as was done previously, for a disc geometry withradius Rmax and surface A = πR2

max: as the state with maximum radius hasRmax = lB

√2M + 1, this yields the number of states in the thermodynamic

limit, M = A/2πl2B = AnB, with nB = eB/h, in agreement with the previousargument about the state minimal surface. Similarly one sees that for thestate |n,m = 0〉 in level n the relative variable η is localized on a circle withradius

RC ≡ 〈|η|〉 = lB√

2n+ 1 (2.32)

which is also called the cyclotron radius.

2.2.2 Coherent states and semi-classical motion

To retrieve the classical trajectory, (2.4), one must construct semi-classicalstates, also called coherent states because they play an important role inquantum optics. For a 1D harmonic oscillator, a coherent state is the eigen-state of the annihilation operator and it is the state with the minimum value


of the product ∆px∆x. Such as state can be built from the displacement op-erator in phase space, which displaces the ground state from 〈x〉 = 0, 〈px〉 = 0to (x0, p0) [Fig. 2.4(a)]. The displacement is done with the operator

D(x0, p0) = e−i(x0p−p0x), (2.33)

where the symbols with hats are operators, not to be confused with variablesx0 and y0. This operator displaces variables, as we can check using formula(2.15)

D†(x0, p0)xD(x0, p0) = eix0pxe−ix0p = x+ x0

andD†(x0, p0)pD(x0, p0) = e−ip0xpeip0x = p+ p0.

The coherent state writes

|x0, p0〉 = D(x0, p0)|n = 0〉, (2.34)

where |n = 0〉 is the 1D harmonic oscillator ground state. Since [D(x0, y0), H] 6=0 the coherent state is not an eigenstate of the Hamiltonian. Indeed the statechanges with time, and that is how we retrieve the trajectory in phase space[Fig. 2.4(a)]. x0 and p0 are not bona fide quantum numbers – this wouldcontradict the fundamental postulates of quantum mechanics, because theassociate operators do not commute. The basis |x0, p0〉 is said to be ”over-complete” [22].

In general, a displacement operator may be constructed from two conju-gate operators, which therefore do not commute. In the case of an electronin a magnetic field in a 2D plane, we have two pairs of non commuting con-jugate operators at our disposal, [X,Y ] = il2B et [ηx, ηy] = −il2B. With thefirst choice, the displacement operator which acts now in real space, writes

D(X0, Y0) = e− i

l2B

(X0Y −Y0X), (2.35)

and the coherent state (in the LLL) is

|X0, Y0;n = 0〉 = D(X0, Y0)|0, 0〉, (2.36)

where |0, 0〉 ≡ |n = 0,m = 0〉. Since the guiding center is a constant of mo-tion, the displacement operator D(X0, Y0) commutes with the Hamiltonian.The state (2.36) remains an eigenstate of the Hamiltonian, which is why thequantum number n is unchanged.


The dynamics enters with the second pair of operators, with the displace-ment operator

D(ηx0 , η

y0) = e

i

l2B

(ηx0 ηy−ηy

0 ηx), (2.37)

which generates a displacement to position η0 = (ηx0 , η

y0), so that a general

semi-classical state may be written as

|X0, Y0; ηx0 , η

y0〉 = D(ηx

0 , ηy0)D(X0, Y0)|0, 0〉. (2.38)

The guiding center is thus centered at R0 = (X0, Y0), and the electron turnsaround that position on a circle of radius r = |η0|. One retrieves thus themotion represented on figure 2.1, in terms of a gaussian wave packet. To provethose dynamic properties, remember that a coherent state is an eigenstate ofthe ladder operator a, and in our case also of b, with

a |X0, Y0; ηx0 , η

y0〉 =

ηx0 − iηy

0√2lB

|X0, Y0; ηx0 , η

y0〉,

b |X0, Y0; ηx0 , η

y0〉 =

X0 + iY0√2lB

|X0, Y0; ηx0 , η

y0〉. (2.39)

This can be checked, for example, when expressing the displacement opera-tors in terms of ladder operators (2.17) and (2.18),

D(X0, Y0) = eβb†−β∗b = e−|β|2/2eβb†e−β∗b,

D(ηx0 , η

y0) = eαa†−α∗a = e−|α|2/2eαa†

e−α∗a, (2.40)

ou l’on a defini

β ≡ X0 + iY0√2lB

, α ≡ ηx0 − iηy

0√2lB

where we used the Baker-Hausdorff formula

eA+B = eAeBe−[A,B]/2, (2.41)

which is valid when [A, [A,B]] = [B, [A,B]] = 0. The coherent state writesthus

|X0, Y0; ηx0 , η

y0〉 = e−(|α|2+|β|2)/2eαa†

eβb†|0, 0〉and we find with formula (2.15)

a |X0, Y0; ηx0 , η

y0〉 = e−(|α|2+|β|2)/2

[

a, eαa†]

eβb†|0, 0〉 = α |X0, Y0; ηx0 , η

y0〉,

b |X0, Y0; ηx0 , η

y0〉 = e−(|α|2+|β|2)/2eαa† [

b, eβb†]

|0, 0〉 = β |X0, Y0; ηx0 , η

y0〉,


which is nothing but equation (2.39).In order to get the time evolution of the coherent state |α, β〉 = |X0, Y0; η

x0 , η

y0〉,

one uses the time evolution operator on state

|α, β〉(t) = e−ih

Ht|α, β〉(t = 0)

= e−(|α|2)/2e−ih

Ht∞∑

n=0

(αa†)n

n!|n = 0, β〉(t = 0)

= e−(|α|2)/2e−ih

Ht∞∑

n=0

(α)n

√n!

|n, β〉(t = 0)

= e−(|α|2)/2e−iωCt/2∞∑

n=0

(αe−iωCt)n

√n!

|n, β〉(t = 0)

= e−iωCt/2|α(t = 0)e−iωCt, β〉, (2.42)

which yields for the eigenvalue time evolution

α(t) = α(t = 0)e−iωCt, β(t) = β(t = 0). (2.43)

Since X0 =√

2lBRe[β], Y0 =√

2lBIm[β], ηx0 (t) =

√2lBRe[α(t)] et ηy

0(t) =−√

2lBIm[α(t)], we retrieve

ηx0 (t) = ηx

0 (t = 0) cos(ωCt), ηy0(t) = ηy

0(t = 0) sin(ωCt)

and thus the trajectory given in equation (2.4), identifyingr = |η0| and R = (X0, Y0), as mentionned earlier.

Chapter 3

Transport properties– IntegerQuantum Hall Effect (IQHE)

This chapter deals with some aspects of the Integer Quantum Hall Effect(IQHE) physics, using the quantum mechanics of an electron in a constantuniform magnetic field, described in the previous chapter. Two main featuresallow to understand the IQHE :

• each completely filled LL (for ν = n) contributes a conductance quan-tum e2/h to the electronic conductivity,

• when additional electrons start populating the next LL at ν 6= n, theyget localized by the impurities disorder potential in the sample, andthey do not contribute to transport. In the absence of impurities, or,more precisely, if translation invariance is not broken in the sample,no plateau can be formed in the Hall resistance, and the classical Hallresult is preserved.

This last feature seems analogous at first sight to Anderson localization in2D in the absence of a magnetic field [23]. It happens that localisation iseven more relevant in a magnetic field.

3.1 Resistance and resistivity in 2D

Theorists calculate resistivity. Experiments measure resistance. For a clas-sical sytem with the shape of a hypercube of edge length L in d dimensions,

33

34 Transport properties– IQHE

the resistance R and the resistivity ρ are related by the well known equation

R = ρL2−d (3.1)

Thus, in two dimensions, the sample resistance is scale invariant. Theproduct R(e2/h) is dimensionless. It is an easy exercise to show that in thecase of a Hall bar geometry, such as shown in Fig. (1.1), the transverse resis-tance and the transverse resistivity are equal in 2D, independent of the Hallbar dimensions. This is a basic ingredient to understand the universality ofthe quantum Hall experimental results. In particular it means that one doesnot have to measure the physical dimensions of a sample to one part in 1010 inorder to obtain the resistivity to that accuracy. The technological progressin semiconductor physics which allowed to manufature 2D Electron Gases(2DEG) with electrical contacts was, in this respect, a decisive one. Eventhe shape of the sample, or the accurate determination of the Hall voltageprobe locations are almost completely irrelevant, in particular, because thedissipation is nearly absent in the QH states.

3.2 Conductance of a completely filled Lan-

dau Level

We first discuss the effect of a constant uniform electric field on the Landaulevel energy structure. We take the electric field along the y direction.

It is convenient to deal with a sample with rectangular shape, and to as-sume in a first step (to be relaxed subsequently) that the system is translationinvariant in the x direction.

An appropriate gauge in this rectangular geometry is the Landau gaugeAL = B(−y, 0, 0), so that the Hamiltonian now writes

H =(px − eBy)2

2m+

p2y

2m− eV (y),

where the potential is V (y) = −Ey (electric field pointing in the y direc-tion). The system is translation invariant in the x direction, so that px, orequivalently hk, is a constant of motion, which corresponds to the quantumnumber m in the symmetric gauge, i.e. to the guiding center eigenvalue. Thelatter, in a state |n, k〉, is delocalized along a straight line in the x direction,

Conductance of a filled LL 35

y=kl

contact contactL R

y’

éner

gie

NL n

µ µLR

B2

maxmink k

Figure 3.1: LL in the Landau gauge, with a voltage bias between the L and the Rcontacts, where chemical potentials are respectively µL and µR. Position yk in the directiony is proportionnal to the wave vector k in direction x : yk = kl2B .

with coordinate kl2B on the y axis. The Hamiltonian is:

H =p2

y

2m+

1

2mω2

C(y + kl2b )2 + eEy,

which can be re-written, completing the square, as:

H =p2

y

2m+

1

2mω2

C(y − yk)2 + hk

E

B+ C,

where we have set px = hk, and yk = −kl2B − eE/mω2C and C is a constant:

C = −12m(

EB

)2. The energy of the state |n, k〉 is

εn,k = hωC

(

n+1

2

)

+ eEyk +1

2mv2 (3.2)

where v ≡ −EB

is the drift velocity ~E ∧ ~B/B2 and is parallel to the x axis.This can be derived by deriving explicitly the current:

⟨

~J⟩

=−em

〈n, k|(p + eA)|n, k〉 .

Thus there is a net current 〈Jx〉 along the x axis.

36 Transport properties– IQHE

We conclude that the energy levels follow the electric potential, whichadds to the energy in zero electric field.

Let us now go one step further by considering a slowly varying electricpotential V (y), which we still assume to be translation invariant along x.We can linearize this potential locally, and repeat the previous analysis: theenergy eigenvalues will not be linear in k any more, but they will roughlyreflect the sum of the LL energy plus the local potential energy. To discusselectrons in a Hall bar, we take into account the sample edges in the ydirection, which create a confinement potential. The latter results in anupward bending of Landau levels in the vicinity of the edges, where contactsallow to measure voltage biases as in figure 3.1.

This justifies the sketch of the LL in figure 3.1, where the LL energyprofile follows the confining potential at the sample edges. The eigenvaluesǫk are not linear in k, but can be linearized locally: it will still reflect thekinetic energy, with the local potential energy added to the LL energy.

In order to compute the level contribution to the conductance (along x),we use formula

In = − e

L

∑

k

〈n, k|vx|n, k〉, (3.3)

where L is the system length along x, and the velocity average value is derivedfrom the energy dispersion relation

1

h

∂εn,k

∂k≃ 1

h

∆εn,k

∆k.

In the last line, we assume that ∆k = 2π/L is very small, which is certainlyvalid if L is very large. Using this, we have

vk =L

2πh∆εn,k =

L

2πh(εn,k+1 − εn,k).

Thus vk has opposite signs on the two edges of the sample. This meansthat in the Hall bar geometry, there are edge currents flowing in oppositedirections. This is not surprising, if we remember the semi-classical pictureof skipping orbits along an edge.

When we sum over vk in equation (3.3, the result depends only on theedge energies, at kmin and kmax, the edge coordinates (see figure ( 3.1)).Thus, provided the electric potential has a slow enough variation in directiony, we may sum over k to get

In = − e

h(εn,kmax − εn,kmin

) .

3.3. LOCALISATION IN A STRONG MAGNETIC FIELD 37

The energies at kmin and kmax are given by the chemical potential at the con-tact points, εn,kmin

= µL and εn,kmax = µR. Since the difference in chemicalpotentials is controlled by a voltage bias ∆µ = (µR − µL) = −eV , we seethat the LL conductance is e2/h, since

In =e2

hV. (3.4)

When n LL are completely filled, we get a conductance

G = ne2

h.

Since this is a transverse conductance (the current is in direction x, is zeroalong y, and the difference in chemical potentials is in direction y), the re-sistance tensor we get is

R = G−1 =

(

0 −RH

RH 0

)

, (3.5)

with the Hall resistance RH = h/e2n. It is important to realize that thisresult, although satisfactory –the Hall resistance only depends on universalconstants e and h, and an integer n–is not sufficient to explain the occur-rence of plateaux. In fact, it is fairly easy to show that the result we havecoincide exactly with the classical Hall value at discrete points in the RH

curve, corresponding to n filled Landau levels; it is enough to remember thatν = hnel/eB = n and to use equation (3.5)to recover the classical valueRH = B/enel. In order for quantized Hall plateaux to be formed, addition-nal electrons or holes injected in the system around a density such that nLL are completely filled must be localized, so as to have no contribution totransport properties. This localization phenomenon is described in the nextsection.

3.3 Localisation in a strong magnetic field

The electric potential Vext(r = R + η) due to impurities is described asa slowly varying function in the xy plane, so that Landau quantization ispreserved. We now do not assume any more that the impurity potential pre-serves translation invariance along the x direction. The potential landscape

38 Localisation in a strong magnetique field

Figure 3.2: Semi-classical motion of an electron in a magnetic field in the presenceof an impurity potential. The guiding center follows the landscape equipotentials. TheHall drift of the guiding center, shown by the arrow is a slow motion compared to thefast electronic cyclotron motion. Electronic transport is possible when an equipotentialconnects the sample edges. If an electronic state is localized within a potential well, itdoes not contribute to transport.

has hills and valleys and fluctuates in space around an average value whichis taken to be zero, with no loss of generality. This potential lifts the LLdegeneracy, because the guiding center is not a constant of motion any more.We see this with the Heisenberg equations of motion

ihX = [X,H] = [X,Vext(X,Y )] = il2B∂Vext

∂Yand ihY = −il2B

∂Vext

∂X,

(3.6)where we used formula (2.15). We see that the guiding center follows theequipotential lines of the impurity potential (Fig. 3.2). In the case we dis-cussed in the previous section, this led to a Hall current in the directionorthogonal to the electric field. Equation (3.6) is a generalization of thisresult. The guiding center motion is perpendicular both to the external fieldand to the local electric field. Quantum states of the LL are thus localizedon equipotential lines corresponding to their energies. The wave functions,(in the shape of rings in zero potential as in Fig. 2.3) are deformed to tuneto their equipotential lines.

Similarly, we get for the ηx et ηy Heisenberg equations of motion

ihηx =[

ηx,1

2mωC(η2

x + η2y) + V (r + η)

]

⇔ ηx = −ωCηy −l2Bh

∂V

∂ηy


et ηy = ωCηx +l2Bh

∂V

∂ηx

.

This provides us with a stability criterion for Landau levels in the presenceof a disorder impurity potenial, since the first terms on the right in the equa-tions above must remain large compare with the terms due to the impuritypotential. The condition reads:

⟨

∂V

∂η

⟩

≪ hωC

lB. (3.7)

This condition is satisfied provided the variation of the potential over a cy-clotron radius is small compared to hωC .

We now have the main ingredients to understand some of the IQHE basicfeatures. The reasoning below is represented in a schematic fashion on figure3.3. We have seen in the previous section that the Hall resistance for integerν = n is exactly RH = h/e2n, while the longitudinal resistance is zero. If themagnetic field intensity is slightly decreased, keeping the electronic densityconstant, since the number of states per LL decreases, some electrons have topromoted to the LL with n+1. They occupy preferentially the lowest energystates available, the bottom of basins in the impurity potential landscape.This is a peculiar form of localisation which is induced by the magnetic field.Localized electrons do not contribute to the transport. Both the transverseand longitudinal resistances stay locked at their value for the completelyfilled level case with ν = n. The fact that the longitudinal resistance is zeroshows that transport is ballistic. Indeed, we have seen that contributionsto the current from the bulk of the sample compensate, so that transportis due to n edge channels (edge states), one per completely filled LL. Thecurrent direction on the edge is determined by the potential gradient, whichrises near the sample edge. Edge currents are thus chiral, forward scatteringat one edge is dissipation less, and dissipation can only occur if an electroncirculating along one edge can be scattered backwards by tunneling to theother edge. This can occur only when edge states trajectories of oppositeschiralities happen to be close to each other. When electronic puddles growbecause more electrons get promoted to level n + 1, they eventually mergeinto one another, until equipotentials connect the two edges and eventuallyan electronic sea extends over the whole sample. When edges get connectedby equipotentials, dissipation occurs, the longitudinal resistance is finite,and the transverse resistance varies rapidly as the magnetic field continues

40 Localisation in a strong magnetique field

n

ε

n

ε

n

ε

(n+1)

ν

NL

(a) (b) (c)

états localisés

états étendus

densité d’états densité d’états densité d’états

RR Rxy xyxyxx xxRR

BBB

Rxx

=n

h/e n2

h/e (n+1)2h/e n2

FE EFEF

Figure 3.3: Quantum Hall effect. In the upper parts of the figure, LL are broadened bythe impurity potential. Their filling is controlled by the Fermi level(EF ). In the middlepart, samples are seen from above, showing equipotential lines, and the gradual filling ofthe n-th level (from left to right). The lowest part of the figure is a sketch of the resistancecurves, as the LL filling factor varies. This figure is to be read column by column, the fillingfactor increasing from the first column to the last one. In the first column (a), we have asituation with completely filled LL, ν = n, where the Fermi level sits exactly between LLn and n+1, the upper level being empty. The Hall resistance is then exactly RH = h/ne2,and the longitudinal resistance is exponentially small (zero at zero temperature). Thesecond column describes a situation where the LL n + 1 has a low filling factor. Electronsoccupy potential wells in the sample and do not contribute to electronic transport. Thissituation occurs when, at fixed electron density, the magnetic field intensity is slightlydecreased from its value for the complete filling of the n-th LL. The resistance values arelocked at their value for ν = n. In the last column on the right, the n + 1 LL is halffilled: equipotential lines connect the two edges, so that dissipation is allowed throughback scattering processes from one edge to the other. The system changes from ballisticregime to a diffusive one, and the Hall resistance varies rapidly towards the next plateau,while the longitudinal resistance reaches it peak value, before decreasing to exponentiallysmall values. Localized states are on the left and on the right of the extended states inthe center of the broadened LL. When the highest occupied LL has a filling larger than1/2, the same reasoning applies in terms of holes.


I I

R ~

56

2 3

41

R ~ µ − µ = µ − µ

3µ − µ = 0

2

35

L

H

2µ = µ

L 3µ = µL

µ = µ = µ6 5 R

R L

Figure 3.4: IQHE measurements at ν = n. The current I is injected through contact1, and extracted at contact 4. Between those two contacts, the chemical potential µL isconstant since (a) there is no backscattering and (b) there are no electrons injected orextracted at contacts 2 and 3 which are used to measure the voltage drop. The chemicalpotential µR stays also constant along the lower edge between contacts 6 and 5. Thelongitudinal voltage drop thus vanishes, so that the longitudinal resistance is zero, RL =(µ3 − µ2)/I = 0. The Hall resistance is determined by the voltage bias between the twoedges µ5 − µ3 = µR − µL.

to decrease until a new conducting channel is formed all along the edges.Thisis reached at half filling of level n+ 1. Above that filling ratio, the evolutiondescribed so far is reproduced in terms of holes.

To understand the IQHE in terms of edge currents, consider the experi-mental set up with six contacts, as shown on figure (3.4).

Electrons are injected through electrode 1 and are extracted through elec-trode 4. The other contacts 2, 3, 5 and 6 are used for voltage bias measure-ments, with no electron injected or extracted. Because back-scattering issuppressed when the filling factor is around ν = n, the chemical potentialsµR and µL are constant along each edge. The chemical potential varies onlyalong input and output electrode 1 and 4. The longitudinal resistance ismeasured for instance between contacts 2 and 3, and is found to vanish:RL = −(µ3 − µ2)/eI = 0. The Hall resistance is determined by the volt-age bias between contacts 3 and 5, RH = −(µ5 − µ3)/eI = −(µR − µL)/eI.This situation is precisely that which was described in the previous section,

42 Transitions between plateaus – percolation

where we computed the resistance for n completely filled LL. We thus findRH = h/ne2.

3.4 Transitions between plateaus – The per-

colation picture

The previous section describes a scenario of transitions between Hall plateauswhich reminds us closely of a percolation mechanism: the resistance jumpsfrom one plateau to the next one when the electron puddles become macro-scopic ones and percolate so as to form an infinite electronic sea which extendsto both edges. Percolation transitions are second order transitions, which ex-hibit critical phenomena, and specific scaling laws for the relevant physicalquantities around the critical point. Those quantities do not depend on mi-croscopic details of the system, they are characterized by critical exponentswhich define a universality class.

The transition is controlled by a ”control parameter” K, which could bethe temperature, or, in the case of quantum phase transitions, at zero tem-perature, by another parameter such as pressure or electronic density [18]. Inour case, the control parameter for transitions between plateaus is the mag-netic field intensity. At the critical field Bc, the correlation length diverges,with a critical exponent ν (not to be confused with the filling parameter)

ξ ∼ |δ|−ν , (3.8)

where δ ≡ (B − Bc)/Bc. Dynamic fluctuations may be similarly describedby a correlation ”time”

ξτ ∼ ξz ∼ |δ|−zν , (3.9)

where z is called the critical dynamic exponent. In a path integral formu-lation, the characteristic time τ is connected to the temperature T throughh/τ = kBT . A finite temperature may be considered as a finite size in thetime direction. [25, 24].

At the critical point, physical quantities follow scaling laws which dependon ratios of dimensionless quantities. One finds for the longitudinal and Hallresistivity

ρL/H = fL/H

(

hω

kBT,τ

ξτ

)


0.10 1.00T(K)

1.0

10.0

100.0

(∆B)−1

(∆B

)−1

N = 1N = 1

N = 0N = 1N = 1dxydB max dxydB max

Figure 3.5: Experiments by Wei et al. [26]. The transition width δB and that ofthe Hall resistivity derivative ∂ρxy/∂B, measured as a funcition of temperature exhibit ascaling law with exponent 1/zν = 0, 42±0, 04, for transitions between filling factors 1 → 2(N = 0 ↓), 2 → 3 (N = 1 ↑) and 3 → 4 (N = 1 ↓).

44 Transitions between plateaus – percolation

= fL/H

(

hω

kBT,δzν

T

)

, (3.10)

where ω is a characteristic measurement frequency, for example in an ac mea-surement, and fL/H(x) are universal functions. In the following, we deal onlywith dc measurements, and ω = 0 properties. For a second order transition,one expects the characteristic width ∆B of the transition as a function ofthe magnetic field intensity, to vary with temperature as

∆B ∼ T 1/zν . (3.11)

This scaling law was actually found in the measurements by Wei et al. [26],who found an exponent 1/zν = 0, 42 ± 0, 04 over a temperature intervalvarying with more than one order of magnitude between 0, 1 and 1, 3K (figure3.5).

The two exponents ν and z may be separetly determined if one takes intoaccount the scaling laws for current fluctuations under applied electric field.One finds

eEℓE ∼ h

τE∼ h

ℓzE,

where τE ∼ ℓzE is the characteristic fluctuation time, which is connectedto a characteristic length ℓE through equation (3.9). One finds thus ℓE ∼E−1/(1+z), and, for the zero frequency resistivity scaling law

ρL/H = gL/H

(

δ

T 1/zν,

δ

E1/ν(1+z)

)

, (3.12)

in terms of universal functions gL,H(x). Other measurements by Wei et al.,dealing with the current scaling laws, find that z ≃ 1, which leads to ν ≃2, 3 ≃ 7/3 [27].

The critical exponent for classical 2D percolation is νp = 4/3, smallerthan the experimental value, close to 7/3. The disagreement is probablydue to quantum tunneling effects between trajectories: such processes allowfor back-scattering before classical trajectories actually touch each other.Chalker and Coddington take into account quantum tunneling in a transfermatrix approach and find a critical exponentν = 2, 5 ± 0, 5 [29]. Numericalsimulations have reproduced the ν = 7/3 exponent [28].

Chapter 4

The Fractional Quantum HallEffect (FQHE)– FromLaughlin’s theory to CompositeFermions.

In the previous chapter, we have seen that the IQHE with ν = n is understoodon the basis of two main ingredients:

(i) Because of LL quantization, there is an excitation gap between theground state with a number of completely filled levels and the next emptylevel (ii) Elementary excitations, obtained by promoting an electron to thenext LL are localized and do not contribute to electronic transport. FilledLandau levels only contribute each a conductance quantum e2/h.

As emphasized in the Introduction, the observation of the FQHE, firstfor a fractional filling of the LLL, with ν = p/(2sp + 1), with integer n andp, was a sign of the complete breakdown of perturbation theory, such asdiagrammatic analysis based on the knowledge of an unperturbed groundstate. For the fractionally filled LLL, the non interacting ground state hasa huge degeneracy, which prohibits using theoretical techniques used so farto take electron-electron interactions into account. We know that certainsuperpositions of ground state configurations must minimize the Coulombinteractions. We know from experiments–the observed activated behaviourof the longitudinal resistance– that there is a gap between the actual groundstate and the first excited states.

The approach described in the previous chapter allows us to deal with

45

46 FQHE – from Laughlin to Composite Fermions

the FQHE, once we have a mechanism which allows to lift the ground statedegeneracy, and to have a gap to the first excited states, be they singlequasi-particles or collective excitations. In that case, we may reproduce thepiece of reasoning of the previous chapter: excited charged quasi-particles arelocalized, so that varying the magnetic field intensity around a given exactfraction of the filling factor leads to plateau formation in the Hall resistivity,for the same reason as in the IQHE. The difficult part is to identify the nondegenerate ground states, and to characterize their properties, the natureof excited states, etc.. Before actually introducing the Laughlin and Jaintrial wave functions which solve this problem, we discuss in the next sectionthe structure of the effective model which describes the electron dynamicswhen we make the approximation that it is restricted to the states of a singlepartially filled LL. This model will be the basis for the Hamiltonian theoryof the FQHE, which will be developped in the following chapters.

4.1 Model for electron dynamics restricted to

a single LL

Since we are interested at first in describing low temperature properties, onlythe lowest excitation energies are of interest. Because we are dealing witha partially filled LL,(ν 6= n), the relevant excitations are restricted to intra-level dynamics. Furthermore, we consider at first that the spins are fullypolarized, and we do not consider spin flip excitations here. Excitations in-volving intra-LL transitions are forbidden by the Pauli principle when thelevel is completely filled (Fig. 4.1). They are allowed for a partially filled LL.In that case, the kinetic energy plays no role, since all single electron statesare degenerate. We omit this constant in the following. Virtual inter-levelexcitations may be considered in a perturbative approach, and give rise toa modified dielectric function ǫ(q), which alters the interaction potential be-tween electrons within the same level [30]. In contrast with screening effectsin metals, which suppress the long distance part of the Coulomb interaction,electronic interactions screening in the presence of a magnetic field alters thepotential only for finite wave vectors: for q → 0 and q → ∞, it vanishes, andǫ(q) → ǫ, where ǫ is the dielectric constant of the underlying semiconductor.Since the electron spin is not flipped during such processes within a spinbranch, it plays no role. Therefore, we deal in the following with spinless

electrons restricted to a single LL 47

h ωC

∆Z

ν = ν =N N(a) (b)

Figure 4.1: Lowest excitations energies. Each LL is separated in two spin branchesbecause of the Zeeman effect. (a) For filling ν = n, excitations which couple states withinthe same LL are forbidden because of the Pauli principle. Only inter level excitations areallowed. (b) For fractional filling of the highest occupied LL, (ν 6= n) excitations withinthe same LL are allowed and provide the lowest excitation energies. Inter LL excitations,with energy hωC or ∆z are neglected in the model.

electrons. A more detailed discussion of spin phenomena in the quantumHall physics–such as the Quantum Hall ferromagnetism– will be given in alater part of these lectures.

