Non-Hermitian systems exhibit the skin effect: eigenstates pile up at boundaries rather than spreading through the bulk. Add disorder, and Anderson localization competes — states localize around defects instead. The two localization mechanisms pull in different directions: the skin effect is boundary-driven, Anderson localization is bulk-driven. Which wins depends on the disorder strength, but the transition between them was poorly understood.
This paper proves that the crossover is topological. The change in the topological invariant associated with an eigenvalue — a winding number in the complex plane — coincides exactly with the eigenvector transition from skin-effect localization to Anderson localization. The proof works through Lyapunov exponents of the Hatano-Nelson model, connecting the spectral topology to the spatial structure of eigenstates.
The result establishes a universal criterion: the localization mechanism is determined not by microscopic details of the disorder but by a topological index. When the winding number changes, the eigenvectors switch from boundary-hugging to defect-hugging. The topological invariant doesn't just classify phases — it diagnoses the localization mechanism in real time, for each individual eigenvalue.