Shape Dependence of Entanglement Entropy in Conformal Field Theories

TL;DR

The paper demonstrates that the non-local part of the entanglement-density for small shape deformations of planar or spherical entangling surfaces in any conformal field theory is universal, fixed solely by the stress-tensor two-point coefficient C_T. Using conformal perturbation theory and a careful relative-entropy analysis, the authors show that the modular Hamiltonian contribution does not affect this non-local term, so the final result is determined entirely by C_T. They apply this to establish the universality of the corner term in d=3 and to derive the Mezei formula for deformed spheres, providing a field-theoretic confirmation of holographic predictions. These results strengthen the link between entanglement structure and stress-tensor data in CFTs and illuminate how geometric deformations encode information about degrees of freedom in conformal theories.

Abstract

We study universal features in the shape dependence of entanglement entropy in the vacuum state of a conformal field theory (CFT) on $\mathbb{R}^{1,d-1}$. We consider the entanglement entropy across a deformed planar or spherical entangling surface in terms of a perturbative expansion in the infinitesimal shape deformation. In particular, we focus on the second order term in this expansion, known as the entanglement density. This quantity is known to be non-positive by the strong-subadditivity property. We show from a purely field theory calculation that the non-local part of the entanglement density in any CFT is universal, and proportional to the coefficient $C_T$ appearing in the two-point function of stress tensors in that CFT. As applications of our result, we prove the conjectured universality of the corner term coefficient $\fracσ{C_T}=\frac{π^2}{24}$ in $d=3$ CFTs, and the holographic Mezei formula for entanglement entropy across deformed spheres.

Figures (5)

• Figure 1: The set-up for the planar case: the original subregion $A$ is the half-space $x^1>0$, with the entangling surface $x^1=0$ (dashed line). We deform the entangling surface to $x^1=-\epsilon\;\chi(x^i)$ (bold line) by glueing on the area elements $\delta A_{a,b}$ at points $x_{a,b}$ along the entangling surface.
• Figure 2: The set-up for the spherical case. We deform the original entangling surface $\mathbf{x}^2=R^2$ (dashed line) by glueing on the area elements $\delta A_{a,b}$ at points $\Omega_{a,b}$ along the entangling surface.
• Figure 3: The shaded region is the domain of dependence $\mathcal{D}(A)$. Analytically continuing in $w=e^{i\tau_c}$ and sending $w\to \infty$ sends the modular Hamiltonian (integrated over the blue line) to the future null boundary of $\mathcal{D}(A)$. Also shown are the future and past tips $E_{\pm}$ of $\mathcal{D}(A)$ and the point at spacelike infinity $E_0$.
• Figure 4: The triangle of recent studies on the shape dependence of entanglement entropy in CFTs. The arrows denote implications.
• Figure 5: The mild logarithmic enhancement comes from the stress tensor in the modular Hamiltonian coming close to one of the other two stress tensor insertions inside $\overline{U_{\delta}}$.