= Solution
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 (physics)> at zero field, a stable quartic model is
$$
F[\phi]=\int d^Dx\left[\frac K2|\nabla\phi|^2+\frac r2\phi^2+\frac u4\phi^4-h\phi\right],\qquad K,u>0,
$$
with $r=a(T-T_c)$ near a mean-field transition. Equilibrium fluctuations have weight $e^{-F/(k_BT)}$. In the <Landau approximation>, minimize the functional. A uniform stationary state obeys $r\phi+u\phi^3=h$. At zero field the minimum is $\phi=0$ for $r>0$ and $\phi=\pm\sqrt{-r/u}$ 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 $r=0$, $\phi=(h/u)^{1/3}$. Gaussian fluctuations around the minimum have inverse <covariance> $(Kq^2+r)/(k_BT)$ above and $(Kq^2+2|r|)/(k_BT)$ 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 exponents>
$$
\boxed{\alpha=0,\quad\beta_{\rm mag}=1/2,\quad\gamma=1,\quad\delta=3,\quad\nu=1/2,\quad\eta=0.}
$$
The <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 $\phi_0^2=|r|/u$. Modes with $q\lesssim\xi^{-1}$ dominate this coarse-grained fluctuation estimate:
$$
\langle(\delta\phi)^2\rangle_\xi\simeq k_BT\int_{q\lesssim\xi^{-1}}\frac{d^Dq}{(2\pi)^D}\frac1{Kq^2+2|r|}
=A_D\frac{k_BT}{K^{D/2}}|r|^{D/2-1},
$$
where the positive dimensionless $A_D$ depends on the averaging window and the factor of two used in $\xi$. Dividing by $\phi_0^2$ gives
$$
\boxed{\mathcal G(r)=A_D\frac{k_BT\,u}{K^{D/2}}|r|^{(D-4)/2}\ll1.}
$$
It 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 $r\to0$, so the asymptotic critical region is fluctuation dominated. Away from it, <mean-field theory> can still work when
$$
|r|\gg\left(A_Dk_BT\,u/K^{D/2}\right)^{2/(4-D)}.
$$
For $D>4$, 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. \b[Mean-field minimization supplies an approximation; the Ginzburg ratio decides whether its neglect of fluctuations is justified.]
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