In second quantized notation, the Hamiltonian restricted to intra LL ex-citations is

H =1

2

∫

d2r d2r′ ψ†n(r)ψn(r)V (r − r′)ψ†

n(r′)ψn(r′). (4.1)

It involves states in the n-th level only, ψn(r) =∑

m〈r|n,m〉en,m et ψ†n(r) =

∑

m〈n,m|r〉e†n,m. Operators en,m and e†n,m are respectively the annihilationand creation operators for an electron in the state |n,m〉. They obey thefermionic anti-commutation rules,

en,m, e†n′,m′

= δn,n′δm,m′ , en,m, en′,m′ = 0. (4.2)

Note that the restricted electron fields ψn(r) are not completely localised.Because the sum over states is restricted to m, we have, with rules (4.2)

ψn(r), ψ†n(r′)

=∑

m

〈r|n,m〉〈n,m|r′〉 ∝ e−|r−r′|2/2l2B 6= δ(r − r′). (4.3)

The field ψ†n(r) creates an electron in the vicinity (4.2) of position r, which

is hardly surprising: this is just another manifestation of the position uncer-tainty when we restrict the dynamics to a single LL. To have a perfect field


localisation, one would have to sum over n, i.e. to superpose a number ofLL.

In reciprocal space, the Hamiltonian writes

H =1

2A

∑

q

v(q)ρn(−q)ρn(q), (4.4)

with the measure∑

q = A∫

d2q/(2π)2 and the Coulomb interaction potentialv(q) = 2πe2/ǫq. The operators ρn(q) are the Fourier components of theelectronic density operator in the n-th level ρn(r) = ψ†

n(r)ψn(r), and one has

ρn(q) =∫

d2r∑

m,m′〈n,m|r〉e−iq·r〈r|n,m′〉e†n,men,m′

=∑

m,m′〈n,m|e−iq·r|n,m′〉e†n,men,m′

= 〈n|e−iq·η|n〉∑

m,m′〈m|e−iq·R|m′〉e†n,men,m′

= Fn(q)ρ(q), (4.5)

where Fn(q) ≡ 〈n| exp(−iq · η)|n〉 is the form factor, and we took advantageof the decomposition r = R + η, which allows to factorize matrix elements

〈n,m|e−iq·r|n′,m′〉 = 〈n| exp(−iq · η)|n′〉 ⊗ 〈m| exp(−iq · R)|m′〉. (4.6)

In the last line of equation (4.5), we have defined the projected density op-erator,

ρ(q) ≡∑

m,m′〈m|e−iq·R|m′〉e†n,men,m′ . (4.7)

4.1.1 Matrix elements

To proceed in practice to actual computations, one needs to compute thematrix elements which enter expression (4.5) for the density operator. Thesimplest way is to use expressions (2.17) and (2.18) for the operators R andη, in terms of a, a†, b and b†. From now on, we take lB ≡ 1 for simplicity.Using complex notation, with q = qx − iqy and q∗ = qx + iqy, we have

q · η =1√2

(

qa+ q∗a†)

, q · R =1√2

(

q∗b+ qb†)

,

electrons restricted to a single LL 49

so that we get for the first matrix element, with n ≥ n′, using the Baker-Hausdorff formula(2.41),

〈n|e−iq·η|n′〉 = 〈n|e−i√2(q∗a†+qa)|n′〉

= e−|q|2/4〈n|e−i√2q∗a†

e− i√

2qa|n′〉

= e−|q|2/4∑

j

〈n|e−i√2q∗a†

|j〉〈j|e−i√2qa|n′〉

= e−|q|2/4

√

n′!n!

(−iq∗√2

)n−n′ n′∑

j=0

n!

(n − j)!(n′ − j)!j!

(

−|q|22

)n′−j

= e−|q|2/4

√

n′!n!

(−iq∗√2

)n−n′

Ln−n′n′

(

|q|22

)

, (4.8)

where we have used

〈n|e−i√2q∗a†

|j〉 =

0 pour j > n√

n!j!

1(n−j)!

(

− i√2q∗)n−j

pour j ≤ n

in the third line and the definition of Laguerre polynomials [21],

Ln−n′n′ (x) =

n′∑

m=0

n!

(n′ −m)!(n− n′ +m)!

(−x)m

m!.

Similarly we find for m ≥ m′

〈m|e−iq·R|m′〉 = 〈m|e−i√2(qb†+q∗b)|m′〉

= e−|q|2/4

√

m′!

m!

(

−iq√2

)m−m′

Lm−m′m′

(

|q|22

)

. (4.9)

Defining functions

Gn,n′(q) ≡√

n′!

n!

(

−iq√2

)n−n′

Ln−n′n′

(

|q|22

)

,

one may also write without the conditions n ≥ n′ et m ≥ m′,

〈n|e−iq·η|n′〉 = [Θ(n− n′)Gn,n′(q∗) + Θ(n′ − n− 1)Gn′,n(−q)] e−|q|2/4 (4.10)


and

〈m|e−iq·R|m′〉 = [Θ(m−m′)Gm,m′(q) + Θ(m′ −m− 1)Gm′,m(−q∗)] e−|q|2/4.(4.11)

For the case n = n′, we find in equation (4.10) the n-th LL form factor:

Fn(|q|) ≡ 〈n|e−iq·η|n〉 = Ln

(

|q|22

)

e−|q|2/4. (4.12)

4.1.2 Projected densities algebra

At first sight, the model defined by (4.4) looks simple. The Hamiltonianis quadratic in density operators. Such models often have exact solutions.It happens that the projection in a single LL generates a non commutativealgebra for operators with different wave vectors, which leads to non trivialquantum dynamics.

Let us compute the commutator [ρ(q), ρ(k)]. For a one particle operatorin second quantized notation, FA(q) =

∑

λ,λ′ fAλ,λ′(q)e†λeλ′ , where fA

λ,λ′(q) =〈λ|fA(q)|λ′〉, the commutation rules in second quantized form follow fromthose in first quantization:

[

FA(q), FB(q′)]

=∑

λ,λ′

[

fA(q), fB(q′)]

λ,λ′ e†λeλ′ . (4.13)

The λ index may comprise a number of different quantum indices. Thisequation follows from the repeated application of

[AB,C] = A[B,C]± − [C,A]±B (4.14)

on electronic operators. Equation (4.14) is valid for commutators as well asanti-commutators.

Using equation (2.16), one finds

[q · R,q′ · R] = qxq′y[X,Y ] + qyq

′x[Y,X]

= i(qxq′y − qyq

′x) = −i(q ∧ q′),

where we have defined q ∧ q′ ≡ −(q × q′)z, and one gets, with the help ofthe Baker-Hausdorff formula (2.41)

[

e−iq·R, e−iq′·R]

= e−i(q+q′)·R(

ei2q∧q′ − e−

i2q∧q′)

= 2i sin

(

q ∧ q′

2

)

e−i(q+q′)·R. (4.15)

4.2. THE LAUGHLIN WAVE FUNCTION 51

This yields, with equation (4.13),

[ρ(q), ρ(k)] = 2i sin

(

q ∧ k

2

)

ρ(q + k), (4.16)

for the algebra of projected density operators. This is isomorphous to themagnetic translation algebra . Indeed, operators which describe electronicdisplacements in the presence of a magnetic field have the same commutationrules. This algebra is closed, and does not depend on the LL n index.

With algebra (4.16), the model is completely defined by the Hamiltonian(4.4), which writes, in terms of projected density operators

H =1

2A

∑

q

vn(q)ρ(−q)ρ(q), (4.17)

where the form factor has been absorbed in the effective interaction potentialin the n-th LL ,

vn(q) =2πe2

ǫ|q| [Fn(q)]2 =2πe2

ǫ|q|

[

Ln

(

|q|22

)]2

e−|q|2/2. (4.18)

The model has the same structure for all LL. The information about thelevel is encoded in the effective potential, which will be discused in the lastsection of this chapter. The LLL physics, which will be the main topic inthe remaining parts of this chapter (except the last section), is thus easilygeneralized to a LL with higher index: one simply has to take into accountthe relevant effective potential, and to replace the filling factor ν by thepartial filling factor of the n-th level, ν = ν − n.

4.2 The Laughlin wave function

In this section, we discuss the arguments used by Laughlin in 1983 to de-rive the almost exact ground state for the fractionally filled LLL (LowestLandau Level), to prove that there is a gap between the ground state andall excited states, and that there exist factionally charged excitations aroundthe fractional filling corresponding to the plateaus observed by Tsui, Stormerand Gossard. Then we will describe Jain’s generalization of Laughlin’s wavefunctions.

52 FQHE – From Laughlin to Composite Fermions

It is a good training to examine first the many-body wave function for thecompletely filled LLL. In that case there is a gap to excited state which is,at first sight, a single particle effect, the Zeeman splitting g∗µbB (see figure4.1).1

4.2.1 The many-body wave function for ν = 1

Laughlin exploited a useful property of the single particle Landau Hamil-tonian eigenfunctions in the symmetric gauge (see equation 2.28) :

φn=0,m(z, z∗) ∝(

z√2lB

)m

e−|z|2/4l2B ,

so that any analytic function f1(z)(defined by ∂∂z∗f1(z, z∗) = 0 ) in the pref-

actor of the gaussian belongs to the LLL. All physical results are of courseindependent of this gauge choice.

Turning now to the many-body wave function for the full LLL (i.e. ν = 1),this means that the most general wave function we are looking for has to beof the form

ψν=1(zi) ∝ fN(zi) exp∑

j

−|zi|24l2B

(4.19)

where zi means (z1, z2, ...., zN ), and fN is analytic in all variables. N isthe total number of electrons, and is equal, since ν = 1 to the total numberof states in the LLL. Since we are dealing with a state where all electronspins are identical, the spin wave function is symmetrical under exchange ofparticles. Since we are dealing with a fermion wave function, the prefactor fN

of the orbital part must be totally antisymmetric under exchange of particles.It can only be a single Slater determinant with all LLL single particle statesoccupied.

This determinant reads:

fN = det

z01 z1

1 ... zN−11

z02 z1

2 ... zN−12

... ... ... ...z0

N z1N ... zN−1

N

(4.20)

1We will show later on that in fact the gap above the ν = 1 ground state is dominatedby exchange effects, and is much larger than the Zeeman gap.

The Laughlin wave function 53

This determinant, called a Vandermonde determinant, is a polynomial in Nvariables, with N zeros. It has a simple expansion as

fN(zi) = Πi<j(zi − zj) (4.21)

Since the highest power of any particle space coordinate zi is N-1, and thiscorresponds to a guiding center eigenvalue mmax = N−1, fN corresponds in-deed to a fully occupied LLL, with all states occupied once (as the expressionof equation 4.20 shows).

A striking remark is that this state being the only LLL eigenstate withν = 1(with fixed center on mass

∑

i zi), it is an eigenstate of the N particleHamiltonian for any interaction potential.

In order to analyze properties of ψν=1, Laughlin resorted to a very originaldetour: the so called ”plasma analogy”. The latter amounts to regard theprobability distribution function of particles in the LLL,(putting lB = 1),

|Ψν=1|2 ∝ ΠNi<j|zi − zj|2 exp−(1/2)

N∑

l

|zl|2 (4.22)

as the Boltzmann weight of a classical statistical mechanics problem, thepartition function Z of which is given by the norm of the wave function. Inother words,

Z =∫

Πid2zi|ψν=1(zi)|2

and|ψν=1|2 = exp−βUclass. (4.23)

Since this is a formal analogy, the inverse temperature β which appears hereis arbitrary. For reasons which will appear later, we choose here β = 2/q,where q will be non trivial later on, but is equal here to 1, so that eventually

Uclass ≡ q2∑

i<j

(− ln |zi − zj|) +q

4

∑

l

|zl|2 (4.24)

Laughlin remarked that Uclass is the internal energy of a 2D classical onecomponent gas of interacting particles with charge q in a uniform neutraliz-ing background. Remember that this is an analogy, which has the advantageof representing an unknown problem, i.e. the probability distribution func-tion of the real integer quantum Hall problem in terms of a different, known,problem, that of the classical statistical properties of a gas of charged 2D


particles. In the equivalent classical problem, particles have logarithmic in-teractions, which are 2D Coulomb interactions, while the real problem hasthe same formulation, as we saw above, for any interaction potential.2

To see that the interactions between the classical particles of the equiv-alent classical problem are 2D Coulomb interactions, remember that in 2Dthe flux of the electric field through a circle of radius R (the sphere S1 of the2D space) is related to the enclosed charge Q by

∫

dx.E = 2πQ

. For a point charge q at the origin, E(r) = qr/r2 so that the electric potentialis Vc = −q ln r/a,3 and Poisson equation in 2D reads:

divE = −∇2Vc(r) = 2πqδ2(r). (4.25)

Thus the first term on the right of equation 4.24 is interpreted as theCoulomb interaction energy among N 2D charge q particles. Because theLLL is filled, N = B/φ0 = 1/(2πl2B) = 1/(2π) particles per unit surface(remember that here lB = 1).

The second term on the right of equation 4.24 represents the potential en-ergy of N particles of charge q interacting with a uniform charged backgroundwith charge density ρB = −1/(2πl2B). Indeed

−∇|z|24l2B

= −1/(l2B) = 2πρB (4.26)

In other words, the uniform background has a charge density which is pre-cisely equal to the density of flux quanta threading the surface. We knowfrom electrostatics that charge neutrality is the condition for thermodynamicequilibrium, which corresponds to the most probable states in the partitionfunction. The overall charge neutrality condition is

nq + ρB = 0 (4.27)

2The actual interaction between particles in the real problem is in fact a 3D Coulombinteraction ∝ 1/r, because the electrons in the 2D potential well are immersed in a 3Dspace. But the analysis of the filled LLL wave function is entirely independent of anyinteraction potential form.

3a is an arbitrary integration constant,which only changes Vc by a constant, and whichwe may later take as a = lB .


Which is satisfied in the ν = 1 LLL ground state, since q = 1. The plasmaanalogy tells us more than the overall neutrality condition: it tells us that thelargest values of the probability distribution function |ψν=1(zi)|2 is whencharge neutrality is realized locally, otherwise huge costs in Coulomb energyreduce drastically the contribution of local density fluctuations to Z.

The conclusion for the filled LLL is that it is a strongly correlated liquid,with random particle positions, but negligible fluctuations on length scalesgreater than lB. This statement holds for any interaction potential, and istrue for non interacting particles, because the ν = 1 Vandermonde determi-nant is the only ground state wave function.

4.2.2 The many-body function for ν = 1/(2s+ 1)

Before the discovery of the FQHE in 1983, the ground state of electrons inthe partially filled LLL had been predicted to be a Wigner crystal: electronswould organise in a triangular cristalline array to reduce their Coulomb inter-actions. There is indeed some experimental evidence that such is the situationat low enough filling of the LLL. It is clear however that the Wigner Crystalcannot produce a FQHE, i. e. a state with a gap above the ground state.The reason is that a crystal is a state with continuous broken translation androtation invariance, so that it has a Goldstone mode, i.e. a collective excita-tion the energy of which goes continuously to zero with the wave vector. Sucha state does not have a gap above the ground state, in contradiction with theFQHE phenomenology. Moreover, the Wigner crystal scenario would haveno particular way of selecting “magic” fractional values of the filling factorobserved to correspond to FQHE plateaus.

The explanation of the FQHE at ν = 1/(2s+ 1) was proposed by Laugh-lin the very year it was discovered [12]. He looked for a trial many bodywave function which would respect the constraints and the symmetries ofthe problem. Here we sketch the essential steps in the construction of theLaughlin wave function, starting with the wave function for two particles.ψ(2)(z, z′).

• The analyticity condition (2.24), for the symmetric gauge imposes thatψ(2)(z, z′) =

∑

m,M αm,M(z+z′)M(z−z′)m exp[−(|z|2+|z′|2)/4l2B], wherem and M are integers.

• Electrons are fermions, with spin polarised electrons, so, as for the


ν = 1 case, the orbital part of the wave function must be antisymmetricwith respect to permutation of the particles. This limits the choiceto odd m integers. The general two particle wave functions is thusrestricted to be a superposition of functions ψ(2)(z, z′) ∝ (z+ z′)M(z−z′)2s+1 exp[−(|z|2 + |z′|2)/4l2B].

• If we take into account the two body problem for electrons with centerof mass angular momentum M and relative angular momentum m thefollowing wave function

ψ(2)Mm(z, z′) = (z + z′)M(z − z′)m exp[−(|z|2 + |z′|2)/4l2B] (4.28)

is unique (aside from normalization factors). It is remarkable that,neglecting LL mixing, this is the exact two body wave function for anycentral potential V (|z−z′|). The powerful restrictions due to analyticity

allow to write ψ(2)Mm(z, z′) without solving any radial equation! There

is only one state in the LLL Hilbert space with center of mass angularmomentum M and relative angular momentum m.

• The corresponding energy eigenvalue V 0m for the two electron problem

in the LL is independent of M and given by the only matrix element:

V 0m =

〈m,M |V |m,M〉〈m,M |m,M〉 . (4.29)

The coefficients V 0m are called the Haldane pseudo-potentials (general-

ized to any LL in a later section in this chapter). The discrete energyeigenstates represent bound states of the (repulsive!) potential. This isunusual: a repulsive potential has no bound states, only a continuousspectrum in the absence of a magnetic field. In the presence of a mag-netic field, the Lorentz force results in quenching the kinetic energy, sowe may have bound states. In zero magnetic field, two electrons con-vert their potential energy in kinetic energy and move away from oneanother. In a magnetic field, the electrons have fixed kinetic energy, sothey are constrained to orbit around one another.

The discrete spectrum for a pair of particles in a repulsive potential isa basic feature in the understanding of the FQHE, as it generates a gapabove the ground state energy for all excitations.


Although the exact solution for the two particle problem cannot be gener-alized to N > 2 in any straightforward fashion, the N particles wave functionproposed by Laughlin obeys the conditions described above. Laughlin gen-eralized the ν = 1 many body wave function, writing

ψL(zj) =∏

i<j

(

zi − zj

lB

)2s+1

e−∑

j|zj |2/4l2B . (4.30)

In the ν = 1 case, s = 0. Note that this is a one variational parameter (theinteger s) trial wave function. The prefactor in Laughlin’s wave function(4.30) is also called the Jastrow factor. Similar wave functions had beenproposed to describe liquid Helium.

The plasma analogy is very useful in the fractional filling case [12]. Fol-lowing this picture, we identify the space integral of the wave function squaremodulus with the partition function of a classical statistical system, describedby a “free energy” Ucl. The partition function is then

Z =∑

Ce−βUcl =

∫

d2z1...d2zN

∣

∣

∣ψL(zj)∣

∣

∣

2, (4.31)

where C represents configurations, so that

−βUcl = 2q∑

i<j

ln∣

∣

∣

∣

zi − zj

lB

∣

∣

∣

∣

+∑

j

|zj|22l2B

,

where q = 2s+ 1. The “ temperature” is, as above, β ≡ 2/q in order to get

Ucl = −q2∑

i<j

ln∣

∣

∣

∣

zi − zj

lB

∣

∣

∣

∣

− q∑

j

|zj|24l2B

. (4.32)

What is different for ν 6= 1 as compared to the previous ν = 1 case? We haveto determine the optimal q knowing that the electronic density is nel = ν B

φ0=

ν2πl2B

. The neutral background charge density is given by the same expression:

ρB = −1/(2πl2B), but the charge neutrality condition of the plasma is now:ρB + qnel = 0 which can be re-written

qν = 1. (4.33)

The only variational parameter of the Laughlin wave function (4.30) is de-termined. The exponent of the fractionally filled LLL ground state wavefunction with ν = 1/(2s+ 1) is 1/ν = 2s+ 1.


4.2.3 Incompressible fluid

In this section, it is shown that the Laughlin wave function is an almostexact ground state of the many-body problem, and that there is a gap toany excited state above this ground state. Then the Laughlin wave functiondescribes an incompressible fluid. Indeed any attempt at altering the volume(here a 2D surface) of the Laughlin liquid by applying an infinitesimal (2D)pressure, thereby effecting an infinitesimal work on the system, should fail,because the excitations needed to describe the change of the system have alower bound, cannot be infinitesimally small. The ν = 1 quantum liquid isalso an incompressible fluid, where the gap to excited states is presumably4

due to the Zeeman effect, or possibly the orbital energy hωC . In the fractionalcase, the gap is obviously determined by the Coulomb energy ∝ e2/(ǫr).

In order to evaluate the ground state energy, we need not take into accountthe kinetic term, since we are restricted to a single LL. This is certainlytrue in the large field limit, since hωc ∝ B, while the Coulomb energy ise2/ǫlB ∝ (B1/2). So we need only take into account the latter term.

Suppose that we write the potential energy, quite generally, in terms ofHaldane pseudo potentials

V =∞∑

m′=0

∑

i<j

vm′Pm′(ij) (4.34)

where Pm(ij) is the projection operator which selects out states in which iand j have relative angular momentum m.(Note that Pm1(ij) and Pm2(jk) donot commute). Suppose we have a potential defined by vm′ = 0 for m′ ≥ m.This is a ”hard core potential”. The Laughlin state with exponent m is anexact energy eigenstate

V ψm(zi) (4.35)

Indeed it is clear that Pm′(ij)ψm = 0 for any m′ < m since every pair hasrelative angular momentum larger than, or equal to m.

Suppose m = 3 (Laughlin state at 1/3 filling ). This model obviously hasa minimum excitation energy v1, which corresponds to allowing at least onepair to have relative angular momentum 1.

The proof that the Laughlin wave function has a gap to excited statesfor the actual Coulomb interaction, follows from the fact that, compared

4It turns out that the gap in the ν = 1 case is also due to Coulomb interactions, as willbe discussed later on in the section of Quantum Hall ferromagnetism.


to the model hard core potential, the additional Haldane pseudo potentialsof the Coulomb potential (i.e. m′ ≥ 3) can be treated perturbatively, be-cause they are all smaller than v1. This proof is valid specifically for theCoulomb potential. Since all Coulomb corrections to the hard core potentialare perturbations, the gap between the ground state and the first excitedstate persists. Thus the Laughlin state, almost the exact ground state of theCoulomb potential Hamiltonian, is that of an incompressible fluid. The ex-citation gap is a necessary condition for zero longitudinal conductivity, andzero resistivity, σxx = 0 = ρxx.

Numerical data show that the overlap between the true ground state forthe Coulomb potential and the Laughlin wave function is extremely good.

4.2.4 Fractional charge quasi-particles

A remarkable property of the Laughlin liquid is that its elementary excita-tions have fractional charge.

Consider the wave function

ψqh(z0, zj) =N∏

i=1

(

zi − z0

lB

)

ψL(zj), (4.36)

where an additional zero sits at position z0. The charge density vanishes atz0. Expanding formally Laughlin’s wave function as

ψL(zj) =∑

miαm1,...,mN

zm11 ...zmN

N e−∑

j|zj |2/4l2B ,

and comparing with the expansion

ψqh(z0 = 0, zj) =∑

miαm1,...,mN

zm1+11 ...zmN+1

N e−∑

j|zj |2/4l2B ,

we see that, compared to Laughlin’s wave function, all particles are displacedfrom one state to the next, mj → mj + 1. In the symmetric gauge where aparticle is found on a ring of radius lB

√2mj + 1, this means that a hole has

been created at the origin z0 = 0 (“ quasi-hole”). Furthermore, this quasi holehas vorticity. If we examine the phase of ψqh(z0 = 0, zj) ∝

∑

j exp(−iθj),where θj = tan−1(yj/xj), we see that a particle circulating on a closed patharound z0 acumulates a phase 2π.


In principle, one can describe a wave function with a “quasi-particle”excitation (with opposite vorticity) in a similar fashion ,

ψqp(z0, zj) = PLLL

N∏

i=1

(

z∗i − z∗0lB

)

ψL(zj), (4.37)

There is a complication here since we are not allowed to use z∗j in a wavefunction which should be analytic in order not to mix in higher LL states. Inorder to remain within the LLL manyfold of states, one should use a projectorPLLL on the LLL. A way of doing this is to divide the polynomial part of thewave function by zj instead of multiplying by z∗j . By partial integration, thisis equivalent to applying ∂zj

to the gaussian factor, which generates z∗j , up toa multiplying factor. Given the complication in handling quasi-particle wavefunctions, we will deal only with quasi-holes in the following, without loss ofphysical generality [32].

In order to check that such excitations have fractional charge, let us useagain the 2D plasma analogy introduced above. The prefactor in the wavefunction (4.33) gives rise to a new term in expression (4.32), Ucl → Ucl + V ,where

V = −qN∑

j

ln∣

∣

∣

∣

zj − z0

lB

∣

∣

∣

∣

.

This is interpreted as the interaction potential between the plasma and acharge 1 ”‘impurity”’ located at z0. This impurity is screened so as to main-tain charge neutrality in the plasma. Since the plasma particles have chargeq, 1/q particles are needed to screen the impurity charge. The quasi-particleof the Laughlin liquid is thus shown to have fractional charge 5

e∗ =e

q=

e

2s+ 1. (4.38)

Another more direct way to see this charge fractionalisation is to introduceq quasi-particles at the same point 6

[ψqh(z0, zj)]q =N∏

j=1

(

zj − z0

lB

)q

ψL(zj) = ψL(zj, z0),

5From now on, we use the generic term ”quasi-particle” for quasi-holes and quasi-particles, except when distinction is necessary.

6The expression on the left is symbolic .


where we find the Laughlin wave function for N+1 electrons, with the addedone at position z0. One needs therefore q quasi-particles to add one electronin a Laughlin liquid, leading to the same conclusion as the plasma analogy(4.38).

It is interesting to give yet another proof that Laughlin quasi-particlescarry fractional charge, in order to show the essential connection between thefractional quantum Hall plateau, with fractional Hall conductivity σxy = ν e2

h,

and the fractionalisation of the quasi-particle charge. Imagine piercing thesample at the origin with an infinitely thin magnetic solenoid and increasingadiabatically the magnetic flux φ from 0 to φ0 = h/e. The time variation ofthe flux inside the solenoid induces an azimuthal electric field, as Faraday’slaw tells us. This field is such that

∮

Cdr.E = −∂φ

∂t. (4.39)

C is a contour surrounding the flux line. If the process is sufficiently slow, theelectric field has low frequency Fourier components only, such that hω ≪ ∆,where ∆ is the energy gap. There is no dissipation. Because the system isin a quantum Hall state, the electric field drives a current density which isradial:

E = ρxy~J ∧ z. (4.40)

So we have

ρxy

∮

C

~J.(z ∧ dr) = −dφdt

(4.41)

The integral on the LHS represents the total current flowing into the regionenclosed by the contour. Thus the charge inside this region obeys

ρxydQ/dt = −dφ/dt (4.42)

At the end of the process, the total charge is

Q = σxyφ0 = σxy(h/e) = νe (4.43)

The final step in the argument is that an infinitesimal flux tube containing aflux quantum is invisible to the particles, and can be removed by a (singular)gauge transformation which has no physical effect.

This derivation underlines the importance of the fact that σxx = 0 andσxy is quantized. The existence of fractionally charged elementary excitationsis a direct consequence of the FQHE.


Numerical data show that there is a finite energy cost to create suchquasi-particles which means that there is a gap between the ground statedescribed by the Laughlin wave function and its lowest elementary excitedstates. This is a necessary condition for the FQHE.

