  In this paper, a non-Hermitian Aubry-Andr\'e-Harper model with power-law
hoppings ($1/s^{a}$) and quasiperiodic parameter $\beta$ is studied, where $a$
is the power-law index, $s$ is the hopping distance, and $\beta$ is a member of
the metallic mean family. We find that under the weak non-Hermitian effect,
there preserves $P_{\ell=1,2,3,4}$ regimes where the fraction of ergodic
eigenstates is $\beta$-dependent as $\beta^{\ell}$L ($L$ is the system size)
similar to those in the Hermitian case. However, $P_{\ell}$ regimes are ruined
by the strong non-Hermitian effect. Moreover, by analyzing the fractal
dimension, we find that there are two types of edges aroused by the power-law
index $a$ in the single-particle spectrum, i.e., an ergodic-to-multifractal
edge for the long-range hopping case ($a<1$), and an ergodic-to-localized edge
for the short-range hopping case ($a>1$). Meanwhile, the existence of these two
types of edges is found to be robust against the non-Hermitian effect. By
employing the Simon-Spence theory, we analyzed the absence of the localized
states for $a<1$. For the short-range hopping case, with the Avila's global
theory and the Sarnak method, we consider a specific example with $a=2$ to
reveal the presence of the intermediate phase and to analytically locate the
intermediate regime and the ergodic-to-multifractal edge, which are
self-consistent with the numerically results.
