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arXiv:1807.08293 · Physical review. D/Physical review. D.

Entanglement entropy in TT¯ -deformed CFT

Bin Chen, Lin Chen, Peng-Xiang Hao

Published Oct 29, 2018• 96 citations
arXivHTMLPDF

Abstract

In this paper, we study the entanglement entropy of a single interval on a cylinder in two-dimensional $T\overline{T}$-deformed conformal field theory (CFT). For such case, the (R\'enyi) entanglement entropy takes a universal form in a CFT. We compute the correction due to the deformation up to the leading order of the deformation parameter in the framework of the conformal perturbation theory. We find that the correction to the entanglement entropy is nonvanishing in the finite temperature case, while it is vanishing in the finite size case. For the deformed holographic large $c$ CFT, which is proposed to be dual to a ${\mathrm{AdS}}_{3}$ gravity in a finite region, we find the agreement with the holographic entanglement entropy via the Ryu-Takayanagi formula. Moreover, we compute the leading order correction to the R\'enyi entropy, and discuss its holographic picture as well.

DOI
10.1103/physrevd.98.086025

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In this paper, we study the entanglement entropy of a single interval on a cylinder in two-dimensional TT‾T\overline{T}TT-deformed conformal field theory (CFT). For such case, the (R'enyi) entanglement entropy takes a universal form in a CFT. We compute the correction due to the deformation up to the leading order of the deformation parameter in the framework of the conformal perturbation theory. We find that the correction to the entanglement entropy is nonvanishing in the finite temperature case, while it is vanishing in the finite size case. For the deformed holographic large ccc CFT, which is proposed to be dual to a AdS3{\mathrm{AdS}}_{3}AdS3​ gravity in a finite region, we find the agreement with the holographic entanglement entropy via the Ryu-Takayanagi formula. Moreover, we compute the leading order correction to the R'enyi entropy, and discuss its holographic picture as well.
We study the entanglement entropy (EE) and the R\'enyi entropy (RE) of multiple intervals in two-dimensional TT‾T\overline{T}TT- deformed conformal field theory (CFT) at a finite temperature by field theoretic and holographic methods. First, by the replica method with the twist operators, we construct the general formula of the RE and EE up to the first order of a deformation parameter. By using our general formula, we show that the EE of multiple intervals for a holographic CFT is just a summation of the single interval case even with the small deformation. This is a nontrivial consequence from the field theory perspective, though it may be expected by the Ryu-Takayanagi formula in holography. However, the deformed RE of the two intervals is a summation of the single interval case only if the separations between the intervals are big enough. It can be understood by the tension of the cosmic branes dual to the RE. We also study the holographic EE for single and two intervals with an arbitrary cutoff radius (dual to the TT‾T\overline{T}TT deformation) at any temperature. We confirm our holographic results agree with the field theory results with a small deformation and high temperature limit, as expected. For two intervals, there are two configurations for EE: disconnected (sss-channel) and connected (ttt-channel) ones. We investigate the phase transition between them as we change parameters: as the deformation or temperature increases the phase transition is suppressed and the disconnected phase is more favored.
Quantum gravity in a finite region of spacetime is conjectured to be dual to a conformal field theory (CFT) deformed by the irrelevant operator TT[over ¯]. We test this conjecture with entanglement entropy, which is sensitive to ultraviolet physics on the boundary, while also probing the bulk geometry. We find that the entanglement entropy for an entangling surface consisting of two antipodal points on a sphere is finite and precisely matches the Ryu-Takayanagi formula applied to a finite region consistent with the conjecture of McGough et al. We also consider a one-parameter family of conical entropies, which are finite and verify a conjecture due to Dong. Since ultraviolet divergences are local, we conclude that the TT[over ¯] deformation acts as an ultraviolet cutoff on the entanglement entropy. Our results support the conjecture that the TT[over ¯]-deformed CFT is the holographic dual of a finite region of spacetime.
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 construct a replica technique to perturbatively compute the odd entanglement entropy (OEE) for bipartite mixed states in TT‾\mathrm{T}\overline{\mathrm{T}}TT deformed CFT2s{\mathrm{CFT}}_{2}\mathrm{s}CFT2​s. This framework is then utilized to obtain the leading-order correction to the OEE for two disjoint intervals, two adjacent intervals, and a single interval in TT‾\mathrm{T}\overline{\mathrm{T}}TT deformed thermal CFT2s{\mathrm{CFT}}_{2}\mathrm{s}CFT2​s in the large central charge limit. The field theory results are subsequently reproduced in the high-temperature limit from holographic computations for the entanglement wedge cross sections in the dual bulk finite cutoff Ba\~nados-Teitelboim-Zanelli geometries. We further show that for finite size TT‾\mathrm{T}\overline{\mathrm{T}}TT deformed CFT2s{\mathrm{CFT}}_{2}\mathrm{s}CFT2​s at zero temperature the corrections to the OEE are vanishing to the leading order from both field theory and bulk holographic computations.
In this work, we study the holographic entanglement entropy in AdS3 gravity with the certain mixed boundary condition, which turns out to correspond to TT¯-deformed 2D CFTs. By employing the Chern-Simons formalism and Wilson line technique, the holographic entanglement entropy in TT¯-deformed BTZ black hole is obtained. We also get the same formula by calculating the RT surface. The holographic entanglement entropy agrees with the perturbation result derived from both TT¯-deformed CFTs and cutoff AdS3. Moreover, our result shows that the deformed entanglement entropy for large deformation parameter behaves like the entanglement entropy of CFT at zero temperature. We also consider the entanglement entropy of two intervals and study the effect of TT¯ deformation on phase transition.
In this paper, we develop a perturbation formulation to calculate the single interval higher spin Rényi and entanglement entropy for two dimensional conformal field theory with W∞(λ) {\mathcal{W}}_{\infty}\left(\lambda \right) W∞​(λ) symmetry. The system is at finite temperature and is deformed by higher spin chemical potential. We manage to compute higher spin Rényi entropy with various spin deformations up to order O(μ2) \mathcal{O}\left({\mu}^2\right) O(μ2) . For spin 3 deformation, we calculate exact higher spin Rényi and entanglement entropy up to O(μ4) \mathcal{O}\left({\mu}^4\right) O(μ4) . When λ = 3, in the large c limit, we find perfect match with tree level holographic higher spin entanglement entropy up to order μ 4 obtained by the Wilson line prescription. We also find quantum corrections to higher spin entanglement entropy which is beyond tree level holographic results. The quantum correction is universal at order μ 4 in the sense that it is independent of λ. Our computation relies on a multi-valued conformal map from n-sheeted Riemann surface Rn {\mathrm{\mathcal{R}}}_n Rn​ to complex plane and correlation functions of primary fields on complex plane. The method can be applied to general conformal field theories with W \mathcal{W} W symmetry.