Quantum Extremal Islands Made Easy, Part II: Black Holes on the Brane
Hong Zhe Chen, Robert C. Myers, Dominik Neuenfeld, Ignacio A. Reyes, Joshua Sandor
PPrepared for submission to JHEP
Quantum Extremal Islands Made Easy, Part II:
Black Holes on the Brane
Hong Zhe Chen, a,b
Robert C. Myers, a Dominik Neuenfeld, a Ignacio A. Reyes c and Joshua Sandor aa Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada b Dept. of Physics & Astronomy, University of Waterloo, Waterloo, ON N2L 3G1, Canada c Max-Planck-Institut f¨ur Gravitationsphysik, Am M¨uhlenberg 1, 14476 Potsdam, Germany
E-mail: [email protected] , [email protected] , [email protected] , [email protected] , [email protected] Abstract:
We discuss holographic models of extremal and non-extremal black holesin contact with a bath in d dimensions, based on a brane world model introduced in[1]. The main benefit of our setup is that it allows for a high degree of analytic controlas compared to previous work in higher dimensions. We show that the appearanceof quantum extremal islands in those models is a consequence of the well-understoodphase transition of RT surfaces, and does not make any direct reference to ensembleaveraging. For non-extremal black holes the appearance of quantum extremal islandshas the right behaviour to avoid the information paradox in any dimension. We furthershow that for these models the calculation of the full Page curve is possible in anydimension. The calculation reduces to numerically solving two ODEs. In the case ofextremal black holes in higher dimensions, we find no quantum extremal islands for awide range of parameters. In two dimensions, our results agree with [2] at leading order;however a finite UV cutoff introduced by the brane results in subleading corrections.For example, these corrections result in the quantum extremal surfaces moving furtheroutward from the horizon, and shifting the Page transition to a slightly earlier time. a r X i v : . [ h e p - t h ] S e p ontents τ Σ = 0 193.3 No-island phase for τ Σ = 0 243.4 No-island phase for χ Σ = 0 243.5 Time-evolution for general χ Σ , τ Σ (cid:54) = 0 283.6 The information paradox 33 d > T = 0 bath 47 T = 0 for d > – 1 – Introduction
Understanding the quantum description of black holes remains a central question intheoretical physics. One unresolved question is the fate of information during blackhole evaporation. In his seminal work, Hawking argued that in a quantum theory blackholes evaporate into a mixed state of radiation, independently of how the black holewas formed [3–5]. Of course, this is in tension with the assumption that to an outsideobserver, the black hole looks like an ordinary, unitary quantum mechanical system, e.g., as suggested by the AdS/CFT correspondence [6, 7]. This tension is colloquiallyknown as the black hole information paradox [8].One way of sharpening the paradox is to consider the von Neumann entropy ofthe Hawking radiation produced during black hole evaporation. Assuming the gravi-tational system begins in a pure state, this entropy gives a measure of the amount ofentanglement between the radiation and the black hole. According to Hawking’s origi-nal calculation, the entanglement increases monotonically throughout the evaporationprocess since the radiation is thermal. On the other hand, unitary evolution wouldrequire that the thermodynamic entropy of the black hole, which is proportional toits horizon area [9–11], set an upper bound on the the entanglement entropy of theradiation. Since the former decreases as the black hole radiates, at some time – knownas the Page time – the thermodynamic entropy of the black hole will equal the entropyof the radiation, and the latter entropy must then decrease in the subsequent evolu-tion, reaching zero when the black hole has disappeared. That is, subtle correlationsbetween the quanta emitted at early and late times must produce a purification of thefinal state, in a unitary evolution of the full system. This qualitative behaviour of theradiation’s entropy as a function of time is known as the Page curve [12] – see also [7].While reconciling Hawkings calculation with the idea that quantum gravity is uni-tary was a longstanding puzzle, recently progress has made it possible to compute thePage curve in a controlled manner [13–15]. The new approach builds on insights comingfrom holographic entanglement entropy [16–19] and its extension to include quantumcontributions [20, 21]. It is best understood in a setting where a black hole is coupledto an auxiliary, non-gravitational reservoir – referred to as the bath – which capturesthe Hawking radiation. This setup can be interpreted as a idealized picture, wherewe split the spacetime into two regions: The first, in which gravity is important, isclose to the black hole while the second region is far away, where gravitational effectsare negligible, at least semi-classically. In this situation, it was argued that instead ofusing Hawking’s calculation, the true entropy of the Hawking radiation captured in a This approach has now also been applied in a variety of different situations involving black holes[22–50] and cosmology [51–58]. – 2 –egion R of the bath should be calculated using the so-called island rule [15] S EE ( R ) = min (cid:26) ext islands (cid:18) S QFT ( R ∪ islands) + A ( ∂ (islands))4 G N (cid:19)(cid:27) . (1.1)This formula instructs us to evaluate the (semiclassical) entanglement entropy of thequantum fields in the bath region R combined with any codimension-two – and possiblydisconnected – subregions in the gravitating region. The boundary of the candidate is-lands also contributes a gravitational term in the form of the usual Bekenstein-Hawkingentropy. One extremizes the right-hand side of eq. (1.1) over all such choices, and ifthe latter yields multiple extrema, the correct choice is the one that yields the smallestentropy for R . If this procedure yields a solution with a nontrivial region ‘islands’, thelatter is called a quantum extremal island – see [37] for a recent review.For an evaporating black hole, an obvious choice for the island region which ex-tremizes the entropy functional is the empty set, in which case the result of eq. (1.1)agrees with Hawking’s calculation. However, if radiation in the region R shares a largeamount of entanglement with the quantum fields behind the horizon, new quantumextremal islands can appear. In particular, this occurs for an old evaporating blackhole, and in this case a quantum extremal island appears just behind the horizon [14].It turns out that after the Page time, this configuration yields the minimal entropy ineq. (1.1). As time evolves further, the entropy of R is controlled by the horizon area ofthe black hole which enters through the second term in eq. (1.1). Hence as the blackhole evaporates, the latter shrinks to zero size and the island rule (1.1) gives a unitaryPage curve.Eq. (1.1) was motivated in part by analyzing a “doubly-holographic” model in[15]. This model provides three different descriptions of the physical phenomena: First,from the boundary perspective , the system consists of two (one-dimensional) quantummechanical systems, which are entangled in a thermofield double state. Further, oneof the quantum mechanical systems is coupled to a two-dimensional holographic CFT,which plays the role of the bath – see figure 1a. With the brane perspective , the quantummechanical systems are replaced by their holographic dual, a two-dimensional black holein JT gravity. The latter has an AdS geometry, which also supports another copy ofthe two-dimensional holographic CFT – see figure 1b. Finally, with the bulk perspective ,the holographic CFT is replaced everywhere with three-dimensional Einstein gravityin an asymptotically AdS geometry. The latter effectively has two boundaries: thestandard asymptotically AdS boundary and the region where JT gravity is supported,which is referred to as the Planck brane – see figure 1c. An advantage of workingin the bulk perspective is that entanglement entropies of subregions in the bath canbe computed geometrically using the usual rules of holographic entanglement entropy– 3 –.d. b.e. f.c. Figure 1 : Illustration of doubly-holographic models: The top row illustrates (a timeslice of) the three perspectives of the model in [15], while the bottom row displays theanalogous descriptions of our construction in higher dimensions [1]. In the latter, weare using the global conformal frame where the boundary CFT lives on R × S d − andthe conformal defect appears on the equator of the ( d − d = 2 and taking a Z quotient across the defect in the boundary or the brane in the bulk. The boundary,brane and bulk gravity perspectives correspond to panels a & d, b & e, and c & f,respectively.[16, 17, 59], taking into account that the RT surfaces that can also end on the Planckbrane [60, 61].One direction for progress is to understand the Page curve and quantum extremalislands in higher dimensions. While limited results have been obtained on this front [14,30–34], we focus here on the holographic model which we introduced in [1]. Our modelallows us to obtain analytic results, while being powerful enough to do calculationsin the regime where the gravitational theory on the brane is well-approximated byEinstein gravity. In our previous paper, we showed that quantum extremal islandscan appear in any spacetime dimension, and clarified several of the properties of thedoubly-holographic model in [15]. Here, we will extend our earlier work and discussthe presence of quantum extremal islands for black holes coupled to bath at a finitetemperature. That is, our analysis provides a higher dimensional extension of thetwo-dimensional scenario considered in [2].The key feature of our holographic model [1] is that it reproduces the three de-– 4 –criptions of the underlying physics discussed above for the doubly-holographic modelof [15]. From the boundary perspective, our system consists of a d -dimensional holo-graphic CFT coupled to codimension-one conformal defect, as shown in figure 1d. Us-ing the standard AdS/CFT dictionary, this description is translated to the bulk gravityperspective. The latter describes the system in terms of ( d + 1)-dimensional Einsteingravity in an asymptotically AdS d +1 geometry coupled to a d -dimensonal brane, whichintersects the boundary at the location of the conformal defect – see figure 1f. Accord-ing to the Randall-Sundrum (RS) scenario [62–64], the gravitational backreaction ofthe brane warps the bulk geometry creating new localized graviton modes in its vicin-ity. This mechanism allows for the brane perspective, shown in figure 1e, where thesystem is described by an effective theory of Einstein gravity coupled to (two copies of)the holographic CFT on the brane, all coupled to the boundary CFT. In [1], we alsoconsidered introducing an intrinsic Einstein term to the brane action, analogous to theconstruction of Dvali, Gabadadze and Porrati (DGP) [65].Hence our construction [1] provides a natural generalization to higher dimensionsof the two-dimensional doubly-holographic setup considered in [15]. Let us also notethat our model resembles the setup in [15] even more closely upon taking a Z orbifoldquotient across the brane. Further, we emphasize that while the three different per-spectives were presented on a more or less equal footing, the fact that the RS gravityon the brane has a finite UV cutoff [62, 63] singles out the brane prespective as aneffective low-energy description, in contrast to the boundary and bulk descriptions. Again, the bulk gravity perspective allows us to calculate entanglement entropiesof boundary regions geometrically with the usual rules of holographic entanglemententropy [16, 17, 59]. From the brane perspective then, quantum extremal islands simplyarise when the minimal RT surfaces in the bulk extend across the brane for certainconfigurations.In this case, the entanglement entropy of the corresponding boundary region R isgiven by S EE ( R ) = min (cid:26) ext Σ R (cid:18) A (Σ R )4 G bulk + A ( σ R )4 G brane (cid:19)(cid:27) (1.2)where Σ R is the usual bulk RT surface, i.e., an extremal codimension-two surface inthe bulk homologous to R . As argued in [1], when the brane supports an intrinsicgravitational action, we must also include a Bekenstein-Hawking area contribution for This does not mean that the bulk description in terms of a(n infinitely thin) brane in AdS d +1 isUV complete. However, it is reasonable to expect that the bulk description can be completed in theUV by a more complicated configuration which can be obtained within string theory, e.g., see [66–68].In contrast, the brane theory has a fundamental cutoff. – 5 –he brane region σ R = Σ R ∩ brane. This intersection of the RT surface with the branebecomes the boundary of the islands seen in the brane prespective.The equivalence between eqs. (1.1) and (1.2) can be easily understood as follows:The bulk term in eq. (1.2) describes the leading planar contributions of the entangle-ment entropy of the boundary CFT, and so matches the first term in eq. (1.1). How-ever, expanding this geometric contribution near the brane also reveals an Bekenstein-Hawking term that matches the induced Einstein term in the effective gravitationalaction on the brane [1]. This contribution combines with the brane term in eq. (1.2)to produce the expected gravitational contribution appearing in eq. (1.1). In fact, theRT contribution also captures higher derivative contributions matching the Wald-Dongentropy [69–72] of the higher curvature terms appearing in the effective gravitationalaction [1]. Further, as discussed in [1], the competition between candidate quantumextremal islands, denoted by the ‘min’ in eq. (1.1) simply becomes the usual competi-tion between different possible RT surfaces in the holographic formula (1.2), e.g., seefigure 2.In the following, we will study the question of quantum extremal islands for blackholes in arbitrary dimensions using the purely geometric description (1.2) of the bulkgravity perspective. As emphasized in [1], the transition between the phase without anisland and that with the island is nothing more than the usual transition between differ-ent classes of RT surfaces [73–75] – see figure 2. In particular, in the island phase, theRT surface crosses the brane so that a portion of the latter, i.e., the island, is includedin the corresponding entanglement wedge. Thus the appearance of quantum extremalislands is simply decribed by a well understood feature of holographic entanglemententropy in a new setting. The main advantage of our construction here and in [1] liesin its simplicity. As we will show, our framework allows us to carry the calculationsremarkably far analytically, complementing previous approaches which heavily reliedon numerics [30]. In our case, the numerics required to extract quantitative results arelimited to solving few ODEs.The remainder of this paper is organised as follows: In section 2, we review thebulk geometry and effective action of our model presented in [1], which is based onthe Karch-Randall setup [64, 76–81] for branes embedded in AdS. We also discussthe addition of a DGP term [65] to the brane action. For the two dimensional bulkgravity case, we summarize the setup of [2], describe the connection to our model andintroduce eternal black holes. In section 3, we construct eternal black holes on thebrane in higher dimensions. As in the d = 2 case, these black holes are in equilibriumwith the bath at finite temperature and so they do not evaporate. Nonetheless, there isa continuous exchange of radiation between the black hole and the bath, which has thepotential to create an information paradox [2]. Hence, we use eq. (1.2) to investigate– 6 – Σ R R Σ R σ R island Figure 2 : The choice of RT surfaces on a constant time slice in the presence of thebrane (coloured green), showing the different ingredients involved in eq. (1.2).under which conditions islands appear. We present the general analysis for the timedependence of the entropy, exploring the island and no-island phases. In section 4,we develop the numerics associated to some integral equations found in the previoussection and explicitly evaluate the Page curve for d = 3, 4 and 5. Section 5 examinesan extremal horizon with a vanishing temperature, and find that in contrast to twodimensions [2], generally islands do not form in higher dimensions. However, this isnot problematic, since at zero temperature the black hole and bath are not actuallyexchanging radiation and thus no information paradox arises. Details for the specialcase d = 2 appear in section 6. We review the induced action on the two-dimensionalbrane, including the introduction of JT gravity terms, given in [1]. We also evaluatethe corresponding quantum extremal surfaces and the Page curve, and show that thebrane cutoff produces subleading corrections compared to the results in [2]. Finally insection 7, we discuss our results and point towards some future directions. Let us review the holographic model discussed in [1]. Beginning with the bulk gravityperspective our setup is described by ( d +1)-dimensional Einstein gravity with a negativecosmological constant, I bulk = 116 πG bulk (cid:90) d d +1 x √− g (cid:20) R ( g ) + d ( d − L (cid:21) , (2.1) Throughout the paper, we ignore surface terms for the gravitational action, e.g., see [82–84]. – 7 –here g ab denotes the bulk metric. We also introduce a codimension-one brane in thebulk with action I brane = − ( T o − ∆ T ) (cid:90) d d x (cid:112) − ˜ g + 116 πG brane (cid:90) d d x (cid:112) − ˜ g ˜ R (˜ g ) , (2.2)where ˜ g ij is the induced metric on the brane. As well as the usual tension term,we have also introduced an intrinsic Einstein-Hilbert term in the brane action, in amanner analogous to Dvali-Gabadadze-Porrati (DGP) braneworld gravity [65]. Wehave separated the brane tension into T o and ∆ T , and will tune ∆ T ∝ /G brane so thatthe brane position is determined entirely by T o . Adding the DGP term is a naturalgeneralisation to higher dimensions of having JT gravity on a two-dimensional brane[15] – see section 6.Since the brane is codimension-one, the bulk geometry away from the brane locallytakes the form of AdS d +1 with the curvature scale set by L . We will work in a regimewhere the induced geometry on the brane will be that of AdS d space – see [1] for details– and so it is useful to consider the following foliation of the AdS d +1 geometry by AdS d slices ds d +1 = L sin θ (cid:0) dθ + ds d (cid:1) . (2.3)The AdS d metric is dimensionless with unit curvature. This metric would cover theentire AdS d +1 vacuum spacetime if we take 0 ≤ θ ≤ π . The solution for the backreactingbrane is constructed by first cutting off the spacetime along an AdS d slice near theasymptotic boundary θ = 0, i.e., at θ = θ B (cid:28) θ B is determined by thebrane tension T o – see below. Then, two such spaces are joined together along thissurface, and the brane is realized as the interface between the two geometries. Withthis construction, the brane divides the bulk spacetime in half, but the backreaction ofthe brane has enlarged the geometry – see figure 3. In this case, the metric (2.3) canbe used to cover a coordinate patch with θ B ≤ θ ≤ π on either side of the brane.With the above construction, the induced geometry on the brane is simply AdS d and using the Israel junction conditions [1, 85], one finds the curvature scale to be1 (cid:96) B = sin θ B L = 2 L ε (1 − ε/ , where ε ≡ (cid:18) − πG bulk LT o d − (cid:19) . (2.4)For the most part, we will be interested in the regime where L /(cid:96) B (cid:28) ε (cid:28)
1. As wewill explain below, this ensures that the gravitational theory on the brane is essentiallyEinstein gravity. Implicitly in eq. (2.4), we have tuned the “shift” ∆ T to produce anembedding of the brane that is independent of the DGP coupling G brane , i.e., the branelocation remains unchanged when we vary G brane . This is achieved by setting∆ T = ( d − d − πG brane (cid:96) B . (2.5)– 8 –dS d +1 AdS d +1 AdS d +1 AdS d CFT d a. b. Figure 3 : A timeslice of our Randall-Sundrum setup. In panel (a), we cut off theAdS d +1 spacetime along an AdS d slice near the asymptotic boundary θ = 0, in themetric (2.3). Two of these spaces are glued together in panel (b) and the brane isrealized as the interface between the two geomeries.The boundary perspective simply considers the dual description of the above grav-itational system using the standard rules of the AdS/CFT correspondence. As de-scribed in [1], when considered in “global” coordinates, the dual solution is naturallythe boundary CFT on a spherical cylinder R × S d − (where the R is the time direction).Further there is also a codimension-one conformal defect positioned on the equator ofthe sphere, where the brane reaches the asymptotic boundary. The central charge ofthe boundary CFT is given by the standard expression c T ∼ L d − /G bulk , e.g., see [86],whereas the ( d − c T ∼ (cid:96) d − eff /G eff (cid:29) c T .Similarly, one can consider the ratio of the couplings in the defect and bath CFTs:˜ λ/λ ∼ (cid:96) eff /L (cid:29) brane perspective by replacing the conformal defect in the boundaryperspective by its gravitational dual. Hence this description includes the boundary CFTon the asymptotic AdS d +1 boundary, but also two copies of the boundary CFT on thebrane, as dictated by the usual Randall-Sundrum (RS) scenario. Of course, the latteris an effective theory with a finite UV cutoff set by the position of the brane, e.g., see[87] and references therein. Further, new (nearly) massless graviton modes localizedin the vicinty of the brane also appear and so the brane also supports a gravitationaltheory. We can think that integrating out the brane CFT (or the bulk gravity) induces In fact, working with the induced metric on the brane (as we do in the following), the short-distancecutoff on the brane is ˜ δ (cid:39) L – see [1] for further details. – 9 –n effective gravitational action on the brane of the form [1] I induced = 116 πG eff (cid:90) d d x (cid:112) − ˜ g (cid:20) ( d − d − (cid:96) eff + ˜ R (˜ g ) (cid:21) (2.6)+ 116 πG RS (cid:90) d d x (cid:112) − ˜ g (cid:20) L ( d − d − (cid:18) ˜ R ij ˜ R ij − d d −
