= Solution
For a ferromagnet the scalar <order parameter> is the spontaneous <spin magnetization>. It distinguishes a symmetry-invariant phase with zero mean from one of two phases with a nonzero mean of opposite signs. At zero field, a finite system with <spin inversion symmetry> has exactly zero mean. The spontaneous value is defined by taking the <thermodynamic limit> first and then sending the symmetry-breaking <conjugate field> to zero from one side.
A <Landau-Ginzburg theory> promotes the coarse magnetization to a scalar field $M(x)$ and uses
$$
\mathcal F[M]=\int d^Dx\left[\frac\kappa2|\nabla M|^2+\frac r2M^2+\frac u4M^4+\frac v6M^6-hM\right],\qquad \kappa>0,\ v>0.
$$
At $h=0$, <spin inversion symmetry> permits only even local powers. The gradient term penalizes spatial variations. The equilibrium statistical weight is $e^{-\mathcal F/(k_BT)}$; the <Landau approximation> chooses a stable minimum of $\mathcal F$, whereas the full functional integral retains fluctuations around it. In a homogeneous phase this gives the <equation of state> $h=rM+uM^3+vM^5$. The field $h$ is conjugate to the <order parameter>, with $M=-\partial f_{\rm eq}/\partial h$.
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