Absence of ferromagnetism or antiferromagnetism in the isotropic Heisenberg model with long-range interactions

It is shown that the isotropic Heisenberg model with an exchange interaction J ( R ) cannot be ferromagnetic or antiferromagnetic if the spherical model with an interaction| J ( R )| does not exhibit a phase transition. From this result it follows that the isotropic Heisenberg model with J ( R ) = A /| R | d +σ (where d is the dimensionality of the lattice, A > 0 and σ > 0) cannot be ferromagnetic for d < 3 if σ >or= d .