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

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 $C^{\ast }$-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.

46 pages. V2: results on localisation expanded and clarified. V3: Further revisions. V4: To appear in Mathematical Physics, Analysis and Geometry

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• Index theory and topological phases of aperiodic lattices
• The Cayley transform in complex, real and graded $K$-theory
• Localised module frames and Wannier bases from groupoid Morita equivalences
• From orbital magnetism to bulk-edge correspondence
• Classifying bulk-edge anomalies in the Dirac Hamiltonian