JHEP12(2020)025 ]. The 1 c Springer October 19, 2020 October 29, 2020 December 3, 2020 : : : , Received Accepted Published Ignacio A. Reyes a Published for SISSA by https://doi.org/10.1007/JHEP12(2020)025 [email protected] , Dominik Neuenfeld, a [email protected] , [email protected] , dimensions, based on a brane world model introduced in [ . d 3 Robert C. Myers, ] at leading order; however a finite UV cutoff introduced by the brane a 2 2010.00018 a,b The Authors. c AdS-CFT Correspondence, Black Holes, Gauge-gravity correspondence

We discuss holographic models of extremal and non-extremal black holes in , [email protected] Waterloo, ON N2L 3G1, Canada Max-Planck-Institut für Gravitationsphysik, Am Mühlenberg 1, 14476 Potsdam, Germany E-mail: [email protected] Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada Department of Physics & Astronomy, University of Waterloo, b c a Open Access Article funded by SCOAP Keywords: ArXiv ePrint: find no quantum extremal islandsresults for agree with a [ wide rangeresults of in parameters. subleading In corrections.extremal two surfaces dimensions, moving further For our outward example, from theto these horizon, a and corrections shifting slightly the earlier result Page time. transition in the quantum islands in those modelssurfaces, and is does a not consequence makeblack any holes of direct the the reference appearance to well-understoodthe of ensemble phase averaging. quantum information extremal For transition paradox non-extremal islands of incalculation has RT of any the the dimension. right full behaviour Pagenumerically to We solving curve avoid two further is ODEs. possible show In in that the any case for of dimension. extremal these The black holes models calculation in reduces the higher to dimensions, we Abstract: contact with a bath in main benefit of our setupto is previous that work it in allows higher for dimensions. a We high show degree that of the analytic appearance control of as quantum compared extremal Hong Zhe Chen, and Joshua Sandor Quantum extremal islands madeholes easy. on Part the II. brane Black JHEP12(2020)025 ]. 34 49 16 20 21 24 5 – 3 41 12 bath = 0 T 57 6= 0 6 – 1 – Σ 9 30 , τ Σ χ 55 28 2 52 = 0 15 2 27 = 0 Σ 6 Σ d > τ χ 26 d > = 0 for Σ τ 49 30 53 = 0 44 64 T 1 64 3.5.1 Island3.5.2 phase No-island phase 6.1 Brane action 6.2 Bulk and6.3 brane geometries Entropies:6.4 island and no-island Page phases curve 5.1 Island phase 5.2 No-island phase 5.3 Islands at 3.6 The information paradox 4.1 General behavior4.2 of the islands The Page curve in 3.3 No-island phase3.4 for No-island phase3.5 for Time-evolution for general 2.1 Braneworlds in2.2 higher dimensions Two dimensions and black holes 3.1 Geometry on3.2 the brane Island phase at 1 Introduction Understanding the quantum description of black holesical remains physics. a central One question unresolved in theoret- questiontion. is In the his fate seminal of work,into information Hawking a during argued black that mixed hole in state a evapora- quantum of theory black radiation, holes independently evaporate of how the black hole was formed [ 7 Discussion 6 Two dimensions revisited 5 Extremal horizon in equilibrium with 4 Numerical results 3 Black hole in equilibrium with an external bath Contents 1 Introduction 2 Braneworld framework JHEP12(2020)025 ] 50 . If – (1.1) 22 R ], set an 11 of the bath – 9 R .  — which captures the N G bath 4 (islands)) ∂ ( A ] 15 [ ] and its extension to include quantum 19 islands) + – ∪ ) over all such choices, and if the latter yields 16 ]. The new approach builds on insights com- R – 2 – 1.1 ( 15 island rule – QFT 13 S  ]. This tension is colloquially known as the black hole 7 , combined with any codimension-two — and possibly dis- 6 ext islands R ].  7 min ]. 8 This setup can be interpreted as a idealized picture, where we split ]. It is best understood in a setting where a black hole is coupled to ) = 1 ]. R 21 ( , 58 – EE 20 ] — see also [ S 51 12 While reconciling Hawkings calculation with the idea that quantum gravity is uni- One way of sharpening the paradox is to consider the von Neumann entropy of the This approach has now also been applied in a variety of different situations involving black holes [ 1 also contributes a gravitational termOne in extremizes the the form right-hand of side themultiple of usual eq. extrema, Bekenstein-Hawking ( entropy. the correct choice is the one that yieldsand cosmology the [ smallest entropy for This formula instructs us totum evaluate the fields (semiclassical) in entanglement entropy the ofconnected bath the — region quan- subregions in the gravitating region. The boundary of the candidate islands black hole while theat second least region semi-classically. is Incalculation, far this the away, true situation, where entropy it gravitational ofshould was effects the be argued are Hawking calculated that radiation negligible, using captured instead the in of so-called a using region Hawking’s ing from holographic entanglementcontributions entropy [ [ an auxiliary, non-gravitational reservoirHawking — radiation. referred to asthe the spacetime into two regions: the first, in which gravity is important, is close to the This qualitative behaviour of thePage radiation’s curve entropy [ as a function of time is knowntary as the was a longstandingPage curve puzzle, in recently a progress controlled has manner made [ it possible to compute the the black hole radiates,entropy at of some the time black —must hole then known will decrease as in equal the theappeared. the Page subsequent evolution, time entropy That reaching of — is, zero subtle the the whenmust correlations the thermodynamic radiation, produce black between and hole the a has the quanta purification dis- emitted latter of at entropy early the and final late state, times in a unitary evolution of the full system. system begins in a purebetween state, the this entropy radiation gives and athe measure the entanglement of black increases the hole. amount monotonicallyradiation of is throughout entanglement According thermal. the to On evaporation thedynamic Hawking’s other process entropy original hand, of since calculation, unitary the evolution the blackupper would hole, bound require which that on is the the proportional thermo- entanglement to entropy its of horizon the area radiation. [ Since the former decreases as hole looks likethe an AdS/CFT ordinary, correspondence unitary [ information quantum paradox mechanical [ system, e.g., as suggestedHawking by radiation produced during black hole evaporation. Assuming the gravitational Of course, this is in tension with the assumption that to an outside observer, the black JHEP12(2020)025 – ]. ]. ]. 15 15 15 30 , 14 ]. ) agrees with 2 1.1 , the holographic ). As time evolves ]. Our model allows 1 1.1 d. Using the standard ’, the latter is called a 1 ]. It turns out that after shares a large amount of 14 R islands -dimensional holographic CFT d bulk perspective , the quantum mechanical systems ], taking into account that the RT ]. 18 – 60 , 16 ] is that it reproduces the three descrip- 59 1 – 3 – brane perspective ) gives a unitary Page curve. ] for a recent review. ). Hence as the black hole evaporates, the latter shrinks b. Finally, with the 37 1.1 1.1 1 c. An advantage of working in the bulk perspective is that 1 is controlled by the horizon area of the black hole which enters a. With the — see [ 1 R , the system consists of two (one-dimensional) quantum mechanical geometry, which also supports another copy of the two-dimensional 2 ) was motivated in part by analyzing a “doubly-holographic” model in [ 1.1 geometry. The latter effectively has two boundaries: the standard asymptotically The key feature of our holographic model [ One direction for progress is to understand the Page curve and quantum extremal Eq. ( For an evaporating black hole, an obvious choice for the island region which extremizes 3 ], we focus here on the holographic model which we introduced in [ tions of the underlyingFrom physics the discussed boundary above perspective, forcoupled our the to system doubly-holographic codimension-one consists model of conformal ofAdS/CFT a defect, [ dictionary, as this shown description in is figure translated to the bulk gravity perspective. The where the gravitational theoryour on previous the paper, brane we showed is thatdimension, well-approximated quantum and extremal by clarified islands Einstein can several gravity. appear ofHere, in In we the any will properties spacetime extend of ourfor earlier the work black doubly-holographic and model holes discuss the in coupledhigher presence [ to of dimensional bath quantum extension extremal at of islands a the two-dimensional finite scenario temperature. considered in That [ is, our analysis provides a surfaces that can also end on the Planck braneislands [ in higher dimensions. While34 limited results have been obtained onus this to front [ obtain analytic results, while being powerful enough to do calculations in the regime AdS AdS boundary and thePlanck region brane where — JT see gravityentanglement is entropies figure of supported, subregions which inusual is the rules bath referred can of to be holographic as computed entanglement the geometrically entropy using [ the of the bath — seeare figure replaced by theirlatter holographic has dual, an a AdS two-dimensionalholographic black CFT hole — inCFT see JT is gravity. figure replaced The everywhere with three-dimensional Einstein gravity in an asymptotically This model provides three differentboundary descriptions perspective of the physical phenomena:systems, first, which from are the entangledmechanical in systems a is thermofield coupled double to a state. two-dimensional holographic Further, CFT, one which of plays the the role quantum can appear. In particular,a this quantum occurs extremal for islandthe an appears Page old just time, evaporating behind this black configurationfurther, the hole, yields the horizon and the entropy [ in minimal of this entropythrough in the case second eq. term ( into eq. ( zero size and the island rule ( quantum extremal island the entropy functional isHawking’s the calculation. empty set, However, inentanglement if with which the radiation case quantum the in fields result the behind of region the eq. horizon, ( new quantum extremal islands this procedure yields a solution with a nontrivial region ‘ JHEP12(2020)025 ]. 64 is UV +1 d ]. In contrast, 67 – 65 orbifold quotient 2 Z ]. Let us also note that our 2 c. f. 15 f. According to the Randall-Sundrum 1 ]. The bottom row reduces to the top upon 1 )-dimensional Einstein gravity in an asymp- and [ – 4 – + 1 and the conformal defect appears on the equator of 2 -dimensonal brane, which intersects the boundary d 1 d ]. In the latter, we are using the global conformal frame − d 1 S ], while the bottom row displays the analogous descriptions × b. e. ] even more closely upon taking a 15 R 15 ] provides a natural generalization to higher dimensions of 1 e, where the system is described by an effective theory of Ein- quotient across the defect in the boundary or the brane in the bulk. 1 ] singles out the brane prespective as an effective low-energy 2 ], we also considered introducing an intrinsic Einstein term to the Z 1 62 , ], the gravitational backreaction of the brane warps the bulk geometry 61 63 – geometry coupled to a 61 +1 d and taking a Illustration of doubly-holographic models: the top row illustrates (a time slice of) the )-sphere — see discussion in section = 2 1 d − d. a. d Hence our construction [ This does not mean that the bulk description in terms of a(n infinitely thin) brane in AdS 2 description, in contrast to the boundary and bulk descriptions. complete. However, it ismore reasonable complicated to configuration expect which thatthe can brane the be theory bulk obtained has description within a can string fundamental be theory, cutoff. completed e.g., in see the [ UV by a the two-dimensional doubly-holographic setup consideredmodel in [ resembles the setupacross in the [ brane. Further,presented we on emphasize a that morea while or the finite less three equal UV different footing, cutoff perspectives the [ were fact that the RS gravity on the brane has perspective, shown in figure stein gravity coupled to (twothe copies boundary of) CFT. the In holographic [ brane CFT action, on analogous the to brane, the all construction of coupled Dvali, to Gabadadze and Porrati (DGP) [ latter describes the systemtotically in AdS terms of ( at the location of the(RS) conformal scenario defect [ — see figure creating new localized graviton modes in its vicinity. This mechanism allows for the brane where the boundary CFTthe lives ( on setting The boundary, brane andrespectively. bulk gravity perspectives correspond to panels a & d, b & e, and c & f, Figure 1. three perspectives of theof model our in construction in [ higher dimensions [ JHEP12(2020)025 ]. ], is 1 1 R (1.2) ) of the bulk 1.2 ]. In our case, the ] of the higher cur- 30 , we review the bulk 71 – ). However, expanding 2 68 1.1  ] lies in its simplicity. As we ) 1 ]. Further, as discussed in [ R 1 σ ], which is based on the Karch- ) to produce the expected grav- brane ( 1 G A 1.2 4 + ) R bulk ) can be easily understood as follows: the (Σ . G 4 . In particular, in the island phase, the RT A 1.2 2 ). In fact, the RT contribution also captures  2 – 5 – ], the transition between the phase without an R 1.1 1 Σ ext  ], when the brane supports an intrinsic gravitational ) and ( 1 1.1 min ) = ] — see figure ), e.g., see figure R ( 74 – ] for branes embedded in AdS. We also discuss the addition 1.2 EE S 72 80 – . As argued in [ ) describes the leading planar contributions of the entanglement en- R ] to the brane action. For the two dimensional bulk gravity case, we 75 , 1.2 64 63 ]. From the brane perspective then, quantum extremal islands simply arise brane. This intersection of the RT surface with the brane becomes the boundary is the usual bulk RT surface, i.e. an extremal codimension-two surface in the 18 ∩ – R R ) simply becomes the usual competition between different possible RT surfaces in 16 Σ 1.1 The remainder of this paper is organised as follows: in section In the following, we will study the question of quantum extremal islands for black The equivalence between eqs. ( In this case, the entanglement entropy of the corresponding boundary region Again, the bulk gravity perspective allows us to calculate entanglement entropies = Σ R numerics required to extract quantitative results are limited togeometry solving and few effective ODEs. action ofRandall our setup model [ presented inof [ a DGP term [ surface crosses the branecorresponding so entanglement that wedge. a portion Thussimply of the decribed the appearance by a latter, of well i.e. quantum understoodsetting. the extremal feature The of island, islands main holographic is is advantage entanglement of entropy includedwill our in in show, construction a the new here our and framework incomplementing allows [ previous us approaches which to heavily carry relied the on numerics calculations [ remarkably far analytically, holes in arbitrarygravity dimensions perspective. using As theisland emphasized purely and in that geometric with [ the descriptionclasses island ( of is nothing RT more surfaces than [ the usual transition between different higher derivative contributions matching the Wald-Dongvature entropy terms [ appearing in thethe effective gravitational competition action [ betweeneq. candidate ( quantum extremal islands,the holographic denoted formula by ( the ‘min’ in tropy of the boundary CFT, andthis so geometric matches contribution the first near termmatches the in the brane eq. ( also induced reveals EinsteinThis an term contribution Bekenstein-Hawking combines in term with the that theitational effective brane gravitational term contribution in action appearing eq. on in ( the eq. brane ( [ action, we must also includeσ a Bekenstein-Hawking area contributionof for the the islands brane seen region in the brane prespective. bulk term in eq. ( given by where bulk homologous to of boundary regions geometricallytropy with [ the usualwhen rules the of minimal RT holographictions. surfaces entanglement in en- the bulk extend across the brane for certain configura- JHEP12(2020)025 ]. ]. 1 2 (2.1) ]. 