= Solution
Write $r_i=\mu_i^2$ and assume $g>0$, as required for quartic stability. In the <mean-field approximation>, the <two-component radial quartic Landau potential> is
$$
V=\frac12(r_1\phi_1^2+r_2\phi_2^2)+g(\phi_1^2+\phi_2^2)^2.
$$
Set $x=\phi_1^2\geq0$, $y=\phi_2^2\geq0$. At fixed $\rho=x+y$, the quartic term is fixed and the quadratic term is minimized by assigning all of $\rho$ to the component with smaller $r_i$. Minimizing over $\rho$ then gives
$$
\boxed{f_{\min}=-\frac{\min(0,r_1,r_2)^2}{16g}}.
$$
The resulting <phase diagram> has these regions:
* If $r_1>0$ and $r_2>0$, $\phi_1=\phi_2=0$: the disordered phase preserves both <Z2 symmetries>.
* If $r_1<\min(0,r_2)$, $\phi_1=\pm\sqrt{-r_1/(4g)}$ and $\phi_2=0$: the first <Z2 symmetry> undergoes <spontaneous symmetry breaking>.
* If $r_2<\min(0,r_1)$, $\phi_2=\pm\sqrt{-r_2/(4g)}$ and $\phi_1=0$: the second <Z2 symmetry> undergoes <spontaneous symmetry breaking>.
On $r_1=r_2=r<0$, the <symmetry> is enhanced to the <orthogonal group> $O(2)$, and the minima form the circle
$$
\boxed{\phi_1^2+\phi_2^2=-\frac{r}{4g}}.
$$
Mixed-component minima occur on this line, rather than in an open coexistence region. Crossing the negative diagonal swaps the selected component discontinuously: this is a <first-order phase transition>, also called a <spin-flop transition>. For example, holding $r_2=r<0$ fixed and moving $r_1$ across $r$, the derivative $\partial f_{\min}/\partial r_1$ jumps from $-r/(8g)$ to zero. The angular degeneracy exactly on the line does not remove this discontinuity.
The positive half-axis $r_1=0$, $r_2>0$ is a <continuous phase transition>, as is $r_2=0$, $r_1>0$. Their endpoint at $(0,0)$ is a <bicritical point>, where the two continuous boundaries meet the first-order negative diagonal. Approaching the origin radially makes the ordered amplitude vanish continuously. The negative half-axes lie inside ordered phases and are not additional transition lines. \b[There are two continuous half-axis boundaries and one first-order negative diagonal, meeting at the origin.]
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-303-phase-diagram.png]
{title=Mean-field phases for a radial quartic interaction}
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