Disordered topological insulators: a non-commutative geometry perspective

Abstract

The progress in the field of topological insulators is impetuous, being sustained by a suite of exciting results on three fronts: experiment, theory and numerical simulation. Very often, the theoretical characterizations of these materials involve advanced and abstract techniques from pure mathematics, leading to complex predictions which nowadays are tested by direct experimental observations. Many of these predictions have already been confirmed. What makes these materials topological is the robustness of their key properties against smooth deformations and onset of disorder. There is quite an extensive literature discussing the properties of clean topological insulators, but the literature on disordered topological insulators is limited. This review deals with strongly disordered topological insulators and covers some recent applications of a well-established analytic theory based on the methods of non-commutative geometry and developed for the integer quantum Hall effect. Our main goal is to exemplify how this theory can be used to define topological invariants in the presence of strong disorder, other than the Chern number, and to discuss the physical properties protected by these invariants. Working with two explicit two-dimensional models, one for a Chern insulator and one for a quantum spin-Hall insulator, we first give an in-depth account of the key bulk properties of these topological insulators in the clean and disordered regimes. Extensive numerical simulations are employed here. A brisk but self-contained presentation of the non-commutative theory of the Chern number is given and a novel numerical technique to evaluate the non-commutative Chern number is presented. The non-commutative spin-Chern number is defined and the analytic theory, together with the explicit calculation of the topological invariants in the presence of strong disorder, is used to explain the key bulk properties seen in the numerical experiments presented in the first part of the review.