The Study On One-dimensional Hubbard Model With Finite Temperature Density-functional Theory

One-dimensional finite temperature many-body systems are usually difficult to study,while density-functional theory provides a way to tackle them.For the density-functional theory,there will be a huge difficulty when dealing a sys-tem with a discontinuity in the exchange-correlation potential.So we propose in this thesis a chemical potential functional theory to solve this problem.First,we numerically solve the thermodynamic Bethe ansatz(TBA)coupled equation-s for a one-dimensional Hubbard model at finite temperature and obtain the second order thermodynamics properties for the specific heat,compressibility,and susceptibility,respectively.Making use of the Bethe ansatz results from the homogeneous Hubbard model,we construct the chemical potential function-al theory(?-BALDA)for the harmonically trapped Hubbard model and obtain the local density profile of the metallic state,the metal-Mott insulating phase,the metal-Mott-metal mixted phase,the metal-Mott-band mixed phase,and the metal-band mixted phase at different temperature.There are five chapters in the thesis as follows.In chapter one,we review the background of cold atomic physics and some background of one-dimensional strongly correlated systems,and simultaneously introduce some new results for the finite temperature research in recently years.Based on these,we introduce our research on the study of the one-dimensional Hubbard model at finite temperature by density-functional theory.In chapter two,we mainly introduce the basic concepts of the density func-tional theory and chemical potential functional theory(?-BALDA).In chapter three,we research on the properties of the Hubbard model in one dimension at finite temperature.At finite temperature,we can obtain the thermodynamics properties for the specific heat,compressibility,susceptibility,chemical potential,and so on.We find that,these three quantities could embody the phase transitions of the system,from the vacuum state to the metallic state,from the metallic state to the Mott insulating phase,from the Mott insulating phase to the metallic state,and from the metallic state to the band-insulating phase at low temperature.With the increasing of the temperature,the thermal fluctuation overwhelms the quantum fluctuations and the phase transition points disappear due to the destruction of the Mott insulating phase.In chapter four,making use of the Bethe ansatz results from the homo-geneous Hubbard model,we construct the chemical potential functional theory for the properties of the harmonically trapped Hubbard model in one dimension at zero and finite temperature.And we show the how to tackle the numerical problem in the chemical potential functional theory.At finite temperature,the local density profile for the metallic phase,the metal-Mott insulating phase,the metal-Mott-metal mixed phase,the metal-Mott-band mixed phase,the metal-band insulating phase,are obtained.We find,at a given interaction strength,all quantum phase are destroyed by thermal fluctuation.At zero temperature,we obtain,at a given different interaction strength,the density profile of the metallic phase,and compare with results which calculated by DMRG method.In chapter five,a summary and outlook for our work are given.