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Entanglement entropy between tangent balls in CFT$_D$

Jiankun Li, Li Song

TL;DR

The paper addresses the problem of computing entanglement entropy between tangent balls in a generic $D$-dimensional CFT by employing the universal solid-torus framework, which recasts the calculation in terms of the CFT partition function on $\mathbb{B}^{D-1}\times S^1$ and uses a replica trick to obtain $S_{\mathrm{disj}}(A:B)$. It derives a general, invariant expression in terms of the vacuum energy density $\mathcal{E}_{vac}$, gamma factors, and ${}_2F_1$ functions that depends on the conformal invariant $\varrho$, with special attention to the tangent, half-space, and nonadjacent half-space limits; the half-space and nonadjacent cases yield explicit, dimension-dependent divergences and finite results that agree with BCFT in $D=2$. In the $D=2$ example, the results are shown to coincide with BCFT predictions, including the standard $S_{vN}=(c/6)\log(4R/\epsilon)$ for a half-infinite interval and a finite-distance formula $S=(c/6)\log\left(1+\tfrac{4R}{d}+2\sqrt{\tfrac{2R}{d}(\tfrac{2R}{d}+1)}\right)$, and can be mapped to an annulus with width $W$, yielding $S_{vN}=(c/6)W$; these checks reinforce the consistency between the general higher-dimensional construction and BCFT/annulus methodologies. Overall, the work provides a unified high-dimensional approach to disjoint-region entanglement with concrete results across dimensions, naturally connecting to the RT framework in the holographic context.

Abstract

We apply the universal method developed in \cite{Jiang:2025jnk} to compute the entanglement entropy between two tangent balls in CFT$_D$. When taking the radius of one ball to infinity, it gives the entanglement entropy between a ball and its tangent half plane. In two-dimensional case, this configuration is equivalent to the entanglement in boundary conformal field theory (BCFT) between the negative half-axis and an interval ending on the boundary.

Entanglement entropy between tangent balls in CFT$_D$

TL;DR

The paper addresses the problem of computing entanglement entropy between tangent balls in a generic -dimensional CFT by employing the universal solid-torus framework, which recasts the calculation in terms of the CFT partition function on and uses a replica trick to obtain . It derives a general, invariant expression in terms of the vacuum energy density , gamma factors, and functions that depends on the conformal invariant , with special attention to the tangent, half-space, and nonadjacent half-space limits; the half-space and nonadjacent cases yield explicit, dimension-dependent divergences and finite results that agree with BCFT in . In the example, the results are shown to coincide with BCFT predictions, including the standard for a half-infinite interval and a finite-distance formula , and can be mapped to an annulus with width , yielding ; these checks reinforce the consistency between the general higher-dimensional construction and BCFT/annulus methodologies. Overall, the work provides a unified high-dimensional approach to disjoint-region entanglement with concrete results across dimensions, naturally connecting to the RT framework in the holographic context.

Abstract

We apply the universal method developed in \cite{Jiang:2025jnk} to compute the entanglement entropy between two tangent balls in CFT. When taking the radius of one ball to infinity, it gives the entanglement entropy between a ball and its tangent half plane. In two-dimensional case, this configuration is equivalent to the entanglement in boundary conformal field theory (BCFT) between the negative half-axis and an interval ending on the boundary.

Paper Structure

This paper contains 6 sections, 30 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Entanglement entropy in the tangent configuration and its half-space limit
  3. Tangent configuration
  4. Half-space limit
  5. Nonadjacent half-space limit
  6. A specific example of computation in CFT$_2$

Figures (7)

  • Figure 1: Solid torus (blue region) hosting the CFT. On a time slice, two disjoint entangling regions $A$ and $B$ are identified.
  • Figure 2: Left panel: Two tangent balls $A$ and $B$ with radii $r$ and $r'$, respectively. The small separation $\epsilon$ between the balls represents a UV cutoff. Right panel: Enclosing configuration of two balls $A$ and $B$, separating by a UV cutoff $\epsilon$. Note that the constructions in the left and right panels lead to different divergence structures in the entanglement entropy.
  • Figure 3: The left panel illustrates the entanglement between a ball and a half-space. In CFT$_{2}$, the configuration corresponds to the right panel, which depicts the entanglement between an interval and the negative half-axis.
  • Figure 4: By considering a time slice of the solid torus in Figure \ref{['fig:solid-torus']}, one obtains two juxtaposed but disjoint $(D{-}1)$-dimensional balls.
  • Figure 5: The left panel illustrates the entanglement between a ball and a half-space. The spherical region $A$ is disjoint with region $B$. By taking the adjacent limit, as shown by the right panel, the inversive product is defined by slightly shifting the radius by $\pm \epsilon$.
  • ...and 2 more figures