Mesoscopic quantum Hall effect

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Thesis

Reference

Mesoscopic quantum Hall effect

LEVKIVSKYI, Ivan

Abstract

Dans cette thèse, nous avons considéré la physique de l'effet Hall quantique à l'échelle mésoscopique. Cette physique est principalement déterminée par les états de bord, qui sont les seules excitations sans gap. Nous avons étudié des effets mésoscopiques tels que le déphasage des états de bord induit par les interactions, le déphasage induit par le bruit et l'équilibration le long des canaux de bord de l'effet Hall quantique. Cette étude nous a permis d'expliquer plusieurs expériences, de faire des propositions expérimentales ainsi que de résoudre certains problèmes théoriques.

LEVKIVSKYI, Ivan. Mesoscopic quantum Hall effect . Thèse de doctorat : Univ. Genève, 2011, no. Sc. 4361

URN : urn:nbn:ch:unige-176326

DOI : 10.13097/archive-ouverte/unige:17632

Available at:

http://archive-ouverte.unige.ch/unige:17632

Disclaimer: layout of this document may differ from the published version.

1 / 1

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UNIVERSITÉ DE GENÈVE FACULTÉ DES SCIENCES

Département de physique théorique Docteur E. Sukhorukov

ÉCOLE POLYTECHNIQUE FÉDÉRAL DE ZURICH DÉPARTEMENT DE PHYSIQUE

Institut de physique théorique Professeur J. Fröhlich

Mesoscopic Quantum Hall Effect

THÈSE

présentée à la Faculté des sciences de l’Université de Genève pour obtenir le grade de Docteur ès sciences, mention physique

par

Ivan Levkivskyi

d’Ukraine

Thèse No. 4361

GENÈVE

Atelier d’impression ReproMail 2011

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Remerciements

Je voudrais remercier mon directeur de thèse Eugene Sukhorukov et mon co-directeur de thèse Juerg Fröhlich pour les nombreuses et fructueuses discussions, leur conseils sagaces et leur assistance permanente tout au long de l’accomplissement de ce travail. Je remercie mes collègues C. W. J. Beenakker, A. Boyarsky, V. Cheianov, I. Chernii, P. Degiovanni, N.

Magnoli, A. Koroliuk et O. Ruchayskiy pour les précieux échanges que j’ai eu avec eux. Je suis reconnaissant envers E. Bieri, M. Heiblum, L. Litvin, S. Oberholzer, F. Pierre, P. Roche et C. Schönenberger pour l’éclaircissement de détails d’expériences. Je voudrais remercier mes jurés de thèse T. Giamarchi, B. Halperin, et L. Levitov pour leurs commentaires et leurs propositions judicieuses.

Je remercie nos secrétaires Cecile, Danièle et Francine pour leur aide dans mes problèmes administratifs. Je voudrais remercier ma femme Ievgeniia pour sa relecture de la thèse et son assistance pendant la réalisation de ce travail. Je remercie Y. Gaponenko pour la traduction du résumé de thèse en langue française.

Ce travail a été soutenu par le Fonds National Suisse de la Recherche Scientifique.

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L’effet Hall Quantique Mésoscopique

Resumée de thèse en langue française

Dans cette thèse, nous avons considéré la physique de l’effet Hall quantique à l’échelle mésoscopique. Cette physique est principalement déterminée par les états de bord, qui sont les seules excitations sans gap. Nous avons étudié des effets mésoscopiques tels que le déphasage des états de bord induit par les interactions, le déphasage induit par le bruit et l’équilibration le long des canaux de bord de l’effet Hall quantique. Cette étude nous a permis d’expliquer plusieurs expériences, de faire des propositions expérimentales ainsi que de résoudre certains problèmes théoriques, comme résumé ci-dessous.

Nous avons développé une nouvelle technique théorique qui permet d’étendre l’approche de la bosonization aux situations de non-équilibre. L’idée principale de cette technique est d’inclure les effets du non-équilibre via les conditions de bord correspondantes dans les équa- tions du mouvement des opérateurs de champ des bosons. Toute l’information concernant l’interaction est donc contenue dans la fonction de Green de ces équations, tandis que l’état de non-équilibre se trouve dans les conditions de bord. Le calcul des fonctions de corrélation électronique pour un système en interaction peut donc être réduit à la recherche des valeurs moyennes sur des électrons libres. Nous avons proposé un modèle physique décrivant le dé- phasage dans les interféromètres électroniques à facteur de remplissage entier et expliquant des résultats expérimentaux récents.

Selon notre modèle, le déphasage dans l’interféromètre à comme origine une forte interaction de Coulomb au bord du gaz d’électrons bidimensionnel. Le caractère à longue portée de l’interaction mène à une séparation du spectre des excitations de bord en modes lent et rapide. La nouvelle échelle d’énergie associée au mode lent détermine la dépendance en tem- pérature de la visibilité ainsi que la période de ses oscillations en fonction de la tension. Nous avons aussi considéré le déphasage dans l’interféromètre de Mach-Zehnder à électrons fortement couplé à un bruit de courant crée par un point de contact quantique. Nous trouvons la visibilité des oscillations de Aharonov-Bohm en fonction de la tension et l’exprimons via une fonction de bruit générant des cumulants. Dans le régime de haute tension, des cumulants de courant d’ordres supérieurs s’ajoutent pour annuler l’effet de dilution d’un point de contact quantique. Ceci mène à un changement abrupt dans la dépendance en tension de la visibilité, à une transparenceT = 1/2du contact.

Nous avons utilisé la technique proposée de bosonization en non-équilibre afin de construire la théorie du processus de la relaxation énergétique aux états de bord de l’effet Hall quantique.

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iv Resumée de thèse en langue française

Nous trouvons que la relaxation de la fonction de distribution de l’énergie passe par plusieurs asymptotes intermédiaires. De plus, la fonction de distribution a courte distance se trouve être fortement asymétrique à cause des effets non Gaussiens du bruit. A des distances plus larges, les culminants d’ordres supérieurs sont supprimés, et la fonction de distribution devient une Lorentzienne. Plus important, la largeur de la Lorentzienne est linéaire avec la transparence T du point de contact quantique, en contraste avec la fonction de distribution de Fermi dont la largeur va comme√

Tà basse transparence. Nous proposons donc d’effectuer des mesures à basses transparences afin de découvrir tous les régimes de relaxation intermédiaires.

