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Lecture Notes, 2006 1 Introduction to the Quantum Hall Effects Lecture notes, 2006 Pascal LEDERER Mark Oliver GOERBIG Laboratoire de Physique des Solides, CNRS-UMR 8502 Universit´ede Paris Sud, Bˆat. 510 F-91405 Orsay cedex 2 Contents 1 Introduction 7 1.1 Motivation............................. 7 1.2 HistoryoftheQuantumHallEffect . 9 1.3 Samples .............................. 14 2 Charged particles in a magnetic field 19 2.1 Classicaltreatment . 19 2.1.1 Lagrangian approach . 20 2.1.2 Hamiltonian formalism . 22 2.2 Quantumtreatment. 23 2.2.1 Wave functions in the symmetric gauge . 25 2.2.2 Coherent states and semi-classical motion . 29 3 Transport properties– Integer Quantum Hall Effect (IQHE) 33 3.1 Resistance and resistivity in 2D . 33 3.2 Conductance of a completely filled Landau Level . 34 3.3 Localisation in a strong magnetic field . 37 3.4 Transitions between plateaus – The percolation picture . 42 4 The Fractional Quantum Hall Effect (FQHE)– From Laugh- lin’s theory to Composite Fermions. 45 4.1 Model for electron dynamics restricted to a single LL . 46 4.1.1 Matrixelements. 48 4.1.2 Projected densities algebra . 50 4.2 The Laughlin wave function . 51 4.2.1 The many-body wave function for ν =1 ........ 52 4.2.2 The many-body function for ν = 1/(2s +1) ...... 55 4.2.3 Incompressiblefluid. 58 3 4 CONTENTS 4.2.4 Fractional charge quasi-particles . 59 4.2.5 Groundstateenergy . 62 4.2.6 Neutral Collective Modes . 67 4.3 Jain’s generalisation – Composite Fermions . 69 4.3.1 The effective potential . 70 5 Chern-Simons Theories and Anyon Physics 75 5.1 Chern-Simonstransformations . 75 5.2 Statistical Transmutation – Anyons in 2D . 78 5.2.1 Anyons and Chern-Simons theories . 80 5.2.2 Fractional charge and fractional statistics . 82 6 Hamiltonian theory of the Fractional Quantum Hall Effect 85 6.1 Miscroscopictheory. 86 6.1.1 Fluctuations of ACS(r).................. 87 6.1.2 Decoupling transformation at small wave vector . 91 6.2 Effective theory at all wave vectors . 95 6.2.1 Approximate treatment of the constraint . 98 6.2.2 Energy gaps computation . 100 6.2.3 Self similarity in the effective model . 103 7 Spin and Quantum Hall Effect– Ferromagnetism at ν = 1 109 7.1 Interactions are relevant at ν =1 ................109 7.1.1 Wavefunctions . .110 7.2 Algebraicstructureofthemodelwithspin . 112 7.3 Effectivemodel ..........................115 7.3.1 Spinwaves.........................117 7.3.2 Skyrmions.........................118 7.3.3 Spin-charge entanglement . 119 7.3.4 Effective model for the energy . 121 7.4 Berryphaseandadiabatictransport . 123 7.5 Applications to quantum Hall magnetism . 127 7.5.1 Spin dynamics in a magnetic field . 127 7.6 Applicationtospintextures . .128 8 Quantum Hall Effect in bi-layers 131 8.1 Introduction............................131 8.2 Pseudo-spinanalogy . .133 CONTENTS 5 8.3 Differences with the ferromagnetic monolayer case . 134 8.4 Experimentalfacts . .137 8.4.1 PhaseDiagram .....................137 8.4.2 Excitation gap . 139 8.4.3 Effect of a parallel magnetic field . 139 8.4.4 The quasi-Josephson effect . 141 8.4.5 Antiparallel currents experiment . 142 8.5 Excitonicsuperfluidity . .145 8.5.1 Collective modes – Excitonic condensate dynamics . 148 8.5.2 Charged topological excitations . 150 8.5.3 Kosterlitz-Thouless transition . 153 8.5.4 Effect of the inter layer tunneling term . 154 8.5.5 Combined effects of a tunnel term and a parallel field B 155 k 8.5.6 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-dimensional electron systems submitted to a perpendicular magnetic field remain a very active field of research, be it experimentally or on the theory level [3]. New surprises arise year after year, exotic states of electronic matter, new ma- terials with fascinating quantum Hall properties keep triggering an intense activity in the field: see for example the number of papers on graphene which appear 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. In heterostructure interfaces such as the GaAs/AlGaAs system, one of the pro- totypical experimental systems, the electronic mobility µ has now reached values up to µ 30 106cm2/Vs, more than two orders of magnitude larger ≃ × than 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 physics regarding electronic