Entanglement entropy and variational methods: Interacting scalar fields

We develop a variational approximation to the entanglement entropy for scalar ϕ4 theory in 1+1, 2+1, and 3+1 dimensions, and then examine the entanglement entropy as a function of the coupling. We find that in 1+1 and 2+1 dimensions, the entanglement entropy of ϕ4 theory as a function of coupling is monotonically decreasing and convex. While ϕ4 theory with positive bare coupling in 3+1 dimensions is thought to lead to a trivial free theory, we analyze a version of ϕ4 with infinitesimal negative bare coupling, an asymptotically free theory known as precarious ϕ4 theory, and explore the monotonicity and convexity of its entanglement entropy as a function of coupling. Within the variational approximation, the stability of precarious ϕ4 theory is related to the sign of the first and second derivatives of the entanglement entropy with respect to the coupling.