Critical behavior study of classical ferromagnetic models with Dzyaloshinskii-Moriya interactions through Monte Carlo simulations

Abstract

Monte Carlo (MC) simulations were intensively employed in the study of the effects of Dzyaloshinskii-Moriya (DM) interactions on phase transitions in the two-dimensional (2d) planar rotator model on a square lattice and the three-dimensional (3d) Heisenberg model on a simple cubic lattice. Both models included isotropic ferromagnetic exchange (J) and DM isotropic exchange (D) interactions between nearest-neighbor classical spins. These interactions together lead to a modulation of the spin vectors along the principal axes of the lattices in a manner that depends upon the ratio of the interactions d = D/J. This makes the spin vectors arrange themselves in a spiral fashion. 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 could be extracted and FSS analyses would not be useful for estimating critical temperatures and critical exponents for arbitrary values of d. The incommensurabilities were circumvented by using both fluctuating boundary conditions (FBC) and a new type of boundary condition, shifted boundary condition (SBC). Internal energy and order parameter data could then be obtained from MC simulations and used with single histogram techniques to calculate thermodynamic functions of interest at convenient temperature intervals. As a result, FSS at the critical points of the models leads to quite precise results. The 2d planar rotator model with DM interactions was found to be equivalent to the pure planar rotator model with a rescaled exchange term which depended upon d. In this case, the modulation of the spin orientations caused by the DM interaction could 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 both in the configuration with the transformed modulation of the spins and in the actual spiral configuration. The 3d Heisenberg model with DM interaction was shown to undergo a second-order phase transition like the pure isotropic Heisenberg model. The spin configurations change from an ordered spiral phase into a disordered paramagnetic phase above the critical temperature. Structure factor calculations for a simple cubic lattice at the transition have additionally shown that the magnetic long-range ordering of the spin vectors does occur, although the universality class of the isotropic Heisenberg model no longer holds for the model with DM interaction. However, the Heisenberg model with DM interactions obeys weak universality. This weak universality class was shown to be the same as the XY model.