fluids in conducting materials. In large band metallic systems, the role of interactions was successfully taken into account until the sixties by the Landau liquid theory [7], which is a perturbation approach: interactions between electrons alter adiabatically the properties of the free electron model, so that the Drude-Sommerfeld model keeps its validity with 7 8 Introduction renormalized coefficients. The theoretical tools corresponding to this physics have involved sophisticated diagramatic techniques such as Feynman dia- grams, which are all based on the existence of a well controlled limit of zero interaction Green’s function. It was realized in the fifties, with the theory of BCS superconductivity [8] that attractive interactions cause a breakdown of the non interacting model, and a spontaneous symmetry breaking (in the superconductivity case a breakdown of gauge invariance), but even in that case Fermi liquid theory seemed an unescapable starting point for conducting systems. 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/seventies or even the Luttinger liquid problem in the eighties did little to suggest the intellectual revolutions which were in store with the discoveries of the frac- tional quantum Hall effects and of superconducting high Tc cuprates [9]. (The heavy fermion physics is somewhat of a hybrid between the former and the latter electronic systems.) The fundamental novelty of both those phenomena, which involve elec- tron systems in two dimensional geometries, is that the largest term in the Hamiltonian is the interaction term, so that perturbation theory is basically useless. If one tries to do perturbation theory from a limiting case, such as the incompletely filled Landau level in the Hall case, or the zero kinetic energy in the High Tc case, one is faced with the macroscopic degeneracy of the starting ground state. There is no way to evolve adiabatically from this degenerate ground state to the physical one. In both cases, interactions do not alter the quantitative properties of a pre-existing ground state. They are essential at determining the symmetry and properties of the ground state which results from the lifting of a formidable degeneracy. Thus the whole aparatus of perturbation theory turned ou to be inadequate for a theoretical understanding of the QHE, as well as of High Tc superconductivity. New methods have had to be devised, and new concepts emerged to account for unexpected exotic phenomena: new particles, new statistics, new ground states,[11] etc.. The most intuitive method turned out to be very useful and fruitful. It amounts to guessing the many-body wave function for 1011 particles for the ground state. This is the incredibly original path chosen≈ in 1983 by Laughlin [12]. More or less the same method led P. W. Anderson to propose the RVB wave function as the basic new object to describe High Tc superconductivity in 1987. That wave function is a resonating superposition of 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 of any site by two electrons. This proposal has been at the center of active discussions over the last twenty years. In the context of Quantum Hall Effects, new ideas such as Chern-Simons theories, 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 strange phenomena such as charge fractionalization, fractional statistics, and so forth, at work among the elementary excitations of the 2D electronic liquid under magnetic 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 phase transitions 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 separates a metallic paramagnetic phase from an insulating antiferromagnetic one. At finite temperature, the “quantum critical regime” involves a broader array of parameter values. Quantum phase transitions, as we shall see, are present in a number of Quantum Hall Effects, as a function of magnetic field or of electron 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 supposed to be a final and complete account of the physical properties of Nature. Hall noted that, contrary to Maxwell’s opinion, if a current I is driven through a thin metallic slab in a perpendicular magnetic field B = Bez, an elec- tronic density gradient develops in the slab, in the direction orthogonal to the current.
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