Localization of generalized Wannier bases implies Chern triviality in non-periodic insulators

TL;DR

This paper proves that in disordered 2D gapped quantum systems, the existence of a localized Wannier basis implies the system has trivial topological properties, specifically zero Chern character.

Contribution

It establishes a direct link between Wannier basis localization and topological triviality in non-periodic insulators, and proposes a new localization dichotomy conjecture.

Findings

Localized Wannier bases imply vanishing Chern character.

The Chern character is proportional to Hall conductivity.

A localization dichotomy conjecture for non-periodic systems is proposed.

Abstract

We investigate the relation between the localization of generalized Wannier bases and the topological properties of two-dimensional gapped quantum systems of independent electrons in a disordered background, including magnetic fields, as in the case of Chern insulators and quantum Hall systems. We prove that the existence of a well-localized generalized Wannier basis for the Fermi projection implies the vanishing of the Chern character, which is proportional to the Hall conductivity in the linear response regime. Moreover, we state a localization dichotomy conjecture for general non-periodic gapped quantum systems.