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

Abstract

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