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arXiv:2604.01638

Mechanism for scale-free skin effect in one-dimensional systems

Shu-Xuan Wang

quant-phcond-mat.mes-hallSubmitted Apr 2, 2026• Announced Apr 3, 2026
v1 · Apr 2, 2026v2 · Aug 28, 2026
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Abstract

Non-Hermitian skin effect (NHSE) is one of the most fascinating phenomena in non-Hermitian systems, which refers to enormous eigenstates localize at the boundary exponentially under open boundary condition (OBC). For typical NHSE, the localization length for a skin mode is independent of the system's size. Recently, some studies have revealed that for specific $1$-dimensional model, the localization length for eigenstates are proportional to the system's length under generalized boundary condition (GBC), and such phenomenon is dubbed as scale-free skin effect (SFSE). Further, SFSE is discovered in $1$-dimensional Hermitian chain with pure imaginary impurity at the end. In this work, we propose a mechanism for SFSE in 1-dimensional systems, which is model-independent. Our work provide a viewpoint for researching SFSE and shed new light on understanding finite size effect in non-Hermitian systems.

Comments
8 pages, 2 figures

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Non-Hermitian skin effect (NHSE) is one of the most fascinating phenomena in non-Hermitian systems, which refers to enormous eigenstates localize at the boundary exponentially under open boundary condition (OBC). For typical NHSE, the localization length for a skin mode is independent of the system's size. Recently, some studies have revealed that for specific 111-dimensional model, the localization length for eigenstates are proportional to the system's length under generalized boundary condition (GBC), and such phenomenon is dubbed as scale-free skin effect (SFSE). Further, SFSE is discovered in 111-dimensional Hermitian chain with pure imaginary impurity at the end. In this work, we propose a mechanism for SFSE in 1-dimensional systems, which is model-independent. Our work provide a viewpoint for researching SFSE and shed new light on understanding finite size effect in non-Hermitian systems.
The non-Hermitian skin effect, namely, that the eigenvalues and eigenstates of a non-Hermitian tight-binding Hamiltonian have significant differences under open or periodic boundary conditions, is a remarkable phenomenon of non-Hermitian systems. Inspired by the presence of the non-Hermitian skin effect, we study the evolution of wave packets in non-Hermitian systems, which can be determined using the single-particle Green's function. Surprisingly, we find that in the thermodynamic limit, the Green's function does not depend on boundary conditions, despite the presence of skin effect. We provide a general proof for this statement in arbitrary dimension with finite hopping range, with an explicit illustration in the non-Hermitian Su-Schrieffer-Heeger model. We also explore its applications in noninteracting open quantum systems described by the master equation. We demonstrate that the evolution of the density matrix is independent of the boundary condition.
The non-Hermitian skin effect is nonreciprocity-induced localization phenomena in which a macroscopic number of eigenstates accumulate anomalously at the boundary, accompanied by the extreme sensitivity to boundary conditions. Here, we develop a geometric characterization of the non-Hermitian skin effect. We demonstrate that the localization length scale associated with the skin effect is encoded in the quantum metric defined solely from right eigenstates, but not in the biorthogonal quantum metric. Moreover, we show that the quantum metrics can exhibit the power-law divergences at gapless points that depend on the different boundary conditions. We also reveal that cusps of the generalized Brillouin zone in non-Bloch band theory are signaled by discontinuities in the quantum metrics. We illustrate these behavior using prototypical non-Hermitian models, such as the Hatano-Nelson model and the non-Hermitian, nonreciprocal Su-Schrieffer-Heeger model.
Symmetry-protected ideal flat bands in one-dimensional (1D) Hermitian lattices are populated by compact localized states (CLS) - a special class of localization with wavefunctions confined within a small region. In this work, we discover that the non-Hermitian skin effect (NHSE) can appear in a flat band. Unlike conventional NHSEs for dispersive bands that are protected by nontrivial point-gap topology, the flat band remains a point on the complex-energy plane and is therefore always topologically trivial. We found that, intriguingly, the flat-band skin effect (FBSE) is associated with the non-trivial spectral topology of the dispersive bands enclosing the flat band on the complex-energy plane, so it only emerges within a finite range of non-Hermitian parameters and can counterintuitively disappear at large non-Hermiticity. Moreover, the gaps between the flat and the dispersive bands can close at higher-order exceptional points under both periodic and open boundary conditions. The flat-band wavefunctions are discontinuous in quantum distance across these exceptional points, signifying that the gap-closing is singular. The FBSE was experimentally observed in a non-Hermitian mechanical lattice. Our work reveals flat-band phenomena unique to non-Hermitian systems and highlights new possibilities in quantum geometry and localization control.
Non-Hermitian skin effects are conventionally manifested as boundary localization of eigenstates in linear systems. In nonlinear settings, however, where eigenstates are no longer well defined, it becomes unclear how skin effects should be faithfully characterized. Here, we propose a Koopman-based characterization of nonlinear skin effects, in which localization is defined in terms of Koopman eigenfunctions in a lifted observable space, rather than physical states. Using a minimal nonlinear extension of the Hatano-Nelson model, we show that dominant Koopman eigenfunctions localize sharply on higher-order observables, in stark contrast to linear skin effects confined to linear observables. This lifted-space localization governs the sensitivity to boundary amplitude perturbations, providing a distinct dynamical signature of the nonlinear skin effect. Our results establish the Koopman framework as a natural setting in which skin effects unique to nonlinear non-Hermitian systems can be identified.
The non-Hermitian skin effect can lead to directional amplification of response, with the associated end-to-end Green's function generally exhibiting size dependence. Any deviation in length or local disorder can drastically alter the amplification factor, rendering the response fragile in practical implementations. In this work, we identify a new type of scale-free, topological, and directionally amplified response in a Hatano-Nelson model under perturbed open boundary conditions. The scale-free response can be attributed to the first order boundary effect and characterized by a winding number defined on a continuous generalization of the finite-size Brillouin zone-a concept introduced in this work. Such scale-free behavior endows the end-to-end Green's function with significant robustness and making it promising for practical applications.
The non-Hermitian skin effect, in which eigenstates of non-Hermitian Hamiltonians are localized at one boundary in the open boundary condition, has attracted great interest recently. In this paper, we investigate the skin effect in one-dimensional dissipative quantum many-body systems, which we call the Liouvillian skin effect (LSE). We rigorously identify the existence of the LSE for generalized boundary conditions by solving the Liouvillian superoperator of an exactly solvable model with the advantage of the Bethe ansatz. The LSE is sensitive to boundary conditions where the signature is reflected in eigenfunctions of the system. We confirm that the LSE is fragile to a tiny coflow boundary hopping with non-Hermitian current but can survive a counterflow boundary hopping in the thermodynamic limit. Our work provides a prototypical example of exactly solvable dissipative quantum many-body lattice systems exhibiting the LSE for generalized boundary conditions. It can be further extended to other integrable open quantum many-body models.