4.2.5 Ground state energy

Beside his ground state wave function proposal, Laughlin showed that ithas lower energy than the Wigner crystal. The latter had been argued tominimize the Coulomb energy. The Laughlin liquid energy is given by

〈ψL|H|ψL〉Z =

1

2Z∑

i6=j

∫

d2zid2zj

e2

ǫ|zi − zj|∫

N∏

k 6=i,j

d2zk

∣

∣

∣ψL(zi, zj; zk)∣

∣

∣

2

=n2

elA

2

∫

d2re2

ǫ|r|g(r), (4.44)

where

g(r) ≡ N(N − 1)

n2elZ

∫

d2z3...d2zN

∣

∣

∣ψL(z1 = 0, z2 = r; z3, ..., zN )∣

∣

∣

2(4.45)

is the pair correlation function. This expression takes advantage of the trans-lation and rotation invariance of the wave function 7 and of the fact that thereare N(N − 1) ways of chosing the zi = z1 and zj = z2 pairs in the first lineof equation (4.44). This expression is usually divided by the total particlenumber N = nelA , and the energy of the homogeneous uncorrelated liquidE0 = (nel/2)

∫

d2re2/ǫr is chosen as energy reference. The Laughlin liquidenergy per particle is written in terms of the pair correlation function.

EL =nel

2

∫

d2re2

ǫ|r| [g(r) − 1], (4.46)

or, in Fourier space

EL =1

2

∑

q

v(q)[s(q) − 1], (4.47)

in terms of the static structure factor

s(q) =1

N〈ρ(−q)ρ(q)〉, (4.48)

7rotation invariance results in r = z = |z|.


which is connected to the pair correlation function by Fourier transformation[25]

[s(q) − 1] = nel

∫

d2r eiq·r[g(r) − 1]. (4.49)

The pair correlation function (or the structure factor ) thus determines theliquid structure and describes possibly a short range order. It can be com-puted from Laughlin’s wave function by Monte Carlo integration [34, 35].

Instead of computing the pair correlation function numerically, Girvinanalysed it in 1984 using symmetries and properties of the 2D one componentplasma [91]. Expanding the Laughlin wave function in terms of z = (z1 −z2)/lB and z+ = (z1 + z2)/lB,

ψL(zj) =∑

M

∼∑

m=1

aM,m(z3, ..., zN )zM+ z

me−(|z+|2+|z|2)/8, (4.50)

where the tilde on the second sum indicates that the sum is on odd integers,one finds for the pair correlation function

g(z) =∼∑

m,m′Am,m′(z+)z∗m

′zme−|z|2/4,

where functions Am,m′(z+) depend only on z+ because the other variablesz3, ...zN have been integrated on. ∂Am,m′(z+)/∂z+ = 0 follows from the liquidtranslation invariance, and rotation invariance imposes Am,m′ = δm,m′bm,which results in

g(z) =∼∑

m=1

bm|z|2me−|z|2/4.

As lim|z|→∞ g(|z|) = 1, and thus lim|z|→∞ ˜∑m=1bm|z|2m = exp(|z|2/4), it is

convenient to rewrite expansion parameters as

bm =2

m!

(

1

4

)m

(1 + cm),

where limits impose limm→∞ cm = 0. The pair correlation function is thusgiven as a sinh plus corrections described by the cm parameters.

g(z) = 2 sinh

(

|z|24

)

e−|z|2/4 +∼∑

m=1

2cmm!

(

|z|24

)m

e−|z|2/4

=(

1 − e−|z|2/2)

+∼∑

m=1

2cmm!

(

|z|24

)m

e−|z|2/4. (4.51)


The Fourier transform yields the static structure factor

[s(q) − 1] = −νe−q2l2B/2 + 4ν∼∑

m=1

cmLm(q2l2B)e−q2l2B , (4.52)

in terms of Laguerre polynomials. The energy (4.47) is finally written as

EL =ν

π

∼∑

m=1

cmV0m − ν

2

∑

q

v0(q), (4.53)

where v0(q) is the effective potential in the LLL (4.18), and we have definedthe Haldane pseudo-potentials

V 0m ≡ 2π

∑

q

v0(q)Lm(q2l2B)e−q2l2B/2. (4.54)

The advantage of the energy expression (4.53) in terms of Haldane pseudo-potentials, defined in terms of the effective potential, allows to describe di-rectly Laughlin liquids in higher index LL (n 6= 0) : pseudo-potentials aregeneralised to LL n, using the appropriate effective potential (4.18),

V nm ≡ 2π

∑

q

vn(q)Lm(q2l2B)e−q2l2B/2. (4.55)

Returning to (4.51), we notice that due to the Laughlin wave functionbehaviour when two particles described by z1 and z2 get close to one another,one has g(z) ∼ |z|2(2s+1) at short distance. This shows that correlations inLaughlin’s wave function are effective in minimizing Coulomb interactions,more so than in any state where fermionic correlations would impose onlyg(z) ∼ |z|2. This short distance behaviour ensures that expansion parametersobey

cm = −1, pour m < s. (4.56)

It is also useful to define the ”moments”

Mn = nel

∫

d2z

(

|z|24

)n

[g(z) − 1]

= 2πnel

[

−n! + 2n+2∼∑

m=1

(n+m)!

m!cm

]

, (4.57)


where the second line is computed with the help of equation (4.51). Followingthe plasma analogy,8 charge neutrality imposes M0 = −1 and thus

∼∑

m=1

cm =1

4

(

1 − ν−1)

= −s2. (4.58)

Perfect plasma screening is expressed by M1 = −1, i.e.

∼∑

m=1

(m+ 1)cm =1

8

(

1 − ν−1)

= −s4. (4.59)

Compressibility properties yield a third sum rule

∼∑

m=1

(m+ 2)(m+ 1)cm =1

8

(

1 − ν−1)2

=s2

2. (4.60)

Those sum rules (4.58-4.60) and (4.56) can be used as constraints on thepair correlations functions (4.51) in connection with Monte Carlo numericalwork. One may use them instead for an approximate determination of thefunction: sum rules form a system of coupled linear equations which can besolved if one sets cm = 0 for m > s+3, which is a reasonable approximation,since limm→∞ cm = 0. In this manner, one finds

cs1 cs3 cs5 cs7 cs9 cs11 cs13s = 1 -1 17/32 1/16 -3/32 0 0 0s = 2 -1 -1 7/16 11/8 -13/16 0 0s = 3 -1 -1 -1 -25/32 79/16 -85/32 0s = 4 -1 -1 -1 -1 -29/8 47/4 -49/8

for s = 1, .., 4.The results for the energy deviate by less than one per cent from nu-

merical results by Levesque et al. [34], as shown on figure 4.2(a). The paircorrelation function apart the correlation hole at small distance, exhibits amaximum at finite distance, where it is most probable to find a second par-ticle.. This maximum is displaced further away from the origin and becomesmore pronounced if the electronic density is lowered [ν = 1/(2s + 1)]. Thismeans an enhanced short range order at low densities where a Wigner crystal


5 10 15 20 25

-0.4

-0.3

-0.2

-0.1

Ene

rgie

Facteur de remplissage

0.1 0.2 0.3 0.4 0.5

−0.4

−0.3

−0.2

−0.1

2 4 6 8 10 12 14

0.2

0.4

0.6

0.8

1

1.2

1.4

r

g (r)s

(b)

s=1

s=2

s=3

Fon

ctio

n de

cor

réla

tion

de p

aire

s

1.0

5

0.5

10 15

Figure 4.2: (a) Comparison of our results for the energy (black segments) of Laughlinstates to numerical results by Levesque et al. (gray line) in units of e2/ǫlB . The line is theresult of an an interpolation formula U(ν) = −0, 782133

√ν(

1 − 0, 211ν0,74 + 0, 012ν1,7)

for Levesque et al.’s results [34]. (b) Pair correlation function for different s. The distancer is mesured in units of the magnetic length lB . The straight dotted line corresponds touncorrelated electrons.


is expected to become more stable. More accurate numerical results confirmthis tendancy (see figure 4.2(b))[39].

4.2.6 Neutral Collective Modes

We have discussed the ground state energy, and analysed elementary excita-tions (fractionally charged quasi-particles) the energy of which is separatedfrom the ground state one by a gap. In order to understand FQHE, we nowhave to show that collective excitations have a dispersion relation with afinite gap above the ground state at all wave vectors. Such collective exci-tations, with wave function |ψq〉 are likely to be well described within the”Single Mode Approximation” (SMA)[35]. 9. In the quantum Hall case,

|ψq〉 = ρ(q)|ψL〉, (4.61)

where ρ(q) is the projected density operator (4.7). Since

ρ(q) =∑

m,m′〈m|e−iq·R|m′〉e†n=0,men=0,m′ ,

the excited state may be interpreted as a superposition of particle-hole exci-tations (particles in the state |n = 0,m〉 and hole in |n = 0,m′〉), an averagedistance ql2B apart. Because of the projection, |ψq〉 has no component in LLn 6= 0. The excitation energy with respect to the ground state is

∆(q) =〈ψL|ρ(−q)Hρ(q)|ψL〉〈ψL|ρ(−q)ρ(q)|ψL〉 − EL

≃ 1

2

〈ψL|[ρ(−q), [H, ρ(q)]|ψL〉〈ψL|ρ(−q)ρ(q)|ψL〉 ≡ f(q)

s(q), (4.62)

where we assumed that the Laughlin state is an eigenstate of the Hamiltonian, H|ψL〉 ≃ EL|ψL〉, which is an excellent approximation. Moreover the liquidstate rotation invariance has been used to get the second line, as well asρ†(q) = ρ(−q). The projected structure factor is connected to the structurefactor (4.52) through

s(q) = s(q) −(

1 − e−q2l2B/2)

.

8Interested readers may find details in the following references [34, 35, 91] (and refer-ences in those papers in particular [38])

9The SMA was used by Feynman in his theory of superfluid He collective modes [40]


Figure 4.3: Dispersion relation for collective excitations [41]. The continuous curves arethe results in the Single Mode Approximation for ν = 1/3, 1/5 et 1/7; the various symbolsare values obtained by exact diagonalisation [42]. Arrows are the expected reciprocallattice parameter moduli for the Wigner crystal at the corresponding filling factors.

Equation (4.62) is precisely the Feynman-Bijl formula, proposed for the de-scription of collective excitations in superfluid He [40]. Using commutationrules for projected density operators (4.16), one finds

∆(q) = 2∑

k

[v0(|k − q|) − v0(k)] sin2

(

q ∧ kl2B2

)

s(k)

s(q). (4.63)

The dispersion relations are shown on figure 4.3, as well as numericalresults of exact diagonalisations of systems with a small number of particles[42]. As expected for an incompressible liquid, dispersion relations havea finite energy gap above the ground state for all wave vectors. They allexhibit a minimum at a finite wave vector. The latter corresponds to thereciprocal lattice parameter modulus for the Wigner crystal at the same fillingfactor. The minimum, called the magneto-roton minimum, in analogy withthe superfluid He case [40], is a sign of short range (crystalline) order. Thecollective mode softening at this wave vector signals a tendancy to Wignercrystal stabilisation when the filling factor is decreased.

The SMA becomes less reliable at large wave vector, where one expects

4.3. JAIN’S GENERALISATION – COMPOSITE FERMIONS 69

the asymptotic behaviour

∆(q ≫ 1/lB) ≃ ∆qp + ∆qh −2πe∗2

ǫql2B,

i.e. the energy to create a pair made with a quasi-particle of energy ∆qp anda quasi-hole with energy ∆qh, with well separated components submitted toCoulomb attraction because of their opposite charges, e∗ and −e∗.

4.3 Jain’s generalisation – Composite Fermi-

ons

The Laughlin wave function describes well the FQHE at ν = 1/(2s+ 1), butit fails to apply to the other fractional states which were discovered subse-quently, such as ν = 2/5, which is one term in the set ν = p/(2sp + 1). Inorder to account for those new states, Haldane [37] and Halperin [43] pro-posed a hierarchy picture. Following the latter, Laughlin quasi-particles withsufficient density condense in an incompressible liquid in order to minimizetheir Coulomb interaction energy due to their charge e∗. The state ν = 2/5would then be a ”daughter” of the Laughlin state at ν = 1/3.

In 1989, Jain proposed an alternative route, the Composite Fermion pic-ture. He first re-interpreted the Laughlin wave function

ψL(zj) =∏

i<j

(

zi − zj

lB

)2s∏

i<j

(

zi − zj

lB

)

e−∑

j|zj |2/4l2B , (4.64)

as a product of two factors : the first one,∏

i<j[(zi − zj)/lB]2s, attaches 2szeros (vortices with 2s flux quanta ) to particles positions, and the secondone,

χν∗=1(zj) =∏

i<j

(

zi − zj

lB

)

, (4.65)

can be interpreted as the wave function of a virtual completely filled LL, witha new (virtual) filling factor ν∗ = 1 [20]. Indeed, it coincides with equations4.20 and 4.21.

Jain’s proposal amounts to generalize equation 4.65 by replacing χν∗=1(zj)by a Slater determinant for p virtual completely filled LL, χν∗=p(zj),

ψJ(zj) = PLLL

∏

i<j

(

zi − zj

lB

)2s

χν∗=p(zj), (4.66)


Projection to the LLL is taken care of by the projector PLLL, since the func-tion χν∗=p(zj) contains, if unprojected, high energy components belongingto LL with p > 1.

What is achieved by this manipulation? The effective number of statesper LL in the virtual levels has been decreased, M → M∗ = M − 2sN ,since the first vortex attachment factor has taken 2sN zeros from the systemwith N electrons. This amounts to renormalize the magnetic field which isnothing but the flux density in terms of flux quanta φ0 = h/e, and the fillingfactor as well, in the following way

B → B∗ = B − 2sφ0nel et ν∗−1 = ν−1 − 2s. (4.67)

With this picture, we may now re-interpret the FQHE at ν = 1/3 as acompletely filled Composite Fermion level, with ν∗ = 1, where a CompositeFermion is an electron with two attached flux quanta. The state at ν = 2/5is re-interpreted as a state with ν∗ = 2 (figure 4.4). The CF picture allows tounderstand the FQHE of electrons at ν = p/(2sp + 1) in terms of an IQHEfor CF at filling factor ν∗ = p, since the CF filling factor ν∗ = hnel/eB

∗ isconnected to the electronic filling factor through

ν =ν∗

2sν∗ + 1, (4.68)

which is equivalent to expression (4.67).In the following chapters, we shall elaborate on the physical meaning of

this picture, which is basically a flux counting device, based on the notionof flux attachment to electrons. It is not an obviously physical approachto renormalize the magnetic field, which is an external object imposed onthe system. Note however that the magnetic field only enters the theorywith the electronic charge e as a multiplying coupling constant. It is thuspermissible to renormalize the charge, which seems especially relevant giventhe fractionalisation of excitation charges discussed above.

4.3.1 The effective potential

Pour mieux comprendre le modele, on discutera dans cette section quelquesproprietes du potentiel d’interaction effectif (4.17). En raison des zeros despolynomes de Laguerre Ln(x), la repulsion coulombienne disparaıt a certainesvaleurs du vecteur d’onde, notamment a q0(n) ≃ 2, 4/

√2n+ 1, ce qui corre-

spond au premier zero x ≃ 1, 2/(2n + 1) [31]. Cela mene a des instabilites

Jains’s generalisation 71

ν = 1/3 :

ν = 2/5 :électron

quantum de flux libre

vortexportant 2s quanta de flux (liés)

fermion composite

ν∗ = 1

ν∗ = 2

Théorie

de FCs

Figure 4.4: Composite Fermions. The electronic state at ν = 1/3 may be understoodas a CF state with integer filling ν∗ = 1. CF are electrons carrying each 2s flux quanta.Similarly, a CF filling factor ν∗ = 2 describes an electronic filling factor = 2/5.

du systeme pour la formation des phases de densite inhomogene avec uneperiodicite caracteristique Λ ≃ 2π/q0(n), car il est energetiquement favorablepour la densite moyenne 〈ρ(q)〉 d’avoir un maximum a q0. La periodicite Λvarie proportionnellement avec le rayon cyclotron RC =

√2n+ 1. Les phases

de densite inhomogene seront discutees plus en detail dans le chapitre 7. Unetransformation dans l’espace reel du potentiel effectif, vn(r) =

∑

q exp(ir ·q)vn(q) confirme la apparition d’une echelle de longueur caracteristique. Pourdes petites valeurs de n, cette transformation peut etre effectuee de facon ex-acte, et l’on trouve une somme finie sur des fonctions de Bessel. Dans desNL plus eleves, n ≫ 1, on peut deduire une loi d’echelle pour le potentiel al’aide de Fn(q) ≃ J0(qRC), ce qui devient exact dans la limite n→ ∞,

vn(r) ≃ v(r/RC)√2n+ 1

, avec v(x) =4e2

πǫxRe

K

1 −√

1 − 4/x2

2

2

,

(4.69)ou J0(x) est la fonction de Bessel d’ordre zero, et K(x) est l’integrale ellip-tique complete de premiere espece [21].

La figure 4.5(a) montre les resultats pour le potentiel effectif dans lesniveaux n = 1, ..., 5. On remarque la formation d’un palier – a part des petitesoscillations – pour des distances moyennes, superpose au potentiel coulom-bien habituel, e2/ǫr, qui est retrouve a grande distance. Ce palier devient pluslarge dans des NL eleves cependant que sa hauteur est diminuee. La forme


2 4 6 8 10

0.2

0.4

0.6

0.8

1

1/r

n=1

n=2n=3

n=4n=5

1.0

0.5

2 4 6 8 10r

v (

r)n

(a)

10 20 30 40 50

0.25

0.5

0.75

1

1.25

1.5

1.75

2 (b)

1.0 2.0 3.0 4.0 5.0

1.5

1.0

0.5

2.0

n=1

n=2

n=3

n=4

n=5

1/r

r/R

v(r)

~

C

Figure 4.5: (a) Potentiel effectif dans l’espace reel pour les NL n = 1, ..., 5, en unitesde e2/ǫ. Le potentiel de Coulomb en 1/r est montre pour comparaison (tirets). (b) Lesresultats pour le potentiel (points) sont traces apres la transformation d’echelle (4.69). Laligne noire represente l’expression approchee v(x) et la ligne grise le potentiel de Coulomb.

(c)(b)(a)

r

Figure 4.6: Les fonctions d’onde des electrons dans un NL n ≥ 1 peuvent etrerepresentees par des anneaux [voir Fig. 2.3(a)]. (a) Si r > 2RC , les anneaux ne serecouvrent par. (b) Pour r <∼ 2RC , les anneaux commencent a avoir un recouvrement,represente par la surface grise foncee. (c) Le recouvrement n’augmente pas de facon sig-nificative lorsque les anneaux sont rapproches davantage.

d’echelle v(x) est mise en evidence apres la transformation des resultats selonl’equation (4.69) : les points, qui representent les resultats exacts, tombentapproximativement sur la meme courbe (noire). L’approximation (4.69), quidevient exacte dans la limite n → ∞, decrit la forme du potentiel de faconsuffisamment appropriee aussi pour de plus bas NL a condition que n > 0.Le point anguleux a r = 2RC dans la forme approchee du potentiel est unartefact mathematique – pour x ≥ 2, l’argument de l’integrale elliptique estreel tandis qu’il devient complexe pour x < 2, ce qui donne lieu a cettediscontinuite. Cet effet pourrait engendrer des divergences artificielles dansd’eventuelles derivees, mais on peut se servir de cette forme du potentieluniquement comme support dans des integrations, ce qui rend la disconti-nuite inoffensive.

La forme du potentiel peut etre illustree dans une image quasi-classique.

Jains’s generalisation 73

Avec la restriction des champs electroniques au n-ieme NL, on a fait unemoyenne sur le mouvement rapide de l’electron, determine par la variable η

qui, sans cette restriction, couplerait des etats de differents n. Les degresde liberte du mouvement des electrons sont donc uniquement leurs centresde guidage. Comme on l’a vu dans la section 2.2.1, la fonction d’onde d’unelectron dans un niveau de Landau n ≥ 1 tient compte de cette moyenne surle mouvement cyclotron et a par consequent une forme d’anneau de rayonRC , representant une densite electronique moyenne [Figs.2.3(b) et 4.6]. Sila distance r entre les centres d’anneaux, qui sont precisement les centresde guidage du mouvement cyclotron de chaque particule, est suffisammentgrande (r > 2RC), la forme des fonctions d’onde de deux particules n’a poureffet qu’une faible correction du potentiel coulombien. A r ∼ RC , les an-neaux commencent a se recouvrir et la repulsion devient donc plus forte. Enrevanche, si l’on rapproche les centres de guidage, ce recouvrement ne devientpas plus grand et la repulsion n’augmente donc pas de facon significative, cequi explique la formation du palier dans le potentiel effectif. La repulsiondevient a nouveau plus importante quand le recouvrement est complet a trespetite distance. Or les centres de guidage etant etales sur la surface minimale2πl2B ne peuvent pas etre approches a des distances plus petites que lB. Pourn = 0, cette image quasi-classique devient plus problematique parce que lesfonctions d’onde sont de forme gaussienne avec une extension spatiale del’ordre de la longueur magnetique. Les electrons devraient donc plutot etrerepresentes par un disque de rayon lB, qui constitue egalement la longueurminimale, comme il a ete decrit dans le chapitre 2 [voir Fig. 2.3(b)].

Chapter 5

Chern-Simons Theories andAnyon Physics

Following the CF theoretical proposal, a field theory was constructed todescribe flux attachment to electrons. Such theories are known as ”Chern-Simons ” theories in the framework of the generalisation of the Maxwelltheory of electromagnetic fields. Lopez and Fradkin were first to point outin 1991 [15] their relevance for the FQHE, followed in 1993 by Halperin, Leeand Read [16], who studied the compressible state at ν = 1/2. The latterfilling factor is the limiting point of p/(2sp+ 1) when p→ ∞ and s = 1.

This chapter aims at introducing some basic notions about Chern-Simonstransformations, but does not pretend to offer a detailed field theoreticaldescription. We describe their connections with anyons, i.e. particles in 2Dwhich obey fractional statistics, and which have a transparent description inthe framework of Chern-Simons theories, the basic notions will be useful inthe follwing chapter.

5.1 Chern-Simons transformations

The Hamiltonian of electrons in a magnetic field writes, in second quantizedform

H = H0 + Hint,

75

76 Chern-Simons theories and anyon physics

where the kinetic term is

H0 =∫

d2rψ†(r)[−ih∇ + eA(r)]2

2mψ(r), (5.1)

and Hint accounts for interactions between electrons. A Chern-Simons trans-formation is a singular unitary transformation,

ψ(r) = e−iφ∫

d2r′θ(r−r′)ρ(r′)ψCS(r), (5.2)

where θ(r) = tan−1(y/x) is the angle formed by vector r and the x axis.This transformation is clearly singular since the angle θ(r) is not defined forr = 0. The density is invariant under this transformation

ρ(r) = ψ†(r)ψ(r) = ψ†CS(r)ψCS(r).

Notice that∫

d2r′θ(r−r′)ρ(r′) (see equation 5.2) is an operator which dependson all electron coordinates. The gradient in expression (5.1) also operates onthe phase factor of the transformation, and one finds

−ih∇ψ(r) = e−iφ∫

d2r′θ(r−r′)ρ(r′)[

−ih∇− φh∇∫

d2r′θ(r − r′)ρ(r′)]

ψCS(r).

We can thus define a new gauge field, the Chern-Simons vector potential,

ACS(r) = − heφ∇

∫

d2r′θ(r − r′)ρ(r′). (5.3)

If this potential obeys the Coulomb gauge, as will be shown later on, ∇ ·ACS(r) = 0, the kinetic Hamiltonian can be re-written as

H0 =∫

d2rψ†CS(r)

[−ih∇ + eA(r) + eACS(r)]2

2mψCS(r). (5.4)

The interaction Hamiltonian is invariant, since it depends only on densityoperators which are invariant.

In order to analyse this new gauge field and its associated magnetic field,BCS(r) = ∇ × ACS(r), it is useful to recall some properties of analyticfunctions. We take here z = x+ iy, unlike our definition in chapter 2. Eachcomplex function may be written as a sum of a real part and an imaginarypart,

f(x, y) = u(x, y) + iv(x, y).

Chern-Simons transformations 77

The analyticity condition ∂z∗f(z) = 0 is expressed, in terms of x and y, byequations known as Cauchy-Riemann differential equations

∂xu(x, y) = ∂yv(x, y), et ∂yu(x, y) = −∂xv(x, y), (5.5)

Instead of using the cartesian notation, one may chose the polar coordinaterepresentation:

f(x, y) = w(x, y)eiχ(x,y),

where w(x, y) and χ(x, y) are real functions. The analyticity condition, (∂x +i∂y)f(x, y) = 0, is now written as

∂xw(x, y) − w(x, y)∂yχ(x, y) + i [∂yχ(x, y) + w(x, y)∂xχ(x, y)]

or, after separation in real parts and imaginary parts, and dividing by w(x, y),by the Cauchy-Riemann equations in the polar representation:

∂x lnw(x, y) = ∂yχ(x, y) et ∂y lnw(x, y) = −∂xχ(x, y). (5.6)

In the simplest case, which is of interest here, f(z) = z = r exp(iθ), thisyields

∂x ln r(x, y) = ∂yθ(x, y) et ∂y ln r(x, y) = −∂xθ(x, y).

With these equations, we compute easily

[∇×∇θ(r)]z = (∂x∂y − ∂y∂x)θ(r) = ∆ ln r = 2πδ(2)(r), (5.7)

where the last step is Poisson equation for a 2D potential. The curl of agradient is usually zero, but this is not the case here, because θ(r) is singularat r = 0, as mentionned above. Similarly, we find

∆θ(r) = −∂x∂y ln(r) + ∂y∂x ln(r) = 0. (5.8)

Together with definition (5.3), this last equation shows that the Chern-Simons field satsifies the Coulomb gauge. Equation (5.7) gives for the corre-sponding magnetic field

BCS = − heφ∫

d2r′∇×∇θ(r − r′)ρ(r′) = −heφρ(r)ez. (5.9)

78 Chern-Simons Theories and Anyon Physics

We notice that this magnetic field is 1)intimately connected to the electronicdensity, and 2) it is a quantum operator, contrary to the usual B field.

In the mean field approximation, the density operator in equation (5.9) isreplaced by the average density 〈ρ(r)〉 = nel, so that the field is renormalized

B → B∗ = B + 〈BCS〉 = B − h

eφnel (5.10)

where, in terms of filling factor,

B → B∗ = B(1 − φν). (5.11)

If we chose φ = 2s, this field renormalisation is precisely that described bythe CF theory c(4.67).

Let us connect with the trial wave function approach. In the first quan-tization language, we can rewrite this field transformation as

ψ(zj) = eiφ∑

i<jθ(zi−zj)ψCS(zj) =

∏

i<j

(

zi − zj

|zi − zj|

)φ

ψCS(zj). (5.12)

We see that the Chern-Simons transformation ties φ flux quanta (singularityof order φ in the phase), leaving off the transformation the vortex modulus,contrary to Jain’s function for φ = 2s (4.66).

5.2 Statistical Transmutation – Anyons in 2D

Chern-Simons theories are especially well fitted to the discussion of anyonphysics. Anyons are particles which live in 2D+1 space, and which obeyfractional statistics, i. e. neither bosonic nor fermionic. All particles knownin the 3D world are either bosons or fermions depending on the behaviourof their wave function upon interchange of two identical particles.This isbasically because in 3D (and higher dimensions), the rotation group is non-abelian. The components of the angular momentum do not commute. Quan-tization of angular momenta is in terms of units of h/2, as can be seen fromthe properties of the Lie algebra for infinitesimal rotations. The classificationin bosons and fermions is not true anymore in 2D, because the rotation groupis a trivial abelian group. Therefore no angular momentum quantization rulesfollow from manipulations of the infinitesimal rotations.

Statistical Transmutation – Anyons in 2D 79

+A B B AA B

(a) (b)

Cz

Figure 5.1: (a) Process for a particle A to follow a path C around a second particle.In 3D, the path can be lifted off from the plane and thus can be reduced to a point(graycurves)). (b) Equivalent processus consisting in two successive exchanges of particles Aand B.

