@article{8d2a703cf6594ac892c00aa9a01ba377,

title = "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∗-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.",

keywords = "Crossed product, Kasparov theory, Topological states of matter",

author = "C. Bourne and A. Rennie",

note = "Funding Information: Acknowledgements CB thanks Hermann Schulz-Baldes and Giuseppe De Nittis for posing the question of continuous Chern numbers to him, which eventually turned into the present manuscript, and for helpful discussions on the topic. The authors also thank Andreas Andersson, Alan Carey, Johannes Kellendonk and Emil Prodan for useful discussions. We would also like to thank an anonymous referee who pointed out an important error in an earlier version of this work. CB is supported by a postdoctoral fellowship for overseas researchers from The Japan Society for the Promotion of Science (No. P16728) and a KAKENHI Grant-in-Aid for JSPS fellows (No. 16F16728). CB also acknowledges support from the Australian Mathematical Society and the Australian Research Council during the production of this work. This work was supported by World Premier International Research Center Initiative (WPI), MEXT, Japan. Lastly, both authors thank the mathematical research institute MATRIX in Australia where part of this work was carried out. Publisher Copyright: {\textcopyright} 2018, Springer Nature B.V.",

year = "2018",

month = sep,

day = "1",

doi = "10.1007/s11040-018-9274-4",

language = "English",

volume = "21",

journal = "Mathematical Physics Analysis and Geometry",

issn = "1385-0172",

publisher = "Springer Netherlands",

number = "3",

}