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

Abstract

<jats:p> 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 non-equivariant homotopy. We prove a Mayer–Vietoris Theorem for manifolds with [Formula: see text]-actions which intertwines Real and [Formula: see text]-equivariant de Rham cohomology groups, and apply it to derive a new localization 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 [Formula: see text]-actions to obtain geometric refinements of this obstruction and localization 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. </jats:p>

Differential geometric invariants for time-reversal symmetric Bloch bundles, II : The low-dimensional “quaternionic” case

“…Proposition 2.18 shows that such an invariant is indeed built through the isomorphism ˆÄ. More precisely, by combining Proposition 2.18 with Theorem 2.11 one obtains (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18) Ä. / WD … ı ˆÄ.…”

“…where the action of Map.X; U .2// 0 Z 2 on Map.X; SU.2// Z 2 is given by the automorphisms (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12).…”

“…The use of the letter † instead of X is motivated to easier connect the results discussed here with the theory developed in Section 3. 4 Proof We will start by proving the injectivity of ˆÄ. Suppose ; 0 2 Map.…”

“…In the case of the involutive torus .T 3 ; TR / described by (1-1) the invariant Ä s .Ᏹ; ‚/ takes values in the first (strong) summand of Z 2 ˚.Z 2 / 3 . For a more recent review of the topological interpretation of the (strong) Fu-Kane-Mele index we refer to [4].…”

’t Hooft Anomalies of Discrete Gauge Theories and Non-abelian Group Cohomology

“…Since their discovery over 30 years ago, immense progress in understanding and classifying these topological phases has been made. For instance, there exists a classification for non-interacting gapped systems in terms of twisted equivariant K-theory [FM13] (see also [BS17]).…”