Anderson Localization Of Hermitian And Non-hermitian Systems

Anderson localization is one of the most focused phenomena in condensed matter physics.In1958,Anderson found three-dimensional disorder electronic systems can undergo phase transitions from the metallic phase with extended states to the insulator phase with localized states.Based on Anderson model,in 1967 Mott proposed the concept of mobility edge,which is the critical energy that separates localized and delocalized states.For a long time,people have paid attention to the Hermitian system.In recent years,in order to study the impact of the environment,people also began to focus on non-Hermitian systems.Some questions are worth further exploring: if there are mobility edges in Hermitian systems,what determines the number of mobility edges;How the boundary conditions and disorders effect on non-Hermitian systems;What are the conditions for the occurrence of reentrant localization.In this paper,Anderson localization will be studied in detail through theoretical derivation and numerical analysis.Firstly,a slowly varying potential model is proposed in Hermitian system.Combinating the asymptotic heuristic argument with the theory of trace map of transfer matrix,mobility edges and pseudo-mobility edges in their energy spectra are solved semi-analytically,and the criterion for judging the mobility edge and pseudo-mobility edge is given.The nature of eigenstates in extended,critical,weakly localized,pseudo-critical and strongly localized is diagnosed by the local density of states,the Lyapunov exponent,the localization tensor and Fractal dimension.Numerical calculation results are in excellent quantitative agreement with theoretical predictions.Secondly,the Anderson localization phenomenon with two different configuration models is considered when the non-Hermitian system changes from periodic boundary to open boundary.The two configurations are with balanced and imbalanced gain and loss,respectively.The averaged inverse participation ratio is used to characterize the localization property of such systems,the real and imaginary parts of the eigenenergy with respect to the boundary conditions are analyzed,and the reason for phase transitions are also given.Finally,the reentrant localization is an interesting phenomenon.It is very important to judge the conditions that its occurrence.We extend an existing model.The uniform disorder and staggered disorder and extended cases in Hermitian and non-Hermitian systems are discussed,respectively.In this paper,the extended state,the intermediate state,and the localized state are distinguished by the normal participation ratio and the inverse participation ratio.A quantity η is introduced to judge whether the reentrant localization phenomenon occurs.The phase diagrams are given.