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arXiv:2205.01137 · Physical review. D/Physical review. D.

Extractable entanglement from a Euclidean hourglass

Takanori Anegawa, Norihiro Iizuka, Daniel Kabat

Published Oct 25, 2022• 2 citations
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Abstract

We previously proposed that entanglement across a planar surface can be obtained from the partition function on a Euclidean hourglass geometry. Here we extend the prescription to spherical entangling surfaces in conformal field theory. We use the prescription to evaluate log terms in the entropy of a conformal field theory in two dimensions, a conformally coupled scalar in four dimensions, and a Maxwell field in four dimensions. For Maxwell we reproduce the extractable entropy obtained by Soni and Trivedi. We take this as evidence that the hourglass prescription provides a Euclidean technique for evaluating extractable entropy in quantum field theory.

DOI
10.1103/physrevd.106.085010

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We previously proposed that entanglement across a planar surface can be obtained from the partition function on a Euclidean hourglass geometry. Here we extend the prescription to spherical entangling surfaces in conformal field theory. We use the prescription to evaluate log terms in the entropy of a conformal field theory in two dimensions, a conformally coupled scalar in four dimensions, and a Maxwell field in four dimensions. For Maxwell we reproduce the extractable entropy obtained by Soni and Trivedi. We take this as evidence that the hourglass prescription provides a Euclidean technique for evaluating extractable entropy in quantum field theory.
We consider entanglement across a planar boundary in flat space. Entanglement entropy is usually thought of as the von Neumann entropy of a reduced density matrix, but it can also be thought of as half the von Neumann entropy of a product of reduced density matrices on the left and right. The latter form allows a natural regulator in which two cones are smoothed into a Euclidean hourglass geometry. Since there is no need to tensor factor the Hilbert space, the regulated entropy is manifestly gauge invariant and has a manifest state-counting interpretation. We explore this prescription for scalar fields, where the entropy is insensitive to a nonminimal coupling, and for Maxwell fields, which have the same entropy as $d\ensuremath{-}2$ scalars.
Entanglement entropy in even-dimensional conformal field theories (CFTs) contains well-known universal terms arising from the conformal anomaly. R\'enyi entropies are natural generalizations of the entanglement entropy that are much less understood. Above two spacetime dimensions, the universal terms in the R\'enyi entropies are unknown for general entangling geometries. We conjecture a new structure in the dependence of the four-dimensional R\'enyi entropies on the intrinsic and extrinsic geometry of the entangling surface. We provide evidence for this conjecture by direct numerical computations in the free scalar and fermion field theories. The computation involves relating the four-dimensional free massless R\'enyi entropies across cylindrical entangling surfaces to corresponding three-dimensional massive R\'enyi entropies across circular entangling surfaces. Our numerical technique also allows us to directly probe other interesting aspects of three-dimensional R\'enyi entropy, including the massless renormalized R\'enyi entropy and calculable contributions to the perimeter law.
We study universal features in the shape dependence of entanglement entropy in the vacuum state of a conformal field theory (CFT) on R1,d−1 {\mathrm{\mathbb{R}}}^{1,d-1} R1,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 σCT=π224 \frac{\sigma }{C_T}=\frac{\pi^2}{24} CT​σ​=24π2​ in d = 3 CFTs, and the holographic Mezei formula for entanglement entropy across deformed spheres.
We consider entanglement entropy in the context of gauge/gravity duality for conformal field theories in even dimensions. The holographic prescription due to Ryu and Takayanagi (RT) leads to an equation describing how the entangling surface extends into the bulk geometry. We show that setting to zero, the timetime component of the Brown-York stress tensor evaluated on the co-dimension 1 entangling surface, leads to the same equation. By considering a spherical entangling surface as an example, we observe that the Euclidean actionmethods in AdS/CFT will lead to the RT area functional arising as a counterterm needed to regularize the stress tensor. We present arguments leading to a justification for the minimal area prescription.
Here, we study universal features in the shape dependence of entanglement entropy in the vacuum state of a conformal field theory (CFT) on R1,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 also 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 CT 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 σ/CT=π2/24 in d = 3 CFTs, and the holographic Mezei formula for entanglement entropy across deformed spheres.
We calculate numerically the logarithmic contribution to the entanglement entropy of a cylindrical region in three spatial dimensions for a Maxwell field. Our result does not agree with the analytical predictions concerning any conformal field theory in four dimensions according to which the coefficient is universal and proportional to the type c conformal anomaly. In cylindrical coordinates the problem decomposes into one dimensional ones along the radial coordinate for each angular momentum. The entanglement entropy of a Maxwell field is equivalent to the one of two identical decoupled scalars with an extra self interaction term.