2d Fu-Kane-Mele invariant as Wess-Zumino action of the sewing matrix

TL;DR

This paper establishes a novel connection between the 2D Fu-Kane-Mele invariant and the Wess-Zumino action of the sewing matrix, providing new insights into topological invariants of insulators and their relation to Berry connections.

Contribution

It demonstrates that the Fu-Kane-Mele invariant can be expressed as a Wess-Zumino action, linking topological invariants to field-theoretic actions and extending to 3D cases.

Findings

Fu-Kane-Mele invariant equals Wess-Zumino action of sewing matrix

Provides a direct proof relating strong invariant to Chern-Simons action

Extends the 2D result to 3D topological insulators

Abstract

We show that the Fu-Kane-Mele invariant of the 2d time-reversal-invariant crystalline insulators is equal to the properly normalized Wess-Zumino action of the so-called sewing matrix field defined on the Brillouin torus. Applied to 3d, the result permits a direct proof of the known relation between the strong Fu-Kane-Mele invariant and the Chern-Simons action of the non-Abelian Berry connection on the bundle of valence states.