Incommensurability and phase transitions in two-dimensional XY models with Dzyaloshinskii-Moriya interactions

Three-dimensional Heisenberg model with Dzyaloshinskii-Moriya interaction: A Monte Carlo study

The three-dimensional classical Heisenberg model on a simple cubic lattice with Dzyaloshinskii-Moriya (DM) interactions between nearest-neighbors in all directions has been studied using Monte Carlo simulations. The Metropolis algorithm, combined with single histogram reweighting techniques and finite-size scaling analyses, has been used to obtain the thermodynamic behavior of the system in the thermodynamic limit. Simulations were performed with the same set of interaction parameters for both shifted boundary conditions (SBC) and fluctuating boundary conditions (FBC). Because of an incommensurability caused by the DM interaction, the SBC incorporated a fixed shift angle at the boundary which varies as a function of the DM interaction and lattice size. This SBC method decreases the simulation time significantly, but the distribution of states is somewhat different than that obtained with FBC. The ground state for nonzero DM interaction is a spiral configuration where the spins are restricted to lie in planes perpendicular to the DM vector. We found that this spiral configuration undergoes a conventional second-order phase transition into a disordered, paramagnetic state with the transition temperature being a function of the magnitude of the DM interaction. The limiting case with only DM interaction in the model has also been considered. The critical exponent ν, the critical exponent ratios α/ν, β/ν, γ/ν, as well as the critical temperature T_{c} and fourth-order cumulant of the order parameter U_{4}^{*} at T_{c} have been estimated for different magnitudes of DM interaction. The critical exponents and cumulants at the transition are different from those for the three-dimensional Heisenberg model, but the ratios α/ν, β/ν, γ/ν, U_{4}^{*}/ν are the same, implying that weak universality is valid for all values of DM interaction. Structure factor calculations for particular cases have been performed considering SBC and FBC in the simulations with different lattice sizes at the critical temperatures.

Incommensurate phases in the two-dimensional XY model with Dzyaloshinskii-Moriya interactions

The two-dimensional XY model with Dzyaloshinskii-Moriya interaction has been studied through extensive Monte Carlo simulations. A hybrid algorithm consisting of single-spin Metropolis and Swendsen-Wang cluster-spin updates has been employed. Single histogram techniques have been used to obtain the thermodynamic variables of interest and finite-size-scaling analysis has led to the phase transition behavior in the thermodynamic limit. Fluctuating boundary conditions have been utilized in order to match the incommensurability between the spin structures and the finite lattice sizes due to the Dzyaloshinskii-Moriya interaction. The effects of the fluctuating boundary conditions have been analyzed in detail in both commensurate and incommensurate cases. The Berezinskii-Kosterlitz-Thouless transition temperature has been obtained as a function of the Dzyaloshinskii-Moriya interaction and the results are in excellent agreement with the exact equation for the transition line. The spin-spin correlation function critical exponent has been computed as a function of the Dzyaloshinskii-Moriya interaction and temperature. In the incommensurate cases, optimal sizes for the finite lattices and the distribution of the boundary shift angle have been extracted. Analysis of the low temperature configurations and the corresponding vortex-antivortex pairs have also been addressed in some regions of the phase diagram.