Long-range hopping in a quasiperiodic potential weakens the non-Hermitian skin effect

In this paper, we investigate a non-Hermitian Aubry-Andr\'e-Harper model characterized by power-law hoppings ($1/{s}^{a}$) and a quasiperiodic parameter $\ensuremath{\beta}$, where $a$ denotes the power-law index, $s$ represents the hopping distance, and $\ensuremath{\beta}$ belongs to the metallic mean family. In the intermediate phases, we find that ergodic states correspond to complex eigenvalues, multifractal states correspond to real eigenvalues, and localized states may exhibit either complex or real eigenvalues. In the localized phase, the energy spectra are complex, with most of the eigenvalues of localized states being real, while a few are complex. Under open boundary conditions, our analysis of fractal dimensions and eigenstates reveals that all ergodic states transform into scale-free localized states. Furthermore, we demonstrate that long-range hoppings weaken the skin effect, offering another perspective for exploring non-Hermitian skin effects.