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

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 ν\approx1.50 and β\approx0.65. Furthermore, we map the global phase diagram, confirming that the critical disorder scales as W_c\propto\sqrtγ, consistent with localization suppression by an imaginary vector potential. Our results establish the rigorous limits of non-Hermitian topological protection in imperfect media.