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Physical Review

Structure of Correlation Functions near the Critical Point

Gerald L. Jones

Published Jul 5, 1968• 18 citations
Publisher

Abstract

An argument is presented for a certain asymptotic form for the correlation functions of two local dynamical variables when the system is near its critical point and in the one phase region. If $A(\mathbf{r})$ and $B(\mathbf{r})$ are local dynamical variables and $\ensuremath{\omega}$ stands for the variables specifying the thermodynamic state, it is found that for $\ensuremath{\omega}$ near the critical point ${\ensuremath{\omega}}_{c}$ and for $r$ large enough, $〈A(\mathbf{r})B(0)〉\ensuremath{-}〈A(\mathbf{r})〉〈B(0)〉\ensuremath{\simeq}\frac{{d}_{A}(\ensuremath{\omega}){d}_{B}(\ensuremath{\omega}){e}^{\ensuremath{-}\ensuremath{\kappa}(\ensuremath{\omega})}}{{r}^{b(\ensuremath{\omega})}}$, where $\ensuremath{\kappa}(\ensuremath{\omega})$ and $b(\ensuremath{\omega})$ are the same for a large class of $A$ and $B$ and $\ensuremath{\kappa}(\ensuremath{\omega})\ensuremath{\rightarrow}0$, $b(\ensuremath{\omega})\ensuremath{\rightarrow}b\ensuremath{\ge}0 \mathrm{as} \ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{c}$. For the Ising model of any dimension in zero field and below the critical temperature, the additional assumption of a single correlation length implies the relationship ${\ensuremath{\gamma}}^{\ensuremath{'}}+2\ensuremath{\beta}=2\ensuremath{-}{\ensuremath{\alpha}}^{\ensuremath{'}}$, which is often conjectured for the critical exponents.

DOI
10.1103/physrev.171.243

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