Phys. Rev. Lett. 134, 176301 (2025)
Unified One-Parameter Scaling Function for Anderson Localization Transitions in Nonreciprocal Non-Hermitian Systems
C. Wang1,*, Wenxun He1,2, X. R. Wang3,4 and Hechen Ren1,2,5,†
1 Center for Joint Quantum Studies and Department of Physics, School of Science, Tianjin University, Tianjin 300350, China.
2 Tianjin Key Laboratory of Low Dimensional Materials Physics and Preparing Technology, School of Science, Tianjin University, Tianjin 300072, China.
3 Physics Department, The Hong Kong University of Science and Technology (HKUST), Clear Water Bay, Kowloon, Hong Kong.
4 School of Science and Engineering, Chinese University of Hong Kong (Shenzhen), Shenzhen 51817, China.
5 Joint School of National University of Singapore and Tianjin University, International Campus of Tianjin University, Binhai New City, Fuzhou 350207, China.
* physcwang@tju.edu.cn
† ren@tju.edu.cn
Abstract
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.
