Entanglement Entropy of Two Spheres

TL;DR

We address the entanglement structure of a massless free scalar field in vacuum for two separated spheres. Using a Bombelli-style lattice approach with Gaussian reduced density matrices, we compute the entanglement entropies and show that the mutual information $S_{A;B}$ between the two spheres is UV-cutoff independent and scales as the product of their surface areas at large separation, i.e., $S_{A;B} \propto R_1^{d-1} R_2^{d-1} / r^{2d-2}$ for dimension $d$. Numerical results in $d=3$ reveal $S_{A;B} \approx - g R_1^2 R_2^2 / r^4$ with $g \approx 0.26$, while in $d=2$ the decay is $-S_{A;B} \approx 0.37 R_1 R_2 / r^2$, and interior regions contribute nonuniformly to mutual information. These findings inform potential connections to black-hole entanglement and yield a scalable method for computing entanglement in arbitrary geometries.

Abstract

We study the entanglement entropy S_{AB} of a massless free scalar field on two spheres A and B whose radii are R_1 and R_2, respectively, and the distance between the centers of them is r. The state of the massless free scalar field is the vacuum state. We obtain the result that the mutual information S_{A;B}:=S_A+S_B-S_{AB} is independent of the ultraviolet cutoff and proportional to the product of the areas of the two spheres when r>>R_1,R_2, where S_A and S_B are the entanglement entropy on the inside region of A and B, respectively. We discuss possible connections of this result with the physics of black holes.

Figures (9)

• Figure 1: The entanglement entropy $S(R)$ of one sphere whose radius is $R$ as a fumction of $R^2/a^2$. The line is the best linear fit.
• Figure 2: $S_{AB} -S_A-S_B$ as a function of $r/a$ for $R_1/a=R_2/a=6,7$, where $S_{AB}$ is the entanglement entropy of two spheres $A$ and $B$ whose radii are $R_1$ and $R_2$. The distance between the centers of the two spheres is $r$.
• Figure 3: $(S_{A} +S_B-S_{AB})^{-1/4}$ as a function of $r/a$ for $R_1/a=R_2/a=6,7$, where $S_{AB}$ is the entanglement entropy of two spheres $A$ and $B$ whose radii are $R_1$ and $R_2$. The distance between the centers of the two spheres is $r$. The straight lines are fitted by the data between $r/a=R_1/a+R_2/a+24$ and $r/a=R_1/a+R_2/a+84$. For $r\gtrapprox 3R(\equiv R_1=R_2)$ the lines are beautifully fitted and the approximate expressions (\ref{['eq:5-1']}) and (\ref{['eq:5-2']}) are precise.
• Figure 4: $G(R_1,R_2)/a^4$ in (\ref{['eq:5-1']}) as a function of $R_{2}^{2}/a^2$ for $R_1/a=4,4.5,\dots,7$. The lines are the best linear fit.
• Figure 5: $g R_1^2/a^2$ as a function of $R_1^2/a^2$, where $g$ is defined as $G(R_1,R_2)=g R_1^2 R_2^2$. The line is the best linear fit.