Studies On The Level Statistics And Dynamics Of Many-body Systems In One Dimension

Many-body localization is a frontier subject in condensed matter fields.As one of the most important problems,the many-body mobility edge is still controversial.In this paper,we study the physical properties of a one-dimensional spinless FermiHubbard model with long-range hopping in the single-particle and many-body cases by exact diagonalization.The key physical quantities we calculated are the ratio of adjacent gap and the time evolution.We find that the model has a clear mobility edge in the single-particle case.In the many-body case,we observe the many-body mobility edges of finite size,and compare with the many-body Aubry-Andr? e model,which are more obvious and easier to observe experimentally.Through the scaling of finite-size analysis,we find that the many-body mobility edges still exist in the thermodynamic limit,so the many-body mobility edges cannot be simply attributed to the finite-size effects.More importantly,we propose an experimentally feasible method for measuring many-body mobility edges.In the first chapter of this paper,we briefly introduce the background related to quantum localization,including many-body localization,the eigenstate thermalization hypothesis,the mobility edges in single-particle and many-body systems,the numerical methods with their advantages and disadvantages,as well as the research progress in the experiment of the many-body localization and many-body mobility edges.In the second chapter,we introduce the statistical methods and dynamic method in the many-body study,including the statistics of the gap and the ratio of adjacent gaps.The method of dynamics concerns a more general time evolution,and the feasibility of this method in experiments.In the third chapter,we study the mobility edges of a one-dimensional spineless Fermi-Hubbard model with long-range hopping in the single-particle case.We compare the extended states with the localized states by calculating the probability density of the eigenstate,the inverse participation ratio(IPR),the ratio of adjacent gap,and the average value of the ratio of adjacent gap.Moreover,we obtain the phase diagram under the influence of the long-range hopping exponent p.In the fourth chapter,we study the mobility edges in one-dimensional spinless Fermi many-body systems.By calculating the average value of the ratio of adjacent gap,we obtain a phase diagram for the energy density and the quasi-disordered potential(?-V)under finite size situations.In addition,we also obtain the phase transition point under the thermodynamic limit by means of the finite scale analysis.Finally,based on the dynamic evolution method proposed in the second chapter,we study the imbalance of the ergodic states and manybody localization states,so as to prove that this method can be used to measure many-body mobility edges experimentally.In the last chapter,we summarise and give an outlook.