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

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.

💡 Research Summary

The manuscript reports a novel type of non‑Hermitian skin effect (NHSE) that is essentially insensitive to boundary conditions, a phenomenon the authors term the “relative skin effect” (RSE). By introducing a spatially varying imaginary scalar potential i Φ(x) into a one‑dimensional lattice, the authors break translational symmetry in the bulk rather than at the physical edges. Consequently, both the energy spectrum and the spatial distribution of eigenstates remain unchanged when the system is switched between open, closed, or periodic boundary conditions. The localization of eigenstates is dictated solely by the points where Φ(x) crosses its spatial average Φₐ, which the authors call imaginary domain walls (IDWs).

To rationalize this behavior, the authors develop an effective description based on a linearized dispersion with an effective velocity vₖ± and a position‑dependent complex potential Φₖ±(x). The probability density of a mode obeys |ψₖ±(x)|² ∝ exp