83 bulk gravity – 81 ,  island 1) 2 − L d ( R d σ R ) + Σ ). . We review the induced action on ]. Beginning with the g 1 ( 6 1.2 R  R g ], generally islands do not form in higher − 2 √ x )-dimensional Einstein gravity with a negative , we develop the numerics associated to some – 6 – +1 4 d + 1 d d Z ) to investigate under which conditions islands appear. appear in section bulk 1.2 1 = 2 πG d 16 examines an extremal horizon with a vanishing temperature, ], describe the connection to our model and introduce eternal black 5 = 2 3 bulk R I , we discuss our results and point towards some future directions. Σ , we construct eternal black holes on the brane in higher dimensions. As 7 3 case, these black holes are in equilibrium with the bath at finite temperature R our setup is described by ( ]. Hence, we use eq. ( The choice of RT surfaces on a constant time slice in the presence of the brane (coloured 2 = 2 d , 4 and 5. Section Throughout the paper, we ignore surface terms for the gravitational action, e.g., see [ 3 = 3 cosmological constant, 2 Braneworld framework 2.1 Braneworlds in higherLet dimensions us reviewperspective the holographic model discussed in [ the two-dimensional brane, includingWe the also introduction evaluate the ofshow corresponding JT that quantum gravity the extremal terms, brane surfaces cutoffFinally given produces and in in subleading the section [ corrections Page compared curve, to and the results in [ d and find that indimensions. contrast to However, this twoand dimensions is bath [ not are problematic, notDetails since actually for at exchanging the zero radiation special temperature and case the thus no black information hole paradox arises. between the black holeparadox [ and the bath,We which present has the the generalisland potential analysis and to for no-island create phases. theintegral an equations time information found In dependence in section the of previous the section and entropy, explicitly exploring evaluate the the Page curve for summarize the setup of [ holes. In section in the and so they do not evaporate. Nonetheless, there is a continuous exchange of radiation Figure 2. green), showing the different ingredients involved in eq. ( JHEP12(2020)025 d — and o (2.4) (2.3) (2.2) d T . spacetime 6 ] for details +1 . d 1 +1  , d o ) g (˜ LT geometry by AdS 1 AdS ˜ R ). Two of these spaces ]. We have separated − ˜ g bulk +1 d d 2.3 − 64 πG p slice near the asymptotic ] — see section 4 so that the brane position space — see [ x d d 15 d . − . We will work in a regime d L 1  brane Z d  /G ≡ 1 2 AdS ε ) can be used to cover a coordinate brane ds ∝ 1 , in the metric ( 2.3 πG + T 2 +1 ∆ = 0 16 . The solution for the backreacting brane d is determined by the brane tension θ dθ where π +  B θ ˜ g ≤ AdS θ − – 7 – θ 2 2 ], one finds the curvature scale to be , L p ≤ sin x 2) 84 d 0 where , d = ε/ 1 1 Z − +1 , and will tune ) d  T (1 T B ε ∆ 2 AdS ∆ θ 2 . Adding the DGP term is a natural generalisation to higher 2 ds d − o L with the curvature scale set by d = o T and θ T = . In this case, the metric ( o ( +1 on either side of the brane. d AdS CFT T B 3 − θ π 2 2 = L +1 ≤ d sin θ , i.e. at brane = ≤ I slice near the asymptotic boundary AdS B 2 B = 0 d 1 ` θ is the induced metric on the brane. As well as the usual tension term, we metric is dimensionless with unit curvature. This metric would cover the entire θ A timeslice of our Randall-Sundrum setup. In panel (a), we cut off the AdS denotes the bulk metric. We also introduce a codimension-one brane in the bulk d vacuum spacetime if we take ij b. ab ˜ g g a. +1 With the above construction, the induced geometry on the brane is simply AdS Since the brane is codimension-one, the bulk geometry away from the brane locally d using the Israel junction conditions [ see below. Then, tworealized such as spaces the are joined interfacedivides together between the along the this bulk two surface, spacetimegeometry geometries. and — in the With see half, brane figure this is patch but construction, with the the backreaction brane of the brane has enlarged the The AdS AdS is constructed by firstboundary cutting off the spacetime along an AdS takes the form ofwhere AdS the induced geometry— on the and brane so will itslices be is that useful of to AdS consider the following foliation of the AdS have also introduced ananalogous intrinsic to Einstein-Hilbert Dvali-Gabadadze-Porrati (DGP) term braneworld inthe gravity brane [ the tension brane into action,is in determined a entirely manner by dimensions of having JT gravity on a two-dimensional brane [ where with action where Figure 3. along an AdS are glued together in panel (b) and the brane is realized as the interface between the two geomeries. JHEP12(2020)025 ] . B ` 86 (2.6) (2.7) (2.5) ' , eff # (at least ` curvature. eff 2 ··· . As we will × 1 /` + )-dimensional 1 2 1  L  2 − ˜ ε R d ), we show the first ), we have or 1) 1 . 2.6 1 1 − d  d ,   ε 4( 2 B ε , to produce an embedding of /L RS /` − brane 2 2 eff G 2 T ij ` G L L ˜ , i.e. the brane location remains ) corresponds to Einstein gravity  ∆ R ∼ ) ≡ = ], whereas the ( . Similarly, one can consider the ij g . T ˜ 2.6 b (˜ R brane 85 2 eff c 1 λ ˜ ` R  2) 2 B ˜ λ/λ G is the time direction). Further there is ` +  − 2) R d eff 2) − ] brane 1 d 1)( − 2 /G πG 2 , L d , e.g., see [ − 4)( ) with − 2 eff – 8 – 16 b d d eff ` 1)( ( − ` bulk λ bulk by replacing the conformal defect in the boundary d − G ( (where the = ∼ /G L d " 1 1 2) 2 ( T T (1 + ˜ − g − ˜  c d d − ∆ − RS ˜ g 1 L S d ( G . This is achieved by setting ] for further details. p − simply considers the dual description of the above gravi- × 1 ∼ x p d ), we have tuned the “shift” ≡ = R T boundary, but also two copies of the boundary CFT on the x d brane c d eff 2.4 RS G d 1 Further, new (nearly) massless graviton modes localized in the 1 Z +1 G — see [ G ). Note that in the regime of interest (i.e. d Z 4 RS L brane perspective 2.4 eff 1 ' πG ˜ δ 1 πG 16 16 + = boundary perspective induced is given in eq. ( I ], when considered in “global” coordinates, the dual solution is naturally the boundary ε We arrive at the The In fact, working with the induced metric on the brane (as we do in the following), the short-distance 1 4 coupled to a negative cosmological constant.of In a(n the infinite) second sequence line of of higher eq.Since curvature corrections, ( the involving gravitational powers of equations of motion set the curvatures to becutoff on roughly the brane is and Hence to leading order, the above gravitational theory ( where and references therein. vicinty of the brane alsocan appear think and so that the integratinggravitational brane out action also the on supports the a brane brane gravitational CFT theory. of (or the We the form [ bulk gravity) induces an effective perspective by its gravitationalon dual. the Hence asymptotic this AdS brane, description as includes dictated the by boundaryan the CFT effective usual theory Randall-Sundrum with (RS) a scenario. finite UV Of cutoff course, set the by latter the is position of the brane, e.g., see [ also a codimension-one conformal defectbrane positioned reaches on the the equator asymptotic of boundary.by the The the sphere, central standard where charge expression the ofCFT the boundary of CFT the is conformalratio given defect of the has couplings in the defect and bath CFTs: tational system using thein standard [ rules ofCFT the on AdS/CFT a correspondence. spherical cylinder As described explain below, this ensures that thegravity. gravitational Implicitly theory in on the eq. brane ( the is brane essentially Einstein that isunchanged independent when of we the vary DGP coupling For the most part, we will be interested in the regime where JHEP12(2020)025 for . and b 0 λ which < 6 b 1 + ), λ ] and thus  2.6 80 . , 2 eff is the standard /` 79 bulk perspective 2 = 0 L ]. For small brane RS b λ 80 /G – 1 = 0 t 75 , = ] that this mass is a crucial 63 for the special case of Rindler Right τ 33 appearing in eq. ( 2 ) ], which we now briefly review. , but required , but we work in a regime such that b 0 2 2) λ > − = 4 d b d ( (1 + λ / 1  for eff 2 1 /` 2  L Lastly, let us note that 2 eff – 9 – 5 /` . 2 1 L  2 eff , we find that the induced brane action is non-local, a ) are irrelevant. /` ] showed that the gravitational theory on the brane was well approx- 6 2 2.6 1 L appear and are problematic in 4) ]. First, there are factors of − 2 defect d ( / 1 Rindler Left Our eternal black hole coupled to the CFT bath, as seen from the . As reviewed in section . However, the latter constraint is replaced by 0 . It turns out that in the case of a brane theory with negative cosmological constant, like A more careful examination in [ Similar factors of < 5 6 = 2 Figure 4. = 3 b imated as semiclassical Einsteinλ gravity with d the curvature squared terms of eq. ( indicate that the bulkd integration analysis leading tosignature this of result must the be tracebrane reconsidered anomaly. action for is In athese addition, changes, JT the we gravitational may two-dimensional relate action analogue our localized setup of directly on the to the that DGP of brane. [ Having accounted for 2.2 Two dimensions and blackIn holes two dimensions, we needto to make connection revisit to our [ setup for an accurate effective brane action and ingredient for islands tolimit exist, in since the which limit islandsit of cannot is vanishing be possible graviton created mass that sinceeffective coincides in their gravitational with the coupling area a Karch-Randall on becomes model,but the infinite. the not brane, graviton responsible Alternatively, and mass for is simply the island. depends thus correlated on the with the island size, RS gravitational coupling induced in the absence of athe DGP one term, we i.e. are consideringangles, here, the the graviton mass graviton is acquires proportionalvanishes a so as mass some we [ power take of the the zero-angle brane limit. angle [ It was suggested in [ for the background of interest),we the work in contribution of the these regime terms where is highly suppressed since JHEP12(2020)025 , . 2 2 ], 2 6 plus (2.8) AdS AdS CFT 2 brane, 2 spacetime) CFT ] described coordinates AdS 2 2 3 , this setup is 1 AdS AdS CFT / 2 , this is essentially the bulk with a horizon and 6 3 AdS , AdS spacetime. In particular, this # 2 1 . The Rinder wormhole, which now belongs to 2 5 ˜ − ρ as exteriors of an eternal non-zero 2 black hole on the brane by taking d 2 AdS spacetime with a brane lying along 2 2 ρ 3 theory spans both the bath and AdS + 2 AdS 2 plus brane theory is dual to a perspective, this entropy is obtained using dτ region. Invoking spacetime just outside the horizon, marking 3 2 CFT 1) 2 2 − – 10 – AdS AdS 2 AdS AdS ρ ( − bulk — this equips the " 3 2 B ` . In the brane perspective, we then have a CFT spanning ρ -quotient across the brane. We may alternatively take the AdS = 2 ∝ 2 Z ] as a thermal BCFT coupled to quantum mechanics. 2 Φ ds quotient across the defect, this, of course, is the alternative 2 Z ) to compute the corresponding entanglement entropies. Analogous ). We may reproduce the , with the former dominating at early times and the latter past a 4 2 1.2 ], up to a 2 . Taking a ] interprets the two Rindler patches of AdS 2 slice (figure 7 ), allowing for the possibility of islands in the 2 ], we choose ‘belt’ subregions consisting of intervals symmetric about the defect. The 1.1 With our setup in place, we can then consider subregions of the boundary CFT and Let us return to our braneworld to see how our setup mimics that of [ Ref. [ 2 details of the resultingThe entropy upshot is calculation that inas we two find sketched dimensions a in are competition figure Page provided between time. a in Notice no-island section that phase thesewith and phases are an now analogous island the to phase, QESs the no-island demarked and by island phases the of intersection [ between our bulk RT surface and the with a defect running throughin its figure middle. We aredescription thus of led the to setup the in boundary [ picture drawn use the RT formula ( to [ which also supports asame theory setup of as JT in gravity. [ boundary Illustrated perspective, in wherein figure thedefect bulk theory. More precisely,bulk the is Euclidean equated to path a integral thermal preparing path the integral Hartle-Hawking preparing a TFD state of two copies of a supporting a dilaton profile the left and right asymptotic boundary regions — the baths — and the Rindler- an AdS Rindler-AdS coordinates in the ‘left’ and ‘right’ exteriorprepared regions. by The the resulting Euclidean picturealso path induce integral is a drawn that horizon in on of figure the a brane. Hartle-Hawking In state fact, the geometry of the brane is itself Rindler- extremal surfaces (QESs) appear inthe the boundaries of an island,the stretching entanglement through wedge the of the bath complements to theabove. intervals. From the bulk perspective, we have an AdS on the boundaries ofof the a half-lines. region The consisting authorsendpoints of then in intervals on compute the both bath the sideseq. regions. entanglement of ( entropy From the the TFD includinggives rise the to defect a and competitiondominating with between a at no-island early phase and times an and island phase, the with the latter former at late times. In the island phase, quantum temperature black hole andspace, subsequently consitituting considers a bath couplingregions region. each and A exterior JT matter to gravity aalternatively is described flat placed on half- by the thehalf-lines thermofield (the double bath regions) (TFD) coupled state to of quantum a mechanics BCFT (dual to living the on two JHEP12(2020)025 ) 1.2 (see that  r 2 + Φ . The result 0 (that is, the B , we also find ˜ Φ θ B B )), whereas for  θ θ 6.38 1 brane G 2 must be accounted for when 2 (see eq. ( ) β (3 . ] at leading order in 3 πct/ 2 4 as πR 7 = -orbifold, hence factors of 2 2 β – 11 – Z brane with an extremal horizon. We then expect , where islands are lacking in the extremal case at 2 3 ], in the limit where the brane approaches the would- ] by a 2 2 ≥ is derived in section AdS ) d πR . Here one would instead take Poincaré coordinates which (2 / ], the appearance of an island caps off the entropy growth at 2 = 2 = 1 bulk and d defect T 3 .) B θ ] for AdS 2 , we also explicitly demonstrate that our bulk RT calculation using eq. ( 6 The Euclidean path integral (orange region) prepares the Hawking-Hartle state. The boundary by slicing through the bulk at a small brane angle corrections which, for instance, push the QES further from the horizon, lower the ). While we find perfect agreement with [ )). Thus, as in [ 3 ) 2 B 24 θ 6.51 It would be straightforward to use our setup to perform the zero-temperature analysis In section ( Recall our setup is related to that of [ AdS 7 O to infinity require thecontrasted inclusion with of our an findings islandsmall in localized brane angle around the horizon. (This is tocomparing be results. also covered in [ would equip the entanglement entropy of large regionsgravitating in the brane. bath to In require particular, the intervals inclusion stretching of from islands on some the location in the bath out corrections to these results dueis to the brane imposingentropy a of UV the cutoff island atno-island finite phase, phase, and no such lead correctionsbrane.) to appear a as hastened the Page bulk transition. RT surface does (Note not that, intersect in the the entropy grows linearly inlate the times no-island it phase iseq. dominated ( by thethe island expected and course-grained given entropy by ofthe two a system copies constant, of from the afigure black hole potential on information the paradox brane, rescuing (the resulting Page curve is shown in the bath region complementary to the belt. precisely reproduces the results ofbe [ high-tension limit of higher dimensions). For early times, we find that the entanglement Figure 5. black hole temperature brane. Namely, it isthe clear island from region the between bulk these picture intersection shown points in belongs the to right the panel entanglement of wedge figure of JHEP12(2020)025 )- 1 this (3.1) (3.2) brane − d = 2 d . ) is dual to . For 3.1 2 − d H R R , CFT 1 − boundary perspective CFT 2 d ]. The general metric is given by dH . 2 88 r , direction. In the following sections, 2 2 − − 2 + denotes the line element on a ( 87 d d − ) 1 r ω d 2 r − ( 2 H d dr − f 1 dH + − 2 2 2 defect – 12 – dt horizon r L 2 2 = 0 L R t ) ) = r r defect ( ( = f f τ − = L 2 L ds ] to higher dimensions. Indeed, it is straigtforward to re-interpret 2 CFT CFT Conformal defects along a CFT bath in the with a suppressed hyperbolic 7 Our eternal black hole coupled to the CFT bath, as seen from the effective , and Figure 7. . Each point in the Penrose diagram represents a hyperbolic space 6 , denotes the AdS curvature scale and 4 L The benefit of our Randall-Sundrum setup is that it allows great flexibility in generaliz- with the blackening factor Here, dimensional hyperbolic plane with unit curvature. This bulk geometry ( are characterized as havingcase nontrivial of neutral horizon black topology, holes with and a we hyperbolic horizon will [ be interested in the we shall apply our setup to extend the results3 mentioned here to higher Black dimensions. hole inIn equilibrium this with section, an weextremal external discuss black bath how holes islands in arise higher-dimensional in brane-world the models. presence Topological of black certain holes topological, non- ing the construction of [ figures is simply a point. Figure 6. perspective JHEP12(2020)025 (3.5) (3.3) (3.4) , the black , where the h 1 r ], where each − d 89 . However, with H ) corresponds to 1 − × 3.5 d R S , we still find a horizon × . Of course, this volume M. 1 R − L R = 0 2 d 1 . ω − . dH d H 1 , πR 1 N πR 2 1 2 ] − 2 d π G = = 1) vol 91 , 16 T ]) is still applicable. Hence, the brane dH ]. In this case, it is straightforward to − 1 2 d 90 92 , ( R ]. The entropy in eq. ( )) corresponds to the curvature scale of the 83 ,T + = 92 1 2 [ 3.1 ! is [ − ) dt – 13 – d 1 to denote the dimensionless volume of the spatial ω L − 1 N − 1 πR − , from the AdS-Rindler coordinates in the bulk. − G = d 2 ) 2 h d (2 4 , and we will see that the brane inherits a black hole r H L ) to investigate the appearance of quantum extremal . After an appropriate Weyl rescaling, the boundary H / 1 πR

