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

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