Extracting Luttinger Liquid Parameters From Geometric Entanglement For Quantum Spin Tube

In this thesis, by mapping to a matrix product state(MPS), the spin 1/2 anisotropic three-leg tube model at a critical region is studied in terms of matrix product state(MPS) algorithm for finite-size systems. First, ground state wave function is obtained by MPS algorithms for five different parameters at the critical region. Then the ge ometric entanglement(GE) per site of the ground state has been obtained by using an efficient algorithm proposed for one-dimensional quantum systems previously. Afterwards, it is found that GE per site of ground states for different system sizes in three-leg tube Heisenberg models show the same finite-size correction relation with the relation observed in one-dimensional systems. Accordingly, Luttinger liquid parameters for five different control parameters in the spin 1/2 anisotropic three-leg Heisenberg tube model are obtained analog to the relations found in one-dimensional systems. From the numerical results, it is believed that the finite-size correction relation for GE per site is universal. Also, finite-size correction coefficient b is universal correspondingly.The thesis consists four parts altogether. The first chapter is the introduction section. In this section, the history of tensor network algorithm, the background of three-leg tube model and the concept of GE are roughly introduced. The second chapter is the algorithms and methods section. In this section, two main algorithms employed during investigation is introduced in detail. One of the two algorithms is the MPS algorithm for periodic boundary conditions, which is used to simulate the ground state wave function. The other algorithm is gradient directed random walk algorithm to calculate GE per site for each ground state. The third chapter is the results and discussion section. In this section, simulation results are presented and discussed. By mapping the three-leg tube to a one-dimensional structure, spin 1/2 anisotropic three-leg tube model is investigated with two MPS based algorithms. Also, the discussion and analysis of numerical results are delivered. The fourth chapter is the summary section for the thesis. A brief summary for the thesis and the further plan for investigations are given in this section.