Hybrid scaling properties of the localization transition in a non-Hermitian disordered Aubry-André model

In this paper, we study the critical behaviors in the non-Hermitian disorder Aubry-Andr\'e (DAA) model, and we assume the non-Hermiticity is introduced by nonreciprocal hopping. We employ the localization length ξ, the inverse participation ratio (IPR), and the energy gap ΔE as the characteristic quantities to describe the critical properties of the localization transition. By performing scaling analysis, the critical exponents of the non-Hermitian Anderson model and the non-Hermitian DAA model are obtained, and these critical exponents are different from their Hermitian counterparts, indicating that the Hermitian and non-Hermitian Anderson and DAA models belong to different universality classes. The critical exponents of the non-Hermitian DAA model are remarkably different from both the pure non-Hermitian AA model and the non-Hermitian Anderson model, showing that disorder is an independent relevant direction at the non-Hermitian AA model critical point. We further propose a hybrid scaling law to describe the critical behavior in the overlapping critical region constituted by the critical regions of the non-Hermitian DAA model and the non-Hermitian Anderson localization.