Numerical determination of entanglement entropy for a sphere

TL;DR

This work numerically determines the universal logarithmic coefficient of entanglement entropy for a free massless scalar field inside a sphere in 3+1 dimensions using Srednicki's regularization. By discretizing the radial direction, expanding in angular momentum, and carefully extrapolating in the infrared and angular quantum numbers, the authors extract the subleading term in $S(R)$ and find $S(R) = s (R/a)^2 + c \log(R/a)^2 + d$ with $c = -1/90$ (and $d$ nonuniversal). The high-precision analysis confirms the analytical prediction that the logarithmic coefficient is fixed by the Weyl anomaly, while the leading area term is nonuniversal. The results support universality of the logarithmic contribution and provide a benchmark for replica-method and AdS/CFT approaches to entanglement entropy in four dimensions.

Abstract

We apply Srednicki's regularization to extract the logarithmic term in the entanglement entropy produced by tracing out a real, massless, scalar field inside a three dimensional sphere in 3+1 flat spacetime. We find numerically that the coefficient of the logarithm is -1/90 to 0.2 percent accuracy, in agreement with an existing analytical result.

Figures (3)

• Figure 1: Plots of $\Delta S_l(n,N)=S_l(n,N)-S_l(n,N_0)$ as a function of $N^{-2l-2}$ for $n=20$ and $l=0$ (top left), $l=1$ (top right), $l=2$ (bottom left), and $l=10$ (bottom right). The gray lines are straight line fits through the data points (black).
• Figure 2: Plot of $S(R)$ as a function of $(R/a)^2$, the gray line is a fit through the last $20$ points.
• Figure 3: Plot of $S_{\rm{log}}(R)-S_{\rm{lin}}(R)=2.5\cdot10^{-5}(R/a)^2-0.005545 \log (R/a)^2-0.03537$ as a function of $(R/a)^2$ (solid gray curve) and computed data points $S(R)-S_{\rm{lin}}(R)=S(R)-0.295406 (R/a)^2$ (black dots). $S_{\rm{log}}(R)$ is obtained from a fit over the last 15 data points, to the right of the vertical dashed line. Error bounds are not visible, being of the order $10^{-8}$.