Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 4 / 32C / iii / Solution

The single-spin average in the effective field is $\tanh (\beta B_{\mathrm{eff}})$. Requiring it to equal the assumed average gives the mean-field self-consistency equationAt $B=0$, the slope of its right side at the origin is $2DJ/(k_{B}T)$. Thus $\boxed{T_c=2DJ/k_B}$. For $T>T_c$, $m=0$ is the only solution and is the stable minimum: thermal fluctuations dominate. For $T<T_c$, two stable nonzero solutions $\pm m(T)$ represent spontaneous magnetization, while the zero solution is unstable. At $T_c$ they merge continuously. Expanding $\tanh x=x-x^3/3+\cdots$ near the transition yields $m^2\sim 3(T_c-T)/T_c$, so the mean-field critical exponent is one-half. These are predictions of the approximation, not a proof of the exact Ising transition in every dimension.