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1 Introduction to the Quantum Hall Effects Lecture notes, 2006 Pascal LEDERER Mark Oliver GOERBIG Laboratoire de Physique des Solides, CNRS-UMR 8502 Université de Paris Sud, Bât. 510 F-91405 Orsay cedex 2 Contents 1 Introduction 7 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 History of the Quantum Hall Effect . . . . . . . . . . . . . . . 9 1.3 Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2 Charged particles in a magnetic field 2.1 Classical treatment . . . . . . . . . . . . . . . . . 2.1.1 Lagrangian approach . . . . . . . . . . . . 2.1.2 Hamiltonian formalism . . . . . . . . . . 2.2 Quantum treatment . . . . . . . . . . . . . . . . . 2.2.1 Wave functions in the symmetric gauge . . 2.2.2 Coherent states and semi-classical motion . . . . . . 19 19 20 22 23 25 29 3 Transport properties– Integer Quantum Hall Effect (IQHE) 3.1 Resistance and resistivity in 2D . . . . . . . . . . . . . . . . . 3.2 Conductance of a completely filled Landau Level . . . . . . . . 3.3 Localisation in a strong magnetic field . . . . . . . . . . . . . 3.4 Transitions between plateaus – The percolation picture . . . . 33 33 34 37 42 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Fractional Quantum Hall Effect (FQHE)– From Laughlin’s theory to Composite Fermions. 4.1 Model for electron dynamics restricted to a single LL . . . . . 4.1.1 Matrix elements . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Projected densities algebra . . . . . . . . . . . . . . . . 4.2 The Laughlin wave function . . . . . . . . . . . . . . . . . . . 4.2.1 The many-body wave function for ν = 1 . . . . . . . . 4.2.2 The many-body function for ν = 1/(2s + 1) . . . . . . 4.2.3 Incompressible fluid . . . . . . . . . . . . . . . . . . . . 3 45 46 48 50 51 52 55 58 4 CONTENTS . . . . . 59 62 67 69 70 . . . . 75 75 78 80 82 6 Hamiltonian theory of the Fractional Quantum Hall Effect 6.1 Miscroscopic theory . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Fluctuations of ACS (r) . . . . . . . . . . . . . . . . . . 6.1.2 Decoupling transformation at small wave vector . . . . 6.2 Effective theory at all wave vectors . . . . . . . . . . . . . . . 6.2.1 Approximate treatment of the constraint . . . . . . . . 6.2.2 Energy gaps computation . . . . . . . . . . . . . . . . 6.2.3 Self similarity in the effective model . . . . . . . . . . . 85 86 87 91 95 98 100 103 7 Spin and Quantum Hall Effect– Ferromagnetism 7.1 Interactions are relevant at ν = 1 . . . . . . . . . 7.1.1 Wave functions . . . . . . . . . . . . . . . 7.2 Algebraic structure of the model with spin . . . . 7.3 Effective model . . . . . . . . . . . . . . . . . . . 7.3.1 Spin waves . . . . . . . . . . . . . . . . . . 7.3.2 Skyrmions . . . . . . . . . . . . . . . . . . 7.3.3 Spin-charge entanglement . . . . . . . . . 7.3.4 Effective model for the energy . . . . . . . 7.4 Berry phase and adiabatic transport . . . . . . . 7.5 Applications to quantum Hall magnetism . . . . . 7.5.1 Spin dynamics in a magnetic field . . . . . 7.6 Application to spin textures . . . . . . . . . . . . 109 109 110 112 115 117 118 119 121 123 127 127 128 4.3 5 4.2.4 4.2.5 4.2.6 Jain’s 4.3.1 Fractional charge quasi-particles . Ground state energy . . . . . . . . Neutral Collective Modes . . . . . . generalisation – Composite Fermions The effective potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chern-Simons Theories and Anyon Physics 5.1 Chern-Simons transformations . . . . . . . . . . . . . 5.2 Statistical Transmutation – Anyons in 2D . . . . . . . 5.2.1 Anyons and Chern-Simons theories . . . . . . . 5.2.2 Fractional charge and fractional statistics . . . at ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . =1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Quantum Hall Effect in bi-layers 131 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 8.2 Pseudo-spin analogy . . . . . . . . . . . . . . . . . . . . . . . 133 CONTENTS 8.3 8.4 8.5 5 Differences with the ferromagnetic monolayer case . . . . . . 134 Experimental facts . . . . . . . . . . . . . . . . . . . . . . . . 137 8.4.1 Phase Diagram . . . . . . . . . . . . . . . . . . . . . 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 Excitonic superfluidity . . . . . . . . . . . . . . . . . . . . . . 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 Bk 155 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 materials 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 factor in the discovery of new electronic states of two dimensional matter. In heterostructure interfaces such as the GaAs/AlGaAs system, one of the prototypical experimental systems, the electronic mobility µ has now reached values up to µ ≃ 30 × 106 cm2 /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 diagrams, 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 insulator problem, the physics of solid or superfluid 3 He 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 fractional 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 electron 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 electrons 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 quantum phase transition and of quantum critical points [18]. Quantum phase transitions occur at zero temperature. They are not controlled by temperature, 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 correction 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 electronic density gradient develops in the slab, in the direction orthogonal to the current. This gradient is equivalent to a transverse voltage V , so that the resulting transverse resistance (the Hall resistance) is proportional to the field, 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 the electron equation of motion: 10 Introduction I −I gaz d’électrons 2D résistance longitudinale résistance de Hall Figure 1.1: Two dimensional electron system under perpendicular magnetic field. The current I is driven through the two black contacts. The longitudinal resistance is measured between two contacts on the same edge, while the Hall resistance is measured across the sample on the two opposite edges. dp p p = −e E + ×B − , dt m τ where E is the electric field, m is the electron (band) mass , p its momentum τ the mean diffusion time due to impurities. The stationary solution for this equation, i.e. that for dp/dt = 0, is py B − m px 0 = −e Ey − B − m 0 = −e Ex + px , τ py . τ With the cyclotron frequency ωC ≡ eB/m and the Drude conductivity σ0 = nel e2 τ /m, one gets py px σ0 Ex = −nel e − nel e (ωC τ ), m m py px σ0 Ey = nel e (ωC τ ) − nel e , m m In terms of the current density j = −nel ep/m, in matrix form E = ρ j, History of the Quantum Hall Effect 11 with the resistivity tensor ρ= σ0−1 enBel − enBel σ0−1 ! 1 = σ0 1 ωC τ −ωC τ 1 ! . (1.1) The conductivity follows by matrix inversion, σ=ρ −1 = σL −σH σH σL ! , (1.2) with σL = σ0 /(1 + ωC2 τ 2 ) et σH = σ0 ωC τ /(1 + ωC2 τ 2 ). In the limit of a pure metal with infinite τ , ωC τ → ∞, one has ρ= 0 − enBel B enel 0 ! , σ= 0 enel B − enBel 0 ! . (1.3) Note that the diagonal (longitudinal ) conductivity is zero together with the longitudinal resistivity. The classical Hall effect, deemed by Hall of purely academic interest, and with no foreseeable application whatsoever is nowadays of current industrial use, and is still useful in condensed matter physics to measure the carrier density 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 of metallic systems, to the quantum treatment of electronic motion in a static uniform magnetic field. He found that problem to be quite analogous in 2D to that of a harmonic oscillator, with an energy structure of equidistant discrete levels, with a distance h̄ωC . Each level is highly degenerate. The surface 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 quanta threading the surface in a perpendicular field B. Because of their fermionic character, 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 This quantum treatment will be reviewed in chapter 2. (1.4) 12 Introduction The Quantum Hall Effect : a macroscopic quantum phenomenon The IQHE, discovered by von Klitzing in 1980 [1] is, at first sight, a direct consequence of Landau quantization, and disorder. In fact, as we shall see, impurity disorder is also a necessary feature: in a tanslationaly invariant system, the Hall resistivity would have the classical value. In fact Hall quantization appears because of the sample impurity potential, not in spite of it. The IQHE appears at low temperature, when kB T ≪ h̄ωC , and is defined by the formation of plateaus in the Hall resistance, which become quantized, for certain ranges of values of B, as RH = (h/e2 )1/n, where n is an integer, the integer part of the filling factor: n = [ν]. Each plateau in the Hall resistance coincides with a zero (exponentially small value in fact) of the longitudinal resistance (Fig. 1.2). A remarkable fact about the resistance quantization is that its value is independent of the sample geometry, of its quality (density and/or distribution of impurities, etc.). The Hall resistance is given entirely in terms of fundamental constants, e and h. The accuracy of the determination of the n = 1 plateau value reaches 1 part in 109 , so that it is now used in metrology as a universal resistance standard, the v. Klitzing constant RK−90 = 25812, 807Ω. Another surprise followed shortly after the discovery of the IQHE. In 1983, D. Tsui, H. Störmer and A. Gossard found the Fractional Quantum Hall Effect (FQHE) [2]. This occurs for ”magical” values of the filling factor, especially within the lowest LL. The first observed fractional plateaus were at ν = 1/3 and ν = 2/3. Since then, a whole series of plateaux have been detected. The remarkable aspect is that for fractional ν values, there is a huge degeneracy of the N body states. Since, apart from impurities, the only relevant energy is the Coulomb repulsion between particles, one is facing a strongly correlated electron system. Our understanding of the FQHE is still to-day essentially based on a revolutionnary theory put forward by Laughlin in 1983: he proposed, by a series of educated guesses, a wave function for N ≈ 1011 particles, written in the first quantization language, which describes an incompressible electronic liquid state, i.e. one such that elementary and collective excitations are separated from the ground state by a gap[12]. Following the discovery of other families of fractional QH plateaus which are not described by the initial Laughlin wave functions, various generalizations have been proposed. B. Halperin generalised in 1983 the Laughlin wave function to the case of an additional discrete degree of freedom, such as History of the Quantum Hall Effect 13 3.0 2.0 Ix Vy 2.5 Vx 1.5 ρxx (kΩ) 2 ρxy (h/e ) 2.0 6 54 3 1.5 1 2 1.0 2/3 3/ 2 3/ 4 1/ 2 3/5 3/7 5/9 4/3 5/3 8/5 7/5 0.5 2/5 4/9 4/7 1.0 5/11 6/11 6/13 0.5 5/7 0.0 8/15 4/5 0 0 4 8 Magnetic Field B (T) 7/13 7/15 12 16 champ magnétique B[T] Figure 1.2: Experimental signature of the quantum Hall effect. Each plateau coincides with 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 the FQHE. 14 Introduction spin [19]. In 1989, Jain generalised the theory to account for observed fractional states with ν = p/(2sp + 1) with s and p integers. He introduced the notion of “Composite Fermions” (CF). The CF theory allows to understand the 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 to the evolution of semiconducting sample preparation to produce 2D electron gases. The order of magnitude of electron densities in thin metallic films was not appropriate for the QHE discovery. The electronic surface density of metallic thin films is of order nel = 1018 m−2 = 1014 cm−2 . As we shall see, the QHE become observable when the electronic surface density is of the order of the magnetic flux density, i. e. nel ∼ nB = eB/h. This would amount to a magnetic field of order ≈ 1000 T, quite out of reach in the laboratory nowadays, when the largest available fields in a dc regime amount to less than 50 T, and less than 80 T for pulsed magnetic fields. More intense fields are available in destructive experiments or nuclear blasts. A useful quantity q which sets a length q scale for the QH physics is the magnetic length, lB = h̄/eB = 25, 7nm/ B[T]. The magnetic length is such that the flux 2 which threads a surface equal to 2πlB is the flux quantum φ0 = h/e Lower electronic densities, typically nel ∼ 1011 cm−2 are reached in semiconducting structures. The samples used at the time of the IQHE discovery were MOSFETs, shown schematically in the figure 1.3. In such a device, a metallic film is separated from a semiconductor, which is doped with acceptors, by an oxyde insulating layer. The metal chemical potential is controlled with a voltage bias VG . When VG = 0, the Fermi level EF lies in the gap between 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 toward the interface with the insulating layer. This results in a downward bending of the semiconductor band close to the interface. Electrons attracted to 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 semi conductor conduction band can be bent below the Fermi level close to the insulating layer, so that electrons which occupy states in that part of the Samples 15 (a) métal oxyde (isolant) semiconducteur I bande de conduction niveaux d’accepteurs EF z métal oxyde semiconducteur V G bande de valence II z (b) E1 E0 (c) métal oxyde (isolant) semiconducteur métal bande de conduction EF VG niveaux d’accepteurs E oxyde (isolant) z électrons 2D bande de conduction niveaux d’accepteurs EF VG bande de valence bande de valence z z Figure 1.3: Metal-Oxyde Field Effect Transistor (MOSFET). The inset I is a schematic view of a MOSFET. (a) Energy level structure. In the metallic part, the band states are occupied up to the Fermi level EF . The oxyde is an insulating film. The Fermi level in the 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 introduction of holes results in a band bending in the semiconducting part and (c) when the gate bias exceeds a certain value, the conduction band is filled close to the insulating interface, and a 2D electron gas is formed. The confining potential has a triangular profile with electric subbands which are represented in the inset II. 16 (a) Introduction AlGaAs (b) GaAs AlGaAs GaAs EF EF dopants (récepteurs) dopants (récepteurs) z électrons 2D z 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 Fermi energy is locked to the dopant levels. The bottom of the GaAs conduction band lies lower than 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 gas forms, on the GaAs side. conduction band form a 2D gas. Electron motion , in spite of a finite extent of the wave function in the z direction is purely 2D if confinement is such that the energy separation between electronic sub-bands E0 (partially filled) and E1 (empty) is significantly larger than kB T (inset II in Fig. 1.3). The problem with MOSFETS is the small distance between the 2D electron gas and the dopants. The latter also act as scattering centers, so that the mean free path is relatively small, and the electron mobility relatively low. This problem is dealt with by forming a 2D electron gas at the interface of a semiconducting heterostructure, such as for example in the III-V heterostructure GaAs/AlGaAs. The two semi-conductors have different gaps between their valence bands and their conduction bands. When the side with the largest gap, Alx Ga1−x As, is doped, the receptor dopant levels are occupied by electrons, and the Fermi level is tied to receptor levels, which may have a higher energy than the bottom of the conduction band in GaAs. The electrons close to the interface migrate in this conduction band [Fig. 1.4(a)]. This polarisation produces a band bending, now on the GaAs side, which is not disordered by the dopants. This spatial separation between the 2D electron gas and the impurities allows to reach larger mobility values than in MOSFETS. Technological progress in the fabrication of semi-conducting heterostructures along the last twenty years has allowed to increase mobilities Samples 17 Density of states Graphene IQHE: R H = h/e2ν at ν = 2(2n+1) Vg =15V T=30mK ∼ 1/ν Usual IQHE: B=9T T=1.6K at ν = 2n (no Zeeman) ∼ν 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 sample with mobility µ ≃ 0, 1 × 106 cm2 /Vs [2] while samples of the same type reach nowadays a mobility of µ ≃ 30 × 106 cm2 /Vs. The discovery of the QHE in graphene in 2005 opens up a new avenue to experiments and theory in the QHE, because graphene is a qualitatively new 2D material, with original electronic structure. See figure 1.5 18 Introduction Chapter 2 Charged particles in a magnetic field Our understanding of integer or fractional quantum Hall effects relies mostly on the quantum mechanics of electrons in a 2D plane, or thin slab, when submitted to a perpendicular magnetic field. There is a notable exception, that of the Integer Quantum Hall Effect (IQHE) observed in anisotropic 3D organic salts such as Bechgaard salts. The IQHE may arise in 3D systems under magnetic field provided the electronic structure of the material under magnetic field exhibits the suitable gap structure. However, in the present lectures, I will adress the main stream of quantum Hall effects physics, that of electrons the dynamics of which is restricted to a plane. The topic of this chapter is the single electron quantum mechanics in a plane under magnetic field. I start with a discussion of the classical mechanics, as a limiting case of the quantum mechanical case. 2.1 Classical treatment The equation of motion of a particle (with charge −e and mass m in a magnetic field B = Bez is as follows: ẍ = −ωC ẏ, ÿ = ωC ẋ, (2.1) This follows from the Lorentz force F = −eṙ×B – By definition, the cyclotron frequency is ωC = eB/m. The equation is solved as: ẋ = −ωC (y − Y ), ẏ = ωC (x − X), 19 (2.2) 20 Charged particle in a static uniform magnetic field B η R r Figure 2.1: Cyclotron motion of an electron in a magnetic field, around the guiding center R. where R = (X, Y ) is a constant of motion. With η = (ηx , ηy ) = r − R, one has η¨x = −ωC2 ηx , η¨y = −ωC2 ηy , (2.3) and the solution is x(t) = X + r sin(ωC t + φ), y(t) = Y + r cos(ωC t + φ), (2.4) where r is the cyclotron motion radius, and φ is an arbitrary angle (constant of 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 of radius r (Fig 2.1). 2.1.1 Lagrangian approach Lagrangian mechanics starts from the energy function L and the minimum action principle, which reproduce the equations of motion of the classical system. This function is defined in configuration space (positions qµ and velocities q̇µ ). The minimum action principle results in the Euler-Lagrange equations d ∂L ∂L − = 0, (2.5) dt ∂ q̇µ ∂qµ Classical approach 21 valid for any index µ. The appropriate function in our case is 1 L(x, y; ẋ, ẏ) = m ẋ2 + ẏ 2 − e [Ax (x, y)ẋ + Ay (x, y)ẏ] , 2 (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 an electromagnetic field, written in a covariant form, with Einstein’s convention, 1 Lrel = mẋµ ẋµ − eẋµ Aµ . 2 The conjugate (or ”canonical”) momenta, which will be needed in the Hamiltonian formulation of clasical or quantum mechanics are px ≡ ∂L = mẋ − eAx , ∂ ẋ py ≡ ∂L = mẏ − eAy . ∂ ẏ (2.7) The Euler-Lagrange equations yield the equations of motion [Eq. (2.1)] mẍ = −eẏ(∂x Ay − ∂y Ax ), mÿ = eẋ(∂x Ay − ∂y Ax ), (2.8) where ∂x ≡ ∂/∂x, ∂y ≡ ∂/∂y, and (∂x Ay − ∂y Ax ) = (∇ × A)z = B is the z component 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 the gauge (it is “gauge invariant”) since ∇ × ∇χ = 0. A usual gauge in non relativistic physics is the Coulomb gauge, ∇ · A = 0.1 The gauge (the gauge function) is not completely determined by the Coulomb gauge condition, which demands only ∆χ = 0, where ∆ = ∇2 is the Laplacian. Gauge transformations in 2D are thus defined as harmonic functions. Two gauge choices are especially useful in the quantum treatment of our problem: the Landau gauge (e.g. for problems defined on a rectangular sample) AL = B(−y, 0, 0) 1 Relativistic mechanics use rather the Lorentz gauge, ∂ µ Aµ = 0. (2.9) 22 Charged particle in a static uniform magnetic field and the symmetric gauge(e.g. for problems defined on a disc) AS = B (−y, x, 0), 2 (2.10) the function which transforms from one of these two gauges to the other is χ = (B/2)xy. Since velocities ẋ and ẏ, are also gauge invariant, it is clear that conjugate (or ”canonical”) momenta in equation (2.7) are not. The gauge invariant momenta (or ”mechanical momenta”) are Πx = mẋ = px +eAx = −mωC ηy , Πy = mẏ = py +eAy = mωC ηx , (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 to use the Hamiltonian formalism of classical mechanics. The Hamiltonian is derived from the Lagrangian by a Legendre transformation, H(x, y; px , py ) = ẋpx + ẏpy − L. It is an energy function defined in phase space (positions/conjugate momenta). One must express velocities in terms of conjugate momenta, using equations (2.7), and one finds for the Hamiltonian H= i 1 h (px + eAx )2 + (py + eAy )2 . 2m (2.12) The Hamiltonian may also be written in a concise fashion, using the ”relative” variables, (ηx , ηy ) (using 2.11), 1 H = mωC2 (ηx2 + ηy2 ), 2 (2.13) where the ”new” variables are nevertheless defined by the variables in phase space, i.e. (x, y, px , py ). 2.2. QUANTUM TREATMENT 2.2 23 Quantum treatment The Hamiltonian formalism allows to introduce the canonical quantization, where one imposes the non commutativity of position with its conjugate momenta, in terms of Planck’s constant h̄, [x, px ] = [y, py ] = ih̄, [x, y] = [px , py ] = [x, py ] = [y, px ] = 0. Since [x, y] = 0, one sees immediately that [ηx , ηy ] = −[X, Y ]. The fact that the guiding center components are constants of motion is expressed by [see also Eq. (2.13)] [X, H] = [Y, H] = 0. (2.14) To compute the commutator between components ηx and ηy , one may use the formula ∂f [A, B]. (2.15) ∂B That formula is valid for two arbitrary operators which commute with their commutator, [A, [A, B]] = [B, [A, B]] = 0. One gets [A, f (B)] = [ηx , ηy ] = e m2 ωC2 ([px , Ay ] − [py , Ax ]) 1 (∂x Ay [px , x] − ∂y Ax [py , y]) eB 2 −ih̄ = eB = or, in terms of magnetic length lB ≡ 2 [ηx , ηy ] = −ilB , q h̄/eB, 2 [X, Y ] = ilB . (2.16) The result is of course gauge invariant. A remarkable point is that the dynamics of a charged particle in a magnetic field is perhaps the simplest example of a non commutative geometry. Notice that, without any knowledge on the energy level structure, the latter has to be degenerate. In any level chosen at random, each state must occupy a minimal surface given by the Heisenberg uncertainty principle, 2 σ = ∆X∆Y = 2πlB . 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” 2πh̄. The level degeneracy may thus be written directly in terms of this minimal surface: the number of states per level and per unit surface being nB = 1/σ = B/φ0 – i.e. the flux density in units of the flux quantum φ0 = h/e. Since electrons follow fermionic statistics, each quantum state is occupied at most by one particle, because of the Pauli principle. When there are many electrons in the system, the filling ν of energy levels is thus described by the ratio between the electron surface density nel and the flux density nB , ν = nel /nB . This ratio is also called the filling 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 conjugate variables. To exhibit explicitly the harmonic oscillator structure, we introduce two sets of ladder operators, (a, a† ) with a = √ 1 (ηx − iηy ), 2lB a† = √ 1 (ηx + iηy ) 2lB lB ηy = √ (a† − a), 2i lB ηx = √ (a† + a), 2 (2.17) and (b, b† ) with b = √ 1 b† = √ (X − iY ) 2lB ilB Y = √ (b† − b), 2 1 (X + iY ), 2lB lB X = √ (b† + b), 2 (2.18) with [a, a† ] = [b, b† ] = 1 et [a, b(†) ] = 0. In terms of ladder operators, the Hamiltonian writes 1 . (2.19) 2 The energy spectrum is thus given by En = h̄ωC (n + 1/2), where n is the eigenvalue of operator a† a. In the context of an electron in a magnetic field the equidistant levels of the oscillator are called ”Landau Levels” (LL, see Fig. 2.2). Formally, in fact, the system may be viewed as a system of two harmonic oscillators, H = h̄ωC a† a + H = h̄ωC a† a + 1 1 + h̄ω ′ b† b + , 2 2 25 niveaux de Landau Quantum treatment 4 3 2 1 n=0 m Figure 2.2: Landau Levels. The quantum number n labels the levels, and m , which is associated to the guiding center, describes the level degeneracy. where the frequency of the second oscillator vanishes, ω ′ = 0. The second quantum 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, mi = n + 1|n + 1, mi, a|n, mi = n|n − 1, mi (pour n >0); √ √ b† |n, mi = m + 1|n, m + 1i, b|n, mi = m|n, m − 1i (pour m >0). When n = 0 ou m = 0, one finds a|0, mi = 0, b|n, 0i = 0, (2.20) and negative numbers are prohibited. An arbitrary state may thus be constructed with the help of ladder operators, starting from the state |0, 0i, (a† )n (b† )m |n, mi = √ √ |0, 0i. n! m! (2.21) The wave functions, which are the state representation in real space, depend on 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) = hx, y|n, mi,we must 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 differential equations, using px = −ih̄∂x and py = −ih̄∂y . With the help of 26 Charged particle in a magnetic field equations (2.11) and (2.17), one finds the representation of ladder operators in the symmetric gauge √ z ¯ a= 2 + lB ∂ , 4lB ! √ z∗ + lB ∂ , b= 2 2 4lB √ z∗ a† = 2 − lB ∂ 2 4lB √ z − lB ∂¯ b† = 2 4lB ! (2.22) where z = x − iy is the electron position in the complex plane2 , z ∗ = x + iy its complex conjugate, ∂¯ = (∂x − i∂y )/2 et ∂ = (∂x + i∂y )/2. A state in the Lowest LL (LLL) is thus determined by the differential equation 2 ¯ z + 4lB ∂ φn=0 (z, z ∗ ) = 0. (2.23) The solution of equation (2.23) is a gaussian multiplied by an arbitrary an¯ (z) = 0, alytic function f (z), with ∂f 2 /4l2 B φn=0 (z, z ∗ ) = f (z)e−|z| , (2.24) Similarly one finds for the state with m = 0 2 z ∗ + 4lB ∂ φm=0 (z, z ∗ ) = 0, (2.25) the solution of which is 2 /4l2 B φm=0 (z, z ∗ ) = g(z ∗ )e−|z| , (2.26) where the function g(z ∗ ) is anti-analytic, ∂g(z ∗ ) = 0. The state |n = 0, m = 0i must thus be represented by a function which is both analytic and antianalytic. The only function which satisfies both requirements is a constant. With the normalisation, one gets φn=0,m=0 (z, z ∗ ) = hz, z ∗ |n = 0, m = 0i = q 2 1 2 2πlB 2 /4l2 B e−|z| , (2.27) The 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 the opposite chirality. Quantum treatment 27 A state corresponding to the quantum number m in the LLL is found from equations (2.20) and (2.22), √ m m 2 z 2 2 ∗ ¯ φn=0,m (z, z ) = q − lB ∂ e−|z| /4lB 2 2πlB m! 4lB = q and 1 2 m! 2πlB ∗ √ n 2 φn,m=0 (z, z ) = q 2 2πlB n! = q 2 2πlB n! 1 z √ 2lB !m 2 /4l2 B e−|z| z∗ − lB ∂ 2 4lB z∗ √ 2lB !n !n , (2.28) 2 /4l2 B e−|z| 2 /4l2 B e−|z| , (2.29) for a state centered at the origin m = 0 in LL n. An arbitrary state writes √ m !n m ∗ 2 z z 2 2 ∗ √ e−|z| /4lB (2.30) φn,m (z, z ) = q − lB ∂¯ 2 2lB 2πlB m!n! 4lB 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 ∗ 2 ∗ 2 |φn=0,m=j (z, z )| = |φn=j,m=0 (z, z )| ∼ |z|2 2 !j 2 2 e−|z| /2lB , j! √ with a probability maximum at radius r0 = 2jlB (Fig. 2.3), they do not represent equal energy states. To conclude the discussion of states |n = 0, mi represented in the symmetric gauge, we compute the average value of the guiding center operator. With the help of equations (2.18), one finds that hRi ≡ hn = 0, m|R|n = 0, mi = 0, but h|R|i = 3 D√ E X 2 + Y 2 = lB It is a Poissonian distribution. q √ 2b† b + 1 = lB 2m + 1. (2.31) 28 Charged particle in a magnetic field 0.4 n=1 n=3 n=5 (a) 0.35 |φn,m=0(z,z*)|2 