Fredholm Homotopies for Strongly-Disordered 2D Insulators

TL;DR

This paper introduces a method to analyze topological indices of 2D topological insulators under strong disorder, proving bulk-edge correspondence and stability of the $$ index, with implications for higher-dimensional systems.

Contribution

It develops a novel interpolation technique for Fredholm operators in disordered topological insulators, establishing bulk-edge correspondence and index stability in the strong disorder regime.

Findings

Proved bulk-edge correspondence for strongly-disordered 2D topological insulators.

Established stability of the $$ index in the mobility gap regime.

Provided an alternative proof for the bulk-edge correspondence in quantum Hall systems.

Abstract

We study topological indices of Fermionic time-reversal invariant topological insulators in two dimensions, in the regime of strong Anderson localization. We devise a method to interpolate between certain Fredholm operators arising in the context of these systems. We use this technique to prove the bulk-edge correspondence for mobility-gapped 2D topological insulators possessing a (Fermionic) time-reversal symmetry (class AII) and provide an alternative route to a theorem by Elgart-Graf-Schenker (2005) about the bulk-edge correspondence for strongly-disordered integer quantum Hall systems. We furthermore provide a proof of the stability of the $\mathbb{Z}_2$ index in the mobility gap regime. These two-dimensional results serve as a model for the study of higher dimensional $\mathbb{Z}_2$ indices.