Renormalization group running of the entanglement entropy of a circle

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.