Let us look at the interchange of two identical particles. A process Tthrough which a particle A is adiabatically displaced around another particleB, is equivalent, modulo a translation, to two exchange processes E (Fig. 5.1).We assume that particles are localized enough so we can neglect their wavefunction overlap. From an algebraic point of view, which takes into accountthe homotopy of the processes, one may write:

E2 = T

modulo a translation. Let us first discuss the 3D case where the path C liesin the x − y plane. Since the third dimension along z is available, we canlift the path C of particle A above particle B, and then shrink it down to apoint, leaving particle B at all times outside the closed path C. .1. We mayassociate a “time” interval to this adiabatic process, such that C(t = 0) = C(the initial path in the x − y plane), and C(t = 1) = 1 ( the position pointof A). The process through which A circles around B(rotation of 2π is thusequivalent to a process which leaves particles unchanged, i.e. the identity, sowe can write

T = 1 et donc E =√1,

where the last equation is written symbollically, meaning that E has twoeigenvalues e1 = exp(2iπ) = 1 and e2 = exp(iπ) = −1 . This superselectionrule shows that in 3D all point particles may be separated in two classes, de-pending on the behaviour of their wave function upon interchange of identicalparticles. e1 = 1 corresponds to bosons and e2 = −1 to fermions.

This piece of reasoning is not valid anymore when the particle dynamicsis restricted to 2D (2 space dimensions). In this case the path C cannot beshrunk to a point without crossing the particle position B. One says that a

1This would be impossible if B were an infinite line in direction z


path which encloses another particle (particle B) is not in the same homo-topy class as a path which encloses none (and can be shrunk to a point. In2D, paths can be classified by the number of enclosed particles, or by thenumber of times it winds around a given particle. From an algebraic point ofview, the physical requirement is that physical quantities must be invariantby a 2π rotation. This requirement does not apply to the wave functions,since only probabilities are eventually observable. Therefore the eigenvaluesλ of the 2π rotation operator R(2π) may be any number of modulus unity,such as eiαπ with α real(whence the name anyon! for particles with gener-alised statistics). It is easy to see that eigenstates of R(2π) with differenteigenvalues λ 6= λ′ are orthogonal. In fact no local observable can connectstates corresponding to different α values. States corresponding to differentλ values are said to belong to different superselection sectors. Schematically,(anti-) commutation rules for the relevant fields must be generalized

ψ(r1)ψ(r2) = ±ψ(r2)ψ(r1) ⇒ ψ(r1)ψ(r2) = eiαπψ(r2)ψ(r1), (5.13)

where απ is called the statistical angle.What about the Pauli principle? Contrary to bosons, fermions occupation

of a state cannot exceed 1. In terms of fields, this is expressed by

2ψ(r)ψ(r) = 0,

for fermionic fields at the same point r1 = r2 = r, where we use equation(5.13) with the minus sign for fermions. In general, for arbitrary angle α ,one finds

(

1 − eiαπ)

ψ(r)ψ(r) = 0. (5.14)

Thus ψ(r)ψ(r) 6= 0 if and only if ψ(r)ψ(r) 6= 0, as is well known for bosons.When α 6= 0 mod(2), we have necessarily ψ(r)ψ(r) = 0, and equation (5.14)is interpreted as generalized Pauli principle.

5.2.1 Anyons and Chern-Simons theories

We now analyse the statistical properties of the fields ψCS(r) which resultfrom the Chern-Simons transformation, using the known properties of elec-tronic fields ψ(r). Defining

τ(r) ≡∫

d2r′θ(r − r′)ρ(r′), (5.15)


to simplify notation in the following expressions one has

ψCS(r1)ψCS(r2) = eiφτ(r1)ψ(r1)eiφτ(r2)ψ(r2)

= eiφτ(r1)eiφτ(r2)e−iφτ(r2)ψ(r1)eiφτ(r2)ψ(r2). (5.16)

With the help of the Hausdorff formula,

eABe−A = B + [A,B] +1

2[A, [A,B]] + ... =

∞∑

n=0

1

n!Cn(A;B), (5.17)

where Cn(A;B) = [A, Cn−1(A;B)] is defined by a recurrence relation, Cn=0(A;B) ≡B, and

[τ(r2), ψ(r1)] =∫

d2r′θ(r2 − r′)[ψ†(r′)ψ(r′), ψ(r1)] = −θ(r2 − r1)ψ(r1),

one finds eventually:

e−iφτ(r2)ψ(r1)eiφτ(r2) = eiφθ(r2−r1)ψ(r1).

This yields for expression (5.16)

ψCS(r1)ψCS(r2) = eiφθ(r2−r1)eiφτ(r1)eiφτ(r2)ψ(r1)ψ(r2)

and similarly, interchanging r1 ↔ r2,

ψCS(r2)ψCS(r1) = eiφθ(r1−r2)eiφτ(r2)eiφτ(r1)ψ(r2)ψ(r1).

With ψ(r1)ψ(r2) = −ψ(r2)ψ(r1), θ(r2−r1) = θ(r1−r2)+π et [τ(r1), τ(r2)] =0, one finds

ψCS(r1)ψCS(r2) = −eiφπψCS(r2)ψCS(r1), (5.18)

and also

ψCS(r1)ψ†CS(r2) + eiφπψ†

CS(r2)ψCS(r1) = δ(r1 − r2). (5.19)

Comparing those expressions to equation (5.13), one sees that φ plays therole of the statistical angle α, and Chern-Simons transformations are found toallow changing particle statistics. Notice moreover that the choice φ = 2s+1transforms fermions into bosons, while φ = 2s does not change the particlestatistics, as is the case for the CF theory discussed above.


5.2.2 Fractional charge and fractional statistics

The topics discussed in the previous section are directly related to the Berryphase, which is a “geometrical” phase the particle wave function acquireswhen the particle is adiabatically displaced along a path in parameter space.An example of Berry phase is that due to the Aharonov-Bohm phase, whichappears when a particle with charge e∗ follows a path ∂Σ = C enclosing asurface Σ

Γ = −e∗

h

∮

∂Σdr · ACS(r) = −e

∗

h

∫

Σd2rBCS(r), (5.20)

where the gauge field is that of the Chern-Simons transformation. In thiscase the Berry phase is an operator, because the “magnetic field” BCS isproportionnal to the density operator. Within the mean field approximationequation (5.10), BCS = hφnel/e, one finds

Γ = 2πe∗

eφN(Σ), (5.21)

where N(Σ) is the number of electrons enclosed within the surface Σ.We now ditinguish three cases:

• In the first case (the most simple one), the particle moving on the pathC around a Laughlin liquid is an electron of charge e∗ = e, it acquiresa phase which is a multiple of Γ = 2πφ. When φ is an integer, as is thecase for Laughlin’s theory, this pahse remains a multtiple of 2π.

• When it is a quasi-particle with a fractional charge, e.g. e∗ = e/φ, theacquired phase acquise is again a multiple of Γ = 2π.

• The most interesting case is that when there is one (or a number of)particle(s) added to the Laughlin liquid within the surface Σ, withinthe path followed by the quasi-particle. Remember that the quasi par-ticle at z0 has a wave function, in Laughlin’s theory, which is obtainedby multiplying by the factor

∏

j(zj − z0). In terms of Chern-Simonstransformations de Chern-Simons, this can be modelled by the trans-formation

UV (r) = eiq∫

d2r′θ(r−r′)ρV (r′),

where ρV (r) is the density of quasi-particles with vorticity q = ±1.For a single quasi-particle at r0, one would have ρV (r) = δ(2)(r − r0).Just as in the case of the Chern-Simons transformation, in contrast


with the Laughlin (or Jain) wave function, this transformation attachesa singular phase to the quasi-particle, without attaching the propermodulus of the zero.

In terms of gauge transformation, one has ACS(r) → ACS(r)−(h/e)∇f(r)when the wave function transforms as ψ(r) → exp[if(r)]ψ(r). The to-tal vector potential is thus

ACS(r) → ACS(r) − hq

e∇∫

d2r′θ(r − r′)ρV (r′),

and the relationship between the magnetic field and the densities (5.9)now writes

BCS(r) = [∇× ACS(r)]z = −heφρ(r) +

hq

eρV (r). (5.22)

The first term gives rise to the same Berry phase Γ as in the case of aquasi-particle circling around a Laughlin liquid enclosed within Σ, andthe second term adds a phase ∆Γ dueto the presence of possible quasi-particles within Σ. Suppose that we have exactly one quasi-particle atposition rV ∈ Σ, ce qui donne

∆Γ =e∗

h

∫

Σd2r

hq

eδ(r − rV ) = 2π

e∗

eq = 2π

q

2s+ 1, (5.23)

for the case where e∗ = e/(2s+ 1) ( Laughlin quasi-particle). One seesthus that cahrge fractionalisation generates engendre egalement unefractional statistic in 2D, since the statistical angle associated with theBerry phase ∆Γ is

α = qe∗

e=

q

2s+ 1. (5.24)

Looking for experimental evidence for fractional statistics is an active fieldof research to this day. The quasi-particle fractional charge has already beenobserved in tunneling experiments [45] as weel as in shot noise experiments[46]. In the latter, one brings the two Hall bar edges close to one another byapplying an adequate gate voltage on electrodes. When the two edges aresufficiently close to one another, a quai-particle may be back-scatterd fromone edge to the other , and its charge gives rise to a characteristic shot noise.The measured quasi-particle charge ν = 1/3 is e/3 [46], and at ν = 2/5 theircharge is e/5.


Because of the close relationship between fractional charge and fractionalstatistics, it is no easy task to find separate evidence for each. Various deviceshave been recently proposed for ν = 5/2, which seems to correspond to astate with non abelian statistics [48]. A discussion of these ideas is beyondthe scope of these lectures.

Chapter 6

Hamiltonian theory of theFractional Quantum Hall Effect

The theory of FQHE follows nowadays distinct and complementary paths.Following Laughlin [12, 20], one appoach concentrates on writing down many-body wave functions, and studying their properties by numerical means, suchas exact diagonalisations, Monte-Carlo computation, Density Matrix Renor-malization Group, and so forth. However, the most powerful computer todate cannot easily handle the large Hilbert space involved with more than12 particles. Tricks may allow to simulate up to 24 particles. Such methodsoften allow to reach definite conclusions about the relative stability of stateswith different symmetries. Sometines, however, the thermodynamic limit isnot available, and doubts linger about the final conclusions of computationson small size particles clusters. It is useful, both for a more transparentunderstanding of the physics at hand, and for possible comparison with nu-merics, to be able to conduct analytical approaches to the theory in thethermodynamic limit. Even though the accuracy may be less satisfactorythan for exact results on small number of particles, it is interesting to havea theory where finite size effects do not blur the conclusions. Such an ana-lytical theory in the thermodynamic limit is the Hamiltonian approach, thefirst challenge of which is to attack the degeneracy problem.

We have discussed some aspects of Chern-Simons theories in the previouschapter. We notice that the transformation (5.2) deals with the kinetic partof the Hamiltonian only, not with the interaction term, which is invariantunder Chern-Simons transformations. However, as we discussed in chapter4, the FQHE is due to a lifting of the fractionally occupied Landau level

85

86 FQHE Hamiltonian theory

degeneracy by interactions among electrons. In fact the model of electronsrestricted to a single LL occupation [equation (4.17)] involves the interactionterm only. There seems to be a contradiction in the theory. This criticism toChern-Simons theory is however too severe. Indeed, the Chern-Simons theoryaims at replacing the true repulsive Couomb interaction by a statisticalinteraction expressed by the concept of flux attachment to electrons. Thisgenerates a singularity in the N particle wave function when two particlesattempt to sit at the same point in space [see equation (5.12)]. The physicalnotion behind the Chern-Simons manipulation is that the non perturbativemany-body effect of Coulomb interactions is to generate collective effects inthe form of flux tubes attached to electrons, in such a way as to renormalizethe effective external magnetic field. The latter renormalisation results inthe degeneracy lifting for fractional filling of the LLL which is a key factor toaccount for the FQHE. Chern-Simons theories have thus to be credited withan interesting step forward.

A difficulty remains however in the mean field version (5.10) which renor-maises the magnetic field, B → B∗ = B(1 − φν): the energy separationbetween the new levels (LL∗) is hω∗

C = heB∗/m. This energy scale whichinvolves the electronic mass m cannot be correct. The physics imposes (4.17)the energy scale e2/ǫlB, which does not depend on m. It is thus appropriateto seek a more satisfatory theory beyond mean field, which should yield thecorrect energy scale in a natural fashion. Various approaches have been pro-posed such as a random phase approximation [16, 49] which renormalises themass m → m∗ to get the right energy scale in the limit ν → 1/2 (p → ∞).An alternative approach, which is discussed in this chapter, is the FQHEHamiltonian theory [50, 51, 105]. We shall concentrate on the formulationdue to Murthy and Shankar [51].

6.1 Miscroscopic theory

This section deals with the connection between Chern-Simons theories andthe effective model described by equation (4.17), in the framework of a mi-croscopic formulation of the Hamiltonian theory. The main focus is on thetreatment of the fluctuations of the vector potential, (4.17), using the a newtheoretical tool, i.e. a new quantum gauge field a(r). The latter objectamounts to resorting to new unphysical degrees of freedom, which have to beremoved in a suitable way, i.e. by imposing constraints on the new system.

Microscopic theory 87

The various steps involved in this approach are fairly involved technically, sowe try and provide as much physical insight as possible. Readers with lessinterest in the mathematical details may directly go to section 6.2 where theend product, the effective theory is discussed in simpler terms. That sectiondoes not rely on the microscopic theory developped in the present one.

6.1.1 Fluctuations of ACS(r)

Using first quantisation 1 the Chern-Simons hamiltonien (5.5) is written asa sum over particles j = 1, ..., Nel

HCS =1

2m

∑

j

[pj + eA∗(rj) + eδACS(rj) + ea(rj)]2 . (6.1)

The mean value of the Chern-Simons vector potential has been absorbedin an effective “external” vector potential, A∗(r) = A(r) + 〈ACS(r)〉, andfluctuations δACS(r) have been singled out. They are connected to densityfluctuations through equation (5.9). Equivalently, in Fourier space, (h ≡ 1)

δACS(q) =2πφ

e|q|δρ(q)e⊥, (6.2)

where δρ(q) =∑

j exp(−iq ·rj)−nel and e⊥ = iez×e‖, with e‖ = q/|q|2. Al-though the Hamiltonian 6.1looks like that of a non interacting particle Hamil-tonian, the presence of δACS(q) introduces the full many body character ofthe problem because δρ(q) depends on all electron co-ordinates. The lastterm in equation (6.1) represents a new transverse gauge field, ∇ · a(r) = 0,which has been introduced, for the time being, artificially. Its quantum char-acter, which corresponds to the quantum character of the operator δρ(q) isensured by the introduction of its conjugate (longitudinal) field P(r), withthe canonical commutation rules

[a(q), P (−q′)] = iδq,q′ , (6.3)

where, in Fourier space

a(q) = a(q)e⊥, P(q) = iP (q)e‖.

1We use here for simplicity, a first quantisation approach.2δACS(q) is indeed transverse to q in order to recover the correct direction for its curl

88 Hamiltonian theory of the FQHE

We have artificially enlarged the Hilbert space with the degrees of freedomof this new gauge field. The physical sub-space is that of states |ϕphys〉 whichare annihilated by a(q),

a(q)|ϕphys〉 = 0. (6.4)

What is the advantage of this formal operation? Just as momentum px isthe translation generator in direction x (see section 2.2.2), the conjugate fieldP(q) is a translation generator for vector potentials. Therefore one may usethe transformation

Ua = ei∑

q′ P(−q′)δACS(q′) (6.5)

to get rid of the Chern-Simons vector potential fluctuations by a vector po-tential translation

U †aa(q)Ua = a(q) − δACS(q), U †

aa(r)Ua = a(r) − δACS(r), (6.6)

The latter equation is derived using (6.3) and Baker-Hausdorff formula(2.41).Because of the constraint (6.2), the transformation is also under the actionof the gradient operator(the momentum in r representation), since

Ua = exp

i2πφ

e

∑

q′P (−q′)

1

|q′|

∑

j

e−iq′·rj − nel

.

One gets therefore

[−i∇j + eA∗(rj) + eδACS(rj) + ea(rj)]Ua

= Ua

[

−i∇j + eA∗(rj) + ea(rj) +2πφ

e∇j

∑

k

P (−q)1

|q|e−iq·rk

]

= Ua

[

−i∇j + eA∗(rj) + ea(rj) +2πφ

eP(rj)

]

,

and for the transformed Hamiltonian

HCP =1

2m

∑

j

[

pj + eA∗(rj) + ea(rj) +2πφ

eP(rj)

]2

. (6.7)

The index CP means that the Hamiltonian is transformed in a basis of Com-posite Particles. In second quantized notation, fields ψCS(r) = UaψCP (r).Notice that the constraint (6.4) is also transformed to


[

a(q) − 2πφ

e|q|δρ(q)

]

|ϕphys〉 = 0. (6.8)

The physical interpretation of this constraint is as follows: instead of treat-ing density fluctuations exactly (or, equivalently, those of the Chern-Simonsvector potential), one describes them by the new quantum field a(q).3 Sincea(q) · P(q′) = 0 (remember that a(q) is a transverse field and P(q) is alongitudinal one), the last two terms of the Hamiltonian (6.8) represent aharmonic oscillator coupled to the sector of charged particles in a field B∗,which is the meaning of the first two terms . To be sure, the Hamiltoniancontains three terms, HCP = HB∗ + Hosc + Hcoupl : that of charged particlesin an effective magnetic field B∗ (this is the expression of the Chern-Simonstheory at the the mean field approximation level ,

HB∗ =1

2m

∑

j

[pj + eA∗(rj)]2 , (6.9)

that of a harmonic oscillator, which represents the density fluctuations

Hosc =1

2m

∑

j

e2a2(rj) +

(

2πφ

e

)2

P 2(rj)

, (6.10)

and a coupling term

Hcoupl =1

m

∑

j

[pj + eA∗(rj)] ·[

ea(rj) +2πφ

eP(rj)

]

. (6.11)

Let us first neglect the coupling term, and discuss the model with the firsttwo terms only. The diagonalization of the Hamiltonian with the couplingterm will be treated in the next section. Consider the harmonic oscillatorterm. It can be re-written, using ρ(r) =

∑

j δ(r − rj) = nel + δρ(r),

Hosc =1

2m

∫

d2rρ(r)

e2a2(r) +

(

2πφ

e

)2

P 2(r)

3This procedure is not new in theoretical physics: in the context of path integrals,there is some similarities with the Hubbard Stratonovich transformation. The theory ofone-dimensional electron sytems resorts to bosonisation, where bosons represent charge orspin fluctuations. Indeed, Murthy and Shankar themselves were influenced, in their ownwords, by the treatment of plasmons in an interacting Fermi liquid, as initially proposedby Bohm and Pines [53].

90 Hamiltonian theory of the FQHE

≃ nel

2m

∑

q

e2a2(q) +

(

2πφ

e

)2

P 2(q)

.

In the last step, we have neglected terms of order 3 in density fluctuations,4

which are taken into account at the level of the harmonic approximationO(δρ(q)2). Now introduce the ladder operators for the harmonic operator

A(q) ≡ 1√

4πφ

[

ea(q) + i2πφ

eP (q)

]

, (6.12)

which obey[

A(q),A†(q′)]

= δq,q′ . (6.13)

The Hamiltonian (6.10) now writes

Hosc = ω0

∑

q

A†(q)A(q), (6.14)

with the characteristic frequency

ω0 = 4πφnel

2m= νφωC . (6.15)

The oscillator ground state is the usual gaussian

χosc = exp

(

− e2

4πφ

∑

q

a2(q)

)

, (6.16)

up to a normalisation factor.In the absence of a coupling term, the N particles wave function is the

product of the oscillator wave function, and that of N charged particles ina field B∗: ψCP (rj) = χosc(rj)φp(rj), where φp(rj) is unique (nondegenerate) for ν∗ = p. Due to the constraint (6.8), the oscillator wavefunction can be written as:

χosc = exp

[

− φ2

∑

q

δρ(−q)2π

|q|2 δρ(q)

]

.

The exponent can be regarded as a Hamiltonian for charged particles inter-acting with a 2D Coulomb potential, v(q) = 2π/q2 ↔ v(r) = − ln |r|. We

4Remember that a(q) ∼ δρ(q),because of the constraint (6.8).


recover here the 2D single component plasma , as discussed in section 4.2.This leads in real space to the expression (with lB ≡ 1)

χosc(rj) = eφ2

∑

j,k

∫

d2rd2r′[δ(r−rj)−nel] ln |r−r′|[δ(r′−rk)−nel]

= C∏

k<j

|rk − rj|φe−φν∑

k|rk|2/4, (6.17)

where C is a proportionnality constant , and we used nel = ν/2π. Notice [seeequation (5.12)], that the singular phase has already been attached throughthe Chern-Simons transformation and the the N electrons wave function thuswrites, in complex coordinates notation ,

ψ(zj) =∏

k<j

(zk − zj)φ e−φν

∑

k|zk|2/4φp(zj). (6.18)

When φ = 2s, i.e. for fermionic statistics we retrieve the Laughlin (4.30), aswell as Jain’s wave function (4.66), up to the LLL projection. The last step,in order to complete the comparison to those wave functions, requires thecorrect magnetic length to be inserted in the gaussian. To this end, noticethat φp(zj) represents the N partciles wave function for an integer fillingfactor ν∗ = p in a field B∗ = [∇ × A∗(r)]z. The characteristic magnetic

length inφp(zj) is l∗B =√

h/eB∗, the gaussian factor in the wave functionis thus

exp

(

−∑

k

|zk|24l∗2B

)

= exp

[

−(1 − νφ)∑

k

|zk|24lB

]

,

where we used l2B/l∗2B = B∗/B = (1− νφ). In the end, the gaussian factor in

the electronic wave function (6.18) is then

exp

(

−∑

k

|zk|24l2B

)

,

as expected for electrons in a magnetic field B.

6.1.2 Decoupling transformation at small wave vector

We are left now with the coupling term between particles (6.9)and oscillators(6.10), the latter term representing collective density modes q. We onlysketch here the main steps of the decouplig derivation for small wave vectors,


|q|lB ≪ 1. The interested reader is refered for more details to the Murthy-Shankar review, especially to cond-mat/0205326v2.

The total Hamiltonian (without the interaction term), now writes

HCP =∑

j

Πj,+Πj,−2m

+ ω0

∑

q

A†(q)A(q) + θω0

∑

q

[

c†(q)A(q) + c(q)A†(q)]

,

(6.19)

where θ =√

πφ/4πnel, c(q) =∑

j q−Πj,+ exp(−iq · rj) et Πj,± = pj,± +eA∗

±(rj). Vector operators are written here in complex notation, V± = Vx ±iVy, et q± = e‖ ·ex± ie‖ ·ey is the unit vector in the direction of q, in complexnotation. To the usual hrmonic oscillator commutation rules (6.13), we haveto add

[Πi,−,Πj,+] = 2eB∗δi,j (6.20)

and

[c(q), c†(q′)] = −2eB∗∑

j

e−i(q−q′)·rj + O(q) ≃ −2eB∗δq,q′ , (6.21)

where the last equation is analogous to a Random Phase Approximation,which is again equivalent to neglecting terms of order 0(q3): density fluctu-ations are taken care of at the gaussian approximation level. Corrections toequation (6.21) would be of higher order than O(δρ(q)2).

In Hamiltonian (6.19), the coupling between particles described by op-erators Πj,± [or c(q) et c†(q)], and the oscillator fields A(q) or A†(q) islinear. The canonical transformation which decouples the Hamiltonian hasthe following form:

U(λ) = eiλS0 = exp

λθ∑

q

[

c†(q)A(q) − c(q)A†(q)]

, (6.22)

Where we want eventually to chose the “flow” parameter λ in a convenientway, leaving it undetermined for the time being. An operator transformedthrough (6.22), Ω(λ) = exp(−iλS0)Ω(λ = 0) exp(iλS0), may be derived fromthe derivative

dΩ

dλ= −ie−iλS0 [S0,Ω] eiλS0 . (6.23)

For operators c(q) and A(q), which occur in Hamiltonian (6.19), this leadsto the flow equations

dA(λ,q)

dλ= −θc(λ,q) et


dc(λ,q)

dλ= −µ

2

θA(λ,q),

with µ2 ≡ 2eB∗nelθ2 = 1/2ν∗, integrated, with initial conditions c(q) =

c(λ = 0,q) et A(q) = A(λ = 0,q), as

A(λ,q) = cos(µλ)A(q) − θ

µsin(µλ)c(q), (6.24)

andc(λ,q) =

µ

θsin(µλ)A(q) + cos(µλ)c(q). (6.25)

This transformation may be interpreted as a rotation in Hilbert space, whichmixes the c(q) degrees of freedom (particles) and A(q) degrees of free-dom (oscillators). The transformed Hamiltonian is thus derived insertingequations(6.24) and (6.25) in Hamiltonian, (6.19), and from the transforma-tion of its first term with the same method (integration of the differentialequation(6.23)). The end result contains many terms, and, even though thecalculation is straightforward, we only show here its global form,

HCP (λ) =∑

j

Πj,+Πj,−2m

+∑

q

α(λ)c†(q)c(q) (6.26)

+β(λ)A†(q)A(q) + γ(λ)[

c†(q)A(q) + c(q)A†(q)]

.

Decoupling is achieved by chosing λ = λ0 such that the implicit equation

γ(λ = λ0) = 0 ⇒ tan(µλ0) = µ. (6.27)

is satisfied. The detailed derivation [51] yields moreover

β(λ0) = ωC et α(λ0) = − 1

2mnel

,

for the other parameters in Hamiltonian (6.26). Finally, the decoupled Hamil-tonian , adding the interaction term, V [ρ(λ0,q)](which depends on the trans-formed density), writes

HCP =∑

j

Πj,+Πj,−2m

− 1

2mnel

∑

j,k

∑

q

Πj,+e−iq·(rj−rk)Πk,− +

∑

j

eB∗

2m

+ωC

∑

q

A†(q)A(q) + V [ρ(λ0,q)]. (6.28)

Some remarks are here in order:


• The oscillator frequency is given by the LL energy difference, ωC , inagreement with Kohn’s theorem which states that collective density ex-citations in the limit q → 0 oscillate at this frequency, in the presenceof translation invariant interactions. In the strong field limit, the os-cillators represent high energy excitations, which condense at T = 0 inthe lowest energy mode. The fourt term then becomes an unimportantconstant, which can be subsequently ignored.

• The third term indicates that particles posess a magnetic momente/2m.

• The sum on wave vectors is limited by the number of oscillators∑

|q|≤Q =nosc. This introduces a cut in Q at large wave vector. In principle, thenumber of operators has not been specified. However, if one focuses ondiagonal terms j = k in the sum in the second term of equation (6.28),this choice influences the effective mass m/m∗ = (1 − nosc/nel) in

∑

j

Πj,+Πj,−2m∗ ,

which lumps together the first and the second term. The natural choiceseems to be nosc = nel, which results in the vanishing of the kineticenrgy term (apart from the non diagonal term j 6= k). This is whatone expects for the dynamics of electrons projected on a single LL[equation (4.17)]. The choice nosc > nel would lead to the unphysicalresult of a negative effective mass, and nosc < nel would not make itvanish. Another justification for the choice nosc = nel (ou Q = kF )willbe given later on, in the discussion of the effective theory (section 6.2).

• There remain non diagonal terms (j 6= k) in the kinetic part. With thecut at Q = kF , those can be rewritten, using

∑

|q|<Q

e−iq·(rj−rk) = δ(rj − rk) −∑

|q|>Q

e−iq·(rj−rk),

and thus as a sum of a term which is zero for j 6= k and a term which isonly relevant at large wave vectors |q| > Q, and which may be neglectedat small wave vectors, |q|lB ≪ 1. Within this approximation, there isno kinetic term any more.

6.2. EFFECTIVE THEORY AT ALL WAVE VECTORS 95

Eventually, the Hamiltonian (6.28), which describes the low energy dy-namics, becomes

HCP (λ) =∑

j

eB∗

2m+ V [ρ(λ0,q)]. (6.29)

This result is fairly satisfactory, since it only contains the interaction term,apart from a constant which plays no dynamical role. To complete the dis-cussion, one must compute the transformed density, ρ(λ0,q), as well as theconstraint, the form of which is also alterd by the transformation. The den-sity is derived following the same procedure as for c(λ,q) and A(λ,q),, fromthe integration of the differential equation (6.23), while the constraint is givenby

ρ(λ,q) =e|q|√

4πφ

[

A(λ,q) + A†(λ,−q)]

.

