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Chern Numbers, Localisation and the Bulk-edge Correspondence for Continuous Models of Topological Phases

Journal Article


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 ℝd-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 the case of the crossed product of a compact disorder space, 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.

Publication Date


  • 2018

Citation


  • Bourne, C. & Rennie, A. (2018). Chern Numbers, Localisation and the Bulk-edge Correspondence for Continuous Models of Topological Phases. Mathematical Physics, Analysis and Geometry, 21 (3), 16-1-16-62.

Scopus Eid


  • 2-s2.0-85049107636

Start Page


  • 16-1

End Page


  • 16-62

Volume


  • 21

Issue


  • 3

Place Of Publication


  • Netherlands

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 ℝd-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 the case of the crossed product of a compact disorder space, 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.

Publication Date


  • 2018

Citation


  • Bourne, C. & Rennie, A. (2018). Chern Numbers, Localisation and the Bulk-edge Correspondence for Continuous Models of Topological Phases. Mathematical Physics, Analysis and Geometry, 21 (3), 16-1-16-62.

Scopus Eid


  • 2-s2.0-85049107636

Start Page


  • 16-1

End Page


  • 16-62

Volume


  • 21

Issue


  • 3

Place Of Publication


  • Netherlands