Hybrid scaling properties of localization transition in a non-Hermitian disorder Aubry-Andr\'{e} model

TL;DR

This paper investigates the critical behavior of a non-Hermitian Aubry-Andre9 model with disorder, revealing new universality classes and proposing a hybrid scaling law for the localization transition.

Contribution

It introduces a hybrid scaling law and identifies distinct critical exponents, showing the non-Hermitian DAA model's unique universality class compared to Hermitian models.

Findings

Critical exponents differ from Hermitian models.

Non-Hermitian DAA model has unique universality class.

Hybrid scaling law describes overlapping critical regions.

Abstract

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 $\xi$, the inverse participation ratio ($\rm IPR$), and the energy gap $\Delta E$ 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.