For the study of cobalt oxide materials, we investigate the spin susceptibility behavior when the temperature and interaction strength vary in the whole filling region, on a triangular lattice. We find the high density of state (DOS) around the van Hove singularity point is important to the existence of a certain itinerant ferromagnetic short-range correlation. Also we study different spin singlet and triplet pairing channels in this region, and find the triplet channels especially the f-wave pairing susceptibility presents the quickest growth via the lowing of temperature. However, it is subtle to stabilize the f-wave pairing since the effective pairing interaction was found to be very small. / In the last part we discuss a special case in quantum phase transition, in which the Green's function of the system can be analytically obtained. We study the entanglement of the system to identify the quantum phase transition in the system. / We investigate several novel strong correlated phenomena in the geometrically frustrated system in this thesis. On a parallel computational grid, we do the state-of-the-art quantum Monte Carlo (QMC) numerical simulation to the extended single-band Hubbard model. We observe a wide range of measurements to clarify and demonstrate the system properties in different electron density, temperature, interaction strength. The results of the single-band Hubbard model on the triangular lattice can help us understand the magnetic property and superconductivity of the cobalt oxide materials. The results of the attractive Hubbard model on the honeycomb lattice can be used to discuss some properties of the "Bardeen-Cooper-Schrieffer (BCS) state to Bose-Einstein condensation (BEC) crossover" picture, such as Dirac Fermions' behaviors and the coexistence of fermionic and bosonic degrees of freedom in the system. / We study the BCS-BEC picture of high Tc superconductor systems. On a honeycomb lattice, we measure a wide range of observables, such as double occupancy, uniform spin susceptibility, on site pair correlation vertex, on-site pair correlation length, potential energy, kinetic energy, to monitor the BEC-type and BCS-type behaviors coexisting in the system. In different regions of the interaction strength-temperature parametric plane, we find that the BEC-type and BCS-type behaviors dominate at different regions of the interaction strength-temperature parametric plane, and they have crossover in the intermediate interaction strength region. / Su, Shiquan. / Adviser: Hai-Qing Lin. / Source: Dissertation Abstracts International, Volume: 70-06, Section: B, page: 3570. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 101-107). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307. / Su, Shiquan.