We investigate a generalized Aubry-André-Harper (AAH) model with non-reciprocal hopping and power-law quasiperiodic potentials $V\left(i\right)=V{[cos\left(2\pi \beta i\right)]}^p$. 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 $p\ge 3$, an intermediate mixed phase emerges, characterized by the coexistence of extended and localized states and the presence of mobility edges. Importantly, we find that prominent high-IPR states associated with well-resolved spectral gaps appear at specific energy levels, whose positions are captured by the relation $x_n=n\beta -\lfloor n\beta \rfloor$, for low-order $n$. 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 $p\ge 3, \mathcal{PT}$-symmetry breaking occurs at the mixed-to-localized phase transition. This work reveals how power-law quasiperiodic potentials and nonreciprocal hopping govern phase transitions, providing new insight into localization phenomena of quasiperiodic systems.

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• Localization
• Phase transitions
• 1-dimensional systems
• Disordered systems
• Non-Hermitian systems