Bulk-boundary correspondence for disordered free-fermion topological phases

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.

v1: 60 pages; v2: References added, 61 pages; v3: Minor improvements, 60 pages

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• Topological zero modes and edge symmetries of metastable Markovian bosonic systems
• The KO-valued spectral flow for skew-adjoint Fredholm operators
• The Cayley transform in complex, real and graded $K$-theory
• Two-Dimensional Time-Reversal-Invariant Topological Insulators via Fredholm Theory
• Locally equivalent quasifree states and index theory
• 'Real' gerbes and Dirac cones of topological insulators
• Classification of topological invariants related to corner states
• Winding Number Statistics for Chiral Random Matrices: Averaging Ratios of Parametric Determinants in the Orthogonal Case
• Absolutely continuous edge spectrum of topological insulators with an odd time-reversal symmetry
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• Topological Anderson insulators by latent symmetry
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• Comparison between two approaches to classify topological insulators using K-theory
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