Bulk-boundary correspondence for disordered free-fermion topological phases Alexander Alldridge1, Christopher Max1,2, Martin R. Zirnbauer2 1 Mathematisches Institut, E-mail:
[email protected] 2 Institut für theoretische Physik, E-mail:
[email protected],
[email protected] Received: 2019-07-12 Abstract: Guided by the many-particle quantum theory of interacting systems, we de- velop 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∗-algebras harbouring the bulk and boundary data of disordered free-fermion ground states. These we connect by a natural bulk-to-boundary short ex- act 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 pic- tures of real operator K-theory (or KR-theory): a bulk class, using Van Daele’s picture, along with a boundary class, using 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. This implies bulk-boundary correspondence, in the presence of disorder, for both the “strong” and the “weak” invariants. 1. Introduction The research field of topological quantum matter stands out by its considerable scope, pairing strong impact on experimental physics with mathematical structure and depth. On the experimental side, physicists are searching for novel materials with technolog- ical potential, e.g., for future quantum computers; on the theoretical side, mathemati- cians have been inspired to revisit and advance areas such as topological field theory and various types of generalised cohomology and homotopy theory.