Koopman Nonlinear Non-Hermitian Skin Effect

TL;DR

The paper addresses the challenge of characterizing nonlinear non-Hermitian skin effects when eigenstates lose meaning by introducing a Koopman-based diagnostic that localizes in a lifted observable space. It demonstrates that dominant Koopman eigenfunctions shift localization from linear to higher-order observables as nonlinearity increases, using a minimal nonlinear Hatano–Nelson model and a boundary-perturbation probe to reveal dynamical signatures. The study shows that boundary-amplitude sensitivity is governed by lifted-space localization and validates the approach with an additional nonlinear Hatano–Nelson variant, arguing that the Koopman framework provides a natural setting for nonlinear skin effects. Together, the results establish a principled, observable-space perspective on nonlinear non-Hermitian skins that complements traditional stationary-state analyses.

Abstract

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.

Figures (5)

• Figure 1: Illustration of the Koopman nonlinear non-Hermitian skin effect. (a) The nonlinear Schrödinger dynamics is recast as a linear evolution in an appropriate observable space within the Koopman framework. (b) The physical state $X(t)$ accumulates at the boundary due to nonreciprocal hopping, giving rise to the skin effect. (c) Unlike the linear skin effect, where localization occurs on the linear observable sector, the Koopman nonlinear skin effect is characterized by the localization of observables on higher-order observable bases.
• Figure 2: Koopman nonlinear non-Hermitian skin effect in the nonlinear Hatano-Nelson model ($L=30$, $t_{\rm R}=0.55$, $t_{\rm L}=0.45$). (a) Linear regime ($\epsilon=0$): dominant Koopman eigenfunctions are localized on the linear observable $\{x_n\}$. (b) Intermediate nonlinearity ($\epsilon=0.8$): the dominant Koopman eigenfunctions acquire a strong weight on the quadratic observables $\{x_n^2\}$, with only subleading contributions from the linear sector. (c) Exactly solvable nonlinear limit ($\epsilon=1$): the dominant Koopman eigenfunctions are fully localized on the higher-order (quadratic) observable $\{x_n^2\}$, demonstrating that the essential dynamical degrees of freedom are shifted to higher-order observables. In all panels, we show the twenty slowest-decaying modes, obtained using EDMD supplement.
• Figure 3: Dynamical manifestation of the Koopman nonlinear skin effect ($L=30$, $t_{\rm R}=0.55$, $t_{\rm L}=0.45$). (a) Schematic illustration of the boundary perturbation protocol. (b) Conceptual comparison between linear ($\epsilon=0$) and nonlinear ($\epsilon=0.8$) dynamics. (c) Time evolution of $\Delta O(t)$ defined in Eq. \ref{['eq:deltaO2']}. Blue and orange curves correspond to perturbations applied at the left ($j=1$) and right ($j=L$) boundaries, respectively, in the linear regime ($\epsilon=0$), while green and red curves show the corresponding results for $\epsilon=0.8$. The left inset shows the same data plotted on a logarithmic scale. The right inset shows the result obtained after reversing the direction of nonreciprocity ($T_{\rm R}\leftrightarrow T_{\rm L}$), ruling out a generic dynamical instability.
• Figure 4: Generality of the Koopman nonlinear skin effect ($L=30,\gamma=0.1$). Distribution of the dominant Koopman eigenfunctions in the lifted observable space for the model defined in Eq. \ref{['eq:KK-model']}. (a) Linear regime ($\epsilon = 0$): the Koopman eigenfunctions are localized on the linear observables. (b) Weak nonlinearity ($\epsilon = 0.05$): the Koopman eigenfunctions acquire finite weight on higher-order observables. (c) Stronger nonlinearity ($\epsilon = 0.3$): the dominant Koopman eigenfunctions are predominantly localized on higher-order observables. In all panels, we show the twenty slowest-decaying modes, obtained using EDMD supplement.
• Figure S1: Validation of the EDMD approximation ($L=30, \gamma=0.1,\epsilon=0.05$). (a) Per-sample relative one-step residual $\|\bm g(Z_m)-K\bm g(Y_m)\|_2/\|\bm g(Z_m)\|_2$ on the training data. (b) Multi-step rollout error $E_{\mathrm{roll}}(n)=\|y_n-\bm g(X_n)\|_2/\|\bm g(X_n)\|_2$ for an unseen initial condition. (c) Eigenfunction consistency check for the dominant Koopman mode: scatter plot of $\Re[\varphi(X_{n+1})]$ versus $\Re[\Lambda\varphi(X_n)]$.