= Solution
A <phase diagram> divides a space of control parameters into equilibrium <thermodynamic phases>, marking <phase coexistence> boundaries and <thermodynamic critical points>. Its dimension counts independently variable controls, not spatial dimensions. An explicit three-dimensional example uses the controls $(r,u,h)$ of a <sextic even Landau potential>:
$$
V(M)=\frac r2M^2+\frac u4M^4+\frac v6M^6-hM,\qquad v>0.
$$
Here $M$ is a scalar <order parameter>, $h$ its <conjugate field>, and $r$ a thermal control. The <equilibrium magnetization> minimizes $V$ globally. At $h=0$, $u>0$, the boundary $r=0$ is a <continuous phase transition>. At $h=0$, $u<0$, the boundary $r=3u^2/(16v)$ is a <first-order phase transition> between $M=0$ and $M=\pm\sqrt{-3u/(4v)}$. The two boundaries meet at the \b[<tricritical point> $r=u=h=0$]. A fixed strictly positive quartic coupling alone cannot produce this <tricritical point>; the stabilizing sextic term is essential when the quartic coefficient is tuned through zero.
The full <phase diagram> also contains the $h=0$ <phase coexistence> sheet between positive and negative ordered phases. Crossing it changes the sign of $M$ discontinuously. For $u<0$, the three-phase line $r=3u^2/(16v)$ is the junction of this sheet and two <tricritical wings>, on which two unequal same-sign <equilibrium magnetizations> coexist at nonzero $h$. The <tricritical wing critical edges> are lines of ordinary <thermodynamic critical points>; they terminate the first-order sheets and meet at the <tricritical point>. Away from these boundaries, varying $h$ produces a smooth crossover.
For an explicit construction of the <tricritical wings>, write their two coexisting <order parameters> as $M_1=s-d$ and $M_2=s+d$, with $s>0$ and $0<d\leq s$. Equality of their <Landau free energies> and first derivatives gives
$$
u=-v\left(\frac{10}{3}s^2+2d^2\right),\qquad
r=v\left(5s^4-\frac23s^2d^2+d^4\right),\qquad
h=\frac83vs^3(s^2-d^2).
$$
Indeed, for these controls the <tricritical wing coexistence factorization> is
$$
V(M)-V(M_1)=\frac v6[(M-s)^2-d^2]^2[M^2+4sM+5s^2-d^2].
$$
The last factor is $(M+2s)^2+s^2-d^2\geq0$, proving that both specified minima are global. The reflected wing has negative $h$ and negative $M$. Letting $d\to0$ gives the <tricritical wing critical edge> $u=-10vs^2/3$, $r=5vs^4$, $h=8vs^5/3$; there $V''(s)=V'''(s)=0$ and $V''''(s)=40vs^2>0$. Letting $d=s$ recovers the three-phase line. This supplies the first-order sheets and their continuous boundary curves, rather than only a zero-field section.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/iii/paper-46-phases.png]
{title=Zero-field section and three-dimensional coexistence wings of a sextic Landau phase diagram}
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