Impact of nonreciprocal hopping on localization in non-Hermitian quasiperiodic systems

We investigate the localization, topological properties, and non-Hermitian skin effect in the non-Hermitian Aubry-Andr\'e-Harper model, incorporating complex quasiperiodic potential and both unmodulated and modulated nonreciprocal hopping. Using Avila's global theory, we calculate the Lyapunov exponent to quantify the localization of eigenstates. The phase diagram of the system, after rescaling the imaginary phase, matches that of the Hermitian counterpart. Loops in the complex spectrum appear only in the localized phase, and we demonstrate that topological transitions coincide with localization transitions. For nonreciprocal hopping, the phase-transition lines obtained by analyzing the asymmetric localization behavior in the localized phase using similarity transformations are consistent with Avila's predictions. Under periodic boundary conditions, the complex spectrum always contains loops, and the topological phase transition aligns with the localization transition. For unmodulated nonreciprocal hopping, the extended phase transitions into a skin phase under open boundary conditions, while the critical phase remains unaffected. In contrast, modulated nonreciprocal hopping drives both the extended and critical phases into skin phases.