0.3 0.25 0.2 0.15 0.1 0.05 0 0 1 2 (b) 3 r/lB=|z|/lB n=0 4 5 n=1 2 y/l B 0 0 -2 -2 -4 -4 -4 -2 0 2 4 -4 -2 x/l B 0 2 4 x/l B n=3 4 n=5 4 2 y/l B 6 4 2 y/l B 4 2 y/l B 0 -2 0 -2 -4 -4 -4 -2 0 x/l B 2 4 -4 -2 0 2 4 x/l B Figure 2.3: Probability density for a state |n, m = 0i for various values √ of n. (a) The density 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. Quantum treatment (a) 29 p p0 p x0 x x (b) y y0 y <x,y|x0 ,y0 > <x,y|n=0,m=0> x x0 x Figure 2.4: Coherent states . This means that both√the particle and its guiding center are located on the 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 with 2 radius Rmax √ and surface A = πRmax : as the state with maximum radius has Rmax = lB 2M + 1, this yields the number of states in the thermodynamic 2 limit, M = A/2πlB = AnB , with nB = eB/h, in agreement with the previous argument about the state minimal surface. Similarly one sees that for the state |n, m = 0i in level n the relative variable η is localized on a circle with radius √ RC ≡ h|η|i = 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-classical states, also called coherent states because they play an important role in quantum optics. For a 1D harmonic oscillator, a coherent state is the eigenstate of the annihilation operator and it is the state with the minimum value 30 Charged particle in a magnetic field of the product ∆px ∆x. Such as state can be built from the displacement operator in phase space, which displaces the ground state from hxi = 0, hpx i = 0 to (x0 , p0 ) [Fig. 2.4(a)]. The displacement is done with the operator D(x0 , p0 ) = e−i(x0 p̂−p0 x̂) , (2.33) where the symbols with hats are operators, not to be confused with variables x0 and y0 . This operator displaces variables, as we can check using formula (2.15) D† (x0 , p0 )x̂D(x0 , p0 ) = eix0 p̂ x̂e−ix0 p̂ = x̂ + x0 and D† (x0 , p0 )p̂D(x0 , p0 ) = e−ip0 x̂ p̂eip0 x̂ = p̂ + p0 . The coherent state writes |x0 , p0 i = D(x0 , p0 )|n = 0i, (2.34) where |n = 0i 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 state changes 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 would contradict the fundamental postulates of quantum mechanics, because the associate operators do not commute. The basis |x0 , p0 i is said to be ”overcomplete” [22]. In general, a displacement operator may be constructed from two conjugate operators, which therefore do not commute. In the case of an electron in a magnetic field in a 2D plane, we have two pairs of non commuting con2 2 jugate operators at our disposal, [X, Y ] = ilB et [ηx , ηy ] = −ilB . With the first choice, the displacement operator which acts now in real space, writes − D(X0 , Y0 ) = e i l2 B (X0 Ŷ −Y0 X̂) , (2.35) and the coherent state (in the LLL) is |X0 , Y0 ; n = 0i = D(X0 , Y0 )|0, 0i, (2.36) where |0, 0i ≡ |n = 0, m = 0i. Since the guiding center is a constant of motion, the displacement operator D(X0 , Y0 ) commutes with the Hamiltonian. The state (2.36) remains an eigenstate of the Hamiltonian, which is why the quantum number n is unchanged. Quantum treatment 31 The dynamics enters with the second pair of operators, with the displacement operator i 2 D̃(η0x , η0y ) = e lB (η0x η̂y −η0y η̂x ) , (2.37) which generates a displacement to position η 0 = (η0x , η0y ), so that a general semi-classical state may be written as |X0 , Y0 ; η0x , η0y i = D̃(η0x , η0y )D(X0 , Y0 )|0, 0i. (2.38) The guiding center is thus centered at R0 = (X0 , Y0 ), and the electron turns around that position on a circle of radius r = |η 0 |. One retrieves thus the motion represented on figure 2.1, in terms of a gaussian wave packet. To prove those dynamic properties, remember that a coherent state is an eigenstate of the ladder operator a, and in our case also of b, with η0x − iη0y √ |X0 , Y0 ; η0x , η0y i, 2lB X + iY0 0 √ |X0 , Y0 ; η0x , η0y i. b |X0 , Y0 ; η0x , η0y i = 2lB a |X0 , Y0 ; η0x , η0y i = (2.39) This can be checked, for example, when expressing the displacement operators in terms of ladder operators (2.17) and (2.18), D(X0 , Y0 ) = eβb † −β ∗ b D̃(η0x , η0y ) = eαa † −α∗ a 2 /2 = e−|β| 2 /2 = e−|α| † ∗ eβb e−β b , † ∗ eαa e−α a , (2.40) où l’on a défini η0x − iη0y X0 + iY0 , α≡ √ β≡ √ 2lB 2lB where we used the Baker-Hausdorff formula eA+B = eA eB e−[A,B]/2 , (2.41) which is valid when [A, [A, B]] = [B, [A, B]] = 0. The coherent state writes thus 2 2 † † |X0 , Y0 ; η0x , η0y i = e−(|α| +|β| )/2 eαa eβb |0, 0i and we find with formula (2.15) 2 +|β|2 )/2 a |X0 , Y0 ; η0x , η0y i = e−(|α| 2 +|β|2 )/2 b |X0 , Y0 ; η0x , η0y i = e−(|α| h † i † a, eαa eβb |0, 0i = α |X0 , Y0 ; η0x , η0y i, † h † i eαa b, eβb |0, 0i = β |X0 , Y0 ; η0x , η0y i, 32 Charged particle in a magnetic field which is nothing but equation (2.39). In order to get the time evolution of the coherent state |α, βi = |X0 , Y0 ; η0x , η0y i, one uses the time evolution operator on state i |α, βi(t) = e− h̄ Ht |α, βi(t = 0) 2 )/2 = e−(|α| i e− h̄ Ht ∞ X (αa† )n |n = 0, βi(t = 0) n! i (α)n 2 √ |n, βi(t = 0) = e−(|α| )/2 e− h̄ Ht n! n=0 n ∞ X (αe−iωC t ) −(|α|2 )/2 −iωC t/2 √ = e e |n, βi(t = 0) n! n=0 n=0 ∞ X = e−iωC t/2 |α(t = 0)e−iωC t , βi, (2.42) which yields for the eigenvalue time evolution α(t) = α(t = 0)e−iωC t , β(t) = β(t = 0). (2.43) √ √ √ x 2l Re[β], Y = 2l Im[β], η 2lB Re[α(t)] et η0y (t) = (t) = Since X = B 0 B 0 0 √ − 2lB Im[α(t)], we retrieve η0x (t) = η0x (t = 0) cos(ωC t), η0y (t) = η0y (t = 0) sin(ωC t) and thus the trajectory given in equation (2.4), identifying r = |η 0 | and R = (X0 , Y0 ), as mentionned earlier. Chapter 3 Transport properties– Integer Quantum 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 constant uniform magnetic field, described in the previous chapter. Two main features allow to understand the IQHE : • each completely filled LL (for ν = n) contributes a conductance quantum e2 /h to the electronic conductivity, • when additional electrons start populating the next LL at ν 6= n, they get localized by the impurities disorder potential in the sample, and they 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 Hall result is preserved. This last feature seems analogous at first sight to Anderson localization in 2D in the absence of a magnetic field [23]. It happens that localisation is even more relevant in a magnetic field. 3.1 Resistance and resistivity in 2D Theorists calculate resistivity. Experiments measure resistance. For a classical 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. The product R(e2 /h) is dimensionless. It is an easy exercise to show that in the case of a Hall bar geometry, such as shown in Fig. (1.1), the transverse resistance and the transverse resistivity are equal in 2D, independent of the Hall bar dimensions. This is a basic ingredient to understand the universality of the quantum Hall experimental results. In particular it means that one does not have to measure the physical dimensions of a sample to one part in 1010 in order to obtain the resistivity to that accuracy. The technological progress in semiconductor physics which allowed to manufature 2D Electron Gases (2DEG) with electrical contacts was, in this respect, a decisive one. Even the shape of the sample, or the accurate determination of the Hall voltage probe locations are almost completely irrelevant, in particular, because the dissipation is nearly absent in the QH states. 3.2 Conductance of a completely filled Landau Level We first discuss the effect of a constant uniform electric field on the Landau level energy structure. We take the electric field along the y direction. It is convenient to deal with a sample with rectangular shape, and to assume in a first step (to be relaxed subsequently) that the system is translation invariant in the x direction. An appropriate gauge in this rectangular geometry is the Landau gauge AL = B(−y, 0, 0), so that the Hamiltonian now writes p2y (px − eBy)2 H= + − eV (y), 2m 2m where the potential is V (y) = −Ey (electric field pointing in the ŷ direction). The system is translation invariant in the x̂ direction, so that px , or equivalently h̄k, is a constant of motion, which corresponds to the quantum number m in the symmetric gauge, i.e. to the guiding center eigenvalue. The latter, in a state |n, ki, is delocalized along a straight line in the x direction, contact L 35 énergie Conductance of a filled LL contact R µL µR NL n k min k max y=kl B2 y’ Figure 3.1: LL in the Landau gauge, with a voltage bias between the L and the R contacts, where chemical potentials are respectively µL and µR . Position yk in the direction 2 y is proportionnal to the wave vector k in direction x : yk = klB . 2 with coordinate klB on the y axis. The Hamiltonian is: H= p2y 1 + mωC2 (y + klb2 )2 + eEy, 2m 2 which can be re-written, completing the square, as: p2y 1 E + mωC2 (y − yk )2 + h̄k + C, H= 2m 2 B 2 where we have set px = h̄k, and yk = −klB − eE/mωC2 and C is a constant: 2 E C = − 21 m B . The energy of the state |n, ki is εn,k = h̄ωC n + 1 1 + eEyk + mv̄ 2 2 2 (3.2) E ~ ∧ B/B ~ 2 and is parallel to the x axis. where v̄ ≡ − B is the drift velocity E This can be derived by deriving explicitly the current: −e J~ = hn, k|(p + eA)|n, ki . m D E Thus there is a net current hJx i along the x axis. 36 Transport properties– IQHE We conclude that the energy levels follow the electric potential, which adds to the energy in zero electric field. Let us now go one step further by considering a slowly varying electric potential V (y), which we still assume to be translation invariant along x. We can linearize this potential locally, and repeat the previous analysis: the energy eigenvalues will not be linear in k any more, but they will roughly reflect the sum of the LL energy plus the local potential energy. To discuss electrons in a Hall bar, we take into account the sample edges in the y direction, which create a confinement potential. The latter results in an upward bending of Landau levels in the vicinity of the edges, where contacts allow to measure voltage biases as in figure 3.1. This justifies the sketch of the LL in figure 3.1, where the LL energy profile follows the confining potential at the sample edges. The eigenvalues ǫk are not linear in k, but can be linearized locally: it will still reflect the kinetic 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 eX In = − hn, k|vx |n, ki, (3.3) L k where L is the system length along x, and the velocity average value is derived from the energy dispersion relation 1 ∂εn,k 1 ∆εn,k ≃ . h̄ ∂k h̄ ∆k In the last line, we assume that ∆k = 2π/L is very small, which is certainly valid if L is very large. Using this, we have L L ∆εn,k = (εn,k+1 − εn,k ). 2πh̄ 2πh̄ Thus vk has opposite signs on the two edges of the sample. This means that in the Hall bar geometry, there are edge currents flowing in opposite directions. This is not surprising, if we remember the semi-classical picture of skipping orbits along an edge. When we sum over vk in equation (3.3, the result depends only on the edge energies, at kmin and kmax , the edge coordinates (see figure ( 3.1)). Thus, provided the electric potential has a slow enough variation in direction y, we may sum over k to get e In = − (εn,kmax − εn,kmin ) . h vk = 3.3. LOCALISATION IN A STRONG MAGNETIC FIELD 37 The energies at kmin and kmax are given by the chemical potential at the contact points, εn,kmin = µL and εn,kmax = µR . Since the difference in chemical potentials is controlled by a voltage bias ∆µ = (µR − µL ) = −eV , we see that the LL conductance is e2 /h, since In = e2 V. h (3.4) When n LL are completely filled, we get a conductance G=n e2 . h Since this is a transverse conductance (the current is in direction x, is zero along y, and the difference in chemical potentials is in direction y), the resistance tensor we get is R̂ = Ĝ −1 = 0 −RH RH 0 ! , (3.5) with the Hall resistance RH = h/e2 n. It is important to realize that this result, although satisfactory –the Hall resistance only depends on universal constants e and h, and an integer n–is not sufficient to explain the occurrence of plateaux. In fact, it is fairly easy to show that the result we have coincide exactly with the classical Hall value at discrete points in the RH curve, corresponding to n filled Landau levels; it is enough to remember that ν = hnel /eB = n and to use equation (3.5)to recover the classical value RH = B/enel . In order for quantized Hall plateaux to be formed, additionnal electrons or holes injected in the system around a density such that n LL are completely filled must be localized, so as to have no contribution to transport properties. This localization phenomenon is described in the next section. 3.3 Localisation in a strong magnetic field The electric potential Vext (r = R + η) due to impurities is described as a slowly varying function in the xy plane, so that Landau quantization is preserved. We now do not assume any more that the impurity potential preserves translation invariance along the x̂ 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 presence of an impurity potential. The guiding center follows the landscape equipotentials. The Hall drift of the guiding center, shown by the arrow is a slow motion compared to the fast electronic cyclotron motion. Electronic transport is possible when an equipotential connects the sample edges. If an electronic state is localized within a potential well, it does not contribute to transport. has hills and valleys and fluctuates in space around an average value which is taken to be zero, with no loss of generality. This potential lifts the LL degeneracy, because the guiding center is not a constant of motion any more. We see this with the Heisenberg equations of motion ∂Vext , ∂X (3.6) where we used formula (2.15). We see that the guiding center follows the equipotential lines of the impurity potential (Fig. 3.2). In the case we discussed in the previous section, this led to a Hall current in the direction orthogonal to the electric field. Equation (3.6) is a generalization of this result. The guiding center motion is perpendicular both to the external field and to the local electric field. Quantum states of the LL are thus localized on 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 tune to their equipotential lines. Similarly, we get for the ηx et ηy Heisenberg equations of motion 2 ih̄Ẋ = [X, H] = [X, Vext (X, Y )] = ilB ih̄η̇x ⇔ η˙x ∂Vext ∂Y and 2 ih̄Ẏ = −ilB 1 = ηx , mωC (ηx2 + ηy2 ) + V (r + η) 2 l2 ∂V = −ωC ηy − B h̄ ∂ηy Conductance of a filled LL et 39 η˙y = ωC ηx + 2 lB ∂V . h̄ ∂ηx This provides us with a stability criterion for Landau levels in the presence of a disorder impurity potenial, since the first terms on the right in the equations above must remain large compare with the terms due to the impurity potential. The condition reads: * ∂V ∂η + ≪ h̄ωC . lB (3.7) This condition is satisfied provided the variation of the potential over a cyclotron radius is small compared to h̄ωC . We now have the main ingredients to understand some of the IQHE basic features. The reasoning below is represented in a schematic fashion on figure 3.3. We have seen in the previous section that the Hall resistance for integer ν = n is exactly RH = h/e2 n, while the longitudinal resistance is zero. If the magnetic field intensity is slightly decreased, keeping the electronic density constant, since the number of states per LL decreases, some electrons have to promoted to the LL with n + 1. They occupy preferentially the lowest energy states 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 transverse and longitudinal resistances stay locked at their value for the completely filled level case with ν = n. The fact that the longitudinal resistance is zero shows that transport is ballistic. Indeed, we have seen that contributions to the current from the bulk of the sample compensate, so that transport is due to n edge channels (edge states), one per completely filled LL. The current direction on the edge is determined by the potential gradient, which rises near the sample edge. Edge currents are thus chiral, forward scattering at one edge is dissipation less, and dissipation can only occur if an electron circulating along one edge can be scattered backwards by tunneling to the other edge. This can occur only when edge states trajectories of opposites chiralities happen to be close to each other. When electronic puddles grow because more electrons get promoted to level n + 1, they eventually merge into one another, until equipotentials connect the two edges and eventually an electronic sea extends over the whole sample. When edges get connected by 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 (a) ε (b) ε (c) EF n EF n ε EF états localisés n états étendus densité d’états densité d’états densité d’états NL (n+1) Rxx R xy Rxx R xy Rxx R xy h/e2 n h/e2 n h/e2 (n+1) ν =n B B B Figure 3.3: Quantum Hall effect. In the upper parts of the figure, LL are broadened by the impurity potential. Their filling is controlled by the Fermi level(EF ). In the middle part, samples are seen from above, showing equipotential lines, and the gradual filling of the n-th level (from left to right). The lowest part of the figure is a sketch of the resistance curves, as the LL filling factor varies. This figure is to be read column by column, the filling factor increasing from the first column to the last one. In the first column (a), we have a situation with completely filled LL, ν = n, where the Fermi level sits exactly between LL n 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). The second column describes a situation where the LL n + 1 has a low filling factor. Electrons occupy potential wells in the sample and do not contribute to electronic transport. This situation occurs when, at fixed electron density, the magnetic field intensity is slightly decreased from its value for the complete filling of the n-th LL. The resistance values are locked at their value for ν = n. In the last column on the right, the n + 1 LL is half filled: equipotential lines connect the two edges, so that dissipation is allowed through back scattering processes from one edge to the other. The system changes from ballistic regime 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 exponentially small values. Localized states are on the left and on the right of the extended states in the center of the broadened LL. When the highest occupied LL has a filling larger than 1/2, the same reasoning applies in terms of holes. Conductance of a filled LL 41 R L ~ µ 3− µ 2= 0 µ2 = µL 2 3 µ3 = µL I I 4 1 6 µ6 = µ5 = µR 5 R ~ µ5− µ3= µR− µL H Figure 3.4: IQHE measurements at ν = n. The current I is injected through contact 1, and extracted at contact 4. Between those two contacts, the chemical potential µL is constant since (a) there is no backscattering and (b) there are no electrons injected or extracted at contacts 2 and 3 which are used to measure the voltage drop. The chemical potential µR stays also constant along the lower edge between contacts 6 and 5. The longitudinal 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 two edges µ5 − µ3 = µR − µL . to decrease until a new conducting channel is formed all along the edges.This is reached at half filling of level n + 1. Above that filling ratio, the evolution described so far is reproduced in terms of holes. To understand the IQHE in terms of edge currents, consider the experimental set up with six contacts, as shown on figure (3.4). Electrons are injected through electrode 1 and are extracted through electrode 4. The other contacts 2, 3, 5 and 6 are used for voltage bias measurements, with no electron injected or extracted. Because back-scattering is suppressed when the filling factor is around ν = n, the chemical potentials µR and µL are constant along each edge. The chemical potential varies only along input and output electrode 1 and 4. The longitudinal resistance is measured for instance between contacts 2 and 3, and is found to vanish: RL = −(µ3 − µ2 )/eI = 0. The Hall resistance is determined by the voltage 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 find RH = h/ne2 . 3.4 Transitions between plateaus – The percolation picture The previous section describes a scenario of transitions between Hall plateaus which reminds us closely of a percolation mechanism: the resistance jumps from one plateau to the next one when the electron puddles become macroscopic ones and percolate so as to form an infinite electronic sea which extends to both edges. Percolation transitions are second order transitions, which exhibit critical phenomena, and specific scaling laws for the relevant physical quantities around the critical point. Those quantities do not depend on microscopic details of the system, they are characterized by critical exponents which define a universality class. The transition is controlled by a ”control parameter” K, which could be the temperature, or, in the case of quantum phase transitions, at zero temperature, by another parameter such as pressure or electronic density [18]. In our case, the control parameter for transitions between plateaus is the magnetic 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 described by a correlation ”time” ξτ ∼ ξ z ∼ |δ|−zν , (3.9) where z is called the critical dynamic exponent. In a path integral formulation, the characteristic time τ is connected to the temperature T through h̄/τ = kB T . A finite temperature may be considered as a finite size in the time direction. [25, 24]. At the critical point, physical quantities follow scaling laws which depend on ratios of dimensionless quantities. One finds for the longitudinal and Hall resistivity ρL/H = fL/H h̄ω τ , kB T ξτ ! Conductance of a filled LL 43 100.0 dxy max dB dxy N=0 N=1 N=1 max 10.0 (∆B) −1 dB −1 (∆B) N=1 N=1 1.0 0.10 1.00 T(K) Figure 3.5: Experiments by Wei et al. [26]. The transition width δB and that of the Hall resistivity derivative ∂ρxy /∂B, measured as a funcition of temperature exhibit a scaling 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 h̄ω δ zν , kB T T ! , (3.10) where ω is a characteristic measurement frequency, for example in an ac measurement, and fL/H (x) are universal functions. In the following, we deal only with dc measurements, and ω = 0 properties. For a second order transition, one expects the characteristic width ∆B of the transition as a function of the 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 interval varying with more than one order of magnitude between 0, 1 and 1, 3K (figure 3.5). The two exponents ν and z may be separetly determined if one takes into account the scaling laws for current fluctuations under applied electric field. One finds h̄ h̄ eEℓE ∼ ∼ z, τE ℓE where τE ∼ ℓzE is the characteristic fluctuation time, which is connected to 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 δ ! , , T 1/zν E 1/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, smaller than the experimental value, close to 7/3. The disagreement is probably due to quantum tunneling effects between trajectories: such processes allow for back-scattering before classical trajectories actually touch each other. Chalker and Coddington take into account quantum tunneling in a transfer matrix approach and find a critical exponentν = 2, 5 ± 0, 5 [29]. Numerical simulations have reproduced the ν = 7/3 exponent [28]. Chapter 4 The Fractional Quantum Hall Effect (FQHE)– From Laughlin’s theory to Composite Fermions. In the previous chapter, we have seen that the IQHE with ν = n is understood on the basis of two main ingredients: (i) Because of LL quantization, there is an excitation gap between the ground state with a number of completely filled levels and the next empty level (ii) Elementary excitations, obtained by promoting an electron to the next LL are localized and do not contribute to electronic transport. Filled Landau levels only contribute each a conductance quantum e2 /h. As emphasized in the Introduction, the observation of the FQHE, first for a fractional filling of the LLL, with ν = p/(2sp + 1), with integer n and p, was a sign of the complete breakdown of perturbation theory, such as diagrammatic analysis based on the knowledge of an unperturbed ground state. For the fractionally filled LLL, the non interacting ground state has a huge degeneracy, which prohibits using theoretical techniques used so far to take electron-electron interactions into account. We know that certain superpositions of ground state configurations must minimize the Coulomb interactions. We know from experiments–the observed activated behaviour of the longitudinal resistance– that there is a gap between the actual ground state 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 state degeneracy, and to have a gap to the first excited states, be they single quasi-particles or collective excitations. In that case, we may reproduce the piece of reasoning of the previous chapter: excited charged quasi-particles are localized, so that varying the magnetic field intensity around a given exact fraction 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 non degenerate ground states, and to characterize their properties, the nature of excited states, etc.. Before actually introducing the Laughlin and Jain trial wave functions which solve this problem, we discuss in the next section the structure of the effective model which describes the electron dynamics when we make the approximation that it is restricted to the states of a single partially filled LL. This model will be the basis for the Hamiltonian theory of 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, only the lowest excitation energies are of interest. Because we are dealing with a partially filled LL,(ν 6= n), the relevant excitations are restricted to intralevel dynamics. Furthermore, we consider at first that the spins are fully polarized, and we do not consider spin flip excitations here. Excitations involving intra-LL transitions are forbidden by the Pauli principle when the level is completely filled (Fig. 4.1). They are allowed for a partially filled LL. In that case, the kinetic energy plays no rôle, since all single electron states are degenerate. We omit this constant in the following. Virtual inter-level excitations may be considered in a perturbative approach, and give rise to a modified dielectric function ǫ(q), which alters the interaction potential between electrons within the same level [30]. In contrast with screening effects in metals, which suppress the long distance part of the Coulomb interaction, electronic interactions screening in the presence of a magnetic field alters the potential only for finite wave vectors: for q → 0 and q → ∞, it vanishes, and ǫ(q) → ǫ, where ǫ is the dielectric constant of the underlying semiconductor. Since the electron spin is not flipped during such processes within a spin branch, it plays no rôle. Therefore, we deal in the following with spinless electrons restricted to a single LL (a) ν= N 47 (b) ν= N ∆Z h ωC Figure 4.1: Lowest excitations energies. Each LL is separated in two spin branches because of the Zeeman effect. (a) For filling ν = n, excitations which couple states within the same LL are forbidden because of the Pauli principle. Only inter level excitations are allowed. (b) For fractional filling of the highest occupied LL, (ν 6= n) excitations within the same LL are allowed and provide the lowest excitation energies. Inter LL excitations, with energy h̄ωC or ∆z are neglected in the model. electrons. A more detailed discussion of spin phenomena in the quantum Hall physics–such as the Quantum Hall ferromagnetism– will be given in a later part of these lectures. In second quantized notation, the Hamiltonian restricted to intra LL excitations is 1Z 2 2 ′ † d r d r ψn (r)ψn (r)V (r − r′ )ψn† (r′ )ψn (r′ ). Ĥ = 2 (4.1) It involves states in the n-th level only, ψn (r) = m hr|n, mien,m et ψn† (r) = P † † m hn, m|rien,m . Operators en,m and en,m are respectively the annihilation and creation operators for an electron in the state |n, mi. They obey the fermionic anti-commutation rules, P n o 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) n o ψn (r), ψn† (r′ ) = X m ′ 2 /2l2 B hr|n, mihn, m|r′ i ∝ e−|r−r | 6= δ(r − r′ ). (4.3) The field ψn† (r) creates an electron in the vicinity (4.2) of position r, which is hardly surprising: this is just another manifestation of the position uncertainty when we restrict the dynamics to a single LL. To have a perfect field 48 FQHE – from Laughlin to Composite Fermions localisation, one would have to sum over n, i.e. to superpose a number of LL. In reciprocal space, the Hamiltonian writes Ĥ = 1 X v(q)ρn (−q)ρn (q), 2A q (4.4) with the measure q = A d2 q/(2π)2 and the Coulomb interaction potential v(q) = 2πe2 /ǫq. The operators ρn (q) are the Fourier components of the electronic density operator in the n-th level ρn (r) = ψn† (r)ψn (r), and one has R P ρn (q) = = Z d2 r X m,m′ X m,m′ hn, m|rie−iq·r hr|n, m′ ie†n,m en,m′ hn, m|e−iq·r |n, m′ ie†n,m en,m′ = hn|e−iq·η |ni = Fn (q)ρ̄(q), X m,m′ hm|e−iq·R |m′ ie†n,m en,m′ (4.5) where Fn (q) ≡ hn| exp(−iq · η)|ni is the form factor, and we took advantage of the decomposition r = R + η, which allows to factorize matrix elements hn, m|e−iq·r |n′ , m′ i = hn| exp(−iq · η)|n′ i ⊗ hm| exp(−iq · R)|m′ i. (4.6) In the last line of equation (4.5), we have defined the projected density operator, X ρ̄(q) ≡ hm|e−iq·R |m′ ie†n,m en,m′ . (4.7) m,m′ 4.1.1 Matrix elements To proceed in practice to actual computations, one needs to compute the matrix elements which enter expression (4.5) for the density operator. The simplest way is to use expressions (2.17) and (2.18) for the operators R and η, 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 1 q · η = √ qa + q ∗ a† , 2 1 q · R = √ q ∗ b + qb† , 2 electrons restricted to a single LL 49 so that we get for the first matrix element, with n ≥ n′ , using the BakerHausdorff formula(2.41), − √i (q ∗ a† +qa) hn|e−iq·η |n′ i = hn|e 2 2 /4 = e−|q| 2 /4 = e−|q| |n′ i − √i q ∗ a† − √i qa hn|e X j e 2 − √i q ∗ a† hn|e 2 −|q|2 /4 s n′ ! n! −iq ∗ √ 2 −|q|2 /4 s n′ ! n! −iq ∗ √ 2 = e = e 2 |n′ i − √i qa |jihj|e 2 |n′ i n−n′ X n′ n! |q|2 − (n − j)!