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arXiv:1407.1358 · Journal of High Energy Physics

Universal thermal corrections to entanglement entropy for conformal field theories on spheres

Christopher P. Herzog

Published Oct 1, 2014• 47 citations
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Abstract

We consider entanglement entropy of a cap-like region for a conformal field theory living on a sphere times a circle in d space-time dimensions. Assuming that the finite size of the system introduces a unique ground state with a nonzero mass gap, we calculate the leading correction to the entanglement entropy in a low temperature expansion. The correction has a universal form for any conformal field theory that depends only on the size of the mass gap, its degeneracy, and the angular size of the cap. We confirm our result by calculating the entanglement entropy of a conformally coupled scalar numerically. We argue that an apparent discrepancy for the scalar can be explained away through a careful treatment of boundary terms. In an appendix, to confirm the accuracy of the numerics, we study the mutual information of two cap-like regions at zero temperature.

DOI
10.1007/jhep10(2014)028

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We consider entanglement entropy of a cap-like region for a conformal field theory living on a sphere times a circle in d space-time dimensions. Assuming that the finite size of the system introduces a unique ground state with a nonzero mass gap, we calculate the leading correction to the entanglement entropy in a low temperature expansion. The correction has a universal form for any conformal field theory that depends only on the size of the mass gap, its degeneracy, and the angular size of the cap. We confirm our result by calculating the entanglement entropy of a conformally coupled scalar numerically. We argue that an apparent discrepancy for the scalar can be explained away through a careful treatment of boundary terms. In an appendix, to confirm the accuracy of the numerics, we study the mutual information of two cap-like regions at zero temperature.
We consider single interval Rényi and entanglement entropies for a two dimensional conformal field theory on a circle at nonzero temperature. Assuming that the finite size of the system introduces a unique ground state with a nonzero mass gap, we calculate the leading corrections to the Rényi and entanglement entropy in a low temperature expansion. These corrections have a universal form for any two dimensional conformal field theory that depends only on the size of the mass gap and its degeneracy. We analyze the limits where the size of the interval becomes small and where it becomes close to the size of the spatial circle.
We compute thermal corrections to Rényi entropies of d dimensional conformal field theories on spheres. Consider the nth Rényi entropy for a cap of opening angle 2θ on S d−1. From a Boltzmann sum decomposition and the operator-state correspondence, the leading correction is related to a certain two-point correlation function of the operator (not equal to the identity) with smallest scaling dimension. More specifically, via a conformal map, the correction can be expressed in terms of the two-point function on a certain conical space with opening angle 2πn. In the case of free conformal field theories, this two-point function can be computed explicitly using the method of images. We perform the computation for the conformally coupled scalar. From the n → 1 limit of our results, we extract the leading thermal correction to the entanglement entropy, reproducing results of arXiv:1407.1358.
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We compute thermal corrections to R enyi entropies of d dimensional conformal eld theories on spheres. Consider the nth R enyi entropy for a cap of opening angle 2 on S d 1 . From a Boltzmann sum decomposition and the operator-state correspondence, the leading correction is related to a certain two-point correlation function of the operator (not equal to the identity) with smallest scaling dimension. More specically, via a conformal map, the correction can be expressed in terms of the two-point function on a certain conical space with opening angle 2n . In the case of free conformal eld theories, this two-point function can be computed explicitly using the method of images. We perform the computation for the conformally coupled scalar. From the n ! 1 limit of our results, we extract the leading thermal correction to the entanglement entropy, reproducing results of arXiv:1407.1358.
We explore entanglement entropy of a cap-like region for a generic quantum field theory residing in the Bunch-Davies vacuum on de Sitter space. Entanglement entropy in our setup is identical with the thermal entropy in the static patch of de Sitter, and we derive a simple relation between the vacuum expectation value of the energy-momentum tensor trace and the RG flow of entanglement entropy. In particular, renormalization of the bare couplings and logarithmic divergence of the entanglement entropy are interrelated in our setup. We confirm our findings by recovering known universal contributions for a free field theory deformed by a mass operator as well as obtain correct universal behaviour at the fixed points. Simple examples of entanglement entropy flows are elaborated in d=2,3,4. Inthreedimensionswefindthatwhiletherenormalizedentanglemententropy is stationary at the fixed points, it is not monotonic. We provide a computational evidence that the universal ‘area law’ for a conformally coupled scalar is different from the known result in the literature, and argue that this difference survives in the limit of flat space. Finally, we carry out the spectral decomposition of entanglement entropy flow and discuss its application to the F-theorem.
A bstract The entanglement entropy of spacetime regions A in odd-dimensional conformal field theories (CFTs) contains a universal constant term, (−1)d−12 F(A) {\left(-1\right)}^{\frac{d-1}{2}}\ F(A) (−1)2d−1​ F(A) − 1 d − 1 2 F A . This quantity can be robustly defined by considering the mutual information of pairs of slightly deformed versions of A . In the case of general three-dimensional CFTs, F ( A ) is positive definite and bounded below by the round disk result, F ( A ) ≥ F 0 ≡ F ( ∂A = 𝕊 1 ). Additionally, strong evidence has been provided that for every region A , F ( A ) /F 0 is maximized, within the space of CFT 3 ’s, by the free scalar field result. In this paper we show that while F ( A ) remains a local minimum around F 0 ≡ F ( ∂A = 𝕊 3 ) for small deformations of the spherical entangling surface, it can take values of arbitrarily large magnitude with either sign for more general regions, and hence it is neither upper- nor lower-bounded in general CFT 5 ’s. We argue that an analogous conjecture regarding the extremization of F ( A ) /F 0 for general regions within the space of theories fails in d = 5. We instead analyze the viability of the weaker bound, F ϵ /F 0 ≤ F ϵ /F 0 | free scalar , ∀CFT 5 for general small geometric deformations of the spherical entangling surface. This is equivalent to a general constraint involving the stress-tensor two-point function C T and the Euclidean partition function on the sphere, namely, C T /F 0 ≤ C T /F 0 | free scalar ≈ 0 . 314, which we show to hold for all known CFT 5 ’s. We also comment on possible extensions of this result to higher dimensions.