Topological invariants for Floquet-Bloch systems with chiral, time-reversal, or particle-hole symmetry

TL;DR

This paper develops $Z_2$-valued bulk invariants for 2+1D driven quantum systems with symmetries, linking bulk properties to boundary states and distinguishing topological phases.

Contribution

It introduces symmetry-adapted $Z_2$-valued invariants for Floquet systems, extending the $W_3$-invariant to classify symmetry-protected topological phases.

Findings

Invariants predict boundary states in driven models.

Successfully applied to chiral, time-reversal, and particle-hole symmetric systems.

Distinguishes between weak and strong topological phases for particle-hole symmetry.

Abstract

We introduce $\mathbb Z_2$-valued bulk invariants for symmetry-protected topological phases in $2+1$ dimensional driven quantum systems. These invariants adapt the $W_3$-invariant, expressed as a sum over degeneracy points of the propagator, to the respective symmetry class of the Floquet-Bloch Hamiltonian. The bulk-boundary correspondence that holds for each invariant relates a non-zero value of the bulk invariant to the existence of symmetry-protected topological boundary states. To demonstrate this correspondence we apply our invariants to a chiral Harper, time-reversal Kane-Mele, and particle-hole symmetric graphene model with periodic driving, where they successfully predict the appearance of boundary states that exist despite the trivial topological character of the Floquet bands. Especially for particle-hole symmetry, combination of the $W_3$ and the $\mathbb Z_2$-invariants allows us to distinguish between weak and strong topological phases.