= Solution
Use the <connected correlation function> $G(x)=\langle M(x)M(0)\rangle-\langle M\rangle^2$, since the disconnected constant in an ordered phase does not measure fluctuation decay. Away from criticality its large-distance exponential defines the <correlation length>:
$$
\xi^{-1}=-\lim_{r\to\infty}\frac1r\log|G(r)|,
$$
when that limit exists. Algebraic prefactors do not affect this definition.
Expand the <Landau-Ginzburg theory> about a stable homogeneous minimum $M_0$. With $A=V''(M_0)>0$, the quadratic fluctuation energy is $\frac12\int d^Dx[\kappa|\nabla\delta M|^2+A(\delta M)^2]$. The <Gaussian functional integral> gives
$$
\widetilde G(k)=\frac{k_BT}{\kappa k^2+A},\qquad \boxed{\xi=\sqrt{\kappa/A}.}
$$
This <Ornstein--Zernike correlation function> shows why the curvature of the free-energy minimum controls correlations. At a <continuous phase transition> that curvature vanishes, allowing long-range fluctuations. In the ordinary quartic theory with $u>0$, $A=r$ above the transition and $A=-2r$ on either ordered branch below, so $\xi\propto|r|^{-1/2}$ on both sides. At an equilibrium first-order transition the competing minima generally retain nonzero curvature, and the <correlation length> remains finite. The limit of local stability at a <spinodal point> is a distinct mean-field phenomenon.
Back to article page
