Structure of Correlation Functions near the Critical Point

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 $A\left(\mathbf{r}\right)$ and $B\left(\mathbf{r}\right)$ are local dynamical variables and $\omega$ stands for the variables specifying the thermodynamic state, it is found that for $\omega$ near the critical point ${\omega }_c$ and for $r$ large enough, $〈A\left(\mathbf{r}\right)B\left(0\right)\mathrm{}〉-〈A\left(\mathbf{r}\right)〉〈B\left(0\right)\mathrm{}〉\simeq \frac{d_A\left(\omega \right)d_B\left(\omega \right)e^{-\kappa \left(\omega \right)}}{r^{b\left(\omega \right)}}$, where $\kappa \left(\omega \right)$ and $b\left(\omega \right)$ are the same for a large class of $A$ and $B$ and $\kappa \left(\omega \right)\to 0$, $b\left(\omega \right)\to b\ge 0 \mathrm{as} \omega \to {\omega }_c$. 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 ${\gamma }^{\prime }+2\mathrm{}\beta =2-{\alpha }^{\prime }$, which is often conjectured for the critical exponents.