Disordered topological insulators via C^*-algebras

¹ University of New Mexico, Department of Mathematics and Statistics - Albuquerque, NM 87131, USA
² Microsoft Research, Station Q, Elings Hall, University of California - Santa Barbara, CA 93106, USA

Received: 16 September 2010
Accepted: 4 December 2010

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 Z₂ 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 Z₂ 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.