Locally equivalent quasifree states and index theory

Abstract

We consider quasifree ground states of Araki's self-dual CAR 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 $K{O}_{\ast }(A^{\mathfrak{r}})$ with $A$ a $C^{\ast ,\mathfrak{r}}$-algebra of allowed deformations. When $A=C^{\ast }(X)$, 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.

arXiv commentv3: Final version to appear in J. Phys A, 31 pages

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