Disordered Topological Insulators: A Brief Introduction

Abstract

It has been some time since non-commutative geometry was proposed by Jean Bellissard as a theoretical framework for the investigation of homogeneous condensed matter systems. Recently, Bellissard's approach has been enthusiastically adopted in the relatively young field of topological insulators, where it facilitated many rigorous results concerning the stability of the topological invariants against disorder. In this work we present a computational program based on the principles of non-commutative geometry and showcase several applications to topological insulators. In the first part we introduce the notion of a homogeneous material and define the class of disordered crystals together with the classification table which conjectures all topological phases from this class. We continue with a discussion of electron dynamics in disordered crystals and we briefly review the theory of topological invariants in the presence of strong disorder. We show how all these can be captured in the language of non-commutative geometry using the concept of non-commutative Brillouin torus, and present a list of known formulas for various physical response functions. In the second part, we introduce auxiliary algebras and develop a canonical finite-volume approximation of the non-commutative Brillouin torus. Explicit numerical algorithms for computing generic correlation functions are discussed. In the third part we derive upper bounds on the numerical errors and demonstrate that the canonical-finite volume approximation converges extremely fast to the thermodynamic limit. Convergence tests and various applications concludes our presentation.