Past exam of the theoretical physics I course of the University of Cambridge / 2022 / 3 / b / Solution

The square lattice has coordination number of a lattice $z=4$. The mean-field decoupling isAccounting for the factor $1/2$ in the directed-neighbour sum gives the Hamiltonian in the mean-field approximation:The constant term is necessary to correct the double counting of the interaction. Let $h=4JS+B$. The one-site partition function isso $Z_{\mathrm{MF}}=e^{-2\beta JN{S}^2}{Z}_1^N$. The Helmholtz free energy is thereforeThe equilibrium mean-field approximation is obtained by minimizing this expression over the allowed $S$. The $s=0$ state supplies the extra $1$ in $Z_1$; omitting it would instead describe the two-state Ising model. This is the Blume-Capel mean-field transition at zero single-ion anisotropy.