Localized Properties Of One Dimensional Long-Range Interaction Systems

Many-body localization is a frontier subject in the field of condensed matter physics,and the influence of long-range interactions in physical systems is one of the important issues.In this thesis,the properties of the one-dimensional single-particle and many-body particle Aubry-André models are studied by using the exact diagonalization method.The statistical properties of energy level difference ratio are analyzed.Through numerical study,we found that the single-particle model has definite mobility edges.Based on the single-particle case,we consider the system with nearest-neighbor interaction,by calculating the statistical distribution of the energy difference ratio and the statistic average of energy difference ?r? related to the energy position.We found that there exists the many-body mobility edge under the limited size of the many-body system for a large parameter range,which is useful for the experimental observations.Through the scaling analysis of the finite size,we find that under the thermodynamic limit,the many-body mobility edge still exists,that is to say,the many-body mobility edge cannot simply be attributed to the size effect.In the first chapter of this thesis we introduce the background of the quantum localization,mainly concentrating on the many-body localization.Mobility edge in the single-particle and many-body system is defined.We briefly introduce the current widely used numerical methods by comparing their advantages and disadvantages,as well as the progress of the present researchon the localization of the long-range interacting system.In the second chapter,we introduce the energy level statistical method and related physical quantities in the many-body system.The energy level difference statistical method and energy level difference ratio statistical method are introduced.For the related physical quantity,the inverse participation ratios(PR)and entanglement entropy are introduced.The feasibility of these methods and physical quantities in experimental observation is analyzed.In the third chapter,the localization of the one-dimensional Aubry-André model with nearest neighbor interaction is studied.We mainly calculate the PR of the extended state and local state,the statistics of energy level difference ratio and the statistical average of the energy level difference ratio,compare the difference between the physical quantities in extended state and local state,and obtain the phase diagram for the system of the quasi-disordered external potential.In the fourth chapter,the local properties of Aubry-Andréunder the one-dimensional many-body long range interaction are studied based on the single-particle case and the repulsive interactions among neighboring lattice sites are considered.By calculating the statistical mean value of energy level difference ratio ?r?,we obtain the phase diagram of energy position and quasi-disordered external potential strength(?-v),and find the many-body mobility edge phenomenon under the finite size condition.At the same time,we also obtain the phase transition point under the thermodynamic limit through the finite scalinganalysis,and it is believed that there exists a many-body mobility edge under the thermodynamic limit.In the last chapter we give the summary and prospect for the future.