= Solution
\b[<Landau theory> expands the <free energy> in a small <order parameter> and keeps terms allowed by the system's symmetries.] For a scalar <order parameter> $q$ with <sign-reversal symmetry in Landau theory> $q\mapsto-q$, a useful free-energy <density> is
$$
f(q,T,h)=f_0(T)+\frac a2(T-T_c)q^2+\frac b4q^4-hq,\qquad a,b>0.
$$
At zero conjugate field $h$, the <quartic Landau equilibrium branches> obtained by minimizing are
$$
\boxed{q_{\rm eq}=0\quad(T>T_c),\qquad q_{\rm eq}=\pm\sqrt{\frac{a(T_c-T)}b}\quad(T<T_c).}
$$
The single minimum becomes two symmetry-related minima: the <order parameter> turns on continuously and a chosen equilibrium phase spontaneously breaks the symmetry. At the <critical temperature> the quadratic restoring term vanishes. The mean-field exponent is $q\propto(T_c-T)^{1/2}$; the susceptibility above $T_c$ is $\partial q/\partial h=1/[a(T-T_c)]$.
\Image[/past-exam-of-the-thermal-and-statistical-physics-course-of-the-university-of-cambridge/2016/paper-1-landau-free-energy.png]
{title=A symmetric quartic Landau free energy changes from one minimum to two as its quadratic coefficient changes sign}
{source=paper-1-landau-free-energy.py}
More general allowed terms describe other transitions. A negative quartic term, stabilized by a positive sixth-order term, can give a first-order transition with a discontinuous <order parameter> and metastable minima. Without the sign-reversal symmetry, a cubic term may also be allowed. The expansion is phenomenological: its coefficients depend on microscopic physics. Minimizing a uniform polynomial neglects important spatial fluctuations, particularly near a <critical point>, so its <critical exponents> need not be exact.<Phase Transitions by David Tong>[Section 5.4: Landau free energy, symmetry and mean-field transitions]{ref}
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