Exact Solutions And Phase Transition Analysis Of The 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 this 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 this exact solution method, cases with 8 bosons on 4 sites, 6 bosons on 6 sites, 10 bosons on 6 sites and 8 bosons on 7 sites, 10, 20, 50 bosons on 3 sites, 10, 15 bosons on 5 sites, and 6 bosons on 5,8,10 sites, respectively, are calculated. From the excitation energies and the corresponding spectra, phase transition behavior of the model are analyzed, which varies with model parameters in different conditions, such as in integer fillings or non-integer fillings, with the same bosons number and different number of sites, or with the same number of sites and different bosons number. As for other related physical quantities in the model, further investigations are expected.