Discrete-time quantum walks (DTQWs) provide a convenient platform for a realization of many topological phases in noninteracting systems. They often offer more possibilities than systems with a static Hamiltonian. Nevertheless, researchers are still looking for DTQW symmetries protecting topological phases and for definitions of appropriate topological invariants. Although the majority of DTQW studies on this topic focus on the so-called split-step quantum walk, two distinct topological phases can be observed in more basic models. Here we infer topological properties of the basic DTQWs directly from the mapping of the Brillouin zone to the Bloch Hamiltonian. We show that for translation-symmetric systems they can be characterized by a homotopy relative to special points. We also propose a topological invariant corresponding to this concept. This invariant indicates the number of edge states at the interface between two distinct phases.
We introduce
Z2-valued bulk invariants for symmetry-protected topological phases in
2+1-dimensional driven quantum systems. These invariants adapt the
W3 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
W3 and the
Z2 invariants allows us to distinguish between weak and strong topological phases.