Disordered Topological Insulators via $C^*$-Algebras

TL;DR

This paper develops a numerical method using $C^*$-algebras to compute topological invariants in disordered topological insulators, enabling analysis of larger systems and revealing phase transitions between metallic and quantum Hall states.

Contribution

It introduces a new numerical procedure for calculating Z_2 invariants in disordered topological insulators using $C^*$-algebra techniques, applicable to larger system sizes.

Findings

Existence of a metallic phase with spin scattering.

Sharp transition to quantum Hall effect when decoupling spin components.

Efficient computation of topological invariants in disordered systems.

Abstract

The theory of almost commuting matrices can be used to quantify topological obstructions to the existence of localized Wannier functions with time-reversal symmetry in systems with time-reversal symmetry and strong spin-orbit coupling. We present a numerical procedure that calculates a Z_2 invariant using these techniques, and apply it to a model of HgTe. This numerical procedure allows us to access sizes significantly larger than procedures based on studying twisted boundary conditions. Our numerical results indicate the existence of a metallic phase in the presence of scattering between up and down spin components, while there is a sharp transition when the system decouples into two copies of the quantum Hall effect. In addition to the Z_2 invariant calculation in the case when up and down components are coupled, we also present a simple method of evaluating the integer invariant in the quantum Hall case where they are decoupled.