2 CFT − vol 1.2 , the bulk geometry is just the AdS vacuum, our previous vol 2 2.2 d (2 = 1 ds − / H d h = r T = 0 S ), the relation between the position of the horizon = 1 . Note that despite the fact that = ω T 2 3.1 − ). In fact, the bulk geometry corresponds to the AdS vacuum (as = 0 (introduced in eq. ( d ω ω 3.2 R ), but we are describing this geometry with the AdS-Rindler coordinates = M = 0 . Since with , and the ‘mass’ parameter M 4 with temperature from eq. ( M . In fact, this bulk geometry provides a higher dimensional generalization of the o 1 L T − d = H Our aim will be to use eq. ( Following the brane world construction outlined in the previous section, we locate In terms of the dual CFT, we are considering a pure state (i.e. the vacuum) in the In the following, we will consider the special case of a topological black hole with Turning back to eq. ( h r corresponds to the time direction in each of the CFTs. × tension construction discussed in section metric with temperature islands, from the brane perspective, where (two copies of) the boundary CFT are supported a codimension-one defect atcorresponds the to center a brane of which each cuts— through CFT. see the figure bulk By and the orthogonallydiscussion intersects holographic of the dictionary, horizon the this braneposition geometry in (above and the in bulk [ is determined precisely as described above in terms of the brane R the entanglement entropy between theinterpreted two as copies the of the entanglementconformal CFT entropy — between frame. and two alternatively, halves can ofcorresponds From be the to the a sphere point fine in tuning of the of original view the temperature of to the CFT, masslessness of the black hole conformal frame where thean boundary appropriate geometry corresponds conformal to transformation, we produce the TFD state on two copies of at expected for where the metric resemblesevaluate that the entropy of and black the hole temperature [ of the black hole Here and in the following,boundary we use geometry, i.e. the volume measuredis by infinite the and metric we must introduce an infrared regulatorvanishing — mass see below. R hole mass resides on a spatialmetric geometry for each CFT reads and hence the scale spatial geometry. The full boundary geometry is then two copies of a thermofield double (TFD) state for two copies of the boundary CFT [ JHEP12(2020)025 , 3.5 = 0 < χ < in the Σ Σ τ χ . While R − Σ Σ τ and Σ χ -dimensional bulk ± = + 1) d χ ( The entanglement entropy at ) and arbitrary 8 R ∂ = 0 , we are considering eternal black Σ χ 2.2 we will argue, using results obtained below, on each side of the defect as a function of time 3.6 – 14 – R ) below. Of course, since the global state which we are 3.7 away form the defect. Σ χ ]. In section . This also implies that from the = of interest consist of those points on a CFT timeslice which R 15 χ R , we will derive expressions for the area of three special cases 3.4 to . The second and third special cases will be RT surfaces -dimensional effective gravity on the brane, i.e. the region bounded by is introduced in eq. ( d χ 3.2 = 0 Σ . The regions τ 2 ], the volume of the corresponding surfaces grows linearly with time. At some 93 We remind the reader that as described in section The calculation of the time-dependence of the area of RT surfaces will proceed in two The coordinate in both CFTs, including the conformal defects. 8 Σ considering is pure, we couldχ equivalently discuss the entanglement entropy of the belt regions therefore continuously exchange radiation.this leads If to island information loss are [ that not also accounted in for higher appropriately, of dimensions the the joint presence system of of black islands holes makes and the radiation entanglement compatible dynamics with unitarity. reduced to one of those three cases. holes which do notblack hole evaporate. on the Nonetheless, brane and from the the fields effective on the brane asymptotic point boundary of are in view, contact, the and can no-island phase which eitheror end end on on entangling entanglingthese surfaces surfaces special located cases naively atreconstruct might the the seem time-evolution defect not of ( that to the the contain time-evolution entanglement enough of entropy, any information we symmetric will to RT surface argue completely in in the section no-island phase can always be steps: in sections of extremal surfaces.Rindler The time first one will be RT surfaces in the island phase anchored at of an RT surface.surfaces We will starts see out in in theknown the remainder [ of no-island this phase, sectionpoint i.e. that its connects at throught volume early the times, willsmaller the horizon. area have RT and grown As gives so the is large, correct well entanglement that entropy. the RT surface in the island phase has they connect through the horizon,through which the we brane, will call whichapparent the from we the no-island will phase, call orthe they the intersection connect of island the phase. RTcontributes surface to and The the the entropy reason brane of perspective, is for a the those quantum appearance names entremal of island, is islands which now is simply explained as a standard phase transition entropy associated to symmetric regions — see figure are further than ais distance evaluated using thein holographic the prescription introduction, of the the corresponding bulk RT perspective surfaces and can as be described in one of two phases. Either in this black hole geometry on the brane. Further, we will compute the entanglement JHEP12(2020)025 1 ). − 2.3 (3.7) (3.8) (3.9) (3.6) = is only r through . We can geometry. o T ) resembles 1 = 0 = 0 − r d 3.6 r H and rescaling the × ) creates additional 0 ) are dimensionless. R 0 , → ≥ 3.6 . ! o 1 T − 2 d = 0 ω, M . χ 1 . dH 2 2 − − r 2 d 2 +  does yields a curvature singularity. B 1 L 0 ` 2 χ dH − .  2 → dr 2 in the bulk spacetime is then given by B r r = θ B , which is tuned in relation to the curvature L B θ ) + θ and an apparent singularity at sin 2 2 + cosh πR 1 – 15 – 2 = dτ ) where ). Lastly, note that since the Rindler wedges are , which is determined by the tension ± (2 B B / 1) dχ ` 3.1 θ = 3.3 = cot − = r = 1 χ = 2 1 2 r θ − T ( 2 d , such that the coordinates in eq. ( − ) by taking the massless limit sinh

