Localization transition in non-Hermitian coupled chain

Abstract

In this paper, we present a study of the properties in a coupled chain modulated by the quasiperiodic complex potential. It is found that as the disorder strength increases, the system undergoes a localization transition from a fully extended phase to an intermediate phase, and then to a fully localized phase. By numerically solving order parameters such as the average inverse participation ratio and the average normalized participation ratio, the existence of the fully extended phase, the intermediate phase with mobility edges, and the fully localized phase during the transition is demonstrated. By the scalar analysis of the normalized participation ratio, we confirm that three distinct localization phases stably exist within the system. Moreover, through analytical derivation, the localization transition points from the extended phase to the intermediate phase and from the intermediate phase to the localized phase can be precisely determined. In addition, the local phase diagram of the system is also obtained by numerical calculation, as shown in Fig.(a). The regions for the extended, intermediate and localized phases are denoted by Ⅰ−a(Ⅰ−b), Ⅱ, and Ⅲ, respectively. The three black solid lines represent the localization transition points determined by the analytical results. One can see that the analytical results match the numerical results.Moreover, we discuss that the relationship between the real-complex spectrum transition and the localization transition. It is found that the energy spectrum of the system can undergo two real-to-complex transitions. Specifically, during the transition from the fully extended phase to the intermediate phase, the first real-complex transition occurs, where part of the energy spectrum changes from the real spectrum to the complex spectrum, while another part spectrum remains real. When the system transitions from the intermediate phase to the fully localized phase, the energy spectrum completely transforms into a complex spectrum. These research results provide a reference for the study of localization transitions and real-complex transitions in one-dimensional coupled chain systems, and also offer a new perspective for the study of localization.