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 ${d}_{c}=2$, 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 ${Y}_{L}$ in nonreciprocal $d$-dimensional systems of size ${L}^{d}$ follows a universal one-parameter scaling form. This feature is encapsulated by the one-parameter $\ensuremath{\beta}$ function $\ensuremath{\beta}[{Y}_{L}]=dln[{Y}_{L}]/dln[L]$, which describes the critical properties of Anderson localization transitions, including the critical exponent $\ensuremath{\nu}$, the fractal dimension $D$, and the critical reduced participation ratio ${Y}_{L}^{*}$. 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.