Entanglement and RG in the $O(N)$ vector model

TL;DR

This paper studies entanglement entropy in the large N $O(N)$ vector model across RG flows in $4- ext{epsilon}$ dimensions, revealing universal area terms and the importance of boundary contributions in replica calculations.

Contribution

It provides a detailed analysis of entanglement entropy along RG flows in the $O(N)$ model, highlighting the role of boundary terms and universal scaling behaviors.

Findings

In 4D, free non-conformal scalars have a universal area term with logarithmic divergence.

In $4- ext{epsilon}$ dimensions, the area term scales as $1/ ext{epsilon}$, showing divergence near four dimensions.

Boundary terms on the entangling surface are essential for consistency in replica trick calculations.

Abstract

We consider the large $N$ interacting vector $O(N)$ model on a sphere in $4-\epsilon$ Euclidean dimensions. The Gaussian theory in the UV is taken to be either conformally or non-conformally coupled. The endpoint of the RG flow corresponds to a conformally coupled scalar field at the Wilson-Fisher fixed point. We take a spherical entangling surface in de Sitter space and compute the entanglement entropy everywhere along the RG trajectory. In $4$ dimensions, a free non-conformal scalar has a universal area term scaling with the logarithm of the UV cutoff. In $4-\epsilon$ dimensions, such a term scales as $1/\epsilon$. For a non-conformal scalar, a $1/\epsilon$ term is present both at the UV fixed point, and its vicinity. For flow between two conformal fixed points, $1/\epsilon$ terms are absent everywhere. Finally, we make contact with replica trick calculations. The conical singularity gives rise to boundary terms residing on the entangling surface, which are usually discarded. Consistency with our results requires they be kept. We argue that, in fact, this conclusion also follows from the work of Metlitski, Fuertes, and Sachdev, which demonstrated that such boundary terms will be generated through quantum corrections.