Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 303 / 1 / a / i

Write $a={\alpha }_2(T)$ and $u={\alpha }_{2n}>0$. At high temperature $a>0$, so $f(m)=a{m}^2+u{m}^{2n}$ has one minimum at $m=0$. At low temperature $a<0$, the origin is a local maximum and two symmetry-related minima appear. The stationary equation isso the equilibrium magnetization isandBecause $a(T)\sim T-T_c$, the two nonzero minima approach zero continuously as $T\uparrow T_c$. The order parameter is continuous while its response becomes singular, so this is a continuous phase transition.