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

被引:0
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作者
C. Bourne
A. Rennie
机构
[1] Friedrich-Alexander-Universität Erlangen-Nürnberg,Department Mathematik
[2] Tohoku University,WPI
[3] iTHEMS Research Group,Advanced Institute for Materials Research (WPI
[4] University of Wollongong,AIMR)
来源
Mathematical Physics, Analysis and Geometry | 2018年 / 21卷
关键词
Crossed product; Kasparov theory; Topological states of matter; Primary: 81R60; Secondary: 19K35, 19K56;
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摘要
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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathbb {R}}^{d}$\end{document}-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.
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