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arXiv:1312.5510 · Journal of High Energy Physics

Holographic Rényi entropy for CFT with W symmetry

Bin Chen, Jiang Long, Jiaju Zhang

Published Apr 1, 2014• 75 citations
arXivHTMLPDF

Abstract

In this paper we investigate the holographic Rényi entropy of two disjoint intervals on complex plane with small cross ratio x for conformal field theory with W symmetry in the ground state, which could be dual to a higher spin AdS3 gravity. We focus on the cases of W 3 and W 4 symmetries. In order to see the nontrivial contributions from the W fields, we calculate the Rényi entropy in the expansion of x to order x 8 in both the gravity and the CFT sides. In the gravity side the classical contributions to the entanglement entropy is still given by the Ryu-Takayanagi area formula under the reasonable assumption, while the 1-loop quantum corrections have to take into account of the contributions not only from massless gravitons, but also from massless higher spin fields. In the CFT side we still use the operator product expansion of twist operators in the small interval limit, but now we need to consider the quasiprimary fields constructed from W fields, besides the ones from Virasoro Verma module. In the large central charge limit, we obtain the classical, 1-loop, 2-loop, and 3-loop parts of the Rényi entropy. The classical and 1-loop results in the gravity and the CFT sides are in exact match. This confirms the higher spin gravity/CFT correspondence, and also supports the holographic computation of Rényi entanglement entropy, including the quantum correction, in both the AdS3 gravity and the higher spin AdS3 gravity.

DOI
10.1007/jhep04(2014)041

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In this paper we investigate the holographic Rényi entropy of two disjoint intervals on complex plane with small cross ratio x for conformal field theory with W symmetry in the ground state, which could be dual to a higher spin AdS3 gravity. We focus on the cases of W 3 and W 4 symmetries. In order to see the nontrivial contributions from the W fields, we calculate the Rényi entropy in the expansion of x to order x 8 in both the gravity and the CFT sides. In the gravity side the classical contributions to the entanglement entropy is still given by the Ryu-Takayanagi area formula under the reasonable assumption, while the 1-loop quantum corrections have to take into account of the contributions not only from massless gravitons, but also from massless higher spin fields. In the CFT side we still use the operator product expansion of twist operators in the small interval limit, but now we need to consider the quasiprimary fields constructed from W fields, besides the ones from Virasoro Verma module. In the large central charge limit, we obtain the classical, 1-loop, 2-loop, and 3-loop parts of the Rényi entropy. The classical and 1-loop results in the gravity and the CFT sides are in exact match. This confirms the higher spin gravity/CFT correspondence, and also supports the holographic computation of Rényi entanglement entropy, including the quantum correction, in both the AdS3 gravity and the higher spin AdS3 gravity.
In this paper, we study the holographic R\'enyi entropy of a large interval on a circle at high temperature for the two-dimensional conformal field theory (CFT) dual to pure AdS3{\mathrm{AdS}}_{3}AdS3​ gravity. In the field theory, the R\'enyi entropy is encoded in the CFT partition function on nnn-sheeted torus connected with each other by a large branch cut. As proposed by Chen and Wu [Large interval limit of R\'enyi entropy at high temperature, arXiv:1412.0763], the effective way to read the entropy in the large interval limit is to insert a complete set of state bases of the twist sector at the branch cut. Then the calculation transforms into an expansion of four-point functions in the twist sector with respect to ${e}^{\ensuremath{-}\frac{2\ensuremath{\pi}TR}{n}}$. By using the operator product expansion of the twist operators at the branch points, we read the first few terms of the R\'enyi entropy, including the leading and next-to-leading contributions in the large central charge limit. Moreover, we show that the leading contribution is actually captured by the twist vacuum module. In this case by the Ward identity the four-point functions can be derived from the correlation function of four twist operators, which is related to double interval entanglement entropy. Holographically, we apply the recipe in [T. Faulkner, The entanglement R\'enyi entropies of disjoint intervals in AdS/CFT, arXiv:1303.7221] and [T. Barrella et al., Holographic entanglement beyond classical gravity, J. High Energy Phys. 09 (2013) 109] to compute the classical R\'enyi entropy and its one-loop quantum correction, after imposing a new set of monodromy conditions. The holographic classical result matches exactly with the leading contribution in the field theory up to ${e}^{\ensuremath{-}4\ensuremath{\pi}TR}$ and l6{l}^{6}l6, while the holographical one-loop contribution is in exact agreement with next-to-leading results in field theory up to ${e}^{\ensuremath{-}\frac{6\ensuremath{\pi}TR}{n}}$ and l4{l}^{4}l4 as well.
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Two-dimensional conformal field theories with a large central charge and a small number of low-dimension operators are studied using the conformal block expansion. A universal formula is derived for the Renyi entropies of N disjoint intervals in the ground state, valid to all orders in a series expansion. This is possible because the full perturbative answer in this regime comes from the exchange of the stress tensor and other descendants of the vacuum state. Therefore, the Renyi entropy is related to the Virasoro vacuum block at large central charge. The entanglement entropy, computed from the Renyi entropy by an analytic continuation, decouples into a sum of single-interval entanglements. This field theory result agrees with the Ryu-Takayanagi formula for the holographic entanglement entropy of a 2d CFT, applied to any number of intervals, and thus can be interpreted as a microscopic calculation of the area of minimal surfaces in 3d gravity.
We investigate the one-loop entanglement entropy of two short intervals with small cross ratio x on a complex plane in two-dimensional conformal field theory (CFT) using operator product expansion of twist operators. We focus on the one-loop entanglement entropy instead of the general order n Rényi entropy, and this makes the calculation much easier. We consider the contributions of stress tensor to order x 10, contributions of W 3 operator to order x 12, and contributions of W 4 operator to order x 14. The CFT results agree with the ones in gravity.
We investigate the short interval expansion of the Rényi entropy for two-dimensional conformal field theory (CFT) on a torus. We require the length of the interval ℓ to be small with respect to the spatial and temporal sizes of the torus. The operator product expansion of the twist operators allows us to compute the short interval expansion of the Rényi entropy at any temperature. In particular, we pay special attention to the large c CFTs dual to the AdS3 gravity and its cousins. At both low and high temperature limits, we read the Rényi entropies to order ℓ6, and find good agreements with holographic results. Moreover, the expansion allows us to read 1/c contribution, which is hard to get by expanding the thermal density matrix. We generalize the study to the case with the chemical potential as well.
Holographic entanglement entropy of two disjoint intervals in AdS3_33​/CFT2_22​
The Ryu-Takayanagi conjecture predicts a holographic dual of the entanglement entropy of a CFT. It proposes that the entanglement entropy is given by the area of the minimal surface in the dual spacetime. In the semi-classical limit, this conjecture is supported by the saddle point approximation. If there are multiple classical solutions, it is assumed that only the minimal action contributes to the entanglement entropy. However, we will point out that these saddles equally contribute to the entanglement entropy in some cases. Therefore, the derivation of the conjecture is incomplete if there are multiple extremal surfaces that extend from a sub-system on the AdS boundary. We will consider two disjoint intervals in CFT1+1_{1+1}1+1​ as the simplest but non-trivial example, and propose another candidate for a holographic dual of the entanglement entropy of this system, which is the sum of all the signed areas of extremal surfaces in the dual spacetime. After that, we will derive it from the CFT calculations and propose the corresponding gravity side action.