Quasiperiodic Skin Criticality in an Exactly Solvable Non-Hermitian Quasicrystal

TL;DR

This work identifies a novel universality class, the quasiperiodic non-Hermitian skin criticality (QNHSC), in a non-Hermitian quasiperiodic lattice derived from a modulated Hatano–Nelson model. By applying a nonunitary gauge transformation, the authors map the problem to a disorder-free chain, enabling exact analytical solutions for the spectrum and eigenstates. They show that all eigenstates share an energy-independent, multifractal spatial structure determined by the global phase $\theta$, with the inverse participation ratio scaling as $\mathrm{IPR} \sim N^{-\beta}$ and $\beta \approx 0.61$, independent of energy. The phenomenon persists in multiband ladders, with a symmetric case $J=t$ yielding universal, energy-independent profiles and unit Bhattacharyya overlap, providing a rigorous analytical benchmark for non-Hermitian quasiperiodic critical phenomena and guiding experimental realizations.

Abstract

Critical states in quasiperiodic systems defy the conventional dichotomy between extended and localized states. In this work, we demonstrate that non-Hermiticity fundamentally reshapes this paradigm by giving rise to an exactly solvable quasiperiodic critical phase with no energy selectivity. We introduce a non-Hermitian quasiperiodic lattice based on a modulated Hatano-Nelson model and uncover a new universality class of quasiperiodic skin criticality, in which all eigenstates share an identical multifractal spatial structure. Through a nonunitary gauge transformation, the system is mapped onto a disorder-free lattice, enabling exact analytical solutions for the full spectrum and eigenstates. As a consequence, the inverse participation ratio is strictly energy-independent and controlled solely by a global phase. We further show that this criticality persists in multiband lattices, establishing a general and analytically controlled framework for non-Hermitian quasiperiodic critical phenomena.

Figures (6)

• Figure 1: Comparison between conventional quasiperiodic criticality and the QNHSC. The panels illustrate the spatial profiles of three representative eigenstates, $|\psi_n|$ (colored lines), emphasizing the distinctive criticality of these two phases. (a) Conventional quasiperiodic criticality: Eigenstates display energy-dependent multifractal fluctuations. Different eigenstates are spatially separated, leading to a partial spatial overlap quantified by $\overline{\mathrm{BC}}<1$. (b) QNHSC: All eigenstates exhibit the same criticality. This collective condensation results in an almost perfect spatial overlap, characterized by $\overline{\mathrm{BC}}\simeq1$.
• Figure 2: (a) Schematic illustration of the quasiperiodically modulated HN lattice. (b) Complex energy spectrum plotted in the complex plane under PBC (blue points) and OBC (red points). Parameters are $N=89$, $J=1$, and $\theta=0$. (c) The $\overline{\mathrm{BC}}$ versus quasiperiodic modulation strength $J$ for system size $N=2584$ and global phase $\theta=0$. Left: Hermitian quasiperiodic model with $J_n^L=J_n^R$. Right: Non-Hermitian quasiperiodic model with $J_n^L = 1/J_n^R$.
• Figure 3: Analysis of eigenstate similarity and scaling behavior in non-Hermitian quasiperiodic HN chains. (a) The spatial distributions of PBC eigenstates $I_n$ for two values of the global phase: $\theta=0$ (red) and $\theta=\pi/2$ (blue). (b) Probability density function of the IPR values for $N=987$, collected over $10^4$ realizations of the global phase $\theta$. (c) Finite-size scaling of the $\overline{\mathrm{IPR}}$ versus system size $N$ on a logarithmic scale. Red points indicate numerical results, which agree with the theoretical prediction (red dashed line). For comparison, the blue points show the $\overline{\mathrm{IPR}}$ in a disorder-free lattice for $\alpha=0$. $\beta$ is the corresponding fractal dimension.
• Figure 4: Properties of the non-Hermitian quasiperiodic HN ladder model. (a) Schematic of the HN ladder model with quasiperiodic hopping amplitudes. (b) Energy spectrum under PBC (blue points) and OBC (red points) for $j=t=1$, $\Delta=2$, and system size $N=89$. (c) Representative eigenstate distributions $I_n$ for two distinct realizations of the global phase $\theta$. (d) The $\overline{\mathrm{BC}}$ as a function of modulation $J$ and $t$, with $N=89$, $\Delta=1$, and $\theta=0$. (e) Finite-size scaling of the $\overline{\mathrm{IPR}}$ versus system size $N$ (red points), obtained from $10^3$ realizations of $\theta$. The dashed line shows the theoretical prediction from the analytical IPR expression, while blue points correspond to the disorder-free lattice ($\alpha=0$).
• Figure 5: (a) Complex energy spectrum in the complex plane for PBC (blue circles) and OBC (red points). Parameters are $N=89$, $J=1$, and $\theta=0$. (b) Finite-size scaling of the $\overline{\mathrm{MIPR}}$ versus system size $N$ on a logarithmic scale. The solid line represent power-law fits. The statistical average is performed over $10^4$ random realizations of the global phase $\theta$.