Desenvolvimento de um algoritmo numérico na técnica do operador diferencial : aplicações em modelos de spins

Abstract

In this work, we present the results obtained for Ising models and Heisenberg spin 1/2, where two and three-dimensional, with disorder or frustration. We apply effective field theory associated with the Operator Differential Technique - TOD. A new formulation of this technique has enabled the development of a numerical implementation where the coefficients are now constructed fully automatically. This allowed growing up the number N of spins of the cluster and thus observe the behavior of the system when it tends to the real case (N→∞), which is bounded by the computational time needed to carry out all operations. We apply this methodology to study the Ising model with random field - RFIM, where we use three probability distributions for the field: bimodal, gaussian and gaussian double-peaked. The phase-diagrams were obtained in t - h plane for the cases Ferromagnetic-F and Antiferromagnetic-AF with the aid of Maxwell's construction procedure (equality of the free energies at line phase transition) identifying the tricritical point - PTC in each case. We present two proposals for obtaining the free energy, and in one of them it was possible to study the behavior of the thermodynamic properties in the regions of 1st and 2nd order. For a second application of numerical implementation, we use the quantum model of anisotropic Heisenberg spin (1/2) (with anisotropy parameter Δ), which lies in the particular cases that are important: one-dimensional Ising (Δ=1) and isotropic Heisenberg (Δ=0), being applied in the study of magnetic thin films formed by monolayers where the presence of free surfaces substantially alters the system behavior. We simulate this case, the spin frustration of considering interactions between the first (J₁) and second (J₂) interactions with neighboring F and AF respectively, being related by the parameter α=J₁/J₂. We studied the influence of increasing the dimensionality of the system, made by increasing the number of layers (L) of the film, the behavior of the phase diagram α - t. Finally, we apply the relations of the Renormalization Group in the Heisenberg Hamiltonian for a thin film to study the behavior of critical exponents as a function of parameters such as temperature and number of layers.