Quantization of Hall Conductance For Interacting Electrons on a Torus

TL;DR

This paper proves that the Hall conductance of interacting electrons on a torus is quantized in integer multiples of e^2/h, using quasi-adiabatic evolution, with extensions to fractional cases under topological order assumptions.

Contribution

It provides a rigorous proof of quantized Hall conductance for interacting systems without averaging assumptions, including fractional cases with topological order.

Findings

Hall conductance is quantized in integer multiples of e^2/h.

Quantization holds up to exponentially small corrections in system size.

Extensions to fractional quantization under topological order are discussed.

Abstract

We consider interacting, charged spins on a torus described by a gapped Hamiltonian with a unique groundstate and conserved local charge. Using quasi-adiabatic evolution of the groundstate around a flux-torus, we prove, without any averaging assumption, that the Hall conductance of the groundstate is quantized in integer multiples of e^2/h, up to exponentially small corrections in the linear size of the system. In addition, we discuss extensions to the fractional quantization case under an additional topological order assumption on the degenerate groundstate subspace.