Critical Behavior of Several Lattice Models with Long-Range Interaction

Abstract

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.