1) ˜ R (cid:19) + · · · (cid:21) , where 1 G eff ≡ G RS (1 + λ b ) with λ b ≡ G RS G brane , G RS = 2 L ( d − G bulk , (cid:96) eff = 2 L ε , (2.7)and ε is given in eq. (2.4). Note that in the regime of interest ( i.e., ε (cid:28) (cid:96) eff (cid:39) (cid:96) B . Hence to leading order, the above gravitational theory (2.6) corresponds toEinstein gravity coupled to a negative cosmological constant. In the second line ofeq. (2.6), we show the first of a(n infinite) sequence of higher curvature corrections,involving powers of L × curvature. Since the gravitational equations of motion set thecurvatures to be roughly 1 /(cid:96) eff (at least for the background of interest), the contributionof these terms is highly suppressed since we work in the regime where L /(cid:96) eff (cid:28) Lastly, let us note that 1 /G RS is the standard RS gravitational coupling induced in theabsence of a DGP term, i.e., λ b = 0.It turns out that in the case of a brane theory with negative cosmological constant,like the one we are considering here, the graviton acquires a mass [64, 76–81]. Forsmall brane angles, the graviton mass is proportional so some power of the brane angle[80, 81] and thus vanishes as we take the zero-angle limit. It was suggested in [33] thatthis mass is a crucial ingredient for islands to exist, since the limit of vanishing gravitonmass coincides with a limit in which islands cannot be created since their area becomesinfinite. Alternatively, it is possible that in the Karch-Randall model, the gravitonmass simply depends on the effective gravitational coupling on the brane, and is thuscorrelated with the island size, but not responsible for the island. In two dimensions, we need to revisit our setup for an accurate effective brane actionand to make connection to [2]. First, there are factors of 1 / ( d −
2) appearing in A more careful examination in [1] showed that the gravitational theory on the brane was wellapproximated as semiclassical Einstein gravity with L /(cid:96) eff (cid:28) λ b >
0, but required L /(cid:96) eff (cid:28) λ b for λ b <
0. However, the latter constraint is replaced by L /(cid:96) eff (cid:28) (1 + λ b ) for the specialcase of λ b < d = 3. – 10 –efectRindler Left Rindler Right τ = t = 0 Figure 4 : Our eternal black hole coupled to the CFT bath, as seen from the bulkperspective .eq. (2.6), which indicate that the bulk integration analysis leading to this result mustbe reconsidered for d = 2. As reviewed in section 6, we find that the induced braneaction is non-local, a signature of the trace anomaly. In addition, the two-dimensionalanalogue of the DGP brane action is a JT gravitational action localized on the brane.Having accounted for these changes, we may relate our setup directly to that of [2],which we now briefly review.Ref. [2] interprets the two Rindler patches of AdS as exteriors of an eternal non-zero temperature black hole and subsequently considers coupling each exterior to a flathalf-space, consitituting a bath region. A matter CFT theory spans both the bathand AdS regions and JT gravity is placed on the AdS region. Invoking AdS / CFT ,this setup is alternatively described by the thermofield double (TFD) state of a BCFTliving on two half-lines (the bath regions) coupled to quantum mechanics (dual tothe AdS spacetime) on the boundaries of the half-lines. The authors then computethe entanglement entropy of a region consisting of intervals on both sides of the TFDincluding the defect and with endpoints in the bath regions. From the AdS perspective,this entropy is obtained using eq. (1.1), allowing for the possibility of islands in theAdS spacetime. In particular, this gives rise to a competition between a no-islandphase and an island phase, with the former dominating at early times and the latter atlate times. In the island phase, quantum extremal surfaces (QESs) appear in the AdS spacetime just outside the horizon, marking the boundaries of an island, stretching Similar factors of 1 / ( d −
4) appear and are problematic in d = 4, but we work in a regime suchthat the curvature squared terms of eq. (2.6) are irrelevant. – 11 – = πR defect Figure 5 : The Euclidean path integral (orange region) prepares the Hawking-Hartlestate. The black hole temperature T = 1 / (2 πR ) is derived in section 3.through the AdS wormhole, which now belongs to the entanglement wedge of thebath complements to the intervals.Let us return to our braneworld to see how our setup mimics that of [2] describedabove. From the bulk perspective, we have an AdS spacetime with a brane lyingalong an AdS slice (fig. 4). We may reproduce the AdS black hole on the brane bytaking Rindler-AdS coordinates in the AdS bulk — this equips the AdS bulk with ahorizon and ‘left’ and ‘right’ exterior regions. The resulting picture is that of a Hartle-Hawking state prepared by the Euclidean path integral drawn in fig. 5. The RinderAdS coordinates also induce a horizon on the brane. In fact, the geometry of the braneis itself Rindler-AdS, ds = (cid:96) B (cid:20) − ( ρ − dτ + d ˜ ρ ρ − (cid:21) , (2.8)supporting a dilaton profile Φ ∝ ρ . In the brane perspective, we then have a CFTspanning the left and right asymptotic boundary regions – the baths – and the Rindler-AdS brane, which also supports a theory of JT gravity. Illustrated in figure 6, thisis essentially the same setup as in [2], up to a Z -quotient across the brane. We mayalternatively take the boundary perspective, wherein the bulk AdS plus brane theoryis dual to a CFT plus defect theory. More precisely, the Euclidean path integralpreparing the Hartle-Hawking bulk is equated to a thermal path integral preparing aTFD state of two copies of a CFT with a defect running through its middle. We arethus led to the boundary picture drawn in figure 7. Taking a Z quotient across thedefect, this, of course, is the alternative description of the setup in [2] as a thermalBCFT coupled to quantum mechanics. – 12 –ith our setup in place, we can then consider subregions of the boundary CFTand use the RT formula (1.2) to compute the corresponding entanglement entropies.Analogous to [2], we choose ‘belt’ subregions consisting of intervals symmetric about thedefect. The details of the resulting entropy calculation in two dimensions are providedin section 6. The upshot is that we find a competition between a no-island phase andan island phase, as sketched in fig. 2, with the former dominating at early times andthe latter past a Page time. Notice that these phases are analogous to the no-islandand island phases of [2], with now the QESs demarked by the intersection between ourbulk RT surface and the brane. Namely, it is clear from the bulk picture shown in theright panel of fig. 2 that the island region between these intersection points belongs tothe entanglement wedge of the bath region complementary to the belt.In section 6, we also explicitly demonstrate that our bulk RT calculation usingeq. (1.2) precisely reproduces the results of [2], in the limit where the brane approachesthe would-be AdS boundary by slicing through the bulk at a small brane angle θ B (that is, the high-tension limit of higher dimensions). For early times, we find thatthe entanglement entropy grows linearly in the no-island phase as 4 πct/ (3 β ) (seeeq. (6.38)), whereas for late times it is dominated by the island and given by a constant, G brane (cid:16) ˜Φ + Φ r (cid:17) (see eq. (6.51)). Thus, as in [2], the appearance of an island caps offthe entropy growth at the expected course-grained entropy of two copies of the blackhole on the brane, rescuing the system from a potential information paradox (theresulting Page curve is shown in fig. 24). While we find perfect agreement with [2] atleading order in θ B , we also find corrections to these results due to the brane imposing aUV cutoff at finite θ B . The result is O ( θ B ) corrections which, for instance, push the QESfurther from the horizon, lower the entropy of the island phase, and lead to a hastenedPage transition. (Note that, in the no-island phase, no such corrections appear as thebulk RT surface does not intersect the brane.)It would be straightforward to use our setup to perform the zero-temperature anal-ysis also covered in [2] for d = 2. Here one would instead take Poincar´e coordinateswhich would equip the AdS bulk and AdS brane with an extremal horizon. We thenexpect entanglement entropy of large regions in the bath to require the inclusion of is-lands on the gravitating brane. In particular, intervals stretching from some location inthe bath out to infinity require the inclusion of an island localized around the horizon.(This is to be contrasted with our findings in d ≥
3, where islands are lacking in theextremal case at small brane angle θ B .)The benefit of our Randall-Sundrum setup is that it allows great flexibility in Recall our setup is related to that of [2] by a Z -orbifold, hence factors of 2 must be accountedfor when comparing results. – 13 –FT R CFT L τ = t = 0horizondefect Figure 6 : Our eternal black hole coupled to the CFT bath, as seen from the effective brane perspective . Each point in the Penrose diagram represents a hyperbolic space H d − . For d = 2 this is simply a point. defectCFT L CFT R Figure 7 : Conformal defects along a CFT bath in the boundary perspective .generalizing the construction of [2] to higher dimensions. Indeed, it is straigtforwardto re-interpret figures 4, 6, and 7 with a suppressed hyperbolic H d − direction. In thefollowing sections, we shall apply our setup to extend the results mentioned here tohigher dimensions. In this section, we discuss how islands arise in the presence of certain topological,non-extremal black holes in higher-dimensional brane-world models. Topological blackholes are characterized as having nontrivial horizon topology, and we will be interestedin the case of neutral black holes with a hyperbolic horizon [88, 89]. The general metricis given by ds = − f ( r ) L R dt + dr f ( r ) + r dH d − , (3.1)– 14 –ith the blackening factor f ( r ) = r L − − ω d − r d − . (3.2)Here, L denotes the AdS curvature scale and dH d − denotes the line element on a( d − H d − . After an appropriate Weyl rescaling,the boundary metric for each CFT reads ds CFT = − dt + R dH d − , (3.3)and hence the scale R (introduced in eq. (3.1)) corresponds to the curvature scale of thespatial geometry. The full boundary geometry is then two copies of R × H d − , wherethe R corresponds to the time direction in each of the CFTs.Turning back to eq. (3.1), the relation between the position of the horizon r h , theblack hole mass M , and the ‘mass’ parameter ω is [84, 91, 92] ω d − = r d − h (cid:18) r h L − (cid:19) = 16 π G N ( d −
1) vol H d − RL M. (3.4)Here and in the following, we use vol H d − to denote the dimensionless volume of thespatial boundary geometry, i.e., the volume measured by the metric dH d − . Of course,this volume is infinite and we must introduce an infrared regulator – see below.In the following, we will consider the special case of a topological black hole withvanishing mass M = ω = 0. Note that despite the fact that ω = 0, we still find ahorizon at r h = L from eq. (3.2). In fact, the bulk geometry corresponds to the AdSvacuum (as expected for M = 0), but we are describing this geometry with the AdS-Rindler coordinates where the metric resembles that of black hole [93]. In this case, itis straightforward to evaluate the entropy and the temperature of the black hole S = vol H d − L d − G N , T = 12 πR . (3.5)In terms of the dual CFT, we are considering a pure state ( i.e., the vacuum) in theconformal frame where the boundary geometry corresponds to R × S d − . However, withan appropriate conformal transformation, we produce the TFD state on two copies of R × H d − with temperature T = 1 / (2 πR ) [93]. The entropy in eq. (3.5) correspondsto the entanglement entropy between the two copies of the CFT – and alternatively,can be interpreted as the entanglement entropy between two halves of the sphere in– 15 –he original conformal frame. From the point of view of the CFT, masslessness of theblack hole corresponds to a fine tuning of the temperature to T = πR .Following the brane world construction outlined in the previous section, we locatea codimension-one defect at the center of each CFT. By the holographic dictionary,this corresponds to a brane which cuts through the bulk and orthogonally intersects thehorizon – see figure 4. Since with ω = 0 , the bulk geometry is just the AdS vacuum, ourprevious discussion of the brane geometry (above and in [1]) is still applicable. Hence,the brane position in the bulk is determined precisely as described above in termsof the brane tension T o . In fact, this bulk geometry provides a higher dimensionalgeneralization of the construction discussed in section 2.2, and we will see that thebrane inherits a black hole metric with temperature T = 1 / (2 πR ), from the AdS-Rindler coordinates in the bulk.Our aim will be to use eq. (1.2) to investigate the appearance of quantum extremalislands, from the brane perspective, where (two copies of) the boundary CFT aresupported in this black hole geometry on the brane. Further, we will compute theentanglement entropy associated to symmetric regions R on each side of the defect asa function of time – see figure 2. The regions R of interest consist of those pointson a CFT timeslice which are further than a distance χ = χ Σ away form the defect. The entanglement entropy is evaluated using the holographic prescription of the bulkperspective and as described in the introduction, the corresponding RT surfaces can bein one of two phases. Either they connect through the horizon, which we will call theno-island phase, or they connect through the brane, which we will call the island phase.The reason for those names is apparent from the d -dimensional effective gravity on thebrane, i.e., the region bounded by the intersection of the RT surface and the braneis a quantum entremal island, which now contributes to the entropy of R . This alsoimplies that from the ( d + 1)-dimensional bulk perspective, the appearance of islandsis simply explained as a standard phase transition of an RT surface. We will see in theremainder of this section that at early times, the RT surfaces starts out in the no-islandphase, i.e., connects throught the horizon. As is well known [94], the volume of thecorresponding surfaces grows linearly with time. At some point its volume will havegrown so large, that the RT surface in the island phase has smaller area and gives thecorrect entanglement entropy.The calculation of the time-dependence of the area of RT surfaces will proceedin two steps: In sections 3.2 to 3.4, we will derive expressions for the area of threespecial cases of extremal surfaces. The first one will be RT surfaces in the island phase The coordinate χ is introduced in eq. (3.7) below. Of course, since the global state which weare considering is pure, we could equivalently discuss the entanglement entropy of the belt regions − χ Σ < χ < χ Σ in both CFTs, including the conformal defects. – 16 –nchored at Rindler time τ Σ = 0. The second and third special cases will be RT surfacesΣ R in the no-island phase which either end on entangling surfaces ∂ R at χ = ± χ Σ and τ Σ = 0, or end on entangling surfaces located at the defect ( χ Σ = 0) and arbitrary τ Σ . While these special cases naively might seem not to contain enough information tocompletely reconstruct the time-evolution of the entanglement entropy, we will arguein section 3.5 that the time-evolution of any symmetric RT surface in the no-islandphase can always be reduced to one of those three cases.We remind the reader that as described in section 2.2, we are considering eternalblack holes which do not evaporate. Nonetheless, from the effective brane point ofview, the black hole on the brane and the fields on the asymptotic boundary are incontact, and can therefore continuously exchange radiation. If island are not accountedfor appropriately, this leads to information loss [15]. In section 3.6 we will argue, usingresults obtained below, that also in higher dimensions the presence of islands makesthe entanglement dynamics of the joint system of black holes and radiation compatiblewith unitarity. To set the stage for the following calculations, we will start by discussing the bulkand brane geometry. As noted above, the bulk metric is described by AdS-Rindlercoordinates ds = L (cid:18) − ( r − dτ + dr r − r dH d − (cid:19) , (3.6)which is obtained from eq. (3.1) by taking the massless limit ω, M → t → R τ and r → L r , such that the coordinates in eq. (3.6) are dimen-sionless. Although the underlying geometry is simply the AdS vacuum, the metric (3.6)resembles a black hole metric with horizons at r = ± r = 0. We can also extend the spacetime at a fixed time-slice through the bifurcationsurface and arrive at a second Rindler wedge. The bulk spacetime thus has two asymp-totic regions, located at r → ∞ , each of which hosts one copy of the boundary CFTon the R × H d − geometry. As noted above (in terms of the dimensionful coordinates),the corresponding TFD state has a (dimensionful) temperature T = 1 / (2 πR ), which istuned in relation to the curvature scale R of the hyperbolic geometries (3.3). Lastly,note that since the Rindler wedges are simply a reparametrization of pure AdS, it isclear that the singularity at r = 0 is only a coordinate singularity. In fact, we canextend the coordinates smoothly through the interior to negative r where we can exitthe region behind the (inner) horizon at r = − This is in contrast to the general metric (3.1) where r → – 17 –or each CFT, we introduce a codimension-one conformal defect (with zero ex-trinsic curvature) at the center of the hyperbolic spatial geometry. It is convenient tochoose slicing coordinates for the hyperbolic boundaries, such that dH d − = dχ + cosh χ dH d − . (3.7)In these coordinates, the location of the conformal defect is χ = 0.From the bulk perspective, the CFT defects are dual to a co-dimension one brane,which spans a slice of constant extrinsic curvature of the bulk spacetime and intersectsthe asymptotic boundary at the location of the CFT defect. In order to describe itstrajectory, it is convenient to write the bulk metric in terms of the slicing coordinates ineq. (2.3). The brane is located at constant θ = θ B , which is determined by the tension T o through eq. (2.4) with (cid:96) B = L sin θ B . (3.8)The trajectory of a hypersurfaces of constant θ B in the bulk spacetime is then given by r sinh χ = cot θ B = (cid:18) (cid:96) B L (cid:19) − . (3.9)As noted in [1], this means that a brane with positive tension ( i.e., T o ≥
0) createsadditional geometry by its backreaction. Of course, the backreaction of a negative-tension brane would remove geometry. However, let us add that there is no (nearly)massless graviton induced on a negative-tension brane and therefore we will onlyconsider positive tensions in the following, i.e., ≤ θ B ≤ π .For such a (positive-tension) brane, the bulk geometry to one side of the brane canbe described by eq. (3.6), with r sinh χ ≤ cot θ B , while the geometry to the other side ofthe brane is given by the same metric with r sinh χ ≥ − cot θ B . We can therefore treateither side of the brane as an AdS-Rindler geometry which is cut off by the brane.Using eq. (3.9), we can determine the induced metric on the brane. After a shortcalculation, one finds ds = (cid:96) B (cid:18) − ( ρ − dτ + dρ ρ − ρ dH d − (cid:19) , (3.10)where we have changed the radial coordinate with (cid:96) B ( ρ −
1) = L ( r − . (3.11) We thank Raman Sundrum for explaining this point to us. – 18 –his brane metric again takes the form of an AdS-Rindler metric, c.f. eq. (3.6). Fur-ther, this demonstrates that the Rindler horizon in the bulk (at r = 1) induces aRindler horizon on the brane (at ρ = 1), as one would expect from the bulk per-spective. From the boundary perspective, this behavior is readily explained by thefact that the conformal defect is in thermal equilibrium with the surrounding CFT.In the effective Randall-Sundrum description of the brane perspective, this behaviourarises because the region of dynamical gravity is coupled to the bath CFT along anaccelerated trajectory, so that the temperature felt by the accelerated boundary agreeswith the temperature of the CFT, e.g., see [95–97]. As already mentioned, this setupgeneralizes the two-dimensional framework presented in [2] to higher dimensions.All calculations below will be done for the case of positive tension branes. However,when it comes to interpretation, we will be particularly interested in the case where1 (cid:29) θ B (cid:39) L(cid:96) B , for which the brane theory is well described as Einstein gravity coupledto two copies of the boundary CFT (with a high cutoff). The reason is that in thislimit, we can interpret the intersection of the brane and the RT surface as boundingan island in this effective gravitational theory. τ Σ = 0We will start our analysis by calculating the area of the RT surface for an entanglingsurface lying in the τ = τ Σ = 0 plane and crossing the Planck brane. In other words,the RT surface is in the connected phase – see figure 8. We are interested in theentanglement entropy of R comprised of the combined regions χ > χ Σ and χ < − χ Σ in both the left and right CFTs. Hence the entangling surfaces of interest have twocomponents (in each CFT) sitting a constant distance away from the defect at χ = ± χ Σ .We note that the induced metric on the latter surfaces is proportional to cosh d − χ Σ .In two dimensions, the analysis of the RT surfaces is simplified because the metric(3.6) has a shift symmetry χ → χ + const, but the latter is absent in higher dimensions.However, we can find a similar simplification by going to a different coordinate systemdefined via [1, 98](1 + ζ ) = r cosh χ , tan ξ = r √ r − χ , (3.12)such that the horizon is located at ξ = ± π . By time-translation invariance, we knowthat the RT surface lies on a constant Rindler time slice and hence we consider the However, it is interesting to note that r = 0 corresponds to ρ = cos θ B = 1 − ( L/(cid:96) B ) , and henceone cannot reach ρ = 0 in the r -coordinate system (unless θ B = π/ – 19 –etric on the τ = 0 slice in the new coordinates, which reads ds E = L (cid:18) dζ ζ + ζ dξ + (1 + ζ ) dH d − (cid:19) . (3.13)Hence the geometry of this spatial slice (or any constant τ slice) is invariant under ξ → ξ + const, which will simplify the following.Making the ansatz ζ = ζ ( ξ ) for the profile of the RT surface, the induced metricon these surfaces takes the form ds = L (cid:34)(cid:32)(cid:18) ∂ζ∂ξ (cid:19) + ζ (1 + ζ ) (cid:33) dξ (1 + ζ ) + (1 + ζ ) dH d − (cid:35) , (3.14)with metric determinantdet( γ ) = L d − (1 + ζ ) d − (cid:32)(cid:18) ∂ζ∂ξ (cid:19) + ζ (1 + ζ ) (cid:33) . (3.15)To obtain the correct RT surface, we now need to extremize the area functional A (Σ R ) = (cid:90) Σ R (cid:112) det( γ ) , (3.16)subject to the correct boundary conditions. Here, a few observations are in order. Theboundary condition is determined by the RT surface ending at the entangling surfaceon both sides of the defect. Alternatively, since our setup is reflection-symmetric acrossthe brane, we can also consider a family of bulk extremal surfaces which end on thebrane and vary with respect to the point of intersection of the brane and the RT surface[1]. Even in higher dimensions, this variation takes a fairly simple form (see eq. (3.22)below), since extremizing the RT surface can be cast as an effectively two-dimensionalproblem with metric ds D = L d − vol H d − (1 + ζ ) d − (cid:18) dζ ζ + ζ dξ (cid:19) . (3.17)Note that the area functional does not explicitly depend on ξ . Rather, ξ plays therole of an angular coordinate and its associated Hamiltonian is conserved. This allowsus to turn the second order equation which determines extremal surfaces into a firstorder expression, dζdξ = ± (cid:115) ζ (1 + ζ ) (cid:18) ζ (1 + ζ ) d − ζ ∗ (1 + ζ ∗ ) d − − (cid:19) , (3.18) Note that the full metric takes the form ds = L ζ cos ξ dτ + ds E , and hence the shift symmetrydoes not extend to the full spacetime metric. – 20 – R ζ ∗ ( ζ QES , ξ
