= Solution
A coarse-grained scalar <order parameter> $\phi(\mathbf x)$ describes a short-range Ising-like transition with sign <symmetry>. Write the dimensionless Landau-Ginzburg functional as
$$
\mathcal H[\phi]=\frac{F[\phi]}{k_BT}
=\int d^dx\left[\frac c2(\nabla\phi)^2+\frac r2\phi^2+\frac u4\phi^4-h\phi\right],\qquad c,u>0.
$$
Here $r=a_t t$, with the relevant critical-mass shift understood when fluctuations are included. Higher local <powers> and <gradients> are negligible only within a justified long-distance expansion. The <partition function> is $Z=\int\mathcal D\phi\,e^{-\mathcal H}$ with a microscopic cutoff. The local functional is not already the fluctuation-integrated thermodynamic <free energy>.
<Mean-field theory> replaces this integral by its stable uniform saddle, $r\phi_0+u\phi_0^3=h$. At $h=0$,
$$
\phi_0=0\quad(r>0),\qquad\phi_0=\pm\sqrt{|r|/u}\quad(r<0),\qquad
f_{\min}-f_0=\begin{cases}0&r>0,\\-r^2/(4u)&r<0.\end{cases}
$$
The branch curvature is $r_L=r+3u\phi_0^2$: it is $r$ above and $2|r|$ below. Hence
$$
\chi_+=r^{-1},\quad\chi_-=(2|r|)^{-1},\quad
\xi=\sqrt{c/r_L},\quad h=u\phi_0^3\text{ at }r=0.
$$
Together with the finite heat-capacity jump from $f_{\min}$, this yields
$$
\boxed{\beta_{\rm MF}=\tfrac12,\quad\gamma_{\rm MF}=1,\quad\delta_{\rm MF}=3,
\quad\nu_{\rm MF}=\tfrac12,\quad\eta_{\rm MF}=0,\quad\alpha_{\rm MF}=0.}
$$
The equal-$|t|$ susceptibility <amplitude> ratio is two. Uniform saddles also miss <domain walls> and spatial texture. For an $n$-component <order parameter>, <symmetry> allows a vector generalization; ordered transverse directions additionally carry Goldstone fluctuations.
To include small fluctuations set $\phi=\phi_0+\delta\phi$. The <Gaussian> action is $\tfrac12\int[c(\nabla\delta\phi)^2+r_L(\delta\phi)^2]$, giving
$$
G(q)=\langle|\delta\phi_q|^2\rangle_{\rm spectral}=\frac1{cq^2+r_L},\qquad
\delta f=\frac12\int^{\Lambda}\frac{d^dq}{(2\pi)^d}\log(cq^2+r_L)+\text{normalization}.
$$
The displayed $G(q)$ is a spectral <covariance> per Fourier delta, avoiding an extra <volume> factor. As $r_L\to0$, a growing <correlation length> lets increasingly large correlated regions fluctuate. The singular <Gaussian> heat correction contains $\int d^dq/(cq^2+r_L)^2$, which is infrared singular for $d<4$ and logarithmic at four. This signals failure of a simple saddle expansion; the <Gaussian> heat exponent obtained by extrapolation is not the exact interacting <critical exponent>.
The <Ginzburg criterion> compares the fluctuation of a correlation-volume average with $\phi_0^2$, rather than using all ultraviolet fluctuations at a point. Keeping $q\lesssim\xi^{-1}$ below the transition gives, up to finite convention-dependent constants,
$$
\langle(\delta\phi)_\xi^2\rangle
\sim\int_{|q|\lesssim\xi^{-1}}\frac{d^dq}{(2\pi)^d}\frac1{cq^2+2|r|}
\sim c^{-d/2}|r|^{(d-2)/2}.
$$
Dividing by $\phi_0^2=|r|/u$ yields
$$
\boxed{\mathcal G\sim u c^{-d/2}|r|^{(d-4)/2}\ll1.}
$$
For $d<4$, writing $r=a_t t$ gives a nonuniversal <Ginzburg reduced-temperature window>
$$
\boxed{|t|\gg\mathrm{Gi},\qquad
\mathrm{Gi}\sim\left[u c^{-d/2}a_t^{(d-4)/2}\right]^{2/(4-d)}.}
$$
Inside $|t|\lesssim\mathrm{Gi}$, interacting fluctuations must be treated. For example, in three dimensions $\mathrm{Gi}\sim u^2/(c^3a_t)$ in this normalization. Above four dimensions the long-wavelength ratio vanishes at criticality; at four it is marginal and interaction-generated logarithms require further analysis.
\Image[/past-exam-of-the-phase-transitions-and-collective-phenomena-course-of-the-university-of-cambridge/2012/paper-1-ptcp2012-ginzburg.png]
{title=Mean-field wells and the Ginzburg test}
{description=Original scalar quartic potentials and dimensionless illustrative Ginzburg ratios. The three-dimensional ratio grows towards criticality, the four-dimensional power estimate is constant and the five-dimensional one falls. A unit-ratio line marks a chosen crossover convention, not measured material data.}
The spatial <gradient> term supplies <correlations> absent from uniform <Landau theory>. The Ginzburg test bounds where its mean-field saddle remains self-consistent; it does not compute nonclassical exponents or a universal material-independent window.<Statistical Field Theory by David Tong>[§2.2.3–§2.2.4: correlation length, upper critical dimension and limits of Gaussian fluctuations]{ref}
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