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Holographic Entanglement Entropy in Boundary Conformal Field Theory PHYSICAL REVIEW D 98, 106016 (2018) Holographic entanglement entropy in boundary conformal field theory † ‡ En-Jui Chang, Chia-Jui Chou, and Yi Yang* Department of Electrophysics, National Chiao Tung University, Hsinchu 300, Republic of China (Received 7 September 2018; published 21 November 2018) We study the holographic entanglement entropy in a (d þ 1)-dimensional boundary quantum field theory at both the zero and finite temperature. The phase diagrams for the holographic entanglement entropy at various temperatures are obtained by solving the entangled surfaces in the different homology. We also verify the Araki-Lieb inequality and illustrate the entanglement plateau. DOI: 10.1103/PhysRevD.98.106016 I. INTRODUCTION entangled regions, there are disconnected minimal surfaces satisfying the homology constraint that leads to the famous Entanglement is a pure quantum mechanics phenomenon phenomenon of the entanglement plateau [17]. inherent in quantum states. It can be measured quantita- On the other hand, BQFT is a quantum field theory tively by the entanglement entropy associated with a defined on a manifold with a boundary where some suitable specified entangled region A located on a Cauchy slice boundary conditions are imposed. It has important appli- of the spacetime by integrating out the degrees of freedom cations in the physical systems with boundaries. For in its complementary region Ac. It has been shown that the example, string theory with various branes and some leading divergence term of the entanglement entropy is condensed matter systems including Hall effect, chiral proportional to the area of the entangling boundary, i.e., the magnetic effect, topology insulator, etc. Several years boundary of the entangled region A. Furthermore, the finite ago, holographic BQFT was proposed by extending the part of the entanglement entropy contains nontrivial infor- manifold where BQFT is defined to a one-dimensional mation about the quantum states. It is well known that the higher asymptotically AdS space, i.e., the bulk manifold, entanglement entropy of a given region and its complement with a geometric boundary [18,19]. The key point of are the same, SA SAc , for a system with only pure states. ¼ holographic BQFT is thus to determine the shape of the However, this is not true anymore for a system with finite geometric boundary in the bulk. For the simple shapes with temperature. The difference δSA SA − SAc has been ¼ high symmetry such as the case of a disk or half plane, conjectured to satisfy the Araki-Lieb inequality δSA ≤ j j many elegant results for BQFT have been obtained in SA∪Ac [1]. [18–20]. Some interesting developments of BQFT can be Calculating entanglement entropy is usually a not easy found in [19–27]. task in QFT. Remarkably, by the AdS=CFT correspondence Since both the entanglement entropy and BQFT can be [2–4], the holographic entanglement entropy (HEE) was studied holographically, it is natural to investigate the HEE proposed to be the area of the minimal entangled surface in in holographic BQFT. The HEE in pure AdS bulk space- [5–8] and was justified later on [9–12]. This prescription time has been studied in [28–30]. It was found that the gives a very simple geometric picture to compute the HEE proper boundary condition gives the orthogonal condition and has been widely studied for the various holographic that requires the minimal entangled surface must be normal setups, for a review see [13]. It was shown that the Araki- to the geometric boundaries if they intersect. The authors Lieb inequality is due to the homology constraint for the – in [28,29] also found an interesting phenomenon that entangled surface [14 16]. For certain choices of the the entanglement entropy depends on the distance to the boundary and carries a phase transition. In addition, the HEE in the (2 þ 1)-dimensional bulk manifold, such as *Corresponding author. [email protected] AdS3 and BTZ black hole, has been considered in [31,32]. † [email protected] In this work, we study the HEE in a (d þ 1)- ‡ [email protected] dimensional BQFT at both the zero and finite temper- atures. In the case of the zero temperature, we consider the Published by the American Physical Society under the terms of (d þ 2)-dimensional pure AdS spacetime as the bulk the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to manifold. We find three phases for the HEE depending the author(s) and the published article’s title, journal citation, on the size and the location of the entangled region A. and DOI. Funded by SCOAP3. Our result is consistent with the conclusion in [28,29]. 