QES ) horizon( ∞ , ξ Σ ) CFT L CFT R ξ R R conformaldefectconformaldefect
Figure 8 : This figure shows the RT surface and various quantities defined in the textfor the RT surface in the connected phase. The entangling region R in the boundaryis composed of the two regions | χ | > χ Σ (where tan ξ Σ = sinh χ Σ ) in both the leftand right CFTs. Note that the right (left) CFT occupies the region on the asymptoticboundary marked in pink (aqua). The conformal defects ( i.e., χ = 0 or ξ = 0 and π )are positioned where the brane (green) reaches these boundary regions. ζ ∗ R R
CFT L CFT R Figure 9 : This figure shows how RT surfaces can intersect the brane before reachingthe turnaround point ζ ∗ , with relatively small brane tension T o , i.e., θ B ∼ O (1), andpositive DGP coupling.where we have introduced ζ ∗ which is the turn-around point for ζ as a function of ξ –see figure 8. The sign depends on whether ζ is going towards (+) or away ( − ) fromthe boundary as ξ increases. In the latter case, where the RT surface does not turnaround before it intersects the brane we have to think of ζ ∗ as a coordinate of vacuumAdS extended past the brane, as shown in figure 9. More generallly, the sign starts outnegative and generally flips after ζ = ζ ∗ has been reached.– 21 –he area functional for the RT surfaces satisfying eq. (3.18) then becomes A (Σ R ) = 4 L d − vol H d − (cid:32)(cid:90) ∞ ζ ∗ ± (cid:90) ζ ∗ ζ QES (cid:33) dζ ζ (1 + ζ ) d − (cid:112) ζ (1 + ζ ) d − − ζ ∗ (1 + ζ ∗ ) d − , (3.19)where here and below, we use the subscript QES to mark coordinates of the intersectionbetween RT surface and brane, which corresponds to a quantum extremal surface in thebrane theory. The upper limit of integration indicated as ∞ must be regulated, sincethe area of the RT surface is infinite. The sign here is the same sign as in eq. (3.18).We have also included a factor of four, since there is one RT surface to each side of thedefect and considering both CFTs, we need to multiply the result by another factor oftwo.Eq. (3.18) yields a family of RT surfaces (parameterized by ζ ∗ ) which are locallyextremal in the bulk away from brane. However, fully extremizing the area functional(3.16) requires that we also extremize over the possible locations where these candidatesurfaces intersect the brane. That is, we consider the extremization condition of the RTsurfaces’ area (plus possibly the area of the QES, should there be extra DGP gravity)with respect to the position of the intersection σ R ,0 = ∂∂ρ QES (cid:18) A (Σ R )4 G bulk + A ( σ R )4 G brane (cid:19) , (3.20)where the two contributions reflect the two contributions in eq. (1.2). Here, ρ QES denotes the location of σ R in coordinates along the brane in eq. (3.10).As described in [1], this extremization leads to a boundary condition restrictingthe angle at which the RT surface meets the brane. Normally, this would be a difficultproblem in higher dimensions. However, here we are leveraging the hyperbolic symme-try along the transverse directions, which reduces the present case to a two-dimensionalproblem. That is, we need only extremize a one-dimensional profile ζ ( ξ ) of the RT sur-face in the effective two-dimensional geometry given by eq. (3.17). Assuming that weconsider an extremal bulk surface which is anchored at the asymptotic boundary, thevariation of the surface’s area with respect to its intersection point with the brane isgiven by δ σ R A (Σ R ) = h ij T i X j | end-point , (3.21)where h ij is the two-dimensional metric (3.17) and T i is a normalized (w.r.t. h ij ) tan-gent vector to the RT surface, which can be obtained from eq. (3.9). The vector X i determines the variation along the brane.In the absence of a DGP gravity term in the action, this variation must vanish for X j along the brane; hence we have a boundary condition which sets the RT surface– 22 –erpendicular to the brane. More generally, we must balance the above variationagainst the variation of the entropy contribution intrinsic to the brane, as can be seenfrom eq. (3.20).The first contribution to eq. (3.20) is then calculated using eq. (3.21) and yields ∂ ρ A (Σ R ) = 4 L d − vol H d − ζ ∗ (1 + ζ ∗ ) d − ζ sin θ B (cid:32)(cid:115) ζ + 1tan θ B ζ − ± (cid:115) ζ (1 + ζ ) d − ζ ∗ (1 + ζ ∗ ) d − − (cid:33) , (3.22)which is evaluated at ζ = ζ QES . Here we have used the brane angle θ B defined ineq. (3.8).If the brane DGP coupling is turned on, the variation of the area also obtains acontribution from the second term in eq. (3.20), ∂ ρ A ( σ R ) = 2 L d − vol H d − ζ ∗ (1 + ζ ∗ ) d − ζ (cid:112) ζ sin θ B − cos θ B ( d − ζ + 1) d − . (3.23)Substituting eqs. (3.22) and (3.23) into eq. (3.20), we obtain the following relationbetween the QES position ζ QES and the deepest point ζ ∗ reached by the RT surface: ζ ∗ (1 + ζ ∗ ) d − = ( ζ + 1) d − (cid:113) ζ sin θ B − cos θ B × (cid:20) λ b cos( θ B ) (cid:113) ζ + (cid:113) ζ − λ (cid:0) ζ sin θ B − cos θ B (cid:1)(cid:21) , (3.24)where λ b was defined in eq. (2.7).A final relation associating ζ QES and the belt width ξ Σ comes from integratingeq. (3.18) from the boundary to the brane, ξ QES = ξ Σ + (cid:90) ∞ ζ ∗ dζ (cid:12)(cid:12)(cid:12)(cid:12) dζdξ (cid:12)(cid:12)(cid:12)(cid:12) − ± (cid:90) ζ QES ζ ∗ dζ (cid:12)(cid:12)(cid:12)(cid:12) dζdξ (cid:12)(cid:12)(cid:12)(cid:12) − . (3.25)After using eq. (3.24), this can then be rewritten as a relation between the location ofthe entangling surface ξ Σ and the QES ζ QES only, if we further use eq. (3.9) together witheq. (3.12) to find the brane trajectory in ζ, ξ coordinates and determine the relationshipbetween ξ and ζ on the brane ζ sin ξ = cot θ B . (3.26)In section 4, we will use eqs. (3.19), (3.24) and (3.25) to produce the late-time part ofthe Page curve for a topological black hole coupled to a bath in higher dimensions.– 23 – .3 No-island phase for τ Σ = 0We can use the result of the previous subsection to obtain a solution for the no-islandphase. The first order equation (3.18) (where we choose the minus sign) again deter-mines the shape of extremal surface. By symmetry, we know that ζ ∗ must lie on thebifurcate horizon and is thus determined by solving (cid:90) ζ ∗ ∞ (cid:12)(cid:12)(cid:12)(cid:12) dζdξ (cid:12)(cid:12)(cid:12)(cid:12) − dζ = − π − ξ Σ . (3.27)Here we have implicitly chosen to perform the calculation in the asymptotic CFT whichsits at negative ξ , i.e., to a particular side of the brane. By symmetry the calculationon the other side of the brane yields the same result. The total area of the two RTsurfaces which connect both CFTs through the horizon is then given by A (Σ R ) = 4 L d − vol H d − (cid:90) ∞ ζ ∗ dζ ζ (1 + ζ ) d − (cid:112) ζ (1 + ζ ) d − − ζ ∗ (1 + ζ ∗ ) d − , (3.28)with ζ ∗ given by eq. (3.27). In the case of small brane angle θ B this phase alwaysdominates at early times. The reason is that the the RT surface in the competing phase, i.e., the phase where the RT surface crosses the brane, has to travel a large distanceto the brane before it can return to the asymptotic boundary across the brane. Thisadditional distance can be made arbitrarily small by choosing a small enough braneangle. We will furthermore see in section 3.5 how the time evolution of an RT surfaceat early times can be mapped to this case. χ Σ = 0Lastly, we will consider the case of a zero-width belt, i.e., the case where the locationof the entangling surface is taken towards the defect, so that the RT surface fallsstraight through the bulk along constant boundary slicing coordinate χ = χ Σ , c.f.figure 10. Note that this setup is essentially the same as considered in [94], whichstudied entanglement entropy of identical half-spaces in the two sides of a time-evolvedTFD.Due to symmetry, the trajectory of the RT surface is determined by its radialcoordinate r as a function of time τ . However, it is convenient to introduce Eddington-Finkelstein coordinates to avoid the coordinate singularity at r = 1. Hence, describingingoing null rays, we have v = τ + r tor ( r ) where r tor ( r ) = 12 log (cid:18) | r − | r + 1 (cid:19) , (3.29)– 24 – R λ UV λ ∗ horizon CFT L CFT R R RFigure 10 : The RT surface of an entangling surface located at the defect in the no-island phase.where r tor ( r ) denotes the usual tortoise coordinate. Note that with the above defini-tions, r tor ( r → ∞ ) → v = τ at the asymptotic AdS boundary. Then themetric becomes ds = L (cid:0) − ( r − dv + 2 dv dr + r dH d − (cid:1) . (3.30)Now the extremal surface will fall from the asymptotic boundary, through the ex-terior, across the Rindler horizon, reaching a minimal radius at r ∗ , within the interior.Then the surface will continue emerging into the second exterior region. Due to reflec-tion symmetry, we need only track the trajectory of the RT surface until it reaches r ∗ .Using eq. (3.30), the area functional can be written as A (Σ R ) = 4 vol H d − L d − (cid:90) λ UV λ ∗ dλ r d − (cid:112) − ( r −
1) ˙ v + 2 ˙ v ˙ r , (3.31)where λ is a radial coordinate intrinsic to the surface, which increases along the surfacemoving from the left asymptotic AdS boundary to the right boundary. The limits ofintegration here correspond to λ ∗ , the value at the minimal radius r ∗ , and λ UV , thevalue at the UV cutoff near the right boundary – see figure 10. We have also includeda factor of 4 to account for the fact that we only integrate from the Z symmetric point λ ∗ out to the right boundary, and the fact that there are two such RT surfaces, one oneither side of the brane. Of course, we have also integrated out the directions alongthe belt, i.e., along the H d − . Now, we fix the reparametrization symmetry of the areafunctional with the following convenient gauge choice (cid:112) − ( r −
1) ˙ v + 2 ˙ v ˙ r = r d − . (3.32) We extend our defintion of r tor ( r ) across the horizon using the standard prescription given in [92]. – 25 –he integrand in eq. (3.31) is independent of v and so we have a conserved ‘ v -momentum’ P v = ∂ L ∂ ˙ v = r d − ( ˙ r − ( r −
1) ˙ v ) (cid:112) − ( r −
1) ˙ v + 2 ˙ v ˙ r = ˙ r − ( r −
1) ˙ v , (3.33)where the second expression results from substituting in the gauge choice (3.32). Usingeqs. (3.32) and (3.33) to solve for ˙ r and ˙ v , we find˙ r [ P v , r ] = (cid:113) ( r − r d − + P v , ˙ v [ P v , r ] = ˙ r − P v r − r − (cid:18) − P v + (cid:113) ( r − r d − + P v (cid:19) . (3.34)Note that we have implicitly chosen a positive sign for ˙ r indicating that r is increasingas we move along the surface out towards the asymptotic boundary.An intuitive picture of the dynamics of the extremal surfaces is given by recastingthe ˙ r equation above as a Hamiltonian constraint,˙ r + U ( r ) = P v , (3.35)where the effective potential is given by U ( r ) = − ( r − r d − . (3.36)In this framework, P v plays the role of the conserved energy and the minimum radius r ∗ corresponds to the turning point where ˙ r = 0, i.e., (1 − r ∗ ) r d − ∗ = P v . (3.37)The area (3.31) of the extremal surface becomes A (Σ R ) = 4 vol H d − L d − (cid:90) r UV r ∗ dr r d − (cid:112) ( r − r d − + P v , (3.38)using eqs. (3.32) and (3.34). Note that r UV denotes the position of the UV cutoff surfacenear the asymptotic AdS boundary.With eq. (3.37), the extremal surface can be specified by the integration constant P v or the boundary condition r ∗ . However, we want to examine the time evolution ofthe entanglement entropy and so we must determine a relation between these constantsand the boundary time. In particular, using eq. (3.34), we can integrate out to the rightboundary to determine v bound − v ∗ = (cid:90) r UV r ∗ dr ˙ v ˙ r = (cid:90) r UV r ∗ dr r − (cid:34) − P v (cid:112) ( r − r d − + P v (cid:35) , (3.39)– 26 –here v ∗ denotes the value of the Eddington-Finklestein time at the turning point.However, because of the Z symmetry of the extremal surface, we know that the turningpoint lies on the surface t = 0, and so we may use eq. (3.29) to write v ∗ = r tor ( r ∗ ) = 12 log (cid:18) − r ∗ r ∗ (cid:19) . (3.40)Further, we know that v bound = τ [ P v ] and hence we find τ [ P v ] = 12 log (cid:18) − r ∗ r ∗ (cid:19) + (cid:90) r UV r ∗ dr r − (cid:34) − P v (cid:112) ( r − r d − + P v (cid:35) . (3.41)Note that the integrand is nonsingular in the vicinity of the horizon, i.e., near r = 1.The time derivative of the area (3.31) admits a very simple form dA (Σ R ) dτ Σ = 4 vol H d − L d − P v = 4 vol H d − L d − r d − ∗ (cid:112) − r ∗ , (3.42)where τ is the boundary time parameter. Further, we also observe that the criticalradius where ∂ r U = 0 is given by r c = d − d − . (3.43)At late times, the turning point is very close to this critical radius, i.e., , the criticalsurface lies near the surface r = r c for a long time, and so we can replace r ∗ → r c intoeq. (3.42). Hence we expect the growth of the area is fixed at late times, i.e., dA (Σ R ) dτ Σ = 4 vol H d − L d − ( d − ( d − / ( d − ( d − / . (3.44)As we will see momentarily, the late time behavior of the entropy of any subregionbounded by constant χ in the no-island phase is determined by a zero-belt widthcalculation. Thus, as in the two-dimensional case studied in [34] (as well as the higherdimensional case [30]), the entropy corresponding to the no-island phase grows withoutbound. A quick derivation of this result follows by considering a small variation of the surface profilein eq. (3.31). The bulk contributions naturally vanishes by the equations of motion determiningthe extremal surface. However, deriving the latter requires an integration by parts which producesboundary terms. These are usually eliminated by fixing the boundary conditions at infinity. In theabove result, we instead allow for a small variation in the boundary time. – 27 – .5 Time-evolution for general χ Σ , τ Σ (cid:54) = 0Given the region R of interest, we can ask how the RT surface changes under timeevolution. If we are in the island phase, the RT surface is completely contained insidethe Rindler patch so that time translations are a symmetry and the entropy is a con-stant. On the other hand, in the no-island phase, the RT surface connects to both bathCFTs. Forward time evolution of both sides is not a symmetry and the area of the RTsurface changes.Obtaining RT surfaces in the no-island phase which are anchored on symmetricentangling surfaces of arbitrary width and at arbitrary times in higher dimensions isgenerally difficult. However, as we will now show, our choice of entangling surfaceswith the hyperbolic symmetry of H d − allows us to map the RT surface at any ( χ Σ , τ Σ )either to some RT surface in the τ = 0 slice, i.e., with ( χ (cid:48) Σ , τ (cid:48) Σ = 0) or to the case wherethe entangling surface is at χ = 0, i.e., with ( χ (cid:48) Σ = 0 , τ (cid:48) Σ ). In particular, this meansthat the solutions obtained in the last two subsections are sufficient to discuss the fulltime evolution of the symmetric entangling surfaces of interest.The strategy we will employ in this chapter is the following. We will performa coordinate change from Rindler space to a particular Poincar´e coordinate systemdefined below. In the new coordinates, the entangling surfaces are straight lines. Byexploiting the boost symmetry of the Poincar´e patch and mapping back to Rindlerspace, the task of calculating entanglement entropy of a subregion with χ Σ at time τ Σ can be reduced to one of the cases discussed in sections 3.3 and 3.4.To understand the required coordinate changes it is convenient to embed AdS d +1 into R d, , i.e., we are looking for a parametrization of (parts of) the hyperboloid definedvia − T − T + X + · · · + X d = − L . (3.45)Our original two Rindler patches correspond to the parametrization T = ± L √ r − τ , T = Lr cosh χ cosh η ,X = ± L √ r − τ , X = ± Lr sinh χ , (3.46) X i = Lr cosh χ sinh η µ i with i = 3 , , . . . , d , where µ i denotes further angular coordinates, e.g., µ = cos φ , µ = sin φ cos φ , . . . ,which, together with η parametrize the H d − slice of the metric (3.7). The AdS bound-ary is located at r → ∞ , and each sign corresponds to one of the two Rindler wedges.On a fixed r slice, we can reach the boundary by taking χ → ±∞ or η → ±∞ . For Recall that R consists of all points more than a distance χ Σ away from the defect in both CFTs. – 28 –ny constant Rindler time ( i.e., fixed τ ), the bifurcation surface reached with r → χ = 0 = X . The entangling surfaces are definedto be at χ = ± χ Σ in both CFTs.We will now consider a particular Poincar´e coordinate system, which covers bothRindler wedges and is defined in terms of embedding coordinates as T = L ˜ t ˜ z , X = L ˜ x ˜ z , X = L ˜ x ˜ z , · · · X d = ˜ z + ˜ x − ˜ t − L z , T = ˜ z + ˜ x − ˜ t + L z , (3.47)where ˜ x = ˜ x + ˜ x + · · · + ˜ x d − . In these coordinates, the bifurcation surface intersectsthe boundary (˜ z →
0) at ˜ x = ˜ t = 0, while the defects are located at ˜ x = 0. The twoCFTs are mapped to the regions ˜ x > x <
0, respectively. We will denote theCFT at ˜ x > x < x = ± sinh χ Σ cosh τ Σ · ˜ x , ˜ t = tanh τ Σ · ˜ x . (3.48)This shows the convenient property of the new Poincar´e coordinates: entangling sur-faces lie along rays ( i.e., straight lines) in the positive half-space with ˜ x >
0, whoseslope depends on the spatial location χ Σ and the Rindler time τ Σ at which the en-tangling surfaces are defined. Further, flipping the sign of ˜ x to − ˜ x in the aboveexpressions yields the entangling surfaces in the left CFT. The relation between theRindler coordinate given in eq. (3.46) and the new Poincar´e coordinates of eq. (3.48)is illustrated in figure 11.We now need to choose cutoffs in order to regulate the area integrals of the RTsurfaces. First, we need to regulate the UV divergence in the entanglement entropy byintroducing maximum radius in both AdS-Rindler patches r UV (cid:29)
1. This translates toa ˜ z -dependent cutoff in the new coordinates,˜ z > ˜ z min = ˜ x − ˜ t r UV − ∼ ˜ x − ˜ t r UV , (3.49)where in the last step, we used that r UV (cid:29) The full Rindler horizons reach the boundary along ˜ x − ˜ t = 0. – 29 –oundaryof H d − defectorigin in Poincar´ecoordinarespoint at infinity inPoincar´e coordinates χ Σ Figure 11 : A time-slice of our setup. The spatial boundary S d − (in global coor-dinates) is split into two hyperbolic discs H d − , shown in pink and aqua, which areglued together at infinity. At the same location, the bifurcate horizon intersects theboundary. The CFT on either disc is dual to a Rindler wedge in the bulk. The defect(green) is a great circle on the global boundary. As indicated in the figure, the Poincar´ecoordinates introduced in this section cover the full sphere, with the point at infinityappearing on the south pole of the sphere. Entangling surfaces are the semi-circlesshown in red.the brane, the transverse directions should just contribute an overall volume factor. Wechoose η max = (cid:96) IR R (cid:29)
1, which translates to˜ z − ˜ t + ˜ x + L (cid:112) ˜ z − ˜ t + ˜ x + ˜ x < L cosh (cid:96) IR R . (3.50)
As a warm-up exercise, we will show that the entropy on the island phase is in fact in-variant under time evolution. This is obviously true, since the RT surface is completelycontained within one Rindler wedge and τ is a Killing coordinate for the correspondingmetric (3.10). Hence the corresponding time evolution of a single Rindler wedge is anisometry of that wedge. In this case, we are looking for an extremal surface which endson the boundary at the location defined by eq. (3.48) for either ˜ x > x < x >
0. Wecan express the problem in a boosted coordinate systems˜ t (cid:48) = γ (˜ t − β ˜ x ) , ˜ x (cid:48) = γ (˜ x − β ˜ t ) , (3.51)with boost parameter β = tanh τ Σ . This is depicted in figure 12. This boost leavesthe cutoffs given in eqs. (3.49) and (3.50) invariant, and changes the equation for the– 30 – t ˜ x ˜ x τ Σ = 0 τ Σ (cid:54) = 0 boost Figure 12 : The left panel shows two components of the entangling surface (red) atnon-zero Rindler time τ Σ in the right CFT in the Poincar´e coordinates (3.47). In theisland phase, these two rays in the boundary geometry are connected by an RT surfacein the bulk. We can perform a boost in ˜ x direction to map this set of entanglingsurfaces to the t = 0 slice, which also corresponds to τ = 0 slice of the hyperbolicboundary geometry. The boost is a symmetry of the defect (green).entangling surface to ˜ x (cid:48) = ± ˜ x (cid:48) sinh χ Σ , ˜ t (cid:48) = 0 . (3.52)This is precisely the entangling surface of the same region at τ = τ (cid:48) Σ = 0 with theappropriate cutoffs. We may thus conclude that entropy of the region R remainsconstant in the island or connected phase, as anticipated. Again because we have apure state globally, we can see that the entropy of the complementary region, i.e., thetwo belts centered on the conformal defects in each of the two CFTs, is independent of τ Σ in this connected phase. For the no-island phase, we focus on the case in which the RT surface connects entan-gling surfaces in the CFTs dual to different Rindler patches. The entangling surfacesare located at ˜ t = − sinh τ Σ sinh χ Σ ˜ x , ˜ x = ± cosh τ Σ sinh χ Σ ˜ x , (3.53)where we have chosen to focus on ˜ x < i.e., to the region on one side of the defect.Similarly to the island phase, we want to go to a new coordinate system in which thecalculation becomes simpler. Now, however, we have to distinguish two cases. Case 1: If τ Σ < χ Σ , we can boost this problem in ˜ x direction with boost parameter β = − sinh τ Σ sinh χ Σ . This is depicted in the upper panel of figure 13. The new entangling We are assuming that both τ Σ and χ Σ are positive (or zero). Let us also note here that τ Σ = χ Σ isa special case, where the entangling surfaces lie in the null plane ˜ t = − ˜ x . Our approach of boostingin the ˜ x direction fails in this case, but the results for the time evolution are smooth across this point. – 31 – t ˜ x ˜ x τ (cid:48) Σ (cid:54) = 0, χ (cid:48) Σ = 0 τ Σ (cid:54) = 0, χ Σ (cid:54) = 0 τ (cid:48) Σ = 0, χ (cid:48) Σ (cid:54) = 0 τ Σ < χ Σ τ Σ > χ Σ Figure 13 : The left panel shows two components of the entangling surface (red) atnon-zero Rindler time τ Σ in the right CFT in the Poincar´e coordinates (3.47). Thesetwo rays are located in different CFTs so that in the no-island phase, they are joingby an RT surface in the bulk which passes through the Rindler horizon. In this case,we can now boost in ˜ x direction to map these two rays to τ (cid:48) Σ = 0 when τ Σ < χ Σ or to χ (cid:48) Σ = 0 when τ Σ > χ Σ .surfaces are then located at˜ t (cid:48) = 0 , ˜ x (cid:48) = ± ˜ x (cid:48) (cid:115) cosh τ Σ sinh χ Σ − sinh τ Σ , (3.54)where ˜ x (cid:48) <
