Quantum Critical Behavior Of The Three-site Bose-Hubbard Model

The study of strongly interacting bosons is a formidable task in the area of theory of condensed-matter physics. The Bose-Hubbard model is a basic model which contains main features in the strongly interacting Bose systems. Various aspects of the model and related models were previously investigated analytically by quantum Monte Carlo Method (QMCM), mean-field approximation method and so on. Those methods provide with a qualitatively proper phase diagram and describe approximately the properties of insulator-superfluid phase transition. Nevertheless, the results provided by these methods are all inaccurate.In this thesis, in order to overcome the drawbacks, we reconstruct the Bose-Hubbard model hamiltonian with differential realization of bosonic operators, and investigate the exact numerical solutions of the one dimension Bose-Hubbard model. Firstly, energy matrices can be generated rapidly by using a Mathematica package. The output are then taken as the input of other diagonalization codes. Then excitation energy eigenvalues and the corresponding wavefuctions are obtained by a Fortran code. By using the exact solutions, some low-lying levels, occupation probabilities of bosons on different sites, the entanglement of the three-site Bose-Hubbard model for boson numbers n=15, 20, 30, 40, respectively, were calculated.From the excitation energies, occupation probalilities, the entanglement, quantum phase transitional behavior of the model is analyzed. These quantities of the model vary with the control parameter x, which clearly show the second order quantum phase transition occurring in the system and the entanglement "can be used as the order parameter.