dH L r 2 2 3.1 r L In fact, we can extend the coordinates smoothly through the → r = 9 2 where we can exit the region behind the (inner) horizon at ds and r R τ → ], this means that a brane with positive tension (i.e. 1 t , each of which hosts one copy of the boundary CFT on the of the hyperbolic geometries ( ) with R 2.4 → ∞ From the bulk perspective, the CFT defects are dual to a co-dimension one brane, For each CFT, we introduce a codimension-one conformal defect (with zero extrinsic This is in contrast to the general metric ( r 9 As noted in [ geometry by its backreaction. Of course, the backreaction of a negative-tension brane would eq. ( The trajectory of a hypersurfaces of constant which spans a slice ofasymptotic constant boundary extrinsic at curvature the of location theit bulk of is the spacetime CFT and convenient defect. intersects to the The In write order brane the to is describe bulk located its metric at trajectory, in constant terms of the slicing coordinates in eq. ( curvature) at the centerslicing coordinates of for the the hyperbolic hyperbolic boundaries, spatial such geometry. that It is convenient toIn choose these coordinates, the location of the conformal defect is simply a reparametrization ofa pure coordinate AdS, singularity. itinterior is to clear negative that theand singularity enter a at new set of Rindler wedges. at a second Rindlerat wedge. The bulk spacetimeAs thus noted has above two (in asymptotichas terms regions, a of located (dimensionful) the temperature dimensionfulscale coordinates), the corresponding TFD state which is obtained fromcoordinates eq. ( Although the underlying geometrya is black simply hole the metric AdS withalso vacuum, horizons extend the at the metric spacetime ( at a fixed time-slice through the bifurcation surface and arrive To set the stagebrane for geometry. the As noted following above, calculations, the we bulk metric will is start described by by AdS-Rindler discussing coordinates the bulk and 3.1 Geometry on the brane JHEP12(2020)025  1 (3.10) (3.11) From the 11 , and hence one 2 ). Further, this ) B 3.6 L/` in both the left and ( Σ , − χ ! − 2 = 1 − 2 d B ) induces a Rindler horizon . We note that the induced θ χ < Σ dH χ . We can therefore treat either 2 = 1 . B ± ρ = cos . θ Σ and r χ ρ = + Σ 1) 2 ). cot 2 1 − χ − d 2 π/ . We are interested in the entanglement − 2 dρ r 2 8 = χ > χ ≥ − ( ρ cosh 2 B χ θ L + , while the geometry to the other side of the 2 B θ – 16 – corresponds to sinh dτ 1) = r cot 1) − = 0 2 ≤ − r ρ 2 ( χ ρ 2 B ( ` plane and crossing the Planck brane. In other words, the − and therefore we will only consider positive tensions in the sinh

r 2 B 10 = 0 = 0 ` Σ Σ . = ), as one would expect from the bulk perspective. τ τ 2 π -coordinate system (unless 2 r = ≤ ), with ds = 1 τ B ), we can determine the induced metric on the brane. After a short ρ θ 3.6 in the ≤ 3.9 ] to higher dimensions. 0 2 comprised of the combined regions = 0 ρ R ]. As already mentioned, this setup generalizes the two-dimensional framework , for which the brane theory is well described as Einstein gravity coupled to two 96 – B L ` 94 All calculations below will be done for the case of positive tension branes. However, Using eq. ( For such a (positive-tension) brane, the bulk geometry to one side of the brane can be We thank Raman SundrumHowever, for it explaining is this interesting point to to us. note that ' 10 11 B right CFTs. Hence the entanglingsitting surfaces a of constant interest have distance twometric away components on from (in the the each latter CFT) defect surfaces at is proportional to cannot reach We will start oursurface lying analysis in by the calculatingRT surface the is in area the connectedentropy of phase of the — see RT figure surface for an entangling θ copies of the boundaryinterpret CFT (with the a intersection high ofeffective cutoff). the gravitational The brane theory. reason and is the that RT in surface this3.2 as limit, bounding we can an Island island phase in at this temperature felt by the accelerated boundarysee agrees [ with the temperature ofpresented the in CFT, [ e.g., when it comes to interpretation, we will be particularly interested in the case where on the braneboundary (at perspective, thisdefect behavior is is readily in explainedSundrum thermal description by equilibrium of the the with factdynamical brane the that gravity perspective, is the surrounding this coupled behaviour conformal CFT. to arises the because bath In the CFT region along the of an effective accelerated trajectory, Randall- so that the where we have changed the radial coordinate with This brane metric again takes thedemonstrates form of that an the AdS-Rindler metric, Rindler cf. horizon eq. ( in the bulk (at side of the brane as an AdS-Rindler geometry whichcalculation, is one cut finds off by the brane. on a negative-tension brane following, i.e. described by eq. ( brane is given by the same metric with remove geometry. However, let us add that there is no (nearly) massless graviton induced JHEP12(2020)025 ) → 3.6 ξ (3.15) (3.16) (3.17) (3.12) (3.13) (3.14) ]. Even 1 plays the ξ , ) below), since # 2 χ , − 3.22 2 d . . ! sinh ! dH 2 . Rather, ) . ) ξ 2 1 2 dξ ζ ! ζ 2 2 − ζ r , and hence the shift symmetry − 2 2 2 E d r + slice) is invariant under (1 + ds √ 2 2 τ dH ζ ζ 2 + + (1 + ) = 2 2 , ) + dζ ζ 2 ξ ) 2 1 + ζ 2 γ ξ dτ 

2 dξ tan 2 ∂ξ ∂ζ cos − det( (1 + d 2 + (1 +  ) ζ q 2 2 !