(n′ − j)!j! 2 j=0 n−n′ n−n′ n′ L |q|2 2 ! , !n′ −j (4.8) where we have used − √i q ∗ a† hn|e 2 0 |ji = q n! pour j > n 1 j! (n−j)! − √i2 q ∗ n−j pour j ≤ n in the third line and the definition of Laguerre polynomials [21], ′ Ln−n (x) n′ ′ = n X (−x)m n! . ′ ′ m! m=0 (n − m)!(n − n + m)! Similarly we find for m ≥ m′ − √i (qb† +q ∗ b) hm|e−iq·R |m′ i = hm|e −|q|2 /4 = e 2 s m′ ! m! |m′ i −iq √ 2 !m−m′ m−m′ Lm ′ |q|2 . 2 ! (4.9) Defining functions Gn,n′ (q) ≡ s n′ ! n! −iq √ 2 !n−n′ ′ Ln−n n′ |q|2 , 2 ! one may also write without the conditions n ≥ n′ et m ≥ m′ , 2 /4 hn|e−iq·η |n′ i = [Θ(n − n′ )Gn,n′ (q ∗ ) + Θ(n′ − n − 1)Gn′ ,n (−q)] e−|q| (4.10) 50 FQHE – from Laughlin to Composite Fermions and 2 hm|e−iq·R |m′ i = [Θ(m − m′ )Gm,m′ (q) + Θ(m′ − m − 1)Gm′ ,m (−q ∗ )] e−|q| /4 . (4.11) For the case n = n′ , we find in equation (4.10) the n-th LL form factor: −iq·η Fn (|q|) ≡ hn|e 4.1.2 |q|2 −|q|2 /4 e . 2 ! |ni = Ln (4.12) Projected densities algebra At first sight, the model defined by (4.4) looks simple. The Hamiltonian is quadratic in density operators. Such models often have exact solutions. It happens that the projection in a single LL generates a non commutative algebra for operators with different wave vectors, which leads to non trivial quantum dynamics. Let us compute the commutator [ρ̄(q), ρ̄(k)]. For a one particle operator P † A A in second quantized notation, F A (q) = λ,λ′ fλ,λ ′ (q)eλ eλ′ , where fλ,λ′ (q) = A ′ hλ|f (q)|λ i, the commutation rules in second quantized form follow from those in first quantization: h i F A (q), F B (q′ ) = Xh i f A (q), f B (q′ ) λ,λ′ λ,λ′ e†λ eλ′ . (4.13) The λ index may comprise a number of different quantum indices. This equation 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 as anti-commutators. Using equation (2.16), one finds [q · R, q′ · R] = qx qy′ [X, Y ] + qy qx′ [Y, X] = i(qx qy′ − qy qx′ ) = −i(q ∧ q′ ), where we have defined q ∧ q′ ≡ −(q × q′ )z , and one gets, with the help of the Baker-Hausdorff formula (2.41) h ′ e−iq·R , e−iq ·R i ′ i ′ i = e−i(q+q )·R e 2 q∧q − e− 2 q∧q q ∧ q′ −i(q+q′ )·R = 2i sin e . 2 ! ′ (4.15) 4.2. THE LAUGHLIN WAVE FUNCTION 51 This yields, with equation (4.13), ! q∧k [ρ̄(q), ρ̄(k)] = 2i sin ρ̄(q + k), 2 (4.16) for the algebra of projected density operators. This is isomorphous to the magnetic translation algebra . Indeed, operators which describe electronic displacements in the presence of a magnetic field have the same commutation rules. 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 Ĥ = 1 X vn (q)ρ̄(−q)ρ̄(q), 2A q (4.17) where the form factor has been absorbed in the effective interaction potential in the n-th LL , 2πe2 2πe2 |q|2 vn (q) = [Fn (q)]2 = Ln ǫ|q| ǫ|q| 2 " !#2 2 /2 e−|q| . (4.18) The model has the same structure for all LL. The information about the level is encoded in the effective potential, which will be discused in the last section of this chapter. The LLL physics, which will be the main topic in the remaining parts of this chapter (except the last section), is thus easily generalized to a LL with higher index: one simply has to take into account the relevant effective potential, and to replace the filling factor ν by the partial filling factor of the n-th level, ν̄ = ν − n. 4.2 The Laughlin wave function In this section, we discuss the arguments used by Laughlin in 1983 to derive the almost exact ground state for the fractionally filled LLL (Lowest Landau Level), to prove that there is a gap between the ground state and all excited states, and that there exist factionally charged excitations around the fractional filling corresponding to the plateaus observed by Tsui, Störmer and Gossard. Then we will describe Jain’s generalization of Laughlin’s wave functions. 52 FQHE – From Laughlin to Composite Fermions It is a good training to examine first the many-body wave function for the completely 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 ∗ µb B (see figure 4.1).1 4.2.1 The many-body wave function for ν = 1 Laughlin exploited a useful property of the single particle Landau Hamiltonian eigenfunctions in the symmetric gauge (see equation 2.28) : ∗ φn=0,m (z, z ) ∝ z √ 2lB !m 2 /4l2 B e−|z| , so that any analytic function f1 (z)(defined by ∂∂z∗ f1 (z, z∗) = 0 ) in the prefactor of the gaussian belongs to the LLL. All physical results are of course independent 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 be of the form X |zi |2 ψν=1 ({zi }) ∝ fN ({zi }) exp − 2 (4.19) 4lB j where {zi } means (z1 , z2 , ...., zN ), and fN is analytic in all variables. N is the total number of electrons, and is equal, since ν = 1 to the total number of states in the LLL. Since we are dealing with a state where all electron spins are identical, the spin wave function is symmetrical under exchange of particles. 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 states occupied. This determinant reads: fN = det 1 z10 z11 z20 z21 ... ... 0 1 zN zN ... z1N −1 ... z2N −1 ... ... N −1 ... zN (4.20) We will show later on that in fact the gap above the ν = 1 ground state is dominated by 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 N variables, 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 this corresponds to a guiding center eigenvalue mmax = N − 1, fN corresponds indeed to a fully occupied LLL, with all states occupied once (as the expression of equation 4.20 shows). A striking remark is that this state being the only LLL eigenstate with P ν = 1(with fixed center on mass i zi ), it is an eigenstate of the N particle Hamiltonian for any interaction potential. In order to analyze properties of ψν=1 , Laughlin resorted to a very original detour: the so called ”plasma analogy”. The latter amounts to regard the probability distribution function of particles in the LLL,(putting lB = 1), 2 |Ψν=1 |2 ∝ ΠN i<j |zi − zj | exp −(1/2) N X l |zl |2 (4.22) as the Boltzmann weight of a classical statistical mechanics problem, the partition function Z of which is given by the norm of the wave function. In other words, Z Z = Πi d2 zi |ψν=1 ({zi })|2 and |ψν=1 |2 = exp −βUclass . (4.23) Since this is a formal analogy, the inverse temperature β which appears here is 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 ≡ q 2 X i<j (− ln |zi − zj |) + qX 2 |zl | 4 l (4.24) Laughlin remarked that Uclass is the internal energy of a 2D classical one component gas of interacting particles with charge q in a uniform neutralizing background. Remember that this is an analogy, which has the advantage of representing an unknown problem, i.e. the probability distribution function 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 54 FQHE – From Laughlin to Composite Fermions particles. In the equivalent classical problem, particles have logarithmic interactions, which are 2D Coulomb interactions, while the real problem has the same formulation, as we saw above, for any interaction potential.2 To see that the interactions between the classical particles of the equivalent classical problem are 2D Coulomb interactions, remember that in 2D the flux of the electric field through a circle of radius R (the sphere S1 of the 2D space) is related to the enclosed charge Q by Z dx.E = 2πQ . For a point charge q at the origin, E(r) = qr/r2 so that the electric potential is Vc = −q ln r/a,3 and Poisson equation in 2D reads: divE = −∇2 Vc (r) = 2πqδ 2 (r). (4.25) Thus the first term on the right of equation 4.24 is interpreted as the Coulomb interaction energy among N 2D charge q particles. Because the 2 LLL is filled, N = B/φ0 = 1/(2πlB ) = 1/(2π) particles per unit surface (remember that here lB = 1). The second term on the right of equation 4.24 represents the potential energy of N particles of charge q interacting with a uniform charged background 2 with charge density ρB = −1/(2πlB ). Indeed −∇ |z|2 2 = −1/(lB ) = 2πρB 2 4lB (4.26) In other words, the uniform background has a charge density which is precisely equal to the density of flux quanta threading the surface. We know from electrostatics that charge neutrality is the condition for thermodynamic equilibrium, which corresponds to the most probable states in the partition function. The overall charge neutrality condition is nq + ρB = 0 2 (4.27) The actual interaction between particles in the real problem is in fact a 3D Coulomb interaction ∝ 1/r, because the electrons in the 2D potential well are immersed in a 3D space. But the analysis of the filled LLL wave function is entirely independent of any interaction potential form. 3 a is an arbitrary integration constant,which only changes Vc by a constant, and which we may later take as a = lB . The Laughlin wave function 55 Which is satisfied in the ν = 1 LLL ground state, since q = 1. The plasma analogy tells us more than the overall neutrality condition: it tells us that the largest values of the probability distribution function |ψν=1 ({zi })|2 is when charge neutrality is realized locally, otherwise huge costs in Coulomb energy reduce 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 scales greater than lB . This statement holds for any interaction potential, and is true for non interacting particles, because the ν = 1 Vandermonde determinant 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 in the partially filled LLL had been predicted to be a Wigner crystal: electrons would organise in a triangular cristalline array to reduce their Coulomb interactions. There is indeed some experimental evidence that such is the situation at low enough filling of the LLL. It is clear however that the Wigner Crystal cannot 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 and rotation invariance, so that it has a Goldstone mode, i.e. a collective excitation the energy of which goes continuously to zero with the wave vector. Such a state does not have a gap above the ground state, in contradiction with the FQHE phenomenology. Moreover, the Wigner crystal scenario would have no particular way of selecting “magic” fractional values of the filling factor observed to correspond to FQHE plateaus. The explanation of the FQHE at ν = 1/(2s + 1) was proposed by Laughlin the very year it was discovered [12]. He looked for a trial many body wave function which would respect the constraints and the symmetries of the problem. Here we sketch the essential steps in the construction of the Laughlin wave function, starting with the wave function for two particles. ψ (2) (z, z ′ ). • The analyticity condition (2.24), for the symmetric gauge imposes that P 2 ψ (2) (z, z ′ ) = m,M αm,M (z+z ′ )M (z−z ′ )m exp[−(|z|2 +|z ′ |2 )/4lB ], where m and M are integers. • Electrons are fermions, with spin polarised electrons, so, as for the 56 FQHE – From Laughlin to Composite Fermions ν = 1 case, the orbital part of the wave function must be antisymmetric with respect to permutation of the particles. This limits the choice to odd m integers. The general two particle wave functions is thus restricted to be a superposition of functions ψ (2) (z, z ′ ) ∝ (z + z ′ )M (z − 2 z ′ )2s+1 exp[−(|z|2 + |z ′ |2 )/4lB ]. • If we take into account the two body problem for electrons with center of mass angular momentum M and relative angular momentum m the following wave function (2) 2 ] ψM m (z, z ′ ) = (z + z ′ )M (z − z ′ )m exp[−(|z|2 + |z ′ |2 )/4lB (4.28) is unique (aside from normalization factors). It is remarkable that, neglecting LL mixing, this is the exact two body wave function for any central potential V (|z−z ′ |). The powerful restrictions due to analyticity (2) allow to write ψM m (z, z ′ ) without solving any radial equation! There is only one state in the LLL Hilbert space with center of mass angular momentum M and relative angular momentum m. • The corresponding energy eigenvalue Vm0 for the two electron problem in the LL is independent of M and given by the only matrix element: Vm0 = hm, M |V |m, M i . hm, M |m, M i (4.29) The coefficients Vm0 are called the Haldane pseudo-potentials (generalized to any LL in a later section in this chapter). The discrete energy eigenstates represent bound states of the (repulsive!) potential. This is unusual: a repulsive potential has no bound states, only a continuous spectrum in the absence of a magnetic field. In the presence of a magnetic field, the Lorentz force results in quenching the kinetic energy, so we may have bound states. In zero magnetic field, two electrons convert their potential energy in kinetic energy and move away from one another. In a magnetic field, the electrons have fixed kinetic energy, so they are constrained to orbit around one another. The discrete spectrum for a pair of particles in a repulsive potential is a basic feature in the understanding of the FQHE, as it generates a gap above the ground state energy for all excitations. The Laughlin wave function 57 Although the exact solution for the two particle problem cannot be generalized to N > 2 in any straightforward fashion, the N particles wave function proposed by Laughlin obeys the conditions described above. Laughlin generalized the ν = 1 many body wave function, writing L ψ ({zj }) = Y zi − zj 2s+1 lB i<j e− P j 2 |zj |2 /4lB . (4.30) In the ν = 1 case, s = 0. Note that this is a one variational parameter (the integer s) trial wave function. The prefactor in Laughlin’s wave function (4.30) is also called the Jastrow factor. Similar wave functions had been proposed to describe liquid Helium. The plasma analogy is very useful in the fractional filling case [12]. Following this picture, we identify the space integral of the wave function square modulus with the partition function of a classical statistical system, described by a “free energy” Ucl . The partition function is then Z= X e−βUcl = ˆ C Z where C represents configurations, so that −βUcl = 2q X i<j 2 d2 z1 ...d2 zN ψ L ({zj }) , (4.31) zi − zj X |zj |2 + , ln 2 lB j 2lB where q = 2s + 1. The “ temperature” is, as above, β ≡ 2/q in order to get Ucl = −q 2 X i<j X |zj |2 zi − zj −q ln . 2 lB j 4lB (4.32) What is different for ν 6= 1 as compared to the previous ν = 1 case? We have to determine the optimal q knowing that the electronic density is nel = ν φB0 = ν 2 . The neutral background charge density is given by the same expression: 2πlB 2 ρB = −1/(2πlB ), 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 determined. The exponent of the fractionally filled LLL ground state wave function with ν = 1/(2s + 1) is 1/ν = 2s + 1. 58 FQHE – From Laughlin to Composite Fermions 4.2.3 Incompressible fluid In this section, it is shown that the Laughlin wave function is an almost exact ground state of the many-body problem, and that there is a gap to any excited state above this ground state. Then the Laughlin wave function describes 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 a lower bound, cannot be infinitesimally small. The ν = 1 quantum liquid is also an incompressible fluid, where the gap to excited states is presumably4 due to the Zeeman effect, or possibly the orbital energy h̄ωC . In the fractional case, the gap is obviously determined by the Coulomb energy ∝ e2 /(ǫr). In order to evaluate the ground state energy, we need not take into account the kinetic term, since we are restricted to a single LL. This is certainly true in the large field limit, since h̄ωc ∝ B, while the Coulomb energy is e2 /ǫlB ∝ (B 1/2 ). So we need only take into account the latter term. Suppose that we write the potential energy, quite generally, in terms of Haldane pseudo potentials V = ∞ X X vm′ Pm′ (ij) (4.34) m′ =0 i<j where Pm (ij) is the projection operator which selects out states in which i and j have relative angular momentum m.(Note that Pm1 (ij) and Pm2 (jk) do not 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 an exact energy eigenstate V ψm ({zi }) (4.35) Indeed it is clear that Pm′ (ij)ψm = 0 for any m′ < m since every pair has relative angular momentum larger than, or equal to m. Suppose m = 3 (Laughlin state at 1/3 filling ). This model obviously has a minimum excitation energy v1 , which corresponds to allowing at least one pair to have relative angular momentum 1. The proof that the Laughlin wave function has a gap to excited states for the actual Coulomb interaction, follows from the fact that, compared 4 It turns out that the gap in the ν = 1 case is also due to Coulomb interactions, as will be discussed later on in the section of Quantum Hall ferromagnetism. The Laughlin wave function 59 to the model hard core potential, the additional Haldane pseudo potentials of the Coulomb potential (i.e. m′ ≥ 3) can be treated perturbatively, because they are all smaller than v1 . This proof is valid specifically for the Coulomb potential. Since all Coulomb corrections to the hard core potential are perturbations, the gap between the ground state and the first excited state persists. Thus the Laughlin state, almost the exact ground state of the Coulomb potential Hamiltonian, is that of an incompressible fluid. The excitation gap is a necessary condition for zero longitudinal conductivity, and zero resistivity, σxx = 0 = ρxx . Numerical data show that the overlap between the true ground state for the 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 excitations have fractional charge. Consider the wave function ψqh (z0 , {zj }) = N Y zi − z0 i=1 lB ψ L ({zj }), (4.36) where an additional zero sits at position z0 . The charge density vanishes at z0 . Expanding formally Laughlin’s wave function as ψ L ({zj }) = X mN − αm1 ,...,mN z1m1 ...zN e {mi } P j 2 |zj |2 /4lB , and comparing with the expansion ψqh (z0 = 0, {zj }) = X {mi } mN +1 − αm1 ,...,mN z1m1 +1 ...zN e P j 2 |zj |2 /4lB , we see that, compared to Laughlin’s wave function, all particles are displaced from one state to the next, mj → mj + 1. In the symmetric gauge where a √ particle 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 hole P has 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 path around z0 acumulates a phase 2π. 60 FQHE – From Laughlin to Composite Fermions 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 ∗ Y zi − z0∗ i=1 lB ψ L ({zj }), (4.37) There is a complication here since we are not allowed to use zj∗ in a wave function which should be analytic in order not to mix in higher LL states. In order to remain within the LLL manyfold of states, one should use a projector PLLL on the LLL. A way of doing this is to divide the polynomial part of the wave function by zj instead of multiplying by zj∗ . By partial integration, this is equivalent to applying ∂zj to the gaussian factor, which generates zj∗ , up to a multiplying factor. Given the complication in handling quasi-particle wave functions, we will deal only with quasi-holes in the following, without loss of physical generality [32]. In order to check that such excitations have fractional charge, let us use again the 2D plasma analogy introduced above. The prefactor in the wave function (4.33) gives rise to a new term in expression (4.32), Ucl → Ucl + V , where N X zj − z0 . ln V = −q l B j This is interpreted as the interaction potential between the plasma and a charge 1 ”‘impurity”’ located at z0 . This impurity is screened so as to maintain charge neutrality in the plasma. Since the plasma particles have charge q, 1/q particles are needed to screen the impurity charge. The quasi-particle of the Laughlin liquid is thus shown to have fractional charge 5 e∗ = e e = . q 2s + 1 (4.38) Another more direct way to see this charge fractionalisation is to introduce q quasi-particles at the same point 6 q [ψqh (z0 , {zj })] = 5 N Y zj − z0 q j=1 lB ψ L ({zj }) = ψ L ({zj }, z0 ), From now on, we use the generic term ”quasi-particle” for quasi-holes and quasiparticles, except when distinction is necessary. 6 The expression on the left is symbolic . The Laughlin wave function 61 where we find the Laughlin wave function for N +1 electrons, with the added one at position z0 . One needs therefore q quasi-particles to add one electron in 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-particles carry fractional charge, in order to show the essential connection between the 2 fractional quantum Hall plateau, with fractional Hall conductivity σxy = ν eh , and the fractionalisation of the quasi-particle charge. Imagine piercing the sample at the origin with an infinitely thin magnetic solenoid and increasing adiabatically the magnetic flux φ from 0 to φ0 = h/e. The time variation of the flux inside the solenoid induces an azimuthal electric field, as Faraday’s law tells us. This field is such that I C dr.E = − ∂φ . ∂t (4.39) C is a contour surrounding the flux line. If the process is sufficiently slow, the electric field has low frequency Fourier components only, such that h̄ω ≪ ∆, where ∆ is the energy gap. There is no dissipation. Because the system is in a quantum Hall state, the electric field drives a current density which is radial: E = ρxy J~ ∧ ẑ. (4.40) So we have ~ ∧ dr) = − dφ (4.41) J.(ẑ dt C The integral on the LHS represents the total current flowing into the region enclosed by the contour. Thus the charge inside this region obeys ρxy I ρxy dQ/dt = −dφ/dt (4.42) At the end of the process, the total charge is Q = σxy φ0 = σxy (h/e) = νe (4.43) The final step in the argument is that an infinitesimal flux tube containing a flux 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 and σxy is quantized. The existence of fractionally charged elementary excitations is a direct consequence of the FQHE. 62 FQHE – From Laughlin to Composite Fermions Numerical data show that there is a finite energy cost to create such quasi-particles which means that there is a gap between the ground state described by the Laughlin wave function and its lowest elementary excited states. This is a necessary condition for the FQHE. 4.2.5 Ground state energy Beside his ground state wave function proposal, Laughlin showed that it has lower energy than the Wigner crystal. The latter had been argued to minimize the Coulomb energy. The Laughlin liquid energy is given by Z N 2 Y hψ L |Ĥ|ψ L i e2 1 XZ 2 2 d2 zk ψ L (zi , zj ; {zk }) d zi d zj = Z 2Z i6=j ǫ|zi − zj | k6=i,j = n2el A Z 2 e2 g(r), dr 2 ǫ|r| (4.44) where 2 N (N − 1) Z 2 L 2 ψ (z = 0, z = r; z , ..., z ) d z ...d z g(r) ≡ 1 2 3 N 3 N 2 nel Z (4.45) is the pair correlation function. This expression takes advantage of the translation and rotation invariance of the wave function 7 and of the fact that there are N (N − 1) ways of chosing the zi = z1 and zj = z2 pairs in the first line of equation (4.44). This expression is usually divided by the total particle number N = Rnel A , and the energy of the homogeneous uncorrelated liquid E0 = (nel /2) d2 re2 /ǫr is chosen as energy reference. The Laughlin liquid energy per particle is written in terms of the pair correlation function. EL = nel Z 2 e2 [g(r) − 1], dr 2 ǫ|r| (4.46) 1X v(q)[s(q) − 1], 2 q (4.47) 1 hρ(−q)ρ(q)i, N (4.48) or, in Fourier space EL = in terms of the static structure factor s(q) = 7 rotation invariance results in r = z = |z|. The Laughlin wave function 63 which is connected to the pair correlation function by Fourier transformation [25] Z d2 r eiq·r [g(r) − 1]. [s(q) − 1] = nel (4.49) The pair correlation function (or the structure factor ) thus determines the liquid structure and describes possibly a short range order. It can be computed from Laughlin’s wave function by Monte Carlo integration [34, 35]. Instead of computing the pair correlation function numerically, Girvin analysed it in 1984 using symmetries and properties of the 2D one component plasma [91]. Expanding the Laughlin wave function in terms of z = (z1 − z2 )/lB and z+ = (z1 + z2 )/lB , ψ L ({zj }) = ∼ XX 2 +|z|2 )/8 M m −(|z+ | aM,m (z3 , ..., zN )z+ z e , (4.50) M m=1 where the tilde on the second sum indicates that the sum is on odd integers, one finds for the pair correlation function g(z) = ∼ X ′ 2 /4 Am,m′ (z+ )z ∗m z m e−|z| , m,m′ where functions Am,m′ (z+ ) depend only on z+ because the other variables z3 , ...zN have been integrated on. ∂Am,m′ (z+ )/∂z+ = 0 follows from the liquid translation invariance, and rotation invariance imposes Am,m′ = δm,m′ bm , which results in ∼ g(z) = X 2 /4 m=1 bm |z|2m e−|z| . As lim|z|→∞ g(|z|) = 1, and thus lim|z|→∞ m=1 bm |z|2m = exp(|z|2 /4), it is convenient to rewrite expansion parameters as P̃ m 2 1 m! 4 bm = (1 + cm ), where limits impose limm→∞ cm = 0. The pair correlation function is thus given as a sinh plus corrections described by the cm parameters. ∼ |z|2 −|z|2 /4 X 2cm g(z) = 2 sinh e + 4 m=1 m! ! = −|z|2 /2 1−e + ∼ X 2cm m=1 m! |z|2 4 !m |z|2 4 !m 2 /4 e−|z| 2 /4 e−|z| . (4.51) 64 FQHE – From Laughlin to Composite Fermions The Fourier transform yields the static structure factor [s(q) − 1] = −νe−q 2 l2 /2 B + 4ν ∼ X 2 cm Lm (q 2 lB )e−q 2 l2 B , (4.52) m=1 in terms of Laguerre polynomials. The energy (4.47) is finally written as EL = ∼ νX ν X cm Vm0 − v0 (q), π m=1 2 q (4.53) where v0 (q) is the effective potential in the LLL (4.18), and we have defined the Haldane pseudo-potentials Vm0 ≡ 2π X 2 v0 (q)Lm (q 2 lB )e−q 2 l2 /2 B . (4.54) q The advantage of the energy expression (4.53) in terms of Haldane pseudopotentials, defined in terms of the effective potential, allows to describe directly Laughlin liquids in higher index LL (n 6= 0) : pseudo-potentials are generalised to LL n, using the appropriate effective potential (4.18), Vmn ≡ 2π X 2 )e−q vn (q)Lm (q 2 lB 2 l2 /2 B . (4.55) q Returning to (4.51), we notice that due to the Laughlin wave function behaviour 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 in Laughlin’s wave function are effective in minimizing Coulomb interactions, more so than in any state where fermionic correlations would impose only g(z) ∼ |z|2 . This short distance behaviour ensures that expansion parameters obey cm = −1, pour m < s. (4.56) It is also useful to define the ”moments” Mn = nel Z |z|2 dz 4 !n " n+2 2 = 2πnel −n! + 2 [g(z) − 1] ∼ X (n + m)! m=1 m! # cm , (4.57) The Laughlin wave function 65 where the second line is computed with the help of equation (4.51). Following the plasma analogy,8 charge neutrality imposes M0 = −1 and thus ∼ X m=1 cm = 1 s 1 − ν −1 = − . 