Critical Correlations in the Ising Model

Abstract

The Ising-model correlation function $C({R}_{12})=〈{\ensuremath{\mu}}_{1}{\ensuremath{\mu}}_{2}〉$ is studied in terms of a novel $N$-fold integral representation. This formula stems from a procedure proposed by Montroll and Berlin. The integral is estimated by maximizing the integrand, an approximation related to the spherical-model assumptions. The correlation function is not of the Ornstein-Zernike type, just above the critical point, but rather $C(R)\ensuremath{\propto}{R}^{\ensuremath{-}1\ensuremath{-}\ensuremath{\eta}}$ for $R\ensuremath{\ll}\frac{1}{\ensuremath{\kappa}}$, and $C(R)\ensuremath{\propto}{\ensuremath{\kappa}}^{\ensuremath{\eta}}{R}^{\ensuremath{-}1}\mathrm{exp}(\ensuremath{-}\ensuremath{\kappa}R)$ for $R\ensuremath{\gg}\frac{1}{\ensuremath{\kappa}}$. The correlation length $\frac{1}{\ensuremath{\kappa}}$ becomes infinite at the critical point. The calculated value $\ensuremath{\eta}=0.646$ is too large, reflecting the omission of important terms in the evaluation of the integral. The unusual mechanism inducing the nonclassical behavior is carefully examined.