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arXiv · 1611.06016

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

Chris Bourne·Adam Rennie
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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.

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Chris Bourne, Adam Rennie. 2016-11-18. Chern numbers, localisation and the bulk-edge correspondence for continuous models of topological phases. https://doi.org/10.1007/s11040-018-9274-4

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