Order by thermal disorder in 2D planar rotator model with dipolar interactions

The energy of a square planar rotator model of spins interacting via dipolar forces is minimized by infinite inequivalent configurations corresponding to spins arranged on four interpenetrating sublattices making angles alpha, -alpha, pi+alpha, pi-alpha with a reference axis, alpha being arbitrary. This infinite degeneracy of the ground state is accidental in nature and one expects that it is removed by thermal fluctuations in agreement with Monte Carlo simulation. Indeed we find that the elementary excitation energies which depend on alpha lead to a free energy which is a function of alpha with minima at alpha = 0 and alpha = pi/2 corresponding to columnar configurations. This selection of columnar configurations out of the infinite ground state manifold is an example of order by thermal disorder.