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

The ordered solution givesso the order-parameter critical exponent isAt $T=T_c$, adding a magnetic term $-hm$ gives $h=2nu{m}^{2n-1}$, henceAbove $T_c$, the zero-field susceptibility is $\chi =(2a{)}^{-1}$. Below $T_c$, the curvature at a minimum is $-4(n-1)a$, so both sides giveAt the ordered minimum, $a=-nu{m}^{2n-2}$ andSince $f_{\mathrm{sing}}\sim |T-T_c{|}^{2-\alpha }$, the heat-capacity critical exponent isThese values obey the Widom scaling relation and Rushbrooke scaling relation at mean-field level.