The final result [51] is, to lowest order in |q|lB

ρ(λ0,q) =∑

j

e−iq·rj

[

1 − il2Bq ∧ Πj

1 + c

]

+c|q|√

4πφ

[

A(q) + A†(−q)]

, (6.30)

where the parameter c is connected to the decoupling parameter, at ν∗ = p,

c2 = cos2(λ0µ) =pφ

pφ+ 1. (6.31)

As regards the constraint, one gets

χ(λ0,q)|ϕphys〉 = 0, with χ(λ0,q) =∑

j

e−iq·rj

[

1 + il2Bq ∧ Πj

c(1 + c)

]

.

(6.32)The constraint does not involve oscillators A(q), only particles. That is anecessary condition for a complete decoupling of Hamiltonian (6.29). Thedensity operators (6.30) however still contain a contribution from oscillators,but the latter vanish on the average when they condense in the ground state〈A(q)〉 = 〈A†(q)〉 = 0, as we assume is the case in the following.

6.2 Effective theory at all wave vectors

The connection with the model (4.17) becomes even clearer when one at-tempts to construct a theory at all wave vectors, which coincides with the


small wave vector theory in the limit |q|lB ≪ 1. This generalisation is basedon a rather daring piece of reasoning: suppose that expressions ( 6.30) and(6.32) represent the first term in the expansion of an exponential. We wouldthen have 5

ρ(q) =∑

j

e−iq·R(e)j et χ(q) =

∑

j

e−iq·R(v)j , (6.33)

with

R(e) =(

x+Πy

1 + c, y − Πx

1 + c

)

et R(v) =

(

x− Πy

c(1 + c), y +

Πx

c(1 + c)

)

.

(6.34)The components of those new operators satisfy the commutation rules

[

X(e), Y (e)]

= il2B, et[

X(v), Y (v)]

= −i l2B

c2. (6.35)

A comparison with equation (2.16) shows that we may interpret R(e) as theguiding center of an electron while R(v) seems to be that of a second particlewith charge −c2, in terms of the electronic charge. The associated densitiesare automatically projected on the LLL, and the final model is 6

H =1

2

∑

q

v0(q)ρ(−q)ρ(q), (6.36)

with the commutation rules for densities

[ρ(q), ρ(k)] = 2i sin

(

q ∧ k

2

)

ρ(q + k),

[χ(q), χ(k)] = −2i sin

(

q ∧ k

2c2

)

χ(q + k), (6.37)

[χ(q), ρ(k)] = 0 et χ(q)|ϕphys〉 = 0.

This is the same model as that discussed in section 4.1.2, where one has addedthe constraint associated with the density χ(q) and its quantum algebra.

5The bars over the symbols mean that we are dealing with generalised densities at allwave vectors.

6we discuss here the LLL, n = 0.To generalise the Hamiltonian theory to higher LL,one needs only substituting the effective potential, v0(q) → vn(q).

Effective theory 97

y

x

(e)

(v)R

R

e x Π lBz2

Figure 6.1: Composite Fermions in teh Hamiltonian theory.

What is the physical content of the model, in the framework of CF pic-ture, supposing that χ(q) describes the density of a second species of particle,which we will call “ pseudo-vortex”? The “pseudo” in this expression indi-cates that this particle lives in a larger Hilbert space, and that projectionto the physical space is necessary, which is the case for states which areannihilated by χ(q). Moreover, this particle, which does not appear in tehHamiltonian, has no dynamics.

• The electron and the pseudo-vortex guiding centers are at a distance∼ |Π|l2B ∼ lB away from one another, which gives rise to a dipolarmoment d = −eez × Πl2B (figure 6.1).

• The CF may be pictured as a composite of an electron and a pseudo-vortex. Notice that the latter has been indirectly introduced by theoscillator degrees of freedom , a(q) or A(q). The choice nosc = nel, aswe discussed earlier, may thus be interpreted as equating the numberof electrons and the number of pseudo-vortices. As a result there are asmany CF as electrons. The CF charge is the sum of the elctron chargeand of the pseudo-vortex one, , i.e. e∗/e = −(1 − c2).

• The pseudo-vortex is an excitation of the electron gas, which is com-posed of the true elementary excitations. This can be checked on equa-tion (6.34): the guiding centers of both particle species are expressed

98 FQHE and the Hamiltonian theory

in terms of electronic co-ordinates x, y and Πx,Πy. This is the physicalmeaning of the constraint

χ(q)|ϕphys〉 = 0.

Notice that we could have started with the model (4.17) (electronsrestricted to a single LL). Instead of resorting to the oscillator gaugefield a(q) and its conjugate field as additional variable, we could haveintroduced the field χ(q) in the effective model (4.17). The microscopictheory discussed in the previous section is nevertheless useful, becauseit has established the connection with Chern-Simons theories.

• The limit p → ∞ or ν → 1/2 yields a charge c2 = 1 for the pseudo-vortex. The CF at ν = 1/2 are electrically neutral, but they have adipolar moment d(ν = 1/2) = −ekF l

2B [105, 54].

6.2.1 Approximate treatment of the constraint

In spite of the relative formal simplicity of the Hamiltonian theory (6.37), itis difficult to treat the constraint explicitly in computations. For a simplertreatment, Murthy et Shankar proposed a ”short-cut” which amounts toreplacing the projected density by a ”preferred combination”

ρCF (q) = ρ(q) − c2χ(q), (6.38)

in the Hamiltonian. A priori, all combinations such as ργ(q) = ρ(q)−γχ(q),with arbitrary γ are equivalent because of the constraint. The advantage ofchosing γ = c2 is that the matrix elements of the projected density operatorobey 〈N |ρCF (q)|0〉 ∝ q2 in the limit q → 0. This is the condition for thestructure factor S(q, ω) =

∑

N |〈N |ρCF (q)|0〉|2δ(ω − EN) to vary as q4 atsmall wave vector, which is demanded by the LLL projection [35, 91].

Notice that the preferred combination short-cut is a valid approximationfor gapped states, such as occur at ν = p/(2sp + 1). The constraint mustbe explicitly dealt with, however, in the sudy of ν = 1/2, which has a com-pressible strange Fermi liquid ground state [16]. Another problem with thepreferred combination is that the Hamiltonian no longer commutes strictlywith the pseudo-vortex density. The model remains however weakly gaugeinvariant because the commutator vanishes in the sub-space defined by theconstraint.


There is another algebraic argument in favor of the preferred combination,i.e. γ = c2 : with this choice, the ρ(q) algebra (6.37) is correctly reproducedto lowest order in q,

[

ρCF (q), ρCF (k)]

≃ i(q ∧ k)ρCF (q + k)

≃ 2i sin

(

q ∧ k

2

)

ρCF (q + k) + O(q3, k3).

Higher order terms in q are suppressed in the Hamiltonian, because of thegaussian in the effective potential (4.18). From now on, the ρCF (q) operatoris interpreted as the CF density. Physically, modes associated with internalCF structure are neglected in this approximation.

The CF basis is introduced by transforming variables R(e) et R(v) in CFguiding center , R(CF ), and cyclotron variable, η

(CF ). This transformationmust ensure that the new variables satisfy commutation rules in terms of theCF magnetic length, l∗B = 1/

√1 − c2,

[

η(CF )x , η(CF )

y

]

= −il∗2B et[

X(CF ), Y (CF )]

= il∗2B , (6.39)

in analogy with the electronic variables (2.16). The appropriate transforma-tion is [51]

R(CF ) =R(e) − c2R(v)

1 − c2et η

(CF ) =c

1 − c2

(

R(e) − R(v))

⇔ R(e) = R(CF ) − η(CF )c et R(v) = R(CF ) − η

(CF )/c. (6.40)

As in the electronic basis (section 4.1), the CF density operator may bewritten in second quantized form as

ρCF (q) =∑

j,j′;m,m′〈m|e−iq·R(CF )|m′〉〈j|ρp(q)|j′〉c†j,mcj′,m′ , (6.41)

where states |j〉 are associated to the operator η(CF ) and states |m〉 to R(CF ).

The first matrix element in this expression is identical to that (4.9) deducedin section 4.1.1, in terms of the CF magnetic length, for m ≥ m′,

〈m|e−iq·RCF |m′〉 = e−|ql∗B |2/4

√

m′!

m!

(

−iql∗B√2

)m−m′

Lm−m′m′

(

|ql∗B|22

)

.


The second one writes , for j ≥ j′,

〈j|ρp(q)|j′〉 ≡ 〈j|eiq·η(CF )c − c2f(q)eiq·η(CF )/c|j′〉

=

√

j′!

j!

(

iql∗Bc√2

)j−j′

e−|ql∗B |2c2/4 (6.42)

×[

Lj−j′

j′

(

|ql∗Bc|22

)

− c2(1−j+j′)e−|q|2/2c2Lj−j′

j′

(

|ql∗B|22c2

)]

.

The gaussian f(q) = exp(−|q|/2c2) in the second term takes into accountthe pseudo-vortex magnetic length, which is different from the electronic one.The operators c†j,m and cj,m, with cj,m, c†j′,m′ = δj,j′δm,m′ and cj,m, cj′,m′ =0, are respectively the CF creation and annihilation operators in state |j,m〉 =|j〉 ⊗ |m〉.

Because of the commutation rule (6.39) for the guiding center co-ordinates,each state occupies a minimal surface 1/nB∗ = 2πl∗2B , in analogy with theelectronic case. There exist thus AnB∗ states per “ CF LL” j, and the fillingfactor for CF LL, ν∗ = nel/nB∗ is connected to the electronic filling factorthrough equation (4.68), ν = ν∗/(2sν∗ + 1). When p CF levels are filled ,ν∗ = p, the ground state can be described by the average

〈c†j,mcj′,m′〉 = δj,j′δm,m′Θ(p− 1 − n), (6.43)

with Heavyside function Θ(x) = 1 for x ≥ 0 and Θ(x) = 0 for x < 0. This isno small progress. There was no way in the electron basis to define a startingstate from which a perturbation treatment might be conducted, except forν = n. In contrast, we only need ν∗ = p, i.e. ν = p/(2sp + 1), in the CFmodel.

6.2.2 Energy gaps computation

The ground state (6.43) can now be used to compute various physical quanti-ties, such as quasi-particle gaps or activation gaps[51]. This is a simple task,once the model is established. As the quasi-particle is a CF added in level pwhen p CF LL are completely filled, its energy relative to the ground stateis

∆qp(s, p) = 〈cp,mHc†p,m〉 − 〈H〉 (6.44)

=1

2

∑

q

v0(q)〈p|ρp(−q)ρp(q)|p〉 −∑

q

v0(q)p−1∑

j=0

|〈p|ρp(q)|j〉|2,

Effective theory 101

where averages, defined with respect to the ground state are computed asWick’s contractions(6.43). Similarly, the CF quasi-hole energy is in levelp− 1

∆qh(s, p) = 〈c†p−1,mHcp−1,m〉 − 〈H〉 (6.45)

= −1

2

∑

q

v0(q)〈p− 1|ρp(−q)ρp(q)|p− 1〉

+∑

q

v0(q)p−1∑

j=0

|〈p− 1|ρp(q)|j〉|2.

The activation gap , i.e. the energy needed to create a non interacting quasi-particle/hole pair,is

∆a(s, p) = ∆qp(s, p) + ∆qh(s, p). (6.46)

The figure 6.2 shows the results for the activation gaps, compared to nu-merical computations, taking into account a finite sample width in directionz. The finite width alters the effective potential, v0(q) → v0(q)f(q), wherevarious prescriptions have been put forward for the correcting factorf(q) :Zhang and Das Sarma haved used Yukawa type potential, which leads to

f(q) = e−qλ, (6.47)

where λ is the width parameter [55]. In this picture two particles cannot getcloser than this width: The distance r in the Coulomb potential is replacedby

√r2 + λ2. Alternatively , on may simulate the finite width by a par-

abolic confinement potential, which leads to a gaussian exp(−z2/4λ2) whichmultiplies the wave function [56], and the correcting factor becomes

f(q) = eq2λ2

[1 − Erf(qλ)] , (6.48)

where Erf(x) is the error function. The general tendency of the factors whichtake into account the sample finite width is to cause a lowering (in modulus)of the characteristic Coulomb energy. This leads to a lowering of activationgaps (se figure 6.2), and also of quasi-particle or quasi-hole gaps. Comparedto numerical results, the Hamiltonian theory overestimates activation gapsby a factor 1, 4 to 2 for λ = 0, but the agreement becomes better for largerλ values.


70x10-3

60

50

40

30

20

10

0

δ

3.02.52.01.51.00.50.0

λ

'PMJ p=3' Hamiltonian Theory p=3' 'PMJ p=4' Hamiltonian Theory p=4'

0.25

0.20

0.15

0.10

0.05

0.00

δ

3.02.52.01.51.00.50.0

λ

'PMJ p=1' 'Hamiltonian Theory p=1' 'PMJ p=2' 'Hamiltonian Theory p=2'

(a)

gaps

d’a

ctiv

atio

nga

ps d

’act

ivat

ion

100x10-3

80

60

40

20

0

Act

ivat

ion

Gap

s

1.41.21.00.80.60.40.20.0

Width parameter b

2/5

Exact diag. 2/5 This theory 2/5 Exact diag. 3/7 This theory 3/7

(b)

paramètre de largeur

gaps

d’a

ctiv

atio

n

Figure 6.2: After reference [51]. Activation gaps as functions of the sample width indirection z, in units of e2/ǫlB . (a) comparison between Hamiltonian theory and numericalcomputation by Park et al. in the framework of Jain’s functions [57]. A Yukawa typepotential has been used to simulate the finite width [equation (6.47)]. (b) comparisonto exact diagonalisation results [56], in the approximation of a parabolic confinementpotential [equation (6.48)].


ν = 4/11 ν = 1+1/3*

Figure 6.3: Second generation Composite Fermions at ν∗ = 1 + 1/3 (ν = 4/11). TheCF2, formed in the partially filled CF level, is a bound state of a “first geberation” CFand a CF vortex carrying two flux quanta. The CF in the lowest level are inert .

6.2.3 Self similarity in the effective model

Until now, we have focused the discussion on correlated electrons at fillingfactors ν = p/(2sp+ 1). We have seen that CF theory allows to understandthe FQHE at those filling factors in terms of quasi-particles, which are CF, inthe mean field approximation of the Hamiltonian theory. Indeed the latterallows to discuss a non degenerate ground state at ν∗ = p , which has pcompletely filled CF levels. This was not possible within the electron model,because of the huge degeneracy in the lowest LL. In that sense, the FQHEof electrons may be interpreted as a CF IQHE at ν∗ = p.

It is natural to ask what is the situation when the CF LL themselves havefractional filling at ν∗ 6= p – here again we face the huge degeneracy problemin the CF model. How is this degeneracy lifted by the residual interactionsbetween CF? The motivation behind the question is the discovery of a FQHEat a fraction ν = 4/11, which corresponds to a CF filling factor ν∗ = 1+1/3.Assume, as is evidenced by experiments that the state is fully spin polarised:7

In that case the first excited CF LL is 1/3 filled, and it is tempting to interpret

7In the case of a partially spin polarised state a FQHE at ν = 4/11 may also beunderstood in the framework of Halperin’s wave function [19].


this state in terms of a FQHE (a Laughlin state) of CF. The CF in the levelp = 1 would bind to a CF vortex carrying two additional flux quanta, givingrise to a “second generation “ CF, see figure 6.3), in analogy with the CFformation in the electron basis and the pseudo-vortex of the electronic liquid.

The Hamiltonian theory is a perfect framework to treat the case of apartial filling of interacting CF level. Formally we may use the same approx-imations as in the deduction of the CF model in the electron basis restrictedto a single LL(section 4.1), i.e. only excitations in the same LL are takeninto account, and inter CF LL excitations belong to a higher energy sector.This approximation is somewhat less justified in the last case because theonly relevant energy scale is e2/ǫlB, both in the CF LL formation and inthe residual interactions. In fact the justification of this approximation willappear later on, due to the appearance of a “small parameter” due to thecharge renormalisation. The restriction to the CF LL then yields for thedensity operator

ρCFp (q) = FCF

p (q)¯ρ(q), (6.49)

where¯ρ(q) =

∑

m,m′〈m|e−iq·RCF |m′〉c†p,mcp,m′ (6.50)

is the projected density operator of CF, and

FCFp (q) ≡ 〈p|ρp(q)|p〉, (6.51)

in terms of matrix elements (6.42), is the “CF form factor” of level p. Asfor the electronic form factor(4.12), it can be absorbed in the effective CFinteraction potential, which leads to

vCFs,p (q) = v0(q)

[

FCFp (q)

]2(6.52)

=2πe2

ǫqe−q2l∗2B /2

[

Lp

(

q2l∗2B c2

2

)

− c2e−q2/2c2Lp

(

q2l∗2B

2c2

)]2

.

Then one finds for the interacting CF Hamiltonian

HCF =1

2

∑

q

vCFs,p (q)¯ρ(−q)¯ρ(q). (6.53)

As in the electronic case the commutator (6.39) between guiding centercomponents R(CF ) induces the operator algebra for ¯ρ(q),

[¯ρ(q), ¯ρ(k)] = 2i sin

(

(q ∧ k)l∗2B

2

)

¯ρ(q + k). (6.54)


This together with Hamiltonian (6.53), defines the interacting CF model.The latter has the same structure as that for electrons restricted to a singlelevel– one has to replace the effective interaction potential by the potentialvCF

s,p (q) and to use the CF magnetic length, l∗B. This self-similarity at thelevel of the model structure can account, under certain conditions, for theexperimental Hall curve self-similarity, which was initially noticed by Maniand v. Klitzing, using a scaling transformation [59]. The difference betweenthe spatial variation of the interaction potentials between CF and betweenelectrons in a LL n (4.18), indicates that this self-similarity of the Hall curve isnot an automatic by-product of the mathematical self similarity of the model.Because of the latter, one expects the formation of incompressible quantumliquids which would be the CF analogues of Laughlin liquids. Formally, suchliquids may be dubbed CF2. The CF2 basis is deduced from the model(6.53), in the same manner as CF are deduced from the electronic model(6.36) and (6.37) : the Hilbert space is enlarged with the CF pseudo vorticesdegrees of freedom ¯χ(q), which carry 2s flux quanta, and which have a chargec2 = 2ps/(2ps+ 1). This leads to a new constraint , ¯χ(q)|ϕphys〉 = 0, for thephysical states |ϕphys〉. The pseudo-vortex operator components satisfy thealgebra

[¯χ(q), ¯χ(k)] = −2i sin

(

q ∧ k

2c2l∗2B

)

¯χ(q + k), (6.55)

induced by the commutation rules for the pseudo-vortex guiding center com-ponents Rv−CF ,

[

Xv−CF , Y v−CF]

= −il∗2B /c2. In order to describe the CF2,

which is built of a first generation CF located at the guiding center R(CF )

and a pseudo-vortex at Rv−CF (se figure 6.3), one introduces a new pre-ferred combination, ρC2F (q) = ¯ρ(q)− c2 ¯χ(q). The CF2 global charge is thuse = (1 − c2)e∗ = (1 − c2)(1 − c2), in units of the electron charge −e. TheCF2 cyclotron variable and guiding center, respectively η

C2F et RC2F , arededuced from the CF guiding center and CF pseudo-vortex in the same man-ner as for CF [see equation (6.40)]. The new preferred combination, whichis interpreted as the CF2 density, is written in second quantized notation

ρC2F (q) =∑

j,j′;m,m′〈m|e−iq·RC2F |m′〉〈j|ρp(q)|j′〉d†j,mdj′,m′ , (6.56)

where operators d†j,m and dj,m, together with dj,m, d†j′,m′ = δj,j′δm,m′ and

dj,m, dj′,m′ = 0, are respectively the CF2 creation and annihilation oper-ators in state |j,m〉. The matrix elements in equation (6.56) are the same


2 4 6 8 10

0.1

0.2

0.3

0.40.40

0.030.30

0.02

0.01

1 3 5 7 9 11 13 15 17 19

2

m

m

électrons

FC

FC

Pse

udo−

pote

ntie

ls

V0.004

0.002

Figure 6.4: Pseudo-potentials for the electronic interaction (black curve), for CF (graycurve) and for CF2 (clear gray curve ), in units of e2/ǫlB . Notice the scale difference onthe z axis.

as for CF [Eqs. (6.41) and (6.42)] if we replace l∗B → lB = l∗B/√

1 − c2 =

1/√

(1 − c2)(1 − c2), the CF2 magnetic length, and c2 → c2. A QuantumHall Effect would then be expected to appear for some CF filling factorsν∗ = p + p/(2sp + 1), where the integer p is the number of completely filledCF2 LL. Such a QHE may be interpreted both as a CF FQHE [58] or as a CF2

IQHE [60]. The filling factors ν∗ are related to the electronic filling factorsthrough relation (4.68), and ν∗ = 1 + 1/3 corresponds thus to ν = 4/11.

The formalism thus described is generic and may be iterated for the nextCF generations. One finds in that manner a hierarchy of states which isdifferent from the Haldane and Halperin hierachies [37, 43], at filling factorswhich are determined by the recurrence relation

νj = pj +νj+1

2sj+1νj+1 + 1, (6.57)

where sj+1 is the number of pairs of flux quanta carried by the pseudo-vortexin the (j+1)-th generation of CF (CFj+1), and pj is the number of completelyfilled FCjs levels. The FCjs IQHE is determined by νj = pj. Formally, theelectronic filling factor corresponds to j = 0 and thus ν0 = ν and ν1 = ν∗.Equation (6.57) is a generalisation of the relation between electronic fillingfactors and those of CF [Eq. (4.68)].

Although the recurrence formula 6.57 for CF hierarchical states suggestsa large number of FQHE, only a limited number are observable in practice.


Indeed, the FQHE due to first generation CF is retsricted to the two lowestLL, n = 0 and 1, while in higher Landau levels, such liquid states competewith electronic solids, as will be discussed in the next chapter. Apart fromthis competition with other phases, one may state certain general stabilitycriteria for higher generation CF (CFj+1). The first one is clearly the stabilitycriterion for the ”parent” state, (CFj), a necessary condition. For n = 2, forexample, the Laughlin liquid is not stable at ν = 1/3, and CF2 are not formedat ν = 4/11. A more restrictive condition for the CFj+1 formation is givenby the form of the effective interaction potential of FCj,

v(j)si,pi(q) =

2πe2

ǫqe−q2/2

j∏

i=1

[

FCiFsi,pi

(qli)]2, (6.58)

in terms of FCj magnetic length lj =√

2sjpj + 1lj−1. Expanded in Haldanepseudo-potentials [see equation (4.54)],

V jm =

∑

q

v(j)si,pi(q)Lm(q2l2j )e

−q2l2j /2,

the interaction must be sufficiently short ranged , i.e. V1/V3 >∼ 1, 2, for aLaughlin state to be stabilized.

Pseudo-potentials with odd indices 8 are plotted in figure 6.4 for electronsin LL n = 0, cf with s = p = 1 and CF2 with s = p = s = p = 1. Noticethe difference in scale on the energy axis: the interaction between CF isroughly one order of magnitude smaller than that for electrons. This is eas-ily understood when looking at the effective CF interaction potential (6.52),which is globally reduced by the CF form factor , [FCF

p (q)]2 ≃ (1 − c2)2, atorder O(q0), compared to the potential between electrons. As two factors ofthis type enter the expression for the CF2 effective interaction potential [seeequation (6.58)], the latter is again an order of magnitude smaller than thatbetween CF. In this sense, (1−c2)2 ≤ 1/9 may be interpreted as the hierachi-cal CF theory small parameter – as discussed earlier, this is a posteriori anindication for the CF LL stability with respect to residual CF interactions.

A second remark is about the specific form of the CF (and CF2) inter-action. Contrary to the electronic case, their pseudo-potentials do not varymonotonically but exhibit a minimum at m = 3. A possible origin of this

8Remember that only odd index pseudo-potentials matter in the case of fully spinpolarised electrons, because of their fermionic statistics (see section 4.2).


peculiarity is the CF dipolar character, due to their internal structure, asalready discussed at the beginning of section 6.2. Since the pair correlationfunction of the Laughlin liquid (with s = 1) is maximum for that value ofthe relative kinetic moment, one may expect this pseudo-potential form tostabilise CF Laughlin liquids.

Chapter 7

Spin and Quantum Hall Effect–Ferromagnetism at ν = 1

Until now we have by-passed all questions connected to the electron spin.We have been satisfied with the notion that the Zeeman effect separatesLL in two spin branches, the energies of which are separated by a gap ∆z

(see figure 4.1). Within this picture, there would be no difference betweenthe IQHE at ν = 2n (both spin branches filled) and that at ν = 2n + 1(only the lower spin branch filled ) – in both cases a plateau would be dueto localisation of additional electrons. The only difference would be themagnitude of the excitation gap. Indeed, since ∆z ≃ hωC/70, the two spinbranches are not resolved in weak magnetic fields, for which only the IQHEat ν = 2n is observed. In that case, the system behaves as if each state wasdoubly degenerate, with |n,m;σ〉 (σ =↑, ↓). However this picture turns outto be incorrect. Interactions between electrons have important effects, evenfor ν = 2n+ 1. A new form of quantum magnetism arises. That is the topicof this chapter.

7.1 Interactions are relevant at ν = 1

Let us start with a discussion of the various energy scales. Notice first that ,since Landau Level quantization only deals with orbital degrees of freedom,the mass which enters the LL energy separation heB/mb is the band mass,mb = 0.068m for GaAs in terms of the bare massm of the electron. The latterdetermines the Zeeman gap ∆z = gheB/m, since the Zeeman effect deals

109

110 Spin and Quantum Hall Effect – Ferromagnetism at ν = 1

with the electron spin, an internal degree of freedom. Moreover the effective gfactor for GaAs is g = −0, 4, which causes the Zeeman gap in this material tobe a factor roughly 70 smaller than the LL separation, as already mentionnedearlier. Expressed in Kelvins, the Zeeman gap is ∆z = 0, 33B[T]K, while theLL separation is hωC = 24B[T]K, where the magnetic field is measured inTesla. On the other hand, the characteristic Coulomb energy is e2/ǫlB =

50√

B[T]K. For a field 6T, which corresponds roughly to filling ν = 1, one

finds thus ∆z ≃ 2K ≪ e2/ǫlB ≃ 120K < hωC ≃ 140K. Interactionsare therefore of the same order of magnitude as the LL separation, and aremore relevant than the Zeeman gap. They must be taken into account whendiscussing effects connected to the electron spin, in particular at ν = 1, whichwe will focus on in the remaining parts of this chapter.

The first problem is to understand why we observe a Quantum Hall Effectat all at this filling factor. The Zeeman gap is so small that each state isalmost degenerate, so we might expect a macroscopic degeneracy at ν = 1 inthe kinetic part of the hamiltonian. Just as for the FQHE, interactions areresponsible for the lifting of this degenaracy. So we are led to this counterintuitive idea that the IQHE at ν = 1 should rather be looked at as a specialcase of FQHE.

7.1.1 Wave functions

Let us start with a two spin 1/2 particles wave function, at ν = 1. Inthe symmetric gauge, the orbital part is built from the single particle wavefunction, in the symmetric gauge φm(z) = zm, (neglecting normalisationfactors) with m = 0, 1 (2.28). As for the spin function, we have four possiblestates for the coupling of two spin 1/2 particles: an antisymettric singlet ,|S = 0,M = 0〉 = (| ↑↓〉 − | ↓↑)/

√2, and a symmetric triplet |S = 1,M〉,

avec |S = 1,M = 1〉 = | ↑↑〉, |S = 1,M = 0〉 = (| ↑↓〉 + | ↓↑〉)/√

2 and|S = 1,M = −1〉 = | ↓↓〉. We are dealing with a problem without explicitspin-dependent potentials. The interaction is SU(2)invariant, the total spinis a good quantum number. Since the fermionic wave function must beantisymmetric, we have

ψS=0(z1, z2) = φs(z1, z2) ⊗ |S = 0,M = 0〉 and

ψS=1(z1, z2) = φa(z1, z2) ⊗ |S = 1,M〉,

Interactions are relevant at ν = 1 111

with φs(z1, z2) = z01z

12 + z0

2z11 = z1 + z2 and φa(z1, z2) = z0

1z12 − z0

2z11 =

z1 − z2. The second choice with an antisymmetric orbital wave functionis energetically favourable if the interaction is sufficiently strongly repulsiveat short range, as is the case for the Coulomb interaction. Thus Coulombinteractions, combined with the Pauli principle, create an exchange forcewhich aligns spins. This is the origin of ferromagnetism in transition metals.