2 ) ζ R 3 L 2 dξ Σ − ζ 2 Z d . By time-translation invariance, we know that = ζ ) – 17 – 2 π which reads 2 2 (1 + + ζ 2 ± ) = ds (1 + − 12 2 2 d R ζ = 2 ζ H ξ (1 + (Σ dζ + χ , for the profile of the RT surface, the induced metric on 1) vol 1 + A 2 2 − ) 1) const, but the latter is absent in higher dimensions. How- 

d ξ − 2 ( 2( d + ∂ξ ∂ζ ζ cosh L L 2( χ  2 L = = r 2 E → ) = ζ " = γ 2 χ ds ) = D L 2 2 2 ζ det( = ds 2 ind (1 + ds slice in the new coordinates, ] 97 = 0 , 1 Note that the area functional does not explicitly depend on Making the ansatz In two dimensions, the analysis of the RT surfaces is simplified because the metric ( τ const, which will simplify the following. Note that the full metric takes the form 12 + role of an angular coordinate and its associated Hamiltoniandoes not is extend conserved. to This the full allows spacetime us metric. in higher dimensions, this variationextremizing takes the a fairly RT simple surfacemetric form can (see be eq. ( cast as an effectively two-dimensional problem with subject to the correctboundary boundary condition conditions. is determinedboth Here, by sides a the of RT few the surface observationsbrane, ending defect. are we at can in Alternatively, also the since order. consider entanglingvary our a surface with The setup family on respect of is bulk to reflection-symmetric extremal the across surfaces point the which end of on intersection the of brane the and brane and the RT surface [ To obtain the correct RT surface, we now need to extremize the area functional with metric determinant Hence the geometry ofξ this spatial slice (or any constant these surfaces takes the form such that the horizon isthe located RT surface at liesthe on a constant Rindler time slice and hence we consider the metric on has a shift symmetry ever, we can find avia similar [ simplification by going to a different coordinate system defined JHEP12(2020)025 — ξ (3.18) L R ) from the R L − CFT CFT CFT CFT as a function of ζ ) or away ( , + ! horizon 1 − 2 2 − − , and positive DGP coupling. d d ) ) (1) 2 2 ∗ in the boundary is composed of the two ζ ζ O R ∼ as a coordinate of vacuum AdS extended ) B ∗ (1 + (1 + θ ζ 2 2 ∗ is going towards ( QES ζ ζ

ζ , ξ , i.e. ) o 2 – 18 – defect defect T ζ QES ζ conformal conformal ) in both the left and right CFTs. Note that the right ( Σ . More generallly, the sign starts out negative and χ (1 + 9 2 ζ v u u t ∗ ∗ ) are positioned where the brane (green) reaches these boundary ζ which is the turn-around point for ζ ± = sinh π ∗ Σ = ζ has been reached. R ξ ξ and ∗ Σ dξ ζ dζ tan = = 0 ζ ξ ) Σ or (where increases. In the latter case, where the RT surface does not turn around , ξ Σ ξ = 0 ∞ R ( R . The sign depends on whether χ This figure shows the RT surface and various quantities defined in the text for the RT R > χ This figure shows how RT surfaces can intersect the brane before reaching the turnaround R 8 | χ , with relatively small brane tension | ∗ ζ see figure boundary as before it intersects the brane wepast have to the think of brane, asgenerally flips shown after in figure where we have introduced Figure 9. point to turn the secondexpression, order equation which determines extremal surfaces into a first order regions (left) CFT occupies the regiondefects on (i.e. the asymptotic boundaryregions. marked in pink (aqua). The conformal Figure 8. surface in the connected phase. The entangling region JHEP12(2020)025 ) ). ). , 3.8 3.16 3.20 (3.22) (3.21) (3.19) (3.20)   1 denotes ) tangent − ij determines ). We have , 2 2 QES h i 2 − − ρ d d − X ) ) d 3.18 2 2 ∗ ) ζ ζ 2 ∗ ) and yields ζ defined in eq. ( ) which are locally ∗ B 3.21 ζ (1 + (1 + ). Here, 5 2 θ 2 2 ∗ (1 + ζ ζ 2 − ∗ 1.2 d ζ ) s ) then becomes 2 − ζ ). The vector ). must be regulated, since the ± , 2 , − 1 3.18  d 3.9 ∞ ) 3.10 ) of the RT surface in the effective − (1 + 2 ζ 2 ζ ) R is a normalized (w.r.t. ζ ξ σ brane ( + 1 ( i B ζ G end-point 2 θ A T | 4 ζ (1 + j 2 to mark coordinates of the intersection 2 + ζ X i tan ) q T ). Assuming that we consider an extremal ) and R QES s ij bulk dζ   h (Σ G ! – 19 – 3.17 3.17 2 4 A 2 − ∗  d ) = ζ QES B ) ) is then calculated using eq. ( ζ θ R , 2 ∗ Z ζ R QES ∂ (Σ ± sin σ 3.20 A 2 ∂ρ ∞ ∗ ζ R ζ (1 + . Here we have used the brane angle σ Z ∗ δ ζ

0 = 2 2 QES − − d ζ d H H = ], this extremization leads to a boundary condition restricting the ζ vol 1 vol 1 1 − in coordinates along the brane in eq. ( − d d L R L σ ) yields a family of RT surfaces (parameterized by ) = 4 ) = 4 is the two-dimensional metric ( R 3.18 R ij (Σ (Σ h A A ρ In the absence of a DGP gravity term in the action, this variation mustThe first vanish contribution for to eq. ( As described in [ Eq. ( The area functional for the RT surfaces satisfying eq. ( along the brane; hence we have a boundary condition which sets the RT surface per- ∂ j which is evaluated at X pendicular to the brane.variation More of generally, the we entropy must contribution balance intrinsic the to the above brane, variation against as can the be seen from eq. ( area with respect to its intersection point with the brane iswhere given by vector to the RT surface,the which variation can along be the obtained brane. from eq. ( in higher dimensions. However, heretransverse we directions, are which leveraging reduces the theis, hyperbolic present we symmetry case need along to only the extremize atwo-dimensional a two-dimensional geometry problem. one-dimensional given profile That bybulk eq. surface ( which is anchored at the asymptotic boundary, the variation of the surface’s where the two contributions reflectthe the location two of contributions in eq. ( angle at which the RT surface meets the brane. Normally, this would be a difficult problem requires that we also extremizeintersect over the the brane. possible That locationsarea is, where (plus we these possibly consider candidate the surfaces the areato of extremization the the position condition QES, of of should the the there intersection RT be surfaces’ extra DGP gravity) with respect area of the RT surfacealso is included infinite. a The factor signconsidering of here both four, is CFTs, since the we there same need is sign to one as multiply RT in the surface eq. resultextremal to ( by in each another the side bulk factor away of from of brane. the two. defect However, fully and extremizing the area functional ( where here and below,between we RT use surface the and subscript brane brane, theory. which The corresponds upper to limit a of quantum integration extremal indicated surface as in the JHEP12(2020)025 ) 3.18 (3.28) (3.26) (3.27) (3.23) (3.24) (3.25) , , . 2 # ) together with 2 −  2 − d B d ) 3.9 θ 2 ∗ 2 ζ . + 1) 1 cos 2 − 5 2 must lie on the bifurcate ζ

(1 + − 2 − ∗ ∗ d B ζ dξ dζ ζ 2)( ) θ

2 2 − ζ − B 2 θ d comes from integrating eq. ( dζ . sin − ( 2 d Σ Σ B ) ξ ξ (1 + θ 2 QES . cos ζ 2 ζ ζ QES ∗ 2 − ) to produce the late-time part of the B ζ ζ − θ Z  2 2 π cos 2 2 B − 2 b reached by the RT surface: ± d θ − 3.25 (1 + λ ), we obtain the following relation between − ) 2 1 2 ∗ 2 ), coordinates and determine the relationship ∗ = ζ ζ B − − ζ

θ sin = cot q 3.20 2 dζ only, if we further use eq. ( ξ dξ dζ 3.20 ζ, ξ – 20 – 1 2

QES dζ ) and ( 2 sin ζ (1 + 2 − QES

∗ ζ 2 ∞ ∗ ζ sin dζ ζ QES ζ 3.24 q ζ dξ dζ Z ) (where we choose the minus sign) again determines 2 1 + and the belt width

