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

If ${\mu }^2<0$ and ${\lambda }^2>0$, the minima areThe discrete $O(1)\cong {\mathbb{Z}}_2$ symmetry is spontaneously broken, while $O(2)$ remains intact. There is no Goldstone mode because only a discrete symmetry is broken.