4 2 (4.58) Perfect plasma screening is expressed by M1 = −1, i.e. ∼ X m=1 (m + 1)cm = 1 s 1 − ν −1 = − . 8 4 (4.59) Compressibility properties yield a third sum rule ∼ X s2 1 −1 2 1−ν = . (m + 2)(m + 1)cm = 8 2 m=1 (4.60) Those sum rules (4.58-4.60) and (4.56) can be used as constraints on the pair correlations functions (4.51) in connection with Monte Carlo numerical work. One may use them instead for an approximate determination of the function: sum rules form a system of coupled linear equations which can be solved if one sets cm = 0 for m > s + 3, which is a reasonable approximation, since limm→∞ cm = 0. In this manner, one finds s=1 s=2 s=3 s=4 cs1 -1 -1 -1 -1 cs3 cs5 cs7 cs9 cs11 cs13 17/32 1/16 -3/32 0 0 0 -1 7/16 11/8 -13/16 0 0 -1 -1 -25/32 79/16 -85/32 0 -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 numerical results by Levesque et al. [34], as shown on figure 4.2(a). The pair correlation function apart the correlation hole at small distance, exhibits a maximum at finite distance, where it is most probable to find a second particle.. This maximum is displaced further away from the origin and becomes more pronounced if the electronic density is lowered [ν = 1/(2s + 1)]. This means an enhanced short range order at low densities where a Wigner crystal 66 FQHE – From Laughlin to Composite Fermions Facteur de remplissage 5 0.1 15 0.3 10 0.2 20 0.4 25 0.5 12 14 15 Energie -0.1 −0.1 -0.2 −0.2 -0.3 −0.3 -0.4 −0.4 Fonction de corrélation de paires (b) 1.4 g s(r) 1.2 s=3 s=2 s=1 1.01 0.8 0.6 0.5 0.4 0.2 2 4 5 6 r8 10 10 Figure 4.2: (a) Comparison of our results for the energy (black segments) of Laughlin states to numerical results by Levesque et al. (gray line) in√units of e2 /ǫlB . The line is the result of an an interpolation formula U (ν) = −0, 782133 ν 1 − 0, 211ν 0,74 + 0, 012ν 1,7 for Levesque et al.’s results [34]. (b) Pair correlation function for different s. The distance r is mesured in units of the magnetic length lB . The straight dotted line corresponds to uncorrelated electrons. The Laughlin wave function 67 is expected to become more stable. More accurate numerical results confirm this tendancy (see figure 4.2(b))[39]. 4.2.6 Neutral Collective Modes We have discussed the ground state energy, and analysed elementary excitations (fractionally charged quasi-particles) the energy of which is separated from the ground state one by a gap. In order to understand FQHE, we now have to show that collective excitations have a dispersion relation with a finite gap above the ground state at all wave vectors. Such collective excitations, with wave function |ψq i are likely to be well described within the ”Single Mode Approximation” (SMA)[35]. 9 . In the quantum Hall case, |ψq i = ρ̄(q)|ψ L i, (4.61) where ρ̄(q) is the projected density operator (4.7). Since ρ̄(q) = X m,m′ hm|e−iq·R |m′ ie†n=0,m en=0,m′ , the excited state may be interpreted as a superposition of particle-hole excitations (particles in the state |n = 0, mi and hole in |n = 0, m′ i), an average 2 distance qlB apart. Because of the projection, |ψq i has no component in LL n 6= 0. The excitation energy with respect to the ground state is ∆(q) = ≃ hψ L |ρ̄(−q)Ĥ ρ̄(q)|ψ L i − EL hψ L |ρ̄(−q)ρ̄(q)|ψ L i f¯(q) 1 hψ L |[ρ̄(−q), [Ĥ, ρ̄(q)]|ψ L i ≡ , 2 hψ L |ρ̄(−q)ρ̄(q)|ψ L i s̄(q) (4.62) where we assumed that the Laughlin state is an eigenstate of the Hamiltonian , Ĥ|ψ L i ≃ E L |ψ L i, which is an excellent approximation. Moreover the liquid state rotation invariance has been used to get the second line, as well as ρ̄† (q) = ρ̄(−q). The projected structure factor is connected to the structure factor (4.52) through s̄(q) = s(q) − 1 − e−q 8 2 l2 /2 B . Interested readers may find details in the following references [34, 35, 91] (and references in those papers in particular [38]) 9 The SMA was used by Feynman in his theory of superfluid He collective modes [40] 68 FQHE – From Laughlin to Composite Fermions Figure 4.3: Dispersion relation for collective excitations [41]. The continuous curves are the results in the Single Mode Approximation for ν = 1/3, 1/5 et 1/7; the various symbols are values obtained by exact diagonalisation [42]. Arrows are the expected reciprocal lattice parameter moduli for the Wigner crystal at the corresponding filling factors. Equation (4.62) is precisely the Feynman-Bijl formula, proposed for the description of collective excitations in superfluid He [40]. Using commutation rules for projected density operators (4.16), one finds ∆(q) = 2 X k [v0 (|k − q|) − v0 (k)] sin 2 2 q ∧ klB 2 ! s̄(k) . s̄(q) (4.63) The dispersion relations are shown on figure 4.3, as well as numerical results of exact diagonalisations of systems with a small number of particles [42]. As expected for an incompressible liquid, dispersion relations have a finite energy gap above the ground state for all wave vectors. They all exhibit a minimum at a finite wave vector. The latter corresponds to the reciprocal lattice parameter modulus for the Wigner crystal at the same filling factor. The minimum, called the magneto-roton minimum, in analogy with the superfluid He case [40], is a sign of short range (crystalline) order. The collective mode softening at this wave vector signals a tendancy to Wigner crystal 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 2πe∗2 , 2 ǫqlB i.e. the energy to create a pair made with a quasi-particle of energy ∆qp and a quasi-hole with energy ∆qh , with well separated components submitted to Coulomb attraction because of their opposite charges, e∗ and −e∗ . ∆(q ≫ 1/lB ) ≃ ∆qp + ∆qh − 4.3 Jain’s generalisation – Composite Fermions The Laughlin wave function describes well the FQHE at ν = 1/(2s + 1), but it fails to apply to the other fractional states which were discovered subsequently, such as ν = 2/5, which is one term in the set ν = p/(2sp + 1). In order to account for those new states, Haldane [37] and Halperin [43] proposed a hierarchy picture. Following the latter, Laughlin quasi-particles with sufficient density condense in an incompressible liquid in order to minimize their Coulomb interaction energy due to their charge e∗ . The state ν = 2/5 would then be a ”daughter” of the Laughlin state at ν = 1/3. In 1989, Jain proposed an alternative route, the Composite Fermion picture. He first re-interpreted the Laughlin wave function Y zi − zj 2s Y zi − zj − P |z |2 /4l2 B j j e , (4.64) ψ L ({zj }) = lB lB i<j i<j as a product of two factors : the first one, i<j [(zi − zj )/lB ]2s , attaches 2s zeros (vortices with 2s flux quanta ) to particles positions, and the second one, Q χν ∗ =1 ({zj }) = Y zi − zj i<j lB , (4.65) can be interpreted as the wave function of a virtual completely filled LL, with a new (virtual) filling factor ν ∗ = 1 [20]. Indeed, it coincides with equations 4.20 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 }), J ψ ({zj }) = PLLL Y zi − zj 2s i<j lB χν ∗ =p ({zj }), (4.66) 70 FQHE – from Laughlin to Composite Fermions Projection to the LLL is taken care of by the projector PLLL , since the function χν ∗ =p ({zj }) contains, if unprojected, high energy components belonging to LL with p > 1. What is achieved by this manipulation? The effective number of states per LL in the virtual levels has been decreased, M → M ∗ = M − 2sN , since the first vortex attachment factor has taken 2sN zeros from the system with N electrons. This amounts to renormalize the magnetic field which is nothing but the flux density in terms of flux quanta φ0 = h/e, and the filling factor as well, in the following way B → B ∗ = B − 2sφ0 nel et ν ∗−1 = ν −1 − 2s. (4.67) With this picture, we may now re-interpret the FQHE at ν = 1/3 as a completely filled Composite Fermion level, with ν ∗ = 1, where a Composite Fermion is an electron with two attached flux quanta. The state at ν = 2/5 is re-interpreted as a state with ν ∗ = 2 (figure 4.4). The CF picture allows to understand the FQHE of electrons at ν = p/(2sp + 1) in terms of an IQHE for CF at filling factor ν ∗ = p, since the CF filling factor ν ∗ = hnel /eB ∗ is connected 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 notion of flux attachment to electrons. It is not an obviously physical approach to renormalize the magnetic field, which is an external object imposed on the system. Note however that the magnetic field only enters the theory with the electronic charge e as a multiplying coupling constant. It is thus permissible to renormalize the charge, which seems especially relevant given the fractionalisation of excitation charges discussed above. 4.3.1 The effective potential Pour mieux comprendre le modèle, on discutera dans cette section quelques propriétés du potentiel d’interaction effectif (4.17). En raison des zéros des polynômes de Laguerre Ln (x), la répulsion coulombienne √ disparaı̂t à certaines valeurs du vecteur d’onde, notamment à q0 (n) ≃ 2, 4/ 2n + 1, ce qui correspond au premier zéro x ≃ 1, 2/(2n + 1) [31]. Cela mène à des instabilités Jains’s generalisation 71 ν = 1/3 : ν∗ = 1 Théorie de FCs électron ν = 2/5 : ν∗ = 2 quantum de flux libre vortex portant 2s quanta de flux (liés) fermion composite Figure 4.4: Composite Fermions. The electronic state at ν = 1/3 may be understood as 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 système pour la formation des phases de densité inhomogène avec une périodicité caractéristique Λ ≃ 2π/q0 (n), car il est énergétiquement favorable pour la densité moyenne hρ̄(q)i d’avoir un maximum à √ q0 . La périodicité Λ varie proportionnellement avec le rayon cyclotron RC = 2n + 1. Les phases de densité inhomogène seront discutées plus en détail dans le chapitre 7. Une P transformation dans l’espace réel du potentiel effectif, vn (r) = q exp(ir · q)vn (q) confirme la apparition d’une échelle de longueur caractéristique. Pour des petites valeurs de n, cette transformation peut être effectuée de façon exacte, et l’on trouve une somme finie sur des fonctions de Bessel. Dans des NL plus élevés, n ≫ 1, on peut déduire une loi d’échelle pour le potentiel à l’aide de Fn (q) ≃ J0 (qRC ), ce qui devient exact dans la limite n → ∞, ṽ(r/RC ) vn (r) ≃ √ , 2n + 1 2 avec ṽ(x) = 4e Re K πǫx 1− q 1 − 4/x2 2 2 , (4.69) où J0 (x) est la fonction de Bessel d’ordre zéro, et K(x) est l’intégrale elliptique complète de première espèce [21]. La figure 4.5(a) montre les résultats pour le potentiel effectif dans les niveaux n = 1, ..., 5. On remarque la formation d’un palier – à part des petites oscillations – pour des distances moyennes, superposé au potentiel coulombien habituel, e2 /ǫr, qui est retrouvé à grande distance. Ce palier devient plus large dans des NL élevés cependant que sa hauteur est diminuée. La forme 72 FQHE – from Laughlin to Composite Fermions 2.02 1.01 (a) 1/r 1.5 1.5 n=1 0.6 n=2 0.5 1/r n=1 ~ v(r) vn (r) 0.8 (b) 1.75 0.4 n=3 n=4 n=5 1.25 n=2 1.01 n=3 0.75 n=4 0.5 0.5 n=5 0.25 0.2 22 4 4 r 6 6 88 10 10 10 1.0 20 2.0 30 3.0 40 4.0 50 5.0 r/R C Figure 4.5: (a) Potentiel effectif dans l’espace réel pour les NL n = 1, ..., 5, en unités de e2 /ǫ. Le potentiel de Coulomb en 1/r est montré pour comparaison (tirets). (b) Les résultats pour le potentiel (points) sont tracés après la transformation d’échelle (4.69). La ligne noire représente l’expression approchée ṽ(x) et la ligne grise le potentiel de Coulomb. (a) (b) (c) r Figure 4.6: Les fonctions d’onde des électrons dans un NL n ≥ 1 peuvent être représentées par des anneaux [voir Fig. 2.3(a)]. (a) Si r > 2RC , les anneaux ne se < 2RC , les anneaux commencent à avoir un recouvrement, recouvrent par. (b) Pour r ∼ représenté par la surface grise foncée. (c) Le recouvrement n’augmente pas de façon significative lorsque les anneaux sont rapprochés davantage. d’échelle ṽ(x) est mise en évidence après la transformation des résultats selon l’équation (4.69) : les points, qui représentent les résultats exacts, tombent approximativement sur la même courbe (noire). L’approximation (4.69), qui devient exacte dans la limite n → ∞, décrit la forme du potentiel de façon suffisamment appropriée aussi pour de plus bas NL à condition que n > 0. Le point anguleux à r = 2RC dans la forme approchée du potentiel est un artéfact mathématique – pour x ≥ 2, l’argument de l’intégrale elliptique est réel tandis qu’il devient complexe pour x < 2, ce qui donne lieu à cette discontinuité. Cet effet pourrait engendrer des divergences artificielles dans d’éventuelles dérivées, mais on peut se servir de cette forme du potentiel uniquement comme support dans des intégrations, ce qui rend la discontinuité inoffensive. La forme du potentiel peut être illustrée dans une image quasi-classique. Jains’s generalisation 73 Avec la restriction des champs électroniques au n-ième NL, on a fait une moyenne sur le mouvement rapide de l’électron, déterminé par la variable η qui, sans cette restriction, couplerait des états de différents n. Les degrés de liberté du mouvement des électrons sont donc uniquement leurs centres de guidage. Comme on l’a vu dans la section 2.2.1, la fonction d’onde d’un électron dans un niveau de Landau n ≥ 1 tient compte de cette moyenne sur le mouvement cyclotron et a par conséquent une forme d’anneau de rayon RC , représentant une densité électronique moyenne [Figs.2.3(b) et 4.6]. Si la distance r entre les centres d’anneaux, qui sont précisément les centres de guidage du mouvement cyclotron de chaque particule, est suffisamment grande (r > 2RC ), la forme des fonctions d’onde de deux particules n’a pour effet qu’une faible correction du potentiel coulombien. A r ∼ RC , les anneaux commencent à se recouvrir et la répulsion devient donc plus forte. En revanche, si l’on rapproche les centres de guidage, ce recouvrement ne devient pas plus grand et la répulsion n’augmente donc pas de façon significative, ce qui explique la formation du palier dans le potentiel effectif. La répulsion devient à nouveau plus importante quand le recouvrement est complet à très petite distance. Or les centres de guidage étant étalés sur la surface minimale 2 2πlB ne peuvent pas être approchés à des distances plus petites que lB . Pour n = 0, cette image quasi-classique devient plus problématique parce que les fonctions d’onde sont de forme gaussienne avec une extension spatiale de l’ordre de la longueur magnétique. Les électrons devraient donc plutôt être représentés par un disque de rayon lB , qui constitue également la longueur minimale, comme il a été décrit dans le chapitre 2 [voir Fig. 2.3(b)]. 74 FQHE – from Laughlin to Composite Fermions Chapter 5 Chern-Simons Theories and Anyon Physics Following the CF theoretical proposal, a field theory was constructed to describe flux attachment to electrons. Such theories are known as ”ChernSimons ” theories in the framework of the generalisation of the Maxwell theory of electromagnetic fields. Lopez and Fradkin were first to point out in 1991 [15] their relevance for the FQHE, followed in 1993 by Halperin, Lee and Read [16], who studied the compressible state at ν = 1/2. The latter filling factor is the limiting point of p/(2sp + 1) when p → ∞ and s = 1. This chapter aims at introducing some basic notions about Chern-Simons transformations, but does not pretend to offer a detailed field theoretical description. We describe their connections with anyons, i.e. particles in 2D which obey fractional statistics, and which have a transparent description in the framework of Chern-Simons theories, the basic notions will be useful in the follwing chapter. 5.1 Chern-Simons transformations The Hamiltonian of electrons in a magnetic field writes, in second quantized form Ĥ = Ĥ0 + Ĥint , 75 76 Chern-Simons theories and anyon physics where the kinetic term is Ĥ0 = Z d2 rψ † (r) [−ih̄∇ + eA(r)]2 ψ(r), 2m (5.1) and Ĥint accounts for interactions between electrons. A Chern-Simons transformation is a singular unitary transformation, ψ(r) = e−iφ̃ R d2 r ′ 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 for r = 0. The density is invariant under this transformation † ρ(r) = ψ † (r)ψ(r) = ψCS (r)ψCS (r). Notice that d2 r′ θ(r−r′ )ρ(r′ ) (see equation 5.2) is an operator which depends on all electron coordinates. The gradient in expression (5.1) also operates on the phase factor of the transformation, and one finds R −ih̄∇ψ(r) = e−iφ̃ R d2 r ′ θ(r−r′ )ρ(r′ ) −ih̄∇ − φ̃h̄∇ Z d2 r′ θ(r − r′ )ρ(r′ ) ψCS (r). We can thus define a new gauge field, the Chern-Simons vector potential, Z h̄ ACS (r) = − φ̃∇ d2 r′ θ(r − r′ )ρ(r′ ). e (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 Ĥ0 = Z 2 d [−ih̄∇ † rψCS (r) + eA(r) + eACS (r)]2 ψCS (r). 2m (5.4) The interaction Hamiltonian is invariant, since it depends only on density operators 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 analytic functions. We take here z = x + iy, unlike our definition in chapter 2. Each complex function may be written as a sum of a real part and an imaginary part, 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, by equations known as Cauchy-Riemann differential equations ∂x u(x, y) = ∂y v(x, y), et ∂y u(x, y) = −∂x v(x, y), (5.5) Instead of using the cartesian notation, one may chose the polar coordinate representation: 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 ∂x w(x, y) − w(x, y)∂y χ(x, y) + i [∂y χ(x, y) + w(x, y)∂x χ(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 ln w(x, y) = ∂y χ(x, y) et ∂y ln w(x, y) = −∂x χ(x, y). (5.6) In the simplest case, which is of interest here, f (z) = z = r exp(iθ), this yields ∂x ln r(x, y) = ∂y θ(x, y) et ∂y ln r(x, y) = −∂x θ(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 a gradient is usually zero, but this is not the case here, because θ(r) is singular at 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 ChernSimons field satsifies the Coulomb gauge. Equation (5.7) gives for the corresponding magnetic field BCS h̄ Z 2 ′ h = − φ̃ d r ∇ × ∇θ(r − r′ )ρ(r′ ) = − φ̃ρ(r)ez . e e (5.9) 78 Chern-Simons Theories and Anyon Physics We notice that this magnetic field is 1)intimately connected to the electronic density, 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) is replaced by the average density hρ(r)i = nel , so that the field is renormalized h B → B ∗ = B + hBCS i = B − φ̃nel e (5.10) where, in terms of filling factor, B → B ∗ = B(1 − φ̃ν). (5.11) If we chose φ̃ = 2s, this field renormalisation is precisely that described by the CF theory c(4.67). Let us connect with the trial wave function approach. In the first quantization language, we can rewrite this field transformation as iφ̃ ψ({zj }) = e P θ(zi −zj ) i<j ψCS ({zj }) = Y i<j zi − zj |zi − zj | !φ̃ ψCS ({zj }). (5.12) We see that the Chern-Simons transformation ties φ̃ flux quanta (singularity of order φ̃ in the phase), leaving off the transformation the vortex modulus, contrary to Jain’s function for φ̃ = 2s (4.66). 5.2 Statistical Transmutation – Anyons in 2D Chern-Simons theories are especially well fitted to the discussion of anyon physics. Anyons are particles which live in 2D+1 space, and which obey fractional statistics, i. e. neither bosonic nor fermionic. All particles known in the 3D world are either bosons or fermions depending on the behaviour of their wave function upon interchange of two identical particles.This is basically because in 3D (and higher dimensions), the rotation group is nonabelian. The components of the angular momentum do not commute. Quantization of angular momenta is in terms of units of h̄/2, as can be seen from the properties of the Lie algebra for infinitesimal rotations. The classification in bosons and fermions is not true anymore in 2D, because the rotation group is a trivial abelian group. Therefore no angular momentum quantization rules follow from manipulations of the infinitesimal rotations. Statistical Transmutation – Anyons in 2D (b) (a) z 79 C A B A B + B A 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(gray curves)). (b) Equivalent processus consisting in two successive exchanges of particles A and B. Let us look at the interchange of two identical particles. A process T through which a particle A is adiabatically displaced around another particle B, 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 wave function overlap. From an algebraic point of view, which takes into account the homotopy of the processes, one may write: E2 = T modulo a translation. Let us first discuss the 3D case where the path C lies in the x − y plane. Since the third dimension along z is available, we can lift the path C of particle A above particle B, and then shrink it down to a point, leaving particle B at all times outside the closed path C. .1 . We may associate 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 point of A). The process through which A circles around B(rotation of 2π is thus equivalent to a process which leaves particles unchanged, i.e. the identity, so we can write √ T =1 et donc E = 1, where the last equation is written symbollically, meaning that E has two eigenvalues e1 = exp(2iπ) = 1 and e2 = exp(iπ) = −1 . This superselection rule shows that in 3D all point particles may be separated in two classes, depending on the behaviour of their wave function upon interchange of identical particles. e1 = 1 corresponds to bosons and e2 = −1 to fermions. This piece of reasoning is not valid anymore when the particle dynamics is restricted to 2D (2 space dimensions). In this case the path C cannot be shrunk to a point without crossing the particle position B. One says that a 1 This would be impossible if B were an infinite line in direction z 80 Chern-Simons Theories and Anyon Physics path which encloses another particle (particle B) is not in the same homotopy class as a path which encloses none (and can be shrunk to a point. In 2D, paths can be classified by the number of enclosed particles, or by the number of times it winds around a given particle. From an algebraic point of view, the physical requirement is that physical quantities must be invariant by a 2π rotation. This requirement does not apply to the wave functions, since only probabilities are eventually observable. Therefore the eigenvalues λ 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 generalised statistics). It is easy to see that eigenstates of R(2π) with different eigenvalues λ 6= λ′ are orthogonal. In fact no local observable can connect states corresponding to different α values. States corresponding to different λ 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 ) = eiαπ ψ(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 result from the Chern-Simons transformation, using the known properties of electronic fields ψ(r). Defining τ (r) ≡ Z d2 r′ θ(r − r′ )ρ(r′ ), (5.15) Statistical Transmutation – Anyons in 2D 81 to simplify notation in the following expressions one has ψCS (r1 )ψCS (r2 ) = eiφ̃τ (r1 ) ψ(r1 )eiφ̃τ (r2 ) ψ(r2 ) = eiφ̃τ (r1 ) eiφ̃τ (r2 ) e−iφ̃τ (r2 ) ψ(r1 )eiφ̃τ (r2 ) ψ(r2 ). (5.16) With the help of the Hausdorff formula, eA Be−A = B + [A, B] + ∞ X 1 1 [A, [A, B]] + ... = Cn (A; B), 2 n=0 n! (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 )] = Z d2 r′ θ(r2 − r′ )[ψ † (r′ )ψ(r′ ), ψ(r1 )] = −θ(r2 − r1 )ψ(r1 ), one finds eventually: e−iφ̃τ (r2 ) ψ(r1 )eiφ̃τ (r2 ) = eiφ̃θ(r2 −r1 ) ψ(r1 ). This yields for expression (5.16) ψCS (r1 )ψCS (r2 ) = eiφ̃θ(r2 −r1 ) eiφ̃τ (r1 ) eiφ̃τ (r2 ) ψ(r1 )ψ(r2 ) and similarly, interchanging r1 ↔ r2 , ψCS (r2 )ψCS (r1 ) = eiφ̃θ(r1 −r2 ) eiφ̃τ (r2 ) eiφ̃τ (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 ) = −eiφ̃π ψCS (r2 )ψCS (r1 ), (5.18) and also † † ψCS (r1 )ψCS (r2 ) + eiφ̃π ψCS (r2 )ψCS (r1 ) = δ(r1 − r2 ). (5.19) Comparing those expressions to equation (5.13), one sees that φ̃ plays the rôle of the statistical angle α, and Chern-Simons transformations are found to allow changing particle statistics. Notice moreover that the choice φ̃ = 2s + 1 transforms fermions into bosons, while φ̃ = 2s does not change the particle statistics, as is the case for the CF theory discussed above. 82 Chern-Simons Theories and Anyon Physics 5.2.2 Fractional charge and fractional statistics The topics discussed in the previous section are directly related to the Berry phase, which is a “geometrical” phase the particle wave function acquires when the particle is adiabatically displaced along a path in parameter space. An example of Berry phase is that due to the Aharonov-Bohm phase, which appears when a particle with charge e∗ follows a path ∂Σ = C enclosing a surface Σ e∗ I e∗ Z 2 Γ=− dr · ACS (r) = − d rBCS (r), (5.20) h̄ ∂Σ h̄ Σ where the gauge field is that of the Chern-Simons transformation. In this case the Berry phase is an operator, because the “magnetic field” BCS is proportionnal to the density operator. Within the mean field approximation equation (5.10), BCS = hφ̃nel /e, one finds Γ = 2π e∗ φ̃N (Σ), e (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 path C around a Laughlin liquid is an electron of charge e∗ = e, it acquires a phase which is a multiple of Γ = 2π φ̃. When φ̃ is an integer, as is the case 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/φ̃, the acquired 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, within the path followed by the quasi-particle. Remember that the quasi particle at z0 has a wave function, in Laughlin’s theory, which is obtained Q by multiplying by the factor j (zj − z0 ). In terms of Chern-Simons transformations de Chern-Simons, this can be modelled by the transformation R 2 ′ ′ ′ UV (r) = eiq d r 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 Statistical Transmutation – Anyons in 2D 83 with the Laughlin (or Jain) wave function, this transformation attaches a singular phase to the quasi-particle, without attaching the proper modulus of the zero. In terms of gauge transformation, one has ACS (r) → ACS (r)−(h̄/e)∇f (r) when the wave function transforms as ψ(r) → exp[if (r)]ψ(r). The total vector potential is thus ACS (r) → ACS (r) − h̄q Z 2 ′ ∇ d r θ(r − r′ )ρV (r′ ), e and the relationship between the magnetic field and the densities (5.9) now writes hq h BCS (r) = [∇ × ACS (r)]z = − φ̃ρ(r) + ρV (r). e e (5.22) The first term gives rise to the same Berry phase Γ as in the case of a quasi-particle circling around a Laughlin liquid enclosed within Σ, and the second term adds a phase ∆Γ dueto the presence of possible quasiparticles within Σ. Suppose that we have exactly one quasi-particle at position rV ∈ Σ, ce qui donne e∗ q e∗ Z 2 hq , d r δ(r − rV ) = 2π q = 2π ∆Γ = h̄ Σ e e 2s + 1 (5.23) for the case where e∗ = e/(2s + 1) ( Laughlin quasi-particle). One sees thus that cahrge fractionalisation generates engendre également une fractional statistic in 2D, since the statistical angle associated with the Berry phase ∆Γ is q e∗ . (5.24) α=q = e 2s + 1 Looking for experimental evidence for fractional statistics is an active field of research to this day. The quasi-particle fractional charge has already been observed 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 by applying an adequate gate voltage on electrodes. When the two edges are sufficiently close to one another, a quai-particle may be back-scatterd from one 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 their charge is e/5. 