Nous avons considéré la classification des modèles effectifs de bord dans le régime de Hall quantique fractionnel. Les principales conditions requises par le modèle effectif - tels que l’invariance de jauge ainsi que l’existence d’excitations électroniques dans le spectre - réduisent considérablement le nombre de modèles permis. Ces modèles de bord sont alors classés selon la matrice à valeurs entières des phases statistiques relatives des excitations élec- troniques. Une forte interaction de Coulomb au bord mène à un nombre d’universalités, ce qui nous permet de réduire encore plus le nombre des modèles de bord. Nous proposons aussi des tests expérimentaux directs sur les modèles effectifs. Nous illustrons la classification générale des modèles de bord avec l’exemple d’un fluide de Hall quantique à un facteur de remplis- sageν= 2/3. Nous focalisons notre attention sur les quatre modèles les plus simples d’états de bord et évaluons les charges et propriétés des dimensions d’échelle des quasi-particules.

Nous étudions le transport à travers un interféromètre de Fabry-Perot à électrons, et montrons que les propriétés échelle des composants Fourier des oscillations d’Aharonov-Bohm dans le courant nous donnent une information par rapport à la charge et les dimensions d’échelle des quasi-particules. Donc, les interféromètres de Fabry-Perot peuvent être utilisés afin de discriminer entre différents modèles effectifs de fluides correspondant au même facteur de remplissage.

Nous proposons une solution au paradoxe de Byers-Yang en considérant un modèle microscopique d’un interféromètre à électrons construit à partir d’un fluide de Hall quantique à facteur de remplissageν = 1/msous forme d’un disque de Corbino. Un niveau fondamen- tal approximé d’un tel interféromètre est décrit par une fonction d’onde de type Laughlin, et les excitations à basse énergie sont des déformations incompressibles de cet état. Nous construisons une théorie effective à basse énergie en projetant l’espace des états du liquide sur l’espace des déformations incompressibles. Un opérateur tunnel des quasi-particules dans notre théorie est trouvé sous la forme d’une fonction injective des coordonnées du point de tunnel, dont la phase dépend de la topologie déterminée par la position des contacts Oh- miques. Nous décrivons à l’aide de l’équation maitresse un fort couplage des états de bord aux contacts Ohmiques et le courant de quasi-particules résultant à travers l’interféromètre.

Nous trouvons que la contribution cohérente au courant de quasi-particules moyen à travers des interféromètres de Mach-Zehnder disparait après la sommation sur les degrés de liberté des quasi-particules. La contribution restante vient du l’effet tunnel des électrons et oscille avec la période électronique, en accord avec le théorème de Byers-Yang.

En conclusion, nous avons appliqué plusieurs approches à l’effet Hall quantique à l’échelle mésoscopique : particule simple, effective et microscopique. Nous avons trouvé que toutes ces approches suggèrent une image commune des états de bord basés sur des modes boso- niques. Dans cette image simple, nous avons explique plusieurs résultats expérimentaux et avons fait des prédictions, dont certaines ont déjà été confirmées. Finalement, nous avons

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Resumée de thèse en langue française v

montré que la physique au croisement de deux domaines, tels que l’effet Hall quantique et la physique mésoscopique, est très riche, intéressante et mérite certainement une investigation intensive et poussée aussi bien du coté théorique qu’expérimental.

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Integer Quantum Hall Effect

2 Equilibrium and non-equibrium bosonization 35 2.1 Bosonisation of 1D fermions . . . 35

2.1.1 1D interacting systems . . . 35

2.1.2 Bosonic fields and Hamiltonian . . . 37

2.1.3 Quantization of boson fields and zero modes . . . 39

2.2 Correlation function at finite temperature . . . 40

2.3 Non-equilibrium bosonization . . . 42

2.3.1 Non-equilibrium boundary conditions and full counting statistics . . . 42

2.3.2 Equilibrium boundary conditions: a simple test . . . 44

2.4 Conclusions . . . 45

3 Interaction induced dephasing of edge states 47

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viii Contents

3.1 Experimental results . . . 47

3.1.1 Only interfering edge channel is biased . . . 47

3.1.2 All edge channels are biased . . . 49

3.2 Model of Mach-Zehnder interferometer . . . 50

3.2.1 Fields and Hamiltonian . . . 50

3.2.2 Bosonization . . . 51

3.2.3 Strong interaction limit and the universality . . . 52

3.3 Visibility and phase shift . . . 54

3.4 Discussion of experiments . . . 57

3.4.1 Only interfering edge channel is biased . . . 59

3.4.2 All edge channels are biased . . . 60

3.4.3 Effects of finite temperature . . . 63

3.5 Conclusion . . . 64

4 Noise induced dephasing of edge states 67 4.1 Experimental setup and the model . . . 68

4.2 Correlation functions . . . 69

4.3 Situation with non-equilibrium Gaussian noise . . . 71

4.4 Noise induced phase transition . . . 73

4.4.1 Non-Gaussian noise in Markovian limit . . . 74

4.4.2 Quantum correction at critical point . . . 75

5 Energy relaxation at the quantum Hall edge 79 5.1 Experimental results . . . 80

5.2 Theoretical model . . . 81

5.2.1 Correlation function . . . 83

5.3 Electron distribution function . . . 85

5.3.1 Short distances . . . 85

5.3.2 Intermediate distances . . . 86

5.3.3 Long distances . . . 88

5.4 Calculation of the energy flux . . . 88

Fractional Quantum Hall Effect

6 Classification of effective edge modelss 95 6.1 Effective theory of QH edges . . . 95

6.1.1 Chern-Simons theory and gauge anomaly . . . 96

6.1.2 Hydrodynamics of incompressible edge deformations . . . 97

6.1.3 Quantization of edge excitations . . . 99

6.1.4 Gauge-invariant formulation . . . 100

6.2 Multi-channel edge models . . . 101

6.2.1 Kinematics of edge models . . . 102

6.2.2 Local excitations . . . 104

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Contents ix

6.2.3 Scaling dimensions of local excitations . . . 105

7 Spectroscopy of quantum Hall edge states at complex filling factors 107 7.1 Minimal models for filling factorν = 2/m . . . 108

