Some results on the shape dependence of entanglement and Rényi entropies

TL;DR

This work analyzes how the universal part of entanglement and Rényi entropies in conformal field theories depends on the shape of the entangling region. Using a combination of conformal perturbation theory, hyperbolic-space mapping, and holographic minimal-surface techniques, it shows that first-order shape deformations do not alter the universal term, while second-order corrections reveal precise, mode-by-mode quadratic contributions for perturbations of circles, and provide analytic and numerical bounds for ellipses in holographic CFT$_3$. The authors argue, supported by Solodukhin's formula in $d=4$, that the sphere minimizes the universal EE across dimensions—becoming a stationary point in $d=3$ holographic theories and a global minimum in $d=4$—and conjecture this holds generally for shapes connected to the sphere. A robust numerical framework is developed to compute the universal term for generic regions in holographic CFT$_3$, enabling detailed exploration of the geometry-EE relationship. Together, these results deepen understanding of how geometry controls universal entanglement properties and suggest the sphere as a canonical reference for EE c-functions.

Abstract

We study how the universal contribution to entanglement entropy in a conformal field theory depends on the entangling region. We show that for a deformed sphere the variation of the universal contribution is quadratic in the deformation amplitude. We generalize these results for Rényi entropies. We obtain an explicit expression for the second order variation of entanglement entropy in the case of a deformed circle in a three dimensional CFT with a gravity dual. For the same system, we also consider an elliptic entangling region and determine numerically the entanglement entropy as a function of the aspect ratio of the ellipse. Based on these three-dimensional results and Solodukhin's formula in four dimensions, we conjecture that the sphere minimizes the universal contribution to entanglement entropy in all dimensions.

Figures (5)

• Figure 1: Perturbed circle (\ref{['garesult']}) as entangling region $V$, $\Sigma = \partial V$. The correction to the universal coefficient $s_{3}$ is of order $\epsilon^2$ and it is given by \ref{['garesult']} in a holographic CFT$_3$.
• Figure 2: Universal coefficient $\tilde{s}_3$ for an elliptic entangling region with semi-axes $a$, $b$, in a $d = 3$ holographic CFT. The blue, solid curve is a tight lower bound obtained numerically. The red, dashed curve $\tilde{s}_3=2 \pi$ is a lower bound set by the area of an ellipsoid (\ref{['eq:ellipsoid_trial']}). The yellow, dotted curve is a lower bound set by the area of a deformed strip \ref{['strip']}. The green, dash-dot curve $\tilde{s}_3 = 2 \pi \left[1 + {3\over8}(1 - b/a)^2\right]$ is an approximation obtained by considering perturbations of a circle \ref{['gaEllipse']}. It is not a bound.
• Figure 3: Geometry of the manifold ${{\mathcal{M}}}$ of \ref{['deltaRho']}. Lines of constant $(u,\Omega)$ are drawn in purple. The entangling surface $\Sigma$ is marked by a blue line, and sits at $u=\infty$. We use a regularization procedure with a cutoff $\delta$ that cuts out a tube centered around $\Sigma$ from ${{\mathcal{M}}}$. $\partial{{\mathcal{M}}}$ is at constant $u=u_m$, and it has topology $S^1\times S^{d-2}$. It is drawn in yellow. When we map to hyperbolic space the Hamiltonian generates time evolution along the purple lines. $\partial{{\mathcal{M}}}$ maps to the boundary of hyperbolic space.
• Figure 4: Actual minimal surface for a perturbed circle (\ref{['eq:perturbed_circle1']}) with $n = 5$ and $A/R_0 = 0.12$, obtained numerically.
• Figure 5: Universal contribution to the EE for a perturbed circle (\ref{['eq:perturbed_circle1']}) in a $d = 3$ holographic CFT$_3$. The blue dots are a tight lower bound obtained numerically, the blue line is the analytic result (\ref{['gaResult']}).