= Solution
For a uniform field with $r^2=\boldsymbol\phi\mathbin\cdot\boldsymbol\phi$, the potential is
$$
V(r)=\frac12\mu^2r^2+gr^4.
$$
For $T>T_c$, one has $\mu^2>0$, so $\boldsymbol\phi_0=0$. 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 $\mu^2<0$, and
$$
|\boldsymbol\phi_0|^2=-\frac{\mu^2}{4g}.
$$
The 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.
Back to article page
