Critical Correlations of Certain Lattice Systems with Long-Range Forces

Abstract

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 $v\left(\mathrm{r}\right)$ having a weak, long-ranged tail that approaches ${\gamma }^d\varphi \left(\gamma \mathrm{r}\right)$ as $\gamma$ 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 $-\frac{v\left(\mathrm{r}\right)}{\mathrm{kT}}$, when $v\left(\mathrm{r}\right)\ll \mathrm{kT}$. We recover by a graph technique the previously known result that in one dimension, quantities of interest such as the inverse correlation length $\kappa$ cannot be expanded in $\gamma$ at the mean-field critical temperature $T_0$ and density ${\rho }_0$ that characterize the critical point in the $\gamma \to 0$ limit. Instead, at ($T_0, {\rho }_0$), $\kappa \sim {\gamma }^{\frac{4}{3}}$ as $\gamma \to 0$. This is true for both the Ising and the spherical models. In the two-dimensional spherical model, we find $\kappa$ to vary as ${\gamma }^2{\left[\ln \right(\frac{1}{\gamma }\left)\right]}^{\frac{1}{2}}$ when $\gamma \to 0$, while in three dimensions $\kappa \sim {\gamma }^{\frac{5}{2}}$. In the Ising case for $d=1\mathrm{}$, we characterize topologically the infinite sum of graphs that contribute to the $\kappa \sim {\gamma }^{\frac{4}{3}}$ 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 $d=3\mathrm{}$ we do not attempt to deal with the correlation length in the Ising case, but instead consider the shift in the critical temperature as $\gamma \to 0$. We find that the spherical-model result that gives a shift of order ${\gamma }^3$ is exact to that order in $\gamma$ for the Ising model. We also find that the lowest-order correction to the spherical-model result is of the form ($\mathrm{const}{\gamma }^6\ln \left(\mathrm{const}\gamma \right)$, in agreement with recent work by Thouless; but in general we expect to find terms of order ${\gamma }^4$ and ${\gamma }^5$ from the spherical-model result dominating this correction.