84 Chern-Simons Theories and Anyon Physics Because of the close relationship between fractional charge and fractional statistics, it is no easy task to find separate evidence for each. Various devices have been recently proposed for ν = 5/2, which seems to correspond to a state with non abelian statistics [48]. A discussion of these ideas is beyond the scope of these lectures. Chapter 6 Hamiltonian theory of the Fractional Quantum Hall Effect The theory of FQHE follows nowadays distinct and complementary paths. Following Laughlin [12, 20], one appoach concentrates on writing down manybody wave functions, and studying their properties by numerical means, such as exact diagonalisations, Monte-Carlo computation, Density Matrix Renormalization Group, and so forth. However, the most powerful computer to date cannot easily handle the large Hilbert space involved with more than 12 particles. Tricks may allow to simulate up to 24 particles. Such methods often allow to reach definite conclusions about the relative stability of states with different symmetries. Sometines, however, the thermodynamic limit is not available, and doubts linger about the final conclusions of computations on small size particles clusters. It is useful, both for a more transparent understanding of the physics at hand, and for possible comparison with numerics, to be able to conduct analytical approaches to the theory in the thermodynamic limit. Even though the accuracy may be less satisfactory than for exact results on small number of particles, it is interesting to have a theory where finite size effects do not blur the conclusions. Such an analytical theory in the thermodynamic limit is the Hamiltonian approach, the first challenge of which is to attack the degeneracy problem. We have discussed some aspects of Chern-Simons theories in the previous chapter. We notice that the transformation (5.2) deals with the kinetic part of the Hamiltonian only, not with the interaction term, which is invariant under Chern-Simons transformations. However, as we discussed in chapter 4, 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 electrons restricted to a single LL occupation [equation (4.17)] involves the interaction term only. There seems to be a contradiction in the theory. This criticism to Chern-Simons theory is however too severe. Indeed, the Chern-Simons theory aims at replacing the true repulsive Couomb interaction by a statistical interaction expressed by the concept of flux attachment to electrons. This generates a singularity in the N particle wave function when two particles attempt to sit at the same point in space [see equation (5.12)]. The physical notion behind the Chern-Simons manipulation is that the non perturbative many-body effect of Coulomb interactions is to generate collective effects in the form of flux tubes attached to electrons, in such a way as to renormalize the effective external magnetic field. The latter renormalisation results in the degeneracy lifting for fractional filling of the LLL which is a key factor to account for the FQHE. Chern-Simons theories have thus to be credited with an interesting step forward. A difficulty remains however in the mean field version (5.10) which renormaises the magnetic field, B → B ∗ = B(1 − φ̃ν): the energy separation between the new levels (LL∗ ) is h̄ωC∗ = h̄eB ∗ /m. This energy scale which involves 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 appropriate to seek a more satisfatory theory beyond mean field, which should yield the correct energy scale in a natural fashion. Various approaches have been proposed such as a random phase approximation [16, 49] which renormalises the mass 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 FQHE Hamiltonian theory [50, 51, 105]. We shall concentrate on the formulation due to Murthy and Shankar [51]. 6.1 Miscroscopic theory This section deals with the connection between Chern-Simons theories and the effective model described by equation (4.17), in the framework of a microscopic formulation of the Hamiltonian theory. The main focus is on the treatment of the fluctuations of the vector potential, (4.17), using the a new theoretical tool, i.e. a new quantum gauge field a(r). The latter object amounts to resorting to new unphysical degrees of freedom, which have to be removed 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, so we try and provide as much physical insight as possible. Readers with less interest in the mathematical details may directly go to section 6.2 where the end product, the effective theory is discussed in simpler terms. That section does 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 as a sum over particles j = 1, ..., Nel ĤCS = 1 X [pj + eA∗ (rj ) + eδACS (rj ) + ea(rj )]2 . 2m j (6.1) The mean value of the Chern-Simons vector potential has been absorbed in an effective “external” vector potential, A∗ (r) = A(r) + hACS (r)i, and fluctuations δACS (r) have been singled out. They are connected to density fluctuations through equation (5.9). Equivalently, in Fourier space, (h̄ ≡ 1) δACS (q) = 2π φ̃ δρ(q)e⊥ , e|q| (6.2) where δρ(q) = j exp(−iq·rj )−nel and e⊥ = iez ×ek , with ek = q/|q|2 . Although the Hamiltonian 6.1looks like that of a non interacting particle Hamiltonian, the presence of δACS (q) introduces the full many body character of the problem because δρ(q) depends on all electron co-ordinates. The last term in equation (6.1) represents a new transverse gauge field, ∇ · a(r) = 0, which has been introduced, for the time being, artificially. Its quantum character, which corresponds to the quantum character of the operator δρ(q) is ensured by the introduction of its conjugate (longitudinal) field P(r), with the canonical commutation rules P [a(q), P (−q′ )] = iδq,q′ , (6.3) where, in Fourier space a(q) = a(q)e⊥ , 1 2 P(q) = iP (q)ek . We use here for simplicity, a first quantisation approach. δACS (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 freedom of this new gauge field. The physical sub-space is that of states |ϕphys i which are annihilated by a(q), a(q)|ϕphys i = 0. (6.4) What is the advantage of this formal operation? Just as momentum px is the translation generator in direction x (see section 2.2.2), the conjugate field P(q) is a translation generator for vector potentials. Therefore one may use the transformation P ′ ′ (6.5) Ua = ei q′ P(−q )δACS (q ) to get rid of the Chern-Simons vector potential fluctuations by a vector potential translation Ua† a(q)Ua = a(q) − δACS (q), Ua† a(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 action of the gradient operator(the momentum in r representation), since Ua = exp i 2π φ̃ X e P (−q′ ) q′ One gets therefore 1 |q′ | X j ′ e−iq ·rj − nel . [−i∇j + eA∗ (rj ) + eδACS (rj ) + ea(rj )] Ua = Ua " = Ua " 2π φ̃ X 1 −i∇j + eA (rj ) + ea(rj ) + ∇j P (−q) e−iq·rk e |q| k ∗ # # 2π φ̃ −i∇j + eA (rj ) + ea(rj ) + P(rj ) , e ∗ and for the transformed Hamiltonian ĤCP " #2 1 X 2π φ̃ P(rj ) = pj + eA∗ (rj ) + ea(rj ) + 2m j e . (6.7) The index CP means that the Hamiltonian is transformed in a basis of Composite Particles. In second quantized notation, fields ψCS (r) = Ua ψCP (r). Notice that the constraint (6.4) is also transformed to Microscopic theory 89 " # 2π φ̃ a(q) − δρ(q) |ϕphys i = 0. e|q| (6.8) The physical interpretation of this constraint is as follows: instead of treating density fluctuations exactly (or, equivalently, those of the Chern-Simons vector potential), one describes them by the new quantum field a(q).3 Since a(q) · P(q′ ) = 0 (remember that a(q) is a transverse field and P(q) is a longitudinal one), the last two terms of the Hamiltonian (6.8) represent a harmonic 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 Hamiltonian contains three terms, ĤCP = ĤB ∗ + Ĥosc + Ĥcoupl : that of charged particles in an effective magnetic field B ∗ (this is the expression of the Chern-Simons theory at the the mean field approximation level , ĤB ∗ = 1 X [pj + eA∗ (rj )]2 , 2m j (6.9) that of a harmonic oscillator, which represents the density fluctuations Ĥosc = and a coupling term Ĥcoupl 1 X 2m j e2 a2 (rj ) + 2π φ̃ e !2 P 2 (rj ) , (6.10) # " 1 X 2π φ̃ = P(rj ) . [pj + eA∗ (rj )] · ea(rj ) + m j e (6.11) Let us first neglect the coupling term, and discuss the model with the first two terms only. The diagonalization of the Hamiltonian with the coupling term will be treated in the next section. Consider the harmonic oscillator P term. It can be re-written, using ρ(r) = j δ(r − rj ) = nel + δρ(r), Ĥosc 3 1 Z 2 2π φ̃ = d rρ(r) e2 a2 (r) + 2m e !2 P 2 (r) This procedure is not new in theoretical physics: in the context of path integrals, there is some similarities with the Hubbard Stratonovich transformation. The theory of one-dimensional electron sytems resorts to bosonisation, where bosons represent charge or spin fluctuations. Indeed, Murthy and Shankar themselves were influenced, in their own words, by the treatment of plasmons in an interacting Fermi liquid, as initially proposed by Bohm and Pines [53]. 90 Hamiltonian theory of the FQHE nel X 2 2 2π φ̃ ≃ e a (q) + 2m q 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 approximation O(δρ(q)2 ). Now introduce the ladder operators for the harmonic operator which obey " # 2π φ̃ ea(q) + i A(q) ≡ q P (q) , e 4π φ̃ 1 h i A(q), A† (q′ ) = δq,q′ . The Hamiltonian (6.10) now writes Ĥosc = ω0 X q (6.12) (6.13) A† (q)A(q), (6.14) with the characteristic frequency ω0 = 4π φ̃ nel = ν φ̃ωC . 2m (6.15) The oscillator ground state is the usual gaussian χosc e2 X 2 a (q) , = exp − 4π φ̃ 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 in a field B ∗ : ψCP ({rj }) = χosc ({rj })φp ({rj }), where φp ({rj }) is unique (non degenerate) for ν ∗ = p. Due to the constraint (6.8), the oscillator wave function can be written as: χosc " # φ̃ X 2π = exp − δρ(−q) 2 δρ(q) . 2 q |q| The exponent can be regarded as a Hamiltonian for charged particles interacting with a 2D Coulomb potential, ṽ(q) = 2π/q 2 ↔ v(r) = − ln |r|. We 4 Remember that a(q) ∼ δρ(q),because of the constraint (6.8). Microscopic theory 91 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 }) = e φ̃ 2 = C P j,k Y k<j R d2 rd2 r ′ [δ(r−rj )−nel ] ln |r−r′ |[δ(r′ −rk )−nel ] |rk − rj |φ̃ e−φ̃ν P k |rk |2 /4 , (6.17) where C is a proportionnality constant , and we used nel = ν/2π. Notice [see equation (5.12)], that the singular phase has already been attached through the Chern-Simons transformation and the the N electrons wave function thus writes, in complex coordinates notation , ψ({zj }) = Y k<j (zk − zj )φ̃ e−φ̃ν P k |zk |2 /4 φp ({zj }). (6.18) When φ̃ = 2s, i.e. for fermionic statistics we retrieve the Laughlin (4.30), as well 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 the correct magnetic length to be inserted in the gaussian. To this end, notice that φp ({zj }) represents the N partciles wave function for an integer filling ∗ ∗ factor ν ∗ = p in a field B q = [∇ × A (r)]z . The characteristic magnetic ∗ length inφp ({zj }) is lB = h̄/eB ∗ , the gaussian factor in the wave function is thus ! " # X |zk |2 X |zk |2 = exp −(1 − ν φ̃) , exp − ∗2 k 4lB k 4lB 2 ∗2 where we used lB /lB = B ∗ /B = (1 − ν φ̃). In the end, the gaussian factor in the electronic wave function (6.18) is then exp − X |zk |2 k 2 4lB ! , 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 only sketch here the main steps of the decouplig derivation for small wave vectors, 92 FQHE Hamiltonian theory |q|lB ≪ 1. The interested reader is refered for more details to the MurthyShankar review, especially to cond-mat/0205326v2. The total Hamiltonian (without the interaction term), now writes ĤCP = X Πj,+ Πj,− j 2m + ω0 X q A† (q)A(q) + θω0 q Xh i c† (q)A(q) + c(q)A† (q) , q (6.19) where θ = π φ̃/4πnel , c(q) = j q̂− Πj,+ exp(−iq · rj ) et Πj,± = pj,± + eA∗± (rj ). Vector operators are written here in complex notation, V± = Vx ± iVy , et q̂± = ek · ex ± iek · ey is the unit vector in the direction of q, in complex notation. To the usual hrmonic oscillator commutation rules (6.13), we have to add [Πi,− , Πj,+ ] = 2eB ∗ δi,j (6.20) P and [c(q), c† (q′ )] = −2eB ∗ X 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 fluctuations are taken care of at the gaussian approximation level. Corrections to equation (6.21) would be of higher order than O(δρ(q)2 ). In Hamiltonian (6.19), the coupling between particles described by operators Πj,± [or c(q) et c† (q)], and the oscillator fields A(q) or A† (q) is linear. The canonical transformation which decouples the Hamiltonian has the following form: iλS0 U (λ) = e ( = exp λθ Xh q † † ) i c (q)A(q) − c(q)A (q) , (6.22) Where we want eventually to chose the “flow” parameter λ in a convenient way, leaving it undetermined for the time being. An operator transformed through (6.22), Ω(λ) = exp(−iλS0 )Ω(λ = 0) exp(iλS0 ), may be derived from the derivative dΩ (6.23) = −ie−iλS0 [S0 , Ω] eiλS0 . dλ For operators c(q) and A(q), which occur in Hamiltonian (6.19), this leads to the flow equations dA(λ, q) = −θc(λ, q) dλ et Microscopic theory 93 dc(λ, q) µ2 = − A(λ, q), dλ θ with µ2 ≡ 2eB ∗ nel θ2 = 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 sin(µλ)c(q), µ (6.24) and µ sin(µλ)A(q) + cos(µλ)c(q). (6.25) θ This transformation may be interpreted as a rotation in Hilbert space, which mixes the c(q) degrees of freedom (particles) and A(q) degrees of freedom (oscillators). The transformed Hamiltonian is thus derived inserting equations(6.24) and (6.25) in Hamiltonian, (6.19), and from the transformation of its first term with the same method (integration of the differential equation(6.23)). The end result contains many terms, and, even though the calculation is straightforward, we only show here its global form, c(λ, q) = ĤCP (λ) = X Πj,+ Πj,− 2m j + Xn α(λ)c† (q)c(q) (6.26) q h io +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 Hamiltonian , adding the interaction term, V [ρ(λ0 , q)](which depends on the transformed density), writes ĤCP = X Πj,+ Πj,− j +ωC 2m X q − X eB ∗ 1 XX Πj,+ e−iq·(rj −rk ) Πk,− + 2mnel j,k q j 2m A† (q)A(q) + V [ρ(λ0 , q)]. Some remarks are here in order: (6.28) 94 FQHE Hamiltonian theory • The oscillator frequency is given by the LL energy difference, ωC , in agreement with Kohn’s theorem which states that collective density excitations in the limit q → 0 oscillate at this frequency, in the presence of translation invariant interactions. In the strong field limit, the oscillators represent high energy excitations, which condense at T = 0 in the lowest energy mode. The fourt term then becomes an unimportant constant, which can be subsequently ignored. • The third term indicates that particles posess a magnetic moment e/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, the number of operators has not been specified. However, if one focuses on diagonal 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 P X Πj,+ Πj,− j 2m∗ , which lumps together the first and the second term. The natural choice seems to be nosc = nel , which results in the vanishing of the kinetic enrgy term (apart from the non diagonal term j 6= k). This is what one expects for the dynamics of electrons projected on a single LL [equation (4.17)]. The choice nosc > nel would lead to the unphysical result of a negative effective mass, and nosc < nel would not make it vanish. Another justification for the choice nosc = nel (ou Q = kF )will be 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 the cut at Q = kF , those can be rewritten, using X |q|<Q e−iq·(rj −rk ) = δ(rj − rk ) − X e−iq·(rj −rk ) , |q|>Q and thus as a sum of a term which is zero for j 6= k and a term which is only relevant at large wave vectors |q| > Q, and which may be neglected at small wave vectors, |q|lB ≪ 1. Within this approximation, there is no kinetic term any more. 6.2. EFFECTIVE THEORY AT ALL WAVE VECTORS 95 Eventually, the Hamiltonian (6.28), which describes the low energy dynamics, becomes X eB ∗ ĤCP (λ) = + V [ρ(λ0 , q)]. (6.29) j 2m This result is fairly satisfactory, since it only contains the interaction term, apart from a constant which plays no dynamical role. To complete the discussion, one must compute the transformed density, ρ(λ0 , q), as well as the constraint, the form of which is also alterd by the transformation. The density is derived following the same procedure as for c(λ, q) and A(λ, q),, from the integration of the differential equation (6.23), while the constraint is given by i e|q| h A(λ, q) + A† (λ, −q) . ρ(λ, q) = q 4π φ̃ The final result [51] is, to lowest order in |q|lB ρ(λ0 , q) = X j 2 e−iq·rj 1 − ilB i q ∧ Πj c|q| h A(q) + A† (−q) , +q 1+c 4π φ̃ (6.30) where the parameter c is connected to the decoupling parameter, at ν ∗ = p, c2 = cos2 (λ0 µ) = pφ̃ . pφ̃ + 1 (6.31) As regards the constraint, one gets " # 2 q ∧ Πj . χ(λ0 , q)|ϕphys i = 0, with χ(λ0 , q) = e−iq·rj 1 + ilB c(1 + c) j (6.32) The constraint does not involve oscillators A(q), only particles. That is a necessary condition for a complete decoupling of Hamiltonian (6.29). The density operators (6.30) however still contain a contribution from oscillators, but the latter vanish on the average when they condense in the ground state hA(q)i = hA† (q)i = 0, as we assume is the case in the following. X 6.2 Effective theory at all wave vectors The connection with the model (4.17) becomes even clearer when one attempts to construct a theory at all wave vectors, which coincides with the 96 FQHE Hamiltonian theory small wave vector theory in the limit |q|lB ≪ 1. This generalisation is based on 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 would then have 5 ρ̄(q) = X (e) e−iq·Rj et χ̄(q) = j X (v) e−iq·Rj , (6.33) j with Πy Πx = x+ ,y − 1+c 1+c ! Πy Πx R et R = x − ,y + . c(1 + c) c(1 + c) (6.34) The components of those new operators satisfy the commutation rules (e) h i 2 X (e) , Y (e) = ilB , (v) et h i X (v) , Y (v) = −i 2 lB . c2 (6.35) A comparison with equation (2.16) shows that we may interpret R(e) as the guiding center of an electron while R(v) seems to be that of a second particle with charge −c2 , in terms of the electronic charge. The associated densities are automatically projected on the LLL, and the final model is 6 Ĥ = 1X v0 (q)ρ̄(−q)ρ̄(q), 2 q (6.36) with the commutation rules for densities ! q∧k [ρ̄(q), ρ̄(k)] = 2i sin ρ̄(q + k), 2 ! q∧k χ̄(q + k), [χ̄(q), χ̄(k)] = −2i sin 2c2 [χ̄(q), ρ̄(k)] = 0 et χ̄(q)|ϕphys i = 0. (6.37) This is the same model as that discussed in section 4.1.2, where one has added the constraint associated with the density χ̄(q) and its quantum algebra. 5 The bars over the symbols mean that we are dealing with generalised densities at all wave vectors. 6 we 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 ez x Π lB2 y (e) R (v) R x Figure 6.1: Composite Fermions in teh Hamiltonian theory. What is the physical content of the model, in the framework of CF picture, supposing that χ̄(q) describes the density of a second species of particle, which we will call “ pseudo-vortex”? The “pseudo” in this expression indicates that this particle lives in a larger Hilbert space, and that projection to the physical space is necessary, which is the case for states which are annihilated by χ̄(q). Moreover, this particle, which does not appear in teh Hamiltonian, has no dynamics. • The electron and the pseudo-vortex guiding centers are at a distance 2 ∼ |Π|lB ∼ lB away from one another, which gives rise to a dipolar 2 moment d = −eez × ΠlB (figure 6.1). • The CF may be pictured as a composite of an electron and a pseudovortex. Notice that the latter has been indirectly introduced by the oscillator degrees of freedom , a(q) or A(q). The choice nosc = nel , as we discussed earlier, may thus be interpreted as equating the number of electrons and the number of pseudo-vortices. As a result there are as many CF as electrons. The CF charge is the sum of the elctron charge and of the pseudo-vortex one, , i.e. e∗ /e = −(1 − c2 ). • The pseudo-vortex is an excitation of the electron gas, which is composed of the true elementary excitations. This can be checked on equation (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 physical meaning of the constraint χ̄(q)|ϕphys i = 0. Notice that we could have started with the model (4.17) (electrons restricted to a single LL). Instead of resorting to the oscillator gauge field a(q) and its conjugate field as additional variable, we could have introduced the field χ̄(q) in the effective model (4.17). The microscopic theory discussed in the previous section is nevertheless useful, because it has established the connection with Chern-Simons theories. • The limit p → ∞ or ν → 1/2 yields a charge c2 = 1 for the pseudovortex. The CF at ν = 1/2 are electrically neutral, but they have a 2 dipolar moment d(ν = 1/2) = −ekF lB [105, 54]. 6.2.1 Approximate treatment of the constraint In spite of the relative formal simplicity of the Hamiltonian theory (6.37), it is difficult to treat the constraint explicitly in computations. For a simpler treatment, Murthy et Shankar proposed a ”short-cut” which amounts to replacing the projected density by a ”preferred combination” ρ̄CF (q) = ρ̄(q) − c2 χ̄(q), (6.38) in the Hamiltonian. A priori, all combinations such as ρ̄γ (q) = ρ̄(q) − γ χ̄(q), with arbitrary γ are equivalent because of the constraint. The advantage of chosing γ = c2 is that the matrix elements of the projected density operator obey hN |ρ̄CF (q)|0i ∝ q 2 in the limit q → 0. This is the condition for the P structure factor S(q, ω) = N |hN |ρ̄CF (q)|0i|2 δ(ω − EN ) to vary as q 4 at small wave vector, which is demanded by the LLL projection [35, 91]. Notice that the preferred combination short-cut is a valid approximation for gapped states, such as occur at ν = p/(2sp + 1). The constraint must be explicitly dealt with, however, in the sudy of ν = 1/2, which has a compressible strange Fermi liquid ground state [16]. Another problem with the preferred combination is that the Hamiltonian no longer commutes strictly with the pseudo-vortex density. The model remains however weakly gauge invariant because the commutator vanishes in the sub-space defined by the constraint. Microscopic theory 99 There is another algebraic argument in favor of the preferred combination, i.e. γ = c2 : with this choice, the ρ̄(q) algebra (6.37) is correctly reproduced to lowest order in q, h i ρ̄CF (q), ρ̄CF (k) ≃ i(q ∧ k)ρ̄CF (q + k) ! q ∧ k CF ≃ 2i sin ρ̄ (q + k) + O(q 3 , k 3 ). 2 Higher order terms in q are suppressed in the Hamiltonian, because of the gaussian in the effective potential (4.18). From now on, the ρ̄CF (q) operator is interpreted as the CF density. Physically, modes associated with internal CF structure are neglected in this approximation. The CF basis is introduced by transforming variables R(e) et R(v) in CF guiding center , R(CF ) , and cyclotron variable, η (CF ) . This transformation must ensure that the new variables satisfy commutation rules in terms of the √ ∗ CF magnetic length, lB = 1/ 1 − c2 , h i ∗2 ηx(CF ) , ηy(CF ) = −ilB h et i ∗2 X (CF ) , Y (CF ) = ilB , (6.39) in analogy with the electronic variables (2.16). The appropriate transformation is [51] R R(e) − c2 R(v) et = 1 − c2 R(e) = R(CF ) − η (CF ) c ⇔ c (e) (v) R − R 1 − c2 (v) R = R(CF ) − η (CF ) /c. (6.40) η (CF ) = (CF ) et As in the electronic basis (section 4.1), the CF density operator may be written in second quantized form as ρ̄CF (q) = X j,j ′ ;m,m′ hm|e−iq·R (CF ) |m′ ihj|ρ̄p (q)|j ′ ic†j,m cj ′ ,m′ , (6.41) where states |ji are associated to the operator η (CF ) and states |mi to R(CF ) . The first matrix element in this expression is identical to that (4.9) deduced in section 4.1.1, in terms of the CF magnetic length, for m ≥ m′ , −iq·RCF hm|e ′ ∗ |2 /4 −|qlB |m i = e s m′ ! m! ∗ −iqlB √ 2 !m−m′ m−m′ Lm ′ ∗ 2 |qlB | . 2 ! 100 FQHE Hamiltonian theory The second one writes , for j ≥ j ′ , hj|ρ̄p (q)|j ′ i ≡ hj|eiq·η s (CF ) c − c2 f˜(q)eiq·η (CF ) /c !j−j ′ |j ′ i ∗ iq̄lB c ∗ 2 2 √ e−|qlB | c /4 = 2 " ! ∗ 2 ′ |qlB c| ′ 2 2 j−j ′ × Lj ′ − c2(1−j+j ) e−|q| /2c Ljj−j ′ 2 j ′! j! (6.42) ∗ 2 |qlB | 2 2c !# . The gaussian f˜(q) = exp(−|q|/2c2 ) in the second term takes into account the 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, mi = |ji ⊗ |mi. Because of the commutation rule (6.39) for the guiding center co-ordinates, ∗2 each state occupies a minimal surface 1/nB ∗ = 2πlB , in analogy with the electronic case. There exist thus AnB ∗ states per “ CF LL” j, and the filling factor for CF LL, ν ∗ = nel /nB ∗ is connected to the electronic filling factor through equation (4.68), ν = ν ∗ /(2sν ∗ + 1). When p CF levels are filled , ν ∗ = p, the ground state can be described by the average hc†j,m cj ′ ,m′ i = 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 is no small progress. There was no way in the electron basis to define a starting state from which a perturbation treatment might be conducted, except for ν = n. In contrast, we only need ν ∗ = p, i.e. ν = p/(2sp + 1), in the CF model. 6.2.2 Energy gaps computation The ground state (6.43) can now be used to compute various physical quantities, 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 p when p CF LL are completely filled, its energy relative to the ground state is ∆qp (s, p) = hcp,m Ĥc†p,m i − hĤi = (6.44) p−1 X X 1X v0 (q)hp|ρ̄p (−q)ρ̄p (q)|pi − v0 (q) |hp|ρ̄p (q)|ji|2 , 2 q q j=0 Effective theory 101 where averages, defined with respect to the ground state are computed as Wick’s contractions(6.43). Similarly, the CF quasi-hole energy is in level p−1 ∆qh (s, p) = hc†p−1,m Ĥcp−1,m i − hĤi 1X = − v0 (q)hp − 1|ρ̄p (−q)ρ̄p (q)|p − 1i 2 q + X v0 (q) q p−1 X j=0 (6.45) |hp − 1|ρ̄p (q)|ji|2 . The activation gap , i.e. the energy needed to create a non interacting quasiparticle/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 numerical computations, taking into account a finite sample width in direction z. The finite width alters the effective potential, v0 (q) → v0 (q)f (q), where various 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 get closer √ than this width: The distance r in the Coulomb potential is replaced by r2 + λ2 . Alternatively , on may simulate the finite width by a parabolic confinement potential, which leads to a gaussian exp(−z 2 /4λ2 ) which multiplies the wave function [56], and the correcting factor becomes f (q) = eq 2 λ2 [1 − Erf(qλ)] , (6.48) where Erf(x) is the error function. The general tendency of the factors which take into account the sample finite width is to cause a lowering (in modulus) of the characteristic Coulomb energy. This leads to a lowering of activation gaps (se figure 6.2), and also of quasi-particle or quasi-hole gaps. Compared to numerical results, the Hamiltonian theory overestimates activation gaps by a factor 1, 4 to 2 for λ = 0, but the agreement becomes better for larger λ values. 