= Solution
The expansion is a local phenomenological description in a small scalar <order parameter> near a transition. Its coefficients summarize microscopic physics and generally depend smoothly on <temperature>. At zero field, spin-reversal <Z2 symmetry> requires even powers of <magnetization>; the conjugate field supplies the term $-hm$. The additive $F_0$ does not affect minimization with respect to $m$.
For the ordinary continuous-transition quartic model, \b[$b>0$ ensures stability], and the quadratic coefficient changes sign, $a=a_0t+O(t^2)$ with $a_0>0$. If $b$ vanishes or is negative, stabilizing higher powers are required and the phase structure can change. Treating one uniform <magnetization> neglects spatial gradients, fluctuations and domains; the resulting exponents are predictions of <mean-field theory>.
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