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 $\ensuremath{\xi}$, the inverse participation ratio ($\mathrm{IPR}$), and the energy gap $\mathrm{\ensuremath{\Delta}}E$ 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.