Unified One-Parameter Scaling Function for Anderson Localization Transitions in Nonreciprocal Non-Hermitian Systems

Using dimensionless conductances as scaling variables, the conventional one-parameter scaling theory of localization fails for nonreciprocal non-Hermitian systems such as the Hatano-Nelson model. Here, we propose a one-parameter scaling function using the participation ratio as the scaling variable. Employing a highly accurate numerical procedure based on exact diagonalization, we demonstrate that this one-parameter scaling function can describe Anderson localization transitions of nonreciprocal non-Hermitian systems in one and two dimensions of symmetry classes AI and A. The critical exponents of correlation lengths depend on symmetries and dimensionality only, a typical feature of universality. Then, we derive a complex-gap equation based on the self-consistent Born approximation to determine the critical disorder at which the point gap closes. The obtained critical disorders perfectly match those given by the one-parameter scaling function. Moreover, we propose a one-parameter β function that can describe the critical properties of such Anderson localization transitions. Finally, we show that the one-parameter scaling function is also valid for Anderson localization transitions in reciprocal non-Hermitian systems such as the two-dimensional class AII^{†} and can, thus, serve as a unified scaling function for disordered non-Hermitian systems.