2470-0010=2018=98(10)=106016(11) 106016-1 Published by the American Physical Society EN-JUI CHANG, CHIA-JUI CHOU, and YI YANG PHYS. REV. D 98, 106016 (2018) In the case of the finite temperature, we consider the and represents a geometric boundary of the bulk. This is (d þ 2)-dimensional Schwazschild-AdS black hole as the our holographic setup for a BQFT living in M with a bulk manifold. The similar three phases are found for boundary P. the HEE. However, due to the presence of the black hole The total action of the system is the sum of the actions of horizon, the HEE for the entangled region A and its the various geometric objects and their boundary terms, complementary region Ac are different. For BQFT at the finite temperature, entanglement entropy is mixed with S ¼ SN þ SGH þ SQ þ SP; ð2:1Þ the thermal entropy given by the Bekenstein-Hawking entropy SBH. We show that the Araki-Lieb inequality where δ ≤ j SAj SBH always holds by a directly calculation. In Z addition, we obtain the entanglement plateau for various pffiffiffiffiffiffi A SN ¼ −gðR − 2ΛN Þ; ð2:2Þ sizes and locations of the entangled region .Because N there is a new phase due to the geometric boundary, the Z entanglement plateau enjoys much richer structure in QFT pffiffiffiffiffiffi SQ ¼ −hðRQ − 2ΛQ þ 2KÞ; ð2:3Þ with boundaries. Q The paper is organized as follows. In Sec. II, we briefly Z review the holographic BQFT and present the solutions ffiffiffiffiffiffi p 0 SM ¼ 2 −γK ; ð2:4Þ which we will use in this work. The HEE in BQFT is M calculated in Sec. III. We discuss the phase structure of the Z HEE at both the zero and finite temperatures. We also pffiffiffiffiffiffi verify the Araki-Lieb inequality and obtain the entangle- SP ¼ 2 −σθ; ð2:5Þ P ment plateau. We summarize our results in Sec. IV. and we have taken 16πG ¼ 1. In the total action (2.1), SN II. HOLOGRAPHIC BOUNDARY QUANTUM is the action of the bulk manifold N with R and ΛN being FIELD THEORY the intrinsic Ricci curvature and the cosmological constant N Q We consider a (d þ 2)-dimensional bulk manifold N of . SQ is the action of the geometric boundary with Λ which has a (d þ 1)-dimensional conformal boundary M RQ, Q and K being the intrinsic Ricci curvature, the as shown in the Fig. 1. The bulk manifold N is either a pure cosmological constant and the extrinsic curvatures of Q AdS spacetime, as in Fig. 1(a), or an asymptotic AdS black embedded in N . SM is the action of the conformal hole with an event horizon, as in Fig. 1(b). In addition, there boundary of M with K0 being the extrinsic curvatures is a (d þ 1)-dimensional hypersurface Q in N that inter- of M embedded in N . We remark that the terms of K and sects the conformal boundary M at a d-dimensional K0 are the Gibbons-Hawking boundary terms for the hypersurface P. A BQFT is defined on M within the boundaries Q and M of the bulk manifold N , respectively. boundary P. The hypersurface Q could be considered as Finally, SP is the common boundary term of Q and M the extension of the boundary P from M into the bulk N with θ ¼ cos−1ðnQ · nN Þ being the supplementary angle FIG. 1. Spacetime setup for the holographic BQFT. (a) The bulk manifold is a pure AdS spacetime. (b) The bulk manifold is an asymptotic AdS black hole with a horizon. 106016-2 HOLOGRAPHIC ENTANGLEMENT ENTROPY IN BOUNDARY … PHYS. REV. D 98, 106016 (2018) Q M between and , which makes a well-defined variational where lAdS is the AdS radius. The conformal boundary of P 1 principle on . Furthermore, gab denotes the metric of the AdSdþ2 is a (d þ )-dimensional Minkowski spacetime N − Q Q γ − N N located at z ¼ 0. bulk manifold , hab ¼gab na nb and ab ¼ gab na nb We propose a simple solution of the geometric boundary denote the induced metric of the boundaries Q and M, σab denotes the metric of P. Here we have defined the unit Q as a (d þ 1)-dimensional hepersurface embedded in the normal vectors of Q and M as nQ and nN . bulk manifold as ab Varying SN with g gives the equation of motion of the l2 Xd N 2 AdS − 2 2 2 bulk , dsQ ¼ 2 dt þ dz þ dxi ; ð2:10Þ z 2 1 i¼ 0 ¼ R − Rg þ ΛN g : ð2:6Þ ab 2 ab ab with a simple embedding x1 ¼ constant. The intrinsic curvature, the extrinsic curvature and the cosmological ab Varying SQ with h gives the equation of motion of the constant on Q can be calculated as geometric boundary Q, 1 dðd þ 1Þ dðd − 1Þ RQ ¼ − ;K¼ 0; ΛQ ¼ − : RQ þ 2K − RQ þ K − ΛQ h ¼ 0; ð2:7Þ 2 ab 2 2 ab ab 2 ab lAdS lAdS ð2:11Þ which is just the Neumann boundary condition originally proposed by Takayanagi in [18] and lately generalized by It is easy to verify that the mixed boundary condition (2.8) Chu et al.
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