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The spectral localizer for estimating bulk gaps and calculating K-theory

Banff International Research Station for Mathematical Innovation and Discovery

For systems where periodic boundary conditions are not an option, the spectral localizer is a tool that can separate edge and bulk effects. This is the tool introduced by Kisil in defining the Clifford spectrum. The localizer can been thought of as an augmented Hamiltonian modeling a finite system and a probe, which picks up the symmetries from the underlying system. In all ten symmetry classes and all dimensions, one can use the localizer to create a K-theory class in either the Trout or Van Daele pictures of K-theory that is expected to correspond to a bulk invariant. As new methods for computing the boundary maps in K-theory are discovered, more and more instances of the index can be rigorously proven to equal a bulk invariant. Specific applications discussed will include numerical studies (by various investigators) of nanowires in class BDI and a Chern insulators built on the vertices of an aperiodic tiling of the plane.

Mathematics; Quantum theory; Mathematical physics

video/mp4

Author affiliation: University of New Mexico

2019-12-03

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/72508

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/