Topological insulators and the Kane-Mele invariant: Obstruction and localization theory

Severin Bunk, Richard J. Szabo

Research output: Contribution to journalArticle

Abstract

We present homotopy theoretic and geometric interpretations of the Kane-Mele invariant for gapped fermionic quantum systems in three dimensions with time-reversal symmetry. We show that the invariant is related to a certain 4-equivalence which lends it an interpretation as an obstruction to a block decomposition of the sewing matrix up to nonequivariant homotopy. We prove a Mayer-Vietoris Theorem for manifolds with Z2-actions which intertwines Real and Z2-equivariant de Rham cohomology groups, and apply it to derive a new localisation formula for the Kane-Mele invariant. This provides a unified cohomological explanation for the equivalence between the discrete Pfaffian formula and the known local geometric computations of the index for periodic lattice systems. We build on the relation between the Kane-Mele invariant and the theory of bundle gerbes with Z2-actions to obtain geometric refinements of this obstruction and localisation technique. In the preliminary part we review the Freed-Moore theory of band insulators on Galilean spacetimes with emphasis on geometric constructions, and present a bottom-up approach to time-reversal symmetric topological phases.
Original languageEnglish
Article number2050017
JournalReviews in Mathematical Physics
Early online date9 Dec 2019
DOIs
Publication statusE-pub ahead of print - 9 Dec 2019

Keywords

  • equivariant and Jandl gerbes
  • Kane-Mele invariant
  • localization
  • time-reversal symmetry
  • Topological phases

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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