Bulk-Boundary Correspondence of Disordered Topological Insulators and Superconductors

In this work we establish a systematic formulation and proof of the bulk-boundary correspondence of disordered topological insulators and superconductors for all symmetry classes of the Tenfold Way. We establish a physically natural description of the ground states in the bulk and at the boundary of these solids for all symmetry classes of the Tenfold Way. This is based on the description of symmetric ground states in terms of so-called quasi-particle vacua with pseudo-symmetries, as introduced by Kennedy--Zirnbauer in the case of no disorder. In doing so, we construct, on the basis of physically natural assumptions, C*-algebras containing the relevant physical quasi-particle vacua of disordered solids in the bulk and in the vicinity of a boundary. We define a physically natural notion of topological phases for the ground states of disordered topological insulators and superconductors in the bulk and at the boundary. The description of ground states in terms of quasi-particle vacua allows us to identify the topological phases with K-classes of the corresponding C*-algebras of observables. An essential aspect of this work is that we have adjusted the description of the K-theory in such a way that the classes are naturally described by the quasi-particle vacua with pseudo-symmetries in the bulk and at the boundary. This leads to a physically natural description of the topological phases and allows us to formulate the bulk-boundary correspondence systematically. The physically natural and explicit form of the bulk-boundary correspondence allows for a deeper insight into its physical properties. To describe the topological phases in the bulk and at the boundary of topological insulators in more detail, we then derive topological invariants quantifying the properties of these topological phases