= Solution
An <order parameter> distinguishes equilibrium <thermodynamic phases>. For a scalar magnetic system, a slowly varying coarse-grained <scalar field> $\phi(x)$ represents local <magnetization>, and its uniform expectation $M=\langle\phi\rangle$ is the <order parameter>. Under <spin inversion symmetry>, $\phi\mapsto-\phi$ and $h\mapsto-h$. The disordered phase has $M=0$ at $h=0$, while a pure ordered phase has $M\ne0$. In a finite symmetric system the average still vanishes at $h=0$; <spontaneous symmetry breaking> means taking the <thermodynamic limit> before $h\to0^+$ or $h\to0^-$ to select a <pure thermodynamic phase>.
<Landau-Ginzburg theory> assigns a local <free-energy functional>
$$
\mathcal F[\phi]=\int d^Dx\left[\frac\kappa2|\nabla\phi|^2+\frac r2\phi^2+\frac u4\phi^4+\frac v6\phi^6-h\phi\right],\qquad \kappa>0.
$$
<Spin inversion symmetry> forbids odd powers at zero <conjugate field>; locality and slow variation motivate the <gradient expansion>. The <gradient energy> penalizes interfaces and determines long-wavelength response. Integrating $e^{-\mathcal F/(k_BT)}$ over fields includes fluctuations, whereas the <Landau approximation> replaces that integral by its stable stationary configuration. For a uniform phase this reduces to minimizing $V(M)$. If $f$ is the equilibrium <free-energy density>, then $M=-\partial f/\partial h$, so the <order parameter> is also the thermodynamic response to its <conjugate field>.
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