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.
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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. in [29] by adding the intrinsic curvature RQ. is satisfied. The crucial problem in the construction of the holo- This simple solution is a special case of the solutions graphic BQFT is to determine the (d þ 1)-dimensional constructed in [28,29] with θ ¼ 0, i.e., the geometric geometric boundary Q that satisfies the boundary condition boundary Q is perpendicular to M at their intersection P. (2.7). However, the boundary condition (2.7) is too strong to have a solution even in the pure AdS spacetime because B. Schwarzschild-AdS black hole there are more constraint equations than the degrees of freedom. In [28,29], the authors proposed the following To study the (d þ 1)-dimensional BQFT at a finite mixed boundary condition, temperature, we consider the bulk manifold N as the (d þ 2)-dimensional Schwarzschild-AdS black hole with ðd − 1ÞðRQ þ 2KÞ − 2ðd þ 1ÞΛQ ¼ 0: ð2:8Þ the metric, Although it is still difficult to obtain a general solution of Q l2 dz2 Xd with the mixed boundary condition (2.8), it is possible to ds2 ¼ AdS −gðzÞdt2 þ þ dx2 ; ð2:12Þ N 2 g z i find solutions in some special cases. A class of solutions z ð Þ i¼1 satisfying the mixed boundary condition (2.8) have been obtained in [28,29]. with In this work, instead of constructing more solutions of the geometric boundary Q, our purpose is to study the dþ1 1 − z boundary effect for the HEE in BQFT. We will thus use an gðzÞ¼ dþ1 ; ð2:13Þ almost trivial solution of Q that is perpendicular to the zH conformal boundary M with a simple embedding. Nevertheless, we will find rich phase structures of the The Hawking temperature and the Bekenstein-Hawking HEE in our simple geometry. entropy density of the (d þ 2)-dimensional Schwarzschild- We present the solutions that will be used to study the AdS black hole are HEE in detail in the following. d þ 1 ld Ld T ¼ ;S¼ AdS : ð2:14Þ 4π BH 4 d A. Pure AdS zH zH To study a (d þ 1)-dimensional BQFT at the vanishing temperature, we consider the bulk manifold N as the Similar to the pure AdS case, we propose a solution of the 2 geometric boundary Q as a (d þ 1)-dimensional hepersur- (d þ )-dimensional pure AdS spacetime AdSdþ2 with the metric, face embedded in the bulk manifold as l2 Xd l2 dz2 Xd ds2 ¼ AdS −dt2 þ dz2 þ dx2 ; ð2:9Þ ds2 ¼ AdS −gðzÞdt2 þ þ dx2 ; ð2:15Þ N 2 i Q 2 g z i z i¼1 z ð Þ i¼2
106016-3 EN-JUI CHANG, CHIA-JUI CHOU, and YI YANG PHYS. REV. D 98, 106016 (2018) with a simple embedding x1 ¼ constant. The intrinsic region A. More precisely, there exists a codimension-1 curvature, the extrinsic curvature and the cosmological region RA ⊂ N , the so called entanglement wedge, which constant on Q are the same as the pure AdS case, is bounded by the minimal surface EA and the entangled region A on the conformal boundary M. dðd þ 1Þ dðd − 1Þ In the presence of the geometric boundary Q, the RQ − ;K0; ΛQ − ; ¼ 2 ab ¼ ¼ 2 2 E lAdS lAdS minimal surface A could end not only on the conformal boundary M but also on the geometric boundary Q as ð2:16Þ showed in [28,29]. We thus propose the following formula which satisfy the mixed boundary condition (2.8). for the HEE in BQFT,
A AreaðEAÞ SEE ¼ min ; X 4 ðdþ2Þ III. HOLOGRAPHIC ENTANGLEMENT ENTROPY GN E E ∂A E P ; ∃ R ⊂ N Without the geometric boundary Q, the prescription to X ¼f Aj AjM ¼ ; AjQ ¼ A A ; compute the entanglement entropy holographically for the ∂RA ¼ EA ∪ A ∪ QAg; ð3:2Þ static situations has been addressed by Ryu and Takayanagi (RT) in [6,7]. It was later generalized in [8] to general states where PA divides the geometric boundary Q into two parts, including arbitrary time dependence. In this work, we focus QA and QAc , with QA having the same homology with A on the static situation. c and QAc having the same homology with A , as shown in We consider a spatial region A with a boundary ∂A lying c c Fig. 2. Requiring the boundary condition (2.8) to be on a Cauchy slice Σ ¼ A þ A ⊂ M, with A the com- E Q A smooth, A should be orthogonal to when they intersect plementary part of , as shown in Fig. 2(a). The HEE is as showed in [28,29]. given by the RT formula [6,7], To be concrete, in this work, we consider a bulk spacetime N with two boundaries QL;R which intersect AreaðEAÞ SA ¼ min ; the conformal boundary M at P ¼ l=2 perpendicularly. X 4 ðdþ2Þ GN We choose the region A ⊂ M as an infinite long strip,
X ¼fEAjEAjM ¼ ∂A; ∃ RA ⊂ N ; ∂RA ¼ EA ∪ Ag; a a ∈ − ∈ Rd−1 2 ð3:1Þ x1 x 2 ;xþ 2 ;xi for i ¼ ; d; where EA is a codimension-2 minimal surface anchored ð3:3Þ on ∂A in the (d þ 2)-dimensional bulk spacetime N . The minimal surface EA is required to satisfy a homology which preserves (d − 1)-dimensional translation invariance constraint: EA is smoothly retractable to the boundary in the directions xi for i ¼ 2; d.
