CRITICAL CORRELATIONS OF CERTAIN LATTICE SYSTEMS WITH LONG-RANGE FORCES

We show the way perturbation procedures originated by Brout and Hemmer and further developed by Lebowitz and co-workers can be used to study the critical behavior of lattice systems with a potential v(r) having a weak, long-ranged tail that approaches γd(γr) as γ approaches zero, where d is dimensionality. We consider both the Ising and the spherical models, and note that the results for the latter model are also those that follow from the Ornstein-Zernike assumption that the direct correlation function behaves like -v(r)kT, when v(r)kT. We recover by a graph technique the previously known result that in one dimension, quantities of interest such as the inverse correlation length κ cannot be expanded in γ at the mean-field critical temperature T0 and density ρ0 that characterize the critical point in the γ→0 limit. Instead, at (T0,ρ0), κ∼γ43 as γ→0. This is true for both the Ising and the spherical models. In the two-dimensional spherical model, we find κ to vary as γ2[ln(1γ)]12 when γ→0, while in three dimensions κ∼γ52. In the Ising case for d=1, we characterize topologically the infinite sum of graphs that contribute to the κ∼γ43 term in the expansion of the pair correlation function. (In the spherical model, a chain graph gives the entire contribution to the pair correlation function for all d). For d=3 we do not attempt to deal with the correlation length in the Ising case, but instead consider the shift in the critical temperature as γ→0. We find that the spherical-model result that gives a shift of order γ3 is exact to that order in γ for the Ising model. We also find that the lowest-order correction to the spherical-model result is of the form (const γ6 ln(const γ), in agreement with recent work by Thouless; but in general we expect to find terms of order γ4 and γ5 from the spherical-model result dominating this correction. © 1969 The American Physical Society.