0. Expressing the result in Rindler coordinates, we are dealing with the caseof an entangling surface in the τ = τ (cid:48) Σ = 0 plane. The new location of the entanglingsurface χ (cid:48) Σ is given by cosh χ (cid:48) Σ = cosh χ Σ cosh τ Σ . (3.55)Note that as cosh τ Σ → cosh χ Σ (and so as | sinh τ Σ sinh χ Σ | → i.e., χ (cid:48) Σ → r (cid:48) UV = r UV cosh τ Σ . (3.56)We should caution the reader that we arrived at eq. (3.56) by substituting the trajectoryof the entangling surface into the boosted cutoff. This means that eq. (3.56) is onlycorrect for a small cutoff. Luckily, the corrections to the new cutoff only change theentanglement entropy at order O (1 /r UV ).– 32 –n conclusion, we found that if τ Σ < χ Σ , the entanglement entropy of the region | χ | > χ Σ at time τ = τ Σ is the same as that of a region | χ | > χ (cid:48) Σ given in eq. (3.55) attime τ = τ (cid:48) Σ = 0 calculated with a different cutoff, given by eq. (3.56). Case 2:
The other case, τ Σ > χ Σ , is shown in the lower panel of figure 13. Nowwe can boost in the ˜ x direction again, but using ˜ β = − sinh χ Σ sinh τ Σ . The new entanglingsurfaces are located at˜ x (cid:48) = 0 , ˜ x (cid:48) = ± ˜ t (cid:48) (cid:115) cosh τ Σ sinh τ Σ − sinh χ Σ . (3.57)While this does not reduce to a surface lying in the τ (cid:48) = 0 plane, in Rindler coordinatesit reduces to an entangling surface for a belt width χ (cid:48) Σ = 0 andcosh τ (cid:48) Σ = cosh τ Σ cosh χ Σ . (3.58)Again, the IR cutoff in eq. (3.50) is unchanged, however, the UV cutoff changes to r (cid:48) UV = r UV cosh χ Σ . (3.59)Let us note that the cutoff location still is continuous. In the previous case, the newcutoff was the old cutoff multiplied by cosh τ Σ . The latter was reliable as long as τ Σ < χ Σ . However, we see here that once τ Σ > χ Σ , the cutoff is no longer time-dependent. Now the preceding results can be combined to give a qualitative description of the timeevolution of the entanglement entropy. Following the discussion in section 2.2 for twodimensions, at time τ = 0, we have a standard thermofield double state of the twoCFTs on hyperbolic spatial geometries, including the conformal defects at χ = 0. If werestrict the observations to either the left or right side, the reduced state is a thermalone and in particular, the bath CFT is in thermal equilibrium with the correspondingconformal defect, with temperature T = 1 / (2 πR ).Using the brane perspective and an appropriate choice of parameters, we candescribe the conformal defects are replaced by (two copies of) the boundary CFT Note that, like above, we have substituted the trajectory of the entangling surface into the boostedexpression. Thus, this equation is only strictly correct in the r UV → ∞ limit, but the corrections aresubleading to the finite part of the entanglement entropy. Recall that we obtain a good aproximation to (semiclassical) Einstein gravity on the brane if wechoose
L(cid:96) B (cid:28) λ b not too close to − – 33 –oupled to Einstein gravity on an AdS d region. For the configuration described above,this yields a topological black hole solution shown in eq. (3.10). We emphasize thatthe latter really describes an AdS d geometry in AdS-Rindler coordinates, and hencethe thermal equilibrium between this ‘black hole’ and the finite temperature CFT onthe asymptotic boundary can be understood as arising because the two systems arecoupled along an accelerated trajectory in the region of dynamical gravity. While theblack hole is in equilibrium with the bath CFT, under time evolution, the two systemsare constantly exchanging thermal quanta. The immediate effect of this process after τ = 0 is to increase the entanglement between one side of the black hole, i.e., one ofthe AdS-Rindler wedges on the brane, and its respective bath CFT.A standard measure for the entanglement between both AdS-Rindler wedges andtheir respective baths is given by the entanglement entropy of the complement of twobelt subregions centered around the conformal defects in the boundary as discussedabove. In section 3.5, we saw that by a judicious change of coordinates (and cutoff),the calculation of the entanglement entropy of these regions can be mapped at latetimes ( i.e., τ Σ ≥ χ Σ ) to the case of a zero-width belt. Further, in section 3.4, we foundthat the entanglement entropy grows linearly in time, as shown in eq. (3.44). As in the two-dimensional case [2, 99], this linear growth of entropy would leadto an information paradox for our eternal black holes, if it was valid for all times.The reason is that the entanglement entropy must be bound from above by the defectentropy, since the defects need to purify the bath system. In the case of interest, thetheory is well approximated by weakly coupled Einstein gravity. This allows us to viewthe quantum fields on the gravitational background as giving a small correction to theentropy and thus, the defect entropy is well-approximated by two times the black holeentropy. The appearance of an island in the effective gravity theory from the brane perspec-tice is simply related to a phase transition of the RT surfaces in the bulk descriptionof our system. The RT surface changes from the no-island phase, in which it connectsboth CFTs through the horizon, to the island phase, in which it connects both sides Implicitly, to apply eq. (3.44), we must also show ∂ τ (cid:48) Σ (cid:39) ∂ τ Σ . The latter follows at late times fromeq. (3.58), which yields ∂∂τ (cid:48) Σ = (cid:18) − sinh χ Σ sinh τ Σ (cid:19) / ∂∂τ Σ . (3.60)Alternatively, the same result also follows by simply observing that eq. (3.58) implies that at latetimes: τ (cid:48) = τ − log (cosh χ Σ ) + O ( e − τ ). Let us add that this linear growth is analogous to that foundfor planar black holes in [94]. The black hole entropy is proportional to the horizon area of the black hole, which in our case isinfinite. Hence to be precise, we must consider an IR regulated entropy, as discussed with eq. (3.50). – 34 –f the defect in a single Rindler wedge. The fact that there will always be an extremalsurface crossing the brane is easy to see: Before we invoke the extremization conditionat the brane, there is an infinite family of candidate RT surfaces, which start in thebath and meet at the brane. To get the correct RT surface, we need only extremize thearea by varying the position of the surface where they meet the brane. Subregion dual-ity and the homology constraint guarantee that there will be one extremal surface forevery belt configuration (although the boundary of the island might sit at the horizonor at the CFT defect).In order to establish unitarity of the Page curve, we still need to argue that theisland appears before the black hole fails to purify the bath region R under consider-ation. In the case of interest here, we have that (cid:96) B L (cid:29)
1. In this approximation, itfollows from eq. (3.24) that ζ QES = ζ hor (cid:32) ζ ∗ (1 + ζ ∗ ) d − ζ (2 d − (1 + λ b ) + . . . (cid:33) . (3.61)In deriving this equation, we have used that the location of the horizon on the brane isat ζ hor ∼ (cid:96) B L (cid:29) ζ ∗ cannot scale with (cid:96) B L at leading order. The reason is that ζ ∗ is bounded from above by a function of the belt width. We can see that the locationof the new quantum extremal surface will always be close to the horizon – see alsothe next section for numerical plots. The leading order contribution to the generalizedentropy is given by the area of the horizon which gives the black hole entropy. While amore involved analysis is needed to demonstrate that the appearance of the island savesunitarity, this shows that the island mechanism has the right qualitative behaviour tounitarize the Page curve. In the previous section, we found a phase transition between the no-island and islandphases that has the right qualitative properties to yield a Page curve consistent withunitarity. The calculations involved differential equations which have no known closedform solution. However, the reader might have realized that all of these equations wereordinary differential equations and are thus easily solved numerically. In this section,we will first present numerical solutions to the equations for the RT surface in theisland phase, and then use the arguments of the previous section to obtain the Pagecurve for massless, topological black holes in equilibrium with a bath.– 35 – .1 General behavior of the islands
As discussed previously, by choosing entangling surfaces with the hyperbolic symme-try of H d − , the problem of finding the corresponding RT surfaces reduces to a two-dimensional problem. Choosing the convenient coordinates in eq. (3.12), we can expressthe profile of the RT surface as ζ ( ξ ). We start here by discussing examples of extremalsurfaces in the island phase for different choices of parameters. Instead of workingwith ζ as a radial coordinate, we conformally compactify the geometry and use thecoordinate (cid:37) = arctan( ζ ) , (4.1)which maps timeslices of AdS to a finite region. In order to calculate the profile of theRT surface, we fix the location of the entangling surface χ Σ at the boundary. Applyingthe large r limit of eq. (3.12), we relate this to ξ Σ , the location of the entangling surfacein ζ, ξ coordinates. We can then use eqs. (3.24) and (3.25) to determine ζ ∗ and ζ QES numerically as a function of ξ Σ . The shape of the RT surface is obtained by integratingeq. (3.18) from the boundary.Figure 14 shows a few examples of RT surfaces in the connected phase for d = 3,4 and 5, i.e., in four, five and six bulk dimensions, respectively. Here, we only showthe geometry on one side of the brane. The other side is determined by a reflectionacross the brane. Since the RT surfaces do not cross the horizon, the configuration isindependent of the choice of Rindler time τ .Figure 14a shows RT surfaces with fixed χ Σ for different values of the dimensionand selected values of the DGP coupling λ b . We can see that positive DGP couplingpushes the point of intersection between brane and RT surface towards the horizon, i.e., it reduces the area of the island’s boundary. Similarly, negative DGP coupling causesthe island to become bigger. This behaviour is readily explained through eq. (2.7)which shows that by increasing (decreasing) the value of λ b , the gravitational couplingin the brane theory, i.e., the effective Newton’s constant, becomes smaller (bigger). Inturn, the coefficient of the Bekenstein-Hawking contribution is bigger (smaller) in theisland rule (1.1) and therefore creating an island of fixed size becomes harder (easier).Figure 14b shows how the RT surface in the island phase behaves as we vary thebrane angle given by sin θ B = L/(cid:96) B (or equivalently the brane tension – see eq. (2.4)).Recall that Einstein gravity is a good approximation when θ B is small. As we departfrom the limit of small brane angle, the island grows.Finally, figure 14c shows that the size of the island varies with χ Σ , the location ofthe entangling surface in the bath. Moreover, as we will discuss momentarily, we seethat an island phase for the RT surface seems to exist for all values of the belt width,although of course it will generally not dominate at early times.– 36 – a) RT surfaces for the island phase in (left to right) d = 3 , ,
5. The DGP coupling λ b ischosen to be 1/0/ − . θ B = π andthe location of the entangling surface is χ Σ = 1.(b) RT surfaces in d = 4 with χ Σ = 1 and brane angle of (left to right) θ B = π, π, π . TheDGP coupling is set to zero.(c) RT surfaces in d = 4 with brane angle θ B = π and (left to right) χ Σ = , ,
3. The DGPcoupling is set to zero.
Figure 14 : RT surfaces in the island phase in higher dimensions. We only show oneside of the brane. The asymptotic boundary of the spacetime is shown in blue, thePlanck brane in green and the RT surfaces in red. The radial coordinate is (cid:37) definedin eq. (4.1). On each side of the horizon (dashed purple line) the angular coordinate ξ runs between − π and π . – 37 – θ B / π ζ Q ES θ B / π ξ Q ES / π θ B / π ρ λ b = λ b = λ b =- (a) The dependence of RT surface parameters on the brane angle θ B for d = 3. θ B / π ζ Q ES θ B / π ξ Q ES / π θ B / π ρ (b) The dependence of RT surface parameters on the brane angle θ B for d = 4. θ B / π ζ Q ES θ B / π ξ Q ES / π θ B / π ρ (c) The dependence of RT surface parameters on the brane angle θ B for d = 5. Figure 15 : The dependece of the RT surface and the quantum extremal surface onthe brane angle θ B for d = 3, 4 and 5. The location of the entangling surface is chosento be χ Σ = 1. – 38 –e can get an even better idea of the qualitative features of the islands in higherdimensions by plotting the turning point ζ ∗ and the QES position ( ζ QES , ξ
QES ) as afunction of the brane angle θ B for different dimensions – see figure 15. A generalfeature is that in the θ B → ξ = π/
2, as discussed around eq. (3.61). In terms of ξ QES and the distancefrom the horizon on the brane, ρ QES , we have ξ QES = π − ζ ∗ (1 + ζ ∗ ) d − λ b θ d − B + O ( θ d B ) (4.2) ρ QES =1 + ζ ∗ (1 + ζ ∗ ) d − λ b ) θ d − B + O ( θ d − B ) , (4.3)where the first terms on the RHSs give the location of the horizon. Granted ζ ∗ tendstowards a finite value as θ B →
0, the above formulas tell us that the QES tends towardsthe horizon on the brane. Applying eq. (4.2) to eq. (3.25) and noting from eq. (3.18)that dζdξ ∼ ∓ ζ d +1 ζ ∗ (1 + ζ ∗ ) d − for ζ (cid:29) , (4.4)we find that ζ ∗ at small θ B is determined by the equation π − ζ ∗ (1 + ζ ∗ ) d − λ b θ d − B + O ( θ d B ) = ξ QES = ξ Σ + 2 (cid:90) ∞ ζ ∗ dζ (cid:12)(cid:12)(cid:12)(cid:12) dζdξ (cid:12)(cid:12)(cid:12)(cid:12) − + O ( θ d B ) , (4.5)with dζ/dξ given by eq. (3.18). At leading order in θ B , the second term on the LHS canbe ignored and the above equation is just the statement that the RT surface shouldstretch from the belt boundary to approximately the bifurcation surface on the brane. d > d -dimensional topological black holes, coupledto a bath on a hyperbolic background, for the cases d = 3 , ,
5. More precisely, weconsider the entropy of the region defined by χ Σ = 1, which is given by4 G bulk S ( τ ) = min (cid:18)(cid:20) A (Σ R ) + 2 Lλ b ( d − A ( σ R ) (cid:21) isl . , [ A (Σ R )] (cid:8)(cid:8) isl . (cid:19) . (4.6)– 39 –
200 400 600 800 1000 1200050010001500 τ G B u l k Δ s τ G B u l k Δ s τ G B u l k Δ s (a) The Page curve for dimensions d = 3 , , χ Σ = 1 and the DGP coupling is set to zero. The brane angle is chosen as θ B = 0 . τ G B u l k Δ s θ B τ P (b) Left: The Page curve for selected brane angles θ B = 0 . , . , .
15 (top to bottom).Right: The Page time τ P as a function of the brane angle θ B . The constant parameters areset to λ b = 0, χ Σ = 1, and d = 4. τ G B u l k Δ s - λ b τ P (c) Left: The Page curve for selected values of the DGP coupling λ b = 0 . , , − . τ P as a function of the DGP coupling λ b . The constantparameters are set to θ B = 0 . χ Σ = 1, and d = 4. Figure 16 : The Page curve in various dimensions. The solid blue line indicates thephysical Page curve. The dashed orange lines correspond to entropies calculated bynon-minimal extremal surfaces. At early times, the RT surface in the no-island phaseis the minimal surface. After some time, the minimal surface transitions to the RTsurface in the island phase. – 40 –ere A (Σ) are the regulated areas of the RT surfaces, and the subscript indicateswhether we consider the extremal surface in the island or no-island phase. Sinceeq. (4.6) is a cutoff dependent quantity, it is convenient to subtract off [ A (Σ R )] (cid:8)(cid:8) isl .,τ =0 .That is, we subtract off the value of the entropy at τ = 0, at which point the minimalRT surfaces in the no-island phase, to define∆ S ( τ ) = S ( τ ) − S ( τ = 0) . (4.7)Even though the UV divergences have been removed, eq. (4.6) would still be infinite,as a result of the infinite extend of the entangling surface. Hence the plots in figure 16show the change in the entropy density,∆ s = ∆ S vol H d − L d − , (4.8)with respect to the entropy at τ = 0. The kinks in the plots of figure 16 indicate thetime at which the island phase of the RT surface begins to dominate. The correspondingtime is, of course, the natural analog of the Page time for eternal black holes coupledto a bath at finite temperature. The slope of the (linearly) rising portion of the Pagecurve has been determined in section 3.4 and is given by4 G N ∆ s/τ ∼ d − ( d − / ( d − ( d − / . (4.9)Moreover, recall that τ is a dimensionless time such that the temperature of the hy-perbolic black hole is π (cf. the discussion in section 3.1). The dimensionful time t isrelated to τ by t = τ R = τ πT , (4.10)where R is the curvature scale for the spatial sections in the bath CFT, as defined ineq. (3.3), and the bath CFT is taken at temperature T = πR .The calculation of the RT surfaces is performed as follows: the area in the islandphase is computed by substituting eqs. (3.24) and (3.26) into eq. (3.25) and numericallysolving for ζ QES . The result is then used together with eq. (3.24) to numerically inte-grate the area in eq. (3.19). There are three different regimes for the calculation of theare in the no-island phase . At early times, τ Σ ≤ χ Σ , the calculation of the entropy of thesubregion with boundaries at ± χ Σ can be translated to the calculation of the entropy Note that we are actually plotting 4 G bulk ∆ s , which is a dimensionless quantity. For the horizontalaxes, also recall that the AdS-Rindler time τ is also dimensionless – see further comments below. – 41 –f a belt with boundary χ (cid:48) Σ = ± arccosh (cid:16) cosh χ Σ cosh τ Σ (cid:17) in the τ = 0 time-slice, as explainedin section 3.5. As also explained in the same section, we need choose a different cutoffon r in this case. However, working in ζ, ξ coordinates, it turns out that the cutoffon ζ does not change. At intermediate times, τ Σ (cid:38) χ Σ , the entropy can be computedby calculating the area of an RT surface for a zero-belt-width entangling surface at atime given in eq. (3.58). Accidentally, the relation between r and ζ works out in such away that the cutoff of r agrees with the cutoff on ζ in the previous calculation. As τ Σ becomes larger, the numerics become less reliable. However, for moderately sized beltwidths we are already well into the regime in which the area of the RT surface growslinearly in time. Therefore, we use a linear fit to extrapolate the last few numeric datapoints to late times, τ Σ (cid:29) χ Σ . We verified that the resulting slope agrees with theanalytic result given in eq. (4.9).In figure 16b, we show how the Page curve and Page time change as we varythe brane angle. As we see, increasing θ B decreases the Page time, or in other wordsdecreases the number of microstates available to the black hole on the brane. This canalso be understood from the CFT point of view where the defect entropy is given interms of an RT surface in the island phase [60, 100]. As the brane angle approacheszero, the Page time diverges. The reason is that in this limit the area of the islanddiverges. The absence of islands in this limit was already noted in [33]. The divergenceas θ B → θ − d B , and in the small-angle approximation we find that τ P ∼ ( d − d − ( d − d θ d − B . (4.11)For example, the numerical coefficient which multiplies θ − d B can be estimated from theabove formula to be 1 .
30 for d = 4. A fit to the numerical data plotted in figure 16bagrees with this value.Figure 16c shows the dependence of the Page curve and Page time on the DGPcoupling. As we decrease the DGP coupling ( i.e., increase G eff ) the Page time goes tozero. The linearity can be easily explained be recalling that in the small θ B regime weare interested in the island sits close to the horizon and thus has a fixed location forvarying values of λ b . The Page transition occurs whenever the area of the RT surfacein the no-island phase exceeds the area of the RT surface in the island phase. Sincethe area in the no-island phase approximately grows linearly with time and the areain the island phase depends approximately linearly on λ b , c.f. eq. (4.6), we obtain alinear relationship between the Page time τ P and λ b . Based on this argument, we can– 42 – τ Σ G B u l k Δ s (a) This figure shows the onset of the Pagecurve for different values of the location ofthe entangling surface χ Σ = 0 . , , , . τ Σ / χ Σ G B u l k Δ s / χ Σ (b) The same plot, but with axes rescaledby χ Σ . The solid lines are numericalresults, while the dashed lines are thebounds explained in the main text. Figure 17 : The initial behaviour of the Page curve in four dimensions (left) and arescaled version of the same plot with bounds (dashed) on the onset (right).estimate the slope of the graph to be τ P /λ b ∼ ( d − d − ( d − d θ d − B , (4.12)which for the parameters in 16c ( i.e., θ B = 0 . d = 4) evaluates to τ P ∼ λ b andagrees with the fitted value of the slope.The Page curve and Page time only depends very weakly on the belt size. In fact,the only significant effect can be seen at very early times of the evaporation. Figure17a shows that for wide belts, the entanglement between the belts and baths startsgrowing convexly ( i.e., ∂ ∆ s/∂τ > i.e., ∂ ∆ s/∂τ <
0) before entering the linear regime.Generally, we can separate the time-dependence of the Page curve into four differentregimes. At times of the order of the thermal scale β ( ∼ .
