Locally equivalent quasifree states and index theory

Abstract

AbstractWe consider quasifree ground states of Araki’s self-dual canonical anti-commutation relation algebra from the viewpoint of index theory and symmetry protected topological (SPT) phases. We first review how Clifford module indices characterise a topological obstruction to connect pairs of symmetric gapped ground states. This construction is then generalised to give invariants inKO*(Ar)withAaC*,r-algebra of allowed deformations. WhenA=C*(X), the Roe algebra of a coarse spaceX, and we restrict to gapped ground states that are locally equivalent with respectX, aK-homology class is also constructed. The coarse assembly map relates these two classes and clarifies the relevance ofK-homology to free-fermionic SPT phases.