= Solution
An <order parameter> distinguishes the equilibrium phases and captures the relevant broken <symmetry>. For an <Ising model>, it is the spontaneous <magnetization>: its sign changes under spin reversal, and it is zero in the symmetric phase. Spontaneous <magnetization> means taking the thermodynamic limit before sending the external field to zero; the finite-volume zero-field mean can vanish even in an ordered phase.
In <Landau-Ginzburg theory>, a slowly varying real <scalar field> $\phi(x)$ represents the coarse-grained local <order parameter>. A useful local <free-energy density> gives the functional
$$
\mathcal F[\phi]=\int d^Dx\left[\frac\kappa2(\nabla\phi)^2+\frac r2\phi^2+\frac u4\phi^4+\frac v6\phi^6-h\phi\right],\qquad\kappa>0,\ v>0.
$$
At zero field, spin-reversal <symmetry> excludes odd powers. The gradient term penalizes rapid spatial variation. The field $h$ is the <field conjugate to an order parameter>. The <mean-field approximation> replaces the fluctuating <scalar field> by the equilibrium minimizer; a uniform minimizer has $\phi=M$ and obeys $rM+uM^3+vM^5=h$.
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