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

TL;DR

This paper studies how entanglement entropy behaves across a quantum phase transition in the $O(N)$ sigma-model, revealing a cusp at the transition and changes in divergence structure due to symmetry breaking.

Contribution

It provides a detailed analysis of entanglement entropy in a non-conformal scalar field system with a quantum phase transition, highlighting the effects of spontaneous symmetry breaking.

Findings

Entanglement entropy follows an area law in both phases.

Spontaneous symmetry breaking alters the subleading divergence from log to log squared.

Entanglement entropy peaks at the quantum phase transition point.

Abstract

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.