# Bulk-boundary correspondence for disordered free-fermion topological phases

## TL;DR

This paper develops a rigorous classification scheme for disordered free-fermion topological phases, proving bulk-boundary correspondence using real C*-algebras and $KR$-theory, applicable to all symmetry classes of the Tenfold Way.

## Contribution

It introduces a mathematically rigorous framework connecting bulk and boundary invariants for disordered free fermion systems via real operator $K$-theory.

## Key findings

- Established bulk-boundary correspondence for disordered systems.
- Constructed real C*-algebras for bulk and boundary data.
- Connected bulk and boundary classes through $KR$-theory maps.

## Abstract

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, 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.