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It has been some time since non-commutative geometry was proposed by JeanBellissard as a theoretical framework for the investigation of homogeneouscondensed matter systems. Recently, Bellissard's approach has beenenthusiastically adopted in the relatively young field of topologicalinsulators, where it facilitated many rigorous results concerning the stabilityof the topological invariants against disorder. In this work we present acomputational program based on the principles of non-commutative geometry andshowcase several applications to topological insulators. In the first part weintroduce the notion of a homogeneous material and define the class ofdisordered crystals together with the classification table which conjecturesall topological phases from this class. We continue with a discussion ofelectron dynamics in disordered crystals and we briefly review the theory oftopological invariants in the presence of strong disorder. We show how allthese can be captured in the language of non-commutative geometry using theconcept of non-commutative Brillouin torus, and present a list of knownformulas for various physical response functions. In the second part, weintroduce auxiliary algebras and develop a canonical finite-volumeapproximation of the non-commutative Brillouin torus. Explicit numericalalgorithms for computing generic correlation functions are discussed. In thethird part we derive upper bounds on the numerical errors and demonstrate thatthe canonical-finite volume approximation converges extremely fast to thethermodynamic limit. Convergence tests and various applications concludes ourpresentation.

A Computational Non-Commutative Geometry Program for Disordered Topological Insulators