It is important to realize that the ν = 1 Laughlin wave function, (4.30)

φν=1(zi) =∏

i<j

(zi − zj)e−∑

k|zk|2/4 (7.1)

is in fact the orbital part of a ferromagnetic N-particles wave function: (4.30),

φν=1(zi) =∏

i<j

(zi − zj)e−∑

k|zk|2/4| ↑, ↑, .... ↑〉. (7.2)

This wave function is the lowest energy state if the Zeeman effect is strong.At first sight, a spin excitation in this state would cost the Zeeman energy.In fact, because of the exchange effect, a strong cost in spin flip-energy ariseseven if the Zeeman effect vanishes (this can actually be done experimentallyby applying external pressure in Ga As samples.) Let us in fact imagine thatto be the case. Then there would be no reason for the total spin to be alongthe z direction, because no external potential breaks the Hamiltonian SU(2)symmetry (if the Zeeman effect vanishes). The most general spin functiondescribing the most general orientation for the total spin is

ψs(θ,ϕ) =∏

m

(

cosθ

2e−iϕ/2e†m,↑ + sin

θ

2eiϕ/2e†m,↓

)

|0〉, (7.3)

where e†m,σ creates an electron in the state |n = 0,m;σ〉. We have chosen theparametrisation in terms of two angles, θ et ϕ which respects the normalisa-tion |σ〉 = u| ↑〉+v| ↓〉 avec |u|2+|v|2 = 1, in order to establish the connectionbetween the SU(2) and O(3) description of the rotation group. This allowsto introduce immediately the magnetisation field at the m Landau site:

n(m) =

sin θm cosϕm

sin θm sinϕm

cos θm

, (7.4)

This will be useful to describe the low energy degrees of freedom within theeffective model (section 8.3).


The wave function in equation (7.2) corresponds to θ = 0. It has aquantum number M = N/2 which is the z component of the total spin.Therefore the total spin is S = N/2, since M ≤ S. Other states, character-ized by θ 6= 0, have −N/2 ≤ M ≤ N/2 [ψs(θ, ϕ), with θ et ϕ arbitrary],are obtained from the former by applying rotation operators in the SU(2)representation (equation 7.3).

In the state described by equations 7:02b, each particle is surrounded byan “exchange hole”, due to the Pauli principle when all spins are aligned.This lowers the Coulomb energy per particle. For filling factor ν = 1

〈VCoulomb〉N

= −√

π

8

e2

ǫlB≈ 200K (7.5)

This is an order of magnitude larger than the Zeeman splitting and is themechanism which strongly stabilizes the ferromagnetic state, and would doso even if the Zeeman effect was zero.

Even though the same mechanism is at work in all simple and transi-tion metals, the latter are not all ferromagnets, because the kinetic energydispersion relation opposes the ferromagnetic polarisation: the broader theband, the larger the kinetic energy cost to produce a finite spin polarisation;only transition metals with the most narrow d-band, Fe, Co and Ni exhibit aferromagnetic state. In the completely filled Landau level, the kinetic energyis frozen; there is no kinetic energy cost in spin polarizing the interactingelectron gas. This is why the ν = 1 IQH state is the “best understood itiner-ant ferromagnetic state” in condensed matter physics. A last remark beforegoing over to the next section: the introduction of the spin degree of fredomis a special aspect of a more general problem: that of the QHE in multi-component sytems [74]. Multi-component systems include systems with spindegrees of freedom, but also iso-spin such as layer index in a bilayer system,or valley index in Si, AlAs or graphene.

7.2 Algebraic structure of the model with spin

In this section, we generalize the model of electrons restricted to a singlelevel , as introduced in section 4.1, to include an internal degree of freedomwith SU(2) symmetry. For the physical spin case, the interaction is SU(2)

Algebraic structure of the model with spin 113

invariant, but we may consider a more general case 1 vσσ′0 (q). For a global

rotation symmetry, v↑↑0 (q) = v↓↓0 (q) et v↑↓0 (q) = v↓↑0 (q). The former moregeneral interaction which may break the SU(2) symmetry will be useful inthe next chapter, where the layer index in a bi-layer will be viewed as aniso-spin index s = 1/2, with | ↑〉 for a state in the upper layer and | ↓〉 forthe lower layer. The interaction Hamiltonian writes now

H =1

2

∑

σ,σ′

∑

q

vσ,σ′0 (q)ρσ(−q)ρσ′(q), (7.6)

where2

ρσ(q) =∑

m,m′fm,m′(q)e†m,σem′,σ (7.7)

is the electron density with spin σ projected on the LLL , with fm,m′(q) =〈m|f(q)|m′〉 the matric element for the projected density operator matrixelement f(q) = exp(−q · R), in first quantised forme [see equation (4.7)].The total electron density is thus written

ρ(q) = ρ↑(q) + ρ↓(q)

=∑

σ,σ′

∑

m,m′fm,m′(q)δσ,σ′e†m,σem′,σ′ , (7.8)

where δσ,σ′ represents the identity 12×2. In comparison with the case withoutinternal degree of freedom , the electron density is replaced by

f(q) → f(q) ⊗ 12×2 ∼ fm,m′(q) ⊗ δσ,σ′ ,

where the right hand part is the matrix representation which is relevantfor the second quantised notation. Similarly, we find the spin densities byreplacing the identity by the SU(2) generators Sµ = τµ/2, where τµ are thePauli matrices ,3 with [τµ, τ ν ] = iǫµνστσ/2 and (τµ)2 = 1. We define

fµ(q) = f(q) ⊗ Sµ, (7.9)

1We have already seen that the interaction potential in a LL with arbitrary n is ob-tained by replacing the gaussian by the form factor, 4.12, exp(−q2/2) → [Fn(q)]2, in theexpression for the effective potential

2We omit for simplicity the index n = 0 in the electron operators for the LLL.3The greek indices refer to the 3D space directions x, y and z. Since we have a Euclidian

space, we do not specify co- or contra-variant vectors, and Einstein summation is the rulefor repeated indices. The symbol ǫµνσ is the unit antisymmetric tensor :1 for µ, ν, σ =x, y, z and all cyclic permutations, −1 for all other permutations and 0 if any index isrepeated.


and the spin densities are written, in second quantized form, as

Sµ(q) =∑

σ,σ′

∑

m,m′

[

fm,m′(q) ⊗ Sµσ,σ′

]

e†m,σem′,σ′ (7.10)

In the case with no internal degree of freedom, the algebraic structure wasderived from equation (4.13) using the commutation rules in first quantisation

[f(q), f(q′)] = 2i sin

(

q ∧ q′

2

)

f(q+q′) → [ρ(q), ρ(q′)] = 2i sin

(

q ∧ q′

2

)

ρ(q+q′).

Now using the same procedure, we have to compute [Sµ, ρ(q′)] and [Sµ(q), Sν(q)],using

[fµ(q), f ν(q′)] = f(q)f(q′) ⊗ SµSν − f(q′)f(q) ⊗ SνSµ (7.11)

=1

2([f(q), f(q′)] ⊗ Sµ, Sν + f(q), f(q′) ⊗ [Sµ, Sν ])

and[fµ(q), f(q′)] = [f(q), f(q′)] ⊗ Sµ. (7.12)

We have

[f(q), f(q′)] = 2i sin

(

q ∧ q′

2

)

f(q + q′),

f(q), f(q′) = 2 cos

(

q ∧ q′

2

)

f(q + q′)

and

[Sµ, Sν ] = iǫµνσSσ,

Sµ, Sν =1

2δµν ,

which yields

[ρ(q), ρ(q′)] = 2i sin

(

q ∧ q′

2

)

ρ(q + q′), (7.13)

[Sµ(q), ρ(q′)] = 2i sin

(

q ∧ q′

2

)

Sµ(q + q′) and (7.14)

[Sµ(q), Sν(q′)] =i

2δµν sin

(

q ∧ q′

2

)

ρ(q + q′) + iǫµνσ cos

(

q ∧ q′

2

)

Sσ(q + q′).

(7.15)

7.3. EFFECTIVE MODEL 115

The equations (7.13)-(7.15) are the SU(2) extensions of the magnetic trans-lation algebra (4.16).

The Hamiltonian (7.6) is written

H =1

2

∑

q

vSU(2)(q)ρ(−q)ρ(q) + 2∑

q

vsb(q)Sz(−q)Sz(q), (7.16)

with potentials

vSU(2)(q) =1

2

[

v↑↑0 (q) + v↑↓0 (q)]

et vsb(q) =1

2

[

v↑↑0 (q) − v↑↓0 (q)]

.

(7.17)The first term in the Hamiltonian (7.16) is SU(2) invariant, while the secondone, if non zero, breaks explicitly the SU(2) symmetry. In the remainingparts of this chapter we discuss ferromagnetism in the physical spin case.The interaction is then SU(2) invariant. Only the (small) Zeeman term

HZ =gheB

mSz(q = 0)

breaks the Hamiltonian SU(2) symmetry.Equation (7.14) exhibits a remarkable property: because of the non com-

mutativity between spin and charge densities, the dynamics of both degreesof freedom are coupled. One can handle spins by acting on charges, and viceversa! This unusual property is connected to the quantum dynamics undermagnetic field, and has a simple expression because of the projection on asingle LL, which results in non commutativity of charge density operatorswith non parallel wave vectors [ equation (7.13)]. Spin-charge entanglementin the LL ferromagnetism is studied in more details in the followingsection.

7.3 Effective model

We discuss two types of spin excitations of the quantum ferromagnetic state:(a) spin waves (magnons) [figure 7.1(a)] and (b) spin textures which havea non zero topological charge (skyrmions) [figure 7.1(b)]. While spin wavesare the Goldstone modes, the energy of which goes to zero, in the limit ofno Zeeman effect, when the wavelength goes to infinity, skyrmions cannot becontinuously deformed to retrieve the ground state. They are topologicallystable objects, classified by an integer which is called their topological charge.


(b)

(a)z

yx

z

yx

Figure 7.1: Excitations in the ferromagnetic state . (a) Spin waves (magnons). Suchan excitation can be continuously deformed to the ground state with the same topologicalproperty of the magnetisation field, as can be seen on the Bloch sphere (right) whichrepresents a mapping of the spin configuration (bold face spins) in the plane on S2 – thegray line can be continuously deformed to a point (b) Skyrmion with non zero topologicalcharge. This excitation has a flipped spin at the origin , r = 0, and the ferromagneticstate is recovered at infinity |r| → ∞. Contrary to spin waves the mapping of this spinconfiguration on the sphere (bold face spins)covers the whole S2 surface once.

Effective model 117

7.3.1 Spin waves

The procedure to derive the spin wave spectrum is analogous to well knownother cases in localized or itinerant electron ferromagnets: one studies thetime evolution of the spin lowering operator:

S−q ≡

N∑

j=1

e−iqµrµj S−

j (7.18)

where rµj are the components of the position operator for the jth particle.

For an excitation in the lowest LL, one has to project this operator, and weobtain

S−q =

∑

m,m′

[

fm,m′(q) ⊗ S−↓,↑]

e†m,↓em′,↑ (7.19)

Then one computes the commutator of this operator with the Coulomb in-teraction Hamiltonian:

[

H, S−q

]

= (1/2)∑

k 6=0

v(k)[

ρ(−q)ρ(q), S−q

]

(7.20)

This is evaluated using the familiar commutator algebra. When applied tothe ground state, which is annihilated by ρk one obtains

[

H, S−q

]

|ψ〉 = ǫqS−q |ψ〉 (7.21)

where

ǫq ≡ 2∑

k 6=0

e−|k|22 v(k) sin2

(

q ∧ k2

)

.

This proves that S−q is an exact excited eigenstate of H with excitation

energy ǫq. In the presence of the Zeeman coupling, ǫq → ǫq + ∆. The onlyassumption is that the ground state at filling factor ν = 1 is fully polarised.

The dispersion relation is quadratic in q at small q:

ǫq ≈ ρsq2

with

ρs ≡1

2

∑

k 6=0

e−|k|22 v(k)|k|2

For very large q, sin2 can be replaced by its average value 1/2 so that

ǫq ≈∑

k 6=0

v(k)e−|k|22

The energy saturates at a constant value for q → ∞.


7.3.2 Skyrmions

For simplicity reasons, let us first discuss a topological excitation in a 2DXY ferromagnet (spins are constrained to lie in a plane). In that case atopological excitation is a vortex type defect which is labelled by the numberof times the spin direction rotates around the origin along a closed pathencircling the defect placed at the origin (figure 7.2). This number is thetopological charge, which does not depend on the path geometry, only on itshomotopy class. Consider the mapping of the circle S1, which parametrizesthe path in the physical plane around the defect we want to characterize, onthe circle S1 which parametrizes the spin orientation in the 2D spin space.The mappings which can be continuously deformed into one another formhomotopy classes. They form a group, the fundamental group, π1(S

1) = Z.The topological charge is Q ∈ Z.

For the XYZ ferromagnet, the parametrization of the most general spintexture maps on the surface of the Bloch sphere, S2. The 2D plane is mappedby stereographic projection on a S2 sphere. Just as for the case of theS1 → S1 mappings, the S2 → S2 mappings which can be continuouslydeformed into one another can be classified in homotopy classes, which forma homotopy group,

π2(S2) = Z.

Elements of the group, integers Q ∈ Z label different topological sectors.Spin waves deform continuously to the ground state, they belong to thetopological sector with topological charge Q = 0, while skyrmions carry atopological charge Q 6= 0 (figure 7.1).

A state with a certain spin texture |ψ[n(r)]〉 may be generated from theferromagnetic ground state by applying a spin rotation operator, using thegenerators (7.10) of the magnetic translation algebra (with internal SU(2)structure) [87]

|ψ[n(r)]〉 = exp

[

−i∑

q

Ωµ−qS

µ(q)

]

|ψFM〉, (7.22)

where |ψFM〉 is the ferromagnetic state(7.2), with a uniform magnetizationalong the z quantization axis. The functions Ωµ

q which enter expression (7.22)are, up to a permutation of the x and y axis, the Fourier transforms of n(r),

Ω(r) =∑

q

Ωqe−ir·q = ez × n(r). (7.23)

Effective model 119

Q = 1 Q = −1

Figure 7.2: Topological excitation in the xy model. The topological charge Q is thenumber of times the spin direction turns by 2π along a path (gray curve) which circlesaround the defect center(black point. Left : topological excitation with charge Q = 1 –spins turn in the same direction as the path indicated by the arrow. Right : topologicalexcitation with charge Q = −1.

This state looks like a coherent state [see section 2.2.2, especially equation(2.34)]. Using the Hausdorff series expansion [equation (5.17)], we have forthe transform of operator Sµ(q)

eiOSµ(k)e−iO = Sµ(k) + δSµk , (7.24)

where we have defined O =∑

q Ωµ−qS

µ(q) et

δSµk = i

[

O, Sµ(k)]

− 1

2

[

O[

O, Sµ(k)]]

+ ... (7.25)

If we limit the Hausdorff series expansion to second order, we find for theaverage magnetization in the spin texture (7.22)

Sµ[n(r)] = 〈Sµ(r)〉 + 〈δSµ(r)〉 =nel

2nµ(r),

as expected for the O(3) magnetization field. This justifies the choice of thefunction Ω(r) [equation (7.23)]. The technical details for the computationleading to this result are not given here. They are closely analogous to thosewe present in the following section for the computation of the charge densityinduced by the spin texture.

7.3.3 Spin-charge entanglement

For a better understanding of the spin-charge entanglement, which we men-tionned above in the discussion of the model algebraic structure (section 7.2),


we now compute the charge density change due to the spin texture (7.22),

δρq ≡ 〈eiOρ(q)e−iO〉 − 〈ρ(q)〉 ≃ i⟨[

O, ρ(q)]⟩

− 1

2

⟨[[

O, O, ρ(q)]]⟩

, (7.26)

where averages are defined with respect to the ferromagnetic ground statewith magnetization along z. Averages are computed using the commutationrules (7.13)-(7.15) and

〈ρ(q)〉 = nelδq,0 et (7.27)

〈Sµ(q)〉 = δµznel

2δq,0. (7.28)

One finds thus[

O, ρ(q)]

=∑

k

Ωµ−k

[

Sµ(k), ρ(q)]

= 2i∑

k

sin

(

k ∧ q

2

)

Ωµ−kS

µ(q + k)

⇒⟨[

O, ρ(q)]⟩

= 0,

since the argument of the sine vanishes for q ‖ k, and

[

O[

O, ρ(q)]]

= 2i∑

k,k′sin

(

k ∧ q

2

)

Ωµ′

−k′Ωµ−k

[

Sµ′(k′), Sµ(q + k)

]

= −∑

k,k′sin

(

k ∧ q

2

)

Ωµ′

−k′Ωµ−k

[

δµµ′sin

(

k′ ∧ (q + k)

2

)

ρ(q + k + k′)

+ 2ǫµ′µν cos

(

k′ ∧ (q + k)

2

)

Sν(q + k + k′)]

.

So the average is

⟨[

O[

O, ρ(q)]]⟩

= −nelǫµ′µz

∑

k

sin

(

k ∧ q

2

)

Ωµ′

k+qΩµ−k.

we get thus for the modified charge density (7.26), with ν = 2πnel,

δρq =ν

4π

∑

k

ǫµ′µz sin

(

k ∧ q

2

)

Ωµ′

k+qΩµ−k

≃ −ν8π

∑

k

ǫµ′µz(q ∧ k)Ωµ′

k+qΩµ−k

=−ν8π

∑

k

ǫµ′µz [i(k + q)] Ωµ′

k+q ∧ (−ik)Ωµ−k, (7.29)

Effective model 121

where we have expanded the sine in the second line, thus resorting to a longwavelength limit, i.e. a slow space varying modulation of the spin density.The Fourier transform back to real space yields the more compact result

δρ(r) =−ν8π

ǫµ′µz∇(2)Ωµ′

(r) ∧∇(2)Ωµ(r)

=−ν8π

ǫijn(r) · [∂in(r) × ∂jn(r)] , (7.30)

where roman indices, i, j = x, y, correspond to 2D space coordinates– not to be confused with the three spin vector components n(r). Com-pare the result (7.30)to the so-called Pontryagin topological charge density (Pontryagin index)

δρtop(r) =1

8πǫijn(r) · [∂in(r) × ∂jn(r)] . (7.31)

We see that the electric charge density is proportional to the Pontryaginindex, δρ(r) = −νδρtop(r). The topological charge is obtained by integrationover the physical plane

∫

d2r δρtop(r) = q ∈ Z. (7.32)

The electric charge of a topological excitation is thus

Q = eνµ. (7.33)

As was the case for the Laughlin quasi-particle, a skyrmion excitation atν = 1/(2s + 1) carries a fractional electric charge, Q = ±e/(2s + 1), for|µ| = 1. The connection to the Berry phase will be discussed in section 7.4.

7.3.4 Effective model for the energy

It is useful to set up a simple energy functional model for the energy of aspin structure. The energy of the state (7.22) with a O(3) magnetizationfield n(r), is computed in the same manner as the charge modulation, usingthe long wavelength expansion,

δE = i⟨[

O,H]⟩

− 1

2

⟨[[

O,H]]⟩

+ ... (7.34)

≃ −1

4

∑

q,q′,k

v0(k)Ωµ′

−k′Ωµ−k〈C〉,


where we have defined

C = 2[

Sµ′(q′), ρ(−k)

]

[

Sµ(q), ρ(k)]

+

ρ(−k),[

Sµ′(q′),

[

Sµ(q), ρ(k)]

]

= 8 sin

(

q′ ∧ k

2

)

sin

(

q ∧ k

2

)

Sµ′(q′ − k)Sµ(q + k)

−δµµ′sin

(

q ∧ k

2

)

sin

(

q′ ∧ (q + k)

2

)

ρ(−k), ρ(q′ + q + k)

−2ǫµ′µσ sin

(

q ∧ k

2

)

cos

(

q′ ∧ (q + k)

2

)

ρ(−k), Sσ(q′ + q + k)

.

The average of this expression is computed, with the help of the structurefactor

1

nel

〈ρ(−q)ρ(q′)〉 = δq,q′ s(q) (7.35)

and of1

nel

〈ρ(−q)Sµ(q′)〉 =δµz

2δq,q′ s(q) (7.36)

and1

nel

〈Sµ(−q)Sµ′(q′)〉 =

δµµ′δµz

4δq,q′ s(q). (7.37)

Finally we find

〈C〉 = 2nelδµ,µ′δq,−q′ sin2

(

q′ ∧ k

2

)

[s(q) − δµz s(q + k)] .

So we find for the energy (7.34)

δE = −nel

2

∑

q,k

v0(k)s(k) sin2

(

q′ ∧ k

2

)

Ωµ−qΩ

µq. (7.38)

In the long wavelength limit, the sine can be linearized, so we get the nonlinear O(3) sigma model

δE =ρS

2

∑

q

(−iq)Ωµ−q(iq)Ω

µq =

ρS

2

∫

d2r[∇n(r)]2, (7.39)

where the exchange stiffness

ρS = − ν

32π2

∫ ∞

0dkk3v(k)s(k), (7.40)

7.4. BERRY PHASE AND ADIABATIC TRANSPORT 123

is

ρS =1

16√

2π

e2

ǫlB(7.41)

at ν = 1, since s(k) = −1. Any magnetization variation incresases the energywith respect to the ground state, which indeed corresponds to ∇n(r) = 0, i.e.a ferromagnetic state with uniform magnetization. The energy dispersion atsmall wave vector

ω(q) ≃ ρS

2q2 (7.42)

is precisely the Goldstone mode (spin wave) energy discussed earlier. Thecomplete analytic expression for the latter is [88] [89]

ω(q) =

√

π

2

e2

ǫlB

[

1 − e−q2/4I0

(

q2

4

)]

, (7.43)

which coincides with expression (7.42) in the limit q ≪ 1.

7.4 Berry phase and adiabatic transport

In the previous parts of this chapter, skyrmions, topological excitations of theQuantum Hall ferromagnet, have been discussed on the basis of the Hamil-tonian theory, using the commutator algebra of the projected density opera-tors. A topological charge density (the Pontryagin density) is associated tospin texture. The properties of the homotopy group Π2(S2) ≡ Z indicatethat the electric charge carried by a topological defect is an integer in termsof the electron charge, at ν = 1.

This result is intimately connected to the Berry phase notion [90, 91]and to the notion of adiabatic transport in quantum mechanics. Considera quantum system, described by a Hamiltonian HR which depends on a setof external controlable parameters. This set is represented by a vector inparameter space R. We assume now that there exists a compact domain inparameter space where the ground state is separated of all excited states bya gap. What is the result of letting the system evolve slowly, with a slowvariation of R(t) along a closed loop in this part of parameter space, duringa time interval T? We have

R(0) = R(T ) (7.44)

124 Spin and Quantum Hall effect – ferromagnetism at ν = 1

If the evolution along the closed path is sufficiently slow, such that h/T <<∆min where ∆min is the minimum energy gap along the loop, the state evolvesadiabatically. This means that at all times, the system remains in the ground

state Ψ(0)R of the Hamiltonian HR(t).

Each point R in parameter space is associated to a complete set of eigenstates:

HRΨ(j)R = ǫ

(j)R Ψ

(j)R . (7.45)

The solution of the time dependent Schrodinger equation

ih∂Φ(r, t)

∂t= HR(t)Φ(r, t) (7.46)

is then:

Φ(r, t) = Ψ(0)R(t)(r)e

iγ(t)e− i

h

∫ t

0dt′ǫ0

R(t′) +∑

j 6=0

aj(t))Ψ(j)R(t) (7.47)

The adiabatic approximation amounts to neglecting the contribution ofexcited states represented by the second term on the right hand side. Thisbecomes exact in the limit of of a very slow variation of R(t) as long as theexcitation gap remains finite. Everything is well known at this point, exceptthe ”Berry Phase” γ(t). γ(t) can be determined by requiring Φ(r, t) to obeythe time dependent Schrodinger equation. The LHS of equation 7.46, if weneglect the aj(t) when j > 0 becomes:

ih∂Φ(r, t)

∂t=[

−hγ(t) + ǫ(0)R(t)

]

Φ(r, t)+ ihRµ

[

∂

∂RµΨ

(0)R(t)(r)

]

eiγ(t)e− i

h

∫ t

0dt′ǫ(0)

R(t′)

(7.48)The RHS of equation 7.46, within the same approximation is:

H~R(t)Φ(r, t) = ǫ(0)R(t)Φ(r, t) (7.49)

Using the completeness relation∣

∣

∣

∣

∣

∂

∂RµΨ

(0)R

⟩

=∞∑

j=0

∣

∣

∣Ψ(j)R (t)

⟩

⟨

Ψ(j)R(t)|

∂

∂RµΨ

(0)R(t)

⟩

. (7.50)

Here again the adiabatic approximation allows to neglect the contributionof excited states, so that equation 7.48 becomes:

ih∂Φ

∂t=

[

−hγ(t) + ihRµ

⟨

Ψ(0)R (t)| ∂

∂RµΨ

(0)R(t)

⟩

+ ǫ(0)R(t)

]

Φ (7.51)

Berry phase and adiabatic transport 125

Equation 7.46 is satisfied if

γ(t) = iRµ

⟨

Ψ(0)R(t)|

∂

∂RµΨ

(0)R(t)

⟩

(7.52)

Thanks to the constraint⟨

Ψ(0)R |Ψ(0)

R

⟩

= 1 we know that γ is real.

At first sight γ(t) could be chosen to vanish. In fact, for each R we havea different set of eigen functions, and one may choose the ground state phasearbitrarily. That is a kind of gauge choice in parameter space: γ(t) and γ inthat sense, are not gauge invariant. Chosing γ = 0 is then a gauge choice.What Berry [90] found is that this is not always possible. In certain casesimplying a closed path in parameter space, there is a finite gauge invariantphase, the Berry Phase,

γBerry ≡∫ T

0γdt = i

∮

ΓdRµ

⟨

Ψ(0)R | ∂

∂RµΨ

(0)R

⟩

. (7.53)

That quantity is ”gauge invariant” because the system returns to thedeparture point in parameter space, so that the arbitrary choice of gauge atthe start has no consequence. This is analogous to electrodynamics whenthe circulation of the vector potential is along a closed path, and equals theenclosed magnetic flux, which is gauge invariant. In fact it is useful to definethe ”Berry connection” , A, in parameter space:

Aµ(R) = i

⟨

Ψ(0)R | ∂

∂RµΨ

(0)R

⟩

(7.54)

which leads toγBerry =

∮

ΓdRµ · Aµ(R) (7.55)

The Berry phase is a purely geometric object, independent of the velocityRµ(t). It only depends on the path in parameter space. It is often easiest toevaluate this expression using Stoke’s theorem, since the curl of A is gaugeinvariant.

It is easy to check that in the case of electromagnetism and the Aharonov-Bohm effect, the Berry connection, A is, up to a multiplicative factor, q

h

(where q is the particle charge), the electromagnetic vector potential A:

Aµ(R) = +q

hAµ

(R) (7.56)


The Berry phase for a loop threaded by a magnetic flux φb is

γBerry =q

h

∮

dRµAµ = 2πφb

φ0

(7.57)

where φ0 is the flux quantum.A second example is of interest for the quantum Hall ferromagnetism.

