Universal Thouless relations for disordered non–Hermitian systems in one dimension

Disordered non-Hermitian systems exhibit rich localization behavior without Hermitian analogs, yet a unified and quantitative framework for spectra and localization has been lacking. We establish universal Thouless relations (UTRs) for one-dimensional disordered non-Hermitian systems that relate spectral densities to Lyapunov exponents. The UTRs apply to arbitrary hopping ranges and disorder types, and determine spectral densities and localization in the thermodynamic limit without large-scale diagonalization. Using the UTRs, we show that the transition between the skin and Anderson regimes is topological, triggered by the closing of a single Lyapunov gap Δ γ . At the transition, we identify a unidirectional multifractal state that is localized in one direction but multifractal in the other. We further derive an exact topological criterion distinguishing the two localization regimes. Our work provides a unified framework (i.e., the UTRs) for studying disordered non-Hermitian systems and opens new avenues for predicting and controlling wave localization.