Topological Quantum Numbers in Quasicrystals

Abstract

We provide an overview on the theory of topological quantum numbers from the point of view of non-commutative topology. Topological phases are described by K-groups of C*-algebras. The algebras are constructed from the set of positions of the nuclei of the materials we want to study. Topological quantum numbers are Chern numbers of K-group elements. Maps between K-groups which are of algebraic topological origin provide the means to obtain relations between different topological quantum numbers as, for instance, in the bulk edge correspondence. We present simple aperiodic examples related to quasicrystals to illustrate the theory.