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arXiv:1602.01380 · Journal of High Energy Physics

Entanglement entropy of excited states in conformal perturbation theory and the Einstein equation

Antony J. Speranza

Published Apr 1, 2016• 22 citations
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Abstract

For a conformal field theory (CFT) deformed by a relevant operator, the entanglement entropy of a ball-shaped region may be computed as a perturbative expansion in the coupling. A similar perturbative expansion exists for excited states near the vacuum. Using these expansions, this work investigates the behavior of excited state entanglement entropies of small, ball-shaped regions. The motivation for these calculations is Jacobson’s recent work on the equivalence of the Einstein equation and the hypothesis of maximal vacuum entropy [ arXiv:1505.04753 ], which relies on a conjecture stating that the behavior of these entropies is sufficiently similar to a CFT. In addition to the expected type of terms which scale with the ball radius as R d , the entanglement entropy calculation gives rise to terms scaling as R 2Δ, where Δ is the dimension of the deforming operator. When $$ \varDelta \le \frac{d}{2} $$ , the latter terms dominate the former, and suggest that a modification to the conjecture is needed.

DOI
10.1007/jhep04(2016)105

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For a conformal field theory (CFT) deformed by a relevant operator, the entanglement entropy of a ball-shaped region may be computed as a perturbative expansion in the coupling. A similar perturbative expansion exists for excited states near the vacuum. Using these expansions, this work investigates the behavior of excited state entanglement entropies of small, ball-shaped regions. The motivation for these calculations is Jacobson’s recent work on the equivalence of the Einstein equation and the hypothesis of maximal vacuum entropy [ arXiv:1505.04753 ], which relies on a conjecture stating that the behavior of these entropies is sufficiently similar to a CFT. In addition to the expected type of terms which scale with the ball radius as R d , the entanglement entropy calculation gives rise to terms scaling as R 2Δ, where Δ is the dimension of the deforming operator. When Δ≤d2\varDelta \le \frac{d}{2}Δ≤2d​ , the latter terms dominate the former, and suggest that a modification to the conjecture is needed.
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.
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.
In this paper we study the scaling of the correction of the Renyi entropy of the excited states in systems described, in the continuum limit, by a conformal field theory (CFT). These corrections scale as L−2ΔnL^{-\frac{2Δ}{n}}L−n2Δ​, where LLL is the system size and ΔΔΔ is the scaling dimension of a relevant bulk operator located around the singularities of the Riemann surface Rn\mathcal{R}_nRn​. Their name is due to their explicit dependence on the Riemann surface Rn\mathcal{R}_nRn​. Their presence has been detected in several works on the entanglement entropy in finite size systems, both in the ground and the excited states. Here, we present a general study of these corrections based on the perturbation expansion on Rn\mathcal{R}_nRn​. Some of the terms in this expansion are divergent and they will be cured with addition cut-offs. These cut-offs will determine how these corrections scale with the system size LLL. Exact numerical computations of the Renyi entropy of the excited states of the XX model are provided and they confirm our theoretical prediction on the scaling of corrections. They allow also a comparison with the other works present in the literature finding that the corrections, for the excited states, have the exact same form of the ones of the ground state case multiplied by a model dependend function of nnn and l/Ll/Ll/L.
For holographic CFT states near the vacuum, entanglement entropies for spatial subsystems can be expressed perturbatively as an expansion in the one-point functions of local operators dual to light bulk fields. Using the connection between quantum Fisher information for CFT states and canonical energy for the dual spacetimes, we describe a general formula for this expansion up to second-order in the one-point functions, for an arbitrary ball-shaped region, extending the first-order result given by the entanglement first law. For two-dimensional CFTs, we use this to derive a completely explicit formula for the second-order contribution to the entanglement entropy from the stress tensor. We show that this stress tensor formula can be reproduced by a direct CFT calculation for states related to the vacuum by a local conformal transformation. This result can also be reproduced via the perturbative solution to a non-linear scalar wave equation on an auxiliary de Sitter spacetime, extending the first-order result in arXiv:1509.00113 .
We use holographic duality to study the entanglement entropy (EE) of Conformal Field Theories (CFTs) in various spacetime dimensions d, in the presence of various deformations: a relevant Lorentz scalar operator with constant source, a temperature T , a chemical potential μ, a marginal Lorentz scalar operator with source linear in a spatial coordinate, and a circle-compactified spatial direction. We consider EE between a strip or sphere sub-region and the rest of the system, and define the “entanglement density” (ED) as the change in EE due to the deformation, divided by the sub-region’s volume. Using the deformed CFTs above, we show how the ED’s dependence on the strip width or sphere radius, L, is useful for characterizing states of matter. For example, the ED’s small-L behavior is determined either by the dimension of the perturbing operator or by the first law of EE. For Lorentz-invariant renormalization group (RG) flows between CFTs, the “area theorem” states that the coefficient of the EE’s area law term must be larger in the UV than in the IR. In these cases the ED must therefore approach zero from below as L→∞. However, when Lorentz symmetry is broken and the IR fixed point has different scaling from the UV, we find that the ED often approaches the thermal entropy density from above, indicating area theorem violation.
We study the entanglement entropy of theories that are derived from relevant perturbation of given CFTs for regions with a singular boundary by using the AdS/CFT correspondence. In the smooth case, it is well known that a relevant deformation of the boundary theory by the relevant operator with scaling dimension Δ=d+22Δ=\frac{d+2}{2}Δ=2d+2​ generates a logarithmic universal term to the entanglement entropy. As the smooth case, when the boundary CFT deformed by a relevant operator, we find that the entanglement entropy of singular surface also contains a new logarithmic term which is due to relevant perturbation of the conformal field theory, and depends on the scaling dimension of relevant operator. We also find for extended singular surfaces, cn×Rmc_{n}\times R^{m}cn​×Rm, as well as logarithmic term, the new universal double logarithmic terms may appear depending on the scaling dimension of relevant operator and spacetime dimensions. These new terms are due to relevant perturbation of the boundary theory.