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arXiv:2203.04325 · Physical Review Letters

Universal Feature of Charged Entanglement Entropy

Pablo Bueno, Pablo A. Cano, Ángel Murcia, Alberto Rivadulla Sánchez

Published Jul 6, 2022• 20 citations
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Abstract

Rényi entropies, S_{n}, admit a natural generalization in the presence of global symmetries. These "charged Rényi entropies" are functions of the chemical potential μ conjugate to the charge contained in the entangling region and reduce to the usual notions as μ→0. For n=1, this provides a notion of charged entanglement entropy. In this Letter, we prove that for a general d(≥3)-dimensional conformal field theory, the leading correction to the uncharged entanglement entropy across a spherical entangling surface is quadratic in the chemical potential, positive definite, and universally controlled (up to fixed d-dependent constants) by the coefficients C_{J} and a_{2}. These fully characterize, for a given theory, the current correlators ⟨JJ⟩ and ⟨TJJ⟩, as well as the energy flux measured at infinity produced by the insertion of the current operator. Our result is motivated by analytic holographic calculations for a special class of higher-curvature gravities coupled to a (d-2) form in general dimensions as well as for free fields in d=4. A proof for general theories and dimensions follows from previously known universal identities involving the magnetic response of twist operators introduced in A. Belin et al. [J. High Energy Phys. 12 (2013) 059.JHEPFG1029-847910.1007/JHEP12(2013)059] and basic thermodynamic relations.

DOI
10.1103/physrevlett.129.021601

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Rényi entropies, S_{n}, admit a natural generalization in the presence of global symmetries. These "charged Rényi entropies" are functions of the chemical potential μ conjugate to the charge contained in the entangling region and reduce to the usual notions as μ→0. For n=1, this provides a notion of charged entanglement entropy. In this Letter, we prove that for a general d(≥3)-dimensional conformal field theory, the leading correction to the uncharged entanglement entropy across a spherical entangling surface is quadratic in the chemical potential, positive definite, and universally controlled (up to fixed d-dependent constants) by the coefficients C_{J} and a_{2}. These fully characterize, for a given theory, the current correlators ⟨JJ⟩ and ⟨TJJ⟩, as well as the energy flux measured at infinity produced by the insertion of the current operator. Our result is motivated by analytic holographic calculations for a special class of higher-curvature gravities coupled to a (d-2) form in general dimensions as well as for free fields in d=4. A proof for general theories and dimensions follows from previously known universal identities involving the magnetic response of twist operators introduced in A. Belin et al. [J. High Energy Phys. 12 (2013) 059.JHEPFG1029-847910.1007/JHEP12(2013)059] and basic thermodynamic relations.
Charged and symmetry-resolved R'enyi entropies are entanglement measures quantifying the degree of entanglement within different charge sectors of a theory with a conserved global charge. We use holography to determine the dependence of charged R'enyi entropies on small shape deformations away from a spherical or planar entangling surface in general dimensions. This dependence is completely characterized by a single coefficient appearing in the two point function of the displacement operator associated with the R'enyi defect. We extract this coefficient using its relation to the one point function of the stress tensor in the presence of a deformed entangling surface. This is mapped to a holographic calculation in the background of a deformed charged black hole with hyperbolic horizon. We obtain numerical solutions for different values of the chemical potential and replica number nnn in various spacetime dimensions, as well as analytic expressions for small chemical potential near n=1n=1n=1. When the R'enyi defect becomes supersymmetric, we demonstrate a conjectured relation between the two point function of the displacement operator and the conformal weight of the twist operator.
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 show that for a d-dimensional CFT in flat space, the Rényi entropy S q across a spherical entangling surface has the following property: in an expansion around q = 1, the first correction to the entanglement entropy is proportional to C T , the coefficient of the stress tensor vacuum two-point function, with a fixed d-dependent coefficient. This is equivalent to a similar statement about the free energy of CFTs living on S 1 × $ \mathbb{H} $ d−1 with inverse temperature β = 2πq. In addition to furnishing a direct argument applicable to all CFTs, we exhibit this result using a handful of gravity and field theory computations. Knowledge of C T thus doubles as knowledge of Rényi entropies in the neighborhood of q = 1, which we use to establish new results in 3d vector models at large N.
The entanglement entropy of a generic d-dimensional conformal field theory receives a regulator independent contribution when the entangling surface 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 (3) characterizing the limit where the surface becomes smooth is proportional to the central charge C T appearing in the two-point function of the stress tensor. In this paper, we prove this relation for general three-dimensional holographic theories, and extend the result to general dimensions. In particular, we define a generalized coefficient σ (d) to characterize the almost smooth limit of a (hyper)conical singularity in entangling surfaces in higher dimensions. We show then that this coefficient is universally related to C T for general holographic theories and provide 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 6.
We derive several new results for Rényi entropy, S n , across generic entangling surfaces. We establish a perturbative expansion of the Rényi entropy, valid in generic quantum field theories, in deformations of a given density matrix. When applied to even-dimensional conformal field theories, these results lead to new constraints on the n-dependence, independent of any perturbative expansion. In 4d CFTs, we show that the n-dependence of the universal part of the ground state Rényi entropy for entangling surfaces with vanishing extrinsic curvature contribution is in fact fully determined by the Rényi entropy across a sphere in flat space. Using holography, we thus provide the first computations of Rényi entropy across non-spherical entangling surfaces in strongly coupled 4d CFTs. Furthermore, we address the possibility that in a wide class of 4d CFTs, the flat space spherical Rényi entropy also fixes the n-dependence of the extrinsic curvature contribution, and hence that of arbitrary entangling surfaces. Our results have intriguing implications for the structure of generic modular Hamiltonians.
We derive several new results for Rényi entropy, S n , across generic entangling surfaces. We establish a perturbative expansion of the Rényi entropy, valid in generic quantum field theories, in deformations of a given density matrix. When applied to even-dimensional conformal field theories, these results lead to new constraints on the n-dependence, independent of any perturbative expansion. In 4d CFTs, we show that the n-dependence of the universal part of the ground state Rényi entropy for entangling surfaces with vanishing extrinsic curvature contribution is in fact fully determined by the Rényi entropy across a sphere in flat space. Using holography, we thus provide the first computations of Rényi entropy across non-spherical entangling surfaces in strongly coupled 4d CFTs. Furthermore, we address the possibility that in a wide class of 4d CFTs, the flat space spherical Rényi entropy also fixes the n-dependence of the extrinsic curvature contribution, and hence that of arbitrary entangling surfaces. Our results have intriguing implications for the structure of generic modular Hamiltonians.