Open Collections

Bulk-boundary correspondence for disordered free-fermion topological phases

Banff International Research Station for Mathematical Innovation and Discovery

Guided by the many-particle quantum theory of interacting systems, we develop a uniform classification scheme for topological phases of disordered gapped free fermions, encompassing all symmetry classes of the Tenfold Way. We apply this scheme to give a mathematically rigorous proof of bulk-boundary correspondence. To that end, we construct real C$^\ast$-algebras harbouring the bulk and boundary data of disordered free-fermion ground states. These we connect by a natural bulk-to-boundary short exact sequence, realising the bulk system as a quotient of the half-space theory modulo boundary contributions. To every ground state, we attach two classes in different pictures of real operator $K$-theory (or $KR$-theory): a bulk class, using Van Daele's picture, along with a boundary class in Kasparov's Fredholm picture. We then show that the connecting map for the bulk-to-boundary sequence maps these $KR$-theory classes to each other.

Mathematics; Quantum theory; Mathematical physics

video/mp4

Author affiliation: University of Cologne

2019-12-04

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/72541

Unreviewed

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