Quantization of conductance in gapped interacting systems

TL;DR

This paper offers a simplified proof of the quantization of Hall conductance in gapped interacting quantum systems on a 2D torus, emphasizing the constancy of Berry's curvature across fluxes.

Contribution

It adapts existing proofs by assuming the Hamiltonian remains gapped under flux threading, simplifying the demonstration of conductance quantization.

Findings

Conductance quantization is proven for gapped systems.

Berry's curvature is shown to be asymptotically constant.

The proof relies on the assumption of persistent spectral gap.

Abstract

We provide a short proof of the quantisation of the Hall conductance for gapped interacting quantum lattice systems on the two-dimensional torus. This is not new and should be seen as an adaptation of the proof of [1], simplified by making the stronger assumption that the Hamiltonian remains gapped when threading the torus with fluxes. We argue why this assumption is very plausible. The conductance is given by Berry's curvature and our key auxiliary result is that the curvature is asymptotically constant across the torus of fluxes.