Topological index for periodically driven time-reversal invariant 2D systems

TL;DR

This paper introduces a new $Z_2$-valued topological index for periodically driven 2D systems with time-reversal symmetry, linking spectral gaps, edge states, and the Kane-Mele invariant.

Contribution

It defines a novel topological index for Floquet systems with time-reversal symmetry and relates it to existing invariants like Kane-Mele and Wess-Zumino amplitude.

Findings

The new index characterizes topological phases in driven systems.

The index relates to the Kane-Mele invariant via spectral gaps.

Numerical simulations confirm the index's relation to edge states.

Abstract

We define a new $Z_2$-valued index to characterize the topological properties of periodically driven two dimensional crystals when the time-reversal symmetry is enforced. This index is associated with a spectral gap of the evolution operator over one period of time. When two such gaps are present, the Kane-Mele index of the eigenstates with eigenvalues between the gaps is recovered as the difference of the gap indices. This leads to an expression for the Kane-Mele invariant in terms of the Wess-Zumino amplitude. We illustrate the relation of the new index to the edge states in finite geometries by numerically solving an explicit model on the square lattice that is periodically driven in a time-reversal invariant way.