The Study Of One-dimensional Spin Chains

In this paper, the ground-state property of the one-dimensional spin chain system is studied. We perform a numerical study by using local density approx-imation (LDA) and density functional theory (DFT) based on exact analytical Bcthc-Ansatz solution. We study the density profile of the ground-state.In chapter1, we introduce the background of this article, including the de-velopment of cold atomic physics, especially the implementation of the optical lattices, which provides experimental platforms for one-dimensional strongly cor-related systems. At the same time we introduce the numerical methods for solving one dimensional spin chain system, such as Bethe-Ansata, local-density approxi-mation methods, Density functional theory, the Density matrix renormalization group and so on.In chapter2, we firstly introduce the single particle Jordan-Wigncr transfor-mation and then the many-body Jordan-Wigner transformation. Making use of the Jordan-Wigner transformation, the spin-1/2XXZ model can be transformed into the one of spinlcss fermions.In chapter3, we study the properties of ground-state of one-dimensional spin-1/2XXZ model. Firstly we discuss the phase diagram without external potential and then the phase diagram under harmonic potential. We get the ground state energy, chemical potential and exchange-correlation(xc) potential.In chapter4, we perform a numerical study on the one-dimensional spinless fermions in optical lattice under the presence of harmonic potential by using local density approximate (LDA) based on exact analytical Bethe-Ansatz solution and Density-matrix renomalization-group method. We study the density profile, the phase diagram, and the thermodynamic stiffness in different phase regions. We find five different phases; A is metal phase, B is the composite metal and insulator phase, C is the composite metal and insulator phase with metal phase in the middle, D is the composite metal and Band-insulator phase, E is Band-insulator phase. We find that the thermodynamic stiffness can be used to characterize the different quantum phase in the presence of the harmonic potential indicated by the nonanalytical points. For small systems and not too strong interactions, LDA in most cases produces reasonable results.In chapter5, we perform a numerical study of one-dimensional spin--1/2XXZ model in optical lattice under the presence of harmonic potential. We give the density distribution of the spin-1/2XXZ model.Conclusions and outlook are given in chapter6.We found that numerical results by using local density approximation based on exact analytical Bethe-Ansatz solution are comparable to the exact results. For small systems and not too strong interactions LDA in most cases produces reasonable results.