On the K-Theoretic classification of topological phases of matter

Abstract

A K-theoretic approach to the study of gapped topological phases was suggested by Kitaev, who produced a Periodic Table of topological insulators and superconductors, modelled on Bott periodicity. We take the algebraic viewpoint, and study gapped phases of free fermions through a twisted crossed product C*-superalgebra associated to the symmetry data of the dynamics. We identify the K-theoretic difference-group of this symmetry algebra, in the sense of Karoubi, as the appropriate point of entry for K-theory. Thus, K-theory provides groups of obstructions between symmetry-compatible gapped Hamiltonians, rather than classification groups for the Hamiltonians themselves. The phenomena of periodicity and dimension-shifts in the difference-groups is shown to be a robust consequence of various isomorphisms in operator K-theory, which have no commutative counterpart.