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