7.2 The role of Coulomb interactions . . . 111

7.3 Experimental determination of scaling dimensions of quasi-particles . . . 113

7.3.1 Charge current through interfeormeter . . . 113

7.3.2 Low-temperature and high-temperature behavior of current . . . 115

8 Microscopic theory of fractional quantum Hall interferometers 119 8.1 The Byers-Yang paradox . . . 120

8.2 Microscopic description of the interferometer . . . 123

8.2.1 Plasma analogy and incompressible states . . . 124

8.2.2 Low-energy subspace . . . 127

8.3 Projection onto the low-energy subspace . . . 129

8.3.1 Edge Hamiltonian . . . 129

8.3.2 Tunneling Hamiltonian . . . 131

8.4 Main results and discussion . . . 133

8.4.1 Role of Ohmic contacts . . . 134

8.4.2 Effects of a singular flux and a modulation gate voltage . . . 135

8.4.3 Current through the interferometer . . . 137

9 Summary of results 143 9.1 Integer quantum Hall effect . . . 143

9.2 Fractional quantum Hall effect . . . 144

Appendices 145 A Solution of equations of motion in the case of a non-linear dispersion . . . 147

B Suppression of the higher order cumulants . . . 148

C Calculation of scaling dimensions. . . 151

D Mathematical aspects of the theory of QH lattices . . . 152

E Commutation relations for quasi-particle operators . . . 155

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Chapter 1 Introduction

The quantum Hall effect (QHE), one of the central subjects of the modern condensed matter physics, continues to attract attention of both experimentalists and theorists after 30 years of its discovery [1]. This effect, observed in two-dimensional electron gas in strong magnetic field, has several exciting features such as precise quantization of Hall conductance in units of e²/h, excitations with fractional charge and fractional statistics, etc. The low energy physics of the quantum Hall effect is determined by the edge excitations, because there exists a gap for excitations in the bulk of the two-dimensional gas. Properties of quantum Hall (QH) edge excitations were investigated in a number of experimental and theoretical works [2].

However, only recently the progress in the fabrication of novel mesoscopic devices [3] made it possible to focus closely on the electronic properties of quantum Hall edge, which were not well understood earlier.

The first experiments with these devices have shown unexpected results, such as strongly non-monotic dephasing in Mach-Zehnder electronic interferometers [4], fast equilibration along quantum Hall edge [5], unusual periodicities of Aharonov-Bohm oscillations [6], etc.

Appearance of such results indicates, that the physics of quantum Hall effect is not fully understood after 30 years of research. Namely, the mesoscopic effects in the quantum Hall regime have not been given necessary attention and need to be studied further. Therefore, the mesoscopic physics of the quantum Hall effect has been chosen to be the subject of our theoretical investigation. In this introduction we recall basics of mesoscopic physics and quantum Hall effect, briefly describe the recent experimental findings, and present the outline of the thesis.

1.1 Mesoscopic physics

We start the introduction with a brief reminder on the mesoscopic physics. This area of the condensed matter physics appeared in early 1980’s as a result of the attempts to answer one simple and important question. It is well known that the conductance of a macroscopic conductor is given by the Ohm’s lawG=σS/L, whereσis the conductivity of a material,Sis the cross-section, andLis the length of the sample. The question is at which length scales the

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2 General introduction

ohmic behavior of the conductance breaks down? First experimental answers to this question launched a series of several theoretical and experimental works which formed an entirely new broad area of the condensed matter physics. In the mesoscopic physics, one studies the electronic transport in conductors whose characteristic length scales are in between the quantum microscopic and classical macroscopic worlds. The physical behavior of conductors at these, so called, mesoscopic scales is very reach. There are several interesting effects such as conductance quantization [7], Coulomb blockade [8], weak localization [9], etc.

Important characteristic length scales which determine behavior of a sample are the following (see Fig. 1.1). The inter-atomic distance,a, is the smallest characteristic distance of a solid [10]. The behavior at such scales is purely quantum and determines the band structure (k)of the material and the value of the Fermi energy,EF. Next, the Fermi wavelength is defined asλF ≡2π/kF, where the Fermi wave-vectorkF is determined from the equation (kF) = EF. AlthoughλF may be larger than the inter-atomic distance, the physics at this scale is still purely quantum, since it is the shortest de-Broigle wavelength among all electrons. The mean free path,lis the length covered by an electron before the initial momentum of the electron is relaxed by elastic scattering off impurities. The corresponding time scale τ_m =l/v_F, where the Fermi velocity is defined as v_F = (1/~)∂(k)/∂k|kF, is called the momentum relaxation time. This scale is very different in different materials. The behavior at this scale might be quantum or classical depending on whether this scale is smaller or larger than another important scale, the phase relaxation length,lφ, since the static disorder itself cannot lead to dephasing. The phase relaxation length is probably the most important length scale, but it is also the most difficult to calculate and measure. At this scale the behavior of a conductor becomes classical. Ohm’s law is typically established when its size is larger than the Fermi wavelength, mean free path, and the phase relaxation length [3]. Processes at the inelastic lengths, such as electron-electron collision length,le−eand electron-phonon collision lengthle−phlead to the thermalization of electrons and the whole sample.

Figure 1.1: The main length scales that determine the physics of a condensed matter system are shown in a typical order. Mesoscopic length scales are the scales at which the system is in transition from quantum to classical behavior. Typical values of these scales are given in the Table 1.1.

