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

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.