Quantum Critical Behavior Of The T-J Model

Strongly correlated systems are of current interest in view of their relevance to thetheory of high-T_c superconductivity, t-J model is an appropriate starting model. The modeldescribes the behavior of hard core electrons on a discrete lattice. There has been previouswork using the Bethe ansatz on one-dimensional models involving both hard core fermionsand bosons. Actually, the t-J model comes from the Hubbard model which was the mainmodel in the theory of metallic magnetic materials. Initially it was introduced to describemagnetism in transitional metals and their compounds. The simplest single-band Hubbardmodel contains two such parameters, the width W of the initial band and the magnitude U ofthe coulomb repulsion between two electrons at the same site. It occurred that for sufficientlylarge U≥U_c～W the ground state of the system is an insulator state, while for U＜U_c it isa metallic state. In the limit U＞＞W in second-order perturbation theory in W/U, we obtainthe following effective Hamiltonian, H=-t sum from ijσ(1-n_i-σ)C_iσ^?C_jσ(1-n_j-σ)+J sum from ij S_iS_j. Themodel with this Hamiltonian is the famous t-J model which is the limiting case of the Hubbardmodel with U＞＞W, but it is sometimes regarded as a separate phenomenological model, inwhich the parameters t and J are independent. The model was developed to describe electronmotion in an antiferromagnetic materials. In this thesis, we investigate the exact numericalsolutions of the one dimension t-J model. Firstly, energy matrices can be generated rapidly byusing a Mathematica package. The output are then taken as the input of other diagonalizationcodes. Then excitation energy eigenvalues and the corresponding wavefunctions are obtainedby a Fortran code. By using the exact solutions, some low-lying levels, the entanglement ofthe one dimension t-J model for fixed total sites and occupied sites by Fermis respectively,were calculated. From the excitation energies, the entanglement, quantum phase transitionalbehavior of the model is analyzed. The quantities of the model vary with the control parameterx, which can clearly differ the system from the kinetic term describing the electron motion fromsite to site and the term of the nearest neighbor interaction.