Lyapunov formulation of band theory for disordered non-Hermitian systems

TL;DR

This paper introduces a real-space Lyapunov approach to analyze disordered non-Hermitian systems, revealing universal relations, topological criteria, and critical states that extend band theory beyond translational symmetry.

Contribution

It develops a Lyapunov-based real-space band theory for disordered non-Hermitian systems, connecting spectral and localization properties with topology.

Findings

Universal non-Hermitian Thouless relations derived

Topological criterion linking skin and Anderson localization modes

Identification of a skin-Anderson transition with critical states

Abstract

Non-Bloch band theory serves as a cornerstone for understanding intriguing non-Hermitian phenomena, such as the skin effect and extreme spectral sensitivity to boundary conditions. Yet this theory hinges on translational symmetry and thus breaks down in disordered systems. Here, we develop a real-space Lyapunov formulation of band theory that governs the spectra and eigenstates of disordered non-Hermitian systems. This framework yields universal non-Hermitian Thouless relations linking spectral density and localization to Lyapunov exponents under different boundary conditions. We further identify an exact topological criterion: skin modes and Anderson-localized modes correspond to nonzero and zero winding numbers, respectively, revealing the topological nature of the skin-Anderson transition. This transition is dictated by an essential Lyapunov exponent and gives rise to novel unidirectional critical states. Our formulation provides a unified and exact description of spectra and localization in generic one-dimensional non-Hermitian systems without translational symmetry, offering new insights into the interplay among non-Hermiticity, disorder, and topology.