Unified one-parameter scaling function for Anderson localization transitions in non-reciprocal non-Hermitian systems

TL;DR

This paper introduces a unified one-parameter scaling function for Anderson localization transitions in non-reciprocal and reciprocal non-Hermitian systems, overcoming limitations of traditional theories and demonstrating universality across dimensions and symmetry classes.

Contribution

The authors propose a new one-parameter scaling function using participation ratio, applicable to non-reciprocal and reciprocal non-Hermitian systems, validated through numerical and analytical methods.

Findings

The scaling function accurately describes localization transitions in 1D and 2D systems.

Critical exponents depend only on symmetry and dimensionality, indicating universality.

The derived disorder thresholds match numerical critical points.

Abstract

By using dimensionless conductances as scaling variables, the conventional one-parameter scaling theory of localization fails for non-reciprocal non-Hermitian systems such as the Hanato-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 non-reciprocal 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. Moreover, we derive a complex-gap equation based on the self-consistent Born approximation that can determine the disorder at which the point gap closes. The obtained disorders match perfectly the critical disorders of Anderson localization transitions from the one-parameter scaling function. Finally, we show that the one-parameter scaling function is also valid for Anderson localization transitions in reciprocal non-Hermitian systems such as two-dimensional class AII$^\dagger$ and can, thus, serve as a unified scaling function for disordered non-Hermitian systems.