FIG. 2. Spacetime setup for the holographic BQFT with an entangled region A. The minimal surface EA could end not only on the conformal boundary M but also on the geometric boundary Q. (a) The bulk manifold is a pure AdS spacetime. The minimal surface for A and Ac have the same homology. (b) The bulk manifold is an asymptotically AdS black hole. The minimal surface for A and Ac have the different homologies due to the black hole horizon.
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In the static gauge, we can write down the ansatz for the A. Pure AdS spacetime E minimal surface A, We first consider the bulk spacetime N as a (d þ 2)- dimensional pure AdS spacetime with the metric (2.9), and a 0 choose the geometric boundary Q as a (d 1)-dimensional z ¼ zðx1Þ;zx 2 ¼ ;zðxÞ¼z0; þ hepersurface with the metric (2.10). This is dual to BQFT at z0ðxÞ¼0; ð3:4Þ the zero temperature. In the case of pure AdS spacetime, the size a can be where x1 ¼ x is the turning point of the minimal integrated to obtain surface EA. Z For a general (d þ 2)-dimensional bulk metric, 1 d ffiffiffi Γ dþ1 v dv p ð 2d Þ a ¼ 2z0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2z0 π : ð3:10Þ 2d Γ 1 Xd 0 1 − v ð2dÞ 2 − 2 2 2 ds ¼ gttðzÞdt þ giiðzÞdxi þ gzzðzÞdz ; i¼1 The HEE is divergent near the boundary at v → 0. We thus i ¼ 1; d; ð3:5Þ need to regulate the HEE by putting a small cut-off ϵ ≪ 1. After the regulation, the HEE can be obtained as A A the size a and the HEE SEE of the entangled region can be calculated as ld L d−1 L d a SA ¼ AdS − ; ð3:11Þ Z EE ðdþ2Þ ϵ 2 1 ˜2 −1=2 2ðd − 1ÞG z0 L 2 g11ðz0vÞ g ðz0vÞ − 1 N a ¼ z0 dv 2 ; ð3:6Þ 0 gzzðz0vÞ g˜ ðz0Þ Z where the divergent term is proportional to the boundary of ld Ld−1 1 A SA ∼ Ld−1 ∼ ∂A A AdS ˜ the entangled region , i.e., EE , as expected. SEE ¼ 2 dvz0gðz0vÞ 2 ðdþ Þ 0 The remaining term is finite. GN In the presence of the geometric boundaries QL;R, the 2 −1=2 g11ðz0vÞ g˜ ðz0Þ E Q 1 − minimal surface A could anchor on L;R in addition to the × 2 ; ð3:7Þ gzzðz0vÞ g˜ ðz0vÞ conformal boundary M. In this work, we only consider the entangled region A being a simple connected region. where v ¼ z=z0 and Under this consideration, there are three types of the vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u minimal surfaces which satisfy the homology constraint, uYd as shown in Fig. 3. If the minimal surface EA is connected ˜ t gðzÞ¼ giiðzÞ; ð3:8Þ and only anchors on the conformal boundary M, the i¼1 entanglement wedge RA takes the shape of the sunset as E and L is the length of the directions in which the translation shown in Fig. 3(a). If the minimal surface A is discon- invariance is preserved, nected and each part anchors on the different geometric Q M Z boundaries L;R in addition to the conformal boundary , R −1 −1 the entanglement wedge A takes the shape of the sky as dd x ¼ Ld : ð3:9Þ E Rd−1 shown in Fig. 3(b). Finally, if the minimal surface A is disconnected and both part anchor on the same geometric Using Eqs. (3.6) and (3.7), the HEE can be solved in term boundary in addition to M, the entanglement wedge RA A of the size a as SEEðaÞ in principle. takes the shape of the rainbow as shown in Fig. 3(c).
