Entanglement Entropy and Variational Methods: Interacting Scalar Fields

TL;DR

This paper develops a variational method to approximate entanglement entropy in scalar  theory across multiple dimensions, revealing how entropy varies with coupling and linking stability to entropy derivatives.

Contribution

It introduces a variational approximation for entanglement entropy in  theory and explores its behavior in different dimensions and coupling regimes, including precarious  theory.

Findings

Entanglement entropy decreases monotonically with coupling in 1+1 and 2+1 dimensions.

In 3+1 dimensions, the entropy's behavior depends on the sign of the bare coupling.

Stability of precarious  theory relates to the derivatives of entanglement entropy.

Abstract

We develop a variational approximation to the entanglement entropy for scalar $\phi^4$ 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 $\phi^4$ theory as a function of coupling is monotonically decreasing and convex. While $\phi^4$ theory with positive bare coupling in 3+1 dimensions is thought to lead to a trivial free theory, we analyze a version of $\phi^4$ with infinitesimal negative bare coupling, an asymptotically free theory known as precarious $\phi^4$ theory, and explore the monotonicity and convexity of its entanglement entropy as a function of coupling. Within the variational approximation, the stability of precarious $\phi^4$ theory is related to the sign of the first and second derivatives of the entanglement entropy with respect to the coupling.