Most of the pioneering works in mesoscopic physics have been done in three dimensional metals. However, all modern experiments are made with the so called two dimensional electron gas, since it has very high mobility. The mobility is an important characteristic of a material defined as follows. In the presence of an external electric field,E, the steady state~ of electrons is reached when the rate of momentum change induced by the field is equal to the rate of its decay due to scattering off impurities:

d~p dt

scattering

= d~p dt

field

. (1.1)

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1.1 Mesoscopic physics 3

The left hand side of the above equation can be rewritten as a ratio of the average momentum to the momentum relaxation time, so that we have:

m~v_d τm

=e ~E, (1.2)

wherevdis the average drift velocity, andmis the effective mass of an electron or hole. Then the mobility of a charge carrier is defined as the ratio of this drift velocity to the value of the applied electric field:

µ≡ v_d E =eτ_m

m . (1.3)

The typical value of the mobility for the three-dimensional metals is∼10⁴cm²/V·s, while for the two-dimensional electron gas it can be as high as∼10⁷ cm²/V·s [11]. In terms of length scales it means that the mean free path for two-dimensional electrons is∼100µm, so that it is much easier to fabricate samples for studies of the mesoscopic effects.

Metals 2DEG

Interactomic distance a∼few Å —- // —-

Fermi wavelength λ_F ∼few Å λ_F ∼40 nm

Mean free path l∼100 nm l∼1µm - 100µm

Electron collision length le−e∼1µm - 10µm le−e∼10µm Electron-phonon scattering length le−ph∼10µm - few mm —- // —-

Table 1.1: Typical values of lengths scales for three-dimensional metals and for two-dimensional electron gas in GaAs-AlGaAs heterostructures.

1.1.1 Two-dimensional electron gas

The two-dimensional electron gas with the best characteristics is obtained at the interface of GaAs and AlGaAs semiconductors (see Fig. 1.2). If these two materials are brought in contact, then the electrons from the wide gapped AlGaAs, which has high Fermi energy, spill over to GaAs, which has a lower Fermi level. These electrons leave a positive charge behind themselves which creates electrostatic potential and bends the bands as shown in Fig. 1.2.

In equilibrium, the Fermi level is constant in space. The electron density is sharply peaked near the interface forming a thin conducting layer which is called two-dimensional electron gas (2DEG). The highest mobility (up toµ = 3·10⁷cm²/V·s) is obtained by the so called modulation doping technique, where the concentrations of Si donors and of Al in AlGaAs

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are varied with the distance from the interface (see chapter 1 of Ref. [12] for details). The typical values of main length scales in 2DEG in comparison to metals are listed in Table 1.1.

Figure 1.2:The two-dimensional electron gas in semiconductor heterostructures.a)The layer of two dimensional electron states which is formed at the interface between n-doped AlGaAs and undoped GaAs is shown in blue color. b)The band structure of these materials before they brought in contact.

EV is the top of the valence band,ECis the bottom of the conductance band,EF is the Fermi energy.

c)The band structure after the equilibrium has been established. The positive charges on Si donors is shown by red pluses, two dimensional electron states are show in blue color.

Importantly, at the temperatures much lower than the level spacing in the effectively formed quantum well inz-direction, only the lowest quantized stateΨ0(z)is occupied, so that the total wave function of each electron factorizes:

Ψ(x, y, z) = Ψ0(z)ψ(x, y). (1.4) This means that from the quantum mechanical point of view, the dynamics of electrons is indeed two-dimensional. This is why the state of electrons at the surface of this semiconductor heterostructure is called two-dimensional electron gas.

The single particle spectrum of 2DEG is quadratic with very small effective massm = 0.067·m_e, so that:

E= ~²

2m(k²_x+k_y²). (1.5)

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1.2 Quantum Hall effect basics 5

To calculate the density of states of two-dimensional electrons, we consider a rectangular sample with dimensions L_xandL_y and areaS = L_xL_y. In such sample with periodical boundary conditions, the wave vectors are quantized with ∆k_i = 2π/L_i, where i = x, y.

This means that the total number of states with the energy less thanEis equal toN(E) = π·(2mE/~²)/(∆kx∆ky). Finally, we come to the result that the density of states per unit area, per spin is constant:

ν(E) = 21 S

dN(E) dE = m

π~²θ(E) (1.6)

in contrast to the three-dimensional case. This result simplifies some calculations in two dimensions. For example, in the degenerate limit,k_BT E_F, one has the relation for the total density of electronsn_s =R

dEν(E)f_F(E) = mE_F/π~². The Fermi wave vector for the two-dimensional electrons is thus given by the expression:

k_F =√

2πn_s, (1.7)

and the Fermi wavelengthλ_F = 40nm for the typical value of electron density isn_s= 10¹² cm⁻². It is interesting, that this length scale, which also determines the average distance between electrons, is much larger than the inter-atomic distance in the bulk GaAs.

1.2 Quantum Hall effect basics

In this section we recall basic notions of the Hall physics and main experimental findings in the regime of the quantum Hall effect.

1.2.1 Classical Hall effect

We start from the description of a simple Drude model of the classical Hall effect [10]. Our starting equation is a generalization of Eq. (1.2) in the presence of the magnetic field perpendicular to the plane of 2DEG:

m~vd

τ_m =e[E~ +~v_d×B/c].~ (1.8) Putting all linear in drift velocity terms on the left hand side, one gets the system of two linear equations for the two components of the velocity:

m/eτm −B/c B/c m/eτm

vx

vy

= Ex

Ey

(1.9) In the next step, we note that the current density vector is given by the product of charge densityensand the average drift velocity:~j =e~vdns. Substituting this equation in the Eq.

(1.9) and using the definition of the resistivity tensorρ:ˆ

E~ = ˆρ~j. (1.10)

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we find that the diagonal and off-diagonal components of the resistivity are given by:

ρxx=ρyy = 1

ensµ (1.11a)

ρxy=−ρyx= B

ensc (1.11b)

This well known result has been used for several decades for the characterization of semiconductor samples. Importantly, the Hall resistivityρ_xydepends only on such characteristic of material as the density of charge carriersn_s. Therefore, one can first find this density mea- suring the Hall resistivity and using Eq. (1.11b), and then restore the value of the mobility from Eq. (1.11a). Here we would like to mention also the relations between resistivityρˆand conductivityσˆ≡ρˆ⁻¹tensors in two dimensions for the further use:

σ_xx=σ_yy = ρ_xx

ρ²_xx+ρ²_xy, (1.12a)

σ_xy=−σ_yx= ρxy

ρ²_xx+ρ²_xy. (1.12b)

Thus, in the limit of a clean system,µ→ ∞, one hasρ_xx= 0andσ_xx= 0at the same time.

