Critical Correlations in the Ising Model

Abstract

The Ising-model correlation function $C\left(R_{12}\right)=〈{\mu }_1{\mu }_2〉$ 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 $C\left(R\right)\mathrm{}\propto R^{-1-\eta }$ for $R\ll \frac{1}{\kappa }$, and $C\left(R\right)\mathrm{}\propto {\kappa }^{\eta }{R}^{-1\mathrm{}}\exp (-\kappa R)$ for $R\gg \frac{1}{\kappa }$. The correlation length $\frac{1}{\kappa }$ becomes infinite at the critical point. The calculated value $\eta =0.646\mathrm{}$ 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.