Entanglement Entropy and Quantum Phase Transition in the $O(N)$ $σ$-model

We investigate how entanglement entropy behaves in a non-conformal scalar
field system with a quantum phase transition, by the replica method. We study
the $\sigma$-model in 3+1 dimensions which is $O(N)$ symmetric as the mass
squared parameter $\mu^{2}$ is positive, and undergoes spontaneous symmetry
breaking while $\mu^{2}$ becomes negative. The area law leading divergence of
the entanglement entropy is preserved in both of the symmetric and the broken
phases. The spontaneous symmetry breaking changes the subleading divergence
from log to log squared, due to the cubic interaction on the cone. At the
leading order of the coupling constant expansion, the entanglement entropy
reaches a cusped maximum at the quantum phase transition point $\mu^{2}=0$, and
decreases while $\mu^{2}$ is tuned away from 0 into either phase.