Research On Anderson Localization Phenomenon Based On Energy Level Statistics

In 1958 Anderson studied the motion of electrons in a disordered system.He found that the electron would be localized in a limited area as the disorder strength increases.This phenomenon is Anderson localization.Until now,Anderson localization phenomenon is still one of the most important subjects in condensed matter physics.On the other hand,the Random Matrix Theory can well characterize the statistical properties of energy levels of complex atomic nuclei.Anderson localization can also be studied by the Random Matrix Theory.Moreover,in recent years,for many-body localization,there are not only the localized phase and the delocalized phase,but also an intermediate phase between these two phases.Based on the Random Matrix Theory,we will use energy level statistics to study the Anderson localization phenomenon.These research results enrich the quantum theory and explain more quantum phenomena,which is of great significance to promote the practical effect of quantum theory.First,an index is introduced,i.e.the non-overlapping higher-order level spacing ratio r(k),k=1,2,3,which represents the first-order,second-order and third-order energy level spacing ratios,respectively.The properties of the relative higher-order level spacing ratio R(k)=1-r(k)in the localized phase and the delocalized phase are studied in the one-dimensional non-periodic potential system.The one-dimensional uniform potential model,Anderson model,Aubry-Andre-Harper model,and Slowly varying potential model are considered,respectively.The results show that in these models,for the localized state,the relationship of the mean of R(k)is(Rn(1)><(Rn(2)><(Rn(3)>,but for the delocalized state,the value of<Rn(1)>is the largest.The properties of R(k)are different in the two phases.Therefore,R(k)can be used to indicate Anderson localization.Secondly,the Anderson localization phenomenon of the near band random matrix ensemble,the far band random matrix ensemble,and the sparse random matrix ensemble is studied by means of energy level statistics.In addition,the existence of the intermediate phase in these three ensembles is also confirmed.The results show that for the near band random matrix ensemble,the critical point of the Anderson transition is B/N1/2<0.5,and the ergodic transition point is B/N1/2≈1.5,where B is the near-band width and N is the size of the matrix.For the far band random Matrix Ensemble,the transition point of the Anderson transition is Bf/N≈0.5,and the ergodic transition point is Bf/N≈0.64,where Bf is the far band width.For the sparse random matrix ensemble,Nη≈2.2 is the critical value of the Anderson transition,and Nη≈3 is the ergodic transition point,where η is a number in the interval[0,1]and Nη represents the number of non-zero matrix elements in each row of the matrixFinally,the Anderson localization phenomenon in the Cayley tree network is studied.The boundary lattice points are connected to the other two boundary lattice points with probabilities p The energy level statistics,inverse participation ratio,Shannon information entropy and ergodic transition parameter are considered.The phase diagram is obtained.Results show that when p>0,the Anderson transition point exists in this system.The larger the value of p,the larger the value of Anderson transition point.When p>0.1,the ergodic transition point exists,and the larger the value of p,the larger the value of the ergodic transition point is.