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Periodically driven non-Hermitian systems can exhibit rich topological bandstructure and non-Hermitian skin effect, without analogs in their static orHermitian counterparts. In this work we investigate the exceptionalband-touching points in the Floquet quasi-energy bands, the topologicalcharacterization of such exceptions points and the Floquet non-Hermitian skineffect (FNHSE). Specifically, we exploit the simplicity of periodicallyquenched two-band systems in one dimension or two dimensions to analyticallyobtain the Floquet effective Hamiltonian as well as locations of the manyexceptional points possessed by the Floquet bulk bands. Two different types oftopological winding numbers are used to characterize the topological features.Bulk-edge correspondence (BBC) is naturally found to break down due to FNHSE,which can be drastically different among different bulk states. Remarkably,given the simple nature of our model systems, recovering the BBC is doable inpractice only for certain parameter regime where a low-order truncation of thecharacteristic polynomial (which determines the Floquet band structure) becomesfeasible. Furthermore, irrespective of which parameter regime we work with, wefind a number of intriguing aspects of Floquet topological zero modes and $\pi$modes. For example, under the open boundary condition zero edge modes and $\pi$edge modes can individually coalesce and localize at two different boundaries.These anomalous edge states can also switch their accumulation boundaries whencertain system parameter is tuned. These results indicate that non-HermitianFloquet topological phases, though more challenging to understand than theirHermitian counterparts, can be extremely rich in the presence of FNHSE.

Non-Hermitian Floquet topological phases: Exceptional points, coalescent edge modes, and the skin effect