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arXiv:1408.1931 · Journal of Physics A Mathematical and Theoretical

On Rényi entropy for free conformal fields: holographic and q-analog recipes

Rodrigo Aros, F. Bugini, Danilo E. Diaz

Published Feb 12, 2015• 6 citations
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Abstract

We describe a holographic approach to explicitly computing the universal logarithmic contributions to entanglement and Rényi entropies for free conformal scalar and spinor fields on even-dimensional spheres. This holographic derivation proceeds in two steps: first, following Casini and Huerta, a conformal mapping to thermal entropy in a hyperbolic geometry; then identification of the hyperbolic geometry with the conformal boundary of a bulk hyperbolic space and use of an AdS/CFT holographic formula to compute the resultant functional determinant. We explicitly verify the connection with the type-A trace anomaly for the entanglement entropy, whereas the Rényi entropy is computed with the aid of the Sommerfeld formula in order to deal with a conical defect. We show that as a by-product, the log coefficient of the Rényi entropy for round spheres can be efficiently obtained as the q-analog of a procedure similar to the one found by Cappelli and D’Appollonio that rendered the type-A trace anomaly.

DOI
10.1088/1751-8113/48/10/105401

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We describe a holographic approach to explicitly computing the universal logarithmic contributions to entanglement and Rényi entropies for free conformal scalar and spinor fields on even-dimensional spheres. This holographic derivation proceeds in two steps: first, following Casini and Huerta, a conformal mapping to thermal entropy in a hyperbolic geometry; then identification of the hyperbolic geometry with the conformal boundary of a bulk hyperbolic space and use of an AdS/CFT holographic formula to compute the resultant functional determinant. We explicitly verify the connection with the type-A trace anomaly for the entanglement entropy, whereas the Rényi entropy is computed with the aid of the Sommerfeld formula in order to deal with a conical defect. We show that as a by-product, the log coefficient of the Rényi entropy for round spheres can be efficiently obtained as the q-analog of a procedure similar to the one found by Cappelli and D’Appollonio that rendered the type-A trace anomaly.
We develop a framework for studying the well-known universal term in the Rényi entropy for an arbitrary entangling region in four-dimensional conformal field theories that are holographically dual to gravitational theories. The shape dependence of the Rényi entropy S_{n} is described by two coefficients: f_{b}(n) for traceless extrinsic curvature deformations and f_{c}(n) for Weyl tensor deformations. We provide the first calculation of the coefficient f_{b}(n) in interacting theories by relating it to the stress tensor one-point function in a deformed hyperboloid background. The latter is then determined by a straightforward holographic calculation. Our results show that a previous conjecture f_{b}(n)=f_{c}(n), motivated by surprising evidence from a variety of free field theories and studies of conical defects, fails holographically.
We compute entanglement entropy for free massive scalar fields in anti-de Sitter (AdS) space. The entangling surface is a minimal surface whose boundary is a sphere at the boundary of AdS. The entropy can be evaluated from the thermal free energy of the fields on a topological black hole by using the replica method. In odd-dimensional AdS, exact expressions of the Rényi entropy S n are obtained for arbitrary n. We also evaluate 1-loop corrections coming from the scalar fields to holographic entanglement entropy. Applying the results, we compute the leading difference of entanglement entropy between two holographic CFTs related by a renormalization group flow triggered by a double trace deformation. The difference is proportional to the shift of a central charge under the flow.
Rényi entropies Sq are useful measures of quantum entanglement; they can be calculated from traces of the reduced density matrix raised to power q, with q ≥ 0. For (d + 1)-dimensional conformal field theories, the Rényi entropies across Sd−1 may be extracted from the thermal partition functions of these theories on either (d+1)-dimensional de Sitter space or R×Hd, where Hd is the d-dimensional hyperbolic space. These thermal partition functions can in turn be expressed as path integrals on branched coverings of the (d+ 1)-dimensional sphere and S1 × Hd, respectively. We calculate the Rényi entropies of free massless scalars and fermions in d = 2, and show how using zeta-function regularization one finds agreement between the calculations on the branched coverings of S3 and on S1 × H2. Analogous calculations for massive free fields provide monotonic interpolating functions between the Rényi entropies at the Gaussian and the trivial fixed points. Finally, we discuss similar Rényi entropy calculations in d> 2.
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.
We study entanglement entropy (EE) in conformal field theories (CFTs) in Minkowski space with a planar boundary or with a planar defect of any codimension. In any such boundary CFT or defect CFT (DCFT), we consider the reduced density matrix and associated EE obtained by tracing over the degrees of freedom outside of a (hemi)sphere centered on the boundary or defect. Following Casini, Huerta, and Myers, we map the reduced density matrix to a thermal density matrix of the same theory on hyperbolic space. The EE maps to the thermal entropy of the theory on hyperbolic space. For boundary CFTs and DCFTs dual holographically to Einstein gravity theories, the thermal entropy is equivalent to the Bekenstein--Hawking entropy of a hyperbolic black brane. We show that the horizon of the hyperbolic black brane coincides with the minimal area surface used in Ryu and Takayanagi's conjecture for the holographic calculation of EE. We thus prove their conjecture in these cases. We use our results to compute the R\'enyi entropies and EE in DCFTs in which the defect corresponds to a probe brane in a holographic dual.
An entanglement Renyi entropy for a spatial partition of a system is studied in conformal theories which admit a dual description in terms of an anti-de Sitter gravity. The divergent part of the Renyi entropy is computed in 4D conformal N=4 super Yang-Mills theory at a weak coupling. This result is used to suggest a holographic formula which reproduces the Renyi entropy at least in the leading approximation. The holographic Renyi entropy is an invariant functional set on a codimension 2 minimal hypersurface in the bulk geometry. The bulk space does not depend on order nnn of the Renyi entropy. The holographic Renyi entropy is a sum of local and non-local functionals multiplied by polynomials of 1/n1/n1/n.