Quantization of the Hall Conductance for General, Multiparticle Schrödinger Hamiltonians

Abstract

We describe a precise mathematical theory of the Laughlin argument for the quantization of the Hall conductance for general multiparticle Schrodinger operators with general background potentials. The quantization is a consequence of the geometric content of the conductance, namely, that it can be identified with an integral over the first Chern class. This generalizes ideas of Thouless et ai, for noninteracting Bloch Hamiltonians to general (interacting and nonperiodic) ones.PACS numbers: 03.65. Bz, 02.40. + m, 72.20.My The integer quantization of the Hall conductance has been explained by Laughlin 1 making clever use of a nontrivial geometry: a ring threaded by a flux tube, combined with a gauge argument. The impact of this work on the development of the subject cannot be overestimated.Our purpose here is to describe a precise mathematical theory of this argument. The two key issues are, first, Laughlin's identification of a physical quantity as the Hall conductance averaged over one unit of quantum flux (of the flux tube that threads the ring). Following Laughlin we shall slightly abuse the terminology and call it the Hall conductance.The second main theme will be the identification of the geometric content of the Hall conductance: Roughly speaking, there is a natural notion of curvature describing how the state of the system is parallel transported in the Hilbert space of states. The Hall conductance is a suitable integral of the corresponding curvature. More precisely, it is an integral over the first Chern class. 2 This was first recognized by Thouless et al in the special case of noninteracting Bloch Hamiltonians. 3 What is shown here is that this holds generally, with electron-electron interaction and general background potential. (No flux averaging is necessary for Bloch Hamiltonians.) Bellissard generalized the result of Thouless et al from rational to real magnetic flux. 4 We shall replace Laughlin's condition that the Fermi energy lies in the region of the localized states (which is not an appropriate condition for multiparticle Hamiltonians) by a condition of nondegeneracy of the multiparticle ground states (for all fluxes).Our work has been independent of, but is nevertheless closely related to, a recent published paper of Niu and Thouless. 5 The general framework is similar, although there are some differences in the details and in the approach. In both works one needs the ground state to be separated from the rest of the spectrum by a finite gap (the nondegeneracy condition). In both approaches one considers time-periodic Hamiltonians (in Niu-Thouless only up to unitary equivalence). In Niu and Thouless the time dependence resides in the substrate potential and it comes from a Galilean transformation that removes the electric field. In our case, the time dependence comes from generating the electromotive force by a flux tube, and so resides in the minimal coupling term in the Hamiltonian. In both, strict quantization is obtained only after a suitable averaging: In Niu-Thouless the averaging is o...

Anomalous Floquet-Anderson Insulator as a Nonadiabatic Quantized Charge Pump

“…In the thermodynamic limit, we expect this dependence to disappear. As in the case of the quantization of the Hall conductance [46], we average over θ x [47]. We therefore get…”

Simultaneous quantization of edge and bulk Hall conductivity

“…Laughlin proposed an adiabatic Gedankenexperiment in order to calculate the Hall conductance [15], Halperin and later on Büttiker studied the conduction by edge channels [13,10], while Thouless, Kohmoto, Nightingale and den Nijs investigated the Hall conductivity as given by the Kubo formula [20]. Laughlin's argument was rigorously analyzed by Avron, Seiler and Simon even for multiparticle Hamiltonians and in presence of a disordered potential [3,4,5]. Bellissard, recently joint by van Elst and Schulz-Baldes, generalized the TKN 2 -work in order to show quantization of the Hall conductivity also in presence of a disordered potential as long as the Fermi level lies in a region of dynamically localized states [6,7], a result that was also obtained by Aizenman and Graf [1].…”

Geometric Properties of Transport in Quantum Hall Systems

“…We start | as an example | with the following Hamiltonian by J. E. Avron, R. Seiler (1985) as already mentioned in the introduction, cf. Fig.…”

“…In this model, interacting particles are considered (J. E. Avron, R. Seiler (1985), Q. Niu, D.J. Thouless (1987), J. E. Avron, R. Seiler, L. G. Ya e (1987)).…”