102 FQHE Hamiltonian theory gaps d’activation (a) δ 0.25 'PMJ p=1' 'Hamiltonian Theory p=1' 'PMJ p=2' 'Hamiltonian Theory p=2' 0.20 0.15 0.10 0.05 0.00 0.0 0.5 1.0 1.5 2.0 2.5 3.0 gaps d’activation λ 70x10 -3 'PMJ p=3' Hamiltonian Theory p=3' 'PMJ p=4' Hamiltonian Theory p=4' 60 50 40 δ 30 20 10 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 λ 100x10 Activation Gaps gaps d’activation (b) 2/5 -3 Exact diag. 2/5 This theory 2/5 Exact diag. 3/7 This theory 3/7 80 60 40 20 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 paramètre de largeur Width parameter b Figure 6.2: After reference [51]. Activation gaps as functions of the sample width in direction z, in units of e2 /ǫlB . (a) comparison between Hamiltonian theory and numerical computation by Park et al. in the framework of Jain’s functions [57]. A Yukawa type potential has been used to simulate the finite width [equation (6.47)]. (b) comparison to exact diagonalisation results [56], in the approximation of a parabolic confinement potential [equation (6.48)]. Effective theory 103 ν = 4/11 ν* = 1+1/3 Figure 6.3: Second generation Composite Fermions at ν ∗ = 1 + 1/3 (ν = 4/11). The CF2 , formed in the partially filled CF level, is a bound state of a “first geberation” CF and 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 filling factors ν = p/(2sp + 1). We have seen that CF theory allows to understand the FQHE at those filling factors in terms of quasi-particles, which are CF, in the mean field approximation of the Hamiltonian theory. Indeed the latter allows to discuss a non degenerate ground state at ν ∗ = p , which has p completely filled CF levels. This was not possible within the electron model, because of the huge degeneracy in the lowest LL. In that sense, the FQHE of 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 have fractional filling at ν ∗ 6= p – here again we face the huge degeneracy problem in the CF model. How is this degeneracy lifted by the residual interactions between CF? The motivation behind the question is the discovery of a FQHE at 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 7 In the case of a partially spin polarised state a FQHE at ν = 4/11 may also be understood in the framework of Halperin’s wave function [19]. 104 FQHE Hamiltonian theory this state in terms of a FQHE (a Laughlin state) of CF. The CF in the level p = 1 would bind to a CF vortex carrying two additional flux quanta, giving rise to a “second generation “ CF, see figure 6.3), in analogy with the CF formation in the electron basis and the pseudo-vortex of the electronic liquid. The Hamiltonian theory is a perfect framework to treat the case of a partial filling of interacting CF level. Formally we may use the same approximations as in the deduction of the CF model in the electron basis restricted to a single LL(section 4.1), i.e. only excitations in the same LL are taken into account, and inter CF LL excitations belong to a higher energy sector. This approximation is somewhat less justified in the last case because the only relevant energy scale is e2 /ǫlB , both in the CF LL formation and in the residual interactions. In fact the justification of this approximation will appear later on, due to the appearance of a “small parameter” due to the charge renormalisation. The restriction to the CF LL then yields for the density operator CF ¯ ρ̄CF (6.49) p (q) = Fp (q)ρ̄(q), where ρ̄¯(q) = X m,m′ hm|e−iq·R CF |m′ ic†p,m cp,m′ (6.50) is the projected density operator of CF, and FpCF (q) ≡ hp|ρ̄p (q)|pi, (6.51) in terms of matrix elements (6.42), is the “CF form factor” of level p. As for the electronic form factor(4.12), it can be absorbed in the effective CF interaction potential, which leads to h i2 CF vs,p (q) = v0 (q) FpCF (q) = ∗2 2 2πe2 −q2 l∗2 /2 q 2 lB c Lp e B ǫq 2 " (6.52) ! − c2 e−q 2 /2c2 Lp Then one finds for the interacting CF Hamiltonian 1 X CF v (q)ρ̄¯(−q)ρ̄¯(q). Ĥ CF = 2 q s,p ∗2 q 2 lB 2c2 !#2 . (6.53) As in the electronic case the commutator (6.39) between guiding center components R(CF ) induces the operator algebra for ρ̄¯(q), ∗2 (q ∧ k)lB ¯ ¯ [ρ̄(q), ρ̄(k)] = 2i sin ρ̄¯(q + k). 2 ! (6.54) Effective theory 105 This together with Hamiltonian (6.53), defines the interacting CF model. The latter has the same structure as that for electrons restricted to a single level– one has to replace the effective interaction potential by the potential CF ∗ vs,p (q) and to use the CF magnetic length, lB . This self-similarity at the level of the model structure can account, under certain conditions, for the experimental Hall curve self-similarity, which was initially noticed by Mani and v. Klitzing, using a scaling transformation [59]. The difference between the spatial variation of the interaction potentials between CF and between electrons in a LL n (4.18), indicates that this self-similarity of the Hall curve is not an automatic by-product of the mathematical self similarity of the model. Because of the latter, one expects the formation of incompressible quantum liquids which would be the CF analogues of Laughlin liquids. Formally, such liquids 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 vortices ¯ (q), which carry 2s̃ flux quanta, and which have a charge degrees of freedom χ̄ 2 ¯ (q)|ϕphys i = 0, for the c̃ = 2p̃s̃/(2p̃s̃ + 1). This leads to a new constraint , χ̄ physical states |ϕphys i. The pseudo-vortex operator components satisfy the algebra ! q ∧ k ∗2 ¯ (q), χ̄ ¯ (k)] = −2i sin ¯ (q + k), [χ̄ l χ̄ (6.55) 2c̃2 B induced by the commutation rules h i for the pseudo-vortex guiding center comv−CF v−CF v−CF ∗2 2 ponents R , X ,Y = −ilB /c̃ . 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 pre2 ¯ (q). The CF2 global charge is thus ferred combination, ρ̄C F (q) = ρ̄¯(q) − c̃2 χ̄ 2 ∗ 2 2 ẽ = (1 − c̃ )e = (1 − c̃ )(1 − c ), in units of the electron charge −e. The 2 2 CF2 cyclotron variable and guiding center, respectively η C F et RC F , are deduced from the CF guiding center and CF pseudo-vortex in the same manner as for CF [see equation (6.40)]. The new preferred combination, which is interpreted as the CF2 density, is written in second quantized notation 2 ρ̄C F (q) = X j,j ′ ;m,m′ hm|e−iq·R C2F |m′ ihj|ρ̃p (q)|j ′ id†j,m dj ′ ,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 operators in state |j, mi. The matrix elements in equation (6.56) are the same 106 FQHE Hamiltonian theory Pseudo−potentiels m V 0.4 0.40 0.30 0.3 0.03 électrons 0.02 0.2 0.004 0.01 0.1 0.002 FC FC2 1 23 5 47 9 m 11 6 13 15 8 17 19 10 Figure 6.4: Pseudo-potentials for the electronic interaction (black curve), for CF (gray curve) and for CF2 (clear gray curve ), in units of e2 /ǫlB . Notice the scale difference on the z axis. √ ∗ ∗ as q for CF [Eqs. (6.41) and (6.42)] if we replace lB → ˜lB = lB / 1 − c̃2 = 1/ (1 − c̃2 )(1 − c2 ), the CF2 magnetic length, and c2 → c̃2 . A Quantum Hall Effect would then be expected to appear for some CF filling factors ν ∗ = p + p̃/(2s̃p̃ + 1), where the integer p̃ is the number of completely filled CF2 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 factors through 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 next CF generations. One finds in that manner a hierarchy of states which is different from the Haldane and Halperin hierachies [37, 43], at filling factors which are determined by the recurrence relation νj = p j + νj+1 , 2sj+1 νj+1 + 1 (6.57) where sj+1 is the number of pairs of flux quanta carried by the pseudo-vortex in the (j +1)-th generation of CF (CFj+1 ), and pj is the number of completely filled FCj s levels. The FCj s IQHE is determined by νj = pj . Formally, the electronic filling factor corresponds to j = 0 and thus ν0 = ν and ν1 = ν ∗ . Equation (6.57) is a generalisation of the relation between electronic filling factors and those of CF [Eq. (4.68)]. Although the recurrence formula 6.57 for CF hierarchical states suggests a large number of FQHE, only a limited number are observable in practice. Effective theory 107 Indeed, the FQHE due to first generation CF is retsricted to the two lowest LL, n = 0 and 1, while in higher Landau levels, such liquid states compete with electronic solids, as will be discussed in the next chapter. Apart from this competition with other phases, one may state certain general stability criteria for higher generation CF (CFj+1 ). The first one is clearly the stability criterion for the ”parent” state, (CFj ), a necessary condition. For n = 2, for example, the Laughlin liquid is not stable at ν̄ = 1/3, and CF2 are not formed at ν̄ = 4/11. A more restrictive condition for the CFj+1 formation is given by the form of the effective interaction potential of FCj , (j) v{si ,pi } (q) = j h i2 2πe2 −q2 /2 Y i e FsCi ,pFi (qli ) , ǫq i=1 (6.58) √ in terms of FCj magnetic length lj = 2sj pj + 1lj−1 . Expanded in Haldane pseudo-potentials [see equation (4.54)], Vmj = X q (j) v{si ,pi } (q)Lm (q 2 lj2 )e−q 2 l2 /2 j , > 1, 2, for a the interaction must be sufficiently short ranged , i.e. V1 /V3 ∼ Laughlin state to be stabilized. Pseudo-potentials with odd indices 8 are plotted in figure 6.4 for electrons in LL n = 0, cf with s = p = 1 and CF2 with s = p = s̃ = p̃ = 1. Notice the difference in scale on the energy axis: the interaction between CF is roughly one order of magnitude smaller than that for electrons. This is easily understood when looking at the effective CF interaction potential (6.52), which is globally reduced by the CF form factor , [FpCF (q)]2 ≃ (1 − c2 )2 , at order O(q 0 ), compared to the potential between electrons. As two factors of this type enter the expression for the CF2 effective interaction potential [see equation (6.58)], the latter is again an order of magnitude smaller than that between CF. In this sense, (1−c2 )2 ≤ 1/9 may be interpreted as the hierachical CF theory small parameter – as discussed earlier, this is a posteriori an indication for the CF LL stability with respect to residual CF interactions. A second remark is about the specific form of the CF (and CF2 ) interaction. Contrary to the electronic case, their pseudo-potentials do not vary monotonically but exhibit a minimum at m = 3. A possible origin of this 8 Remember that only odd index pseudo-potentials matter in the case of fully spin polarised electrons, because of their fermionic statistics (see section 4.2). 108 FQHE Hamiltonian theory peculiarity is the CF dipolar character, due to their internal structure, as already discussed at the beginning of section 6.2. Since the pair correlation function of the Laughlin liquid (with s = 1) is maximum for that value of the relative kinetic moment, one may expect this pseudo-potential form to stabilise 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 separates LL 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 between the 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 due to localisation of additional electrons. The only difference would be the magnitude of the excitation gap. Indeed, since ∆z ≃ h̄ωC /70, the two spin branches are not resolved in weak magnetic fields, for which only the IQHE at ν = 2n is observed. In that case, the system behaves as if each state was doubly degenerate, with |n, m; σi (σ =↑, ↓). However this picture turns out to be incorrect. Interactions between electrons have important effects, even for ν = 2n + 1. A new form of quantum magnetism arises. That is the topic of 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 h̄eB/mb is the band mass, mb = 0.068m for GaAs in terms of the bare mass m of the electron. The latter determines the Zeeman gap ∆z = gh̄eB/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 g factor for GaAs is g = −0, 4, which causes the Zeeman gap in this material to be a factor roughly 70 smaller than the LL separation, as already mentionned earlier. Expressed in Kelvins, the Zeeman gap is ∆z = 0, 33B[T]K, while the LL separation is h̄ωC = 24B[T]K, where the magnetic field is measured in Tesla. On the other hand, the characteristic Coulomb energy is e2 /ǫlB = q 50 B[T]K. For a field 6T, which corresponds roughly to filling ν = 1, one finds thus ∆z ≃ 2K ≪ e2 /ǫlB ≃ 120K < h̄ωC ≃ 140K. Interactions are therefore of the same order of magnitude as the LL separation, and are more relevant than the Zeeman gap. They must be taken into account when discussing effects connected to the electron spin, in particular at ν = 1, which we will focus on in the remaining parts of this chapter. The first problem is to understand why we observe a Quantum Hall Effect at all at this filling factor. The Zeeman gap is so small that each state is almost degenerate, so we might expect a macroscopic degeneracy at ν = 1 in the kinetic part of the hamiltonian. Just as for the FQHE, interactions are responsible for the lifting of this degenaracy. So we are led to this counter intuitive idea that the IQHE at ν = 1 should rather be looked at as a special case of FQHE. 7.1.1 Wave functions Let us start with a two spin 1/2 particles wave function, at ν = 1. In the symmetric gauge, the orbital part is built from the single particle wave function, in the symmetric gauge φm (z) = z m , (neglecting normalisation factors) with m = 0, 1 (2.28). As for the spin function, we have four possible states for the coupling of two spin√1/2 particles: an antisymettric singlet , |S = 0, M = 0i = (| ↑↓i − | ↓↑)/ 2, and a symmetric triplet |S =√1, M i, avec |S = 1, M = 1i = | ↑↑i, |S = 1, M = 0i = (| ↑↓i + | ↓↑i)/ 2 and |S = 1, M = −1i = | ↓↓i. We are dealing with a problem without explicit spin-dependent potentials. The interaction is SU(2)invariant, the total spin is a good quantum number. Since the fermionic wave function must be antisymmetric, we have ψS=0 (z1 , z2 ) = φs (z1 , z2 ) ⊗ |S = 0, M = 0i ψS=1 (z1 , z2 ) = φa (z1 , z2 ) ⊗ |S = 1, M i, and Interactions are relevant at ν = 1 111 with φs (z1 , z2 ) = z10 z21 + z20 z11 = z1 + z2 and φa (z1 , z2 ) = z10 z21 − z20 z11 = z1 − z2 . The second choice with an antisymmetric orbital wave function is energetically favourable if the interaction is sufficiently strongly repulsive at short range, as is the case for the Coulomb interaction. Thus Coulomb interactions, combined with the Pauli principle, create an exchange force which 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 }) = Y i<j (zi − zj )e− P k |zk |2 /4 (7.1) is in fact the orbital part of a ferromagnetic N-particles wave function: (4.30), φν=1 ({zi }) = Y i<j (zi − zj )e− P k |zk |2 /4 | ↑, ↑, .... ↑i. (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 arises even if the Zeeman effect vanishes (this can actually be done experimentally by applying external pressure in Ga As samples.) Let us in fact imagine that to be the case. Then there would be no reason for the total spin to be along the z direction, because no external potential breaks the Hamiltonian SU(2) symmetry (if the Zeeman effect vanishes). The most general spin function describing the most general orientation for the total spin is ψs ({θ, ϕ} ) = Y m ! θ θ cos e−iϕ/2 e†m,↑ + sin eiϕ/2 e†m,↓ |0i, 2 2 (7.3) where e†m,σ creates an electron in the state |n = 0, m; σi. We have chosen the parametrisation in terms of two angles, θ et ϕ which respects the normalisation |σi = u| ↑i+v| ↓i avec |u|2 +|v|2 = 1, in order to establish the connection between the SU(2) and O(3) description of the rotation group. This allows to introduce immediately the magnetisation field at the m Landau site: sin θm cos ϕm n(m) = sin θm sin ϕm , cos θm (7.4) This will be useful to describe the low energy degrees of freedom within the effective model (section 8.3). 112 Spin and Quantum Hall Effect – Ferromagnetism at ν = 1 The wave function in equation (7.2) corresponds to θ = 0. It has a quantum 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, characterized 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 by an “exchange hole”, due to the Pauli principle when all spins are aligned. This lowers the Coulomb energy per particle. For filling factor ν = 1 hVCoulomb i π e2 =− ≈ 200K N 8 ǫlB r (7.5) This is an order of magnitude larger than the Zeeman splitting and is the mechanism which strongly stabilizes the ferromagnetic state, and would do so even if the Zeeman effect was zero. Even though the same mechanism is at work in all simple and transition metals, the latter are not all ferromagnets, because the kinetic energy dispersion relation opposes the ferromagnetic polarisation: the broader the band, 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 a ferromagnetic state. In the completely filled Landau level, the kinetic energy is frozen; there is no kinetic energy cost in spin polarizing the interacting electron gas. This is why the ν = 1 IQH state is the “best understood itinerant ferromagnetic state” in condensed matter physics. A last remark before going over to the next section: the introduction of the spin degree of fredom is a special aspect of a more general problem: that of the QHE in multicomponent sytems [74]. Multi-component systems include systems with spin degrees 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 single level , as introduced in section 4.1, to include an internal degree of freedom with 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 v0σσ (q). For a global rotation symmetry, v0↑↑ (q) = v0↓↓ (q) et v0↑↓ (q) = v0↓↑ (q). The former more general interaction which may break the SU(2) symmetry will be useful in the next chapter, where the layer index in a bi-layer will be viewed as an iso-spin index s = 1/2, with | ↑i for a state in the upper layer and | ↓i for the lower layer. The interaction Hamiltonian writes now H= 1 X X σ,σ′ v (q)ρ̄σ (−q)ρ̄σ′ (q), 2 σ,σ′ q 0 (7.6) where2 ρ̄σ (q) = X fm,m′ (q)e†m,σ em′ ,σ (7.7) m,m′ is the electron density with spin σ projected on the LLL , with fm,m′ (q) = hm|f (q)|m′ i the matric element for the projected density operator matrix element f (q) = exp(−q · R), in first quantised forme [see equation (4.7)]. The total electron density is thus written ρ̄(q) = ρ̄↑ (q) + ρ̄↓ (q) X X fm,m′ (q)δσ,σ′ e†m,σ em′ ,σ′ , = (7.8) σ,σ ′ m,m′ where δσ,σ′ represents the identity 12×2 . In comparison with the case without internal 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 relevant for the second quantised notation. Similarly, we find the spin densities by replacing the identity by the SU(2) generators S µ = τ µ /2, where τ µ are the Pauli matrices ,3 with [τ µ , τ ν ] = iǫµνσ τ σ /2 and (τ µ )2 = 1. We define f µ (q) = f (q) ⊗ S µ , 1 (7.9) We have already seen that the interaction potential in a LL with arbitrary n is obtained by replacing the gaussian by the form factor, 4.12, exp(−q 2 /2) → [Fn (q)]2 , in the expression for the effective potential 2 We omit for simplicity the index n = 0 in the electron operators for the LLL. 3 The 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 rule for 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 is repeated. 114 Spin and Quantum Hall Effect – Ferromagnetism at ν = 1 and the spin densities are written, in second quantized form, as S̄ µ (q) = X X h σ,σ ′ m,m′ i µ † fm,m′ (q) ⊗ Sσ,σ ′ em,σ em′ ,σ ′ (7.10) In the case with no internal degree of freedom, the algebraic structure was derived from equation (4.13) using the commutation rules in first quantisation q ∧ q′ q ∧ q′ [f (q), f (q )] = 2i sin f (q+q′ ) → [ρ̄(q), ρ̄(q′ )] = 2i sin ρ̄(q+q′ ). 2 2 ′ Now using the same procedure, we have to compute [S̄ µ , ρ̄(q′ )] and [S̄ µ (q), S̄ ν (q)],using [f µ (q), f ν (q′ )] = f (q)f (q′ ) ⊗ S µ S ν − f (q′ )f (q) ⊗ S ν S µ (7.11) 1 ([f (q), f (q′ )] ⊗ {S µ , S ν } + {f (q), f (q′ )} ⊗ [S µ , S ν ]) = 2 and [f µ (q), f (q′ )] = [f (q), f (q′ )] ⊗ S µ . We have (7.12) q ∧ q′ [f (q), f (q )] = 2i sin f (q + q′ ), 2 ! q ∧ q′ ′ f (q + q′ ) {f (q), f (q )} = 2 cos 2 ! ′ and [S µ , S ν ] = iǫµνσ S σ , 1 µν {S µ , S ν } = δ , 2 which yields q ∧ q′ ρ̄(q + q′ ), (7.13) [ρ̄(q), ρ̄(q )] = 2i sin 2 ! q ∧ q′ µ ′ [S̄ (q), ρ̄(q )] = 2i sin S̄ µ (q + q′ ) and (7.14) 2 ! ! ′ ′ q ∧ q q ∧ q i δ µν sin ρ̄(q + q′ ) + iǫµνσ cos S̄ σ (q + q′ ). [S̄ µ (q), S̄ ν (q′ )] = 2 2 2 (7.15) ′ ! 7.3. EFFECTIVE MODEL 115 The equations (7.13)-(7.15) are the SU(2) extensions of the magnetic translation algebra (4.16). The Hamiltonian (7.6) is written H= X 1X vSU (2) (q)ρ̄(−q)ρ̄(q) + 2 vsb (q)S̄ z (−q)S̄ z (q), 2 q q (7.16) with potentials i 1 h ↑↑ v0 (q) + v0↑↓ (q) 2 i 1 h ↑↑ v0 (q) − v0↑↓ (q) . 2 (7.17) The first term in the Hamiltonian (7.16) is SU(2) invariant, while the second one, if non zero, breaks explicitly the SU(2) symmetry. In the remaining parts of this chapter we discuss ferromagnetism in the physical spin case. The interaction is then SU(2) invariant. Only the (small) Zeeman term vSU (2) (q) = HZ = et vsb (q) = gh̄eB z S̄ (q = 0) m breaks the Hamiltonian SU(2) symmetry. Equation (7.14) exhibits a remarkable property: because of the non commutativity between spin and charge densities, the dynamics of both degrees of freedom are coupled. One can handle spins by acting on charges, and vice versa! This unusual property is connected to the quantum dynamics under magnetic field, and has a simple expression because of the projection on a single LL, which results in non commutativity of charge density operators with non parallel wave vectors [ equation (7.13)]. Spin-charge entanglement in 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 have a non zero topological charge (skyrmions) [figure 7.1(b)]. While spin waves are the Goldstone modes, the energy of which goes to zero, in the limit of no Zeeman effect, when the wavelength goes to infinity, skyrmions cannot be continuously deformed to retrieve the ground state. They are topologically stable objects, classified by an integer which is called their topological charge. 116 Spin and Quantum Hall Effect – Ferromagnetism at ν = 1 z (a) y x (b) z y x Figure 7.1: Excitations in the ferromagnetic state . (a) Spin waves (magnons). Such an excitation can be continuously deformed to the ground state with the same topological property of the magnetisation field, as can be seen on the Bloch sphere (right) which represents a mapping of the spin configuration (bold face spins) in the plane on S2 – the gray line can be continuously deformed to a point (b) Skyrmion with non zero topological charge. This excitation has a flipped spin at the origin , r = 0, and the ferromagnetic state is recovered at infinity |r| → ∞. Contrary to spin waves the mapping of this spin configuration on the sphere (bold face spins)covers the whole S2 surface once. Effective model 7.3.1 117 Spin waves The procedure to derive the spin wave spectrum is analogous to well known other cases in localized or itinerant electron ferromagnets: one studies the time evolution of the spin lowering operator: N X Sq− ≡ µ e−iqµ rj Sj− (7.18) j=1 where rjµ 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 we obtain i X h − S̄q− = fm,m′ (q) ⊗ S↓,↑ e†m,↓ em′ ,↑ (7.19) m,m′ Then one computes the commutator of this operator with the Coulomb interaction Hamiltonian: h i H, S̄q− = (1/2) X k6=0 h v(k) ρ̄(−q)ρ̄(q), S̄q− i (7.20) This is evaluated using the familiar commutator algebra. When applied to the ground state, which is annihilated by ρ̄k one obtains where h i H, S̄q− |ψi = ǫq S̄q− |ψi ǫq ≡ 2 X − e |k|2 2 v(k) sin 2 k6=0 (7.21) ! q∧k . 2 This proves that S̄q− is an exact excited eigenstate of H with excitation energy ǫq . In the presence of the Zeeman coupling, ǫq → ǫq + ∆. The only assumption is that the ground state at filling factor ν = 1 is fully polarised. The dispersion relation is quadratic in q at small q: ǫ q ≈ ρs q 2 with ρs ≡ 1 X − |k|2 e 2 v(k)|k|2 2 k6=0 For very large q, sin2 can be replaced by its average value 1/2 so that ǫq ≈ X v(k)e− |k|2 2 k6=0 The energy saturates at a constant value for q → ∞. 118 7.3.2 Spin and Quantum Hall Effect – Ferromagnetism at ν = 1 Skyrmions For simplicity reasons, let us first discuss a topological excitation in a 2D XY ferromagnet (spins are constrained to lie in a plane). In that case a topological excitation is a vortex type defect which is labelled by the number of times the spin direction rotates around the origin along a closed path encircling the defect placed at the origin (figure 7.2). This number is the topological charge, which does not depend on the path geometry, only on its homotopy class. Consider the mapping of the circle S 1 , which parametrizes the path in the physical plane around the defect we want to characterize, on the circle S 1 which parametrizes the spin orientation in the 2D spin space. The mappings which can be continuously deformed into one another form homotopy 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 spin texture maps on the surface of the Bloch sphere, S 2 . The 2D plane is mapped by stereographic projection on a S 2 sphere. Just as for the case of the S 1 → S 1 mappings, the S 2 → S 2 mappings which can be continuously deformed into one another can be classified in homotopy classes, which form a homotopy group, π2 (S 2 ) = Z. Elements of the group, integers Q ∈ Z label different topological sectors. Spin waves deform continuously to the ground state, they belong to the topological sector with topological charge Q = 0, while skyrmions carry a topological charge Q 6= 0 (figure 7.1). A state with a certain spin texture |ψ[n(r)]i may be generated from the ferromagnetic ground state by applying a spin rotation operator, using the generators (7.10) of the magnetic translation algebra (with internal SU(2) structure) [87] " |ψ[n(r)]i = exp −i X q # Ωµ−q S̄ µ (q) |ψF M i, (7.22) where |ψF M i is the ferromagnetic state(7.2), with a uniform magnetization along the z quantization axis. The functions Ωµq which enter expression (7.22) are, up to a permutation of the x and y axis, the Fourier transforms of n(r), Ω(r) = X q Ωq e−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 the number of times the spin direction turns by 2π along a path (gray curve) which circles around 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 : topological excitation 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 for the transform of operator S̄ µ (q) eiŌ S̄ µ (k)e−iŌ = S̄ µ (k) + δSkµ , where we have defined Ō = P q (7.24) Ωµ−q S̄ µ (q) et ii 1h h (7.25) Ō Ō, S̄ µ (k) + ... 2 If we limit the Hausdorff series expansion to second order, we find for the average magnetization in the spin texture (7.22) nel µ n (r), S µ [n(r)] = hS µ (r)i + hδS µ (r)i = 2 as expected for the O(3) magnetization field. This justifies the choice of the function Ω(r) [equation (7.23)]. The technical details for the computation leading to this result are not given here. They are closely analogous to those we present in the following section for the computation of the charge density induced by the spin texture. h i δSkµ = i Ō, S̄ µ (k) − 7.3.3 Spin-charge entanglement For a better understanding of the spin-charge entanglement, which we mentionned above in the discussion of the model algebraic structure (section 7.2), 120 Spin and Quantum Hall Effect – Ferromagnetism at ν = 1 we now compute the charge density change due to the spin texture (7.22), iiE 1 Dhh Ō, Ō, ρ̄(q) , (7.26) 2 where averages are defined with respect to the ferromagnetic ground state with magnetization along z. Averages are computed using the commutation rules (7.13)-(7.15) and δρq ≡ heiŌ ρ̄(q)e−iŌ i − hρ̄(q)i ≃ i iE Dh Ō, ρ̄(q) − hρ̄(q)i = nel δq,0 et n el hS̄ µ (q)i = δ µz δq,0 . 2 (7.27) (7.28) One finds thus h i Ō, ρ̄(q) X = k = 2i h X k iE Dh Ō, ρ̄(q) ⇒ i Ωµ−k S̄ µ (k), ρ̄(q) ! k∧q sin Ωµ−k S̄ µ (q + k) 2 = 0, since the argument of the sine vanishes for q k k, and Ō Ō, ρ̄(q) h ′ i ′ k∧q = 2i sin Ωµ−k′ Ωµ−k S̄ µ (k′ ), S̄ µ (q + k) 2 k,k′ X = − X sin k,k′ µ′ µν + 2ǫ ′ k′ ∧ (q + k) k∧q ′ Ωµ−k′ Ωµ−k δ µµ sin ρ̄(q + k + k′ ) 2 2 k′ ∧ (q + k) cos S̄ ν (q + k + k′ ) . 2 So the average is Dh h iiE Ō Ō, ρ̄(q) = −nel ǫ µ′ µz X k ! ′ k∧q Ωµk+q Ωµ−k . sin 2 we get thus for the modified charge density (7.26), with ν = 2πnel , δρq ! ′ ν X µ′ µz k∧q = ǫ sin Ωµk+q Ωµ−k 4π k 2 ′ −ν X µ′ µz ǫ (q ∧ k)Ωµk+q Ωµ−k ≃ 8π k ′ −ν X µ′ µz = ǫ [i(k + q)] Ωµk+q ∧ (−ik)Ωµ−k , 8π k (7.29) Effective model 121 where we have expanded the sine in the second line, thus resorting to a long wavelength limit, i.e. a slow space varying modulation of the spin density. The Fourier transform back to real space yields the more compact result −ν µ′ µz (2) µ′ ǫ ∇ Ω (r) ∧ ∇(2) Ωµ (r) 8π −ν ij = ǫ n(r) · [∂i n(r) × ∂j n(r)] , 8π δρ(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). Compare the result (7.30)to the so-called Pontryagin topological charge density ( Pontryagin index) δρtop (r) = 1 ij ǫ n(r) · [∂i n(r) × ∂j n(r)] . 