Non-Hermitian bulk-boundary correspondence in a periodically driven system

Bulk-boundary correspondence, connecting the bulk topology and the edge states, is an essential principle of the topological phases. However, the conventional bulk-boundary correspondence is broken down in general non-Hermitian systems. In this paper, we construct a one-dimensional non-Hermitian Su-Schrieffer-Heeger model with periodic driving that exhibits the non-Hermitian skin effect: all the eigenstates are localized at the boundary of the systems, whether they are the bulk states or the zero and the $\ensuremath{\pi}$ modes. To capture the topological properties, the non-Bloch winding numbers are defined by the non-Bloch periodized evolution operators based on the generalized Brillouin zone. Furthermore, the non-Hermitian bulk-boundary correspondence is established: the non-Bloch winding numbers $({W}_{0,\ensuremath{\pi}})$ characterize the edge states with quasienergies $\ensuremath{\epsilon}=0,\ensuremath{\pi}$. In our non-Hermitian system, a novel phenomenon can emerge: the robust edge states can appear even when the Floquet bands are topological trivial with zero non-Bloch band invariant, which is defined in terms of the non-Bloch effective Hamiltonian. We also show the relation between the non-Bloch winding numbers $({W}_{0,\ensuremath{\pi}})$ and the non-Bloch band invariant $(\mathcal{W})$: $\mathcal{W}={W}_{0}\ensuremath{-}{W}_{\ensuremath{\pi}}$.