Infinite-fold energy degeneracy in 2D square lattices of magnetic spheres

We show that a two-dimensional square lattice of magnets can be studied by placing small cylindrical neodymium magnets inside plastic spherical shells and floating them on water, leaving their magnetic moments free to re-orient within the plane. Experimentally, anti-correlated dipole orientations between nearest neighbors appear to be favored energetically. This motivates the construction of a simplified single-variable energy function for a 2D square lattice of magnetic dipoles. For odd numbers of spheres, this ansatz yields a continuum of dipole configurations with the same energies, matching the observed behavior that the orientation of the dipoles in these lattices can be rotated freely. The behavior of square lattices with even numbers of spheres is strikingly different, showing strongly preferred orientations. While the energy calculated in this simplified model is larger than that of the actual ground state for finite size clusters, its asymptotic value in the limit where the number of spheres goes to infinity is in good agreement with the literature value. Additionally, rectangular arrangements of magnetic spheres with and without a defect are analyzed within the class of the single variable energy function. Simple experimental demonstrations qualitatively reproduce several interesting results obtained from all these analyses.