FIG. 3. The minimal surfaces in the pure AdS bulk spacetime. (a) the entanglement wedge RA has the shape of the sunset when the entangled region A is very small. (b) the entanglement wedge RA has the shape of the sky when the entangled region A is very large. (c) the entanglement wedge RA has the shape of the rainbow when the entangled region A is very closed to the boundary.
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The HEE corresponding to the different minimal surfaces boundary Q. When jxj is beyond the critical points at ( xc, can be calculated as ac), the sky phase disappears, the sunset phase and the A A rainbow phase compete until x reaches the boundaries Ssunset ¼ SEEðaÞ; ð3:12Þ at x ¼ l=2. In the pure AdS case, it is easy to see that, for a entangled 1 1 c SA SA l − a 2 x SA l − a − 2 x ; region A, and its complementary A , the associated sky ¼ 2 EEð þ j jÞ þ 2 EEð j jÞ minimal surfaces EA and EAc have the same homology. ð3:13Þ Therefore, A and Ac share the same minimal surface E E A Ac 1 1 A ¼ Ac , as well as the same HEE SEE ¼ SEE. SA SA l a − 2 x SA l − a − 2 x : rainbow ¼ 2 EEð þ j jÞ þ 2 EEð j jÞ B. Schwarzschild-AdS black hole ð3:14Þ We next consider the bulk spacetime N as a (d þ 2)- Although each of the HEE in the above three cases is the dimensional Schwarzschild-AdS spacetime with the metric local minimum, the global minimum depends on the size a (2.12), and choose the geometric boundary Q as a (d þ 1)- and the location x of the entangled region A. dimensional hypersurface embeded in N with the metric For a small enough A, the entanglement wedge RA (2.15). This is dual to BQFT at the finite temperature. The takes the shape of the sunset. While for a large enough A, temperature in BQFT is identified with the Hawking R the entanglement wedge A takes the shape of the sky. temperature of the black hole by the holographic corre- A If the location of the region is very close to one of spondence. The temperature and the entropy density of the Q E the geometric boundaries L;R, the minimal surface A black hole were given in Eq. (2.14). would be inclined to the boundary and break into two In the case of AdS black hole spacetime, the size a of parts, the entanglement wedge RA will take the shape of the entangled region A can be expressed as the following the rainbow as shown in Fig. 3(c). integral The HEE transfers among the three phases as the size a Z 1 d and the location x of the entangled region A varying, which v dv a ¼ 2z0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð3:15Þ correspond to the quantum phase transitions at the zero 0 ð1 − ðbvÞdþ1Þð1 − v2dÞ temperature in the dual BQFT. The phase diagram is shown where we have defined the parameter b ¼ z0=z , which in Fig. 4 (We set l ¼ 1=2 in the figures of this paper). At the H measures how close the minimal surface EA is from the middle, x ¼ 0, there is a critical value a0 for the size of the horizon. entangled region A.Fora
