Locally equivalent quasifree states and index theory

We 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 in KO*(A(t)) with A a C*(,t)-algebra of allowed deformations. When A = C*(X), the Roe algebra of a coarse space X, and we restrict to gapped ground states that are locally equivalent with respect X, a K-homology class is also constructed. The coarse assembly map relates these two classes and clarifies the relevance of K-homology to free-fermionic SPT phases.