Long-range dipole-dipole interactions in a two-dimensional Heisenberg ferromagnet

It has previously been suggested by Yafet, Kwo, and Gyorgy that dipole-dipole interactions can induce long-range ferromagnetic order in the infinitely extended, isotropic two-dimensional Heisenberg ferromagnet. A crucial, though problematic, step employed in this previous investigation is the long-wavelength expansion of certain dipole sums, which are treated in a continuum approximation by replacing the sums over atomic sites by integrals. Without appealing to a continuum approximation, we are able to derive the exact leading terms of the long-wavelength expansion of the dipole sums of a square lattice. We show that the earlier approximations of Yafet, Kwo, and Gyorgy are correct, apart from terms characteristic of the symmetry of the square lattice, which are unimportant to a spin-wave theoretic discussion of ferromagnetic order, such as considered by these earlier authors. Our result supports recent studies of the effect of dipole interactions in two-dimensional spin systems which have utilized the dipole sum approximations of Yafet and co-workers.