Discuss the zero-field phase boundaries, taking $h=0$ and writing $r=\mathcal A_2$, $q=\mathcal A_4$, $s=\mathcal A_6>0$. This qualification matters because a conjugate field generally rounds the zero-field continuous transition. Nonzero <stationary points> satisfy $r+qm^2+sm^4=0$.

For $q>0$, the <continuous phase transition> occurs at
$$
\boxed{r=0,\qquad q>0.}
$$
On the $r>0$ side the only stable phase is $m=0$; on the $r<0$ side the equilibrium <global minima> are $\pm m_0$ with $m_0^2\sim-r/q$. The <magnetization> therefore grows continuously from zero as $r$ changes sign.

For $q<0$, the <first-order phase transition> instead occurs while $r$ is positive. At <phase coexistence>, $m=0$ and $m=\pm m_0$ are equally deep <global minimum> points. With $a=m_0^2>0$, stationarity and equality of <free energies> give
$$
r+qa+sa^2=0,\qquad \frac12ra+\frac14qa^2+\frac16sa^3=0.
$$
Eliminating $r$ gives $-qa^2/4-sa^3/3=0$, hence
$$
\boxed{m_0^2=-\frac{3\mathcal A_4}{4\mathcal A_6},\qquad \mathcal A_2=\frac{3\mathcal A_4^2}{16\mathcal A_6},\qquad \mathcal A_4<0.}
$$
At this coexistence value, the <sextic even Landau potential> factors as $f(m)=s m^2(m^2+3q/(4s))^2/6\geq0$. This verifies that the three equal stationary values are <global minima>. Above this coexistence value of $r$ the disordered phase wins; below it the ordered phase wins, with a finite jump in the phase-selected <magnetization>. The values $r=0$ and $r=q^2/(4s)$ are the disordered and ordered <spinodal points>, respectively, not the coexistence boundary: the latter follows from the repeated root of $r+qa+sa^2=0$.

The two transition loci meet at the <tricritical point>,
$$
\boxed{\mathcal A_2=\mathcal A_4=0,\qquad \mathcal A_6>0.}
$$
Two independent control parameters such as $T$ and $g$ can tune these two conditions. In a generic local phase diagram their coefficient map has a <Jacobian matrix> of <rank> two at the intersection; unspecified functions $\mathcal A_2(T,g)$ and $\mathcal A_4(T,g)$ do not by themselves determine the geometric orientation of the boundaries. The sextic term stabilizes this intersection and the negative-quartic first-order side.

Solution

Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 303 / 1 / b / Solution

For the <Sextic even Landau potential>
$$
G(\phi)=\frac16\phi^6+\frac14g\phi^4+\frac12\varepsilon\phi^2,
$$
put $q=\phi^2$. The stationary-point equation is
$$
G'(\phi)=\phi(q^2+gq+\varepsilon)=0,
$$
so the nonzero stationary points have
$$
q_\pm=\frac{-g\pm\sqrt{g^2-4\varepsilon}}2.
$$
At such a point, $\varepsilon=-q^2-gq$, and hence
$$
G=-\frac13q^3-\frac14gq^2
=-q^2\left(\frac q3+\frac g4\right).
$$
A first-order transition occurs when a nonzero minimum has the same free energy as the disordered minimum $G(0)=0$. Therefore
$$
q=-\frac{3g}{4},
\qquad
\boxed{\varepsilon=\frac{3g^2}{16},\quad g<0.}
$$
At $g=0,\varepsilon=0$, the first-order line meets the continuous line and forms the <tricritical point>. For $g>0$, the transition at $\varepsilon=0$ is continuous.

For $g<0$, the disordered state $\phi=0$ is a local minimum when $\varepsilon>0$ and loses stability at the disordered <spinodal point> $\varepsilon=0$. The ordered minima exist while
$$
g^2-4\varepsilon\geq0,
$$
and merge with the intervening maxima at the ordered spinodal $\varepsilon=g^2/4$. Consequently the two spinodals enclose a region of <metastability>. If increasing $\varepsilon$ corresponds to heating, the ordered phase can persist above the coexistence curve throughout
$$
\boxed{\text{superheating:}\qquad
g<0,\quad \frac{3g^2}{16}<\varepsilon<\frac{g^2}{4},}
$$
while the disordered phase can persist below it throughout
$$
\boxed{\text{supercooling:}\qquad
g<0,\quad 0<\varepsilon<\frac{3g^2}{16}.}
$$

Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 4 / 34D / ii / Solution

{title2=$G(\phi)=\phi^6/6+g\phi^4/4+\varepsilon\phi^2/2$}

For the <Landau free energy> $G(\phi)=\phi^6/6+g\phi^4/4+\varepsilon\phi^2/2$, positive $g$ gives a <continuous phase transition> at $\varepsilon=0$. For negative $g$, <phase coexistence> between the disordered <global minimum> $\phi=0$ and the ordered <global minima> occurs on $\varepsilon=3g^2/16$, with $\phi^2=-3g/4$. Indeed, on that locus the <polynomial> is $G(\phi)=\phi^2(\phi^2+3g/4)^2/6\geq0$, so these three zeros are <global minima>. For $g<0$, the disordered and ordered <spinodal points> are respectively $\varepsilon=0$ and $\varepsilon=g^2/4$; equality of stationary <free energies> determines <phase coexistence>, while loss of a <local minimum> determines a <spinodal point>.

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 Incoming links: Sextic even Landau potential