Construction and properties of a topological index for periodically driven time-reversal invariant 2D crystals

TL;DR

This paper develops a mathematical framework for a topological invariant in periodically driven 2D lattice systems with time-reversal symmetry, extending the Kane-Mele invariant to include additional information relevant for quasienergy gaps.

Contribution

It introduces a new $\,\mathbb Z_2$-valued topological index for driven systems, relating it to Kane-Mele invariants and providing new expressions involving Wess-Zumino amplitudes.

Findings

Defined a gap-dependent topological invariant for driven 2D systems.

Connected the invariant to Kane-Mele invariants and boundary anomalies.

Provided mathematical expressions relating invariants to Wess-Zumino amplitudes.

Abstract

We present mathematical details of the construction of a topological invariant for periodically driven two-dimensional lattice systems with time-reversal symmetry and quasienergy gaps, which was proposed recently by some of us. The invariant is represented by a gap-dependent $\,\mathbb Z_2$-valued index that is simply related to the Kane-Mele invariants of quasienergy bands but contains an extra information. As a byproduct, we prove new expressions for the two-dimensional Kane-Mele invariant relating the latter to Wess-Zumino amplitudes and the boundary gauge anomaly.