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 R d -action. The spectral triple allows us to employ the nonunital 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.

Electron Dynamics: Concrete Physical Models

“…The recent work [11] by Bourne and Rennie extended these result to continuum models. Below we briefly state the main results for the discrete case, which are covered extensively in [26].…”

Observation of Topological Edge Modes in a Quasiperiodic Acoustic Waveguide

“…• The proof [34] of the quantization and stability of the topological invariants for continuum models also proceeds quite differently from the one for discrete models [29].…”

“…The aim of this note is to walk the reader through [34], as adopted to the acoustic waveguide analyzed in the main text.…”

“…), Bulk-boundary correspondence. According to [34], the topological class of P ω is mapped into the K 1 -class of the torus T 2 generated by the function e ıφ . Since it involves only the vertical coordinate of the torus 6, the horizontal coordinate plays no role in the bulk-boundary correspondence treated in our work.…”

“…To assess the topological character of the gaps, we employ the bulk-boundary correspondence for continuum models established in [34]. The bulk-topological invariant is supplied by the non-commutative Chern number of the gap projection P G = χ (−∞,G] ∆ m (φ) − G :…”