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

Understand

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.