A Unified Framework for the Non-Hermitian Localization: Boundary-Insensitive Modes and Electric-Magnetic Analogy

Abstract

The non-Hermitian skin effect is fundamentally characterized by its sensitivity to boundary conditions, reflected in changes to the energy spectrum and boundary-localized eigenstates. Here, we demonstrate that a spatially inhomogeneous imaginary scalar potential field induces a skin effect that is insensitive to boundary conditions. Both the spectrum and eigenstate distribution remain invariant, a behavior not captured by existing theories. We attribute this anomaly to translational symmetry breaking induced by spatially varying imaginary potentials in finite systems. We further formulate a theory that universally predicts localization in single-particle non-Hermitian systems. This framework classifies skin effects into two fundamental types: electric, driven by imaginary scalar potentials, and magnetic, driven by imaginary vector potentials, and reveals a phase transition between them, where eigenstates become fully delocalized. Our work provides a unified theory for non-Hermitian localization, allowing full control over skin modes via potential engineering in various platforms like photonic crystals and cold-atom systems.

Figures (6)

• Figure 1: Comparison of the nonreciprocal skin effect and the relative skin effect in terms of spectra and eigenstate distributions. CBC, OBC, and PBC denote closed, open, and periodic boundary conditions, respectively. (a)-(b) Spectrum and eigenstate distributions for the nonreciprocal skin effect under different boundary conditions, with the Bloch Hamiltonian $H(k) = 2t\cos k - 2\mathrm{i}\gamma\sin k$, where the strength of asymmetric hopping $\gamma=0.1$ and the nearest-neighbor hopping amplitude $t=1$. PBC denote periodic boundary conditions. (c)-(d) Spectrum and eigenstate distributions for the relative skin effect under different boundary conditions, with $\Phi(x) = \operatorname{sgn}((x - L/4)(x - 3L/4))$ and $\lambda = 0.02$. Black: RSE skin modes; red and blue: excluded localized states du_NonHermitianTearingDissipation_2024. Inset: numerical fitting of the localization length, using only $\operatorname{Re}(E) > 0$ modes ordered by decreasing energy. For (a)-(d), $N=200$.
• Figure 2: Eigenstate distributions under different boundary conditions and numerical fitting of the localization length for $\Phi(x) = \operatorname{sgn}((x - L/2)(x - L))$. (a) Eigenstate distribution when the imaginary domain wall coincides with the boundary. Under open boundary conditions, the boundary imaginary domain wall vanishes, and the associated localization disappears. Here $N=200,\lambda = 0.02$. (b) Numerical fitting of the localization length, using only modes with $\operatorname{Re}(E) > 0$, ordered by decreasing energy.
• Figure 3: The Hatano–Nelson model within the generalized non-Hermitian skin effect framework. (a) Real-space schematic of the lattice model. Arrows of different colors indicate distinct directions of nonreciprocal hopping, and red peaks denote the localization positions. (b) Eigenstate distribution of the Hatano–Nelson model, where the localization positions for the $k_+$ and $k_-$ modes coincide. $N=200,\gamma=0.05$. (c) Schematic of the field $\Phi_{k_+}(x)$.
• Figure 4: Classification of known non-Hermitian skin effects (NHSEs) within the generalized non-Hermitian skin effect (GNHSE) framework. Any single-particle Hamiltonian can be analyzed within the GNHSE framework to determine its localization properties, requiring only the effective velocity $\mathbf{v}_{k_\pm}$ and potential field $\Phi_{k_\pm}(x)$. Existing NHSEs can be universally constructed from distinct types of imaginary vector or scalar potential fields. The effective velocity $\mathbf{v}_{k_\pm} = \frac{\partial \mathrm{Re}(E(k))}{\partial k}|_{k_\pm}$ represents the linearized dispersion near mode $k_\pm$, such that $\operatorname{sgn}(\mathbf{v}_{k_+}\cdot\mathbf{v}_{k_-})<0$. Notably, both imaginary scalar li_ScalefreeLocalizationPT_2023 and vector li_ImpurityInducedScalefree_2021 potentials can independently induce SFL. For instance, SFL induced by a Dirac delta scalar potential $\delta(x-d)$li_ScalefreeLocalizationPT_2023, representing a type of NHESE, differs fundamentally from the RSE in its boundary sensitivity: $d$ is the distance from the boundary, which significantly influences the localization length of both the energy spectrum and the eigenstates.
• Figure 5: Two verification methods for the phase transition between the non-Hermitian magnetic skin effect (encompassing nonreciprocal skin effect) and the non-Hermitian electric skin effect (including relative skin effect). (a) Phase diagram of the order parameter $\eta$ for the nonreciprocal skin effect (NSE) and relative skin effect (RSE), where the white dashed and blue solid lines indicate numerical and theoretical phase boundaries for the $\sin k_+ =1$ mode, respectively. The warm-colored region corresponds to RSE dominance; the cold-colored region, to NSE dominance. At $\eta=0$, $\xi^{\text{E-M}}_{k_+}\rightarrow\infty$, so that $\operatorname{sgn}(1/\xi^{\text{E-M}}_{k_+}\xi^{\text{E-M}}_{k_-})=0$, resulting in fully delocalized eigenstates. Parameters: $\gamma\in[10^{-4},0.05]$, $\lambda\in[0.01,0.075]$, $j\in[N+1,2N]$, $N=50$. (c) For pure NSE ($\lambda = 0$), $\operatorname{sgn}(\xi^ {\text{M}}_{k_+}\xi^{\text{M}}_{k_-})>0$, so the $k_+$ and $k_-$ modes localize at the same edge. Moreover, since the localization length of the red and blue circles are reciprocal ($|\beta_{red}|=1/|\beta_{blue}|$), they form three concentric circles together with the black circle representing the Brillouin zone (BZ). (d) Upon introducing RSE ($\lambda = 0.05$), $\operatorname{sgn}(\xi^{\text{E}}_{k_+}\xi^{\text{E}}_{k_-})<0$ implies opposite localization for $k_+$ and $k_-$ modes. Consequently, one of $\xi^{\text{M}}_{k_+}$ or $\xi^{\text{M}}_{k_-}$ is enhanced while the other is suppressed. This splits both the red and blue circles into two trajectories: the mode with strengthened localization exhibits $|\beta|$ further from BZ (upper branches), while the weakened mode has $|\beta|$ closer to the BZ (lower branches). Some strongly suppressed modes ($|\sin k_\pm| \ll 1$) become plane waves (overlap of red/blue and black points). (e) At the phase transition point for $\sin k_+=1$ ($\lambda = 2\gamma$), even the mode with weakest localization coincides with the BZ, exhibiting plane-wave behavior.