= Solution
An <order parameter> distinguishes phases by a macroscopic quantity that changes at the transition. For an <Ising model> ferromagnet it is the <magnetization> per <Ising spin>, $M=N^{-1}\sum_n\langle\sigma_n\rangle$. At zero <magnetic field> the <statistical Hamiltonian> is invariant under <spin inversion symmetry>. The disordered phase preserves this <symmetry>, whereas a selected ordered phase has $M\ne0$ and a spin-reversed partner with $-M$. At finite volume with symmetric <boundary conditions> the <expected value> is zero even below the transition; <spontaneous magnetization> means taking the <thermodynamic limit> before sending a symmetry-breaking field to zero.
<Landau-Ginzburg theory> introduces a coarse-grained local <order parameter> $\phi(x)$ and a symmetry-constrained <free-energy functional>. For a short-range system with scalar <spin inversion symmetry>, a useful expansion is
$$
\mathcal F[\phi]=\int d^Dx\left\{\frac\kappa2|\nabla\phi|^2+\frac r2\phi^2+\frac u4\phi^4+\frac v6\phi^6-h\phi\right\},\qquad \kappa>0.
$$
The coefficients vary smoothly with microscopic control parameters at the level of the <Landau approximation>; normally $r=A(T-T_c)$ with $A>0$. The <gradient> term penalizes spatial variation. The <magnetic field> is conjugate to the <order parameter>, and odd powers at $h=0$ are excluded by <spin inversion symmetry>. With $u>0$, the sextic term may be omitted locally; with $u<0$, a stabilizing positive $v$ is essential. The equilibrium uniform <order parameter> minimizes the local potential $V(M)$, giving
$$
h=rM+uM^3+vM^5.
$$
This <equation of state> follows from minimization, rather than being imposed as an independent assumption. Vector or tensor <order parameters> and their allowed invariants describe other <symmetry> classes.
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