Dynamical localization in non-Hermitian quasicrystals

We study the localization transition in periodically driven one-dimensional non-Hermitian lattices where piece-wise two-step drive is constituted by uniform coherent tunneling and incommensurate onsite gain loss. find that system can be localized, delocalized, or mixed-phase depending on driving frequency phase shift of complex potential. Two critical frequencies are identified, first one corresponds to largest potential so quasi-energy spectrum still real all states extended, second disappear full spectrum, very weak leads emergence localized when lower than this frequency. In high limit, we separates two regions with respectively tends a constant value captured an effective Hamiltonian.