16 in figure 17a) theentanglement growth increases until it enters a phase of fast growth between τ Σ ∼ O ( β )and τ Σ ∼ O ( χ Σ ). This fast growth depends on the belt size. At time τ Σ ∼ O ( χ Σ ) auniversal, linear behavior takes over, which is independent of the belt width. Theentanglement keeps growing until at the Page time τ P it saturates and stays constant.In the following we will explain the region of fast growth and its transition intothe region of universal linear growth. To understand the behaviour of the Page curve,first consider a few characteristics of our belt geometries. As can be seen from themetric in eq. (3.7), points on any of our entangling surfaces are a fixed distance χ Σ – 43 –rom the surface at χ = 0, where the defect is located, i.e., where the bath is coupledto the black hole. However, the extrinsic curvature of the entangling surfaces whichwe consider depends on this distance. Similarly, the entangling surfaces with larger χ Σ have a larger regulated volume.In [101], it was proposed that the growth of entropy S [Σ] for an arbitrary entanglingsurface Σ is bound by 1 R dS [Σ] dτ = dS [Σ] dt ≤ s th v ent A (Σ) , (4.13)where A (Σ) is the area of the entangling surface Σ, as measured by the boundary metricin eq. (3.7). The thermal entropy density s th and the entanglement velocity v ent areregion independent constants. The entropy density is given by the black hole entropy( i.e., G bulk times horizon area) divided by the CFT volume of the spatial slices (again,measured by the metric (3.7)): s th = 14 G bulk L d − R d − . (4.14)In [101] which primarily considers flat space, v ent is defined such that eq. (4.13)is saturated at times just above the thermal scale for sufficiently straight entanglingsurfaces – this definition is well-defined in the sense that v ent turns out to be independentof the shape of the entangling surface, provided it is sufficiently straight [102, 103]. Inhyperbolic space, v ent can be similarly defined by demanding that the straight surface χ = 0 saturates eq. (4.13) – we shall justify this choice further below – specifically, v ent = ( d − d − ( d − d − , (4.15)obtained by comparison of eq. (4.13) with the zero-width belt result in eq. (3.44).It is clear that (4.13) cannot be tight at late times for belts of finite width. Thereason is that the area factor on the right hand side A [ χ >
0] is exponentially largecompared to A [ χ = 0], while, as can be seen from figure 17a, all belts share the samerate of entanglement growth at late times. To more tightly bound the late time behaviorof finite width belts, we will therefore need to combine eq. (4.13) with the monotonicityof mutual information. It will turn out that the optimal bound obtained in this way forfinite-width belts uses eq. (4.13), but always evaluated for the χ = 0 surface Σ at latetimes; thus we will find that the χ = 0 surface acts as a bottleneck for entanglementgrowth even for finite width belts. The proper distance would be Rχ Σ in the boundary metric (3.3). – 44 –o see why the surface at χ Σ = 0 acts as a bottleneck, let us formulate the morerefined combined bound now, following closely [101]. To this end, it will be less helpfulto consider the entanglement entropy of the bath intervals R ; instead we will considertheir complement ¯ R , i.e., belts surrounding the defects, whose entropy is the sameas R since the state of both Rindler patches is pure. Considering ¯ R instead of R isequivalent to looking at the Page curve of the black hole instead of that of the radiation.It is useful to rewrite the entropy displayed in the Page curve as∆ S = I [ ¯ R L : ¯ R R ](0) − I [ ¯ R L : ¯ R R ]( τ ) , (4.16)where I [ ¯ R L : ¯ R R ]( τ ) = S ( ¯ R L ) + S ( ¯ R R ) − S ( ¯ R L ∪ ¯ R R ) is the mutual informationbetween the regions ¯ R in the left ( L ) and right ( R ) CFT at time τ .Similar to [101], we now assume that information is only transported with thebutterfly velocity v but or less. For the hyperbolic geometries considered here and thetemperature T = πR , this velocity is given by [105, 106] v but = 1 d − . (4.17)This implies that a belt region ¯ R (cid:48) at time τ (cid:48) can be considered a subsystem of theoriginal belt ¯ R at τ if χ Σ − χ (cid:48) Σ v but ≥ | τ − τ (cid:48) | . We can then use monotonicity of mutualinformation I [ ¯ R L : ¯ R R ]( τ ) ≥ I [ ¯ R (cid:48) L : ¯ R (cid:48) R ]( τ (cid:48) ) = S [ ¯ R (cid:48) L ]( τ (cid:48) ) + S [ ¯ R (cid:48) R ]( τ (cid:48) ) − S [ ¯ R (cid:48) L ∪ ¯ R (cid:48) R ]( τ (cid:48) ) . (4.18)In our setup, we have that the one-sided entropies are time-independent, S [ ¯ R (cid:48) R/L ]( τ (cid:48) ) = S [ ¯ R (cid:48) R/L ](0). Using eq. (4.13) we can then bound S [ ¯ R (cid:48) L ∪ ¯ R (cid:48) R ]( τ (cid:48) ) from above S [ ¯ R (cid:48) L ∪ ¯ R (cid:48) R ]( τ (cid:48) ) ≤ Rs th v ent A ( ∂ ¯ R (cid:48) ) τ (cid:48) + S [ ¯ R (cid:48) L ∪ ¯ R (cid:48) R ](0) . (4.19)Collecting everything, we find a bound on the Page curve of the black hole,∆ S [ ¯ R L ∪ ¯ R R ] ≤ Rs th v ent A ( ∂ ¯ R (cid:48) ) τ (cid:48) + I [ ¯ R L : ¯ R R ](0) − I [ ¯ R (cid:48) L : ¯ R (cid:48) R ](0) . (4.20)To find a tightest bound this has to be minimized over all choices of χ (cid:48) Σ , see below.For any fixed χ (cid:48) Σ it is sufficient to focus on the case where τ (cid:48) < τ , which will alwaysgive the smaller bound. The mutual information appearing on the right hand side areevaluated on the initial time slice and can be obtained numerically by using the resultsof section 3.3. The butterfly velocity is defined as the spread of the region in which the commutator of an operator O ( t ) with O ( t ) is bigger than 1 [104]. – 45 –rom eq. (4.20), it is now easy to see why the entanglement growth becomes uni-versal at late times. Note that eq. (4.20) is in fact a family of inequalities, parametrizedby a choice of regions ¯ R (cid:48) . The time τ (cid:48) is chosen such that ¯ R (cid:48) at τ (cid:48) is just barely asubsystem of ¯ R at τ , in the sense described below eq. (4.17). For times before τ (cid:48) weassume that the mutual information of subregions ¯ R (cid:48) is allowed to decrease as fast aspossible, while still compatible with eq. (4.13). Since the regions ¯ R (cid:48) at time τ (cid:48) aresubregions of ¯ R at time τ , their mutual information bounds the mutual information ofregions ¯ R . We can find a tight bound on the Page curve by minimizing over all choicesof ¯ R (cid:48) , or in other words, by minimizing over all χ (cid:48) Σ with τ (cid:48) = τ − χ Σ − χ (cid:48) Σ v but . It turns outthat, for sufficiently large τ , the tightest bound is obtained for χ (cid:48) Σ = 0, yielding theprescription stated below eq. (4.15). We thus see from the first term on the right handside of eq. (4.20) that this surface acts as a bottle neck for information transfer andthus controls the late time growth of entropy. Matching this behaviour to the late timerate of growth of the exact Page curve provides further justification, a posteriori , forthe choice of the entanglement velocity stated in eq. (4.15).The bounds found in this way are presented in figure 17b. We see that a fastgrowth at early times is allowed by the bounds, before the linear growth phase isentered. Further, as can be seen from the figure, these bounds are fairly loose. Itwould be interesting to understand how to make them tighter. Note that the bluecurve in figure 17b behaves qualitatively different than the other curves. The reasonis that the early convex onset of the curve is controlled by the thermal scale and thuslasts for roughly ∆ τ ∼ O ( β ), independent of the belt width. The rescaling in figure17b magnifies the early time behavior of belts with χ Σ < χ Σ <
1. Thus, while all other curves show the linearentanglement spreading for time scales τ ∼ O ( χ Σ ) > O ( β ), the behavior of the bluecurve is dominated by entanglement spreading through thermalization, since the beltwidth is of order of the thermalization scale. The quadratic growth at times belowthe thermal scale is reminiscent of the ‘pre-local-equilibration growth’ described in[102, 103].Let us end with a few observations regarding the structure of entanglement spread-ing in our system. First, we note that the entanglement velocity (4.15) for Rindlerspacetime with hyperbolic spatial slices differs from the analogous velocity √ d ( d − Note that time-reflection symmetry demands that the Page curve have an early time expansioncontaining only even powers of τ . For the zero-width, it is easily verified, at least numerically, fromeq. (3.41) that √ − r ∗ ∼ τ so that the growth is indeed quadratic by eq. (3.42). For finite-widthbelts, plugging eqs. (3.55) and (3.56) into eq. (3.12) shows that early time evolution is equivalent toholding the cutoff at fixed ζ and shifting the ξ of the entangling surface by ∼ τ , again leading toquadratic entanglement growth. – 46 –) − d / [2( d − − d in flat space [94] dual to AdS planar black holes. Furthermore,for d >
3, the entangling velocity for a CFT on hyperbolic space exceeds the butterflyvelocity, eq. (4.17). Typically, whenever v ent > v but , one might worry about contra-dictions to entanglement monotonicity laws [101, 107] which apply above the thermalscale. However, no immediate contradictions appear in the present case, as we nowexplain.For concreteness, let us interpret eq. (3.44) as describing the entanglement growthin hyperbolic space without defects, specifically, computing the entropy for a regionconsisting of half-spaces χ > This growth saturateseq. (4.13) with v ent > v but in d > v but permitted by operator commutator growth [104]. Specifically,by applying an analysis similar to the one reviewed around eq. (4.17) to thermal relativeentropies, [101, 107] argue that, for regions and times above the thermal scale, entan-glement growth must be bounded by the thermal entropy density s th times the volumebetween the entangling surface and a tsunami wavefront propagating with speed v but away from the entangling surface (in either direction). Said differently, the rate dS/dt ofentanglement growth is bounded by s th v but times the area of the tsunami wavefront —this is essentially eq. (4.13) with v but replacing v ent and the tsunami wavefront replacingthe entangling surface. In flat space, the tsunami wavefront can be typically chosen topropagate in a direction away from the entangling surface such that it shrinks or doesnot grow in time ( e.g., propagating inward from a spherical entangling surface). Thus,for the flat space equivalent of eq. (4.13) to be saturated, one must require v ent < v but .In hyperbolic space however, it is possible for the tsunami wavefront to grow in bothdirections away from the entangling surface. Indeed, this is precisely what happens forthe hyperbolic half-space which has an entangling surface χ = 0 of minimal area; withina few thermal times, the tsunami wavefront propagating in either direction grows to anarea exponentially large compared to the entangling surface. We thus see that, thoughthe hyperbolic half-space saturates eq. (4.13) with v ent > v but , this does not contradictthe bound on entanglement spreading due to the butterfly velocity. T = 0 bath Here we turn our attention to extremal black holes. In particular, we consider thesame bulk geometry described in section 2, i.e., a backreacting codimension-one braneextending across the spacetime which locally has the geometry of AdS d +1 . However, we To be precise, we should multiply eq. (3.44) by · G bulk with the factor of 1 / χ = 0 — one on either side of the TFD. – 47 – igure 18 : The Poincare patch models a zero temperature extremal black hole. Thebrane intersects the CFT Poincar´e patch at the origin and infinity.replace the AdS-Rindler coordinates introduced in eq. (3.6) with Poincar´e coordinates, ds = L z (cid:0) dz − dt + dx + · · · + dx d − (cid:1) . (5.1)Of course, the coordinate singularity at z → ∞ corresponds to an extremal T = 0horizon. Figure 18 illustrates the Poincar´e patch in our bulk geometry.For the most part, we will be interested in limit of large tension ( i.e., (cid:96) B (cid:29) L ), forwhich the brane theory can be described as Einstein gravity coupled to two copies ofthe boundary CFT. As we describe in a moment, the brane geometry naturally inheritsa Poincar´e metric from the bulk geometry. Hence the brane supports an extremalblack hole which is equilibrium with the T = 0 bath CFT on the asymptotic AdSboundary. We note that with Poincar´e coordinates, we are examining the system ina new conformal frame where the bath CFT is living on flat d -dimensional Minkowskispace, ds CFT = − dt + dx + · · · + dx d − . (5.2)This brane perspective is illustrated in figure 19a.Of course, we may also have the boundary perspective where the d -dimensionalCFT in Minkowski space is coupled to a codimension-one conformal defect. For sim-plicity, we insert the latter at x = 0 for the metric in eq. (5.2) and so the inducedgeometry on the defect is also flat, i.e., ( d –1)-dimensional Minkowski space. The Pen-rose diagram for this perspective is shown in figure 19b. Note that in contrast to the– 48 – = 0horizondefect t = 0defecta. b. Figure 19 : The brane and boundary perspectives of the extremal black hole setup.finite temperature TFD state (entangling two copies of the bath CFT) in section 3, herefor the T = 0 scenario, we only have a single copy of the bath CFT, e.g., compare theabove to figures 6 and 7. Of course, at T = 0, we are simply studying the vacuum stateof the defect CFT in flat space (analogous to what was done in [1] but in a differentconformal frame). We may recall from [2] that for the extremal case in d = 2, one always finds islandsfor the analogous belt regions. This result is a consequence of two features which holdfor d = 2: firstly, there always exists a bulk RT surface intersecting the brane to producean island; secondly, the alternative no-island RT candidate surface has an additionalIR divergence and this surface is therefore subdominant. However, neither of thesestatements hold in d ≥
3. Indeed, we will find in higher dimensions that quantumextremal islands do not appear in the large tension limit. Nonetheless, no informationparadox arises since extremal black holes do not radiate, i.e., the black hole and the bathare not exchanging radiation. This contrasts with the non-extremal case in section 3,where the information paradox for the eternal black hole in the effective d -dimensionalgravity theory arises because of the continuous exchange of quanta between the blackhole and the bath. Of course, the paradox is avoided by the appearance of quantumextremal islands.The remainder of this section is organized as follows. We shall begin by first ex-plicitly constructing the bulk and brane metrics to be used in the extremal case and byintroducing the entanglement entropy calculation which we wish to consider. Then, in Of course, this is a pure state, as is manifest in bulk since the Poincar´e time slices constitutecomplete Cauchy slices. Coming from integrating the length of the surface down to the extremal horizon. – 49 –ubsections 5.1 and 5.2, we carry out this calculation using RT surfaces correspondingto island and no-island phases, respectively. Finally, we collect these results in subsec-tion 5.3 to determine when each phase dominates.The Poincar´e coordinates (5.1) cover a wedge of the AdS d +1 vacuum geometry.However, in the present geometry with a backreacting brane, a portion of two suchwedges would appear on either side of the brane – see figure 18. If we consider thecoordinate transformation z = y sin θ, x = y cos θ , (5.3)the metric (5.1) is transformed to the form given in eq. (2.3), where the AdS d sliceseach inherit a Poincar´e metric. As described in section 2, the brane spans one suchslice at a fixed θ = θ B determined by the brane tension T o according to eq. (2.4), i.e., sin θ B = 2 ε (1 − ε/ . (5.4)The induced metric on the brane then becomes ds d = L y sin θ B (cid:0) dy − dt + dx + · · · + dx d − (cid:1) , (5.5)and we may then read off the curvature scale of the brane as (cid:96) B = L/ sin θ B , as expectedfrom eq. (2.4). Here, y is interpreted as the radial Poincar´e coordinate running alongthe brane, and the Poincar´e horizon on the brane, located at y → ∞ , is inherited fromthe bulk. As usual, we wish to work in the regime L /(cid:96) B (cid:28)
1, or alternatively θ B (cid:28) θ = θ B and the CFT of the flat asymptotic boundary at z = 0 becomes the zerotemperature bath. This then provides a direct extension of the extremal scenario in [2]to d dimensions. The question which interests us here is then whether the entanglementwedge of certain subregions in the bath includes islands residing on the brane.Specifically, we consider the entanglement entropy calculation for a boundary region R that is the complement of a “belt” geometry centered on the defect at x = 0, i.e., theboundary subregion R = ( −∞ , − b ] ∪ [ b, ∞ ). According to the RT formula we shouldconsider codimension-two surfaces V sharing the same boundary ∂ V = ∂ R ≡ Σ CFT .To determine RT surface candidates among these surfaces, we must search for surfaceswhich extremize their area. As we discussed in the introduction, there are generally twosets of surfaces which achieves this extremization; the RT prescription then instructs– 50 – igure 20 : The bulk dual to a d -dimensional Minkowski CFT with a defect (green dot)along a line x = 0. The CFT lives on the asymptotic boundary of a Poincar´e AdS d +1 spacetime with a brane (green line) running through it. We consider the entanglemententropy of the complement R = ( −∞ , − b ] ∪ [ b, ∞ ) of a belt geometry in the CFT. Asconsidered in section 5.1, one candidate RT surface Σ R , shown in red, intersects thebrane at a QES σ R , forming an island on the brane belonging to the entanglementwedge of R . Various quantities defined in section 5.1 are marked in this figure.us to choose the one with the smallest area. The first class of surfaces are those whichintersect the brane, forming a quantum extremal island on the brane which belongs tothe entanglement wedge of R – see figure 20. We will say that this RT surface is in the island phase . The second set of surfaces fall trivially into the bulk and do not produceislands on the brane, i.e., these surfaces are in the no-island phase . As a starting point, let us review the calculation for RT surfaces of belt geometries inpure AdS [16]. That is, we are considering the complement of R , but the RT calcula-tions for this region and for its complement, R = [ − b, b ], are equivalent. Integratingout the x , . . . , x d − directions in which the brane is constant, the area functional of acodimension-2 surface V becomes A ( V ) = L d − vol ⊥ d − (cid:90) V dx (cid:114) (cid:16) dzdx (cid:17) z d − , (5.6)where vol ⊥ d − is the volume of transverse directions { x , . . . , x d − } . The RT surface Σ R is obtained by extremizing the area functional (5.6) with respectto the profile z ( x ). This functional, viewed as a Lagrangian, contains no explicit Note that in contrast to vol H d − introduced in section 3, vol ⊥ d − has the dimensions of length d − and so is essentially given by (cid:96) d − IR where (cid:96) IR is an IR cutoff in the x , . . . , x d − directions. – 51 – a) Heading into the bulk ± = +. (b) Heading out of the bulk ± = − . Figure 21 : Definitions for the choice of ± in eq. (5.7) and for the corresponding∆ x ( >
0) from eq. (5.8) on the two branches of the RT surface.dependence on x and hence the corresponding Hamiltonian is a constant along Σ R ,allowing us to deduce dzdx = ± (cid:113) z d − ∗ − z d − z d − (5.7)for some constant z ∗ . Further, the sign ± above is determined by whether we are onthe portion of the RT surface heading into the bulk (+) or heading out of the bulk ( − )with increasing x – see figures 20 and 21. From eq. (5.7), we see that dz/dx = 0at z = z ∗ and therefore z ∗ is the maximal z -value attained by Σ R . We can integrateeq. (5.7) to obtain the trajectory of the RT surface:∆ x = z d d z d − ∗ F (cid:34) , d d −
1) ; d d −
1) + 1; (cid:18) zz ∗ (cid:19) d − (cid:35) (5.8)Here ∆ x > x -separation between a point on the RT trajectoryand the initial (final) endpoint on the asymptotic boundary, on the portion of the RTsurface heading into (out of) the bulk — see figure 21. If we evaluate this expression at z = z ∗ , we obtain half of the width of the boundary strip (in the x direction) definedby the RT surface. Denoting this width as D , which we emphasize is in the empty AdSvacuum (see figure 20), we have D √ π Γ (cid:104) d d − (cid:105) Γ (cid:104) d − (cid:105) z ∗ . (5.9) As noted previously, if we restrict our attention to positive tension T o , we will have 0 < θ B < π/ x as one heads away from the boundary ( z, x ) =(0 , − b ), in order for the RT surface to meet the brane. – 52 –ow returning to the geometry with the backreacting brane, each half of the RTsurface Σ R on either side of the brane will follow the trajectory given in eq. (5.8) forpure AdS prior to meeting the brane. We have placed the defect at x = 0 and theRT surface begins on the asymptotic boundary at x = − b . Further, if we were toextend the RT surface past the brane, it would hit the asymptotic boundary again at x = − b + D . In terms of eq. (5.8), x along the trajectory is then given by x = − b + (cid:40) ∆ x when heading into bulk (towards z = z ∗ ) D − ∆ x when heading out of bulk (away from z = z ∗ ) . (5.10)In general, as illustrated in figure 20, D (cid:54) = b , rather, the relation between D (or z ∗ ) and b must be determined by demanding that the choice of the intersection σ R ofthe RT surface with the brane should extremize the RT surface’s area (plus the areaof the QES, when brane action includes an extra DGP term). As described in [1] andreviewed around eq. (3.21), this extremization leads to a boundary condition restrictingthe angle at which the RT surface meets the brane. Again, we may reduce this to atwo-dimensional problem where we view the RT surface as a geodesic in an effectivetwo-dimensional geometry ds D = L d − (vol ⊥ d − ) dz + dx z d − , (5.11)and the area becomes the length of the geodesic in this geometry.As before, we may use eq. (3.21) to determine the variation of the RT surface areaunder perturbations of σ R , the QES on the brane. Here, h ij is given by eq. (5.11), thedeviation vector X j is chosen to be ∂ y , and the tangent T i determined from eq. (5.7),with both X j and T i normalized with respect to h ij . Hence, upon perturbing theintersection of the RT surface with the brane, the RT area varies as ∂A (Σ R ) ∂y QES = 2 L d − vol ⊥ d − z d − cos θ QES = 2 L d − vol ⊥ d − (cid:32) cos θ B z d − ∗ ± (cid:115) z d − − z d − ∗ sin θ B (cid:33) , (5.12)where θ QES is the angle between the RT surface and the brane, y QES is the y coordinateof σ R – see figure 20 — and the ± sign is the same one as introduced in eq. (5.7)and illustrated in figure 21. An extra factor of 2 is included to account for the twocomponents of the RT surface on either side of the brane. From eq. (5.5), we read offthe area of σ R : A ( σ R ) =vol ⊥ d − (cid:18) Ly QES sin θ B (cid:19) d − , ∂A ( σ R ) ∂y QES = − ( d − ⊥ d − L d − sin θ B z d − . (5.13)– 53 –he extremality condition 0 = ∂∂y QES (cid:18) A (Σ R )4 G bulk + A ( σ R )4 G brane (cid:19) (5.14)is satisfied ifcos θ QES = λ b sin θ B ⇐⇒ z QES = z ∗ (cid:20) sin θ B (cid:18) λ b cos θ B + (cid:113) − λ sin θ B (cid:19)(cid:21) d − , (5.15)where λ b is defined in eq. (2.7). The relationship between z ∗ and b may then be deter-mined by substituting ( x , z ) = ( z QES cot θ B , z QES ) into eq. (5.10), and using eqs. (5.9),(5.8) and (5.15) to find b = ± ∆ x + 1 ∓ D − z QES cot θ B = F ( d, λ b , θ B ) z ∗ (5.16) F ( d, λ b , θ B ) ≡ ± z d QES d z d ∗ F (cid:34) , d d −
1) ; d d −
1) + 1; (cid:18) z QES z ∗ (cid:19) d − (cid:35) + (1 ∓ √ π Γ (cid:104) d d − (cid:105) Γ (cid:104) d − (cid:105) − z QES z ∗ cot θ B (5.17)where the top (bottom) signs chosen above if the RT surface intersects the brane tothe left (right) of the extremal point z = z ∗ . We have noted in the second equality ofeq. (5.16) that all terms of the previous expression are linear in z ∗ ; in particular, notein eq. (5.17) that the ratio z QES /z ∗ is determined by eq. (5.15). In figure 22a, we haveplotted the position of the intersection σ R between the RT surface and the brane as afunction of the brane angle θ B for various λ b and d = 3. In section 5.3, we shall discussthe fact that, for θ B below some critical angle θ c , the extremal surfaces discussed herefail to exist. That is, y QES , the position of the QES on the brane, runs off to infinityas θ B → θ c from above.Having determined the profile of the RT surfaces, we may proceed to evaluate theircorresponding entropies using the RT formula (1.2) – keeping in mind that we havenot shown that these surfaces minimize the entropy functional yet. Inserting eqs. (5.7)and (5.13) into the generalized entropy functional, we find that the entropy of the belt– 54 – = θ B π y QES b λ b = - λ b = λ b = λ b =- λ b =- + (a) Position of σ R as a function of brane an-gle - - λ b θ c π d = = = = = = (b) Critical brane angle as a function of theratio of G bulk to G brane . Figure 22 : Plots of the position of σ R , the intersection of the RT surface with thebrane, and the critical brane angle at which this surface runs off to y QES → + ∞ .geometry R and hence of the complementary bath region R is given by A (Σ R )4 G bulk + A ( σ R )4 G brane = L d − G brane vol ⊥ d − z d − + L d − G bulk (cid:40) (1 ∓ √ π Γ (cid:104) − d d − (cid:105) ( d − (cid:104) d − (cid:105) vol ⊥ d − z d − ∗ + 2 d − (cid:34) vol ⊥ d − δ d − ∓ vol ⊥ d − z d − F (cid:32) , d d − − d d −
1) ; (cid:18) z ∗ z QES (cid:19) d − (cid:33)(cid:35) (cid:41) (5.18)where z = δ defines the UV cutoff surface near the asymptotic AdS boundary, and z QES and z ∗ are linearly related to b by eqs. (5.15) and (5.17). For z QES (cid:28) z ∗ , thehypergeometric function becomes 1 + O [( z QES /z ∗ ) d − ], giving A (Σ R )4 G bulk + A ( σ R )4 G brane = L d − G bulk (cid:40) d − ⊥ d − δ d − − d − √ π Γ (cid:104) d d − (cid:105) ( d −
2) Γ (cid:104) d − (cid:105) d − vol ⊥ d − D d − (cid:41) + vol ⊥ d − (cid:18) Lz QES (cid:19) d − (cid:40) G eff + O (cid:34) G bulk (cid:18) z QES z ∗ (cid:19) d − (cid:35)(cid:41) , (5.19)where we have used eq. (5.9) to replace z ∗ with D in the first line. Note from eq. (5.15)that z QES /z ∗ ∼ [( λ b + 1) θ B ] / ( d − so the correction is indeed smaller than the otherterms shown here in high tension limit.Using the brane perspective, let us examine the various contribution to the gener-alized entropy on the right-hand side of eq. (5.19). Beginning with the leading term of– 55 –he second line in eq. (5.19), we find that it corresponds to Bekenstein-Hawking of theQES, i.e., G eff times the area of σ R . It is interesting to note that that there are nohigher curvature corrections to the generalized entropy of the QES as might have beenexpected from the Wald-Dong entropy formula. Turning to the first term in the firstline of eq. (5.19), we have the area law divergence associated with the two componentsof the entangling surface Σ
CFT at x = ± b . This leaves us with the second term in thefirst line. Upon closer examination can be recognized as the finite contribution to theentanglement entropy for a belt of width D , up to an additional factor of 2, e.g., see[16, 17]. Further, we note that both contributions on the first line of eq. (5.18) containa prefactor proportional to L d − /G bulk ∼ c T , which measures the number of degrees offreedom in the boundary CFT, e.g., [86].We can see that these results correspond approximately to the expected entropyfrom the brane perspective as follows: We begin by considering the contribution fromthe CFT to one side of the conformal defect, say x <
0. Imagine we begin with a singlecopy of the CFT in flat space (5.2), and evaluate the entropy of a belt of width D withentangling surfaces at x = − b and x = D − b . For this geometry, the holographicentanglement entropy becomes [16, 17] S EE = L d − G bulk (cid:40) d − ⊥ d − δ d − + 1 d − ⊥ d − δ d − − d − √ π Γ (cid:104) d d − (cid:105) ( d −
2) Γ (cid:104) d − (cid:105) d − vol ⊥ d − D d − (cid:41) , (5.20)where we have separated the area law contributions of the two components of theentangling surface. Now from the brane perspective in our system, the bath CFTreside in flat space for x < d geometry of the brane for x >
0. However the latter can be produced by makinga local Weyl transformation in the positive x domain: ds = δ x sin θ B ds CFT = δ x sin θ B (cid:0) − dt + dx + · · · + dx d − (cid:1) . (5.21)Note that this is geometry is not the induced metric (5.5) but rather we are consideringthe standard conformal frame where one strips off the factor of ( L/δ ) from the bulk One can argue that all of the higher curvature corrections to the Wald-Dong entropy must cancelagainst one another as follows: In the present case, these terms would arise from integrating outthe boundary CFT on the gravitating brane and so should be conformally invariant, e.g., see [108].However, by a simply Weyl transformation, the brane metric becomes flat and further both the intrinsicand the extrinsic curvatures of σ R vanish. Hence in this flat conformal frame, the higher curvaturecorrections to the Wald-Dong entropy individually vanish. Hence while these curvatures do not vanishin the original conformal frame, the higher curvature entropy corrections must all cancel against oneanother. – 56 –etric. Now the net effect of this Weyl transformation on the entanglement entropy(5.20) is to modify the cutoff appearing in the area law contribution for the surface at x = D − b , i.e., δ → ( D − b ) sin θ B (cid:39) z QES , where the latter assumes that θ B (cid:28) S (cid:48) EE (cid:39) L d − G bulk (cid:40) d − ⊥ d − δ d − − d − √ π Γ (cid:104) d d − (cid:105) ( d −
2) Γ (cid:104) d − (cid:105) d − vol ⊥ d − D d − (cid:41) + 14 L ( d − G bulk (cid:18) Lz QES (cid:19) d − vol ⊥ d − δ d − . (5.22)Now using eq. (2.7), the term on the second line can be recognized as the contribution ofone of the boundary CFTs to the Bekenstein-Hawking entropy of the quantum extremalsurface on the brane. Hence combining the above contribution (5.22) with that fromthe other copy of the boundary CFT (which extends to the bath for x >
0) and theDGP contribution to the Bekenstein-Hawking entropy, we precisely recover the leadingcontributions in eq. (5.19). Hence this simple CFT argument allows us to match theleading contributions in the holographic result with the expected entanglement entropy.
