Quasiperiodic and periodic extended Hatano-Nelson model: Anomalous complex-real transition and non-Hermitian skin effect

We study the effect of quasiperiodic and periodic onsite potentials in a Hatano-Nelson model with next-nearest-neighbor hopping. By considering a nonreciprocal next-nearest-neighbor hopping and a quasiperiodic onsite potential under periodic boundary conditions, we show a breakdown of the typical correspondence between the delocalization-localization and complex-real transitions as a function of the potential strength. Moreover, we reveal that in the delocalized regime, when the potential strength increases, the eigenstates under open boundary conditions exhibit a bidirectional non-Hermitian skin effect, i.e., they tend to localize on both edges instead of localizing on either of the edges. However, when a periodic onsite potential is considered, the system not only exhibits a bidirectional skin effect but also shows a complete direction reversal of the skin effect as a function of the onsite periodic potential.