Anomalous localization and duality in non-Hermitian quasiperiodic models

Abstract

Boundary conditions can have dramatic impact in non-Hermitian systems, as exemplified by the non-Hermitian skin effect. Focusing on one-dimensional non-Hermitian quasiperioidic lattices, we show that the interplay of quasiperiodicity and the non-Hermitian skin effect leads to counterintuitive localization properties. On the one hand, for Anderson localized states under the periodic boundary condition, we find that their localization features can be boundary-sensitive, which originates from the incompatibility of the periodic boundary condition with quasiperiodicity. On the other hand, for non-localized states, the well-known extended-localized duality relation can break down, as their counterparts in the dual model can also be nonlocal. We discuss how these remarkable phenomena can be engineered and analyzed from the perspective of Lyapunov exponents. Our findings shed new light on localization in non-Hermitian quasiperiodic systems.

Figures (5)

• Figure 1: Schematics of spatial profiles of eigenstates under the OBC and PBC for different sign patterns of the Lyapunov exponents. Green profiles denote non-localized states, red ones denote Anderson localized states, blue denotes boundary-localized skin modes, and yellow denotes states that are extended in one direction and localized in the other sun2025. The quantity $1/|\gamma_i|$ indicates the corresponding localization length.
• Figure 2: (a)(b) Eigenspectra of $\hat{H}_{1}(g,0)$ under (a) PBC and (b) OBC, respectively, where the color indicates the FD of the corresponding eigenstate. The parameters are (a1)(b1) $g=0$ and (a2)(b2) $g=1$. (c) Spatial distributions of typical localized states, marked in (a2) with matching colors. (d) Spatial distributions of every tenth eigenstate in (b2). These eigenstates are all localized toward the boundary $n=1$. The system size for all calculations is $N=4181$.
• Figure 3: Spatial distributions for eigenstates of $\hat{H}_{1}(g,0)$, with $E=0.025$, and (a) $g=0.5$, (b) $g=1$, (c) $g=1.5$, (d) $g=2$. Black dots represent numerical results, and red lines denote linear fits. The numbers outside the brackets represent the localization strengths obtained from the linear fits, and those within the brackets represent the corresponding $\gamma_{3}(E)$ and $\gamma_{4}(E)$. For all calculations, the system size is $N=10946$ and the PBC is imposed.
• Figure 4: (a)(b) Eigenspectra of (a) $\hat{H}_{1}(1,h)$ and (b) $\hat{H}_{2}(1,h)$, respectively, under the PBC, with color contours indicating the FD of the corresponding eigenstates. The parameters are (a1)(b1) $h=0.1$, (a2)(b2) $h=0.4$, and (a3)(b3) $h=0.6$. (c1-c3)(d1-d3) Spatial distributions of the eigenstates marked by circles on the real axis in (a1–a3) and (b1–b3), respectively. The corresponding sign patterns of $\gamma_{i}(E)$ are indicated in the upper-right corners of (c)(d). For all calculations, the system size is fixed at $N=4181$ under the PBC, with the irrational number $\tau$ is approximated by $\tau_{\rm RA}=2584/4181$.
• Figure S1: (a) Lyapunov exponents of the boundary-sensitive localized states in Fig. 2(a3) of the main text, plotted as functions of the inverse logarithm of the system size, $1/\ln N$. Points of different colors correspond to $\gamma_1$ (red), $\gamma_2$ (yellow), $\gamma_3$ (green), and $\gamma_4$ (blue), respectively. Dashed lines show linear fittings for the smallest $\gamma_1$, the largest $\gamma_2$, the smallest $\gamma_3$, and the largest $\gamma_4$ for each $N$. The sign pattern $(---+)$ of the Lyapunov exponents remains unchanged with increasing system size. (b) Scaling analysis of the FD for a dual pair of eigenstates of the models $H_1$ and $H_2$, marked by circles on the real axis in Fig. 4(a3) and (b3) of the main text. Black dots are numerical results, and red lines are linear fittings. Both FDs approach $1$ as $N$ approaches inifinity, confirming that these states are extended.