Estudo analítico de modelos tipo XY tridimensionais puros e diluídos via Princípio Variacional de Bogoliubov

Abstract

In this work, we have analytically studied, by means of the Bogoliubov Variational Principle (BVP), the pure Heisenberg and XY with bond dilution models, both ferromagnetic, classical (continuous spins), anisotropic, three-dimensional and in the presence of a crystalline field. Furthermore, we have shown the equivalence between the BVP and theSelf-Consistent Harmonic Approximation (SCHA) a trial harmonic Hamiltonian. In the study of anisotropic Heisenberg model, the magnetization and phase diagrams are obtained as a function of parameters of the Hamiltonian. Limiting cases, such as the isotropicHeisenberg, XY and Planar Rotor models in two and three dimensions are analyzed and compared to previous results obtained from analytical approximations and Monte Carlo simulations. In the study of anisotropic XY model with bond dilution, we have obtained the critical concentration and compared with the exact result as well as from other approximation methods, besides reviewing the Planar Rotor diluted system as a limitingcase. By showing the equivalence between the BVP and SCHA, we arrive at a system of coupled equations for the variational parameters, which arise by employing the BVP, and the system of coupled equations for the coupling constants that appear employing the SCHA, and we found that such systems are identical.