On the RG running of the entanglement entropy of a circle

TL;DR

The paper investigates RG monotonicity of entanglement-related quantities in three-dimensional spacetime by leveraging strong subadditivity (SSA) and Lorentz invariance. By constructing an infinite family of boosted circles and applying a symmetric SSA bound, it proves the circle entanglement entropy in 2+1 dimensions is concave in the radius, implying the constant term increases while the area-law coefficient decreases from UV to IR, in line with the F-theorem and holographic c-theorems. It also analyzes how intrinsic definitions of the fixed-point constant term may be ill-defined and discusses universality issues, with higher-dimensional generalizations facing additional divergences and surface-geometry subtleties. The results provide a field-theory corroboration of holographic monotonicity properties and outline key open problems, including a potential reliance on mutual information to define a universal c-like quantity. The work highlights the special role of surface geometry (corners, trihedral angles) in higher dimensions and sets a path for refining inequalities to obtain robust c-theorem candidates beyond 2+1D.

Abstract

We show, using strong subadditivity and Lorentz covariance, that in three dimensional space-time the entanglement entropy of a circle is a concave function. This implies the decrease of the coefficient of the area term and the increase of the constant term in the entropy between the ultraviolet and infrared fixed points. This is in accordance with recent holographic c-theorems and with conjectures about the renormalization group flow of the partition function of a three sphere (F-theorem). The irreversibility of the renormalization group flow in three dimensions would follow from the argument provided there is an intrinsic definition for the constant term in the entropy at fixed points. We discuss the difficulties in generalizing this result for spheres in higher dimensions.

Figures (3)

• Figure 1: Two interval $A$ and $B$ of size $\sqrt{r R}$ have their end-points on the light-cone. The size of $Y$ is $r$ while the distance between the upper end-points of $A$ and $B$ is $R$.
• Figure 2: Circles on a plane obtained by rotating a single circle around a point different from its center. The marked contour shows a typical set involved in the series of the right hand side of (\ref{['ecu']}). This set approaches a circle of specific radius as the number $N$ of rotated circles increases, but the perimeter of the resulting shape is greater than the one corresponding to a circle. Also, since the slope of the wiggles around the limit circle does not change with $N$, the contribution of the corners to the entropy does not vanish in the large $N$ limit.
• Figure 3: The figure shows two parallel circles of radius $r$ and $R$ obtained by cutting the light-cone with two parallel spatial planes. Another spatial plane cuts the light-cone in a circle of radius $\sqrt{r R}$ which is tangent to the two parallel circles.