The Z_2-valued spectral flow of a symmetric family of Toeplitz operators

TL;DR

This paper introduces a secondary $oldsymbol{bZ}_2$-valued spectral flow for symmetric families of operators, establishing a new index theorem and applying it to Toeplitz operators, with implications for topological insulators.

Contribution

It develops a secondary $oldsymbol{bZ}_2$-valued spectral flow theory, proves an index theorem relating it to the suspension operator, and applies it to Toeplitz operators and topological insulators.

Findings

Secondary spectral flow equals secondary index of the suspension operator.

Graded secondary spectral flow of Toeplitz operators matches the secondary index of a Callias-type operator.

Recovers the bulk-edge correspondence for 2D topological insulators of type AII.

Abstract

We consider families $A(t)$ of self-adjoint operators with symmetry that causes the spectral flow of the family to vanish. We study the secondary $\mathbb{Z}_2$-valued spectral flow of such families. We prove an analog of the Atiyah-Singer-Robbin-Salamon theorem, showing that this secondary spectral flow of $A(t)$ is equal to the secondary $\mathbb{Z}_2$-valued index of the suspension operator $\frac{d}{dt}+A(t)$. Applying this result, we show that the graded secondary spectral flow of a symmetric family of Toeplitz operators on a complete Riemannian manifold equals the secondary index of a certain Callias-type operator. In the case of a pseudo-convex domain, this leads to an odd version of the secondary Boutet de Monvel's index theorem for Toeplitz operators. When this domain is simply a unit disc in the complex plane, we recover the bulk-edge correspondence for the Graf-Porta module for 2D topological insulators of type AII.