A New Numerical Method for $\mathbb{Z}_2$ Topological Insulators with Strong Disorder

TL;DR

This paper introduces a novel numerical method to efficiently compute $$ indices in disordered topological insulators, leveraging spectral properties of projection differences, applicable to models in two and three dimensions.

Contribution

The paper presents a new numerical approach based on spectral analysis of projection differences to determine $$ indices in disordered topological insulators, enhancing computational efficiency.

Findings

Successfully applied to Bernevig-Hughes-Zhang model

Effective for Wilson-Dirac model in 3D

Demonstrates high efficiency and accuracy

Abstract

We propose a new method to numerically compute the $\mathbb{Z}_2$ indices for disordered topological insulators in Kitaev's periodic table. All of the $\mathbb{Z}_2$ indices are known to be derived from the index formulae which are expressed in terms of a pair of projections introduced by Avron, Seiler, and Simon. For a given pair of projections, the corresponding index is determined by the spectrum of the difference between the two projections. This difference exhibits remarkable and useful properties, as it is compact and has a supersymmetric structure in the spectrum. These properties make it possible to numerically determine the indices of disordered topological insulators highly efficiently. The method is demonstrated for the Bernevig-Hughes-Zhang and Wilson-Dirac models whose topological phases are characterized by a $\mathbb{Z}_2$ index in two and three dimensions, respectively.