Universal Critical Scaling and Phase Diagram of the Non-Hermitian Skin Effect under Disorder

TL;DR

This paper reveals that non-Hermitian topology can protect transport against disorder up to a critical point, where a phase transition occurs, differing from traditional localization in Hermitian systems, and maps the phase diagram with critical exponents.

Contribution

It introduces a new understanding of how non-Hermitian topology alters disorder-induced localization, identifying a critical transition and universality class in such systems.

Findings

Identifies a sharp phase transition from skin phase to Anderson localization.

Determines critical exponents nd or the transition.

Maps the phase diagram showing  scaling with disorder and non-Hermiticity.

Abstract

Standard scaling theory dictates that disorder leads to immediate localization in one-dimensional Hermitian systems. We demonstrate that non-Hermitian topology fundamentally alters this paradigm, protecting transport up to a substantial critical disorder strength. By employing a numerically stable log-space transfer matrix approach up to thermodynamic scales (N=1000), we identify a sharp phase transition from the topological skin phase to the Anderson localized phase. Finite-size scaling analysis reveals that this transition belongs to a unique universality class with critical exponents \nu\approx1.50 and \beta\approx0.65. Furthermore, we map the global phase diagram, confirming that the critical disorder scales as W_c\propto\sqrt\gamma, consistent with localization suppression by an imaginary vector potential. Our results establish the rigorous limits of non-Hermitian topological protection in imperfect media.