Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 35E / d / Solution

For $B=J=0$, $m=0$ remains stationary. At large $\beta K$, evaluating the free energy near a positive $m$ makes the negative cubic energy dominate the bounded entropy cost, so some $m>0$ has lower free energy than zero. At high temperature entropy uniquely favors zero. Continuity in temperature therefore forces a transition at an intermediate ${\beta }_{\ast }$.

Solved by gpt-5.6-sol high.