8π (7.31) We see that the electric charge density is proportional to the Pontryagin index, δρ(r) = −νδρtop (r). The topological charge is obtained by integration over the physical plane Z d2 r 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 at ν = 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 a spin structure. The energy of the state (7.22) with a O(3) magnetization field n(r), is computed in the same manner as the charge modulation, using the long wavelength expansion, δE = i Dh ≃ − Ō, H iE − iiE 1 Dhh Ō, H + ... 2 ′ 1 X v0 (k)Ωµ−k′ Ωµ−k hCi, 4 q,q′ ,k (7.34) 122 Spin and Quantum Hall Effect – Ferromagnetism at ν = 1 where we have defined h ′ i n h ′ io C = 2 S̄ µ (q′ ), ρ̄(−k) S̄ µ (q), ρ̄(k) + ρ̄(−k), S̄ µ (q′ ), S̄ µ (q), ρ̄(k) q′ ∧ k q∧k ′ sin S̄ µ (q′ − k)S̄ µ (q + k) 2 2 ′ q ∧ (q + k) q∧k ′ sin ρ̄(−k), ρ̄(q′ + q + k) −δ µµ sin 2 2 ′ q ∧ (q + k) q∧k µ′ µσ cos ρ̄(−k), S̄ σ (q′ + q + k) . −2ǫ sin 2 2 = 8 sin The average of this expression is computed, with the help of the structure factor 1 hρ̄(−q)ρ̄(q′ )i = δq,q′ s̄(q) (7.35) nel and of δ µz 1 (7.36) δq,q′ s̄(q) hρ̄(−q)S̄ µ (q′ )i = nel 2 and ′ 1 µ δ µµ δ µz µ′ ′ δq,q′ s̄(q). (7.37) hS̄ (−q)S̄ (q )i = nel 4 Finally we find hCi = 2nel δµ,µ′ δq,−q′ sin 2 q′ ∧ k [s̄(q) − δ µz s̄(q + k)] . 2 ! So we find for the energy (7.34) q′ ∧ k nel X v0 (k)s̄(k) sin2 Ωµ−q Ωµq . δE = − 2 q,k 2 ! (7.38) In the long wavelength limit, the sine can be linearized, so we get the non linear O(3) sigma model δE = ρS X ρS Z 2 (−iq)Ωµ−q (iq)Ωµq = d r[∇n(r)]2 , 2 q 2 (7.39) where the exchange stiffness ρS = − ν Z∞ dkk 3 v(k)s̄(k), 32π 2 0 (7.40) 7.4. BERRY PHASE AND ADIABATIC TRANSPORT is ρS = 1 e2 √ 16 2π ǫlB 123 (7.41) at ν = 1, since s̄(k) = −1. Any magnetization variation incresases the energy with respect to the ground state, which indeed corresponds to ∇n(r) = 0, i.e. a ferromagnetic state with uniform magnetization. The energy dispersion at small wave vector ρS (7.42) ω(q) ≃ q 2 2 is precisely the Goldstone mode (spin wave) energy discussed earlier. The complete analytic expression for the latter is [88] [89] ω(q) = r q2 π e2 2 1 − e−q /4 I0 2 ǫlB 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 the Quantum Hall ferromagnet, have been discussed on the basis of the Hamiltonian theory, using the commutator algebra of the projected density operators. A topological charge density (the Pontryagin density) is associated to spin texture. The properties of the homotopy group Π2 (S2 ) ≡ Z indicate that the electric charge carried by a topological defect is an integer in terms of 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. Consider a quantum system, described by a Hamiltonian HR which depends on a set of external controlable parameters. This set is represented by a vector in parameter space R. We assume now that there exists a compact domain in parameter space where the ground state is separated of all excited states by a gap. What is the result of letting the system evolve slowly, with a slow variation of R(t) along a closed loop in this part of parameter space, during a 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 evolves adiabatically. This means that at all times, the system remains in the ground (0) state ΨR of the Hamiltonian HR(t) . Each point R in parameter space is associated to a complete set of eigen states: (j) (j) (j) HR ΨR = ǫR ΨR . (7.45) The solution of the time dependent Schrodinger equation ih̄ ∂Φ(r, t) = HR(t) Φ(r, t) ∂t is then: − h̄i (0) Φ(r, t) = ΨR(t) (r)eiγ(t) e Rt 0 dt′ ǫ0R(t′ ) + (7.46) X (j) aj (t))ΨR(t) (7.47) j6=0 The adiabatic approximation amounts to neglecting the contribution of excited states represented by the second term on the right hand side. This becomes exact in the limit of of a very slow variation of R(t) as long as the excitation gap remains finite. Everything is well known at this point, except the ”Berry Phase” γ(t). γ(t) can be determined by requiring Φ(r, t) to obey the time dependent Schrödinger equation. The LHS of equation 7.46, if we neglect the aj (t) when j > 0 becomes: " # i i ∂Φ(r, t) h ∂ (0) (0) iγ(t) − h̄ ih̄ = −h̄γ̇(t) + ǫR(t) Φ(r, t) + ih̄Ṙµ Ψ (r) e e R(t) ∂t ∂Rµ Rt 0 (0) dt′ ǫR(t′ ) (7.48) The RHS of equation 7.46, within the same approximation is: (0) HR(t) ~ Φ(r, t) = ǫR(t) Φ(r, t) (7.49) Using the completeness relation * + + ∞ ∂ E X ∂ (j) (0) (0) (j) ΨR(t) | µ ΨR(t) . ΨR (t) = Ψ ∂Rµ R ∂R (7.50) j=0 Here again the adiabatic approximation allows to neglect the contribution of excited states, so that equation 7.48 becomes: " * + # ∂Φ ∂ (0) (0) (0) ih̄ = −h̄γ̇(t) + ih̄Ṙµ ΨR (t)| µ ΨR(t) + ǫR(t) Φ ∂t ∂R (7.51) Berry phase and adiabatic transport 125 Equation 7.46 is satisfied if γ̇(t) = iṘ D µ * (0) ΨR(t) | (0) (0) ∂ (0) ΨR(t) µ ∂R + (7.52) E Thanks to the constraint ΨR |ΨR = 1 we know that γ̇ is real. At first sight γ̇(t) could be chosen to vanish. In fact, for each R we have a different set of eigen functions, and one may choose the ground state phase arbitrarily. That is a kind of gauge choice in parameter space: γ̇(t) and γ in that sense, are not gauge invariant. Chosing γ̇ = 0 is then a gauge choice. What Berry [90] found is that this is not always possible. In certain cases implying a closed path in parameter space, there is a finite gauge invariant phase, the Berry Phase, γBerry ≡ Z T 0 γ̇dt = i I Γ dR µ * (0) ΨR | + ∂ (0) ΨR . µ ∂R (7.53) That quantity is ”gauge invariant” because the system returns to the departure point in parameter space, so that the arbitrary choice of gauge at the start has no consequence. This is analogous to electrodynamics when the circulation of the vector potential is along a closed path, and equals the enclosed magnetic flux, which is gauge invariant. In fact it is useful to define the ”Berry connection” , A, in parameter space: µ A (R) = i * γBerry = I which leads to (0) ΨR | Γ ∂ (0) ΨR µ ∂R dRµ · Aµ (R) + (7.54) (7.55) The Berry phase is a purely geometric object, independent of the velocity Ṙµ (t). It only depends on the path in parameter space. It is often easiest to evaluate this expression using Stoke’s theorem, since the curl of A is gauge invariant. It is easy to check that in the case of electromagnetism and the AharonovBohm effect, the Berry connection, A is, up to a multiplicative factor, h̄q (where q is the particle charge), the electromagnetic vector potential A: q Aµ (R) = + Aµ(R) h̄ (7.56) 126 Spin and Quantum Hall effect – ferromagnetism at ν = 1 The Berry phase for a loop threaded by a magnetic flux φb is γBerry = qI φb dRµ Aµ = 2π h̄ 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 circuit in parameter space The gap to the first excited state is h̄|∆|. ~ = 0 where the spectrum has a degeneracy. During must avoid the origin ∆ the adiabatic evolution of the ground state, one has D (0) (0) ~ Ψ∆ ~ |S|Ψ∆ ~ E = h̄S ~ ∆ ~ |∆| (7.59) ~ is defined by the polar angle θ and the Thus if the orientation of ∆ ~ >. For a spin S = 1/2, an azimuthal angle φ, the same must be true for < S appropriate set of states is: cos 2θ sin 2θ eiφ |Φθ,φ >= ! (7.60) since these obey: θ θ hΦθ,φ |S |Φθ,φ i = h̄S cos − sin2 2 2 z 2 and D ! = h̄S cos θ E hΦθ,φ |S x + iS y |Φθ,φ i = Φθ,φ |S + |Φθ,φ = h̄S sin θeiφ (7.61) (7.62) ~ around axis What is the Berry phase in the case of a slow rotation of ∆ ẑ, at constant θ? γBerry = i =i Z 0 2π dφ cos θ 2 sin θ −iφ e 2 Z 0 2π * ∂ dφ Φθ,φ | Φθ,φ ∂φ 0 i sin 2θ eiφ ! = −S + Z 0 2π (7.63) dφ(1 − cos θ) (7.64) 7.5. APPLICATIONS TO QUANTUM HALL MAGNETISM = −S Z 0 2π dφ 1 X cos θ d cos θ′ = −SΩ 127 (7.65) where Ω is the solid angle subtended by the path as viewed from the origin of the parameter space. This is precisely the Aharonov-Bohm phase one expects for a charge −S particle traveling on the surface of a unit sphere surrounding a magnetic monopole. The degeneracy of the spectrum at the origin is precisely the cause for presence of the magnetic monopole [90] The definition of the connection A implies the existence of a singularity at the south pole, θ = π. A ”Dirac string” (i.e. an infinitely thin solenoid carrying one flux quantum) is attached to the monopole. The singularity would 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 the spin the Hamiltonian of which is given by 7.58, the Lagrangian must be: n o L = h̄S −ṁµ Aµ + ∆µ mµ + λ(mµ mµ − 1) (7.67) where m is the spin coordinate on the unit sphere, λ is a Lagrange multiplier which enforces the length constraint, and the Berry connection A obeys: ~ ×A=m ∇ ~ (7.68) This Lagrangian reproduces correctly the equations for the spin dynamics which describe its precession. 7.5 7.5.1 Applications to quantum Hall magnetism Spin dynamics in a magnetic field In the following, we show that the Lagrangian above allows to describe the quantum spin dynamics in an effective field. The equations of motion are: Using 7.67 we have d δL δL = µ dt δ ṁ δmµ (7.69) δL = −Aµ δ ṁµ (7.70) 128 and Spin and Quantum Hall effect – ferromagnetism at ν = 1 δL = −ṁν ∂µ Aν + ∆µ + 2λmµ , δmµ (7.71) ∆µ + 2λmµ = Fµν ṁν , (7.72) so that where Fµν = ∂µ Aν − ∂ν Aµ The previous section on the Berry phase shows we must chose: Fµν = ǫαµν mα (7.73) ~m which is equivalent to ∇ ~ ∧ A[m] = m. The equation of motion becomes: δµ + 2λmµ = ǫαµν mα ṁν (7.74) Multiplying both members of equation 7.74 by ǫγβµ mβ , then applying on both sides of this equation the identity: ǫναβ ǫνλη = δαλ δβη − δαη δβλ , we get: ~m ~ − ∆∧ γ = ṁγ − mγ (ṁβ mβ ) (7.75) The last term vanishes, because of the constraint on the length of m. Using Euler-Lagrange equations, we retrieve the spin precession equations in a magnetic field. Compare 7.67 with the Lagrangian of a particle of mass m, and charge −e in a magnetic field with vector potential A: 1 L = mẋµ ẋµ − eẋµ Aµ 2 (7.76) We see that the Lagrangian in 7.67 is equivalent to a Lagrangian of a zero mass object, with charge −S, placed at m, ~ moving on a unit sphere containing a magnetic monopole. The Zeeman term is analogous to a constant electric ~ which exerts a force S ∆ ~ on the particle. The Lorentz force due to field −∆, the monopole field drives the particle on a constant latitude orbit on the unit sphere. The absence of a kinetic term in ṁµ ṁµ in the Lagrangian indicates that 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 an electron 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 has a velocity ẋµ , the variation rate of the local spin orientation seen by the electron is ṁν = ẋµ ∂x∂µ mν . This induces a non trivial Berry phase in the presence of a spin texture. Indeed, the one particle Lagrangian contains an additional term with a time derivative of first order, wich adds to the term due to the field-matter minimal coupling term: L′ = −eẋµ Aµ − h̄S ṁν Aν [m] ~ (7.77) L′ = −eẋµ (Aµ + aµ ) (7.78) The first term is the field-matter coupling, the second one gives rise to the Berry phase. We have for the latter ∇m ∧ A = m. ~ That can be re-written, ∂ ν ν µ using ṁ ≡ x˙ ∂xµ m . Then with ! ∂ ~ mν Aν [m] 2πa = φ0 S ∂xµ µ (7.79) a is the Berry connection, a vector potential which adds to the magnetic field vector potential. The curl of a thus contributes a ”Berry” flux which adds to the magnetic field flux: b = ǫαβ ∂aβ ∂xα (7.80) ! 1 ∂ ∂ ~ mν Aν [m] α β ∂x ∂x 2π ! ∂ ∂ ~ mν Aν [m] = φ0 Sǫαβ [ α β ∂x ∂x ! ∂ ∂mγ ∂Aν ν + m ] ∂xβ ∂xα ∂mγ = (φ0 S/2π)ǫαβ The first term of the last equation vanishes by symmetry, which results in: b = φ0 Sǫαβ ∂mν ∂mγ (1/2)F νγ ∂xβ ∂xα (7.81) with F µν = ǫαµν mα . We used the symmetry ν ↔ γ in the last surviving term . With S = 1/2 one gets b = φ0 ρ̃ (7.82) 130 Spin and Quantum Hall effect – ferromagnetism at ν = 1 with 1 αβ abc a ǫ ǫ m ∂α m b ∂β m c 8π 1 αβ = ǫ m ~ · ∂α m ~ ∧ ∂β m ~ 8π ρ̃ ≡ (7.83) (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 then causes this spin texture to attract (or repel) at the end of the process a charge density ν ρ̃. Since the skyrmion topological charge is an integer, R Qtop = d2 rρ̃(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 in bi-layers 8.1 Introduction In the previous chapters, we have emphasized the importance of Coulomb interactions in the Quantum Hall Effect physics, including for the ν = 1 filling factor of the LLL. Even if the Zeeman coupling vanishes, Coulomb interactions stabilize a ferromagnetic order, which has important consequences on the 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 rôle of spin is played by the isospin index of each layer [87, 93, 94]. The analogy with the ferromagnetic monolayer system at ν = 1 will be extensively used in the following. Quantum Hall bilayer physics is quite rich, and involves coupling between layers at various equal, or different, filling factors. This chapter focuses on the particular case of two layers at ν = 1/2, for which exciting results have been obtained in the last few years. Modern MBE techniques have allowed in the recent years to manufacture 2D 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 electron gases organized in parallel layers, at a distance d from each other which is comparable with the magnetic length, and to the average distance between electrons in the layer (i.e. d ∼ 10nm ). We know that correlations 131 132 Quantum Hall Effect in bi-layers W W 2t d Figure 8.1: Sketch of the conduction band profile for a two dimensional electrons system in a bi-layer. The order of magnitude for the width, as well as for the distance, of the two layers is W ∼ d ∼ 10nm. In the presence of a tunneling term t, the band splits into a bonding, 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 oppose the ferromagnetic polarisation. The FQHE results from gap formation between the ground state and excited states, resulting in an incompressible state. Theory predicts that gaps appear for certain fractional fillings in the bilayer system when inter-layer interactions are strong enough [19, 96]. Such predictions have been backed by experiments [97]. Recently [98], theory has predicted that inter-layer correlations could induce unusual broken symmetry states, with a new type of inter-layer coherence. This new interlayer coherence appears even in the absence of inter-layer tunneling, when the coupling between layers is of purely Coulombic origin. What appears here is excitonic superfluidity, which is the unexpected realization, in the bi-layer physics, of the phenomenon of excitonic superfluidity predicted in 3D semiconductors since 1962 [99]. This phenomenon has been looked for without unquestionable experimental success ever since [100] until it eventually 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 8.2 133 Pseudo-spin analogy We assume that the Zeeman effect saturates the “real” spin, so that it does not play any rôle 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 pseudospin order, in a direction defined by the polar angle θ, and azimuthal angle φ. In the Landau gauge, such a state writes [see also equation (7.3)] |ψi = Y k ! θ θ cos c†k↑ + sin eiφ c†k↓ |0i . 2 2 (8.1) In the state described by this wave function, each state k is occupied by one electron, and has an amplitude cos(θ/2) to be in the ↑ layer, and amplitude sin(θ/2) exp(iφ) to be in layer ↓. Physically the ratio between the squared amplitude may be altered by applying a voltage between the layers, so as to charge one layer at the expense of the other one, the total filling factor remaining equal to 1. Remember that in the Landau gauge, a state k labels 2 a state localized on a line at guiding center position Xk ≡ klB . We discuss the following cases: Spins along the z axis. For θ = 0 this wave function describes a spin alignment along the ẑ axis, |ψz i = Πk c†k,↑ |0i = Πi<j (zi − zj ) |↑, ↑, ... ↑i . The choice θ = 0 describes a situation where all particles are in the layer labelled by ↑. Spins along the x̂ axis. One has θ = π/2, φ = 0, which yields the state c† + c† |ψx i = Πk k↑√ k↓ |0i . 2 Obviously, this wave function which describes a symmetric superposition of electronic states in the two layers must have a low energy, compared to θ = 0, as soon as the Coulomb energy plays a rôle, as we shall see later on, and when both 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 c† + eiφ c† |ψxy i = Πk k↑ √ k↓ |0i . 2 That state, as the previous one describes a symmetric superposition of states with equal amplitude in each layer, but the resulting pseudo-spin magnetization 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 index for each electron is undetermined except when θ = 0. The amplitude for an electron 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 weight remains equal to 1, since sin2 (θ/2) + cos2 (θ/2) = 1. Even in the absence of quantum tunneling between layers (i.e. physical transfer of electron between layers), quantum mechanics, with the superposition principle, allows to describe the possibility of the simultaneous presence of an electron in both layers. In the ferromagnetic monolayer situation (ν = 1), we have seen in the previous chapter that in the absence of Zeeman coupling, the ferromagnetic exchange coupling, due to its Coulomb origin, does not break the Hamiltonian SU(2) symmetry. A different situation arises in the bi-layer case. In the next section, we list the various physical parameters which give its originality to the bi-layer pseudo-spin ferromagnetism 8.3 Differences with the ferromagnetic monolayer 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), interactions between electrons in one layer and electrons in the other are negligible. The layers are single layers with filling ν = 1/2, there are two flux quanta per electron. A Composite Fermion construction, whereby two flux quanta are attached to each electron (see chapter 4 and 6 and reference [16]) results in a problem where CF evolve in zero effective 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 CF carries a dipole [105]. Although this an interesting object in its own right, we are not going to discuss this limit in these lectures. In any case, there is a continum of particle-hole excitations above the Fermi liquid ground state, and there is no QHE • At a distance comparable to the magnetic length, i.e. d ∼ lB , intralayer and inter-layer Coulomb interactions, although different, become comparable, especially if d → 0, in which case they become equal. Using the pseudo-spin analogy, we can write the interaction Hamiltonian [see equation (7.6) in the previous chapter] Hcoul = 1 X X σ,σ′ v (q)ρ̄σ (q)ρ̄σ′ (−q). 2 σ,σ′ q 0 (8.2) we have therefore v0↑↑ (q) = v0↓↓ (q) ≡ v A (q) v0↑↓ (q) = v0↓↑ (q) ≡ v E (q) where v A (q) [v E (q)] is the Fourier transform with respect to the planar coordinates of the intra-layer [inter-layer ]interaction between electrons. Neglecting the physical width of the sample , we have v A (q) = (2πe2 /q) exp(−q 2 /2) et v E (q) = e−qd v A (q). Letting vSU (2) (q) = [v A (q) + v E (q)]/2 and vsb (q) = [v A (q) − v E (q)]/2, one may separate, in the interaction Hamiltonian, a part which is independent of the pseudo spin, v0 (q), and one which is not , as we have in equations (7.16) and (7.17) of the previous chapter. One finds then H= X 1X vSU (2) (q)ρ̄(−q)ρ̄(q) + 2 vsb (q)S̄ z (−q)S̄ z (q), 2 q q h (8.3) i with S̄ z (q) = [ρ̄↑ (q) − ρ̄↓ (q)]/2. Note that H, S̄ µ (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 v A (q) > v E (q), for all wave vectors, the second term in equation (8.3) creates an easy magnetization plane, perpendicular to ẑ, in the bi-layer plane. The pseudo-spin Hamiltonian symmetry is reduced from SU(2) to U(1). 136 Quantum Hall bi-layers • A third difference with the mono-layer at ν = 1 arises from the interlayer tunnelling term. This term has the following expression: T = − t X † ck,↑ ck,↓ + c†k,↓ ck,↑ 2 k = −t X S̄kx . (8.4) (8.5) k It acts as a pseudo-magnetic field applied along x̂. It stabilises the pseudo-spin orientation in direction x̂. Indeed the symmetric combination of states in each potential √ well (bonding combination) corresponds −x would to the spinor ψb = (1, 1)/ 2 (bonding state). The direction √ correspond to an anti-bonding state, ψa−b = (1, −1)/ 2 It is feasible experimentally to have widely different values of t between 10−3 et 10−1 × e2 /ǫlB . Note that the tunnel term, since it creates a preferred direction in the xy plane, breaks the U (1) symmetry in the Hamiltonian. • Apply a voltage bias between the layers. This generates a term −e(N↑ − N↓ )V = −2eS z V which is analogous to a magnetic field applied along the z quantization axis of the pseudo-spin. Minimizing the anisotropy term added to the electric term, we see that V creates a charge unbalance between the two layers, which, in pseudo-spin language is a finite value for S z = (N↑ − N↓ )/2 • One may apply a magnetic field Bk parallel to the plane of the bi-layer. Such a parallel field has no effect (except in terms of Zeeman gap) in the monolayer case. One may expect new effects in the bi-layer case, which should be orbital effects (the real spin is expected to be entirely polarised). Chosing a gauge for the associated vector potential, the presence of Bk can be taken into account as follows: Axk = 0 y A|| = Ak = 0 , Azk = Bx (8.6) where ẑ is the direction perpendicular to the layers. In that gauge, the gauge invariant tunnel term becomes 2π iφ t → te 0 R d/2 −d/2 dzAzk i2π Bxd φ = te 0 ≡ teiQx , 8.4. EXPERIMENTAL FACTS 137 with Q = 2πBd/φ0 = 2π/Lk , which implies Lk = φ0 . Bd in the presence of Bk , the total tunneling term becomes thus T = t X eiQx c†k,↑ ck,↓ + e−iQx c†k,↓ ck,↑ k h i = −t eiQx (S x + iS y ) + e−iQx (S x − iS y ) = 2tS cos(φ + Qx) (8.7) This term is thus equivalent to a field rotating around the x axis, uniform along y, which is causing the pseudo-spin magnetization to oscillate along x. In fact this term competes with the exchange term, as we shall see later on. • When a Bk field is applied, the tunnelling current between the layers becomes J↑↓ = it X k eiQx c†k,↑ ck,↓ − e−iQx c†k,↓ ck,↑ h (8.8) i = it eiQx (S x + iS y ) − e−iQx (S x − iS y ) = 2tS sin(φ + Qx). 8.4 8.4.1 Experimental facts Phase Diagram As discussed above, the energy difference ∆SAS = 2t between symmetric and antisymmetric superpositions of layer states of the two wells may vary, depending on the sample, from a few millidegrees to a few hundreds of degrees Thus ∆SAS may be much smaller, or much larger than the inter-layer Coulomb interactions. Experiments are able to scan a large array of the characteristic ratio ∆SAS /(e2 /ǫd), from a weak electronic correlation regime to a strong one. When the layers are far from each other (d ≫ lB ), there are no inter-layer correlations, each layer is in the ν = 1/2 metallic ground state, there is no 138 Quantum Hall bi-layers Figure 8.2: Phase diagram for the bi-layer QHE (after Murphy et al. ??). The samples with parameters below the dotted line exhibit the IQHE and an excitation gap. QHE. When the inter-layer separation decreases, an excitation gap is found to 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, since things look as if the two layers were like a single one, with total filling factor ν = 1. All symmetric states are occupied, we have the usual QHE. A far more 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 gap survives in this limit because of a spontaneous breaking of the U (1) gauge symmetry associated with the phase degree of freedom –the azimuthal angle φ in expression (8.1) [87, 98]. Figure 8.2 shows the experimental determination of the QHE part of phase diagram, below the dotted line. This change from single particle to collective behaviour is analogous to the ferromagnetic behaviour of a monolayer at ν = 1. In the latter case, the excitation gap remains finite even when the Zeeman effect vanishes, because of the exchange forces connected to the Coulomb interaction. The remarkable fact is that the IQHE at ν = 1 survives when ∆SAS → 0 provided the inter-layer distance between layers is smaller than a critical value d/lB ∼ 2. In that case, the gap is a purely collective effect due to interactions. As we shall see, it is due to a pseudo-ferromagnetic quantum Halll state, which posesses a spontaneous inter-layer coherence. Experiments 139 Figure 8.3: Experiment by Murphy et al. [94]. The thermal excitation gap ∆ is plotted 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. The dotted line is a rough estimate of the tunnel amplitude renormalized by the parallel magnetic field. This single particle effect exhibits a slow negative variation, compared to the observed effect. The inset is an Arrhenious plot of the dissipation, measured by the longitudinal resistance. The low temperature activation energy is ∆ = 8.66K. The gap however decreases sharply at a much lower temperature, roughly 0.4K. 8.4.2 Excitation gap An additional indication of the collective nature of excitations is provided by 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 to temperatures of the order of ∆/kB . Instead, the gap decreases sharply as soon as A T ∼ 0, 4K. This suggests that the order responsible for the collective excitation gap is vanishing . 