Numerical results obtained from Monte Carlo simulation in the ground state and at finite temperatures as well as the exact diagonalization and the transfer matrix method are reported. Efficiency of the Monte Carlo method in the ground state is ex-amined. Spin, charge and superconducting correlations are investigated for the Hubbard and the t-J model in one and two dimensions. The momentum distribution in the one-dimensional Hubbard model shows fermi-liquid-like behavior at least in two orders of magnitude smaller energy scale than the band width. The short-range incommensurate spin correlation is observed and analyzed in the doped Hubbard and the t-J model both in one and two dimensions. In the ground state, the superconducting correlation shows the absence of system size dependence in two dimensions at the filling smaller than 0.8 and in one dimension at any filling. The binding energy of two fermions and the spin and charge distortion around an itinerant fermion are also discussed in the t-J and the coupled spin-fermion model.
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