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We make a large-scale Monte Carlo (MC) simulation of a ferromagnetic Heisenberg model with dipolar interactions on a two-dimensional square lattice L × L with open boundaries. We find that, as L is increased, the spin structure changes from a ferromagnetic one to a novel one characterized by a circular arrangement of spins. We suggest that, in the thermodynamic limit, a long-range circular phase occurs in the cooperation of the exchange and the dipolar interactions and that the phase transition between the paramagnetic phase and the circular phase is of the second order. In this study, we develop a discrete update MC method which reduces the CPU time from of the order N 2 to N log 2 N per one MC sweep, where N is the number of the spins.
Sasaki et al. (Tue,) studied this question.