Power law hopping of single particles in one-dimensional non-Hermitian quasicrystals

In this paper, a non-Hermitian Aubry-Andr\'e-Harper model with power law hoppings (1/sᵃ) and quasiperiodic parameter β is studied, where a is the power law index, s is the hopping distance, and β is a member of the metallic mean family. We find that the number of the quasiperiodic parameter β-dependent regimes depends on the strength of the non-Hermiticity. Under a particularly weak non-Hermitian effect, there preserves P_=1,2,3,4 regimes where the fraction of ergodic eigenstates is β dependent as β^L (L is the system size), similar to those in the Hermitian case. However, P_ 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-localized edge, which are self-consistent with the numerically results.