Universal entanglement for higher dimensional cones

TL;DR

The paper investigates universal, regulator-free contributions to entanglement and Rényi entropies from (hyper)conical singularities in entangling surfaces across dimensions. By leveraging Mezei's deformation analysis, it proves in holographic CFTs that the cone coefficient $\sigma^{(d)}$ in the almost-smooth limit is proportional to the stress-tensor central charge $C_T$, with dimension-dependent prefactors, and it extends this relation to general dimensions and to Rényi entropies via $\sigma_n^{(d)}$ expressed in terms of the twist scaling dimension $h_n$. The authors provide explicit formulas for odd and even $d$, verify consistency with Ryu-Takayanagi results in several dimensions, and supply concrete results for $d=3,4,6$ including free-field and holographic theories. They conjecture universality for general CFTs beyond holography, and they propose a structured framework linking geometric singularities, $C_T$, and twist data to organize the Rényi cone coefficients across dimensions.

Abstract

The entanglement entropy of a generic $d$-dimensional conformal field theory receives a regulator independent contribution when the entangling region contains a (hyper)conical singularity of opening angle $Ω$, codified in a function $a^{(d)}(Ω)$. In arXiv:1505.04804, we proposed that for three-dimensional conformal field theories, the coefficient $σ$ characterizing the smooth surface limit of such contribution ($Ω\rightarrow π$) equals the stress tensor two-point function charge $C_{ T}$, up to a universal constant. In this paper, we prove this relation for general three-dimensional holographic theories, and extend the result to general dimensions. In particular, we show that a generalized coefficient $σ^{ (d)}$ can be defined for (hyper)conical entangling regions in the almost smooth surface limit, and that this coefficient is universally related to $C_{ T}$ for general holographic theories, providing a general formula for the ratio $σ^{ (d)}/C_{ T}$ in arbitrary dimensions. We conjecture that the latter ratio is universal for general CFTs. Further, based on our recent results in arXiv:1507.06997, we propose an extension of this relation to general Rényi entropies, which we show passes several consistency checks in $d=4$ and $d=6$.

Figures (3)

• Figure 1: In panel (a), we show an entangling region $V$ whose boundary contains a sharp corner of opening angle $2\Omega$. In panel (b), we show the analogous surface in $d=4$, i.e., a region whose boundary contains a conical singularity of opening angle $\Omega$. The smooth limit is found in both cases for $\Omega\rightarrow \pi/2$.
• Figure 2: An entangling region $V$ whose boundary corresponds to a deformed $S^1$ in eq. (\ref{['rb1']}). Two infinitesimal corner singularities appear at $\phi=0$ and $\pi$, with opening angle $2\Omega=\pi-\epsilon$.
• Figure 3: In panel (a), we show a two-sphere deformed according to eq. (\ref{['defor']}) with $\epsilon=0.3$. We observe the appearance of two conical singularities in the poles. In panel (b), we plot a cross-section of the same surface. The deficit angle of the conical singularities is determined by $\epsilon=\pi/2-\Omega$ for small $\epsilon$.