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^*-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.
Highlights
The research field of topological quantum matter stands out by its considerable scope, pairing strong impact on experimental physics with mathematical structure and depth
Our formulation turns out to be well suited for the introduction of K -theoretical invariants, and we show how a disordered IQPV J ∈ A naturally defines a class in the real K -theory of the bulk algebra A (Definition 16)
The factor 2(Λ) is the Hilbert space of a so-called tight-binding model, where the lattice Λ ∼= Zd reflects the organisation of the crystal by atomic sites. The finitedimensional factors V± account for the spin and orbital degrees of freedom active in the conduction bands (V+) and the valence bands (V−) near the chemical potential
Summary
The research field of topological quantum matter stands out by its considerable scope, pairing strong impact on experimental physics with mathematical structure and depth. Building on the pioneering work of Bellissard et al [6,7,8,9], Schulz-Baldes et al [31, 43,50], we give a complete and uniform proof of bulk-boundary correspondence for symmetry-protected topological (SPT) phases of free fermions with disorder. Since for free fermions, the appearance of gapless edge modes signals a topologically non-trivial bulk, it is essential that the picture of K -theory used for the boundary classes allows the spectral gap to close at the boundary This is an important aspect not present in previous approaches to the bulk-boundary correspondence using K -theory.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
