Past exam of the theoretical physics I course of the University of Cambridge / 2015 / Paper 1 / 6 / b

Let the coordination number be $z_c=4$, and choose the spin-space $z$ axis along $\mathbf{S}=S\hat{\mathbf{z}}$. The mean-field decoupling of a bond isThe original directed neighbor sum counts each bond twice, so its factor $1/2$ cancels that double counting. The resulting energy isThe constant prevents double counting of the mean interaction energy and is necessary if free energy is minimized with respect to $S$. With the source's unnormalized solid-angle measure, the single-site integral isThereforeAt $S=0$ take $\sinh x/x\to 1$. A normalized solid-angle measure would remove the factor $(4\pi {)}^N$ without changing the self-consistency equation. These are three-component classical spins on a two-dimensional positional lattice, not spins constrained to a plane.

The PDF's hint is a correct angular-integration identity for a term with coefficient $\beta J/2$. That coefficient is not the full single-site molecular field after summing all four neighbors and decoupling both entries of each bond. The net coefficient is $\beta z_{c}J$, as follows from the explicitly stated Hamiltonian; inserting the hint's coefficient as the total field would lose the coordination and double-counting factors.