Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 303 / 1 / a

For the partition function $Z={\sum }_{\{S\}}{e}^{-\beta E}$, differentiation with respect to the external field givesTherefore the magnetization per site is

The mean-field approximation replacesby its first three terms. Since every site has $q$ neighbours,The sites then decouple, and their three possible weights sum toHence $c_1=1$, $c_2=2$, and

Using the field derivative while imposing the self-consistency equation givesAt $B=0$, $m=0$ is always a solution. Linearizing the right-hand side for small $m$ gives slopeA continuous phase transition occurs when this slope crosses one, so, with $k_B=1$ and ${\kappa }_c=e^{-g/T_c}$,On the side where the slope exceeds one, the nonzero solutions lower the mean-field free energy and the trivial solution is unstable. The equation has a physical continuous-transition solution only in the corresponding parameter range of this spin-one model.