Disordered Topological Insulators: A Non-Commutative Geometry Perspective

TL;DR

This review explores how non-commutative geometry can define and compute topological invariants in strongly disordered topological insulators, extending beyond traditional Chern numbers, with numerical methods applied to specific models.

Contribution

It introduces a non-commutative geometric framework for topological invariants in disordered insulators and presents a novel numerical technique for their evaluation.

Findings

Non-commutative Chern number can be computed in disordered regimes.

Topological invariants remain well-defined under strong disorder.

Numerical simulations confirm the robustness of topological properties.

Abstract

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 (NCG) 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 2-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 are used to explain the key bulk properties seen in the numerical experiments presented in the first part of the review.