Phase diagram of the two-dimensional dipolar Heisenberg model with Dzyaloshinskii-Moriya interaction and Ising anisotropy

We study phase transitions in the two-dimensional Heisenberg model with Dzyaloshinskii-Moriya interaction, Ising anisotropy η, and dipolar interaction under zero and finite magnetic fields H. For three typical strengths (zero, weak, and strong) of the dipolar interaction, we present the H-η phase diagrams by estimating order parameters for skyrmion-lattice and helical phases and in-plane magnetization using a Monte Carlo method with an O(N) algorithm. We find in the phase diagrams three types of skyrmion-lattice phases, i.e., two square lattices and a triangular lattice, helical phases with diagonal and vertical (or horizontal) stripes, a canted ferromagnetic phase, and a polarized ferromagnetic phase. The effect of the dipolar interaction varies the types of the skyrmion and helical phases in a complex manner. The dipolar interaction also expands the regions of the ordered phases accompanying shifts of the phase boundaries to the positive H and η directions and causes an increase of the density of skyrmions and shortening of the pitch length (stripe width) of helical structures. We discuss the details of the features of the phase transitions.