| The Hubbard model is the most popular model for studying strongly correlated quantum systems. According to the studying system is boson or fermion system, the Hubbard model can be divided into the Fermi-Hubbard model and the Bose-Hubbard model. Researches on high-temperature superconducting properties in cuprate using the Fermi-Hubbard model, and researches on quantum phases of ultracold boson gases in op-tical lattices using the Bose-Hubbard model, are both the most popular research topics. For both the Fermi-and Bose-Hubbard models, though their hamiltonians have simple format, they are only analytically solvable in one-dimensional system; in two-or three-dimensional system, more simulations of the Hubbard model are made numerically based on the computer. With the enhancement of the performance of computer, and the develop-ment of the numerical algorithms especially the quantum Monte Carlo (QMC) algorithm, the Hubbard model can be used to explore the exotic properties and physical mechanisms of high-temperature superconductivity in cuprate and quantum phases of ultracold bo-son gases more deeply. The main research work of this thesis focus on two-dimensional inhomogeneous systems.By introducing stripe term into the two-dimensional Fermi-Hubbard model, we con-struct the stripe Hubbard model. And with the solution of this model using the deter-minant quantum Monte Carlo (DQMC) algorithm, we mainly study the variation of the superconducting properties with stripe potential energy inhomogeneity, under different electron densities, different electron interactions and different stripe periods. The results show that, when using the stripe period P=4which is found in the experiments of high-temperature superconductivity in cuprate, the high-temperature superconducting proper-ties are enhanced with the increase of the stripe potential energy, and we also observe the phenomena such as π phase shift which is consistent with the experiment. By contrast, the other potential energy inhomogeneities such as stripe with period P=2and checker-board suppress the high-temperature superconducting properties. At the same time, we also study how the stripe strength and period affect the electron distribution, spin correla-tion and the system energy, and propose the idea that the stripe potential energy inhomo-geneity can enhance the electron spin correlation, which is a physical mechanism of the enhancement of high-temperature superconducting properties. By studying the variation of the system energy with stripe period, we explain why the P=4stripe can be observed in the experiments of high-temperature superconductivity in cuprate.With the solution of the two-dimensional plaquette Hubbard model using the DQMC algorithm, we study the variation of the superconducting properties with hopping strength inhomogeneity, for the half-filling and doping electron densities. The results show that, for different electron densities, there exists optimal hopping strength inhomogeneity, i.e., for the intra-plaquette hopping strength t and inter-plaquette hopping strength t’, there is an intermediate t’/t～0.4where the high-temperature superconducting properties are strongest. By studying the variations of intra-plaquette and inter-plaquette nearest neigh-bor electron spin correlation with hopping strength inhomogeneity, we propose the phys-ical mechanism of the optimal hopping strength inhomogeneity. By studying the spin correlation and d-wave pairing correlation under different electron densities, we verify that, at half-filling, anti-ferromagnetic correlation is the dominant status of the system; while at doping densities,d-wave pairing correlation is dominant. And by comparing the d-wave and s-wave pairing correlation, we prove that the d-wave pairing is still the dominant high-temperature superconducting factor, even in the hopping strength inhomo-geneous system.With the solution of the anisotropic hardcore bose-Hubbard model using the stochas-tic Green function (SGF) QMC algorithm, we study the influence of the supersolid phase of unltracold hardcore boson gases in square optical lattice by anisotropy. By calculating the measurements such as structure factor, superfluid density and ground state energy, we analyze the stability of the supersolid phase under hopping strength anisotropy and nearest neighbor interaction anisotropy, separately. The results show that, for both the two kinds of anisotropy, the supersolid phase is unstable, phase separation happens. With the variation of particle interactions and distribution in real space, we also analyze the variation of the quantum phases such as superfluid phase and Mott insulator phase, and the influence of anisotropy.With the solution of the hopping strength anisotropic hardcore bose-Hubbard mod-el using the stochastic series expansion (SSE) QMC algorithm, we study that, with the increase of the longitudinal hopping strength ty, the energy gaps of unltracold hardcore boson gases without neighbor interaction in three-and four-leg ladder optical lattices. The results show that, the energy gaps appear at some special particle densities:in the three-leg ladder lattice, energy gaps appear at particle densities ρ=1/3and ρ=2/3; in the four-leg ladder lattices,energy gaps appear at particle densities ρ=1/4,ρ=1/2and C=3/4.We confirm that the corresponding superfluid densities disappear when energy gaps appear.By analyzing the exponential decay behavior of gaps and correlation length-es with ty,we obtain the critical tyc≈1at ρ=1/3and ρ=2/3in the three-leg ladder lattice；while in the four-leg ladder lattice,the critical tyc≈1.8atρ=1/4and ρ=3/4,and tyc≈1.9at ρ=1/2.We propose that,in ladder optical lattice,particle-holes resonance happens at some special particle densities,then the particle-hole pairs form,which is the reason why energy gaps appear. |
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