Fate of Wannier-Stark localization and skin effect in periodically driven non-Hermitian quasiperiodic lattices

The eigenstates of one-dimensional Hermitian and non-Hermitian (NH) tight-binding systems (in the presence/absence of quasiperiodic potential) and an external electric field undergo complete localization with equally spaced eigenenergies, known as the Wannier-Stark (WS) localization. In this paper, we demonstrate that when the electric field is slowly modulated with time, nontrivial phases with multiple mobility edges emerge in place of the WS localized phase, which persists up to a certain strength of the non-Hermiticity. On the other hand, for a large driving frequency, we retrieve the sharp delocalized to the usual (no WS) localized phase transition, similar to the static NH Aubry-Andr\'e-Harper type without any electric field. This vanishing of WS localization can be attributed solely to the time-periodic drive and occurs irrespective of the non-Hermiticity. Interestingly, under the open boundary condition, we find that contrary to the undriven systems where an external electric field destroys the skin effect (SE) completely, the SE appears in certain regimes of the parameter space when the electric field is temporally driven. This appearance of SE is closely related to the absence of extended unitarity. In addition, in the presence of the drive, the skin states are found to be multifractal, contrary to its generic nature in such NH systems. An in-depth understanding about the behavior of the states in the driven system is established from the long-time dynamics of an initial excitation.