106016-6 HOLOGRAPHIC ENTANGLEMENT ENTROPY IN BOUNDARY … PHYS. REV. D 98, 106016 (2018) which is exact the same as in the pure AdS case as it should The HEE corresponding to the minimal surfaces EAc can be be. While for a finite size a, the HEE in the black hole case calculated as dramatically deviates from that in the pure AdS case. Ac A In the case of the black hole spacetime, the associated SR−sunset ¼ SEEðaÞþSBH; ð3:21Þ minimal surfaces EA and EAc for the entangled region A c and its complementary A have different homology due to c 1 1 SA SA l − a 2 x SA l − a − 2 x ; A R−sky ¼ 2 EEð þ j jÞ þ 2 EEð j jÞ the presence of the black hole horizon, so that the HEE SEE A Ac for a region is generically not the same as the HEE SEE ð3:22Þ for its complementary Ac. This is the crucial difference 1 1 between the cases of the AdS black hole and the pure AdS Ac A A S − ¼ S ðl þ a − 2jxjÞ þ S ðl − a −2jxjÞ þ S : spacetime. R rainbow 2 EE 2 EE BH As in the pure AdS case, there are three types of the ð3:23Þ minimal surfaces depending on the size a and the location x c of the region A, and similarly for its complementary Ac,as For a small enough A, hence the large enough A , the shown in Fig. 5. The HEE corresponding to the different minimal surface EA only anchors on ∂N , and the entan- R minimal surfaces EA can be calculated as glement wedge A takes the shape of the sunset, similar to the case of the pure AdS; while the minimal surface EAc A A includes both EA and the black hole horizon, and the Ssunset ¼ SEEðaÞ; ð3:18Þ entanglement wedge RAc takes the shape of the reversed- sunset, as shown in Figs. 5(a) and 5(d). For a large enough 1 1 c A A − 2 A − − 2 A, hence the small enough A , the minimal surface EA S ¼ S ðl a þ jxjÞ þ S ðl a jxjÞ þ SBH; sky 2 EE 2 EE breaks into two parts plus the horizon and the entanglement ð3:19Þ wedge RA takes the shape of the sky; while the minimal surface EAc is EA minus the horizon, and the entanglement 1 1 wedge RAc takes the shape of the reversed-sky, as shown SA SA l a − 2 x SA l − a − 2 x : rainbow ¼ 2 EEð þ j jÞ þ 2 EEð j jÞ in Figs. 5(b) and 5(e). When the location of the entangled region A is very close to one of the geometric boundaries, : ð3 20Þ the minimal surface EA would be inclined to the boundary
FIG. 5. The minimal surfaces in the asymptotically AdS black hole bulk spacetime. The thick black line at the top indicates the black hole horizon. (a/d) the entanglement wedge RA=RAc has the shape of the sunset/reversed-sunset when the entangled region A is very small. (b/e) the entanglement wedge RA=RAc has the shape of the sky/reversed-sky when the entangled region A is very large. (c/f) the entanglement wedge RA=RAc has the shape of the rainbow/reversed-rainbow when the entangled region A is very closed to the boundary.
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The HEE transfers among the different phases corre- sponding to the phase transitions at the finite temperature in the dual BQFT. These phase transitions are the mixture of the quantum phase transition and the thermal phase transition. Figure 6 shows the phase diagram of the HEE for the entangled region A at the horizon zH ¼ 1.5, i.e., at the temperature T ¼ 0.212 in the dual BQFT. At the middle, x ¼ 0, there is a critical value a0 for the size of the region A.Fora
FIG. 7. The phase diagrams of the holographic entanglement entropy in the Schwarzschild-AdS black hole with different horizons. i.e., different temperatures. The critical points ( xt, at) and ( xc, ac) change with the temperatures. (a, b, c, d) The holographic A A Ac entanglement entropy SEE for the entangled region . (e, f, g, h) The holographic entanglement entropy SEE for its supplementary region Ac A Ac . For the large horizon or low temperature, the phase diagrams of SEE and SEE approach to each other and become the same as the phase diagram in the pure AdS bulk spacetime.
