Equality of bulk and edge Hall conductance revisited

Abstract

The integral quantum Hall effect can be explained either as resulting from bulk or edge currents (or, as it occurs in real samples, as a combination of both). This leads to different definitions of Hall conductance, which agree under appropriate hypotheses, as shown by Schulz-Baldes et al. by means of K-theory. We propose an alternative proof based on a generalization of the index of a pair of projections to more general operators. The equality of conductances is an expression of the stability of that index as a flux tube is moved from within the bulk across the boundary of a sample. The model and the result The simultaneous quantization of bulk and edge conductance is essential to the QHE in finite samples, as explained in [8, 13]. In these two references that property is established in the context of an effective field theory description, resp. of a microscopic treatment suitable to the integral QHE. The present paper is placed in the latter setting as well. In our model H is a discrete Schrödinger operator on the single-particle Hilbert space l2(Z×N) over the upper half-plane. It is obtained from the restriction (with e.g. Dirichlet boundary conditions) of a ‘bulk’ Hamiltonian HB acting on l 2(Z×Z). These assumptions are spelled out in detail at the end of this section. The spectrum of HB (but not that of H , as a rule) has an open gap ∆ containing the Fermi energy: ∆ ∩ σ(HB) = ∅ . (1) Let PB be the Fermi projection: PB = E(−∞,μ](HB) for any μ ∈ ∆. A real-valued function g ∈ C(R) with g(λ) = 1 (resp. 0) for λ large and negative (resp. positive) will be called a switch function. We remark that PB = g(HB) if the switch function has supp g ⊂ ∆.

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Cite this paper

@inproceedings{Elbau2002EqualityOB, title={Equality of bulk and edge Hall conductance revisited}, author={Peter Elbau}, year={2002} }