Entanglement Entropy of Conformal Field Theory in All Dimensions

TL;DR

This work develops a field-theoretic method to compute entanglement entropy in $\mathrm{CFT}_D$ for all dimensions using a solid-torus replica construction, mapping the problem to a thermal partition function with $\beta=2\pi$ and expressing disjoint-ball entanglement in terms of the ground-state (Casimir) energy $\langle H\rangle$ and the Casimir density $\mathcal{E}_{\rm vac}$. It yields an explicit, conformally invariant formula for the entanglement entropy between two disjoint balls, and shows that in the adjacent limit the familiar area-law divergences emerge, reproducing known results in $D=2$ and holographic predictions in $D=4$. The RT formula arises naturally within this framework, with the disjoint entropy linked to the hyperbolic volume of the entangling region and the EWCS, while shape-independence holds for reflection-symmetric configurations and can be extended to arbitrary shapes via conformal mappings or numerical torus partition functions. The approach thus unifies finite disjoint entropies across dimensions, clarifies their geometric interpretation in holography, and points to extensions to massive QFTs and broader shape analyses.

Abstract

We provide a field-theoretic method to calculate entanglement entropy of CFT in all dimensions. This method works for entangling surfaces of arbitrary shape. The formalism manifests a field-theoretic proof of the Ryu-Takayanagi formula.

Figures (6)

• Figure 1: The Euclidean pure state density matrix $\rho =|\psi\rangle \langle\psi|$ for the annular CFT$_2$. Two distinct intervals $A$ and $B$ have finite entanglement entropy $S(A:B)$.
• Figure 2: As shown in Jiang:2024ijxJiang:2025tqu, after subtracting segments $C$ and $D$ with two discs in the infinite system, we obtain an annular region in which $A$ and $B$ are in a pure entangled state $\psi_{AB}$. It is identical to the annular CFT$_2$ in Figure \ref{['fig:density']}.
• Figure 3: The Euclidean pure state density matrix $\rho = |\psi\rangle \langle\psi|$ for the solid torus CFT$_D$. The lower half solid torus corresponds to $|\psi\rangle$. The upper half annulus corresponds to $\langle\psi|$. We calculate the entanglement entropy between two disjoint co-dimensional one balls $A$ and $B$ (colored regions) in the bounded region $\mathcal{B}$ (blue region).
• Figure 4: The cut-and-glue procedure in the replica trick to construct $\mathcal{B}_{n}$. Each solid torus is cut along the $(D-1)$-dimensional ball $A$ and is glued with others cyclically. Red lines represent gluing operations. Note that the resulted manifold is also a solid torus with period $2n\pi$.
• Figure 5: Entangling regions (colored regions) are bounded by ($D-2$)-spheres (black circles), respectively. Left panel (the cavity configuration): A ($D-1$)-ball living in a spheric cavity is entangled with the region outside the cavity. Right panel (the juxtaposed configuration): Two disjoint ($D-1$)-balls with different radii are entangled.