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

The self-consistency equation requires that the mean spin computed using the single-site Boltzmann distribution equals the spin $S$ used to form the effective field. With $h=4JS+B$,ThusEquivalently, differentiating part (b) gives $N^{-1}\partial F_{\mathrm{MF}}/\partial S=4J[S-⟨s{⟩}_h]$, so stationarity of the Helmholtz free energy in the mean-field approximation is exactly this condition. Among multiple stationary points one must select the stable minimum.