Abstract
A continuous manifold of inequivalent spin configuration minimize the energy of the square planar rotator model when the spins are coupled by dipolar interactions. This continuous degeneracy is accidental in nature and causes a soft mode at $\mathbf{q}=(2\ensuremath{\pi}/a)(1/2,1/2)$ in the simple spin-wave spectrum. Consequently no long-range order is obtained in simple spin-wave approximation in contrast with the expectation based on Monte Carlo simulation. The solution of the puzzle is found going beyond the harmonic approximation. Indeed a careful treatment of the magnon-magnon interaction replaces the accidental soft mode by a temperature-dependent gap so that long-range order is restored. Satisfactory agreement between renormalized spin-wave theory and Monte Carlo simulation is obtained in the low-temperature regime. Critical properties investigated by finite size scaling of Monte Carlo data differ from both mean-field and two-dimensional Ising model critical behavior.
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