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

TL;DR

This paper introduces a unified theory for non-Hermitian localization, revealing boundary-insensitive skin effects caused by imaginary scalar potentials and classifying skin effects into electric and magnetic types with a phase transition.

Contribution

It develops a universal framework predicting localization in non-Hermitian systems and classifies skin effects into two fundamental types, explaining boundary-insensitive phenomena.

Findings

Imaginary scalar potentials induce boundary-insensitive skin effects.

A phase transition exists between electric and magnetic skin effects.

Eigenstates can be fully delocalized at the phase transition.

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.