Non-Commutative Tools for Topological Insulators

TL;DR

This paper reviews non-commutative mathematical tools used to analyze topological insulators, emphasizing their robustness and physical implications for bulk and edge states in disordered systems.

Contribution

It introduces and discusses non-commutative calculus and geometry tools for topological insulators, highlighting their ability to handle disorder and relate bulk invariants to edge phenomena.

Findings

Bulk topological invariants are robust in disordered systems.

A general relation links observable currents to edge indices.

Tools reveal physical consequences of topological robustness.

Abstract

This paper reviews several analytic tools for the field of topological insulators, developed with the aid of non-commutative calculus and geometry. The set of tools includes bulk topological invariants defined directly in the thermodynamic limit and in the presence of disorder, whose robustness is shown to have non-trivial physical consequences for the bulk states. The set of tools also includes a general relation between the current of an observable and its edge index, relation that can be used to investigate the robustness of the edge states against disorder. The paper focuses on the motivations behind creating such tools and on how to use them.