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.

By performing atomistic simulations, we have studied some features of classical Heisenberg model using the statistical Monte Carlo method MC under the Hinzke–Nowak algorithm. First, we have deeply explored magnetic and thermal properties of a core–shell nanosphere model and investigate the behaviors of the temperature-dependent magnetization, magnetic susceptibility and phase diagrams for different possible exchange interactions. The obtained results show the existence of diverse -types behaviors in the Néel classification nomenclature. Then, we have applied the same computational method to the real (Fe, Ni) nanostructure using experimental values of magnetic parameters for iron and nickel. It is demonstrated that (Fe, Ni) nanoparticle exhibits a compensation phenomenon compatible with those found in the experimental studies.