1.2.2 Experimental results

There are several experimental methods of determination of the resistivity tensor of a material. A popular experimental setup for the determination of this tensor for a two-dimensional sample, the so called Hall bridge, is shown in Fig. 1.3. In this setup one drives a constant current, typically of the order of a few nA, through the sample inxdirection and measures potentials at four voltage probes.

Figure 1.3:The setup for the measurement of longitudinal and Hall resistivity of 2DEG sample. A weak current is driven inxdirection, while the potentials at four voltage probes are measured.

The quantities which one is interested in are the longitudinal voltage dropV_x=V₁−V₂, and the Hall voltage V_H = V₂−V₃. The resistivity tensor can be then found form the relations:

ρ_xx= V_x I

W

L, ρ_xy =V_H

I . (1.13)

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Following common convention, we will refer to all off-diagonal components as to Hall components, i.e., we denoteσ_xy≡σ_H,R_xy≡R_H, etc. Importantly, the expression for the Hall resistivity isuniversal, i.e. it is independent of the sample dimensions. This means that one does not need to measure these dimensions with high accuracy to get the precise value of the Hall conductivity. Such situation is possible only in two dimensions, where the resistivity and resistance have the same dimensionality.

Figure 1.4:Experimental results for the longitudinal and Hall resistance in the strong perpendicular magnetic field. The longitudinal resistance shows strong deeps at the values of magnetic field which correspond to rational values of Landau levels filling factor. The Hall resistance has precisely quantized plateaus near these points. The effect is strongly pronounced near the integer values of the filling factor and near the simple fractions such as, e.g.,ν= 1/3. The fractional plateaus are observed only in very clean samples.

The first measurements of the longitudinal and Hall resistance of the high quality 2DEG in strong magnetic field have given exciting results. Namely, around the values of the magnetic field which correspond to the integer values of Landau levels filling factor (1.26) longitudinal resistance becomes suppressed by a factor as high as10¹³, while at the same time the Hall resistance becomes equal to

RH= h

ne², (1.14)

wherenis the integer closest to the value of the filling factor [1]. The precision of quantization of the Hall resistance at these so called Hall plateaus is extremely high, and can be as high asδR_H/R_H '10⁻¹⁰in the modern samples. Such a high precision made it possible to use the quantum Hall effect for metrology applications and for the precise determination of the fine structure constant. This is possible in particular due to a very small value of the

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longitudinal conductivity of the sample. Namely, if the longitudinal conductivity is zero, one can write the current trough arbitrary cross-sectiona−bas

Ia−b= Z

d~n·~j=σH

Z

d~l·E~ =σHVa−b. (1.15) So that the voltage drop between pointsaandbis independent of the geometry at all.

Among other interesting results it is worth mentioning that the conductivity in the regime of integer QHE has a very simple scaling behavior [13]. Namely, around every plateau, numbered by an integern, the conductivities satisfy the following relation

(σxx/σ_n⁽⁰⁾)²+ (hσH/e²−n−1/2)²= 1, (1.16) whereσ⁽⁰⁾n depends only on the temperature. It is interesting, that in spite of such precise quantization and universal behavior, distribution of currents in a sample at Hall plateaus is highly inhomogeneous and changes between the plateaus and even at the same plateau [14].

Later, it has been found that in more clean samples and at lower temperatures Hall plateaus appear near the values of magnetic field which correspond to the fractional values of Landau levels filling factor with odd denominator, such asν = 1/3,2/3,2/5, etc. [15] This effect is now called fractional quantum Hall effect. One of the most exciting features of the fractional QHE is the presence of the excitations with fractional charge. Existence of such excitations has been confirmed recently by measurements of the shot noise of backscattering currents in samples with a narrow constriction [16]. The Shottky formula for the powerSof this shot noise gives:

S=e^∗hIi, (1.17)

wherehIiis the average current ande^∗is the elementary charge. Experiment shows that at, e.g.,ν = 1/3, the charge ise^∗=e/3. Finally, there has been observed a Hall plateau at the filling factor with even denominator, namely atν = 5/2[17]. It is now suggested, that some excitations at this value of the filling factor might have the so callednon-Abelian statistics [18]. All experiments at the moment do not contradict this suggestion.

To conclude, we see that the quantum Hall effect exhibits extremely reach physics. Two Nobel prizes have been awarded for the works on quantum Hall effect. Nevertheless, after more than thirty years of research, some experiments, in particular those discussed in this thesis, bring unexpected results. Most of them utilize complex mesoscopic devices, fabrication of which has become possible recently. These experiments are described below in details.

However, before we proceed with the discussion of these results, we consider some basic theoretical aspects of quantum Hall effect.

1.2.3 Landau quantization

We start by considering the quantum mechanical problem of motion of a single electron in two dimensions in a homogeneous magnetic field directed perpendicular to the plane [19]. The Hamiltonian for an electron in the electromagnetic field is given byH = (~p−e ~A/c)²/2m, so that the Schrödinger equation has the following form:

(~p−e ~A/c)²

2m ψ(x, y) =Eψ(x, y), (1.18)

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where A~ is the vector potential of the magnetic field, and~p = −i~∇ is the momentum operator. It is useful to choose the so called Landau gauge for the vector potential of the homogeneous magnetic field directed alongz-axis:

Ax=−By, Ay= 0, (1.19)

whereB is the magnitude of the field. Substituting this equation in Eq. (1.18) we find that the Schrödinger equation acquires the following form:

(i~∂x−eBy/c)²

2m −~²∂_y² 2m

!

ψ(x, y) =Eψ(x, y). (1.20) The differential operator on the left hand side of Eq. (1.20) is translationary invariant in thex-direction, so that one can use the wave function of the following form:

ψ(x, y) = exp[ikx]χ(y). (1.21) Substituting this equation in Eq. (1.20) we get

(~k+eBy/c)²

2m −~²∂_y² 2m

!

