Applications Of Clock And Shift Operators For Anderson Localization Phenomena

Anderson localization(AL)is a common phenomenon in nature.As early as 1958,Anderson proposed the concept in the study of electron behaviors in disorder or amorphous systems.Subsequently,the phenomenon has been further researched in many areas,for example,in optics,acoustics,microwaves,Bose-Einstein condensate,and photonic crystals.Until today,AL transition is still a hot research area in the condensed state physics and various disordered systems,but there are still some controversies.On the other hand,the principle of uncertainty is one of the basic principles of quantum mechanics.Recently,based on the clock operator (?) and the shift operator (?),a novel uncertainty inequality is proposed,which has a lowest bound LB.In recent literatures people have pointed they are related to state localization properties.However,they do not give the corresponding explicit relation.At the same time,there exist open problems for AL transition in non-Hermitian systems.Considering the above mentioned,with the help of (?),(?) and LB,we study AL transition in a single electronic lattice(Hermitian and non-Hermitian)systems.Firstly,the one-dimensional nonuniform lattice(Hermitian)system is studied.The slowly varying potential model,the nearest neighbor tight-binding model and the random dimer potential model are taken as examples.Numerical results show that the uncertainty ?U2 is greater than,equal to and less than ?V2 for the extended,critical and localized states,respectively.At the same time,for the extended and localized states,the lowest limit LB respectively approach to 1.0 and 0.0 as the system sizes become larger.Therefore,they can be used as a novel index to characterize AL transition.Secondly,the one-dimensional quasi-periodic lattice(non-Hermitian)system is studied.A lattice model with incommensurate complex potential is taken as an example.Numerical results show that the AL transition can also be characterized by the uncertainty ?U2,the uncertainty ?V2 and reduced relative uncertainty U_r.Results are good consistent with the existing results.Therefore,they can also be used to distinguish the extended from localized states in the non-Hermitian system.In short,all the above researches provide that these quantities can be as fruitful indexes to reflect AL.Comparisons with other measures,for instance,Thouless exponent,participation ratio and quantum information entropy,they are very nice complement to existing measures of localization.