Defesa de Tese de Doutorado #433 – Gabriel de Albuquerque Silva – 07/08/2024

Monte Carlo (MC) simulations have been employed in the study of the two-dimensional planar rotator model on a square lattice and the three-dimensional Heisenberg model on a simple cubic lattice, both models set with isotropic ferromagnetic (FM) and Dzyaloshinskii- Moriya (DM) interactions between classical spins. These interactions together lead to a modulation on the spin vectors along the principal axes of the lattice. Such modulation depends on the ratio of the DM to the FM interaction intensities (d). This makes the spin vectors arrange themselves in a spiral way. Moreover, an incommensurability, depending on the value of d, may take place in these models when finite lattices are used together with the standard periodic boundary condition (PBC). As a consequence, no meaningful results from finite-size scaling (FSS) analyses can be extracted. Then, to make FSS analysis useful, so that critical temperatures and critical exponents for arbitrary values of d could be extracted, the incommensurabilities have been circumvented by using fluctuating boundary condition (FBC) and a new type of boundary condition, named shifted boundary condition (SBC). Coherent energy and order parameter data could then be obtained from MC simulations and be used with single histogram techniques to calculate thermodynamic functions of interest at convenient temperature intervals. As a result, FSS at the criticality of the models have led to quite conclusive results. The two-dimensional planar rotator model with the DM interaction has been seen to be equivalent to the pure planar rotator model with a rescaled exchange term which depends on d. In this case, the modulation on the spin orientations caused by the DM interaction can accordingly be undone to recover the typical spin arrangement of the pure planar rotator model. As a result, the critical behavior of the model is dictated by the Berezinskii-Kosterlitz-Thouless (BKT) phase transition. The topological structure of vortices of the pure planar rotator model is present either in the configuration with the undone modulation on the spins or in the actual spiral configuration. The three-dimensional Heisenberg model with DM interaction has shown to undergo a second-order phase transition like the pure isotropic Heisenberg model. The spin configurations change from spiral phase into disordered paramagnetic phase when the critical temperature is overcome. Structure factor calculations in a simple cubic lattice at the transition have additionally shown that the magnetic long-range ordering of the spin vectors does occur, though the universality class of the isotropic Heisenberg model has no longer held for the model with DM interaction. More precisely, the Heisenberg model has presented what is called weak universality in the presence of the DM interactions. This weak universality class has shown to be the same as the XY model.