Chiral symmetry and bulk-boundary correspondence in periodically driven one-dimensional systems

In periodically driven lattice systems, the effective (Floquet) Hamiltonian can be engineered to be topological; then, the principle of bulk-boundary correspondence guarantees the existence of robust edge states. However, such setups can also host edge states not predicted by the Floquet Hamiltonian. The exploration of such edge states and the corresponding unique bulk topological invariants has only recently begun. In this work we calculate these invariants for chiral symmetric periodically driven one-dimensional systems. We find simple closed expressions for these invariants, as winding numbers of blocks of the unitary operator corresponding to a part of the time evolution. This gives a robust way to tune these invariants using sublattice shifts. We illustrate our ideas on the periodically driven Su-Schrieffer-Heeger model, which, as we show, can realize a discrete-time quantum walk; this opens a useful connection between periodically driven lattice systems and discrete-time quantum walks. Our work helps interpret the results of recent simulations where a large number of Floquet Majorana fermions in periodically driven superconductors have been found.