8.4.3 Effect of a parallel magnetic field Another experimental finding suggests strongly a collective order phenomenon: the strong sensitivity of the system to a relatively weak Bk magnetic field, applied in a direction parallel to the layers plane. The figure 8.3 shows that the activation gap decreases rapidly when B|| , the parallel component 140 Quantum Hall bi-layers Figure 8.4: Example of electronic process in a 2D bi-layer, such that the flux of Bk produces decoherence effects. In this process, an electron tunnels at point A from the upper layer to the lower one. The electron pair thus created moves coherently, then annihilates at point b where the particle tunnels in the other direction. The amplitude for such a process depends on the flux of Bk through the path . of the field, increases. Assume that the electronic gas in each layer is stricly 2D (in other words neglect the physical width of the potential wells). Then the orbital effect of Bk can only be due to electronic processes between the layers with closed loops containing some flux from Bk . Such loops will cause B|| to be felt if there is phase coherence over the whole loop. Such a loop is shown on figure 8.4. An electron tunnels from one layer to the other at point A, travels a distance L|| , tunnels back to the departure layer, then back to point A. The magnetic field parallel component, Bk , contributes to the amplitude of this process a (gauge invariant) Aharonov-Bohm phase factor, exp(2πiφ/φ0 ), where φ is the flux of Bk threading this circuit. Such loops contribute significantly to correlations, since one observes a rapid decrease of the activation gap as a function of Bk : the decrease is by a factor 2 up to a critical field Bk∗ ∼ 0.8T , beyond which the gap remains roughly constant. This value is remarkably small. Let Lk be the length such that the flux through the loop is one flux quantum: Lk Bk∗ d = φ0 ⇔ Lk [Å] = 4, 14×105 /d[Å]Bk [T]). With Bk∗ = 0, 8T and d = 150Å, one has Lk = 2700Å, i.e. roughly twenty times the average distance between electrons in a layer, and thirty times the magnetic length corresponding to B⊥ . A significant decrease 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 simultaneously in both layers, without tunnel between layers. (b) It is possible to inject current in one layer and to extract it in another. The tunneling current then behaves as a superfluid current . which implies enormous coherence lengths. This is again a hint of the strongly collective 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 theoretical ideas about excitonic superfluidity in bi-layers. In the standard transport experiments on bi-layers, a current JHQ is injected in both layers simultaneously, and is also extracted from both layers simultaneously. In the experiment by Spielman et al., a current JHQ is injected in one layer, and extracted from the other one (Fig. 8.5). Qualitative differences arise in the tunnel conductance when the ratio d/lB is varied, for example varying the electronic density at constant filling of the LLL. Below a critical value of the ratio (the critical value which corresponds to the transition line between the QHE regime at ν = 1 and the metallic regime) a giant anomaly appears at zero bias, as shown on fig. 8.6. The qualitative understanding of this experiment is as follows: for d/lB ≫ 1, electronic liquids in different layers are uncorrelated. At zero inter-layer bias, the Coulomb repulsion between electrons in one layer, and an electron in the other will inhibit the inter-layer tunneling process of the latter: the zero bias conductance is strongly suppressed. Only a finite bias, of the order of the Coulomb repulsion e2 /(ǫd), can supersede the latter. When d/lB ≃ 1 a coherent state is established, such that an 142 Quantum Hall bi-layers electron in one layer is bound to a hole in the other one at the same Landau site. Coulomb repulsion is strongly suppressed by this collective structure for inter-layer tunneling events, and the tunneling conductance increases by two orders of magnitude. At the time of this writing, as far as the author knows, there is yet no general consensus on the intrinsic, or extrinsic character of the zero bias conductance finiteness. Is it impossible, for fundamental reasons, to ever observe a divergent conductance at zero bias, which would be the signature of a complete analogy with the superconducting Josephson junction? Is the conductance finiteness due to experimental limitations, (impurities, etc.), or to the order parameter topological defects at finite temperature? Those questions 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 bilayer system at ν = 1, one needs an experimental proof of electron-hole pair transport. How can one couple to and detect electrically neutral objects by electric transport? The solution is to notice that electron-hole pair transport in a bi-layer implies an anti-parallel circulation of currents in different layers. Experiments have allowed, in the last few years to get independent electric connections to each layer [108]. It has thus been possible to inject equal intensity currents with opposite flow direction in both layers, to test the contribution of excitons to particle transport [6]. The figure 8.7 is a schematic representation of what one expects from such an anti-aparallel current experiment. The two traces represent the expected Hall voltage in each layer, neglecting all quantum phenomena except the excitonic condensation. Because of the Lorentz force, the Hall voltage is proportional to the magnetic field. In a bi-layer system driven by two oppositely directed parallel currents, the Hall voltage will have opposite signs in the two layers. If the two layers are sufficiently coupled, and the magnetic field has the relevant intensity (so that ν↑ = ν↓ ≃ 1/2 the inter-layer electronhole pairs which form will carry anti-parallel currents. The Hall voltage in both layers must then vanish, as suggested by the figure 8.7. Two experimental groups have confirmed those predictions [104]. 143 (a) A) NT=10.9 D) (d) B) (b) NT=6.9 NT=5.4 z -7 -1 Tunneling =1 /dV (10 W ) Conductance at nTdJ Conductance tunnel Experiments 0.5 NT=6.4 (c) C) -5 0 5 -5 0 5 Tension intercouches V (mV) Interlayer Voltage (mV) Figure 8.6: Quasi-Josephson effect [102]. Plot of the tunneling conductance dJz /dV as a function of voltage bias between the two layers, for various electronic densities, NT in units of 1010 cm−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 at zero bias. In the low density sample (d) there is a finite tunneling conductance peak at zero bias. The current at zero bias vanishes, contrary to the superconducting Josephson junction current. Whence the expression ”quasi-Josephson effect”. 144 Quantum Hall bi-layers tension de Hall 10 + − + − + − + − + − + − + − + − − + − + − 0 + + − − + + − ν=1 −10 5 10 champ magnétique Figure 8.7: Antiparallel currents experiment (After ref. [6]). A Hall voltage measurement detects the exciton condensation. The two traces are schematic representations of the Hall voltage in each layer when electric currents flow in opposite directions. Quantum effects other than the excitonic condensation are ignored in this figure. When currents flow 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 exitonic condensation occurs, in a certain span of magnetic field values, the opposite currents in the layers will be carried by a uniform exciton current density in one direction. Since excitons are electrically neutral, they are not submitted to Lorentz forces, and the Hall voltage must vanish in both layers, as observed experimentally by Kellogg et coll. [104]. 8.5. EXCITONIC SUPERFLUIDITY 8.5 145 Excitonic superfluidity Within the pseudo-spin analogy(section 9.2), the Coulomb interaction between layers favours a ferromagnetic state with an easy magnetization plane when 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)] Y c†k↑ + eiφ c†k↓ √ |ψφ i = |0i . 2 (8.9) k In other words, θ/2 = π/4, the magnetization is in the bi-layer plane, and one has hSz i = 0. The amplitude is equal for opposite pseudo-spin states, which means that, for the time being, we consider a situation with zero bias between the two layers. The total occupation of each k state is 1. When the tunnel term t vanishes, φ has any value, provide it is the same all over the bi-layer plane. When the system chooses a particular φ value, among the continuous 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 family of equivalent ground states, with the phase φ as parameter. This phase is conjugate to the difference in particle number between the two layers. In equation (8.9), the phase is well defined, but the number of particles of each pseudo-spin (i.e. the number of particles is each layer) is completely undetermined. Similarly, one may construct a state such that the phase is undetermined, while the number of particles in each layer is specified exactly. To do this, integrate 8.9 over the phase. This yields |ψS z i = Z dφ −i(N↑ −N↓ )φ e |ψφ i . 2π (8.10) We obtain thus a wave function with exactly N↑ particles in the ↑ layer, and N↓ = N − N↑ in the ↓ layer, N being the total number of guiding centers. The angle φ and S z 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 be written as i Yh † |ψφ i = ck↑ + eiφ(Xk ,t) c†k↓ |0i , (8.12) k where φ is the superfluid phase of the system. The long wavelength superfluid mode corresponds to equal intensity currents of opposite signs propagating in the two layers. To understand better why the state described by (8.9) breaks the gauge symmetry associated to the charge difference between layers, consider the θ gauge transformation induced by the unitary operator U− (θ) = ei 2 (N↑ −N↓ ) . This transformation acts on electron creation operators as θ U−† (θ)c†k↑ U− (θ) = e−i 2 c†k↑ (8.13) . (8.14) U−† (θ)c†k↓ U− (θ) i θ2 =e c†k↓ 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. D E D E nel iφ S x (Xk ) ≡ c†k↑ ck↓ = S̄ x (Xk ) = e , 2 2 with the total density nel = 1/2πlB . (Here I have defined the x̂ direction as the arbitrary direction of the sponaneous pseudo-spin orientation in the plane x, y). This order parameter is not gauge invariant, D E S x (Xk ) → U−† (θ)c†k↑ ck↓ U− (θ) = eiθ S x (Xk ) . (8.16) That is a more formal way to show that the state has less symmetry than the Hamiltonian, and breaks the U (1) symmetry associated with the conservation of the charge difference between layers N↑ − N↓ . 1 In a superconductor, the order parameter χ(r) = hc†↑ (r)c†↓ (r)i transforms in a non trivial way under the gauge transformation associated with the total charge conservation, Ũ+ (θ) = exp[iθ(N↑ + N↓ )/2]. The pseudo-spin bi-layer order parameter is invariant under this transformation: this expresses simply the fact that the total particle number N↑ + N↓ is conserved in the excitonic superfluid. 1 Excitonic superfluidity 147 We can write an expression for the inter-layer tunneling current operator as a function of position in space, J↑↓,Xk = −it c†k↑ ck↓ − c†k↓ ck↑ , the average of which (8.17) hJ↑↓,Xk i = −it [S x (Xk )∗ − S x (Xk )] = t sin(φ). This expression is similar to the Josephson current expression: it depends only on the order parameter phase, not on the inter-layer voltage bias. The pseudo-spin language expresses the conjugate character of phase and charge difference between layers through the commutation relations of the spin density operators. With the order parameter along x, [S y , S z ] = iS x ≃ i. As S y ∝ sin φ ≈ φ, this leads to [φ, S z ] = i. As a consequence, the current associated to the phase gradient Jzz = 2ρE ∇φ h̄ 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 this difference should be conserved in the limit t = 0 . This is analogous to the superconducting BCS wave function, which has a fluctuating total number of particles, while it is in fact strictly conserved for an isolated sample.The solution of this apparent paradox is that each macroscopic piece of the sample may be subdivided in smaller macroscopic parts, between which particle exchanges are numerous and rapid, so that phase coherence is established in each macroscopic part of the sample, the total particle number remaining constant. Furthermore, in the thermodynamic limit, the ratio δN/N is of order N −1/2 → 0. A similar reasoning holds in the bi-layer case. Examine a slighlty more complicated object than the order parameter, D GXk ,Xk′ = c†k↑ ck↓ c†k′ ↓ ck′ ↑ E . (8.18) This object conserves the total particle number in each layer. It is equal to hS̄ x (Xk )S̄ x (Xk′ )i, 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. To see that, define the state |f erro ↑i as the state where all electrons are in the Q ↑ layer , |f erro ↑i = k c†k↑ |0i. Then the state (8.9) may be re-written as |ψφ i ≡ Y k 1 + eiφ c†k↓ ck↑ |f erro ↑i . √ 2 (8.19) This can be again re-written in a form reminiscent of a bosonic coherent state, Y exp eiφ b†k |f erro ↑i , (8.20) |ψφ i ≡ k where b†k = c†k↑ ck↓ is the excitonic boson. This is the reason why one may speak of a ”coherent” state (see section 2.2.2). One also speaks of ”spontaneous phase coherence”, when the tunneling term is absent. Indeed in that case the coherent state is entirely due to Coulomb interactions. On the contrary, when the tunneling term is finite, the symmetric combination of layer states is the most stable, even in the absence of interactions. This is analogous to the magnetization induced by an external magnetic field in the case of ”real” spins. 8.5.1 Collective modes – Excitonic condensate dynamics As mentionned above, a consequence of the breaking of a continuous symmetry by the phase coherence is the existence of the collective excitation mode (Goldstone mode) the energy of which goes continuously to zero as the wavelength goes to infinity. The Hamiltonian formalism was used above to derive collective mode energies in the ferromagnetic monolayer case. Here we use the Lagrangian formulation, with the inclusion of the Berry connexion term discussed in the previous chapter. The Lagrangian which describes the long wavelength physics, in the absence of applied inter-layer voltage bias, and with zero tunneling term, is ν Z 2 L = d rṁ · A[m] (8.21) 2 4πlB Z i ρA ρE h − d2 r β(mz )2 + . |∇mz |2 + |∇mx |2 + |∇my |2 2 2 Excitonic superfluidity 149 Coefficients β, ρA and ρE may be evaluated with a microscopic approach, as we have seen in section 8.3.2 Let us write the Euler-Lagrange equations of motion, d δL δL = . (8.22) µ dt δ ṁ δmµ Here the ground state is taken with the (vector) order parameter of length 1 aligned along the x̂ axis. For small variations of the order parameter away from x̂, one may linearize, considering only first order deviations in my and mz . m = [1 − O(m2y + m2z ), my , mz ], and one chooses the Berry connexion A = (0, −mz /2, my /2), which yields ν δL ν mz , = Ay [m] = − 2 2 δ ṁy 4πlB 4πlb 2 ν ṁz δL = + ρE ∆my , 2 δmy 4πlB 2 (8.23) and ν ν my , Az [m] = − 2 2 4πlB 4πlb 2 ν ṁy + ρA ∆my − 2βmz , = 2 4πlB 2 δL δ ṁz δL δmz = (8.24) where ∆ = ∇ · ∇ is the Laplacian. In Fourier space, applying 8.22, one finds the system of linear equations iω 4π 2 q ρE ν 4π (2β ν + q 2 ρA ) −iω ! my mz ! = 0. (8.25) So finally the collective mode dispersion relation is given by 4π ω (q) = ν 2 2 (2β + q 2 ρ2A )q 2 ρE . (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 = 2 4π 0 2 ρq . ν The expansion in gradients of mz is not stricly correct, because the long range nature of he Coulomb interaction induces a non local term which we do not take into account here. The latter term is smaller than the terms considered here. 150 QHE bi-layers Q=+1/2 v=+1 Q=+1/2 v=−1 Q=−1/2 v=+1 Q=−1/2 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, lim ω(q)|β6=0 q→0 4π q = 2βρE q . ν That is analogous to the bosonic superfluid collective mode (with weak repulsive interactions). But here the order parameter represents the condensation of neutral bosons, which carry no charge. 8.5.2 Charged topological excitations For a system in the same universality class as that of the 2D XY model, there must exist a Kosterlitz-Thouless (KT) transition at TKT = (π/2)ρS /kB . The essence of this transition is the ionisation (dissociation) of vortex-antivortex pairs. In our case, the order parameter symmetry group is U (1), but the pseudo-spin direction is not confined to the xy plane, so that the pseudo-spin vortex 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 has the approximate following form q m = ± 1 − m2z cos θ, q 1 − m2z sin θ, mz (r) , (8.27) Excitonic superfluidity 151 where the ± sign refers to the vorticity (left or right) and θ is the azimuthal angle 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 core singularity. The local topological charge is computed using the Pontryagin density expression [see equation (7.31)] δρ = − 1 ij ǫ (∂i m × ∂j m) · m. 8π With expression (8.27), this density writes δρ(r) = 1 dmz . 4πr dr The total charge is Q = d2 rδρ(r) = 21 [mz (∞) − mz (0)]. For a meron, the spin at the core is either ↑ or ↓, and gradually gets oriented in the xy plane as the distance from the core increases. It lies in the xy plane far from the meron core. The topological charge is thus ±1/2 depending on the core spin polarity. The general result for the topological charge is R Q= 1 [mz (∞) − mz (0)] nv 2 (8.28) where nv is the vortex winding number. The electric charge is ±νe/2, half that 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 † † M E Y c + c m,↑ √ m+1,↓ |0i . ψnv =+1,−1/2 = 2 m=0 (8.29) In this expression, c†m,↑(↓) creates an electron in layer↑ (↓), in the state of angular momentum m in the LLL, and M is the corresponding moment on the 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 an electron has been suppressed at the center, in the ↓ layer: all states have 1/2 occupation, except m = 0 which is empty. The meron charge can be changed without changing the vorticity, as we see with the wave function † † M E Y c + c † m,↑ √ m+1,↓ |0i . ψnv =+1,+1/2 = c0,↓ 2 m=0 This state has charge −1/2, because an electron has been created in the state m = 0 in the ↓ layer. It is useful to examine a meron pair wave function, to check wether the meron is a half skyrmion. Examine the case of a pair of merons with opposite vorticities, but equal charges, placed at points z̄1 and z̄2 . The following wave function seems to obey our requirements, ψλ = eiφ (zj − z̄1 ) (zj − z̄2 ) Y 1 j √ 2 ! Φf erro , (8.30) j where φ is an arbitrary angle and ()j is a spinor for the j-th particle. At large distance from z̄1 and z̄2 , the spinor for each particle becomes eiφ 1 zj ! . (8.31) This corresponds to a fixed spin orientation in the xy plane, with an angle φ with the x axis. Vorticity is thus zero. By construction, the spin orientation is purely ↑ for an electron at z̄2 , and purely ↓ for an electron at z̄1 . Moreover, the net charge must be νe since, asymptotically, the factor zj is the same as for the Laughlin quasi-particle in the spin polarised state. For symmetry reasons, one might think that a charge νe/2 is asociated to each localised state near z̄1 or z̄2 . The fact is that this wave function (8.30) is nothing but a different representation for the skyrmion! Choose z̄1 = λ and z̄2 = −λ, and suppose for simplicity that the asymptotic orientation of spins is in the x direction, so that φ = 0. Now rotate all spins by a global rotation around the ŷ axis, with an angle −π/2. Using π 1 exp i σ y √ 4 2 zj − λ zj + λ ! , Excitonic superfluidity 153 one finds the variational skyrmion wave function. The previous wave function is well adapted to the U (1) symmetry, because t describes spins oriented mainly in the xy plane. 8.5.3 Kosterlitz-Thouless transition The presence of topological defects of the vortex type may spoil the phase coherence of the XY ground state. This may happen at zero temperature, because of quantum fluctuations, if the distance between layers exceeds a critical distance d∗ . Here we are discussing thermal effects. The effective model at finite temperature is given by ρS Z 2 E= d r |∇φ|2 . 2 For typical experimental parameter values in the AsGa bi-layers, the HartreeFock estimate of the exchange stiffness ρS goes from 0, 1K to 0, 5K. The Kosterlitz-Thouless transition is due to ionisation of vortices in the XY model, at a temperature TKT approximately given by the exchange stiffness ρS . Free vortices induce a discontinuous renormalisation of the exchange stiffness, which vanishes at TKT . The classical action generates a logarithmic interaction between vortices. A meron gas has an energy of the form E = M Ecore − 2πρS M X i<j ni nj ln M X e2 Rij qi qj + , Rcore 4ǫRij i<j (8.32) where Ecore is the meron core energy, Rcore its size, and Rij is the separation between the i-th and the j-th meron. The last term is new. It is specific of the QHE bi-layer physics: it is due to Coulomb interactions between the merons fractional charges. qi = ±1 is the electric charge (±e/2) sign, of the i-th meron. The origin of the logarithmic term is not, as in the superconducting case, the kinetic energy stored in the supercurrents. It comes from the loss of exchange energy due to the phase gradients associated to vortices. The Coulomb interaction is irrelevant at TKT because it decreases faster with distance than the logarithmic interactions. It may cause a shift of TKT , but the transition is not qualitatively altered. The phase diagram however becomes richer, with chiral phases, with an order parameter hni qi i where vorticity and electric charge are no longer independent[109]. 154 QHE bi-layers v=−1 ξ Λ v=+1 Figure 8.9: In the presence of an inter-layer tunneling term, meron pairs of opposite vorticities 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 Hef f = Z t ρs |∇φ|2 − cos φ . dr 2 2πρ2 2 (8.33) Here ρs is the iso-spin exchange stiffness, which may be computed microscopically with the same techniques used to compute the equivalent parameter in the ”true” ferromagnetic case. For a finite t value, the collective mode acquires a mass (just as spin wave in a SU (2) ferromagnet acquire a mass in an external field, because of the Zeeman effect). Quantum fluctuations are thereby decreased, which explains the upwards curvature of the phase transition line in the phase diagram. The tunneling term, because it breaks the U(1) symmetry, and gives a larger energy cost to vortex pairs configurations, destroys the KT transition. Excitonic superfluidity 155 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 are oriented 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 is proportional to its length Λ, so that we have a vortex confinement mechanism analogou to quark confinement in elementary particles such as hadrons or mesons. The line tension (the energy per unit length) may be estimated by examining the infinitely long domain wall parallel to the y axis. The optimal form is given in that case by φ(r) = 2 sin −1 x tanh ξ ! , (8.34) 1/2 2 where the characteristic wall width is ξ = (2πlB ρS /t) thus (see [110]) !1/2 8ρs tρS = . T0 = 8 2 2πlB ξ . The line tension is (8.35) If the wall is long enough (Λ ≫ ξ), the total energy of a segment of length Λ will be approximately Epair = 2Ecore + e2 + T0 Λ. 4ǫR (8.36) Minimising, we conclude that Epair is optimal for Λ = Λ′0 = (e2 /4ǫT0 )1/2 , whence !1/2 e2 T0 ′ Epair = 2Ecore + . ǫ Thus, except for meron core energies, the charge gap at fixed d (i.e. at 1/4 1/2 fixed ρs ), is proportional to T0 ∝ t1/4 ∼ ∆SAS . This is in contrast with the free electron case. The exponent 1/4 is small. The charge gap increases quickly as soon as the tunneling term is non negligible. The cross-over regime between the pseudo-spin meron pair textures and the domain wall texture is established at a finite t value. 8.5.5 Combined effects of a tunnel term and a parallel field Bk We have seen that the vector potential corresponding to a parallel field Bk 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 tunneling matrix 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 saw in section 9.3), one finds a spatial variation of the phase, φ → φ − Qx. The tunneling term competes with the spin stiffness one. The latter is minimized for a uniform magnetization, while the former favours a rotating one, when Bk 6= 0, ) Z ( St 2 ρS |∇φ| − H= cos(φ − Qx) d2 r . (8.37) 2 2πlB This Hamiltonian, known in the 2D physics of commensurate-incommensurate transitions as the Pokrovski-Talapov model, has a phase diagram structure which depends on the relative values of ρs and Q. If Q → 0, the energy is minimised by a phase φ = Qx, the exchange energy loss being ρs Q2 . One has then a commensurate state (the term phase here 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 at the value dictated by the periodic potential minima. When Q increases, minimising the periodic term with a linear variation of the phase becomes too costly in exchange energy. The conflict between those two terms results in the appearance of solitons which are phase defects. The latter which are solutions of a Sine Gordon equation, express the compromise between the ”elastic energy” (the exchange term, quadratic in Q, and the periodic ”potential energy”, (the sinusoidal tunneling term). The limiting ~ ∼ 0 and the average value of cos(φ − Qx) vanishes. behaviour is when ∇φ For Bk larger than a critical value Bkc periodically ordered topological defects start being formed, in a uniaxial 2D anisotropic environment. At zero temperature, the critical value is Bkc 2lB 2t 1/2 = B⊥ . πd πρs With ∆SAS = 0.45K, one finds Bkc ≈ 1, 3T, slightly larger than the observed value 0.8T. The corresponding value of Lk is large. In the commensurate phase, the order parameter tumbles more and more rapily as Bk incrases, since φ = Qx. In the incommensurate phase, the system state becomes roughly independent of Bk , so that the excitation gap saturates at a fixed value. In the presence of the tunneling term, the lowest energy charged excitations are meron pairs with opposite vorticities and Excitonic superfluidity 157 equal charges (i.e. ±1/2, connected to one another by a domain wall with a constant line energy. For Bk = 0, the energy is independent of the wall orientation. The effect of Bk 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, the Hamiltonian becomes H= Z ( ) i ρS h t 2 2 dr (∂x ϕ + Q) + (∂y ϕ) − cos ϕ . 2 2 2πlB 2 (8.38) Thus Bk defines a preferred direction of this problem. Domain walls align in this direction and involve a phase change, in terms of ϕ, with a preferred sign (negative for Q > O). One can show that the energy per unit soliton length of the wall, i.e. the line tension, decreases linearly with Q, and thus with Bk , i.e. Bk T = T0 1 − ∗ , Bk where T0 is given by equation (8.35). There is a transition when T becomes negative. We have seen in section 7.5.5 that the charge excitation q 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 + s 1/2 Bk e2 T0 1 − ∗ ǫ Bk . Thus, as Bk 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 the charge gap at Bk = 0 and that when Bk → ∞ should be roughly ∆0 t 1/4 (e2 /ǫlB )1/2 t1/4 . = ≃ 3/4 ∆∞ tcr 8ρs Using typical values for t and ρs , the former expression yields values between 1.5 and 7, in qualitative agreement with experiment. 158 8.5.6 QHE bi-layers 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 Z n d2 r ei[φ(r)+Qx] − e−i[φ(r)+Qx] n o = F eiφ(r) qy =0,qx =Q o n o − F e−iφ(r) (8.39) qy =0,qx =−Q (8.40) where F{f (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 ) = 2πet2 L2 [S(Q, eV ) − S(−Q, −eV )] . h̄ (8.41) where S(q, h̄ω) is the fluctuations spectral density of the opeator eiφ at wave vector q et and frequency ω, i.e. the transform of heiφ(r,t) e−iφ(0,0) i. A striking prediction follows: when disorder is weak, the spectral density, and thus It (V ) exhibit a peak centered at eV = h̄ωQ where ωQ is the collective frequency at wave vector Q. Thus as Bk varies, the conductance peak position varies according to the low energy collective mode dispersion. The parallel field only allows tunneling events between states which differ by their momentum Q. Energy conservation ensures that the 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 direct measurement of the collective mode dispersion relation, which is found to be linear in Q, as predicted by theory. As shown in figure 8.10, the tunneling current in the presence of a parallel field Bk exhibits a peak which corresponds directly to the collective mode dispersion of the superfluid phase. The figure shows the tunneling conductance at T = 25mK for an electronic density of 5.2 × 1010 cm−2 , for a series of parallel magnetic field values between Bk = 0 and 0.6T. The insert is a blow-up of curves for Bk values between 0, 07 and 0, 35T. The dots indicate the position of satellite resonances for dI/dV . (After [111] Excitonic superfluidity 159 2.5 -7 1.5 10 Ω -1 -6 -1 dI/dV (10 Ω ) 2.0 -200 0 200 B|| = 0 V (µV) 1.0 0.5 B|| = 0.6T 0.0 -200 0 200 V (µV) Figure 8.10: Experimental determination of the Goldstone mode dispersion in a transport experiment [111]. The tunneling current in the presence of a parallel field Bk exhibits a peak which corresponds directly to the collective mode dispersion of the superfluid phase. The figure shows the tunneling conductance at T = 25mK for an electronic density of 5, 2 × 1010 cm−2 , for a series of parallel magnetic field values between Bk = 0 and 0.6T. The insert is a blow-up of curves for Bk values between 0, 07 and 0, 35T. The dots indicate the position of satellite resonances for dI/dV . (After reference [111] 160 QHE bi-layers eV* (meV) 0.2 0.1 0 0 10 20 6 30 -1 q (10 m ) 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 = eBk d/h̄ in the presence of a parallel magnetic field, for different electronic densities (cross : nel = 6.4 × 1010 cm−2 ; squares : nel = 6.0 × 1010 cm−2 ; black dots : nel = 5, 2 × 1010 cm−2 . The dotted line is a theoretical estimate by Girvin for the Goldstone mode dispersion at small q [87]. The continuous line is a guide for the eye, and corresponds to a collective mode velocity of 1, 4 × 104 m/s. 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