Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 43 / 1 / b

Use the connected correlation function $G(x)=⟨M(x)M(0)⟩-⟨M{⟩}^2$, since the disconnected constant in an ordered phase does not measure fluctuation decay. Away from criticality its large-distance exponential defines the correlation length:when that limit exists. Algebraic prefactors do not affect this definition.

Expand the Landau-Ginzburg theory about a stable homogeneous minimum $M_0$. With $A=V^{\prime \prime }(M_0)>0$, the quadratic fluctuation energy is $\frac{1}{2}\int d^{D}x[\kappa |\nabla \delta M{|}^2+A(\delta M{)}^2]$. The Gaussian functional integral givesThis 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.