Above, we studied the set of candidate RT surfaces which intersect the brane. In fact(for θ B < π/ x planesanchored on the entangling surface Σ CFT on the asymptotic boundary and fall straightinto the bulk. By reflection symmetry about x = ± b , these planes trivially extremizethe area functional, which becomes A ( V ) = 2 L d − vol ⊥ d − (cid:90) V dzz d − . (5.23)A factor of 2 has been included above to account for the two planes at x = ± b . Unlike the surfaces considered in section 5.1, these planes do not intersect the braneand thus no islands are formed on the brane. The entropy in this no-island phase iseasily obtained from evaluating the area functional (5.23), which then yields A (Σ R )4 G bulk = L d − d − G bulk vol ⊥ d − δ d − , (5.24)where δ is again the UV cutoff in the boundary CFT. Further, we are only performing the Weyl transformation (5.21) for x > δ/ tan θ B , which corre-sponds to the intersection of the brane with the UV cutoff surface z = δ . Further, let us note that for the special case d = 2, the integral produces an IR divergence at z → ∞ . However, there is no such IR divergence for d ≥ – 57 – .3 Islands at T = 0 for d > θ B < π/
2, an island can only possiblyexist when − < λ b <
1; more specifically, for this range of the DGP parameter λ b ,there is a critical angle θ c < π/ θ B that supports the islandphase – recall that this critical angle was plotted in figure 22b. For θ B > θ c , the islandphase exists and is dominant. At θ B = θ c the entropies computed by the island and no-island RT surfaces equalize, leading to a transition to the no-island phase below θ c . Aswe shall find that θ c scales as (1 + λ b ) d − at its smallest, this precludes the possibility ofislands in the regime where the brane is well-described by QFT on semiclassical gravity– see footnote 5. This differs from the d = 2 case, where the island phase alwaysexists; furthermore, while the no-island RT surface in d > d = 2 produce an IR divergence and thus are never dominant.Let us begin our analysis by constraining the parameter space in which each type ofRT candidate surface exists. It is easy to see that the extremal planes of the no-islandphase exist if and only if θ B ≤ π/ It is slightly more involved to determine when theextremal surfaces in the island phase exists. For a start, the first equality of eq. (5.15)indicates that for θ B < π/
2, sensible extremal surfaces intersecting the brane can onlypossibly exist when − < λ b < From figure 22a, we see that this is the range of λ b for which there exists some θ B < π/ y = 0 or tothe horizon y = ∞ . Of course, this was our regime of interest, as this was the regime where a (nearly) massless gravitonis induced on the brane. Specifically, this can be seen as follows: Let us take the extreme case of λ b = 1 ( λ b = − θ QES = θ B − π/ θ QES = θ B + π/ λ b = 1, this implies that when θ B > π/
2, the RT surface falls straight into the bulk until it hits the brane, i.e., z ∗ = ∞ – see figure20. Now as θ B → π/ θ B < π/
2. For λ b = −
1, one can argue that for θ B < π/
2, the QES is stuck to the defect, i.e., z QES = ∞ . As increasing (decreasing) λ b beyond 1 ( −
1) means the DGP entropy contributionexerts a greater force pushing the QES towards the horizon (the defect), it follows that no QES existsfor θ B < π/ λ b > λ b < − i.e., have the RT surface anchored in the unphysical region behind the brane. – 58 –o be more precise, we must consider properties of the F function introduced ineq. (5.17). For − < λ b <
1, some (numerically deduced) facts about F ( d, λ b , θ B ) arethat it is decreasing in λ b and increasing in θ B . Moreover, F ( d, λ b , θ B close to 0) = − (1 + λ b ) d − θ − d − d − [1 + O ( θ B )] + Dz ∗ (5.25) F ( d, λ b , θ B close to π ) = (1 − λ b ) d − ( π − θ B ) − d − d − { O [( π − θ B ) ] } . (5.26)Since the former diverges negatively while the latter diverges positively, it follows thatthere exists a critical angle θ c for which F ( d, λ b , θ c ) = 0. For − < λ b <
1, we have0 < θ c < π/ θ c → , π/ λ b → − ,
1, respectively. The physical significanceof θ c can be seen from the second equality of eq. (5.16): for θ B above θ c , there existextremal surfaces which intersect the brane; as θ B → θ c from above, z ∗ , z QES , y
QES runoff to + ∞ as ∼ ( θ B − θ c ) − ; finally, for θ B < θ c , no extremal surfaces exist whichintersect the brane. In Figure 22b, we plot the critical angle θ c as a function of λ b forvarious d .Before continuing, let us briefly note a number of peculiarities which arise when | λ b | >
1. First, for λ b >
1, there exists a range of θ B (cid:38) π/ i.e., the RT prescription fails completely. This may indicatethat there is no CFT plus defect theory which can be dual to a bulk with this rangeof parameters – of course, the brane has a negative tension in this regime and so thereis no effective gravitational theory on the brane. Second, recall that as λ b → − λ b < − − < λ b < θ B ≤ π/
2, extremal planesanchored on the entangling surfaces to either side of the brane, which correspond to ano-island phase; and, for θ B > θ c , extremal surfaces which intersect the brane, corre-sponding to an island phase. As both types of surfaces exist for θ c < θ B < π/
2, theRT formula instructs us to choose the surface with the smallest area in this parameterspace. Thus, we consider the area difference: (cid:20) A (Σ R )4 G bulk + A ( σ R )4 G brane (cid:21) isl . − (cid:20) A (Σ R )4 G bulk (cid:21) (cid:8)(cid:8) isl . = − L d − vol ⊥ d − d − G bulk z d − ∗ F ( d, λ b , θ B ) (5.27)where we have used eqs. (5.17), (5.18), (5.24), and the hypergeometric function iden- In particular then, no islands form with λ b > θ B ≤ π/ – 59 –ity F (cid:20) , d d − − d d −
1) ; w (cid:21) = √ − w + (cid:16) wd (cid:17) F (cid:20) , d d −
1) ; d d −
1) + 1; w (cid:21) . (5.28)From eq. (5.27), we see that whenever the island- and no-island-type surfaces coexist,the island-type surface always gives a lower area and is thus the surface picked out bythe RT formula. Moreover, we see that entropy transitions continuously between theisland and no-island phases at the critical angle θ c where F ( d, λ b , θ c ) = 0. Altogether,we find that, for θ B < θ c , we are in the no-island phase where the RT surface is given byplanes falling straight into the bulk, and, for θ B > θ c , we transition to an island phasewhere the RT surface is given by extremal surfaces which intersect the brane and forman island.To gain intuition for the critical angle θ c from the brane perspective, we note fromeq. (5.19) that eq. (5.27) can may be approximated as (cid:20) A (Σ R )4 G bulk + A ( σ R )4 G brane (cid:21) isl . − (cid:20) A (Σ R )4 G bulk (cid:21) (cid:8)(cid:8) isl . = − L d − G bulk √ π Γ (cid:104) d d − (cid:105) ( d −
2) Γ (cid:104) d − (cid:105) vol ⊥ d − z d − ∗ + vol ⊥ d − (cid:18) Lz QES (cid:19) d − (cid:40) G eff + O (cid:34) G bulk (cid:18) z QES z ∗ (cid:19) d − (cid:35)(cid:41) (5.29)in the small θ B limit. Building upon the discussion given below eq. (5.19), we interpretthe RHS as giving a change in generalized entropy due to the introduction of the islandin the effective theory of the asymptotic boundary and brane. Namely, comparing withthe island rule (1.1), the first term on the RHS of eq. (5.29) gives the change in S QFT due to the introduction of the island, and the second term gives Bekenstein-Hawkingentropy of the QES. Hence, for θ B > θ c , the island phase is favoured as the introductionof the island reduces generalized entropy. For θ B < θ c , the QES ceases to exist andonly the no-island phase is possible.We briefly comment that, unlike for the CFT region considered in [1], the addi-tion of topological terms to the bulk gravity theory does not change the favourabilitybetween the island and no-island phases of the belt geometry. This is because sucha modification can only effect a topological contribution to the Wald-Dong entropy This can be proven using eq. (15.1.8) and (15.2.25) of [109]. – 60 –ormula and, for the belt geometry, the RT surfaces in both phases have vanishingEuler characteristic. Namely, the RT surface of the island phase has the topology ofan infinite strip while the RT surface of the no-island phase consists of two half-planes.Thus, the topological contribution would not favour one phase over the other.In closing, we note that, unlike the d = 2 case [2], we have found that in thesmall θ B limit, where an effective theory of gravity plus quantum matter emerges onthe brane, islands typically do not exist for extremal black holes in d ≥
3. To be moreprecise, eq. (5.25) and figure 22b suggest that θ d − c ∼ λ b . It is still possible to stayin the island phase by tuning 1 + λ b to scale as ∼ θ d − B . However, from eq. (2.7), wesee that this limit λ b → − + corresponds to G eff → + ∞ , leading to a breakdown ofthe semiclassical description of the effective brane theory [1] (as mentioned in footnote5.). We remark that, unlike for non-extremal black holes to be discussed in section 3,there is no immediate information paradox that arises as a result of the lack of islandsin the extremal case here. In this section, we specialize to the case of d = 2 which, as mentioned in the maintext, requires a slightly different treatment. We begin with a discussion of the inducedaction on the brane, supplemented with JT gravity. Next, we review the bulk AdS and brane AdS geometries. Finally, we study extremal surfaces serving as candidateRT surfaces to determine the entropy in the two phases, with and without an island,leading to the Page curve. At leading order in an expansion in terms of small braneangles, i.e., θ B →
0, our results precisely agree to those of [2]. However, we can alsoretain the subleading terms, which produce corrections due to the finite UV cutoff onthe brane.
We begin by briefly reviewing the modifications for the induced brane action in twodimensions – a more complete discussion can be found in [1].Let us start in the absence of JT gravity, considering only the brane action I induced induced by the bulk Einstein-Hilbert action (with cosmological constant) given ineq. (2.1), its corresponding Gibbons-Hawking action on the brane, and the brane ten-sion term I brane = − T o (cid:90) d x (cid:112) − ˜ g. (6.1)As we saw in section 2, the induced action evaluated for higher dimensions containscoefficients with factors of ( d −
2) (see eq. (2.6)), which prevent a naive substitution– 61 – →
2. Instead, redoing the calculation specifically in two dimensions, the inducedbrane action is found to be I induced = 116 πG eff (cid:90) d x (cid:112) − ˜ g (cid:104) (cid:96) eff − ˜ R log (cid:18) − L R (cid:19) + ˜ R + L R + · · · (cid:105) . (6.2)where the two effective scales are (cid:18) L(cid:96) eff (cid:19) =2 (1 − πG bulk LT o ) , G eff = G bulk /L . (6.3)Notice that while the first equality follows the same definition as in higher dimensions,the second one must be redefined for d = 2 (c.f. eq. (2.7)). The unusual logarithmic termabove arises from the nonlocal Polyakov action [110], which appears from integratingout the two-dimensional CFT on the brane – see the discussion in [1]. In the absenceof any DGP terms in the brane, extremization of I induced leads to an AdS brane withradius of curvature (cid:96) B related to (cid:96) eff in the same way as in higher dimensions ( i.e., through eqs. (2.4) and (2.7)): L (cid:96) eff = f (cid:18) L (cid:96) B (cid:19) ≡ (cid:32) − (cid:115) − L (cid:96) B (cid:33) . (6.4)Thus, as in the higher dimensional case, the large tension limit leads to (cid:96) B (cid:29) L and asmall brane angle θ B in eq. (2.4). In this limit, the brane moves towards the would-beAdS boundary at θ = 0, giving rise to a logarithmic UV divergence in eq. (6.2) as L/(cid:96) B → e.g., see recent discussions of quantum extremalislands in d = 2, e.g., [2, 13, 15, 113]). Following [1], we then choose the brane actionas I brane = I JT − πG bulk L (cid:90) d x (cid:112) − ˜ g , (6.5)with the JT action taking the usual form (again as in section 2, we are omittingboundary terms), I JT = 116 πG brane (cid:90) d x (cid:112) − ˜ g (cid:20) Φ ˜ R + Φ (cid:18) ˜ R + 2 (cid:96) JT (cid:19)(cid:21) . (6.6)The Einstein-Hilbert term, though topological, still contributes to the generalized en-tropy with weight Φ . With the addition of JT gravity on the brane in eq. (6.5), we– 62 –rrive at the following induced action on the brane, I induced = 116 πG eff (cid:90) d x (cid:112) − ˜ g (cid:104) − ˜ R log (cid:18) − L R (cid:19) + L R + · · · (cid:105) + 116 πG brane (cid:90) d x (cid:112) − ˜ g (cid:20) ˜Φ ˜ R + Φ (cid:18) ˜ R + 2 (cid:96) JT (cid:19)(cid:21) , (6.7)where we have redefined the topological part of the dilaton upon collecting the coeffi-cients multiplying an Einstein-Hilbert terms, i.e., ˜Φ = Φ + G brane /G eff . (6.8)Note that we have discarded the usual tension coefficient T o in eq. (6.5) and instead cho-sen the tension such that no cosmological constant appears in the first line of eq. (6.7)for simplicity. In eq. (6.7), it is clear that varying Φ yields an equation of motion sim-ply setting the radius of curvature on the brane to (cid:96) B = (cid:96) JT . The limit of small braneangle θ B , related to (cid:96) B still through the first equality of eq. (2.4), is therefore obtainedby taking (cid:96) JT (cid:28) L . Note that this leads to a logarithmic UV divergence in eq. (6.7)similar to the non-JT case, as mentioned below eq. (6.4). Similarly, the source-freeequations of motion for the dilaton can then be obtained by varying the metric andfurther shifting the dilaton, as discussed in [1].The above reviews our discussion of the induced action in [1]. However, we wouldlike to compare our results for the quantum extremal surfaces and the Page curveto those derived in [2]. To facilitate this comparison, we make the following fieldredefinitions φ = Φ G brane , φ = φ + Φ4 G brane , (6.9)˜ φ = φ − G eff log (cid:18) L(cid:96) JT (cid:19) , ˜ φ = φ − G eff log (cid:18) L(cid:96) JT (cid:19) , (6.10)giving the bare and renormalized values of the dilaton — we shall clarify the meaningof this renormalization shortly. In terms of the latter, induced action (6.7) now reads I induced = 116 πG eff (cid:90) d x (cid:112) − ˜ g (cid:20) − ˜ R log (cid:18) − (cid:96) JT R (cid:19) + ˜ R + L R + · · · (cid:21) + 14 π (cid:90) d x (cid:112) − ˜ g (cid:20) ˜ φ ˜ R + 2 (cid:96) JT ( ˜ φ − ˜ φ ) (cid:21) . (6.11)Here, the first line eq. (6.11) may be interpreted as the renormalized effective actionproduced by integating out the brane CFT, and the second line contains the renormal-ized JT action, which can be compared to eq. (2) in [2]. Here, ‘renormalized’ means– 63 –hat we have absorbed the logarithmic UV divergence that would otherwise appearin the induced action as L/(cid:96) B → φ → ˜ φ in eq. (6.10).As before, the dilaton ˜ φ acts as a Lagrange multiplier which fixes the brane geom-etry to be locally AdS with radius of curvature (cid:96) B = (cid:96) JT . The equation of motion forthe induced metric ˜ g ij , on the other hand, yields the dilaton equation of motion − ∇ i ∇ j ˜ φ + ˜ g ij (cid:32) ∇ ˜ φ − ˜ φ − ˜ φ (cid:96) JT (cid:33) = 2 π (cid:101) T CFT ij = − ˜ g ij L G eff f (cid:18) L (cid:96) JT (cid:19) . (6.12)In the final expression, we evaluated the renormalized CFT stress tensor (cid:101) T CFT ij usingthe function f defined in eq. (6.4). The standard discussions of JT gravity ( e.g., [13, 114]) refer to the source-free dilaton equation, i.e., the RHS vanishes, but this iseasily accommodated by a further shift ˆ φ = ˜ φ + (cid:96) JT L G eff f (cid:18) L (cid:96) JT (cid:19) . (6.13) Let us now review the geometry for our current setup. Due to the simplicity of AdS , wewill find it convenient to describe RT surfaces using global coordinates, even though wewill be considering Rindler time evolution, as in the main text. In global coordinates,we may write the bulk AdS metric as ds = L cos ˜ r (cid:2) − d ˜ τ + d ˜ r + sin ˜ r dϕ (cid:3) (6.14)where ˜ τ ∈ R , ˜ r ∈ [0 , π/
2] and ϕ ∈ [ − π, π ].In the AdS-Rindler coordinates, the AdS geometry becomes ds = L (cid:18) − ( r − dτ + dr r − r dχ (cid:19) , (6.15) Recall that we also removed the power law divergence corresponding to the induced cosmologicalconstant term by introducing a counterterm in eq. (6.5). As noted in [1], f ( L /(cid:96) JT ) = L /(cid:96) JT + O ( L /(cid:96) JT ) and hence this expression yields the expectedtrace anomaly (cid:104) ( (cid:101) T CFT ) ii (cid:105) = 2 × c π ˜ R to leading order in L/(cid:96) JT . But the latter also receives additionalcorrections due to the finite UV cutoff on the brane – see eq. (2.45) in [1]. Recall that the centralcharge of the boundary CFT is given by c = 3 L/ G bulk and the extra factor of two in the trace anomalyarises because the brane supports two copies of this CFT. Note that implementing this shift in the action (6.11) introduces a new cosmological constantterm. Hence an alternative approach would be to introduce a general brane tension T o in eq. (6.5)and then tune the latter to absorb both the corresponding (power law) UV divergence in the inducedaction and the RHS of the dilaton equation (6.12). – 64 –hich is just the special case of eq. (3.6) for d =2. Here, τ, χ ∈ ( −∞ , ∞ ) and oneexterior region is given by r >
1. As described in section 2, the AdS-Rindler coordinatesare useful for the description of vacuum AdS as a topological black hole, such that theboundary CFT is in a thermofield double state. The inverse temperature with respect totime τ is 2 π , giving the periodicity of iτ necessary for a smooth Euclidean continuation— we shall also define a dimensionful time and temperature shortly. Indeed, thesecoordinates describe a horizon at r = 1. Note that in d = 2, the boundary geometryis flat, i.e., it is simply two copies of R . The AdS-Rindler coordinates ( τ, r, χ ) arerelated to the global coordinates (˜ τ , ˜ r, ϕ ) in eq. (6.14) bytanh τ = sin ˜ τ cos ϕ sin ˜ r , tanh χ = sin ϕ sin ˜ r cos ˜ τ , r = cos ˜ τ − sin ϕ sin ˜ r cos ˜ r . (6.16)As described above in section 6.1, extremizing the brane action in eq. (6.7) withrespect to Φ (or eq. (6.11) with respect to ˜ φ ) fixes the intrinsic brane geometry to beAdS with radius of curvature (cid:96) B = (cid:96) JT . This becomes the θ = θ B slice of the AdS metric written as in eq. (2.3), where θ B is determined bysin θ B = L(cid:96) JT , (6.17)as in eq. (3.8). We write the induced metric on the brane as (cid:96) JT ds = (cid:96) JT (cid:18) − ( ρ − dτ + dρ ρ − (cid:19) = − π (cid:96) JT β dy + dy − sinh (cid:16) π ( y + − y − ) β (cid:17) . (6.18)The first line element with ( τ, ρ ) is simply the special case of AdS-Rindler coordinatesgiven in eq. (3.10) with d = 2. The light-cone coordinates ( y + , y − ) in the secondline element are those used by [2], whose results we wish to compare against. Therelationship between ( τ, ρ ) and ( y + , y − ) is given by τ = π ( y + + y − ) β = 2 πtβ , ρ = coth (cid:20) π ( y + − y − ) β (cid:21) . (6.19)Given that the TFD has temperature π with respect to dimensionless time τ , we haveintroduced the dimensionful time t = y + + y − where the temperature becomes T = 1 /β . On the brane, eq. (6.12) is easily solved for the dilaton profile in terms of ρ or y ± :˜ φ = ˆ φ + 2 πφ r β ρ = ˆ φ + 2 πφ r β coth (cid:20) π ( y + − y − ) β (cid:21) , (6.20) This is the same time coordinate introduced below eq. (3.6), though the relation β = 2 π R losesits meaning as there is no spatial curvature in d = 2. – 65 –here φ r is a constant introduced in [2] (see eq. (18) and discussion below (2) there).In the AdS-Rindler metric given in eq. (6.15), we introduce a surface of largeconstant r = r UV which will serve as the UV cutoff surface. Then following [2], we takethe induced metric on this surface as the background metric for the bath CFT, i.e., ds CFT = L r ( − dτ + dχ ) , (6.21)with the conformal defect at χ = 0. Now the light-cone coordinates y ± can be ex-tended to describe the geometry of AdS bulk, and in particular the bath region onthe asymptotic boundary near θ = π as well as the brane geometry given in eq. (6.18)at θ = θ B , by taking an AdS metric in the form eq. (2.3). Indeed, on the asymptoticboundary, with metric given in eq. (6.21), y ± are related to ( τ, χ ) with y ± = β ( τ ∓ χ )2 π , ds CFT = − (cid:18) πLr UV β (cid:19) dy + dy − . (6.22)As in higher dimensions, we are interested in computing the entanglement entropyof a boundary region R comprised of all of the points with | χ | ≥ χ Σ in the two baths(associated with the two copies of the CFT entangled in the TFD state). That is, thisregion is the complement of two intervals (‘belts’) centered on the conformal defectsin the two boundaries (which corresponds to the intersection of the brane with theasymptotic boundary – see figure 23). Focusing on a single Rindler wedge and on oneside of the brane, the entangling surface is located at a fixed χ = − χ Σ <
0, which wedefine as y + − y − b > b = β π χ Σ , (6.23)for all Rindler times τ . Similar assignments apply for the patches covering the otherportions of the boundary.Finally, we note that going to the asymptotic boundary (with ˜ r → π/ r → ∞ ),eq. (6.16) yields the relation of the global and Rindler coordinates on the boundary:tan ϕ = sinh χ cosh τ , tan ˜ τ = sinh τ cosh χ , (6.24)which allow us to simplify some calculations below. It will be useful to denote the(time-dependent) global coordinate angle of the entangling surface at χ = − χ Σ as ϕ Σ . We should note that the geometry in [2] can be seen as a Z orbifold of our setup (see section2.2). Hence they would only consider χ < y + − y − > y + − y − < y + − y − is always positive and θ = π, θ B correspond respectivelyto the bath and brane. – 66 – .3 Entropies: Island and no-island phases Now we turn to the problem of computing entropies using the RT formula in thebackground of the hyperbolic AdS black hole coupled to the AdS brane with JTgravity. Specifically, we wish to compute the entropy of the region R complementaryto belts centered on the defects, as described at the end of subsection 6.2. In the islandand no-island phases the RT formula equates the entropy to: (cid:20) A (Σ R )4 G bulk + φ QES (cid:21) isl . , (cid:20) A (Σ R )4 G bulk (cid:21) (cid:8)(cid:8) isl . . (6.25)The RT variational problem instructs us to consider extremal co-dimension two sur-faces Σ R in the bulk, which in AdS are simply geodesics. Although we are primarilyconcerned with evolution in Rindler time, the boundaries of the entangling surface aresimply four points; these can always be simultaneously placed on a surface of constantglobal time. This property, not present in higher dimensions, allows us to simplify theanalysis by using global coordinates as seen below.Now just as in higher dimensions, the minimization procedure yields two competingphases. At early times, the minimal surfaces cross the Rindler horizon avoiding thebrane and the entropy is given purely by the bulk length of the RT surface, as in thesecond of the expressions (6.25). This length stretches with Rindler time and leads toa growing entropy. At late times the RT surfaces go across the brane instead, leadingto an island where the contribution of the dilaton becomes important, as shown in thefirst of the expressions (6.25). As in the rest of the paper, we restrict to the regime ofsmall brane angle θ B .We begin by considering geodesics and their lengths in global coordinates. As iswell known, a convenient way to parametrize the RT surfaces on constant global time˜ τ is by using two anchoring points ϕ , ϕ , where geodesics are given bysin(˜ r ) cos (cid:18) ϕ − ϕ + ϕ (cid:19) = cos (cid:18) ϕ − ϕ (cid:19) . (6.26)such that the curves hit the boundary ˜ r → π/ ϕ and ϕ . The area (length in The fact that the endpoints reside at constant global time, together with the conservation ofthe charge associated with the global time Killing vector (obtained by dotting with the RT tangent)implies that the RT surfaces themselves must reside on constant global time slices. – 67 – = 2) of an RT surface with this trajectory is given by A = L (cid:88) i ∈{ , } tanh − (cid:34) csc (cid:18) ∆ ϕ (cid:19) (cid:115) − cos (cid:18) ∆ ϕ r i (cid:19) cos (cid:18) ∆ ϕ − ˜ r i (cid:19)(cid:35) (6.27)= L log (cid:20) (∆ ϕ/ (cid:15) (cid:15) (cid:21) − L (cid:18) ∆ ϕ (cid:19) ( (cid:15) + (cid:15) ) + O ( (cid:15) ) + O ( (cid:15) ) , (6.28)where ∆ ϕ = | ϕ − ϕ | , (cid:15) i = π − ˜ r i ( i ∈ { , } ) (6.29)are respectively the opening angle of the RT surface and the UV cutoffs (in the globalradial coordinate) at which the area integral is terminated, see figure 23.The leading order term in eq. (6.28) corresponds to the standard entanglemententropy formula of an interval on the circle [17, 115] (but allowing now for two differentUV cutoffs). We have also included the next-to-leading order terms as these will beimportant for computing corrections to entropy formulas on the brane.Now as usual, one must appropriately regularize the areas of the RT surfaces. Asexplained above, we place the cutoff surface at a large holographic radius r = r UV inthe Rindler radial coordinate. In terms of global coordinates, this describes the surfacesin (˜ τ ) = (sin ˜ r cos ϕ ) − ( r −
1) cos (˜ r ) . (6.30)Expanding to leading order in r UV , one finds that the UV cutoff is associated with alength in eq. (6.29) given by (cid:15) = 1 r UV (cid:115) τ ) + cosh(2 χ ) + O ( r − ) . (6.31)where we have used eq. (6.24). Here and below, we shall use (cid:15) to denote the cutoffat the end-point of the RT surface at the asymptotic boundary; (cid:15) , on the other hand,will either be a cutoff at the asymptotic boundary or due to the brane, depending onwhether we are in the no-island or island phase. Note that although the entropiesdiverge with the regulator r UV , these contributions will cancel once we consider thedifference between the island and no island phases, as seen below.Equipped with this, we can now compute the generalized entropy in the two phasesand reproduce the Page curve found in [2].– 68 – (cid:15) ϕr = r UV UV cutoff ϕ QES ϕ ϕ = − ϕ b Figure 23 : A slice of constant global time in AdS , showing the two phases of thegeneralized entropy. The two cutoffs (cid:15) , involved in the computation are associatedto the UV cutoff at the asymptotic boundary and the brane, respectively. The globalcoordinate angles ϕ , ϕ relate to the RT surface opening angle, while ϕ QES is the angleat which the RT surface intersects the brane and corresponds to the boundary of theisland. Recall the geometry is cut at the brane and continued by gluing it to anothercopy.
