Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 303 / 3 / a / i / Solution

When all masses are equal, the free energy depends on the real vector $\mathbf{\varphi }$ only through inner products, so its symmetry is the orthogonal group $O(N)$. The mean-field potential isFor ${\mu }_0^2>0$, its unique minimum is $\mathbf{\varphi }=0$, which preserves $O(N)$. For ${\mu }_0^2<0$, the minima form the sphereChoosing one minimum leaves the subgroup $O(N-1)$ that fixes its direction, so spontaneous symmetry breaking gives $\boxed{O(N)\to O(N-1)}$.