Consider a quantum spin coupled to a magnetic field, with a Hamiltonian:

H = −~∆(t) · ~S (7.58)

The gap to the first excited state is h|~∆|. The circuit in parameter space

must avoid the origin ~∆ = 0 where the spectrum has a degeneracy. Duringthe adiabatic evolution of the ground state, one has

⟨

Ψ(0)~∆|~S|Ψ(0)

~∆

⟩

= hS~∆

|~∆|(7.59)

Thus if the orientation of ~∆ is defined by the polar angle θ and theazimuthal angle φ, the same must be true for < ~S >. For a spin S = 1/2, anappropriate set of states is:

|Φθ,φ >=

(

cos θ2

sin θ2eiφ

)

(7.60)

since these obey:

〈Φθ,φ|Sz|Φθ,φ〉 = hS

(

cos2 θ

2− sin2 θ

2

)

= hS cos θ (7.61)

and〈Φθ,φ|Sx + iSy|Φθ,φ〉 =

⟨

Φθ,φ|S+|Φθ,φ

⟩

= hS sin θeiφ (7.62)

What is the Berry phase in the case of a slow rotation of ~∆ around axisz, at constant θ?

γBerry = i∫ 2π

0dφ

⟨

Φθ,φ|∂

∂φΦθ,φ

⟩

(7.63)

= i∫ 2π

0dφ(

cos θ2

sin θ2e−iφ

)

(

0i sin θ

2eiφ

)

= −S∫ 2π

0dφ(1− cos θ) (7.64)

7.5. APPLICATIONS TO QUANTUM HALL MAGNETISM 127

= −S∫ 2π

0dφ

1∑

cos θ

d cos θ′ = −SΩ (7.65)

where Ω is the solid angle subtended by the path as viewed from the originof the parameter space. This is precisely the Aharonov-Bohm phase oneexpects for a charge −S particle traveling on the surface of a unit spheresurrounding a magnetic monopole. The degeneracy of the spectrum at theorigin is precisely the cause for presence of the magnetic monopole [90]

The definition of the connection A implies the existence of a singularityat the south pole, θ = π. A ”Dirac string” (i.e. an infinitely thin solenoidcarrying one flux quantum) is attached to the monopole. The singularitywould be attached to the north pole if we had chosen the basis

e−iφ|Φθ,φ > (7.66)

In order to reproduce correctly the Berry phase in a path integral for thespin the Hamiltonian of which is given by 7.58, the Lagrangian must be:

L = hS

−mµAµ + ∆µmµ + λ(mµmµ − 1)

(7.67)

where m is the spin coordinate on the unit sphere, λ is a Lagrange multiplierwhich enforces the length constraint, and the Berry connection A obeys:

~∇×A = ~m (7.68)

This Lagrangian reproduces correctly the equations for the spin dynamicswhich describe its precession.

7.5 Applications to quantum Hall magnetism

7.5.1 Spin dynamics in a magnetic field

In the following, we show that the Lagrangian above allows to describe thequantum spin dynamics in an effective field.

The equations of motion are:

d

dt

δLδmµ

=δLδmµ

(7.69)

Using 7.67 we haveδLδmµ

= −Aµ (7.70)


andδLδmµ

= −mν∂µAν + ∆µ + 2λmµ, (7.71)

so that∆µ + 2λmµ = Fµνmν , (7.72)

where Fµν = ∂µAν − ∂νAµ

The previous section on the Berry phase shows we must chose:

Fµν = ǫαµνmα (7.73)

which is equivalent to ~∇~m ∧ A[m] = m. The equation of motion becomes:

δµ + 2λmµ = ǫαµνmαmν (7.74)

Multiplying both members of equation 7.74 by ǫγβµmβ, then applying onboth sides of this equation the identity: ǫναβǫνλη = δαλδβη − δαηδβλ, we get:

−(

~∆ ∧ ~m)

γ= mγ −mγ(mβmβ) (7.75)

The last term vanishes, because of the constraint on the length of m. Us-ing Euler-Lagrange equations, we retrieve the spin precession equations in amagnetic field.

Compare 7.67 with the Lagrangian of a particle of mass m, and charge−e in a magnetic field with vector potential A:

L =1

2mxµxµ − exµAµ (7.76)

We see that the Lagrangian in 7.67 is equivalent to a Lagrangian of a zeromass object, with charge −S, placed at ~m, moving on a unit sphere containinga magnetic monopole. The Zeeman term is analogous to a constant electricfield −~∆, which exerts a force S~∆ on the particle. The Lorentz force due tothe monopole field drives the particle on a constant latitude orbit on the unitsphere. The absence of a kinetic term in mµmµ in the Lagrangian indicatesthat the particle has zero mass, and is in the lowest LL of the monopole field.

7.6 Application to spin textures

Consider a ferromagnet with a local static spin orientation m(r). When anelectron is displaced, one may assume that the strong exchange field forces

7.6. APPLICATION TO SPIN TEXTURES 129

the electron spin to follow the local orientation of m(r). If the electron hasa velocity xµ, the variation rate of the local spin orientation seen by theelectron is mν = xµ ∂

∂xµmν . This induces a non trivial Berry phase in the

presence of a spin texture. Indeed, the one particle Lagrangian contains anadditional term with a time derivative of first order, wich adds to the termdue to the field-matter minimal coupling term:

L′ = −exµAµ − hSmνAν [~m] (7.77)

The first term is the field-matter coupling, the second one gives rise to theBerry phase. We have for the latter ∇m ∧ A = ~m. That can be re-written,using mν ≡ xµ ∂

∂xµmν . Then

L′ = −exµ(Aµ + aµ) (7.78)

with

2πaµ = φ0S

(

∂

∂xµmν

)

Aν [~m] (7.79)

a is the Berry connection, a vector potential which adds to the magnetic fieldvector potential. The curl of a thus contributes a ”Berry” flux which addsto the magnetic field flux:

b = ǫαβ ∂aβ

∂xα(7.80)

= (φ0S/2π)ǫαβ ∂

∂xα

(

∂

∂xβmν

)

Aν [~m]1

2π

= φ0Sǫαβ[

(

∂

∂xα

∂

∂xβmν

)

Aν [~m]

+

(

∂

∂xβmν

)

∂mγ

∂xα

∂Aν

∂mγ]

The first term of the last equation vanishes by symmetry, which results in:

b = φ0Sǫαβ ∂m

ν

∂xβ

∂mγ

∂xα(1/2)F νγ (7.81)

with F µν = ǫαµνmα. We used the symmetry ν ↔ γ in the last surviving term.

With S = 1/2 one getsb = φ0ρ (7.82)


with

ρ ≡ 1

8πǫαβǫabcma∂αm

b∂βmc (7.83)

=1

8πǫαβ ~m · ∂α ~m ∧ ∂β ~m (7.84)

We recognize in 7.83 the Pontryagin topological density.If now, starting from a uniform magnetization, we deform the ground

state magnetization adiabatically into a spin texture, everything happens,for orbital degrees of freedom, as if flux from b(r) was injected adiabatically.In a quantum Hall state with ρii = 0 and ρxy = ν, the Faraday law thencauses this spin texture to attract (or repel) at the end of the process acharge density νρ. Since the skyrmion topological charge is an integer,

Qtop =∫

d2rρ(r) = integer,the charge associated to a skyrmion in the IQHE is δρ = −νe× (integer).We have thus recovered, as a result of the Berry phase, the result obtained

earlier by the Hamiltonian approach.

Chapter 8

Quantum Hall Effect inbi-layers

8.1 Introduction

In the previous chapters, we have emphasized the importance of Coulomb in-teractions in the Quantum Hall Effect physics, including for the ν = 1 fillingfactor of the LLL. Even if the Zeeman coupling vanishes, Coulomb interac-tions stabilize a ferromagnetic order, which has important consequences onthe excitations spectrum. Instead of a spin degenerate metal at νσ = 1/2,we have a quantum Hall ferromagnetic state with a gap.

An analogous effect occurs for bilayers ( a system of two coupled layers),where each layer has filling ν = 1/2. In that case, the role of spin is played bythe isospin index of each layer [87, 93, 94]. The analogy with the ferromag-netic monolayer system at ν = 1 will be extensively used in the following.Quantum Hall bilayer physics is quite rich, and involves coupling betweenlayers at various equal, or different, filling factors. This chapter focuses onthe particular case of two layers at ν = 1/2, for which exciting results havebeen obtained in the last few years.

Modern MBE techniques have allowed in the recent years to manufacture2D electron gases with high mobility, in bi-layers or multi-layers structures[95]. As shown on the figure 8.1, a bi layer is a system of two 2D electrongases organized in parallel layers, at a distance d from each other whichis comparable with the magnetic length, and to the average distance be-tween electrons in the layer (i.e. d ∼ 10nm ). We know that correlations

131

132 Quantum Hall Effect in bi-layers

W W

d

2t

Figure 8.1: Sketch of the conduction band profile for a two dimensional electrons systemin a bi-layer. The order of magnitude for the width, as well as for the distance, of the twolayers is W ∼ d ∼ 10nm. In the presence of a tunneling term t, the band splits into abonding, and an anti-bonding band, with a separation ∆SAS = 2t.

are especially important at high fields, when electrons occupy the LLL only,because the kinetic energy is then frozen out of the problem, and cannot op-pose the ferromagnetic polarisation. The FQHE results from gap formationbetween the ground state and excited states, resulting in an incompress-ible state. Theory predicts that gaps appear for certain fractional fillings inthe bilayer system when inter-layer interactions are strong enough [19, 96].Such predictions have been backed by experiments [97]. Recently [98], the-ory has predicted that inter-layer correlations could induce unusual brokensymmetry states, with a new type of inter-layer coherence. This new inter-layer coherence appears even in the absence of inter-layer tunneling, whenthe coupling between layers is of purely Coulombic origin. What appearshere is excitonic superfluidity, which is the unexpected realization, in thebi-layer physics, of the phenomenon of excitonic superfluidity predicted in3D semiconductors since 1962 [99]. This phenomenon has been looked forwithout unquestionable experimental success ever since [100] until it eventu-ally appeared in a spectacular manner in the Quantum Hall bi-layer system[87, 93, 94, 101, 102, 103, 104, 6].

8.2. PSEUDO-SPIN ANALOGY 133

8.2 Pseudo-spin analogy

We assume that the Zeeman effect saturates the “real” spin, so that it doesnot play any role any more. Each layer is given a “pseudo-spin” label ↑ or +for one layer, and ↓ or − for the other. In the situation we chose to discuss,ν↑ + ν↓ = 1.

A state having inter-layer coherence is a state with ferromagnetic pseudo-spin order, in a direction defined by the polar angle θ, and azimuthal angleφ. In the Landau gauge, such a state writes [see also equation (7.3)]

|ψ〉 =∏

k

(

cosθ

2c†k↑ + sin

θ

2eiφc†k↓

)

|0〉 . (8.1)

In the state described by this wave function, each state k is occupied by oneelectron, and has an amplitude cos(θ/2) to be in the ↑ layer, and amplitudesin(θ/2) exp(iφ) to be in layer ↓. Physically the ratio between the squaredamplitude may be altered by applying a voltage between the layers, so asto charge one layer at the expense of the other one, the total filling factorremaining equal to 1. Remember that in the Landau gauge, a state k labelsa state localized on a line at guiding center position Xk ≡ kl2B. We discussthe following cases:

Spins along the z axis. For θ = 0 this wave function describes a spin align-ment along the z axis,

|ψz〉 = Πkc†k,↑ |0〉 = Πi<j(zi − zj) |↑, ↑, ... ↑〉 .

The choice θ = 0 describes a situation where all particles are in the layerlabelled by ↑.

Spins along the x axis. One has θ = π/2, φ = 0, which yields the state

|ψx〉 = Πk

c†k↑ + c†k↓√2

|0〉 .

Obviously, this wave function which describes a symmetric superposition ofelectronic states in the two layers must have a low energy, compared to θ = 0,as soon as the Coulomb energy plays a role, as we shall see later on, and whenboth layers have the same potential.

Spins along a general direction in the xy plane. In that case, we have θ =

134 Quantum Hall bi-layers

π/2, φ 6= 0 and thus

|ψxy〉 = Πk

c†k↑ + eiφc†k↓√2

|0〉 .

That state, as the previous one describes a symmetric superposition of stateswith equal amplitude in each layer, but the resulting pseudo-spin magneti-zation has been rotated by an angle φ with respect to the x axis

In any case, the total occupation of each k state is 1, but the layer indexfor each electron is undetermined except when θ = 0. The amplitude for anelectron to be in layer ↑ is cos θ/2, the amplitude to be in layer ↓ is eiφ sin θ/2.The most general choice is to have neither θ = π/2 nor 0. The total weightremains equal to 1, since sin2(θ/2) + cos2(θ/2) = 1.

Even in the absence of quantum tunneling between layers (i.e. physicaltransfer of electron between layers), quantum mechanics, with the superposi-tion principle, allows to describe the possibility of the simultaneous presenceof an electron in both layers.

In the ferromagnetic monolayer situation (ν = 1), we have seen in theprevious chapter that in the absence of Zeeman coupling, the ferromagneticexchange coupling, due to its Coulomb origin, does not break the HamiltonianSU(2) symmetry. A different situation arises in the bi-layer case. In the nextsection, we list the various physical parameters which give its originality tothe bi-layer pseudo-spin ferromagnetism

8.3 Differences with the ferromagnetic mono-

layer case

What are the main differences between the bi-layer physics (with ν↓ = ν↑ =1/2) and the monolayer at ν = 1?

• When the two layers are far away from each other (d/lB ≫ 1), in-teractions between electrons in one layer and electrons in the otherare negligible. The layers are single layers with filling ν = 1/2, thereare two flux quanta per electron. A Composite Fermion construction,whereby two flux quanta are attached to each electron (see chapter 4and 6 and reference [16]) results in a problem where CF evolve in zeroeffective magnetic field, B⋆ = 0. This is a metallic state, CF organize

Differences with the ferromagnetic monolayer 135

in a strange Fermi Liquid, with circular Fermi surface, where each CFcarries a dipole [105]. Although this an interesting object in its ownright, we are not going to discuss this limit in these lectures. In anycase, there is a continum of particle-hole excitations above the Fermiliquid ground state, and there is no QHE

• At a distance comparable to the magnetic length, i.e. d ∼ lB, intra-layer and inter-layer Coulomb interactions, although different, becomecomparable, especially if d→ 0, in which case they become equal. Us-ing the pseudo-spin analogy, we can write the interaction Hamiltonian[see equation (7.6) in the previous chapter]

Hcoul =1

2

∑

σ,σ′

∑

q

vσ,σ′0 (q)ρσ(q)ρσ′(−q). (8.2)

we have therefore

v↑↑0 (q) = v↓↓0 (q) ≡ vA(q)

v↑↓0 (q) = v↓↑0 (q) ≡ vE(q)

where vA(q) [vE(q)] is the Fourier transform with respect to the pla-nar coordinates of the intra-layer [inter-layer ]interaction between elec-trons. Neglecting the physical width of the sample , we have vA(q) =(2πe2/q) exp(−q2/2) et vE(q) = e−qdvA(q).

Letting vSU(2)(q) = [vA(q) + vE(q)]/2 and vsb(q) = [vA(q) − vE(q)]/2,one may separate, in the interaction Hamiltonian, a part which is in-dependent of the pseudo spin, v0(q), and one which is not , as we havein equations (7.16) and (7.17) of the previous chapter. One finds then

H =1

2

∑

q

vSU(2)(q)ρ(−q)ρ(q) + 2∑

q

vsb(q)Sz(−q)Sz(q), (8.3)

with Sz(q) = [ρ↑(q) − ρ↓(q)]/2. Note that[

H, Sµ(q = 0)]

6= 0 for

µ = x, y, because of the second term, which breaks the SU(2) symmetry.

Since, for a finite separation between layers, we always have vA(q) >vE(q), for all wave vectors, the second term in equation (8.3) createsan easy magnetization plane, perpendicular to z, in the bi-layer plane.The pseudo-spin Hamiltonian symmetry is reduced from SU(2) to U(1).


• A third difference with the mono-layer at ν = 1 arises from the inter-layer tunnelling term. This term has the following expression:

T = − t

2

∑

k

(

c†k,↑ck,↓ + c†k,↓ck,↑)

(8.4)

= −t∑

k

Sxk . (8.5)

It acts as a pseudo-magnetic field applied along x. It stabilises thepseudo-spin orientation in direction x. Indeed the symmetric combina-tion of states in each potential well (bonding combination) correspondsto the spinor ψb = (1, 1)/

√2 (bonding state). The direction −x would

correspond to an anti-bonding state, ψa−b = (1,−1)/√

2

It is feasible experimentally to have widely different values of t between10−3 et 10−1 × e2/ǫlB. Note that the tunnel term, since it creates apreferred direction in the xy plane, breaks the U(1) symmetry in theHamiltonian.

• Apply a voltage bias between the layers. This generates a term −e(N↑−N↓)V = −2eSzV which is analogous to a magnetic field applied alongthe z quantization axis of the pseudo-spin. Minimizing the anisotropyterm added to the electric term, we see that V creates a charge unbal-ance between the two layers, which, in pseudo-spin language is a finitevalue for Sz = (N↑ −N↓)/2

• One may apply a magnetic field B‖ parallel to the plane of the bi-layer.Such a parallel field has no effect (except in terms of Zeeman gap) inthe monolayer case. One may expect new effects in the bi-layer case,which should be orbital effects (the real spin is expected to be entirelypolarised). Chosing a gauge for the associated vector potential, thepresence of B‖ can be taken into account as follows:

A|| =

Ax‖ = 0

Ay‖ = 0

Az‖ = Bx

, (8.6)

where z is the direction perpendicular to the layers. In that gauge, thegauge invariant tunnel term becomes

t→ tei 2π

φ0

∫ d/2

−d/2dzAz

‖ = tei2π Bxd

φ0 ≡ teiQx,

8.4. EXPERIMENTAL FACTS 137

with Q = 2πBd/φ0 = 2π/L‖, which implies

L‖ =φ0

Bd.

in the presence of B‖, the total tunneling term becomes thus

T = t∑

k

(

eiQxc†k,↑ck,↓ + e−iQxc†k,↓ck,↑)

= −t[

eiQx(Sx + iSy) + e−iQx(Sx − iSy)]

= 2tS cos(φ+Qx) (8.7)

This term is thus equivalent to a field rotating around the x axis, uni-form along y, which is causing the pseudo-spin magnetization to oscil-late along x. In fact this term competes with the exchange term, as weshall see later on.

• When a B‖ field is applied, the tunnelling current between the layersbecomes

J↑↓ = it∑

k

(

eiQxc†k,↑ck,↓ − e−iQxc†k,↓ck,↑)

(8.8)

= it[

eiQx(Sx + iSy) − e−iQx(Sx − iSy)]

= 2tS sin(φ+Qx).

8.4 Experimental facts

8.4.1 Phase Diagram

As discussed above, the energy difference ∆SAS = 2t between symmetricand antisymmetric superpositions of layer states of the two wells may vary,depending on the sample, from a few millidegrees to a few hundreds of de-grees Thus ∆SAS may be much smaller, or much larger than the inter-layerCoulomb interactions. Experiments are able to scan a large array of thecharacteristic ratio ∆SAS/(e

2/ǫd), from a weak electronic correlation regimeto a strong one.

When the layers are far from each other (d≫ lB), there are no inter-layercorrelations, each layer is in the ν = 1/2 metallic ground state, there is no


Figure 8.2: Phase diagram for the bi-layer QHE (after Murphy et al. ??). The sampleswith parameters below the dotted line exhibit the IQHE and an excitation gap.

QHE. When the inter-layer separation decreases, an excitation gap is foundto appear, together with a quantized Hall plateau with σxy = e2/h [94, 101].If ∆SAS/Ec ≫ 1 (with Ec = e2/(ǫd), this is fairly easy to understand, sincethings look as if the two layers were like a single one, with total filling factorν = 1. All symmetric states are occupied, we have the usual QHE. A farmore interesting situation arises when the ν = 1 QHE is found in the limit∆SAS → 0. In this limit, the excitation gap is clearly a collective effect,since it may be as large as 20 K while ∆SAS < 1K. The excitation gapsurvives in this limit because of a spontaneous breaking of the U(1) gaugesymmetry associated with the phase degree of freedom –the azimuthal angle φin expression (8.1) [87, 98]. Figure 8.2 shows the experimental determinationof the QHE part of phase diagram, below the dotted line. This changefrom single particle to collective behaviour is analogous to the ferromagneticbehaviour of a monolayer at ν = 1. In the latter case, the excitation gapremains finite even when the Zeeman effect vanishes, because of the exchangeforces connected to the Coulomb interaction.

The remarkable fact is that the IQHE at ν = 1 survives when ∆SAS → 0provided the inter-layer distance between layers is smaller than a criticalvalue d/lB ∼ 2. In that case, the gap is a purely collective effect due tointeractions. As we shall see, it is due to a pseudo-ferromagnetic quantumHalll state, which posesses a spontaneous inter-layer coherence.

Experiments 139

Figure 8.3: Experiment by Murphy et al. [94]. The thermal excitation gap ∆ is plot-ted as a function of the magnetic field tilt angle, in a bilayer with small tunnel term(∆SAS = 0.8K). The black dots correspond to filling factor ν = 1, the triangles to ν = 2/3.The arrow shows the critical angle θc. The continuous line is a guide to the eye. Thedotted line is a rough estimate of the tunnel amplitude renormalized by the parallel mag-netic field. This single particle effect exhibits a slow negative variation, compared to theobserved effect. The inset is an Arrhenious plot of the dissipation, measured by the longi-tudinal resistance. The low temperature activation energy is ∆ = 8.66K. The gap howeverdecreases sharply at a much lower temperature, roughly 0.4K.

8.4.2 Excitation gap

An additional indication of the collective nature of excitations is providedby the excitation gap variation with temperature, as shown on figure 8.3.The activation energy ∆ at low temperature is clearly larger than ∆SAS.If ∆ was a single particle gap, one would expect an Arrhenius law up totemperatures of the order of ∆/kB. Instead, the gap decreases sharply assoon as A T ∼ 0, 4K. This suggests that the order responsible for thecollective excitation gap is vanishing .

8.4.3 Effect of a parallel magnetic field

Another experimental finding suggests strongly a collective order phenom-enon: the strong sensitivity of the system to a relatively weak B‖ magneticfield, applied in a direction parallel to the layers plane. The figure 8.3 showsthat the activation gap decreases rapidly when B||, the parallel component


Figure 8.4: Example of electronic process in a 2D bi-layer, such that the flux of B‖

produces decoherence effects. In this process, an electron tunnels at point A from the upperlayer to the lower one. The electron pair thus created moves coherently, then annihilatesat point b where the particle tunnels in the other direction. The amplitude for such aprocess depends on the flux of B‖ through the path

.

of the field, increases. Assume that the electronic gas in each layer is stricly2D (in other words neglect the physical width of the potential wells). Thenthe orbital effect of B‖ can only be due to electronic processes between thelayers with closed loops containing some flux from B‖. Such loops will causeB|| to be felt if there is phase coherence over the whole loop. Such a loop isshown on figure 8.4.

An electron tunnels from one layer to the other at point A, travels adistance L||, tunnels back to the departure layer, then back to point A.The magnetic field parallel component, B‖, contributes to the amplitude ofthis process a (gauge invariant) Aharonov-Bohm phase factor, exp(2πiφ/φ0),where φ is the flux of B‖ threading this circuit.

Such loops contribute significantly to correlations, since one observes arapid decrease of the activation gap as a function of B‖: the decrease is bya factor 2 up to a critical field B∗

‖ ∼ 0.8T , beyond which the gap remainsroughly constant. This value is remarkably small. Let L‖ be the length suchthat the flux through the loop is one flux quantum: L‖B

∗‖d = φ0 ⇔ L‖[A] =

4, 14×105/d[A]B‖[T]). With B∗‖ = 0, 8T and d = 150A, one has L‖ = 2700A,

i.e. roughly twenty times the average distance between electrons in a layer,and thirty times the magnetic length corresponding to B⊥. A significantdecrease of the excitation gap is already observed in a parallel field of 0.1T,

Experiments 141

Figure 8.5: (a) In a standard experiment, the Hall current is transported simultaneouslyin both layers, without tunnel between layers. (b) It is possible to inject current in onelayer and to extract it in another. The tunneling current then behaves as a superfluidcurrent .

which implies enormous coherence lengths. This is again a hint of the stronglycollective nature of the observed order in quantum Hall bi-layers.

8.4.4 The quasi-Josephson effect

A spectacular experiment by Spielman et al. [102], confirmed the theoreticalideas about excitonic superfluidity in bi-layers. In the standard transportexperiments on bi-layers, a current JHQ is injected in both layers simultane-ously, and is also extracted from both layers simultaneously. In the experi-ment by Spielman et al., a current JHQ is injected in one layer, and extractedfrom the other one (Fig. 8.5). Qualitative differences arise in the tunnel con-ductance when the ratio d/lB is varied, for example varying the electronicdensity at constant filling of the LLL. Below a critical value of the ratio(the critical value which corresponds to the transition line between the QHEregime at ν = 1 and the metallic regime) a giant anomaly appears at zerobias, as shown on fig. 8.6. The qualitative understanding of this experimentis as follows: for d/lB ≫ 1, electronic liquids in different layers are uncorre-lated. At zero inter-layer bias, the Coulomb repulsion between electrons inone layer, and an electron in the other will inhibit the inter-layer tunnelingprocess of the latter: the zero bias conductance is strongly suppressed. Onlya finite bias, of the order of the Coulomb repulsion e2/(ǫd), can supersedethe latter. When d/lB ≃ 1 a coherent state is established, such that an


electron in one layer is bound to a hole in the other one at the same Landausite. Coulomb repulsion is strongly suppressed by this collective structurefor inter-layer tunneling events, and the tunneling conductance increases bytwo orders of magnitude.

At the time of this writing, as far as the author knows, there is yet nogeneral consensus on the intrinsic, or extrinsic character of the zero biasconductance finiteness. Is it impossible, for fundamental reasons, to everobserve a divergent conductance at zero bias, which would be the signatureof a complete analogy with the superconducting Josephson junction? Is theconductance finiteness due to experimental limitations, (impurities, etc.),or to the order parameter topological defects at finite temperature? Thosequestions are still being discussed among specialists.

8.4.5 Antiparallel currents experiment

In order to check the ideas about bosonic superfluid exciton liquid in the bi-layer system at ν = 1, one needs an experimental proof of electron-hole pairtransport. How can one couple to and detect electrically neutral objects byelectric transport? The solution is to notice that electron-hole pair transportin a bi-layer implies an anti-parallel circulation of currents in different layers.

Experiments have allowed, in the last few years to get independent electricconnections to each layer [108]. It has thus been possible to inject equalintensity currents with opposite flow direction in both layers, to test thecontribution of excitons to particle transport [6].

The figure 8.7 is a schematic representation of what one expects fromsuch an anti-aparallel current experiment. The two traces represent the ex-pected Hall voltage in each layer, neglecting all quantum phenomena exceptthe excitonic condensation. Because of the Lorentz force, the Hall voltage isproportional to the magnetic field. In a bi-layer system driven by two oppo-sitely directed parallel currents, the Hall voltage will have opposite signs inthe two layers. If the two layers are sufficiently coupled, and the magneticfield has the relevant intensity (so that ν↑ = ν↓ ≃ 1/2 the inter-layer electron-hole pairs which form will carry anti-parallel currents. The Hall voltage inboth layers must then vanish, as suggested by the figure 8.7.

Two experimental groups have confirmed those predictions [104].

Experiments 143

-5 0 5-5 0 5

Interlayer Voltage (mV)

Tu

nn

eli

ng

Con

du

cta

nce

at

nT=

1

(1

0-7

W-1

)

0.5

A)

B)

C)

D)

NT=6.9

NT=6.4

NT=5.4N

T=10.9

(b)

(c)

(a) (d)

Tension intercouches V (mV)

Con

duct

ance

tunn

el d

J /d

Vz

Figure 8.6: Quasi-Josephson effect [102]. Plot of the tunneling conductance dJz/dVas a function of voltage bias between the two layers, for various electronic densities, NT

in units of 1010cm−2. In the samples [from (a) to (c)] with larger electronic density (i.e.,smaller lB) the system does not exhibit any QHE, tunnelling processes are suppressed atzero bias. In the low density sample (d) there is a finite tunneling conductance peak atzero bias. The current at zero bias vanishes, contrary to the superconducting Josephsonjunction current. Whence the expression ”quasi-Josephson effect”.


