Is Renormalized Entanglement Entropy Stationary at RG Fixed Points?

TL;DR

The paper investigates whether the Renormalized Entanglement Entropy $F(R)$ acts as a stationary c-function at RG fixed points in (2+1)D. It combines strip-entropy relations, direct disk-EE numerics for a free massive scalar, and a holography-free toy model on $H^2\times S^1$ to probe stationarity. The main result is that $F'(0)$ is negative (and possibly divergent) for small mass perturbations, and the toy model likewise shows nonstationarity, indicating REE is not generally stationary at UV fixed points, though monotonicity under RG remains ensured by strong subadditivity. These findings caution against interpreting $F(R)$ as a gradient-flow c-function in 3D and motivate seeking stationary alternatives or regime-specific interpretations.

Abstract

The renormalized entanglement entropy (REE) across a circle of radius R has been proposed as a c-function in Poincaré invariant (2+1)-dimensional field theory. A proof has been presented of its monotonic behavior as a function of R, based on the strong subadditivity of entanglement entropy. However, this proof does not directly establish stationarity of REE at conformal fixed points of the renormalization group. In this note we study the REE for the free massive scalar field theory near the UV fixed point described by a massless scalar. Our numerical calculation indicates that the REE is not stationary at the UV fixed point.

Figures (4)

• Figure 1: The entropic $c$-function $c \equiv R \, \partial_R \, S^{(1+1)}_{\text{interval}}$ for the $(1+1)$-dimensional massive scalar field as a function of $t \equiv m R$, where $m$ is the mass and $R$ is the length of the interval. The black curve comes from a numerical calculation using the prescription in Casini:2005zv. The blue and orange curves are the analytic approximations in \ref{['smallLarge']} at small and large values of $t$, respectively. The dotted red line marks the conformal value $c(0) = 1/3$.
• Figure 2: The function $C_{\text{strip}} \equiv R^2 \, \partial_R \, \hat{S}^{(2+1)}_{\text{strip}}$ for the free massive scalar field in black, where $\hat{S}^{(2+1)}_{\text{strip}} \equiv S^{(2+1)}_{\text{strip}} / L$ is the entanglement entropy per unit length across the strip of width $R$. The orange curve is the IR approximation in \ref{['IRstrip']}. The dotted red line is the initial value at $t = mR = 0$ given in \ref{['C0']}.
• Figure 3: The first derivative $C_{\text{strip}}'(t)/t$ as $t = mR \to 0$. The black curve comes from the numerical calculation, and the dotted orange curve comes from fitting the numerics to the function $C_{\text{strip}}'(t)/t \approx a/(t \log^2 t)$ as $t \to 0$. We find $a \approx -0.25$, which agrees with the analytic result in \ref{['Cstder']}. This means that $C_{\text{strip}}$ is not stationary for the massive scalar field.
• Figure 4: The renormalized entanglement entropy ${\cal F}$ across the circle of radius $R$ for the massive real free scalar plotted in black as a function of $(mR)^2$. In this plot it can clearly be seen that $\partial_{(mR)^2} {\cal F}$ is negative and nonzero at $(mR)^2 = 0$, which implies that the REE ${\cal F}$ is not stationary at the UV fixed point of a free massless scalar field. The dotted red line is the zero mass value ${\cal F}_{\rm UV}={\ln 2\over 8}- {3\zeta(3)\over 16 \pi^2}$.