3 p ∞ ∗ 2 ζ 2 ζ 2 − ∗ r − ), ( d ζ Z ζ d ) into eq. ( ∞ 3.18 2 = 0 + H ). QES Z + − ζ d Σ 3.19 Σ + 1) vol 2.7 τ H 3.23 ξ 1 2 QES ζ − = vol d 2 QES 2 L ζ − and the deepest point 1 + d and the QES ) and ( QES L q = ( ξ ), this can then be rewritten as a relation between the location of the Σ ) ) = 4 ξ 2 QES B 3.22 2 − ζ θ R , i.e. to a particular side of the brane. By symmetry the calculation on d on the brane 3.24 ) = 2 ξ ) ζ (Σ 2 ∗ R ζ cos( A σ b ( , we will use eqs. ( λ and A " was defined in eq. ( 4 ρ (1 + ξ ) to find the brane trajectory in ∂ ∗ b × ζ λ 3.12 A final relation associating If the brane DGP coupling is turned on, the variation of the area also obtains a the other side ofwhich the connect brane both yields CFTs the through same the result. horizon is The then total given area by of the two RT surfaces Here we have implicitlysits chosen at to negative perform the calculation in the asymptotic CFT which 3.3 No-island phaseWe for can use thephase. result The of first order thethe equation previous shape ( subsection of to extremalhorizon surface. obtain and is a By thus solution symmetry, determined we for by know solving the that no-island In section Page curve for a topological black hole coupled to a bath in higher dimensions. After using eq. ( entangling surface eq. ( between where from the boundary to the brane, the QES position contribution from the second term in eq. ( Substituting eqs. ( JHEP12(2020)025 ), ]. 3.30 (3.29) (3.30) 91 R L . Note that this . Using eq. ( CFT CFT 10 , ∗ r  . | 1  1 − + 1 − 2 d r r horizon this phase always dominates | , cf. figure B  dH , within the interior. Then the Σ θ ∗ 2 χ r . Hence, describing ingoing null r log = + 1 2 = 1 Note that with the above definitions, χ r ) = 13 r dv dr ], which studied entanglement entropy of ( tor 93 r + 2 2 – 21 – dv 1) at the asymptotic AdS boundary. Then the metric − 2 τ r ( R across the horizon using the standard prescription given in [ = 0 = ) where − ) Σ r Σ  r v ( ( χ 2 tor ∗ tor L r r λ . However, it is convenient to introduce Eddington-Finkelstein = UV ). In the case of small brane angle + τ λ 2 τ ds 3.27 = and hence v how the time evolution of an RT surface at early times can be mapped 0 R → 3.5 denotes the usual tortoise coordinate. R The RT surface of an entangling surface located at the defect in the no-island phase. ) ) r ( given by eq. ( tor r → ∞ ∗ ζ Now the extremal surface will fall from the asymptotic boundary, through the exterior, Due to symmetry, the trajectory of the RT surface is determined by its radial coordinate r We extend our defintion of ( as a function of time 13 tor Figure 10. across the Rindler horizon, reachingsurface a will continue minimal emerging radius into at thewe second need exterior only region. track Due the to trajectory reflection symmetry, of the RT surface until it reaches where r becomes r coordinates to avoid therays, coordinate we have singularity at Lastly, we will consider the caseentangling surface of is a taken zero-width towards belt, thethe i.e. defect, bulk the so along case that where constant the the RT boundarysetup surface location is slicing falls of essentially straight coordinate the through the sameidentical half-spaces as in considered the in two [ sides of a time-evolved TFD. see in section to this case. 3.4 No-island phase for with at early times. Thewhere reason the RT is surface that crossesit the the can RT brane, return surface has to in tobe the the travel made asymptotic competing a arbitrarily boundary large phase, small distance across i.e. to by the the the choosing brane. phase brane a This before small additional enough distance brane can angle. We will furthermore JHEP12(2020)025 - ∗ v r (3.38) (3.34) (3.35) (3.36) (3.37) (3.31) (3.32) (3.33) out to ∗ ). Using λ , the value 3.32 UV λ . is increasing as r  , ˙ r , 2 v 2 v v P , and P ∗ + r v , + 2 ˙ + 2 2) 1) ˙ v 2) − symmetric point d − − 1) ˙ d 2) 2( 2 2 − 2( r r − Z d ( r 2 . 2( 1) r . We have also included a factor − r indicating that 1) 2 . ( and so we have a conserved ‘ − − r ˙ r . − − 2) d 10 2 r 2 v − v 2 = ˙ r q d , ( r 2 P ˙ ( r = 2 ) 2( v − q v ˙ r v d r = q P v + 1) ˙ , 1) 2) + 2 ˙ v dr 2 v − dλ r − ) = , i.e. 2 − P + 2 ˙ d P r v 2 2 UV 2 ( 2( − ∗ r UV r ∗ r + v r ( 1) ˙ ∗ r – 22 – λ  ( = 0 U , we find ) λ ˙ Z v ˙ 1) ˙ 2) r 2 1 − ∗ − − Z 1 + r − 1 2 r − − d 2 − 1 ( ˙ r ˙ d − r − 2 2 2( ( 2 d ) = and L r r r r − ( L 2 − ˙ (1 r ( d 2 , the value at the minimal radius − r 1) − ∗ d = p − ) is independent of U d λ H q v − 1 H = P 2 3.31 − r ˙ v L ( − 2 ∂ ∂ ˙ r r q = plays the role of the conserved energy and the minimum radius ) = 4 vol ) = 4 vol v 2 R v ] = ] = ) to solve for P R P , r , r (Σ v v (Σ A 3.33 P P A [ [ ) of the extremal surface becomes ˙ ˙ r v 3.31 . Now, we fix the reparametrization symmetry of the area functional with the ) and ( is a radial coordinate intrinsic to the surface, which increases along the surface 2 − λ d 3.32 An intuitive picture of the dynamics of the extremal surfaces is given by recasting the The integrand in eq. ( H equation above as a Hamiltonian constraint, ˙ The area ( In this framework, corresponds to the turning point where r where the effective potential is given by Note that we have implicitlywe chosen move a along positive the sign surface for out towards the asymptotic boundary. where the second expressioneqs. results ( from substituting in the gauge choice ( following convenient gauge choice momentum’ at the UV cutoff nearof the 4 right to boundary account — forthe see the right figure boundary, fact and that the wethe fact only brane. that integrate there Of from are course, the twothe we such RT have surfaces, also one integrated on out either the side directions of along the belt, i.e. along where moving from theintegration left here asymptotic correspond AdS to boundary to the right boundary. The limits of the area functional can be written as JHEP12(2020)025 ). 3.42 (3.42) (3.43) (3.44) (3.39) (3.40) (3.41) . = 1 r , .     into eq. ( 2 v 2 , v c P 2 ∗ r P r + + → − 2) 2) ∗ 1 − r − d d q . v 2( 2 v 2( r 2 2 P r − P / / d ∗ 1) r 2) 1) 1) . 1 − − − d d − −  ( ( 2 d 2 ∗ ∗ r r ] (as well as the higher dimensional 2) 1) L r r ( ( 2 ) to write − 34 − − − q q d 1 1 + d d H ( ( − . −  3.29 1 1 2 1 1 −     log d − − 1 1 L ), we can integrate out to the right boundary 1 2 d d denotes the position of the UV cutoff surface 2 = 4 vol − Further, we also observe that the critical radius 1 − − 1 v d – 23 – 2 = 2 UV ) admits a very simple form P 3.34 r H ) = r 14 r 2 c 1 ∗ r r − dr and hence we find ( dr d 3.31 ] L tor v UV 2 UV r r ∗ = 4 vol r ∗ P − r [ r d . However, we want to examine the time evolution of the Z ) = τ Z ∗ H r R ∗ = + Σ = v ˙ ˙ (Σ for a long time, and so we can replace r v  dτ ). Note that ∗ ∗ c r r r dA dr = 4 vol bound in the no-island phase is determined by a zero-belt width calcula- , and so we may use eq. ( symmetry of the extremal surface, we know that the turning point v − 3.34 ) = 2 UV χ 1 1 + R r ∗ r Z = 0 r Σ ), the extremal surface can be specified by the integration constant  Z t (Σ dτ = log ) and ( dA is given by 3.37 ∗ 1 2 v 3.32 − = 0 ] = denotes the value of the Eddington-Finklestein time at the turning point. How- v U is the boundary time parameter. r P ∗ ]), the entropy corresponding to the no-island phase grows without bound. ). The bulk contributions naturally vanishes by the equations of motion determining the ex- [ bound v ∂ τ τ v 30 As we will see momentarily, the late time behavior of the entropy of any subregion The time derivative of the area ( With eq. ( 3.31 A quick derivation of this result follows by considering a small variation of the surface profile in or the boundary condition 14 v case [ eq. ( tremal surface. However,terms. deriving These the are latter usuallyinstead eliminated requires allow by for an fixing a integration the small by boundary variation in parts conditions the at which boundary infinity. produces time. In boundary the above result, we bounded by constant tion. Thus, as in the two-dimensional case studied in [ At late times, the turninglies point near is the very close surface toHence this we critical expect radius, the i.e. , growth the of critical the surface area is fixed at late times, i.e. where where Note that the integrand is nonsingular in the vicinity of the horizon, i.e. near ever, because of the lies on the surface Further, we know that to determine where near the asymptotic AdS boundary. P entanglement entropy and so we mustboundary determine a time. relation between In these particular, constants using and eq. the ( using eqs. ( JHEP12(2020)025 , in into Σ ,... χ 2 (3.45) (3.46) +1 φ ± d = cos 1 χ φ either to some η , ) Σ . For any constant = sin . The defect in the , τ cosh χ , 1 Σ 4 χ ). The AdS boundary χ ( → 3.7 sinh , µ r → ±∞ 1 , . . . , d , cosh φ η 4 Lr . , 2 Lr ± or L = = = 3 − i = cos 2 2 can be reduced to one of the cases away from the defect in both CFTs. = 3 Σ or to the case where the entangling Σ d τ µ → ±∞ χ with X χ + = 0) 0 Σ ··· slice of the metric ( at time 6= 0 , τ 2 + Σ . In particular, this means that the solutions 0 Σ − ) Σ 2 – 24 – χ 1 d 0 χ Σ ( , τ X H , τ Σ + χ 2 i 2 we can ask how the RT surface changes under time τ , X τ , T = 0 T η µ 0 Σ 15 − . χ . The entangling surfaces are defined to be at 2 ( 1 2 1 sinh 1 cosh T 3.4 sinh X − − − χ slice, i.e. with 2 2 allows us to map the RT surface at any ), the bifurcation surface reached with r r and τ 2 parametrize the p p − cosh of interest, η d = 0 = = 0 L L , and each sign corresponds to one of the two Rindler wedges. On a 3.3 , i.e. with H ± ± Lr χ τ R = = = consists of all points more than a distance = 0 → ∞ i 1 1 R T χ r X X denotes further angular coordinates, e.g., i slice, we can reach the boundary by taking µ r To understand the required coordinate changes it is convenient to embed AdS The strategy we will employ in this chapter is the following. We will perform a coordi- Obtaining RT surfaces in the no-island phase which are anchored on symmetric entan- , i.e. we are looking for a parametrization of (parts of) the hyperboloid defined via Recall that 2 15 d, fixed Rindler time (i.e. fixed CFT is located at both CFTs. where which, together with is located at Our original two Rindler patches correspond to the parametrization R nate change from Rindler spaceIn to the a new particular coordinates, Poincaré thesymmetry coordinate of entangling system the surfaces Poincaré defined are patch below. and straightentanglement mapping lines. entropy back of By to a Rindler exploiting subregion space, the with discussed the boost in task of sections calculating RT surface in the surface is at obtained in the lastsymmetric two subsections entangling surfaces are of sufficient interest. to discuss the full time evolution of the changes. gling surfaces of arbitrarydifficult. width However, and as at we arbitrarybolic will times now symmetry in show, of higher our dimensions choice of is entangling generally surfaces with the hyper- Given the region evolution. If wethe are Rindler patch in so the thatOn island time the translations phase, other are the hand, aForward symmetry RT in time and surface the evolution the is no-island of entropy phase, is completely both a the contained sides constant. RT inside is surface not connects a to symmetry both and bath the CFTs. area of the RT surface 3.5 Time-evolution for general JHEP12(2020)025 ) 3.46 (3.47) (3.48) ··· ) and the . The two 3.46 , = 0 2 2 ˜ z ˜ x ˜ x L = 2 , whose slope depends on . 0 . 