Emergent multi-loop nested point gap in a non-Hermitian quasiperiodic lattice

TL;DR

This work introduces a non-Hermitian, geometric-series-modulated quasiperiodic lattice (GSM) and analyzes its localization and topological properties. By increasing the number of geometric-series terms $N$, the system exhibits multi-loop nested point gaps and mobility edges with higher winding numbers; in the limit $N\to\infty$, all mobility edges merge into a single edge with winding number $1$, as demonstrated via Avila's global theory. A dual-space transformation reveals opposite localization behavior and a skin effect, linking the lattice-space topology to dual-space dynamics. The results point to realizations in artificial quantum systems, such as Rydberg arrays, offering a route to engineer and probe high-winding-number non-Hermitian topologies and their localization transitions.

Abstract

We propose a geometric series modulated non-Hermitian quasiperiodic lattice model, and explore its localization and topological properties. The results show that with the ever-increasing summation terms of the geometric series, multiple mobility edges and non-Hermitian point gaps with high winding number can be induced in the system. The point gap spectrum of the system has a multi-loop nested structure in the complex plane, resulting in a high winding number. In addition, we analyze the limit case of summation of infinite terms. The results show that the mobility edges merge together as only one mobility edge when summation terms are pushed to the limit. Meanwhile, the corresponding point gaps are merged into a ring with winding number equal to one. Through Avila's global theory, we give an analytical expression for mobility edges in the limit of infinite summation, reconfirming that mobility edges and point gaps do merge and will result in a winding number that is indeed equal to one.

Figures (11)

• Figure 1: The scheme diagram of GSM lattice model (a) and multi-loop nested point gaps spectra (b).
• Figure 2: (a) The phase diagram of fractal dimension ${\Gamma}$ in the $\lambda-E$ plane. Eigenenergies in the complex plane with $\lambda=0.2$ (b), $\lambda=0.7$ (c), $\lambda=1$ (d) and $\lambda=1.7$ (e). (f) Arg[det($H-E_{b}$)] versus $\theta$, where we set $E_{b}=1,~1,~1,~-0.95$ [marked as rhombus in (b)-(e)]. Throughout, we set $L=610$, $q=0.2$, $\alpha=377/610$ and $N =2$.
• Figure 3: (a) The phase diagram of fractal dimension ${\Gamma}$ in the $\lambda-E$ plane. Eigenenergies in the complex plane with $\lambda=0.8$ (b), $\lambda=1.2$ (c), $\lambda=1.9$ (d). (f) Arg[det($H-E_{b}$)] versus $\theta$, where we set $E_{b}=0,~0,~-1.85$ [marked as rhombus in (b)-(d)]. The other parameters $L=610$, $\alpha=377/610$, $q=0.2$ and $N =3$.
• Figure 4: (a) The phase diagram of fractal dimension ${\Gamma}$ in the $\lambda-E$ plane, red dashed line denotes the analytical ME. (b) The winding number versus $\lambda$ for $E_{b}=-2$ (red line) and $E_{b}=2$ (blue line). The insets exhibit point gaps in the complex plane, where the base point $E_{b}=-2,~2$ [the rhombus in the insets]. Parameters $L=610$, $\alpha=377/610$, $q=0.2$ and $N = \infty$.
• Figure 5: (a) The dual space phase diagram. Eigenenergies in the complex plane with $\lambda=1.7$ under PBCs (b) and OBCs (c). (d) The dual space wave function of the eigenstate corresponding to an eigenenergy of $-0.9217$. The other parameters $L=610$, $\alpha=377/610$, $q=0.2$ and $N =2$.