Chern numbers, localisation and the bulk-edge correspondence for continuous models of topological phases

TL;DR

This paper develops mathematical tools to analyze continuous models of disordered topological phases, enabling the calculation of higher Chern numbers and establishing a bulk-boundary correspondence.

Contribution

It constructs unbounded Kasparov modules and spectral triples for continuous disordered models, extending topological invariants and bulk-boundary relations to this setting.

Findings

Higher Chern numbers computed using non-unital local index formula

Extension of pairing to larger algebra related to dynamical localisation

Bulk-boundary correspondence established for continuous disordered models

Abstract

In order to study continuous models of disordered topological phases, we construct an unbounded Kasparov module and a semifinite spectral triple for the crossed product of a separable $C^*$-algebra by a twisted $\mathbb{R}^d$-action. The spectral triple allows us to employ the non-unital local index formula to obtain the higher Chern numbers in the continuous setting with complex observable algebra. In addition, the pairing can be extended to a larger algebra closely related to dynamical localisation, as in the tight-binding approximation. The Kasparov module allows us to exploit the Wiener-Hopf extension and the Kasparov product to obtain a bulk-boundary correspondence for continuous models of disordered topological phases.