Critical Behavior of Several Lattice Models with Long-Range Interaction

We consider a one-dimensional model with infinite-range interaction, a two-dimensional model, and a three-dimensional model, whose free energies can be expressed in terms of the largest eigenvalue of an integral equation. High- and low-temperature expansions in powers of the reciprocal of the range of the exponential part of the interaction, with the classical Curie-Weiss theory as leading term, are developed and studied in the critical region. We find that to leading order in the critical region the resummed high- and low-temperature expansions are analytic at the classical critical point but are nonanalytic at a displaced critical point. The modified singularities, which are no longer of Curie-Weiss type, give critical exponents which are identical with those obtained by Brout and others, and are almost surely not the true exponents. The technique, however, suggests a possible general method of successive approximation to true critical behavior.