χ(y) =Eχ(y). (1.22) One can easily see that this is nothing but the Schrödinger equation for the harmonic oscillator whose equilibrium positiony0and the frequencyωcare given by

y₀=~ck

eB, ω_c =eB

mc. (1.23)

Thus we conclude that the energy levels for the electron in the magnetic field, the so called Landau levels, are given by the expression:

E(n, k) =~ωc(n+1

2), (1.24)

wherenis a positive integer number. Importantly, these energies are independent of quantum numberk, which implies a strong degeneracy.

To calculate the number of states in a single Landau level, we consider a rectangular sample with dimensionsLxandLyand with periodic boundary conditions. The wave vector kis then quantized in multiples of∆k = 2π/Lx. The maximum value of the wave vector kmaxcan be found from the condition that the position of the center of oscillatory0(kmax) takes the valueLyand from Eq. (1.23). Thus, the total number of states isN =kmax/∆k= LxLy·eB/2π~c, and the density of states per unit of area is given by

nLL= eB 2π~ ≡ 1

2πl²_B, (1.25)

where we have introduced the so called magnetic length: lB =p

eB/~c. In fact, the mag- netic flux through the area2πl²_Bis equal to the flux quantumΦ0=hc/e. Next, we can define

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the Landau levels filling factor as the ration of electron density to the Landau level density of states:

ν ≡ ns

n_LL. (1.26)

From a different point of view, the filling factor is the number of electrons per flux quantum.

One can check, as well, that the density of states at a Landau level equals to the density of states of 2DEG in the absence of the magnetic field integrated over the energy interval~ωc:

2nLL= m

π~² ·~ωc, (1.27)

where we have taken into account the fact that there is one state for each spin projection.

The energy splitting between two states of an electron with different projection of spin along the magnetic field is given by the Zeeman energy. It is typically smaller than the cyclotron energy, so that the density of states of 2DEG in magnetic field looks as shown in Fig. 1.5.

Figure 1.5:Density of states of the two dimensional electron gas without (left panel) and with (right panel) magnetic field. The cyclotron gap is typically larger than Zeeman splitting. The Landau levels in magnetic field are highly degenerate, so that the total number of states in a level is equal to the number of states of 2DEG in corresponding marked interval of energies.

Naively, the Landau quantization might explain the appearance of at least integer QHE.

Namely, one could argue that when one increases the magnetic field, the density of states at Landau levels grows and the levels move up. So that when each level crosses the Fermi level one would observe conductivity peak, while when the Fermi level is between the Landau levels, one would see a plateau. In fact, such situation is not possible, because there are no states in between the Landau levels and therefore the Fermi level is always stuck at the last partially filled level until it gets completely empty. Indeed, straightforward linear response calculations (see chapter 11 of Ref. [12]) for the conductivity of non-interacting 2D electrons

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1.3 Theoretical approaches to quantum Hall effect 11

give a simple linear dependence on the magnetic field. Moreover, one can show that the resistivity tensor should be equal to

ˆ ρ=

0 −B/ensc B/en_sc 0

(1.28) in arbitrarytranslational invariantsystem. In order to do so one just needs to note that in the reference frame moving with a velocity~vwith respect to the lab frame there appears an electric fieldE~ =−~v×B/c~ =−~j×B/en~ _sc. This is a very robust picture, valid in classical and quantum situations. Thus, for the quantum Hall effect to be observed, one needs to break the translational invariance, e.g., by adding disorder. To explain the fractional Hall effect one needs to consider the electron-electron interactions as well.

1.3 Theoretical approaches to quantum Hall effect

In this section we consider three most important approaches to the theoretical description of the quantum Hall effect. We start with the single particle description of the integer QHE, then we discuss the effective theory of quantum Hall effect and finally the microscopic variational function approach to fractional QHE

1.3.1 Single particle picture

Let us consider the classical motion of an electron in strong magnetic filed and an external potential. If this potential is constant, then the motion of an electron is described by the equation~r(t) =R~+~r⁰(ω_ct), where~r⁰(ω_ct)describes the circular motion around the center R~ which does not depend on time. In the presence of a non-trivial external potentialφit is instructive to consider the motion of this guiding center, defined in general situation as:

R~ =~r+B~ ×~r/Bω˙ c. (1.29) The equation of motion of an electron in terms of this coordinate isB~ ×R~˙ =∇~ϕ(~r). In the strong magnetic field the radius of orbiting motion is small and one can replaceϕ(~r)byϕ(R)~ assuming a smooth potential. Then the guiding center drifts along the equipotential lines:

~

v_d≡R~˙ = c B²

B~ ×∇~ϕ, (1.30)

so that ϕ(X(t), Y(t)) = const. The typical trajectories in the disorder potential in a 2D sample is shown in Fig. 1.6. Electrons drift clockwise around hills and counter-clockwise around valleys. There are also trajectories which go through the whole sample.

In the quasi-classical picture, these drift trajectories correspond to wave functions localized along the equipotential lines. The energy of each orbital is shifted from the Landau level energy by an amountδE, approximately equal to the value of the potential at the position of the orbital. For example, although in a homogeneous electric fieldϕ(x, y) =Eyeigenfunc- tions (1.21) are not changed, their energies are no longer degenerate: They depend on the wave vectork, so that Eq. (1.24) takes the form

E(n, k) =~ω_c(n+1

2) +eϕ(y₀(k)), y₀=kl²_B. (1.31)

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Importantly, the states that are far from a Landau level are all localized, since they drift around the hills or valleys of the potential, while the states near the Landau level go through the whole sample and thus might contribute to transport. The actual density of states in a disordered sample thus looks as shown in Fig. 1.7. The localized states create a “reservoir”

for the Fermi level, so that the scenario described at the end of Sec. 1.2.3 becomes possible.

One may wonder, however, whether the quasi-classical picture gives a reliable description in the extreme quantum limit of strong fields, whereEF ' ~ωc. The point is that the commutation relation for the guiding center coordinates are given by

[X, Y] =il²_B. (1.32)

Therefore, in strong magnetic field limit,lB → 0, they become effectively classical. In- tuitively, this can be understood by considering the expression for magnetic length l²_B =

~c/eB. The Plank constant enters this equation in the form of the ratio to the magnetic field, thus the strong magnetic field suppresses quantum effects. Moreover, the disorder in real samples is typically very smooth, because the impurities are separated from the 2DEG plane by a distance of order of 100 nm, while the magnetic length is of order of 10 nm. Therefore, the quasi-classical approximation is applicable.

