Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 303 / 3 / a / Solution

For a uniform field with $r^2=\mathbf{\varphi }\cdot \mathbf{\varphi }$, the potential isFor $T>T_c$, one has ${\mu }^2>0$, so ${\mathbf{\varphi }}_0=0$. The Hessian has $N$ positive equal eigenvalues, and all $N$ modes are gapped. The full $O(N)$ symmetry is unbroken.

For $T<T_c$, one has ${\mu }^2<0$, andThe vacuum manifold is $S^{N-1}$. Choosing one point breaks $O(N)$ spontaneously to $O(N-1)$. The radial fluctuation has squared mass $-2{\mu }^2$, while the $N-1$ tangent directions are gapless Nambu-Goldstone bosons, one for each broken continuous generator modulo the unbroken subgroup.