Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 50 / 1 / b / Solution

Use the connected correlation function $G(x)=⟨M(x)M(0)⟩-⟨M{⟩}^2$ in a selected pure thermodynamic phase. The exponential correlation length is defined by ${\xi }^{-1}=-{\lim }_{|x|\to \infty }|x{|}^{-1}\log |G(x)|$, when this limit exists in the chosen direction. Linearizing the Landau-Ginzburg theory around a stable uniform minimum $M_0$ givesHence the Landau scalar correlation length is $\boxed{\xi =\sqrt{K/V^{\prime \prime }(M_0)}}$. Away from criticality the massive Gaussian field correlation tail behaves as $|x{|}^{-(D-1)/2}{e}^{-|x|/\xi }$, up to its amplitude. At a continuous transition $V^{\prime \prime }$ tends to zero, so $\xi$ diverges and fields vary on scales much longer than the lattice spacing. This motivates a gradient expansion, but the simultaneous growth of correlated fluctuations is precisely why the Ginzburg criterion must be checked.