Finite volume calculation of $K$-theory invariants

TL;DR

This paper introduces a finite volume method to compute $K$-theory invariants for operators on lattices, connecting index theory with practical calculations relevant to topological insulators.

Contribution

It presents a novel finite-dimensional matrix approach, the spectral localizer, for calculating $K$-theory invariants and secondary $ ext{Z}_2$-invariants in topological insulators.

Findings

Index computed as the signature of the spectral localizer.

Secondary $ ext{Z}_2$-invariants obtained via Pfaffian sign.

Reconciliation of two approaches to topological insulator invariants.

Abstract

Odd index pairings of $K_1$-group elements with Fredholm modules are of relevance in index theory, differential geometry and applications such as to topological insulators. For the concrete setting of operators on a Hilbert space over a lattice, it is shown how to calculate the resulting index as the signature of a suitably constructed finite-dimensional matrix, more precisely the finite volume restriction of what we call the spectral localizer. In presence of real symmetries, secondary $\mathbb{Z}_2$-invariants can be obtained as the sign of the Pfaffian of the spectral localizer. These results reconcile two complementary approaches to invariants of topological insulators.