The Study On The Properties Of The Critical Phase In A One-dimensional Quasi-disordered System

Quantum localization has always been an issue of interest to researchers.Based on the numerical method of diagonalization,we use the level spacing statis-tics,the inverse participation rate(IPR),the mean of inverse participation rate(MIPR)and the probability density of wave function to study the Aubry-Andre model and off-diagonal Aubry-Andre model with quasi-disordered potential in one-dimensional systems.In the Aubry-Andre model,we find that under different boundary conditionS,the average value of the ratio of adjacent gap<?>in the the critical phase is not the same.At the same time,we propose a method under open boundary conditions,using energy level degeneracy,calculating the average value of ratio of adjacent gap to determine the different phases when factor(?)w ? 0 This approach saves resources and time more than adding random factor.In the off-diagonal Aubry-Andre model,we examine the methods proposed in the Aubry-Andre model and study the extended phase,critical phase and localized phase.We find that under the same model,the average value of the ratio of adjacent gap in the critical phase is approximately the same,while in different models,it is not a fixed value.According to the inverse participation rate and the average value of the ratio of the adjacent gap,we get the phase diagram of commensurate and incommensurate modulations.In addition,we add the incommensurate potential in the model,and find that the critical phase changes rapidly into the localized phase,the extended phase is relatively stable,and all phases become the localized phase,when the disorder strength is large enough.In the first chapter of this paper we introduce the background knowledge of the quantum localization,and describe the present status in the field,which leads to our research topic:the study of the critical phase properties in one-dimensional quasi-disordered system.In this chapter,we describe single particle systems and many-body systems.For the single particle system,we introduce the properties of the extended state,the localized state,the critical state and the mobility edge of single particle.And for the many-body system,we mainly introduce the concept of the many-body localization and mobility edge.In the second chapter,we mainly introduce the three commonly used meth-ods of characterizing the system in the single particle system and many-body system:the ratio of adjacent gap,the inverse participation rate,and the mean of inverse participation rate.In this chapter,we select several topics,and introduce three methods in details.Some results in references are tested.In the third chapter,we study the Aubry-Andre model with quasi-disordered potential in one-dimensional system.It is found that the average value of the ratio of adjacent gap can still be used to judge the extended phase,the localized phase,and the critical phase in the one-dimensional single particle system after adding the random offset.Under the different boundary conditions,the average value of the ratio of adjacent gap between the extended phase and the critical phase is not the same.In the fourth chapter,we study the off-diagonal Aubry-Andre in two cases:without potential or with incommensurate potential.First,we prove that the method proposed in the Aubry-Andre model is still applicable here.Compared to the results in the Aubry-Andre model,we find that in the same model,the average value of the ratio of adjacent gap in the critical phase is approximately equal.But in different models,the critical phase has a different value.When there is an incommensurate potential in the system,the critical phase becomes localized phase with the increase of incommensurate potential,while the extended phase is robust.In addition,we also obtain the phase diagram without potential and with incommensurate potential.In the last chapter,we summarise and give an outlook.