1 ˜ (in global coordinates) is split x > · 1 11 1 defect , − Σ ˜ x d origin in Poincaré coordinares τ 2 . S Σ L χ = 0 + 2 2 ˜ t ˜ t , respectively. We will denote the CFT = tanh as the left CFT. Comparing eqs. ( z − 0 ˜ t − 2˜ 0 2 1 ˜ 2 x < at which the entangling surfaces are defined. x < ,X 1 ˜ 1 x Σ 1 + ˜ ˜ , τ ˜ z x ˜ x 2 1 – 25 – ˜ z L ˜ x while the defects are located at and · = = 0 Σ Σ 16 ) is illustrated in figure in the above expressions yields the entangling surfaces 1 2 , τ χ > 1 ˜ x 1 3.48 = 0 ˜ . In these coordinates, the bifurcation surface intersects x − 1 cosh sinh ˜ t − to 2 d ± ,T = 1 , shown in pink and aqua, which are glued together at infinity. At x 2 1 ˜ 1 = x − ˜ L + ˜ x 1 2 d ˜ x − − H d ··· and the Rindler time 2 ˜ t H ) at Σ + 0 z χ − 2˜ boundary of 2 2 2 x → x ˜ + ˜ z ,X + ˜ 2 1 ˜ t ˜ z 2 x point at infinity in Poincaré coordinates A time-slice of our setup. The spatial boundary ˜ z L as the right CFT, and the one at = ˜ ) in the boundary limit, it is easy to see that the entangling surfaces in the right = = 0 2 1 d ˜ x T > 3.47 X We now need to choose cutoffs in order to regulate the area integrals of the RT surfaces. We will now consider a particular Poincaré coordinate system, which covers both 1 The full Rindler horizons reach the boundary along ˜ x 16 in the left CFT.new The Poincaré relation coordinates between of the eq. Rindler ( coordinate given inFirst, we eq. need ( to regulate the UV divergence in the entanglement entropy by introducing This shows the convenient property ofalong the rays new (i.e. Poincaré straight coordinates: lines) in entanglingthe surfaces the spatial lie positive location half-space with Further, flipping the sign of CFTs are mapped to theat regions and ( CFT get mapped to where the boundary ( Rindler wedges and is defined in terms of embedding coordinates as into two hyperbolic discs the same location, theto bifurcate a horizon Rindler intersects wedge theindicated in boundary. in the the The bulk. figure, CFTwith on the The the either Poincaré point defect disc coordinates at (green)semi-circles is infinity introduced is shown dual appearing in in a on red. this great the section circle south cover on pole the the of full global the sphere, boundary. sphere. As Entangling surfaces are the Figure 11. JHEP12(2020)025 ). 3.10 (3.50) (3.51) (3.52) (3.49) -dependent ˜ z with the appro- = 0 0 Σ τ remains constant in the . This boost leaves the = R , 12 τ ) . slice, which also corresponds to ˜ t ). In the island phase, these two , depending on which Rindler , β 0 IR . R 2 ` ˜ t − = 0 3.47 < 1 t . This translates to a 2 = 0 − 1 UV x 1 r ˜ 0 x . We can express the problem in a 2 1 cosh (˜ ˜ t 0 ˜ x γ L  or 2 = > ∼ = 0 0 0 1 UV 1 < 1 2 , Σ r ˜ x ˜ t ˜ > x τ 2 2 Σ − 1 2 . x − χ ˜ x 1 L 2 1 2 UV + ˜ – 26 – ˜ x r , + 2 1 boost  ) sinh 2 x 1 = 1 0 x ˜ x UV ˜ + ˜ x r β + ˜ 2 2 min ± ) invariant, and changes the equation for the entangling . This is depicted in figure ˜ t ˜ 2 z − Σ ˜ t = τ ˜ t − ) for either > is a Killing coordinate for the corresponding metric ( ( − 0 2 3.50 2 γ 2 ˜ τ ˜ x z 2 ˜ z ˜ z 3.48 = q 0 ˜ t = tanh ) and ( β 3.49 in the right CFT in the Poincaré coordinates ( , which translates to 1 Σ The left panel shows two components of the entangling surface (red) at non-zero 6= 0 τ 1 2  Σ ˜ x ˜ x τ IR R ` slice of the hyperbolic boundary geometry. The boost is a symmetry of the defect (green). direction to map this set of entangling surfaces to the = Second, we need an IR cutoff which we impose in the transverse directions along the ˜ t 1 ˜ x = 0 max cutoffs given in eqs. ( surface to This is precisely thepriate entangling cutoffs. surface We of may the thus same conclude that region entropy at of the region wedge we are interestedboosted in. coordinate systems Here, we choose with boost parameter under time evolution. Thiswithin is one obviously Rindler true, wedge since and theHence RT the surface corresponding is completely timewedge. contained evolution In of this a case, we singlethe are location Rindler looking defined wedge for by is an eq. extremal an ( surface isometry which of ends that on the boundary at 3.5.1 Island phase As a warm-up exercise, we will show that the entropy on the island phase is in fact invariant where in the last step, we used that entangling surface. Sincebrane, the the solution transverse is directions should independentη just of contribute an shifts overall in volume factor. all We directions choose along the maximum radius in bothcutoff AdS-Rindler in patches the new coordinates, Figure 12. Rindler time rays in the boundary geometryin are connected by an RT surfaceτ in the bulk. We can perform a boost JHEP12(2020)025 is a Σ (3.53) (3.54) (3.55) χ 6= 0 = 0 = 0 Σ 0 Σ Σ χ χ τ , , = 0 6= 0 0 0 Σ Σ , τ τ , Σ 2 τ direction to map these ˜ x 2 2 Σ Σ ˜ x τ χ Σ in this connected phase. τ sinh Σ 2 τ − ). These two rays are located in . Our approach of boosting in the cosh sinh . The new entangling surfaces 2 Σ ˜ x ± direction with boost parameter χ cosh − 13 3.47 2 . 2 = ˜ Σ = x 1 ˜ t . ˜ x Σ sinh Σ Σ > χ τ χ s Σ < χ 0 2 τ ˜ x Σ τ cosh cosh ± when = , i.e. to the region on one side of the defect. = Σ 0 0 1 – 27 – 0 Σ ˜ x plane. The new location of the entangling surface χ = 0 < > χ 0 Σ 2 Σ are positive (or zero). Let us also note here that χ τ ˜ x = 0 cosh Σ , 0 χ Σ 2 τ ˜ x or to = and Σ Σ Σ τ χ τ Σ τ < χ we can boost this problem in sinh sinh Σ τ , − 17 , Σ = = 0 ˜ 6= 0 t 0 when ˜ t Σ < χ in the right CFT in the Poincaré coordinates ( χ Σ = 0 , 1 Σ . Expressing the result in Rindler coordinates, we are dealing with the case of The left panel shows two components of the entangling surface (red) at non-zero τ 2 . This is depicted in the upper panel of figure τ 0 ˜ 0 Σ x ˜ x Σ τ Σ if τ χ 6= 0 < Σ 0 2 τ ˜ sinh x sinh ˜ t − is given by We are assuming that both direction fails in this case, but the results for the time evolution are smooth across this point. = 17 0 Σ 2 ˜ special case, where thex entangling surfaces lie in the null plane where an entangling surface in the χ calculation becomes simpler. Now, however, we haveCase to 1: distinguish two cases. β are then located at located at where we have chosenSimilarly to to focus the on island phase, we want to go to a new coordinate system in which the conformal defects in each of the two CFTs,3.5.2 is independent of No-island phase For the no-island phase,gling we surfaces focus in the on CFTs the dual case to in different Rindler which patches. the The RT entangling surface surfaces connects are entan- different CFTs so thatpasses in through the the no-island Rindler phase,two horizon. they rays to are In joing this by case, an we RT can surface now in boost theisland in bulk or which connected phase, ascan anticipated. see Again that because the we have entropy a of pure the state globally, complementary we region, i.e. the two belts centered on the Figure 13. Rindler time JHEP12(2020)025 . Σ (3.57) (3.58) (3.59) (3.56) < χ 18 for two Σ ) at time τ 2.2 3.55 . Now we can . If we restrict ) is only correct 13 = 0 3.56 χ . limit, but the corrections are given in eq. ( Σ ). χ 0 Σ 2 ) changes along the trajectory 3.56 → ∞ plane, in Rindler coordinates it Σ ) by substituting the trajectory > χ τ and sinh | 2 3.49 ), the new entangling surface gets UV . The new entangling surfaces are χ | − 1 r = 0 3.56 Σ Σ . . . 0 τ χ Σ = 0 Σ τ τ cosh Σ Σ Σ 0 Σ τ τ χ 2 | → χ sinh sinh χ Σ Σ τ χ − , the entanglement entropy of the region sinh cosh cosh cosh Σ cosh = sinh sinh s | 0 ˜ = UV β UV ˜ t . . The latter was reliable as long as r – 28 – r < χ 0 0 Σ ± Σ , the cutoff is no longer time-dependent. τ τ = = Σ Σ = 2 → τ 1 0 0 UV 0 UV 0 Σ ˜ x r r cosh > χ χ , is shown in the lower panel of figure cosh Σ Σ (and so as ) is unchanged, however, the UV cutoff changes to τ Σ , > χ χ 3.50 Σ . τ , we have a standard thermofield double state of the two CFTs = 0 ) 0 2 is the same as that of a region cosh ˜ UV x = 0 Σ τ → /r τ Σ (1 = τ O τ direction again, but using 2 cosh ˜ x calculated with a different cutoff, given by eq. ( the other case, at time ) remains unchanged, but the UV cutoff in eq. ( = 0 Σ 0 Σ 3.50 τ In conclusion, we found that if Importantly, the cutoffs are not boost invariant in this case. The IR cutoff given in Note that, like above, we have substituted the trajectory of the entangling surface into the boosted > χ | = 18 χ on hyperbolic spatial geometries,the including observations the to conformal defects either at the left or rightexpression. side, Thus, the thissubleading reduced to equation the state is finite is only part of a strictly the correct thermal entanglement entropy. in one the and 3.6 The informationNow paradox the preceding resultsevolution can of be the combineddimensions, entanglement to at entropy. give time a Following qualitative the description discussion of the in time section Let us note that thewas cutoff the location still old is cutoffHowever, continuous. we multiplied In see the by here previous that case, once the new cutoff reduces to an entangling surface for a belt width Again, the IR cutoff in eq. ( located at While this does not reduce to a surface lying in the | τ Case 2: boost in the We should caution the readerof the that entangling we surface arrived into atfor the boosted a eq. small cutoff. ( cutoff. This meansentropy Luckily, the at that corrections eq. order to ( the new cutoff only change the entanglement closer and closer to the defect, i.e. eq. ( of the entangling surface to Note that as JHEP12(2020)025 ) Σ χ (3.60) ≥ Σ τ 21 ). we can describe 3.50 19 ) implies that at late times: 3.58 . Σ . The latter follows at late times from , we found that the entanglement ∂ Σ ∂τ τ 20 ∂ 2 ). 3.4 / 1 '  0 Σ 3.44 Σ τ Σ τ ∂ χ is to increase the entanglement between one 2 2 . 5 = 0 sinh sinh – 29 – ], this linear growth of entropy would lead to an τ − 1 . 98 ). We emphasize that the latter really describes an , )  2 = πR 3.10 0 Σ (2 ∂ ∂τ / — see footnote 1 ), we must also show = 1 . Let us add that this linear growth is analogous to that found for planar − ) τ T 2 3.44 − e ( O region. For the configuration described above, this yields a topological d ) + Σ χ ]. not too close to 93 b λ , we saw that by a judicious change of coordinates (and cutoff), the calculation log (cosh ), which yields and 3.5 geometry in AdS-Rindler coordinates, and hence the thermal equilibrium between − 1 The appearance of an island in the effective gravity theory from the brane perspectice As in the two-dimensional case [ A standard measure for the entanglement between both AdS-Rindler wedges and their Using the brane perspective and an appropriate choice of parameters, τ d 3.58 The black hole entropy is proportional to the horizon area of the black hole, which in our case is infinite. Recall that we obtain a good aproximation to (semiclassical)Implicitly, Einstein gravity to on apply the brane eq. if ( we choose  = 21 19 20 0 B L ` Hence to be precise, we must consider an IR regulated entropy, as discussed with eq. ( Alternatively, the same resultτ also follows by simplyblack holes observing in that [ eq. ( single Rindler wedge. Thebrane fact is that easy there to will see: always be before an we extremal invoke surface the crossing extremization the condition at theeq. brane, ( there is the defect entropy is well-approximated by two times theis black hole simply entropy. related tosystem. a The phase RT surface transition changesthrough of from the the the horizon, no-island RT to phase, surfaces the in which island in it phase, the connects in both bulk which CFTs it description connects of both our sides of the defect in a information paradox for ouris eternal that black the holes, entanglement ifthe entropy it must defects be was need valid boundapproximated to for from by all purify above weakly times. by the the coupledfields bath The defect Einstein on reason system. entropy, the gravity. since gravitational background This In as giving allows the a us small case correction to of to view the interest, the entropy and the quantum thus, theory is well section of the entanglement entropyto of the these case regions of canentropy a be grows zero-width mapped linearly belt. in at time, late Further, as times in shown (i.e. section in eq. ( side of the black hole,bath i.e. CFT. one of the AdS-Rindler wedges on therespective brane, baths and is its given by respective regions the centered entanglement entropy around of the the conformal complement of defects two in belt sub- the boundary as discussed above. In AdS this ‘black hole’understood and as the arising because finite thein temperature two the systems CFT region are coupled of onCFT, along dynamical the an under gravity. accelerated time asymptotic trajectory While evolution, boundaryThe the immediate the black can effect two of hole be this systems is process are after in constantly equilibrium exchanging with thermal the quanta. bath defect, with temperature the conformal defects are replaced bygravity (two copies on of) an the AdS boundary CFTblack coupled hole to Einstein solution shown in eq. ( in particular, the bath CFT is in thermal equilibrium with the corresponding conformal JHEP12(2020)025 ) ∗ ζ (4.1) 3.24 (3.61) as a radial ζ . ! under consideration. In the ... ), we can express the profile R + 2 3.12 ) 2 b − λ d ) , 2 ∗ ) at leading order. The reason is that ζ ζ (1 + B L 4) ` − (1 + d 2 ∗ (2 hor ζ ζ – 30 – . In this approximation, it follows from eq. ( 2 1 = arctan( %  1 +

