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arXiv:2603.17404

Anomalous localization and duality in non-Hermitian quasiperiodic models

Wenzhi Wang, Tianyu Li, Wei Yi

quant-phcond-mat.dis-nnSubmitted Mar 18, 2026• Announced Mar 19, 2026
v1 · Mar 18, 2026v2 · Mar 21, 2026v3 · Oct 6, 2026
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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.

Journal reference
W. Wang, T. Li, and W. Yi, Phys. Rev. B 114, 144205 (2026)
DOI
10.1103/dqsx-y2r3
Comments
10 pages, 5figures

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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.
Quasiperiodic lattices can host critical eigenstates, whereas nonreciprocal hopping in non-Hermitian lattices can induce non-Hermitian skin effect. In this work, we investigate localization phenomena in a Hatano--Nelson model with quasiperiodically modulated hopping amplitudes, where nonreciprocity arises from unequal modulation strengths of the right and left hoppings. Using a non-unitary gauge transformation, we map the non-Hermitian system into a Hermitian quasiperiodic system and obtain an exact analytical expression for the Lyapunov exponent in the thermodynamic limit. Under periodic boundary conditions, inverse participation ratios and finite-size scaling analysis are used to identify the quasiperiodic critical regimes. The comparison shows that parameter regimes hosting quasiperiodic critical states under periodic boundary conditions can exhibit the non-Hermitian skin effect under open boundary conditions. Furthermore, the non-Hermitian skin effect associated with quasiperiodic critical regimes is also observed in representative long-range hopping models and multiband extensions. Our results provide an analytically controlled perspective on how quasiperiodicity, modulated nonreciprocity, and boundary conditions jointly shape the non-Hermitian skin effect in critical regimes.
The non-Hermitian skin effect (NHSE), which drives bulk states toward system boundaries, modifies bulk-boundary correspondence and complicates the identification of topological edge modes. Although breaking translational symmetry is known to influence this behavior, a systematic comparison of different structural classes remains limited. Here we investigate periodic, random, and quasiperiodic (Fibonacci) systems using a one-dimensional non-Hermitian quantum walk model. By matching the local scattering parameters in a topologically nontrivial regime, we isolate the role of spatial structure in the presence of the NHSE. We find that periodic systems exhibit pronounced boundary accumulation of bulk states. Random systems suppress this accumulation through Anderson localization, but the topological gap becomes partially filled with localized in-gap states. In contrast, the Fibonacci quasiperiodic system suppresses large-scale boundary accumulation while maintaining a well-defined topological gap. Analysis of the wave functions suggests that the hierarchical quasiperiodic structure fragments bulk states across multiple length scales, thereby mitigating the NHSE. These results identify deterministic quasiperiodicity as a distinct structural regime for controlling non-Hermitian skin dynamics and isolating topological boundary modes.
Localization in non-Hermitian quasicrystals can differ fundamentally from its Hermitian counterpart when non-reciprocity is spatially disordered. Here we study a one-dimensional non-Hermitian Aubry-André-Harper chain with a Bernoulli imaginary gauge field and quasiperiodic onsite modulation. In the nearest-neighbor limit, we identify an anomalous transition from a fully erratic non-Hermitian skin effect (ENHSE) phase to a fully localized phase. Although the fractal dimension vanishes in both regimes, the Lyapunov exponent and the fluctuation of the eigenstate center of mass sharply distinguish them. For generic finite-size realizations, this transition is further accompanied by a complex-to-real spectral change under periodic boundary conditions and a change of spectral winding from nontrivial to trivial. With weak next-nearest-neighbor hopping, we uncover an anomalous mobility edge at the same location as in the Hermitian generalized Aubry-André-Harper model, but separating Anderson-localized states from ENHSE-type macroscopic-accumulation states rather than extended states. We further show that this anomalous localization structure is reflected in spectral winding and wave-packet dynamics: single realizations exhibit winding-dependent drift, winding-resolved averaging preserves opposite directional responses, and full disorder averaging largely restores Hermitian-like transport. Our results establish practical diagnostics of anomalous localization and mobility edges in non-Hermitian quasicrystals.
The mobility ring (MR), a critical boundary in the complex energy plane separating extended and localized states, is fundamental to understanding the Anderson transition in non-Hermitian (NH) disordered systems. While MRs have been extensively studied in one-dimensional (1D) NH quasiperiodic models, rigorous analytical frameworks beyond 1D remain critically scarce. Here, we investigate a class of two-dimensional (2D) quasiperiodically decorated Lieb lattices (QDLLs) featuring complex incommensurate potentials selectively applied to the lattice vertices. By exactly mapping these 2D structures onto NH generalized Aubry-Andr{é}-Harper (AAH) models and leveraging extended-localized transition point, we analytically derive the Lyapunov exponents and obtain exact expressions for the MRs. These exact theoretical boundaries are strongly corroborated by numerical computations of wavefunction fractal dimensions and real-space probability distributions. Furthermore, we reveal distinct evolutionary behaviors of the MRs driven by the quasiperiodic potential strength: systems characterized by κ=2κ=2κ=2 possess a single MR, whereas systems with κ=3κ=3κ=3 undergo a dynamic sequential evolution from a single integrated ring into two independent rings. We hope that our exact results of MRs in 2D will benefit the study of Anderson localizations and MRs in high-dimensional NH systems.
Non-Hermitian systems exhibit unique spectral properties, including the non-Hermitian skin effect and exceptional points, often influenced by boundary conditions. The modulation of these phenomena by generalized boundary conditions remains unexplored and not understood. Here, we analyze the Hatano-Nelson model with generalized boundary conditions induced by complex hopping amplitudes at the boundary. Using similarity transformations, we determine the conditions yielding real energy spectra and skin effect, and identify the emergence of exceptional points where spectra transition from real to complex. We demonstrate that tuning the boundary hopping amplitudes precisely controls the non-Hermitian skin effect, i.e., the localization of eigenmodes at the lattice edges. These findings reveal the sensitivity of spectral and localization properties to boundary conditions, providing a framework for engineering quantum lattice models with tailored spectral and localization features, with potential applications in quantum devices.