No-island phase.
We begin with the no-island phase. Here once again due to thesimplicity of AdS , the minimal surfaces lie on constant global time slices. The RTsurface consists of two pieces, one connected piece on either side of the brane withtrajectory given by eq. (6.26) where ϕ = − ϕ Σ and ϕ = − π + ϕ Σ (recall the definitionof ϕ Σ below eq. (6.24)). The total RT length is given by double eq. (6.28) (due to thetwo pieces) with both cutoffs (cid:15) , (cid:15) given by eq. (6.31). Substituting this into eq. (6.28)with ∆ ϕ = π − ϕ Σ and using eq. (6.24), the associated entanglement entropy in theno-island phase is (cid:20) A (Σ R )4 G bulk (cid:21) (cid:8)(cid:8) isl . = LG bulk log (2 r UV cosh τ )= 2 c (cid:20) βπδ cosh (cid:18) πtβ (cid:19)(cid:21) , (6.32)where we used the Brown-Henneaux central charge c = 3 L G bulk . (6.33)– 69 –n the second line of eq. (6.32), we have expressed the answer in terms of the dimen-sionful time t , as in eq. (6.19) (see also below eq. (3.6)) and the short-distance cutoffin the boundary CFT δ = β π r UV (6.34)in the y ± coordinates on the boundary . Eq. (6.32) matches the entropy from eq. (29)of [2], accounting for the fact that here the central charge is doubled since we includethe regions on both sides of the brane . For times much larger than the thermal scale, (cid:20) A (Σ R )4 G bulk (cid:21) (cid:8)(cid:8) isl . = 2 c (cid:20) log (cid:18) β πδ (cid:19) + 2 πtβ (cid:21) + O (cid:0) ce − πt/β (cid:1) , (6.38)which corresponds to the linear growth predicted by Hawking. Island phase.
Let us next consider the island phase. As explained in section 2, sincetranslations in Rindler time are an isometry, we can use this symmetry to bring theproblem to the ˜ τ = 0 = τ slice. Notice that this is also a symmetry of the dilatonprofile as is clear from eq. (6.20).We will leave point 1 anchored on the cutoff surface near the asymptotic boundaryat global coordinate ϕ = − ϕ Σ , as in the no-island phase. But, the RT surface willnow intersect the brane at its other endpoint. Here it is important to distinguishbetween two different angles appearing in the island calculation – see figure 23. First, To be precise, eq. (6.32) computes the entropy of R in a CFT with metric − dy + dy − and shortdistance cutoff δ — here, δ is both the proper distance cutoff and the cutoff in y ± . We may equivalentlytake the CFT metric to be the induced metric − (cid:0) Lδ (cid:1) dy + dy − , in eq. (6.22), with coordinate cutoff δ in y ± , corresponding to a proper distance cutoff L as measured by the induced metric. There is a typo in eq. (29) of [2]: inside the logarithm, it should be β/π rather than π/β . TheUV cutoff δ is also hidden. The full answer is obtained by applying the conformal transformation w ± = tanh (cid:18) πy ± R β (cid:19) = − coth (cid:18) πy ± L β (cid:19) (6.35)(mapping the vacuum to a TFD) to the entropy formula S [ − dw + dw − , δ ] = c (cid:20) − ( w + R − w + L )( w − R − w − L ) δ (cid:21) (6.36) → S [ − dy + dy − , δ ] = S [ − dw + dw − , δ ] − c
12 log( ∂ y + R w + R ∂ y − R w − R ∂ y + L w + L ∂ y − L w − L ) , (6.37)where y ± R = t ± b and y ± L = t ∓ b are the entangling surfaces on the R and L sides respectively. Wehave used the notation S [ ds , δ ] to denote entropy in a CFT living in ds with proper distance cutoff δ as measured by ds . Eq. (6.37) gives the length of the piece of the RT surface to one side of thebrane; eq. (6.32) is then exactly double eq. (6.37). – 70 – (together with ϕ ) characterize the trajectory of the RT surface, as in eq. (6.26),such that the trajectory, when maximally extended (even behind the brane), reachesthe asymptotic boundary at ϕ and ϕ . The opening angle ∆ ϕ appearing in eq. (6.28)is defined in terms of ϕ and ϕ as per eq. (6.29). Second, there is the global angularcoordinate ϕ = ϕ QES of the QES where the RT surface intersects the brane. In thelimit of vanishing brane angle θ B → ϕ QES → ϕ but, at finite θ B , ϕ QES (cid:54) = ϕ .While (cid:15) is still given by eq. (6.31), the regulator (cid:15) is now provided by the braneposition and is given by (cid:15) = tan − [tan( θ B ) sin( ϕ QES )] (6.39)= θ B sin( ϕ QES ) + θ B ϕ QES ) cos ( ϕ QES ) + O ( θ B ) , (6.40)which we use below perturbatively in the regime of θ B (cid:28)
1. From eq. (6.28), the areaof the RT surface (including the pieces to either side of the brane and to either side ofthe horizon) is given in terms of ϕ Σ and ϕ QES by (cid:20) A (Σ R )4 G bulk (cid:21) isl . = LG bulk log (cid:15) θ B sin (cid:16) ϕ Σ + ϕ QES (cid:17) sin( ϕ QES ) + Lθ B G bulk −
13 + sin ϕ QES (cid:16) ϕ Σ + ϕ QES (cid:17) + O (cid:18) Lθ B G bulk (cid:19) . (6.41)We can also write this in terms of the y ± coordinates of [2], reviewed around eqs. (6.18)and (6.22) (see also footnote 42). Placing the belt boundary at θ = π , y + − y − = b andthe QES at θ = θ B , y + − y − = a (matching the a and b of [2]), we find (cid:20) A (Σ R )4 G bulk (cid:21) isl . = LG bulk log r UV θ B sinh (cid:104) π ( a + b ) β (cid:105) sinh πaβ + Lθ B G bulk (cid:104) π ( a − b ) β (cid:105) sinh (cid:104) π ( a + b ) β (cid:105) − + O (cid:18) Lθ B G bulk (cid:19) (6.42)= 2 c β(cid:96) JT πδ ˜ δ sinh (cid:104) π ( a + b ) β (cid:105) sinh πaβ − c ˜ δ (cid:96) JT sinh (cid:16) πaβ (cid:17) sinh (cid:16) πbβ (cid:17) sinh (cid:104) π ( a + b ) β (cid:105) + O (cid:32) c ˜ δ (cid:96) JT (cid:33) , (6.43)where, in the second line, we have written the answer in terms of the CFT centralcharge c and cutoff δ (in y ± ) in the bath, given in eqs. (6.33) and (6.34); we have– 71 –lso used the proper distance UV cutoff ˜ δ = L on the brane (hinted at earlier beloweq. (6.12)) with the induced metric given in eq. (6.18) — see discussion in [1]. Usingeq. (6.17) (as well as (cid:96) B = (cid:96) JT ), we can write θ B = ˜ δ(cid:96) JT (cid:32) δ (cid:96) JT + O (cid:32) ˜ δ (cid:96) JT (cid:33)(cid:33) . (6.44)Eq. (6.43) is to be interpreted as the von Neumann entropy of the effective CFT span-ning the asymptotic boundary and the brane. The first term of eq. (6.43) preciselyrecovers the expected CFT result , while the higher orders in ˜ δ/(cid:96) JT may be inter-preted as corrections due to the finite UV cutoff on the brane. Curiously, the leadingorder correction in eq. (6.43) vanishes for the case of a zero-width belt b = 0, i.e., when R completely contains the baths. We may add eq. (6.43) to the bare dilaton profile φ ,given by eqs. (6.10) and (6.20), evaluated at the QES, to obtain the generalized entropy (cid:20) A (Σ R )4 G bulk + φ QES (cid:21) isl . =2 ˆ φ + 4 πφ r β coth (cid:18) πaβ (cid:19) + 2 c βπδ sinh (cid:104) π ( a + b ) β (cid:105) sinh πaβ − c ˜ δ (cid:96) JT sinh (cid:16) πaβ (cid:17) sinh (cid:16) πbβ (cid:17) sinh (cid:104) π ( a + b ) β (cid:105) + O (cid:32) c ˜ δ (cid:96) JT (cid:33) , (6.47)where we have included dilaton contributions from the QES points on both the left andthe right of the TFD. Recall that ˆ φ conveniently absorbs the part of eq. (6.43) whichbecomes logarithmically divergent on the brane as we take the UV limit ˜ δ/(cid:96) JT → To see this, we may apply the transformation between w and y R written in eq. (6.35) to S − L dw + dw − δ bath − (cid:96) JT dw + dw − ( w + − w − ) brane , L = c (cid:34) (cid:96) JT L ( w +QES − w − QES ) − ( w +Σ − w +QES )( w − Σ − w − QES ) δ (cid:35) (6.45) → S − L dy + dy − δ bath − (cid:96) JT dw + dw − ( w + − w − ) brane , L = S − L dw + dw − δ bath − (cid:96) JT dw + dw − ( w + − w − ) brane , L − c
12 log( ∂ y +Σ w +Σ ∂ y − Σ w − Σ ) , (6.46)where we have used the notation S [ • , • ] introduced in footnote 45, and y ± Σ = t ± b and y ± QES = t ∓ a correspond to the entangling surface and the QES respectively. (In this footnote, we have swapped thesign of y + − y − on the AdS brane relative to the main text, so that here y + − y − > y + − y − < – 72 –erivable from the renormalized matter effective action, and that the renormalizationof φ → ˜ φ ∼ ˆ φ is precisely designed to eliminate the UV divergence of the mattereffective action on the brane. The first line of eq. (6.47) matches exactly eq. (19)of [2], accounting for the doubling and quadrupling of the dilaton and von Neumannentropies here (since eq. (19) of [2] considers only one side of the TFD and they workwith an end of the world brane with bulk spacetime only to one side). The terms ofhigher order in ˜ δ/(cid:96) JT are the corrections due to the UV cutoff, inherited from the vonNeumann entropy in eq. (6.43).To find the location y + − y − = a of the QES, the RT prescription instructs us toextremize the generalized entropy given in eq. (6.47). Symmetry has already allowedus to restrict the QES to the same slice of Rindler time τ ∝ t = y + + y − as the anchoringpoint on the asymptotic boundary. It thus remains only to extremize eq. (6.47) in thespacial direction. Setting the derivative of eq. (6.47) in y + − y − = a to zero, we obtainthe extremization condition:6 πφ r cβ = sinh (cid:16) πaβ (cid:17) sinh (cid:104) π ( a − b ) β (cid:105) sinh (cid:104) π ( a + b ) β (cid:105) δ (cid:96) JT sinh (cid:16) πaβ (cid:17) sinh (cid:16) πbβ (cid:17) sinh (cid:104) π ( a + b ) β (cid:105) + O (cid:32) ˜ δ (cid:96) JT (cid:33) . (6.48)At leading order in ˜ δ/(cid:96) JT , this matches eq. (20) in [2] accounting for the fact that wehave two copies of the CFT versus a single copy of JT gravity. This equation can besolved for the QES position a in terms of the belt width b numerically or analyticallywith an additional expansion in φ r cβ (cid:29) a = b + β π (cid:34) log (cid:18) πφ r cβ (cid:19) − ˜ δ (cid:96) JT (cid:16) − e − πbβ (cid:17) + O (cid:32) ˜ δ (cid:96) JT (cid:33) + O (cid:18) cβφ r (cid:19)(cid:35) , (6.49)matching eq. (21) in [2] at leading order in ˜ δ/(cid:96) JT , again accounting for the doubling ofthe CFT. We see that the leading order correction due to finite ˜ δ/(cid:96) JT is to push theQES further from the bifurcation point at y + − y − = + ∞ .Having found the location of the QES, we may re-evaluate the generalized entropy In fact, the match between the first line if eq. (6.47) and (19) in [2] is exact even after keepingall terms collected in their “constant”. This can be checked by keeping all constant terms in the vonNeumann entropy calculation, described in eq. (6.46), as well as the topological dilaton contributionˆ φ . – 73 –f the island phase by substituting eq. (6.49) into eq. (6.47), obtaining (cid:20) A (Σ R )4 G bulk + φ QES (cid:21) isl . = 2 (cid:18) ˆ φ + 2 πφ r β (cid:19) + 2 c (cid:18) βπδ (cid:19) + 4 πcb β − c ˜ δ (cid:96) JT (cid:16) − e − πbβ (cid:17) − c β πφ r e − πbβ + c (cid:104) O (cid:16) ˜ δ /(cid:96) JT (cid:17) + O (cid:0) c β /φ r (cid:1)(cid:105) . (6.50)(We have also dropped terms of order c β ˜ δ φ r (cid:96) as these are inherently smaller than eitherthe c ˜ δ /(cid:96) JT or c β /φ r corrections.) The first line simply evaluates the generalizedentropy, given in eq. (6.47), at the bifurcation surface, i.e., taking a → + ∞ . Inparticular, we recognize the first term as giving the Bekenstein-Hawking result for thecourse-grained entropy of two black holes2 S BH = 2 (cid:18) ˆ φ + 2 πφ r β (cid:19) . (6.51)This classical contribution dominates eq. (6.50) in the limit S BH (cid:29) c and corresponds toeq. (30) in [2]. The other terms on the first line of eq. (6.50) evaluate the von Neumannentropy, given in eq. (6.43), after re-absorbing the UV divergence on the brane into ˆ φ .Specifically, the second term gives the UV contribution from the entangling surface onthe asymptotic boundary (also appearing in the no-island phase in eq. (6.32)), and thethird and fourth terms give finite contributions to the renormalized entropy includinga ˜ δ /(cid:96) JT correction. Moving to the second line in eq. (6.50), we have a correction due tothe displacement of the QES location a from the bifurcation point. Here, the dilatonand von Neumann components of generalized entropy both receive contributions atorder φ r β · c β φ r ∼ c βφ r . Note that there are no dilaton corrections at orders φ r β · cβφ r and φ r β · cβ ˜ δ φ r (cid:96) because the bifurcation point extremizes the dilaton profile . The order ˜ δ /(cid:96) JT correction in the QES location given in eq. (6.49) is not visible at the order shown ineq. (6.50). Collecting together the results of the previous subsection, we have two phases. At earlytimes, we have the no-island phase, with generalized entropy given by eq. (6.32). Over It is helpful to consider the coordinate (cid:37) = (cid:112) ρ −
1, in terms of which eq. (6.20) reads ˜ φ =ˆ φ + πφ r β (cid:112) (cid:37) and the brane metric (cid:96) JT ds = (cid:96) JT (cid:16) − (cid:37) dτ + d(cid:37) (cid:37) +1 (cid:17) , near the horizon (cid:37) = 0,resembles the standard flat metric − (cid:37) dτ + d(cid:37) in polar coordinates with (cid:37) the usual radial coordinate.The dilaton and the von Neumann entropy in eq. (6.47) should then have an expansion in terms ofnon-negative integer powers of (cid:37) QES . Eq. (6.49) gives the first corrections to (cid:37)
QES = 0 at orders cβφ r and cβ ˜ δ φ r (cid:96) JT , leading to the corrections mentioned in the main text. – 74 – S S BH t Figure 24 : Page curve for the equilibration of our topological black hole in d = 2. Weplot the entropy ∆ S = S ( t ) − S (0) of the subregion on the CFT which is associated tothe radiation, where we subtract the value of the entropy at t = 0.time, this entropy grows at a rate proportional to the temperature 1 /β and the number c of matter degrees of freedom participating in Hawking radiation, as emphasized ineq. (6.38). This growth, however is capped off by an island phase, where quantumextremal surfaces on the brane just outside the black hole horizon surround an island,containing a portion of the black hole interior, now belonging to the entanglementwedge of the bath. In this latter phase, generalized entropy is given by the constantvalue written in eq. (6.50) which is dominated by double the Bekenstein-Hawking blackhole entropy, as given in eq. (6.51). Viewing eq. (6.51) as the course-grained entropyfor the two sides of the black hole, this is precisely the expected maximal entropy ofthe system.To find the Page time τ P = 2 πt P /β marking the transition between the two phases,we equate the corresponding generalized entropies given in eqs. (6.38) and (6.50): τ P = 2 πt P β = 3 c (cid:18) ˆ φ + 2 πφ r β (cid:19) + log(2) + 2 πbβ − ˜ δ (cid:96) JT (cid:16) − e − πbβ (cid:17) − cβe − πbβ πφ r + O (cid:16) ˜ δ /(cid:96) JT (cid:17) + O (cid:0) c β /φ r (cid:1) . (6.52)Overall, we recover a Page curve, with entropy growing linearly in a no-island phaseup to the Page time, and saturating to a constant maximal value in an island phaseafter the Page time. In figure 24, we plot the Page curve after subtracting off the initialentropy (which includes the UV divergences from the asymptotic boundary). In this paper, we applied the framework introduced in [1], which uses Randall-Sundrumplus DGP gravity, to extend the discussion of quantum extremal islands in [2] to higher– 75 –imensional black holes. As reviewed in section 2, this setup precisely realizes thethree different perspectives of the holographic system described in [15]. From theboundary perspective, the system is described in terms of the d -dimensional boundaryCFT coupled to a conformal defect. The usual holographic dictionary then yields thebulk perspective, where the dual description is Einstein gravity in a ( d + 1)-dimensionalAdS bulk spacetime bi-partitioned by a d -dimensional brane. The brane perspectiveis an intermediate characterization of this system given by the d -dimensional effectivetheory induced by the bulk theory on the asymptotic boundary and the brane. Thatis, in this description, the boundary CFT spans the asymptotic boundary, which isnon-gravitational, and the brane which supports a gravitational theory by the usualRandall-Sundrum mechanism.We have considered the vacuum state of the system with respect to global time,which simplifies the bulk geometry to be pure AdS. However, as discussed in sections3 and 4, by viewing this setup in AdS-Rindler coordinates, the global vacuum can bere-interpreted as in terms of a massless hyperbolic black hole. This induces a similardescription of the brane geometry as a black hole of one lower dimension. The ‘two’asymptotic boundaries then play the role of bath CFTs in equilibrium with the blackhole on the brane at a finite temperature T = πR . Similarly, as explained in section 5,viewing our setup in Poincar´e coordinates, we have an extremal horizon in the bulk andon the brane. The latter was coupled to a (single) T = 0 bath CFT on the asymptoticboundary.While islands have been numerically studied previously in [30], our approach pro-vides a relatively simple setting in which analytic calculations are possible. In par-ticular, the doubly-holographic nature of our model reduces the entropy calculationsinvolving islands in the presence of massless hyperbolic, or extremal black holes of arbi-trary dimension to holographic entanglement entropy calculations in (locally) pure AdSin one dimension higher. From the d -dimensional brane perspective, when computingthe entropy of a boundary region R in the island phase, a quantum extremal surface σ R marks the boundary of an island on the brane stretching to the horizon; this islandbelongs to the entanglement wedge of the bath region R . From the bulk perspective,the RT surface of R runs into the bulk from its anchoring surface Σ CFT = ∂ R andintersects the brane at σ R . As noted in [30], the entanglement wedge of σ R stretchesthrough the bulk and is manifestly connected to the island on the brane in this higher-dimensional picture, despite the apparent disconnection in the effective d -dimensionaltheory. To determine the RT surface in an island phase, we must not only extremizethe area functional locally within the bulk, but also extremize with respect to the in-tersection of the RT surface and the brane. Since the deep bulk (IR) and near-brane(UV) contributions (further modified by DGP contributions) to the RT area, respec-– 76 –ively, can be interpreted as renormalized von Neumann and gravitational Wald-Dongentropies [1], this bulk calculation is equivalent to the island prescription of extremizinggeneralized entropy over candidate quantum extremal surfaces.The most striking difference between our holographic construction and the two-dimensional model of [2] is that, as detailed in section 6, JT gravity does not appearautomatically but has to be added by hand to the brane theory for d = 2, in analogyto the DGP terms in higher dimensions. However, this may be contrasted with theinduced gravity on the branes in higher dimensions, where adding a DGP term providesfiner control over the model, but is not strictly necessary for interpreting the braneperspective as an effective CFT coupled to gravity. Having added JT gravity as aDGP term, we showed in section 6 that applying the RT formula in the AdS bulk andincluding the DGP entropy, as in the d = 2 analogue of eq. (1.2), correctly reproducesthe results of [2] at leading order in an expansion in terms of small brane angles, i.e., θ B (cid:28)
1. A finite θ B imposes a finite UV cutoff in the effective brane theory, as shown ineq. (6.44), and therefore subleading corrections to entropy formulas appear in the islandphase – see eq. (6.50). Of course, with a finite UV cutoff, we would not, for instance,expect the holographic entropy to precisely satisfy the CFT transformation rules of theentanglement entropy used by [2] in deriving their results [1]. These corrections havethe effect of pushing the QES slightly further from the horizon, lowering the entropyof the island phase, and shifting the Page transition to an earlier time.As discussed extensively in [1], our braneworld construction clarifies further con-ceptual puzzles that appeared early discussions of quantum extremal islands in a holo-graphic framework, e.g., [2, 15, 113]. One particularly confusing feature of the islandrule is the (implicit) appearance of the entanglement entropy of the QFT degrees offreedom in the region R on both sides of eq. (1.1). Our model puts the explanation ofthis fact given in [2] on solid footing. The entanglement entropy in the left hand sideof eq. (1.1) computes the full entanglement entropy in the UV complete picture (theboundary perspective), while the entropy on the right hand side is to be interpreted inan effective, semiclassical theory (our brane perspective). In partiular, as noted in sec-tion 2, the interpretation of the brane perspective as d -dimensional Randall-Sundrumgravity coupled to a CFT only holds for the low energy physics at scales longer than theshort distance cutoff ˜ δ (cid:39) L . At shorter distance scales, gravity is no longer localizedto the brane. In contrast, the boundary perspective or the bulk perspective gives acomplete description of quantum state. By the standard rules of the AdS/CFT correspondence, the boundary and bulk perspectives givean equivalent descriptions of the physical phenomena. – 77 – on-extremal black holes in higher dimensions