0

10

−10

champ magnétique

5 10

tens

ion

de H

all

ν = 1

− − −− − − −+ + + +

+ + +

− − − −

− − −

+ + +

+++ +

Figure 8.7: Antiparallel currents experiment (After ref. [6]). A Hall voltage measure-ment detects the exciton condensation. The two traces are schematic representations ofthe Hall voltage in each layer when electric currents flow in opposite directions. Quantumeffects other than the excitonic condensation are ignored in this figure. When currentsflow in an uncorrelated manner between both layers, one must observe finite Hall voltages,which balance the Lorentz force in each layer. As the currents flow in opposite directions,Hall voltages must have opposite signs in each layer compared to the other. If exitoniccondensation occurs, in a certain span of magnetic field values, the opposite currents inthe layers will be carried by a uniform exciton current density in one direction. Sinceexcitons are electrically neutral, they are not submitted to Lorentz forces, and the Hallvoltage must vanish in both layers, as observed experimentally by Kellogg et coll. [104].

8.5. EXCITONIC SUPERFLUIDITY 145

8.5 Excitonic superfluidity

Within the pseudo-spin analogy(section 9.2), the Coulomb interaction be-tween layers favours a ferromagnetic state with an easy magnetization planewhen the inter-layer distance is small enough to stabilize a correlated state.The ground state wave function is then of the form [see equation (8.1)]

|ψφ〉 =∏

k

c†k↑ + eiφc†k↓√2

|0〉 . (8.9)

In other words, θ/2 = π/4, the magnetization is in the bi-layer plane, andone has 〈Sz〉 = 0. The amplitude is equal for opposite pseudo-spin states,which means that, for the time being, we consider a situation with zero biasbetween the two layers. The total occupation of each k state is 1. When thetunnel term t vanishes, φ has any value, provide it is the same all over thebi-layer plane. When the system chooses a particular φ value, among thecontinuous infinity of choices, the original U(1) symmetry of the Hamiltonian(in the presence of the pseudo-spin anisotropy) is broken by the ground state.

In the limit of zero tunneling term, we have thus a one parameter familyof equivalent ground states, with the phase φ as parameter. This phase isconjugate to the difference in particle number between the two layers. Inequation (8.9), the phase is well defined, but the number of particles ofeach pseudo-spin (i.e. the number of particles is each layer) is completelyundetermined. Similarly, one may construct a state such that the phase isundetermined, while the number of particles in each layer is specified exactly.To do this, integrate 8.9 over the phase. This yields

|ψSz〉 =∫ dφ

2πe−i(N↑−N↓)φ |ψφ〉 . (8.10)

We obtain thus a wave function with exactly N↑ particles in the ↑ layer, andN↓ = N − N↑ in the ↓ layer, N being the total number of guiding centers.The angle φ and Sz are canonical conjugate variables,

[φ, Sz = N↑ −N↓] = 1, (8.11)

whence the uncertainty relation δ(N↑ −N↓) × δφ > 1.Since we are dealing with a continuous broken symmetry, there must exist

a Goldstone mode the energy of which goes to zero in the limit of infinite

146 QHE bi-layers

wavelength. A state such that the phase varies in time and space may bewritten as

|ψφ〉 =∏

k

[

c†k↑ + eiφ(Xk,t)c†k↓]

|0〉 , (8.12)

where φ is the superfluid phase of the system. The long wavelength superfluidmode corresponds to equal intensity currents of opposite signs propagatingin the two layers.

To understand better why the state described by (8.9) breaks the gaugesymmetry associated to the charge difference between layers, consider thegauge transformation induced by the unitary operator U−(θ) = ei θ

2(N↑−N↓).

This transformation acts on electron creation operators as

U †−(θ)c†k↑U−(θ) = e−i θ

2 c†k↑ (8.13)

U †−(θ)c†k↓U−(θ) = ei θ

2 c†k↓ . (8.14)

The Hamiltonian is invariant under this transformation,

U †−(θ)HU−(θ) = H, (8.15)

since [H, (N↑ −N↓)] = 0, in the absence of inter-layer tunneling terms.In contrast, expression (8.9) shows that the coherent phase exhibits a non

trivial order parameter.

Sx(Xk) ≡⟨

c†k↑ck↓⟩

=⟨

Sx(Xk)⟩

=nel

2eiφ,

with the total density nel = 1/2πl2B. (Here I have defined the x directionas the arbitrary direction of the sponaneous pseudo-spin orientation in theplane x, y).

This order parameter is not gauge invariant,

Sx(Xk) →⟨

U †−(θ)c†k↑ck↓U−(θ)

⟩

= eiθSx(Xk) . (8.16)

That is a more formal way to show that the state has less symmetrythan the Hamiltonian, and breaks the U(1) symmetry associated with theconservation of the charge difference between layers N↑ −N↓.

1

1In a superconductor, the order parameter χ(r) = 〈c†↑(r)c†↓(r)〉 transforms in a non

trivial way under the gauge transformation associated with the total charge conservation,U+(θ) = exp[iθ(N↑ +N↓)/2]. The pseudo-spin bi-layer order parameter is invariant underthis transformation: this expresses simply the fact that the total particle number N↑ +N↓

is conserved in the excitonic superfluid.

Excitonic superfluidity 147

We can write an expression for the inter-layer tunneling current operatoras a function of position in space,

J↑↓,Xk= −it

(

c†k↑ck↓ − c†k↓ck↑)

, (8.17)

the average of which

〈J↑↓,Xk〉 = −it [Sx(Xk)

∗ − Sx(Xk)] = t sin(φ).

This expression is similar to the Josephson current expression: it dependsonly on the order parameter phase, not on the inter-layer voltage bias.

The pseudo-spin language expresses the conjugate character of phase andcharge difference between layers through the commutation relations of thespin density operators. With the order parameter along x,

[Sy, Sz] = iSx ≃ i.

As Sy ∝ sinφ ≈ φ, this leads to [φ, Sz] = i. As a consequence, the currentassociated to the phase gradient

Jzz =2ρE

h∇φ

is indeed the difference of the electric currents in the two layers.An apparent conceptual difficulty is that the wave function (8.9) describes

a state where the difference between the layer charges fluctuates, while thisdifference should be conserved in the limit t = 0 . This is analogous to thesuperconducting BCS wave function, which has a fluctuating total numberof particles, while it is in fact strictly conserved for an isolated sample.Thesolution of this apparent paradox is that each macroscopic piece of the samplemay be subdivided in smaller macroscopic parts, between which particleexchanges are numerous and rapid, so that phase coherence is establishedin each macroscopic part of the sample, the total particle number remainingconstant. Furthermore, in the thermodynamic limit, the ratio δN/N is oforder N−1/2 → 0.

A similar reasoning holds in the bi-layer case. Examine a slighlty morecomplicated object than the order parameter,

GXk,Xk′ =⟨

c†k↑ck↓c†k′↓ck′↑

⟩

. (8.18)

This object conserves the total particle number in each layer. It is equal to〈Sx(Xk)S

x(Xk′)〉, and it is non zero in the wave function (8.9).

148 QHE bi-layers

Notice that the wave function (8.9) is indeed an exciton condensate. Tosee that, define the state |ferro ↑〉 as the state where all electrons are in the↑ layer , |ferro ↑〉 =

∏

k c†k↑ |0〉. Then the state (8.9) may be re-written as

|ψφ〉 ≡∏

k

1 + eiφc†k↓ck↑√2

|ferro ↑〉 . (8.19)

This can be again re-written in a form reminiscent of a bosonic coherentstate,

|ψφ〉 ≡∏

k

exp(

eiφb†k)

|ferro ↑〉 , (8.20)

where b†k = c†k↑ck↓ is the excitonic boson. This is the reason why one mayspeak of a ”coherent” state (see section 2.2.2). One also speaks of ”spon-taneous phase coherence”, when the tunneling term is absent. Indeed inthat case the coherent state is entirely due to Coulomb interactions. On thecontrary, when the tunneling term is finite, the symmetric combination oflayer states is the most stable, even in the absence of interactions. This isanalogous to the magnetization induced by an external magnetic field in thecase of ”real” spins.

8.5.1 Collective modes – Excitonic condensate dynam-ics

As mentionned above, a consequence of the breaking of a continuous sym-metry by the phase coherence is the existence of the collective excitationmode (Goldstone mode) the energy of which goes continuously to zero as thewavelength goes to infinity. The Hamiltonian formalism was used above toderive collective mode energies in the ferromagnetic monolayer case. Here weuse the Lagrangian formulation, with the inclusion of the Berry connexionterm discussed in the previous chapter. The Lagrangian which describes thelong wavelength physics, in the absence of applied inter-layer voltage bias,and with zero tunneling term, is

L =ν

4πl2B

∫

d2rm · A[m] (8.21)

−∫

d2r

β(mz)2 +

ρA

2|∇mz|2 +

ρE

2

[

|∇mx|2 + |∇my|2]

.


Coefficients β, ρA and ρE may be evaluated with a microscopic approach, aswe have seen in section 8.3.2 Let us write the Euler-Lagrange equations ofmotion,

d

dt

δLδmµ

=δLδmµ

. (8.22)

Here the ground state is taken with the (vector) order parameter of length1 aligned along the x axis. For small variations of the order parameter awayfrom x, one may linearize, considering only first order deviations in my andmz. m = [1 − O(m2

y + m2z),my,mz], and one chooses the Berry connexion

A = (0,−mz/2,my/2), which yields

δLδmy

=ν

4πl2BAy[m] = − ν

4πl2b

mz

2,

δLδmy

=ν

4πl2B

mz

2+ ρE∆my , (8.23)

and

δLδmz

=ν

4πl2BAz[m] = − ν

4πl2b

my

2,

δLδmz

=ν

4πl2B

my

2+ ρA∆my − 2βmz , (8.24)

where ∆ = ∇ ·∇ is the Laplacian. In Fourier space, applying 8.22, one findsthe system of linear equations

(

iω 4πν

(2β + q2ρA)4πνq2ρE −iω

)(

my

mz

)

= 0. (8.25)

So finally the collective mode dispersion relation is given by

ω2(q) =(

4π

ν

)2

(2β + q2ρ2A)q2ρE . (8.26)

When d = β = 0, and ρA = ρE = ρ0 one retrieves the collective mode(pseudo-spin wave ) of the ferromagnetic SU(2) phase,

ω(q)|B=0 =4π

νρ0q2.

2The expansion in gradients of mz is not stricly correct, because the long range natureof he Coulomb interaction induces a non local term which we do not take into accounthere. The latter term is smaller than the terms considered here.

150 QHE bi-layers

Q=+1/2 Q=+1/2

Q=−1/2 Q=−1/2

v=+1 v=−1

v=+1 v=−1

Figure 8.8: Four meron ”flavours”. With two possibilities for the choice of the vorticity,and two additional ones for the choice of the pseudo-spin orientation at the vortex core,merons have a topological charge Q = ±1/2, and exist in four possible ”flavours”.

The mass term β 6= 0 changes qualitatively the collective mode dispersion,which becomes linear in q at small q,

limq→0

ω(q)|β 6=0 =4π

ν

√

2βρEq .

That is analogous to the bosonic superfluid collective mode (with weak repul-sive interactions). But here the order parameter represents the condensationof neutral bosons, which carry no charge.

8.5.2 Charged topological excitations

For a system in the same universality class as that of the 2DXY model, theremust exist a Kosterlitz-Thouless (KT) transition at TKT = (π/2)ρS/kB. Theessence of this transition is the ionisation (dissociation) of vortex-antivortexpairs. In our case, the order parameter symmetry group is U(1), but thepseudo-spin direction is not confined to the xy plane, so that the pseudo-spinvortex is in fact a ”meron” , which may be considered as a half skyrmion.

The system order parameter in the presence of a vortex at the origin hasthe approximate following form

m =

±√

1 −m2z cos θ,

√

1 −m2z sin θ,mz(r)

, (8.27)


where the ± sign refers to the vorticity (left or right) and θ is the azimuthalangle of the position vector r. At large distance from the meron center, mz(r)tends to zero to minimise the capacitive energy. At the vortex core, however,we have mz = ±1,mx = my = 0, to avoid the large energy cost of a coresingularity.

The local topological charge is computed using the Pontryagin densityexpression [see equation (7.31)]

δρ = − 1

8πǫij(∂im × ∂jm) · m.

With expression (8.27), this density writes

δρ(r) =1

4πr

dmz

dr.

The total charge is Q =∫

d2rδρ(r) = 12[mz(∞) −mz(0)]. For a meron, the

spin at the core is either ↑ or ↓, and gradually gets oriented in the xy planeas the distance from the core increases. It lies in the xy plane far from themeron core. The topological charge is thus ±1/2 depending on the core spinpolarity.

The general result for the topological charge is

Q =1

2[mz(∞) −mz(0)]nv (8.28)

where nv is the vortex winding number. The electric charge is ±νe/2, halfthat of a skyrmion, which comes as a support of the meron as a half skyrmion,as mentionned above.

One may write a meron variational wave function. The simplest one is

∣

∣

∣ψnv=+1,−1/2

⟩

=M∏

m=0

c†m,↑ + c†m+1,↓√2

|0〉 . (8.29)

In this expression, c†m,↑(↓) creates an electron in layer↑ (↓), in the state ofangular momentum m in the LLL, and M is the corresponding moment onthe sample edge. The vorticity is +1, since far from the core, the spinor is

χ(θ) = (1/√

2)

(

eiθ

1

)

,

152 QHE bi-layers

where θ is the polar angle of the vector r. The charge is +1/2 because anelectron has been suppressed at the center, in the ↓ layer: all states have 1/2occupation, except m = 0 which is empty. The meron charge can be changedwithout changing the vorticity, as we see with the wave function

∣

∣

∣ψnv=+1,+1/2

⟩

= c†0,↓

M∏

m=0

c†m,↑ + c†m+1,↓√2

|0〉 .

This state has charge −1/2, because an electron has been created in the statem = 0 in the ↓ layer.

It is useful to examine a meron pair wave function, to check wether themeron is a half skyrmion. Examine the case of a pair of merons with oppositevorticities, but equal charges, placed at points z1 and z2. The following wavefunction seems to obey our requirements,

ψλ =∏

j

1√2

(

eiφ(zj − z1)(zj − z2)

)

j

Φferro , (8.30)

where φ is an arbitrary angle and ()j is a spinor for the j-th particle.At large distance from z1 and z2, the spinor for each particle becomes

zj

(

eiφ

1

)

. (8.31)

This corresponds to a fixed spin orientation in the xy plane, with an angle φwith the x axis. Vorticity is thus zero. By construction, the spin orientationis purely ↑ for an electron at z2, and purely ↓ for an electron at z1. Moreover,the net charge must be νe since, asymptotically, the factor zj is the sameas for the Laughlin quasi-particle in the spin polarised state. For symmetryreasons, one might think that a charge νe/2 is asociated to each localisedstate near z1 or z2.

The fact is that this wave function (8.30) is nothing but a different rep-resentation for the skyrmion! Choose z1 = λ and z2 = −λ, and suppose forsimplicity that the asymptotic orientation of spins is in the x direction, sothat φ = 0. Now rotate all spins by a global rotation around the y axis, withan angle −π/2. Using

exp(

iπ

4σy)

1√2

(

zj − λzj + λ

)

,


one finds the variational skyrmion wave function. The previous wave functionis well adapted to the U(1) symmetry, because t describes spins orientedmainly in the xy plane.

8.5.3 Kosterlitz-Thouless transition

The presence of topological defects of the vortex type may spoil the phasecoherence of the XY ground state. This may happen at zero temperature,because of quantum fluctuations, if the distance between layers exceeds acritical distance d∗. Here we are discussing thermal effects.

The effective model at finite temperature is given by

E =ρS

2

∫

d2r |∇φ|2 .

For typical experimental parameter values in the AsGa bi-layers, the Hartree-Fock estimate of the exchange stiffness ρS goes from 0, 1K to 0, 5K.

The Kosterlitz-Thouless transition is due to ionisation of vortices in theXY model, at a temperature TKT approximately given by the exchange stiff-ness ρS. Free vortices induce a discontinuous renormalisation of the exchangestiffness, which vanishes at TKT . The classical action generates a logarithmicinteraction between vortices.

A meron gas has an energy of the form

E = MEcore − 2πρS

M∑

i<j

ninj ln(

Rij

Rcore

)

+M∑

i<j

qiqje2

4ǫRij

, (8.32)

where Ecore is the meron core energy, Rcore its size, and Rij is the separationbetween the i-th and the j-th meron. The last term is new. It is specific of theQHE bi-layer physics: it is due to Coulomb interactions between the meronsfractional charges. qi = ±1 is the electric charge (±e/2) sign, of the i-thmeron. The origin of the logarithmic term is not, as in the superconductingcase, the kinetic energy stored in the supercurrents. It comes from the lossof exchange energy due to the phase gradients associated to vortices. TheCoulomb interaction is irrelevant at TKT because it decreases faster withdistance than the logarithmic interactions. It may cause a shift of TKT ,but the transition is not qualitatively altered. The phase diagram howeverbecomes richer, with chiral phases, with an order parameter 〈niqi〉 wherevorticity and electric charge are no longer independent[109].

154 QHE bi-layers

ξ

Λ

v=+1

v=−1

Figure 8.9: In the presence of an inter-layer tunneling term, meron pairs of oppositevorticities are bound by a string, or domain wall, of length Λ and characteristic width ξ.The meron confinement energy varies linearly with Λ.

8.5.4 Effect of the inter layer tunneling term

As already discussed above, an inter-layer tunneling term breaks the U(1)Hamiltonian symmetry

Heff =∫

d2r(

ρs

2

)

|∇φ|2 − t

2πρ2cosφ . (8.33)

Here ρs is the iso-spin exchange stiffness, which may be computed microscop-ically with the same techniques used to compute the equivalent parameterin the ”true” ferromagnetic case. For a finite t value, the collective modeacquires a mass (just as spin wave in a SU(2) ferromagnet acquire a massin an external field, because of the Zeeman effect). Quantum fluctuationsare thereby decreased, which explains the upwards curvature of the phasetransition line in the phase diagram.

The tunneling term, because it breaks the U(1) symmetry, and gives alarger energy cost to vortex pairs configurations, destroys the KT transition.


To lower the energy, the system deforms the spin deviations in domain walls,or strings, which connect vortex cores, as shown on figure 8.9. Spins areoriented in direction x, which is imposed by the tuneling term everywhere,except in the wall region, where they rotate quickly of 2π. The wall energy isproportional to its length Λ, so that we have a vortex confinement mechanismanalogou to quark confinement in elementary particles such as hadrons ormesons. The line tension (the energy per unit length) may be estimated byexamining the infinitely long domain wall parallel to the y axis. The optimalform is given in that case by

φ(r) = 2 sin−1

(

tanhx

ξ

)

, (8.34)

where the characteristic wall width is ξ = (2πl2BρS/t)1/2

. The line tension isthus (see [110])

T0 = 8

(

tρS

2πl2B

)1/2

=8ρs

ξ. (8.35)

If the wall is long enough (Λ ≫ ξ), the total energy of a segment of lengthΛ will be approximately

Epair = 2Ecore +e2

4ǫR+ T0Λ. (8.36)

Minimising, we conclude that Epair is optimal for Λ = Λ′0 = (e2/4ǫT0)

1/2,whence

Epair = 2E ′core +

(

e2T0

ǫ

)1/2

.

Thus, except for meron core energies, the charge gap at fixed d (i.e. at

fixed ρs), is proportional to T1/20 ∝ t1/4 ∼ ∆

1/4SAS. This is in contrast with

the free electron case. The exponent 1/4 is small. The charge gap increasesquickly as soon as the tunneling term is non negligible. The cross-over regimebetween the pseudo-spin meron pair textures and the domain wall texture isestablished at a finite t value.

8.5.5 Combined effects of a tunnel term and a parallelfield B‖

We have seen that the vector potential corresponding to a parallel field B‖may be chosen as A = (0, 0, Az

|| = Bx), where z is the direction perpendicular

156 QHE bi-layers

to the layers. Equation (8.7) shows that the expression of the tunnelingmatrix element changes with the field, so as to respect gauge invariance:instead of a constant phase φ = 0 (spin alignment along the x axis, as we sawin section 9.3), one finds a spatial variation of the phase, φ → φ−Qx. Thetunneling term competes with the spin stiffness one. The latter is minimizedfor a uniform magnetization, while the former favours a rotating one, whenB‖ 6= 0,

H =∫

ρS |∇φ|2 −St

2πl2Bcos(φ−Qx)

d2r . (8.37)

This Hamiltonian, known in the 2D physics of commensurate-incommensuratetransitions as the Pokrovski-Talapov model, has a phase diagram structurewhich depends on the relative values of ρs and Q.

If Q → 0, the energy is minimised by a phase φ = Qx, the exchangeenergy loss being ρsQ

2. One has then a commensurate state (the term phasehere is used for the order parameter phase φ, not to be confused with a”thermodynamic phase”) : for any x, the order parameter phase is locked atthe value dictated by the periodic potential minima.

When Q increases, minimising the periodic term with a linear variation ofthe phase becomes too costly in exchange energy. The conflict between thosetwo terms results in the appearance of solitons which are phase defects. Thelatter which are solutions of a Sine Gordon equation, express the compromisebetween the ”elastic energy” (the exchange term, quadratic in Q, and theperiodic ”potential energy”, (the sinusoidal tunneling term). The limiting

behaviour is when ~∇φ ∼ 0 and the average value of cos(φ − Qx) vanishes.For B‖ larger than a critical value Bc

‖ periodically ordered topological defectsstart being formed, in a uniaxial 2D anisotropic environment.

At zero temperature, the critical value is

Bc‖ = B⊥

2lBπd

2t

πρs

1/2

.

With ∆SAS = 0.45K, one finds Bc‖ ≈ 1, 3T, slightly larger than the observed

value 0.8T. The corresponding value of L‖ is large.In the commensurate phase, the order parameter tumbles more and more

rapily as B‖ incrases, since φ = Qx. In the incommensurate phase, the sys-tem state becomes roughly independent of B‖, so that the excitation gapsaturates at a fixed value. In the presence of the tunneling term, the low-est energy charged excitations are meron pairs with opposite vorticities and


equal charges (i.e. ±1/2, connected to one another by a domain wall witha constant line energy. For B‖ = 0, the energy is independent of the wallorientation. The effect of B‖ is more clearly seen with a variable change.Let ϕ(r) = φ(r)−Qx. This variable is constant in the commensurate phase,and varies in the incommensurate one. In terms of this new variable, theHamiltonian becomes

H =∫

d2r

ρS

2

[

(∂xϕ+Q)2 + (∂yϕ)2]

− t

2πl2Bcosϕ

. (8.38)

Thus B‖ defines a preferred direction of this problem. Domain walls alignin this direction and involve a phase change, in terms of ϕ, with a preferredsign (negative for Q > O).

One can show that the energy per unit soliton length of the wall, i.e. theline tension, decreases linearly with Q, and thus with B‖, i.e.

T = T0

1 − B‖B∗

‖

,

where T0 is given by equation (8.35). There is a transition when T becomesnegative. We have seen in section 7.5.5 that the charge excitation gap is

given by the vortex pair energy with an optimal separation Λ =√

e2/4ǫT .

The equation (8.36) for the meron pair energy is equally affected by the T0

renormalisation, which yields

Epair ≃ 2Ecore +

√

e2T0

ǫ

1 − B‖B∗

‖

1/2

.

Thus, as B‖ increases, the line tension decreases and the line gets longer.On the whole, the energy is lowered. Far into the incommensurate phase,the inter layer tunneling term becomes negligible. Therefore, the ratio of thecharge gap at B‖ = 0 and that when B‖ → ∞ should be roughly

∆0

∆∞=

t

tcr

1/4

≃ (e2/ǫlB)1/2t1/4

8ρ3/4s

.

Using typical values for t and ρs, the former expression yields values between1.5 and 7, in qualitative agreement with experiment.

158 QHE bi-layers

8.5.6 Effect of an inter-layer voltage bias

What is the tunneling curent in the presence of an inter-layer voltage bias?The total tunneling current is

It ∝ et∫

d2r

ei[φ(r)+Qx] − e−i[φ(r)+Qx]

(8.39)

= F

eiφ(r)∣

∣

∣

qy=0,qx=Q− F

e−iφ(r)∣

∣

∣

qy=0,qx=−Q(8.40)

where Ff(r)|q is the 2D Fourier transform of f(r) at wave vector q.Experimental results show that the tunneling current vanishes at zero

inter-layer voltage bias, so that the current can be computed perturbatively.To second order in t, one has

It(V ) =2πet2L2

h[S(Q, eV ) − S(−Q,−eV )] . (8.41)

where S(q, hω) is the fluctuations spectral density of the opeator eiφ at wavevector q et and frequency ω, i.e. the transform of 〈eiφ(r,t)e−iφ(0,0)〉. A strikingprediction follows: when disorder is weak, the spectral density, and thusIt(V ) exhibit a peak centered at

eV = hωQ

where ωQ is the collective frequency at wave vector Q. Thus as B‖ varies,the conductance peak position varies according to the low energy collectivemode dispersion. The parallel field only allows tunneling events betweenstates which differ by their momentum Q. Energy conservation ensures thatthe state energies differ by eV . This has been fully confirmed by experiment[111]. See figures 8.10, and 8.11. A transport experiment allows thus a directmeasurement of the collective mode dispersion relation, which is found to belinear in Q, as predicted by theory.

As shown in figure 8.10, the tunneling current in the presence of a paral-lel field B‖ exhibits a peak which corresponds directly to the collective modedispersion of the superfluid phase. The figure shows the tunneling conduc-tance at T = 25mK for an electronic density of 5.2 × 1010cm−2, for a seriesof parallel magnetic field values between B‖ = 0 and 0.6T. The insert is ablow-up of curves for B‖ values between 0, 07 and 0, 35T. The dots indicatethe position of satellite resonances for dI/dV . (After [111]


2.5

2.0

1.5

1.0

0.5

0.0

dI/

dV

(10

-6Ω

-1)

-200 0 200

V (µV)

B|| = 0

B|| = 0.6T

-200 0 200

V (µV)

10-7

Ω-1

Figure 8.10: Experimental determination of the Goldstone mode dispersion in a trans-port experiment [111]. The tunneling current in the presence of a parallel field B‖ exhibitsa peak which corresponds directly to the collective mode dispersion of the superfluidphase. The figure shows the tunneling conductance at T = 25mK for an electronic densityof 5, 2× 1010cm−2, for a series of parallel magnetic field values between B‖ = 0 and 0.6T.The insert is a blow-up of curves for B‖ values between 0, 07 and 0, 35T. The dots indicatethe position of satellite resonances for dI/dV . (After reference [111]

160 QHE bi-layers

0.2

0.1

0

eV* (

meV

)

3020100

q (106m

-1)

Figure 8.11: Goldstone mode dispersion determined by Spielman et al. [111] [111].Energy eV ∗ of the resonance peaks, as a function of the wave vector Q = eB‖d/h inthe presence of a parallel magnetic field, for different electronic densities (cross : nel =6.4 × 1010cm−2; squares : nel = 6.0 × 1010cm−2; black dots : nel = 5, 2 × 1010cm−2. Thedotted line is a theoretical estimate by Girvin for the Goldstone mode dispersion at smallq [87]. The continuous line is a guide for the eye, and corresponds to a collective modevelocity of 1, 4 × 104m/s.

In figure 8.11, the energy eV ∗ of the resonance peaks is shown as a func-tion of the wave vector Q = eB‖d/h in the presence of a parallel magneticfield, for different electronic densities (cross : nel = 6.4 × 1010cm−2; squares: nel = 6.0 × 1010cm−2; black dots : nel = 5.2 × 1010cm−2. The dottedline is a theoretical estimate by Girvin for the Goldstone mode dispersion atsmall q [87]. The continuous line is a guide for the eye, and corresponds toa collective mode velocity of 1.4 × 104m/s.

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