Multiple windows of eigenstate-selective delocalization in a non-Hermitian quasiperiodic lattice

We investigate localization in a non-Hermitian quasiperiodic lattice with a real quasiperiodic onsite potential of strength $W$ and a staggered imaginary quasiperiodic modulation of strength $λ$. Using the inverse participation ratio (IPR), normalized participation ratio (NPR), fractal dimension $D_2$, transfer-matrix Lyapunov exponent, and wave-packet dynamics, we find a strongly non-monotonic and state-dependent response. At fixed $λ$, varying $W$ yields two distinct delocalization windows, whereas varying $λ$ at fixed $W$ produces a single re-entrant delocalization window. These regimes arise within selected eigenstate sectors, where localized states coexist with extended or critical states, leading to mobility-edge-like behavior. The corresponding suppression of the Lyapunov exponent and enhancement of the dynamical RMS spreading are consistent with the IPR, NPR, and $D_2$ results. Finite-size scaling confirms the persistence of the re-entrant features, while the complex-energy spectrum resolves their state-selective structure. The resulting phase boundaries delineate distinct localization regimes in the $(W,λ)$ plane. Our results show that the interplay between quasiperiodicity and a staggered non-Hermitian modulation sharing the same quasiperiodic spatial structure can generate re-entrant, state-dependent localization, with implications for localization and dynamics in non-Hermitian quasiperiodic lattices.