First-order transition in a cubic-quartic Landau potential ($A_2=2A_3^2/(9A_4),\Delta m=-2A_3/(3A_4)$)

For a real unrestricted scalar order parameter and $A=A_2{m}^2/2+A_3{m}^3/3+A_4{m}^4/4$ with $A_3\ne 0$ and $A_4>0$, the disordered global minimum is replaced discontinuously by a nonzero minimum at the displayed phase coexistence condition. At coexistence $A=A_4{m}^2[m+2A_3/(3A_4){]}^2/4$. Ordered stationary points first appear at the spinodal point $A_2=A_3^2/(4A_4)$, while the disordered point loses local stability at $A_2=0$; neither local limit is the equilibrium transition condition.

At coexistence, the landau free energy can factor as $F(\varphi )=\lambda {\varphi }^2(\varphi -v{)}^2/4$. The minima at $0,v$ have equal free energy. Differentiating shows the intermediate maximum at $v/2$, with barrier height $\lambda v^4/64$. The barrier is a local free-energy barrier; predicting spatial nucleation also requires interfacial or gradient information.

For $a,c>0$, $b\le 0$ and the molecular bound $Q\in [-1/2,1]$, coexistence with $Q=0$ occurs at $b_t=-{\min }_{0<Q\le 1}(a/Q+3cQ)$. Negative $Q$ has positive free energy and cannot be the stable phase in this coefficient range. Thus $a\le 3c$ gives $b_t=-\sqrt{12ac}$ and jump $0\to \sqrt{a/(3c)}$, whereas $a>3c$ gives $b_t=-(a+3c)$ and jump $0\to 1$. The interior case is the usual small-order-parameter Landau theory result. A formal unconstrained quartic minimization can violate the molecular bounds when extrapolated too far.