One-parameter scaling function for Anderson localization transitions in high-dimensional lattices with nonreciprocity

The applicability of the one-parameter scaling hypothesis to nonreciprocal high-dimensional disordered systems, specifically, those in dimensions higher than the critical dimension d₂=2, as predicted by the standard one-parameter scaling theory of Anderson localization, remains unresolved. In this work, we propose that the size dependence of the reduced participation ratio Y₋ in nonreciprocal d-dimensional systems of size L^d follows a universal one-parameter scaling form. This feature is encapsulated by the one-parameter function Y₋=dlnY₋/dlnL, which describes the critical properties of Anderson localization transitions, including the critical exponent, the fractal dimension D, and the critical reduced participation ratio Y₋^*. We demonstrate the applicability of our one-parameter scaling theory across the AI, A, and AII classes of three-dimensional nonreciprocal systems under periodic boundary conditions and further extend our analysis to higher dimensions in the presence of nonreciprocity, specifically d=4, 5, and 6. Our one-parameter scaling theory also applies to nonreciprocal systems with open boundary conditions, where all states are non-Hermitian skin modes. The scaling analysis indicates that open nonreciprocal systems do not exhibit genuine extended states extending throughout the system in the thermodynamic limit.