Figure 1.6:Schematic illustration of the quasi-classical picture of electron motion in a typical potential profile. The value of the potential is indicated by the intensity of the gray shadow. The electrons drift along the equipotentials shown by thin black lines. The direction of the drift velocity is determined by the local electric filed and is indicated by the black arrows. Electrons encircle the hills and valleys of the potential landscape, but there are also equipotentials which go through the whole sample. In the quantum case, the electron wave functions are localized along these equipotential lines.

Next important effect which is necessary to understand is the high precision of the conductance quantization. The key feature of the motion in a magnetic field which is responsi- ble for such result is the fact that the coordinate of orbital is proportional to its momentum (1.23). This means that the states with opposite momenta are located at the opposite sides of a macroscopic sample. This, in turn, leads to strongly suppressed backscattering, and, as follows from the Landauer-Büttiker formalism, to the quantization of conductance. Namely, let us consider the situation where the Fermi level is in the mobility gap as shown in Fig. 1.7.

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The only states that contribute to the current are the states near the Fermi surface, i.e. near the actual edge of the sample. Then each state gives a contribution to the current proportional to its group velocity:

I=eX

n

Z dk

2πv(n, k) =eX

n

Z dk1

h

∂E(n, k)

∂k . (1.33)

Performing the change of variables in the above integral we come to the conclusion that the current is given by

I= e h

X

n

Z ∆µ 0

dE=ne²

hV, (1.34)

wherenis the number of filled Landau levels andV is the potential difference between the two edges.

Figure 1.7:Electron density of states in a disordered sample. The gray shadow indicates the localized states, while the dashed region indicates the conducting, delocalized, states near the center of Landau levels. The right panel shows positions of the corresponding states in a real space. The value of the Fermi energy in this figure corresponds to the situation at a Hall plateau.

The picture described above, is, of course, an idealization. According to the experiment [14], there always exist some delocalized states in bulk, which also carry some current density. But it is important, that scattering between the states in the bulk does not spoil the quantization of the Hall conductance. When the Fermi level is close to the Landau level, the edge states percolate through the whole sample, as show in Fig. 1.8, and may approach each other. This means that in between the Hall plateaus the backscattering is no longer suppressed, and the longitudinal conductivityσ_xxhas a large peak, exactly in agreement with the experimental results. Moreover, the quantum percolation picture predicts the scaling behavior of the length of the edgeLas a function of the detuningδfrom the Landau level:

L∼ |δ|⁻^7/3 (1.35)

is also in good agreement with experiments [20].

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Figure 1.8: Schematic illustration of the percolation transition in random potential landscape. Left panel: Most of the states are empty, so that the electrons fill only a few “lakes” of localized states.

This is an insulating state. Middle panel: At half filling the shoreline percolates, so that the states from opposite edges approach each other. In this state the conductance is finite because of the strong backscattering.Right panel:Most of the states are filled, so that there are only few “dry islands”. This state is also insulating. However, two opposite edge states contribute to a quantized Hall conductance.

The idea that the transitions between different Hall plateaus exhibit scaling behavior has shown to be very fruitful. Namely, the theoretical studies of RG flow of the conductivity tensor showed good agreement with the experimental behavior (1.16). The RG flow found in these works for the non-interacting 2D system in magnetic filed and with Gaussian correlated disorder is shown in Fig. 1.9. The longitudinal conductivityσ_xxgoes to zero as the size of the system grows, while there are several fixed points atσ_xx = 0andσ_H = ne²/h. This result agrees with the experimental finding that the most precise quantization is obtained for a conductance of large samples.

Figure 1.9:Two-parametric RG flow of the conductivity, in units ofe²/h, of a non-interacting electron gas in random delta correlated potential, after Ref. [21]. The electron gas becomes insulating in the infrared limit. The Hall conductivity in this limit takes integer values at corresponding IR fixed points.

It turns out that the fixed points of the RG flow in QHE has a topological nature. To show this, we follow the ideas proposed in works of Thouless (see chapter 11 of Ref. [12]

and references therein). Namely, we consider a 2DEG sample with a geometry where the voltage source and the ammeter are replaced with two fluxes, as described in Fig. 1.10. In this geometry the current operators are conjugated to the corresponding fluxesIi=∂H/∂Φi,

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i=x, y, whereHis the Hamiltonian of the system. Taking this into account, we come to the conclusion that the Kubo formula for the Hall conductance reads:

GH =i~hΨ0|∂H

∂Φx

P₀ (H−E0)²

∂H

∂Φy − ∂H

∂Φy

P₀ (H−E0)²

∂H

∂Φx|Ψ0i, (1.36) whereP0projects off the ground state, andE0is the ground state energy. Next, we rewrite this equation using the expression for the derivatives of the wave function,|∂Ψ0/∂Φii = P0(H−E0)⁻¹∂H/∂Φi|Ψ0i, given by the first order perturbation theory, so that:

GH= e² 2πh

ZZ Φ₀ 0

dΦxdΦy

h∂Ψ0

∂Φx|∂Ψ0

∂Φyi − h∂Ψ0

∂Φy|∂Ψ0

∂Φxi

. (1.37)

Here we formally integrated Eq. (1.36) in the range of one flux quantum and divided it by Φ²₀, since the conductance should not depend on the boundary conditions set byΦi, if the Fermi level is inside the mobility gap. The wave function is periodic with respect to the singular fluxes, thus|Ψ0i(Φx,Φy)can be considered as mapping from the torus(Φx,Φy)to the complex projective space of wave functions. The integral in (1.37) is thus the integral of the Jacobian of this mapping. Such integral is topologically invariant and can take only integer values. Mathematically speaking, the Hall conductance is given by the first Chern numberof this mapping.