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FIG. 8. Phase Diagram of the holographic entanglement en- Ac A tropy SEE for a entangled region in the Schwarzschild-AdS blackhole bulk spacetime with the horizon zH ¼ 1.5. FIG. 9. The phase diagrams of the holographic entanglement A Ac A entropy SEE and SEE for the entangled region and its A Ac HEE SEE at the different temperatures. In the low- complementary are combined together. There are five zones, temperature limit, i.e., the large zH, the phase diagram in marked with different colors, that are associated to the different the black hole case is asymptotic to the phase diagram phases. The vertical lines represent the different phase transition in the pure AdS case as expected. While in the high- tracks at the different location x ¼ 0, 0.13, 0.23, 0.30, 0.40. temperature limit, i.e., the small zH, both the sky and the rainbow phases shrink to zero, only the sunset phase survives. However, the critical points ( x˜t, a˜ t) and ( x˜c, a˜ c) for The behaviors of the HEE in these two limits are easy to Ac shift with the temperature in the opposite way of understand from the viewpoint of the holographic principle. ( xt, at) and ( xc, ac) by a similar argument for A.In In the low-temperature limit, the horizon is far away from the low-temperature limit with zH → ∞, the system is the the conformal boundary M at z ¼ 0 where the BQFT lives, same as that in the pure AdS spacetime. While in the high- so that the horizon hardly affects the shape of the minimal temperature limit with zH → 0, only the reversed-sky phase surface. In addition, the Bekenstein-Hawking entropy den- exists. The critical points and the phase boundaries in the sity in Eq. (2.14) approaches to zero in this limit and can be Ac phase diagrams of the HEE SEE at the different temper- neglected. Therefore, the system in the low-temperature atures are shown in Figs. 7(e)–7(h). limit is the same as that in the pure AdS spacetime. On the other hand, in the high-temperature limit, the C. Entanglement plateau horizon is very close to the conformal boundary so that the rainbow phase is impossible. In addition, the Bekenstein- It was conjectured that the HEE satisfies the Araki-Lieb Hawking entropy density is divergent in this limit so inequality that the sky phase, which includes the horizon, is Δ A A − Ac ≤ disfavored. Therefore, only the sunset phase exists in the j SEEj¼jSEE SEEj SBH; ð3:24Þ high-temperature limit. in the holographic BQFT. To show that, we explore the The HEE of Ac has the similar behavior with the critical HEE for A and Ac in more details by plotting their phase points ( x˜ , a˜ ) and ( x˜ , a˜ ). The phase diagram for the t t c c diagrams together in the Fig. 9. In the phase diagram, there HEE of Ac is shown in Fig. 8. At the middle x 0, there is ¼ are five zones as marked in the plot. The associated phase in a critical value a˜ 0.Fora
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FIG. 10. (a) The three-dimensional plot of the holographic entanglement plateau vs the location x and the size a of the entangled region A. The different phase transition tracks at the different location x ¼ 0, 0.13, 0.23, 0.30, 0.40 are outlined and plotted in (b).
Δ A We see that j SEEj¼SBH for the zones I, II and V. This in both cases. For the pure AdS spacetime, we found three induces the well-known entanglement plateau. ΔSA at phases depending on the size and the location of the EE A different location x is plotted in Fig. 10(b).Forjxj ≤ xt, entangled region . We obtained the phase diagram of Δ A the HEE in the holographic BQFT. It is easy to see that the by increasing the size a, SEE goes through the zones I-III-V A A Ac and plots the typical entanglement plateau. For xt ≤ jxj ≤ xc, HEE SEE for a region is always the same as the HEE SEE by increasing the size a, ΔSA goes through the zones I-III– for its complementary Ac. For the Schwarzschild-AdS EE A IV-V and plots the plateau with a defected corner. For black hole spacetime, we found that the HEE SEE is ≤ ≤ ˜ Δ A Ac xc jxj xt, by increasing the size a, SEE goes through generically not the same as the HEE SEE due to the the zones I-III–IV with the upper plateau disappearing. homology constraint. Three new phases were found for ˜ ≤ ≤ ˜ Δ A Ac For xt jxj xc, by increasing the size a, SEE goes the region . We obtained the phase diagrams of the HEE A Ac through the zones I-II-IV. Finally, for x˜c ≤ jxj ≤ l=2,by for both and and showed that both of them are Δ A asymptotic to that in the pure AdS case in the low- increasing the size a, SEE goes through the zones I-II − temperature limit as expected. and always takes the constant value SBH. The three- dimensional graph of the entanglement plateau is plotted Furthermore, we verified the Araki-Lieb inequality Δ A A − Ac ≤ in Fig. 10(a). j SEEj¼jSEE SEEj SBH and obtained the entangle- ment plateau by combining the phase diagrams of the IV. SUMMARY HEE for both A and Ac together. We plotted the three- dimensional entanglement plateau vs the size a and the In this paper, we studied the HEE in a (d 1)- þ location x of the entangled region A. dimensional holographic BQFT. We considered two simple solutions for the geometric boundary Q embedded in the (d þ 2)-dimensional bulk manifolds in the holographic ACKNOWLEDGMENTS BQFT. The AdS 2 bulk manifold corresponds to BQFT dþ We would like to thank Chong-Sun Chu and Rong-Xin at the zero temperature, and the (d þ 2)-dimensional Miao for useful discussions. This work is supported by the Schwarzschild-AdS black hole bulk manifold corresponds Ministry of Science and Technology (MOST 106-2112-M- to BQFT at the finite temperature. 009 -005 -MY3) and in part by National Center for We generalized the Ryu and Takayanagi formula by Theoretical Science (NCTS), Taiwan. including the geometric boundaries and calculated the HEE
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