Topological invariants for Floquet-Bloch systems with chiral, time-reversal, or 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 nonzero 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, the combination of the ${W}_{3}$ and the ${\mathbb{Z}}_{2}$ invariants allows us to distinguish between weak and strong topological phases.