= Solution
An <order parameter> distinguishes phases macroscopically. In an <Ising model> it is the local coarse-grained <magnetization>; the uniform mean is $M=V^{-1}\int\phi(x)d^Dx$. In a selected ordered phase $M\ne0$, whereas the disordered phase has $M=0$ at zero <conjugate field>. The ordered signs are related by <spin inversion symmetry>. A finite zero-field system may average both signs to zero, so <spontaneous symmetry breaking> is defined using the <thermodynamic limit> before removing a symmetry-selecting field. For other systems the <order parameter> can be a vector, a density difference, or a tensor rather than a single magnetization.
<Landau-Ginzburg theory> makes the <order parameter> a spatial field and expands a local <free energy> consistent with the symmetries. For a short-range scalar system with spin inversion symmetry at $h=0$, a useful expression 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 linear source is the <field conjugate to an order parameter>. Odd terms at zero field are forbidden by spin inversion symmetry. The coefficients depend smoothly on control parameters away from singular fluctuation effects. With $u>0$ the quartic term suffices locally; if $u<0$, take $v>0$ for stability. The <Landau approximation> minimizes this functional; statistical field theory also integrates fluctuations about those minima.
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