Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 303 / 1 / e

This is the Blume–Capel model. In the mean-field approximation, write $m=⟨{\sigma }_i⟩$ and replaceBecause every site has coordination number $q$, the resulting energy isThe one-site partition function is thereforeWith $\kappa =e^{-\beta g}$, the mean-field partition function is $Z_{\mathrm{MF}}=e^{-\beta NJq{m}^2/2}{Z}_1^N$. Taking $F=-T\log Z_{\mathrm{MF}}$ gives

Set $B=0$, $h=\beta Jqm$, and $A=1+2\kappa$. The required power series is obtained fromThe quadratic Landau coefficient isIts vanishing gives the line of continuous transitionsThe quartic coefficient isAt a tricritical point, both $a_2$ and $a_4$ vanish. Since $A=1+2\kappa$, the condition $a_4=0$ gives $A=6\kappa$ and hence $\kappa =1/4$. Substitution into the critical line yieldsThe sextic coefficient is positive there, so the sixth-order term stabilizes the free energy. Equivalently, $g_{\mathrm{tri}}=T_{\mathrm{tri}}\log 4$.