Advances in Theoretical and Mathematical Physics

Volume 26 (2022)

Number 3

Cobordism invariance of topological edge-following states

Pages: 673 – 710



Matthias Ludewig (School of Mathematical Sciences, University of Adelaide, Australia; and Fakultät für Mathematik, Universität Regensburg, Germany)

Guo Chuan Thiang (School of Mathematical Sciences, University of Adelaide, Australia; and Beijing International Center for Mathematical Research, Peking University, Beijing, China)


We prove that a spectral gap-filling phenomenon occurs whenever a Hamiltonian operator encounters a coarse index obstruction upon compression to a domain with boundary. Furthermore, the gapfilling spectra contribute to quantised current channels, which follow and are localised at the possibly complicated boundary. This index obstruction is shown to be insensitive to deformations of the domain boundary, so the phenomenon is generic for magnetic Laplacians modelling quantum Hall systems and Chern topological insulators. A key construction is a quasi-equivariant version of Roe’s algebra of locally compact finite propagation operators.

The authors thank U. Bunke, N. Higson and P. Hochs for their suggestions, and the Australian Research Council for financial support under grants FL170100020, DE170100149, and DP200100729. G.C.T. acknowledges H. Wang and Q. Wang for their hospitality at East China Normal University where part of this work was done, as well as G. De Nittis and E. Prodan for helpful discussions.

Published 22 February 2023