Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 69 / 3 / c

The Landau-Ginzburg theory for a scalar order parameter supplements a local symmetry-allowed potential with a gradient penalty. For a ${\mathbb{Z}}_2$ symmetry at zero field, a stable quartic model iswith $r=a(T-T_c)$ near a mean-field transition. Equilibrium fluctuations have weight $e^{-F/(k_{B}T)}$. In the Landau approximation, minimize the functional. A uniform stationary state obeys $r\varphi +u{\varphi }^3=h$. At zero field the minimum is $\varphi =0$ for $r>0$ and $\varphi =\pm \sqrt{-r/u}$ for $r<0$. The latter pair is spontaneous symmetry breaking of the discrete symmetry.

The uniform free-energy density at its minimum is zero above the transition and $-r^2/(4u)$ below, up to regular backgrounds. Differentiating the equation of state gives ${\chi }_{+}=1/r$ and ${\chi }_{-}=1/(2|r|)$. At $r=0$, $\varphi =(h/u{)}^{1/3}$. Gaussian fluctuations around the minimum have inverse covariance $(K{q}^2+r)/(k_{B}T)$ above and $(K{q}^2+2|r|)/(k_{B}T)$ below. Their correlation lengths are ${\xi }_{+}=(K/r{)}^{1/2}$ and ${\xi }_{-}=(K/(2|r|){)}^{1/2}$. These results give the usual mean-field critical exponentsThe heat capacity has a finite jump rather than a power-law divergence. Negative quartic coefficient requires higher stabilizing terms and can produce a first-order transition; a vanishing quartic coefficient requires a tricritical analysis. Thus the assumed positivity of $u$ matters.

The Ginzburg criterion tests whether this saddle description is self-consistent. Below the transition compare fluctuations of the order parameter averaged over a correlation volume with its squared mean-field value ${\varphi }_0^2=|r|/u$. Modes with $q\lesssim {\xi }^{-1}$ dominate this coarse-grained fluctuation estimate:where the positive dimensionless $A_D$ depends on the averaging window and the factor of two used in $\xi$. Dividing by ${\varphi }_0^2$ givesIt is the fluctuation of a correlation-volume average, not the full cutoff-dominated microscopic variance, that should be compared with the mean-field order parameter.

For $D<4$, $\mathcal{G}$ diverges as $r\to 0$, so the asymptotic critical region is fluctuation dominated. Away from it, mean-field theory can still work whenFor $D>4$, the long-wavelength ratio tends to zero near criticality, consistently with Gaussian control and mean-field thermodynamic exponents. At $D=4$, quartic coupling is marginal; scale-dependent logarithms require a renormalization-group analysis rather than interpreting a constant bare ratio as a proof of no corrections. This identifies four as the upper critical dimension of the short-range scalar quartic model.

The Landau-Ginzburg theory is valuable because symmetries restrict a manageable list of local operators, the gradient term describes correlations and interfaces, and the Ginzburg criterion identifies when neglected fluctuations become decisive. Below the upper critical dimension an interacting renormalization-group fixed point, rather than the bare saddle, determines universal exponents. For continuous order parameters the manifold of ordered minima also has Goldstone modes, and lower-critical-dimension fluctuations require separate attention. Mean-field minimization supplies an approximation; the Ginzburg ratio decides whether its neglect of fluctuations is justified.