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arXiv · 0911.4706

Quantization of Hall Conductance For Interacting Electrons Without Averaging Assumptions

Matthew B. Hastings·Spyridon Michalakis
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Abstract

We consider two-dimensional Hamiltonians on a torus with finite range, finite strength interactions and a unique ground state with a non-vanishing spectral gap, and a conserved local charge, as defined precisely in the text. Using the local charge operators, we introduce a boundary magnetic flux in the horizontal and vertical direction and evolve the ground state quasi-adiabatically around a square of size one magnetic flux, in flux space. At the end of the evolution we obtain a trivial Berry phase, which we compare, via a method reminiscent of Stokes' Theorem, to the Berry phase obtained from an evolution around a small loop near the origin. As a result, we prove, without any averaging assumption, that the Hall conductance for interacting electron systems is quantized in integer multiples of e^2/h up to small corrections bounded by a function that decays as a stretched exponential in the linear size L. Finally, we discuss extensions to the fractional case under an additional topological order assumption to describe the multiple degenerate ground states.

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Matthew B. Hastings, Spyridon Michalakis. 2009-11-24. Quantization of Hall Conductance For Interacting Electrons Without Averaging Assumptions. https://arxiv.org/abs/0911.4706

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