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

TL;DR

This paper investigates how power-law hopping influences the localization and ergodic properties of single particles in a non-Hermitian quasiperiodic one-dimensional model, revealing robust spectral edges and phase transitions.

Contribution

It introduces a detailed analysis of power-law hopping effects in non-Hermitian quasicrystals, identifying new spectral edges and phase regimes with analytical and numerical methods.

Findings

Existence of ergodic, multifractal, and localized phases depending on the power-law index.

Identification of ergodic-to-multifractal and ergodic-to-localized spectral edges.

Robustness of spectral edges against non-Hermitian effects.

Abstract

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.