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