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arXiv:1806.07444 · Physical Review Letters

Entanglement Entropy and TT¯ Deformation

William Donnelly, Vasudev Shyam

Published Sep 26, 2018• 145 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.121.131602

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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.
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.
Entanglement entropy in holographic conformal field theories with boundaries of generic spacetime signature cannot be described entirely within the standard holographic duality to a real bulk spacetime. We show, both in static and time-dependent setups, that an exact match with the conformal field theory predictions in two boundary dimensions entails an extension of the dual spacetime to complex coordinates. As a consequence, relevant bulk objects such as Ryu-Takayanagi surfaces and branes are in general complex, yet anchored on a real asymptotic boundary preserving the physical meaning of field-theoretical quantities. We also provide a third numerical check of the holographic prediction with a two-dimensional system of free Gaussian fermions exhibiting the same asymptotic regimes across different entanglement phases.
It was proposed by Ryu and Takayanagi that the entanglement entropy in conformal field theory (CFT) is related through the AdS/CFT correspondence to the area of a minimal surface in the bulk. We apply this holographic geometrical method of calculating the entanglement entropy to study the vacuum case of a CFT which is holographically dual to empty anti-de Sitter (AdS) spacetime. We present all possible minimal surfaces spanned on one or two spherical boundaries at AdS infinity. We give exact analytical expressions for the regularized areas of these surfaces and identify finite renormalized quantities. In the case of two disjoint boundaries the existence of two different phases of the entanglement entropy is confirmed. A trivial phase corresponds to two disconnected minimal surfaces, while the other one corresponds to a tube connecting the spherical boundaries. A transition between these phases is reminiscent of the finite temperature deconfinement transition in the CFT on the boundary. The exact analytical results are thus consistent with previous numerical and approximate computations. We also briefly discuss the character of a spacetime extension of the minimal surface spanned on two uniformly accelerated boundaries.
A central insight of holography is that spacetime geometry emerges from the quantum entanglement of a boundary theory. This work proposes and validates a precise, local manifestation of this principle: in a 1+1D boundary conformal field theory (CFT) dual to an asymptotically bulk, the Ricci scalar curvature is proportional to the negative second derivative of the entanglement entropy density. This hypothesis, motivated by the Ryu-Takayanagi formula and the geodesic deviation equation, posits that local fluctuations in boundary entanglement directly encode local bulk curvature. The relation is tested numerically using a tensor network toy model inspired by the Multi-scale Entanglement Renormalization Ansatz (MERA). Localized suppressions of boundary entanglement, analogous to removing disentanglers in MERA, are shown to induce localized positive curvature spikes in the emergent bulk geometry. The results provide direct computational evidence for the entanglement-curvature correspondence, reinforcing the paradigm that spacetime geometry is built from quantum entanglement.
We consider fine-grained probes of the entanglement structure of two-dimensional conformal field theories deformed by the irrelevant double-trace operator TT‾T\overline{T}TT and its closely related but nonetheless distinct single-trace counterpart. For holographic conformal field theories, these deformations can be interpreted as modifications of bulk physics in the ultraviolet region of anti-de Sitter space. Consequently, we can use the Ryu-Takayanagi formula and its generalizations to mixed state entanglement measures to test highly nontrivial consistency conditions. In general, the agreement between bulk and boundary quantities requires the equivalence of partition functions on manifolds of arbitrary genus. For the single-trace deformation, which is dual to an asymptotically linear dilaton geometry, we find that the mutual information and reflected entropy diverge for disjoint intervals when the separation distance approaches a minimum, finite value that depends solely on the deformation parameter. This implies that the mutual information fails to serve as a geometric regulator which is related to the breakdown of the split property at the inverse Hagedorn temperature. In contrast, for the double-trace deformation, which is dual to anti-de Sitter space with a finite radial cutoff, we find all divergences to disappear including the standard quantum field theory ultraviolet divergence that is generically seen as disjoint intervals become adjacent. We furthermore compute reflected entropy in conformal perturbation theory. While we find formally similar behavior between bulk and boundary computations, we find quantitatively distinct results. We comment on the interpretation of these disagreements and the physics that must be altered to restore consistency. We also briefly discuss the TJ‾T\overline{J}TJ and JT‾J\overline{T}JT deformations.
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.