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

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, $g_{\mathrm{tri}}=T_{\mathrm{tri}}\log 4$.