 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 usesAt $h=0$, spin inversion symmetry permits only even local powers. The gradient term penalizes spatial variations. The equilibrium statistical weight is $e^{-\mathcal{F}/(k_{B}T)}$; 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 $h=rM+u{M}^3+v{M}^5$. The field $h$ is conjugate to the order parameter, with $M=-\partial f_{\mathrm{eq}}/\partial h$.