As noted above, in section 3, we considered AdS-Rindler coordinates in the bulk, pro-viding a description of the pure AdS spacetime as a two-sided massless non-extremalblack hole. A similar black hole geometry is induced on the brane, coupled to and inequilibrium with bath regions on the asymptotic boundary in both Rindler wedges. Weconsidered the entropy of bath regions R complementary to belts centered around thedefects in the two Rindler wedges. This setup, from the perspective of the effectivetheory on the brane and asymptotic baths, is analogous to the two-dimensional setupat finite temperature considered in [2].We find, in particular, that the information paradox for eternal black holes andits resolution studied in [2] makes an expected re-appearance in higher dimensions, asreviewed in section 3.6. Again, this information paradox is resolved by the appearanceof a quantum extremal island when a second quantum extremal surface minimizes thegeneralized entropy in the island rule (1.1). Our holographic construction translatesthis competition between quantum extremal surfaces to the usual competition betweendifferent possible RT surfaces in the holographic formula (1.2). In particular, at latetimes, the minimal RT entropy is provided by a second extremal surface with compo-nents which cross the brane, as illustrated in figure 2. From the brane perspective, theintersection of this RT surface with the brane becomes the quantum extremal surfacesbounding the island in the black hole background. The island belongs to the entangle-ment wedge of the bath region R . Without the appearance of islands, the entropy ofbath subregions would grow ad-infinitum. With the islands however, the ever-growingentropy of the no-island phase is eventually capped off by the constant finite entropyof this island phase at late times. Further, our higher-dimensional discussion providesa simple explanation for the saturation of entropy: the connected pieces of the RT sur-face in the island phase are isolated to individual Rindler wedges and are thus invariantunder time translation ( i.e., forward boosts in both wedges).Recall that the global state is pure, i.e., from the boundary perspective, it is athermofield double state of two copies of the boundary CFT plus conformal defect.Hence the entropy of R is identical to that of its complement R . This gives a usefulalternative view of the evolution of the entropy. The region R consists of a belt regioncentered on the conformal defect in the two bath regions. Hence from this point ofview, we are considering the entanglement entropy of two isolated boundary regions A and B on either side of the corresponding eternal black hole in the bulk. This isessentially the same system studied in [94], except that here the spatial sections of thebath geometry are hyperbolic in the present case. As in [94], the entropy grows atearly times but then quickly thermalizes. In this case, the growth of the entropy stops,– 78 –ecause it is bounded by subadditivity, i.e., S ( A ∪ B ) ≤ S ( A ) + S ( B ). In fact, forthe holographic system, the late time entropy saturates this inequality which erases themutual information between two boundary subregions. The primary difference betweenthe framework studied in [94] and our setup, is the addition of a backreacting branewhich creates extra spacetime geometry for the RT surfaces to traverse in this late-time island phase and so delays the onset of this phase where the entropy is saturated.From the boundary perspective, this longer thermalization time relative to [94] can beunderstood as a consequence of the large number of degrees of freedom introduced bythe conformal defect.Further as in [2], we find that the island extends outside the event horizon, i.e., thequantum extremal surfaces appear outside of the horizon. If we focus on the entropyof R as above, this feature again has a simple explanation in our holographic setup,in terms of entanglement wedge nesting. Recall in the island phase, the individualcomponents of the RT surface yield the entropy of the individual belt regions on theboundary of either Rindler wedge. Since these belts are subregions of the full hyperbolicslice on which the corresponding CFT resides, the RT surface must remain within thecorresponding Rindler wedge. That is, the bifurcation surface of the Rindler horizon inthe bulk is the RT surface corresponding to either of the copies of the CFT in the TFDstate [93], and the Rindler wedge is the corresponding entanglement wedge. Hence, byentanglement wedge nesting [116, 117], the RT surface and entanglement wedge for anysubregion of H d − on the boundary must lie within the corresponding Rindler wedge.Finally it was straightforward to see from eq. (3.11) that the horizon on the brane isprecisely the intersection of the Rindler horizon in the bulk with the brane. Hence thequantum extremal surface on the brane, i.e., the intersection of RT surface with thebrane, must lie outside of the black hole horizon. This also means that if we considerregions R far away from the defect, the RT surface will pass close to the horizon. Thus,analogously to the situation discussed in [2], information about the horizon seems tobe contained in the entanglement of CFT regions of the bath which are furthest fromthe black hole. Extremal black holes in higher dimensions
In section 5, by taking a Poincar´e patch of the bulk, we considered an extremal blackhole on the brane coupled to a (single) bath CFT in a flat background. As in [2], wecalculated the entanglement entropy for a bath region R which corresponded to pointsgreater than some distance b from the conformal defect. In the case of extremal blackholes, we did not find a transition as the system was time evolved, but instead foundthat the appearance of an island is linked to the choice of brane angle θ B (or branetension) and the DGP coupling. – 79 –ue to the scale invariance of Poincar´e coordinates, it is clear that as we push theentangling surface out in the bath region, i.e., increase b , we proportionately reduce thesize of the island. Again, this behaviour reproduces the intuition suggested in [2] thatthe region near the extremal horizon deep in the gravitating region (our brane) is canbe contained within the far-away portion of the bath. Actually, our higher-dimensionalpicture shows that these regions are not far from each other at all — they are bothclose to the spatial infinity of the Poincar´e patch which corresponds to a single pointin the global frame. In the other extreme b →
0, we find that regions of the branearbitrarily close to the asymptotic boundary can be recovered by portions of the bathsufficiently close to the defect. This is in contrast to the two dimensional JT model,where a maximum island size exists.Interestingly, a further qualitative deviation from the two-dimensional case is seenat small brane angles θ . Recall that, in the two-dimensional JT model, the island phaseis always dominant for belt geometries in the extremal case [2]. In contrast, we havefound in d > θ B below some critical θ c >
0. As θ B approaches θ c from above, the quantum extremal surface of the island phase runs offinfinity ( i.e., towards the extremal horizon). For θ B < θ c , no quantum extremal surfaceexists on the brane and the bulk RT surface is simply given by two planes on eitherside of the brane running straight into the bulk. Since the area of these latter surfacesis IR finite in d >
2, their candidacy for RT surfaces must be considered even when thealternative island-phase surfaces exist. In fact, we find that θ c is precisely the angle atwhich the entropies of the no-island-type and island-type surfaces match – above thisangle, the island-type surfaces remain favourable as RT surfaces. The relevance of small θ B (and in particular θ B < θ c ) is that in this limit, the effective theory on the braneis described by Einstein gravity with small higher curvature corrections, which is themost interesting parameter regime. While the lack of islands for θ B < θ c is strikinglydifferent from the two-dimensional case, we remark that, in the extremal case, islandsare not required from an information-theoretic standpoint and their absence shouldperhaps not be terribly surprising. This is to be contrasted with the non-extremalcase, where islands are necessary, at all brane angles, to tame the otherwise unboundedgrowth of black hole entropy at late times and avoid the information paradox.Of course, an interesting question may be to examine how varying the geometry ofthe entangling surface affects the appearance of quantum extremal islands at T = 0.For example, rather than belt geometries, one might consider spherical regions bisectedby the conformal defect. – 80 – ot an ensemble In order to derive the island formula, a crucial ingredient was the appearance of worm-holes in the replica trick. In the two-dimensional models involving JT gravity studiedso far [39, 41], the existence of wormholes follows from the fact that JT gravity is de-fined by averaging over an ensemble of Hamiltonians. For example, JT gravity emergesas the low energy effective description of the SYK model [118–121], or has a definitionin terms of a matrix model [122].On the contrary, our construction relies only on the standard holographic rules ofthe AdS/CFT correspondence where there is no such averaging of the couplings in theboundary theory. This is in line with the general expectations for higher dimensionalholography. This lack of averaging characterizes the UV-complete description of thesystem, i.e., the boundary perspective. Nonetheless, quantum extremal islands appearin the effective description of the brane perspective and once again one likes to un-derstand them as remnants of replica wormholes in the limit n → i.e., replica wormholes in theeffective theory. In fact considering Renyi entropy calculations in the boundary theory,one sees that the corresponding bulk geometry induces connections between the differ-ent copies of the brane theories, i.e., replica wormholes on the brane [1]. This becomesparticularly clear in our setup where the brane lives in the bulk and does not serve asa boundary of spacetime. We emphasize that here this discussion implicitly relies onthe standard derivation of the RT prescription for holographic entanglement entropy[123, 124] in the bulk perspective, where again we assume that there is no ensembleaveraging. Following the logic of [52], one might be tempted to turn the logic around and, giventhe appearance of wormholes in the brane description of our model, conclude that thereis some form of ensemble averaging in the dual boundary theory. However, this line ofargument implicitly assumes a precise equivalence between the boundary theory andthe ‘bulk’ gravity theory (containing wormholes). We stress that this equivalence doesnot hold in our construction. Rather the gravitational theory on the brane is an effectivetheory and so the arguments of [52] do not extend to this situation. Instead, in our Ref. [56] formulates a point of view where integrating out the bath CFT generates an averagingover couplings in the theory of the conformal defect. – 81 –ituation replica wormholes appear, but wormholes connecting independent instances ofthe boundary theory do not play a role. For example, this implies that higher powers ofthe partition function of the boundary CFT with a conformal defect will still factorize.Nonetheless, this issue is certainly worth further examination since in two dimen-sions, replica wormholes have now been shown to play an important role in a varietyof situations, e.g., calculations of Renyi entropies [41, 50], the spectral form factor[122, 125], correlation functions [90, 126], and overlap of black hole microstate wave-functions [41, 50]. Apart from Renyi entropies, it is not clear how to reproduce theseeffects in our construction, or in higher dimensions more generally. Furthermore, itwas suggested in [41, 127] that in non-averaged theories wormholes might appear as aresult of some diagonal approximation. To obtain a full quantum gravitational answer,additional off-diagonal terms need to be added. Given that we have a system, wherewormholes appear in an approximate formulation, while at the same time having somecontrol over a UV complete description, one might hope that studying our system willgive an idea of how this suggestion might be realized.
Future directions
Having produced a setup in which quantum extremal islands can be studied with rel-ative ease, some possible avenues of further investigation were suggested above, but anumber of other possible extensions to the present work also come to mind.For example, one may consider information-theoretic questions similar to thoseraised in [2]. There, the authors investigated whether a protocol can be implementedto retrieve information from the island. In particular, the entanglement wedge of thecomplete left system plus an interval of the right bath contains an island that naivelyappears causally disconnected from the left and the right bath interval. However, byacting with operators in the left and right baths, it was argued that sufficient negativenull energy can be generated to pull information from this region into the left exterior,to be picked up by the left defect and bath. One could try to reproduce this protocolin our higher-dimensional setup using insertions of operators on the left and rightasymptotic boundaries. The negative null energy produced would then shift the bulkhorizon and hence the induced horizon on the brane.Recall that above, we described how in the present discussion the appearance of thequantum extremal surfaces outside of the horizon was a simple result of the nesting ofentanglement wedges from the bulk perspective. However, another question raised by[2] is whether this protrusion of islands outside the horizon violates causality. In par-ticular, the portion of the island of the baths outside the horizon appears to be causallyconnected to the defects. Naively, this appears to allow communication between thebaths and defects even if the coupling between these systems is severed. The resolution– 82 –f this paradox comes from noting that a splitting quench between the defect and bathsystems would inevitably create a positive energy shock causing an outward shift of thehorizon. It was argued in [2], using a JT version of the quantum focusing conjecture[128, 129], that this shift would have the final event horizon swallow the island, prevent-ing post-quench communication between the bath and defect. It would be interesting tore-create this problem in our setup to probe the quantum focusing conjecture in higherdimensions. From the bulk perspective, a splitting quench would be implemented by abulk end-of-the-world brane anchored asymptotically on the splitting surface [130]. In d = 2, the splitting surface on the asymptotic boundary can be obtained by a conformaltransformation from a full plane; in the bulk, the end-of-the-world brane can similarlybe obtained by a diffeomorphism from a planar brane in pure AdS. In d >
3, however,the calculations will become more complicated, e.g., the end-of-the-world brane will, ingeneral, backreact on the geometry such that the bulk is no longer locally pure AdS.Returning to the issue of extracting information from the island, entanglementwedge reconstruction [24, 131–137] allows us to recover information about the islandwith data from the boundary CFT in the corresponding boundary subregion. One in-teresting question would be to evaluate the expectation value of various CFT operatorsin the island, e.g., reconstructing (cid:104) T ij (cid:105) in the vicinity of the horizon. The latter is par-ticular interesting because while the appearance of quantum extremal islands pointedout a new resolution of the information paradox, this does not directly address theissue of firewalls [138, 139]. Here asking if the black hole horizon develops a firewallin the late time phase of the Page curve can be addressed by evaluating (cid:104) T ij (cid:105) on thehorizon. While a direct boundary reconstruction of the latter remains to be done, weare confident that no singularities arise in our framework. The reason is that in thebulk, the system is in the vacuum state and we are simply examining this state from aRindler frame of reference. Hence in fact, we expect that (cid:104) T ij (cid:105) = 0 on the horizon andthroughout the black hole solution on the brane. This is related to the fact that in the present paper, for the sake of simplicity, wehave chosen to work with a bulk that is pure AdS, i.e., the temperature was tuned to T = πR . The Rindler horizon in this geometry consequently corresponds to a masslesshyperbolic black hole. An obvious extension would then be to consider massive black We thank Ahmed Almheiri for raising this question. The vanishing of the stress tensor on the brane is an essential feature of our construction as theAdS d brane geometry must be a solution of the corresponding gravitational equations. That is, theCFT on the brane cannot provide a source in these equations (at least to leading order for large c T )otherwise the geometry would deviate from AdS space. Recall that while the brane CFT is in itsvacuum state, the bath CFT is coupled to the brane along an accelerating trajectory – see discussionunder eq. (3.11). This acceleration allows the bath CFT to achieve equilibrium at a finite temperature. – 83 –oles. Again, calculations will be made difficult by the fact that the brane and bulkequations of motion must be solved simultaneously with the former back-reacting onthe latter. In particular, the equilibrium configuration will now involve excitations ofthe CFT on the brane, i.e., the effective Einstein equations on the brane will be sourcedby the stress tensor of the boundary CFT residing there.Yet another direction to take would be to consider our setup from the perspective oftensor networks and error correction codes [135, 140–142]. For instance, as noted in [94],the MERA-like tensor network constructing the time-evolved CFT thermofield doublestate on the asymptotic boundary shares a similar geometry to codimension-one bulkspatial slices stretching through the bulk wormhole. One might then be motivated, as in[141], to view these spatial slices as supporting tensor networks implementing quantumerror correction codes between the bulk and boundary. It would be interesting to seewhat such a network would tell us about the effective theory (see e.g., [143, 144]) onthe brane and how information on the brane is ultimately encoded in the asymptoticCFT and defect theory. On a related note, one might also study the complexity ofthese brane configurations, for example, using the higher-dimensional bulk to probeholographic complexity conjectures [145–148], e.g., see [149].Above, we emphasized the effective character of the gravitational theory on thebrane with the appearance of a short distance cutoff in Randall-Sundrum gravity. How-ever, as discussed in [1], the brane perspective also provides an effective description ofthe coupling of the bath CFT to the conformal defect. In particular, it only accountsfor the couplings localized at the defect, which dominate at low energies, but ignoresthe subtle ‘nonlocal’ couplings, which can seen as coming through the AdS d +1 geometrywith the bulk description. Given the simplicity of our construction, it may provide auseful framework in which to further understand these nonlocal couplings, which im-plicitly provide subtle correlations between the island degrees of freedom and those inthe bath CFT [1, 150].Lastly, in order to explain the fast growth of entanglement at early times for largeregions, in section 4.2 we computed bounds on entanglement growth in hyperbolicspace. While they display the expected qualitative behavior, they are not particularytight. Instead, the difference between bounds and numerical data becomes bigger as χ Σ grows. It would be interesting to improve these bounds. Acknowledgments
We would like to thank Ahmed Almheiri, Raphael Bousso, Xi Dong, Roberto Emparan,Netta Engelhardt, Zach Fisher, Greg Gabadadze, Juan Hernandez, Don Marolf, Shan-Ming Ruan, Edgar Shaghoulian, Antony Speranza and Raman Sundrum for useful– 84 –omments and discussions. Research at Perimeter Institute is supported in part by theGovernment of Canada through the Department of Innovation, Science and EconomicDevelopment Canada and by the Province of Ontario through the Ministry of Collegesand Universities. RCM is supported in part by a Discovery Grant from the NaturalSciences and Engineering Research Council of Canada, and by the BMO FinancialGroup. HZC is supported by the Province of Ontario and the University of Waterloothrough an Ontario Graduate Scholarship. RCM and DN also received funding fromthe Simons Foundation through the “It from Qubit” collaboration. The work of IRis funded by the Gravity, Quantum Fields and Information group at AEI, which isgenerously supported by the Alexander von Humboldt Foundation and the FederalMinistry for Education and Research through the Sofja Kovalevskaja Award. IR alsoacknowledges the hospitality of Perimeter Institute, where part of this work was done.
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