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 γᵈφ(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~γ4/3 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)]1/2 when γ→0, while in three dimensions κ~γ5/2. In the Ising case for $d=1$, we characterize topologically the infinite sum of graphs that contribute to the κ~γ4/3 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 (constγ⁶ln(constγ), 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.