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The self consistent random phase approximation (SCRPA) and a direct analytical (DA) method are proposed to solve the Extended Hubbard Model (EHM) in one dimension (1D). We have considered an EHM including on-site and off-site interactions for closed chains in 1D with periodic boundary conditions. The comparison of the SCRPA results with the ones obtained by a DA approach shows that the SCRPA treats the problem of these closed chains in a rigorous manner. The analysis of the nearest-neighbour repulsion effect on the dynamics of our closed chains shows that this repulsive interaction between the electrons of the neighbouring atoms induces supplementary conductivity, since, the SCRPA energygap vanishes when these closed chains are governed by a strong repulsive on-site interaction and intermediate nearest-neighbour repulsion.

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Using the finite cluster approximation (FCA) based on a single-site cluster theory and Monte Carlo simulation (MCS) under the Metropolis algorithm, we study the critical properties of two-dimensional (2D) spins-½ system by analyzing the effect of the anisotropic parameter [Formula: see text] and next-nearest-neighbor (NNN) interaction [Formula: see text] on the critical temperature [Formula: see text] of the 2D anisotropic square lattice. It is found that the MCS is more reliable than the FCA method in the resolution of the Ising model. Moreover, MC calculations reveal that [Formula: see text] is strongly correlated with the external magnetic field [Formula: see text], [Formula: see text], and [Formula: see text], where the competition between these parameters can generate several rich phase diagrams.