Dipolar interaction and long-range order in the square planar rotator model

A continuous manifold of inequivalent spin configuration minimize the energ y 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 q=(2 pi/a)(1/2,1/2) in the simple spin-wave spectrum. Conse quently no long-range order is obtained in simple spin-wave approximation i n contrast with the expectation based on Monte Carlo simulation. The soluti on 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 resto red. Satisfactory agreement between renormalized spin-wave theory and Monte Carlo simulation is obtained in the low-temperature regime. Critical prope rties investigated by finite size scaling of Monte Carlo data differ from b oth mean-field and two-dimensional Ising model critical behavior.