Validity of mean-field theories for infinitely long-range forces

Lattice models with pair interactions consisting of a sum of a short-range interaction and a weak long-range interaction, are studied. The long-range potentials are such that it is possible to divide the lattice into sublattices with an interaction of the Kac form ${\varphi }^{rs}={\gamma }^{\nu }{F}^{rs}\left(\gamma r_{ij}\right)$ between each pair $i$, $j$ of spins, where $r$ and $s$ characterize the sublattices to which the spins belong. It is shown that under certain specified conditions a mean-field theory with several mean fields is exact in the limit $\gamma \to 0$. The special case in which the long-range interaction do not depend upon the sublattice variables is well known and is, in contradistinction to the general case, covered by the Lebowitz-Penrose theorem.