Figure 1.10:Topological arguments for the integer quantization of the Hall conductance.Left panel:

The Corbino disk is threaded by a singular flux tube shown in blue color. When the flux is adiabatically increased, the induced electric field leads to the charge current shown by red arrows. The flux quantum can be gauged away, which leads to the conclusion that the corresponding transferred charge should be integer.Right panel:One can replace the voltage source (V) and the ammeter (A) in the classical Hall bar geometry with the two fluxes. Then one can use the fluxΦxto generate voltage and the fluxΦy

to monitor the current. The topology of 2DEG in such gedanken setup is equivalent to the torus with a single hole. The edge state at the edge of this hole does not contribute to the current and can be shrank.

The conductance of the 2DEG in such setup is given by the topological invariant (1.37).

Another simple and robust argument has been proposed by Laughlin in [22] and devel- oped by Halperin in [23]. Namely, they consider the Corbino disc threaded by the singular magnetic flux tube, as shown in Fig. 1.10. When one changes the flux, the induced electric field around the flux induces the Hall current and therefore the charge transfer form one edge

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of the disk to another:

∆Q= Z

dtI= Z

dtσ_H∂Φ

∂t =σ_H∆Φ. (1.38)

After the insertion of a flux quantum, the system should move from one eigenstate to another, since the singular flux quantum is a pure gauge. Since the charge of arbitrary excitation is integer times the electron charge, we conclude that the Hall conductance should be integer timese²/h.

All our considerations in this section concern the non-interacting electrons. In real samples electrons do interact, and one may wonder about applicability of discussed results. How- ever, the last two considerations, in fact, do not rely on the absence of interactions. As soon as the Fermi level lies in a mobility gap, the ground state on torus is non-degenerate, and only integer charged excitations are present in the system, these two argumentshold, so that the Hall conductance must be quantized asσ_H =n·e²/h, with integern. The discovery of fractional values of the Hall conductance number was therefore surprising. It means that both mentioned conditions might be broken in the system with strong interaction. And indeed, the minimal charge of excitations was experimentally found to be less than electron charge in the regime of fractional QHE. Explanation of this striking phenomenon and other features of the FQHE are discussed in the next two sections.

1.3.2 Effective theory

In this section we start our consideration of the fractional QHE with the most simple and probably the most powerful approach to the quantum Hall physics: the effective theory. The main idea of this approach is to construct the low energy action of the quantum Hall effect from the general considerations, without recurse to any microscopic calculations, as in, e.g., the Landau-Ginzburg theory of superconductivity. Let us consider an infinitely extended quantum Hall liquid. Such a liquid can be described by a conserved current,J^µ. The con- tinuity equation,∂_µJ^µ = 0, is solved in two dimensions by introducing potentialsB_µ, so that

J^µ= 1

2π^µνλ∂νBλ. (1.39)

Here and below, we use units wheree = ~ = 1, and adopt the Einstein summation convention, unless specified otherwise. The current is invariant under the gauge transformations Bµ→Bµ+∂µβ. This means that the action for the fieldBµshould be gauge-invariant with respect to these transformation. Generally, one can thus write

S[B] =α0

Z

d³r^µνλBµ∂νBλ+α1

Z

d³r∂µBν∂^µB^ν+. . . (1.40) By counting dimensions, it is easy to see that the first term in the above action, the so called Chern-Simons action,

S[B] =α₀ Z

d³r^µνλB_µ∂_νB_λ, (1.41)

has zero dimension, while all other possible terms have lower dimensions, i.e., are irrelevant at large distance and low energy scales. For example, the Maxwell-like term has dimension

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−1. Importantly, the term (1.41) breaks time reversal symmetry, but this is not prohibited for the system in an external magnetic field. It is also important to notice, that the action (1.41) istopological, i.e., it does not depend on the metric of a manifold on which it is defined.

Next, the interaction of the QH current with an external electromagnetic field, described by a vector potentialAµ, is given by the term:

Sint[A, B] = Z

d³rAµJ^µ= 1 2π

Z

d³rAµ^µνλ∂νBλ. (1.42) Integrating out the fieldsBµ, we arrive at an effective action for the electromagnetic field in the Chern-Simons form again:

Seff[A] = 1 α0(4π)²

Z

d³r^µνλAµ∂νAλ. (1.43) The average currenthJ^µi=δS_eff[A]/δA_µis then given by Hall’s law:

hJ^µi=σH^µνλ∂µAλ, (1.44)

whereσ_H = 1/8π²α₀ is the Hall conductivity. Thus, we conclude that for an infinitely extended liquid the action (1.41) correctly reproduces the behavior of current in the regime of QH effect. Already at this point we must notice, that the ground state of topological theories is typically degenerate on compact manifolds [24], so that the assumption made in the Thouless argument may in fact be incorrect.

Having constructed a gauge invariant low-energy action for an incompressible QH liquid, we proceed to analyze the spectrum of local excitations. It has been found that all the states in the topological field theory with the Chern-Simons action (1.41) are described by Wilson lines [24]. For instance, a general local excitation at the pointris created by the operator:

ψq(r) = exp

iq Z r

dr^µBµ

, (1.45)

whereqis a constant. The charge operator in the theory with action (1.41) is given by an integral over a space-like plane of the density:

Q_em= Z

J⁰d²r= (1/2π) Z

d²r^νλ∂_νB_λ. (1.46) Thus, one sees that the operator (1.45) indeed creates a local excitation with a chargeQem= q/4πα0. The statistical phase which is acquired after permutation of two excitations of the type (1.45) is determined by braiding the corresponding Wilson lines [24]. Considering two excitations labeled byq₁andq₂one arrives after a simple calculation at the following expression for the statistical phase: θ₁₂ = πq₁q₂/4πα₀. The obvious constraint on the effective theory is that the spectrum of excitations in this theory must contain electrons. This means that the excitation (1.45) with unit charge has to have the fermion statistics. Thus we have π(4πα₀)²/4πα₀ = πm, wheremis an odd integer. This constraint gives usα₀ =m/4π, therefore the Hall conductivity isσH = 1/8π²α0 = (1/m)e²/hin physical units, which corresponds to the so called Laughlin series of the FQHE states.