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arXiv:1506.03206 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Holographic calculation for large interval Rényi entropy at high temperature

Bin Chen, Jie-qiang Wu

Published Nov 2, 2015• 29 citations
arXivHTMLPDF

Abstract

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 ${\mathrm{AdS}}_{3}$ gravity. In the field theory, the R\'enyi entropy is encoded in the CFT partition function on $n$-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 ${l}^{6}$, 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 ${l}^{4}$ as well.

DOI
10.1103/physrevd.92.106001

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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−2\ensuremathπTRn{e}^{\ensuremath{-}\frac{2\ensuremath{\pi}TR}{n}}e\ensuremath−n2\ensuremathπTR​. 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\ensuremathπTR{e}^{\ensuremath{-}4\ensuremath{\pi}TR}e\ensuremath−4\ensuremathπTR 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−6\ensuremathπTRn{e}^{\ensuremath{-}\frac{6\ensuremath{\pi}TR}{n}}e\ensuremath−n6\ensuremathπTR​ 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.
The Monster CFT plays an important role in moonshine and is also conjectured to be the holographic dual to pure gravity in AdS3. We investigate the entanglement and Rényi entropies of this theory along with other extremal CFTs. The Rényi entropies of a single interval on the torus are evaluated using the short interval expansion. Each order in the expansion contains closed form expressions in the modular parameter. The leading terms in the q-series are shown to precisely agree with the universal corrections to Rényi entropies at low temperatures. Furthermore, these results are shown to match with bulk computations of Rényi entropy using the one-loop partition function on handlebodies. We also explore some features of Rényi entropies of two intervals on the plane.