Generalized Aubry-Andr\'{e}-Harper model with power-law quasiperiodic potentials

We investigate a generalized Aubry-Andr\'{e}-Harper (AAH) model with non-reciprocal hopping and power-law quasiperiodic potentials $V(i) = V\left[ \cos(2\pi \beta i) \right]^p$. Our study reveals that the interplay between nonreciprocity, quasiperiodicity, and the power-law exponent $p$ gives rise to a variety of phase transitions and localization phenomena. In the Hermitian case, the system undergoes a direct transition from extended to localized phases for $p=1, 2$, while for $p \geq 3$, an intermediate mixed phase emerges, characterized by the coexistence of extended and localized states and the presence of mobility edges. Importantly, we find that high inverse participation ratio (IPR) states appear at specific energy levels, whose positions are accurately described by the universal relation $x_n = n\beta - \lfloor n\beta \rfloor$, with a mirror-symmetric spatial distribution. In the non-Hermitian regime, the energy spectrum becomes complex and the $\mathcal{PT}$ transition coincides with the extended-to-localized phase boundary for $p = 1, 2$, whereas for $p \geq 3$, $\mathcal{PT}$-symmetry breaking occurs at the mixed-to-localized phase transition. This work reveals how power-law quasiperiodic potentials and non-reciprocal hopping govern phase transitions, providing new insight into localization phenomena of quasiperiodic systems.