B L ` hor ζ = cannot scale with ∗ . We start here by discussing examples of extremal surfaces in ζ ) QES ξ ζ ( ζ and that 1  B L ` , the problem of finding the corresponding RT surfaces reduces to a two-dimensional ∼ In order to establish unitarity of the Page curve, we still need to argue that the island 2 − d hor of the RT surfacethe as island phase forcoordinate, different choices we of conformally parameters. compactify the Instead geometry of and working use with the coordinate 4.1 General behavior ofAs the discussed previously, islands by choosingH entangling surfaces with theproblem. hyperbolic symmetry Choosing of the convenient coordinates in eq. ( However, the reader might have realized thatequations all of and these equations are were ordinarynumerical differential thus solutions easily to solved thethe equations numerically. for arguments the of In RT the this surfaceblack previous holes in section, section in the we equilibrium to island with will obtain phase, a and the first bath. then Page present use curve for massless, topological 4 Numerical results In the previous section, we foundthat a phase has transition between the the right no-islandThe and qualitative calculations island phases involved properties differential to equations which yield have a no known Page closed curve form consistent solution. with unitarity. section for numerical plots.given by The the leading area of orderanalysis the contribution horizon is to which needed the gives to the generalizedshows black demonstrate that entropy hole the that is entropy. island While the mechanism acurve. appearance has more of the involved right the qualitative island behaviour saves to unitarize unitarity, this the Page In deriving this equation, weζ have used that theis location bounded of from the horizon abovethe on by new the a quantum brane extremal function is surface of at will the always belt be close width. to We the can horizon — see see that also the the location next of appears before the black holecase fails of to interest here, purify we the havethat bath that region To get the correctof RT the surface, surface we where they needguarantee meet only that the extremize there brane. the willthe Subregion area be boundary duality by and one of the varying the extremal homology island the surface constraint might position for sit every at the belt horizon configuration or (although at the CFT defect). an infinite family of candidate RT surfaces, which start in the bath and meet at the brane. JHEP12(2020)025 ζ, ξ , as 2 (4.2) (4.3) tends ) from , 4 and π/ ∗ ζ = )). Recall 3.18 = 3 ξ numerically d as a function 2.4 ) ) and therefore QES ζ QES 1.1 , ξ , the location of the Σ and ) which shows that by , χ ∗ QES ) ζ ζ 2.7 1) ( − . A general feature is that d 2( B ) θ 15 d B ( θ at the boundary. Applying the ( O is small. As we depart from the Σ O B + χ θ + 2) 2 − ) to determine − d d B for different values of the dimension and 2( B θ and the distance from the horizon on the θ Σ 2 3.25 2 2 χ − , the location of the entangling surface in − d 2 d ) ) Σ b ) QES and the QES position b 2 ξ ∗ 2 λ ξ ∗ – 31 – ζ λ ∗ ζ ζ . We can see that positive DGP coupling pushes ) and ( b , the gravitational coupling in the brane theory, i.e. 1 + λ b (1 + (1 + λ ∗ 3.24 2(1 + 2 ∗ ζ ζ , the above formulas tell us that the QES tends towards − 0 2 (or equivalently the brane tension — see eq. ( π → B = = 1 + ). In terms of B θ L/` . ), we relate this to QES QES τ 3.61 for different dimensions — see figure shows that the size of the island varies with ξ ρ = B B 3.12 θ θ . The shape of the RT surface is obtained by integrating eq. ( 14c Σ ξ sin shows RT surfaces with fixed shows how the RT surface in the island phase behaves as we vary the brane limit, the QES always approaches the horizon on the brane at shows a few examples of RT surfaces in the connected phase for 0 , we have 14b 14 14a → QES B limit of eq. ( ρ θ r We can get an even better idea of the qualitative features of the islands in higher Finally, figure Figure Figure Figure where the first termstowards a on finite the value r.h.s.s as give the location of the horizon. Granted discussed around eq. ( brane, course it will generally not dominate at early times. dimensions by plotting the turning point of the brane angle in the that Einstein gravity islimit a of good small brane approximation angle, when the island grows. entangling surface in theisland bath. phase Moreover, for as the we RT surface will seems discuss to momentarily, exist we for see all that values an of the belt width, although of the effective Newton’s constant,Bekenstein-Hawking becomes contribution smaller is (bigger). bigger (smaller)creating In in an turn, the island the island of fixed coefficient rule size ( of becomes the harder (easier). angle given by selected values of thethe DGP point coupling of intersection betweenthe brane area and RT of surface the towardsbecome island’s the bigger. horizon, boundary. i.e. This Similarly, it behaviour negativeincreasing reduces is DGP (decreasing) readily the coupling explained value through causes of eq. the ( island to 5, i.e. in four, fiveon and one six side bulk dimensions, ofSince respectively. the the Here, brane. we RT only surfaces Thechoice show do other of the geometry Rindler not side time is cross determined the by horizon, a the reflection configuration across is the independent brane. of the surface, we fix thelarge location of thecoordinates. entangling surface We can then useas a eqs. function ( of the boundary. which maps timeslices of AdS to a finite region. In order to calculate the profile of the RT JHEP12(2020)025 . The DGP π 1 2 is chosen to be b π, λ 1 4 π, 1 8 . The DGP coupling is 3 , = 1 , B 1 3 θ . 2 π = ). On each side of the horizon and the location of the entangling Σ χ 4 4.1 π and = 2 π B . The DGP coupling − θ 5 , 4 , = 3 d defined in eq. ( % and (left to right) 4 runs between π – 32 – ξ = B θ and brane angle of (left to right) = 1 Σ χ with with brane angle = 4 = 4 d d . RT surfaces in the island phase in higher dimensions. We only show one side of the = 1 Σ χ for the dashed/solid/dotted curves. The brane angle is 9 . 0 RT surfaces in − RT surfaces for the island phase in (left to right) RT surfaces in / 0 / brane. The asymptotic boundarythe of RT the surfaces spacetime in is red.(dashed shown purple in The line) blue, radial the the coordinate Planck angular is brane coordinate in green and (c) set to zero. Figure 14. (b) coupling is set to zero. 1 surface is (a) JHEP12(2020)025 0.5 0.5 0.5 . . . /π /π /π B B B . 0.25 0.25 0.25 θ θ θ = 4 = 3 = 5 d d = 1 d Σ for for for χ B B B θ θ θ 0 0 0

1.1 1.1 1.1 1.5 1.4 1.3 1.2 1.0 1.5 1.4 1.3 1.2 1.0 1.5 1.4 1.3 1.2 1.0

ρ ρ ρ 0.5 0.5 0.5 =-0.99 b λ /π /π /π B B B 0.25 0.25 0.25 θ θ θ =0 b λ – 33 – 0 0 0 0 0 0 0.5 0.5 0.5 0.25 0.25 0.25 =1

b QES QES QES

π ξ π ξ π ξ / / / λ 0.5 0.5 0.5 . The location of the entangling surface is chosen to be 5 /π /π /π B B B 0.25 0.25 0.25 θ θ θ The dependence of RT surface parameters on the brane angle The dependence of RT surface parameters on the brane angle The dependence of RT surface parameters on the brane angle and 4 The dependece of the RT surface and the quantum extremal surface on the brane angle (c) (a) (b) , = 3 d 0 0 0 2 0 8 6 4 2 0 8 6 4 0 8 6 4 2

10 10 10

for

QES QES QES ζ ζ ζ B Figure 15. θ JHEP12(2020)025 3.5 (4.6) (4.7) (4.8) (4.4) (4.5) show ) that 16 3.18 ) is a cutoff , ) indicate the d B 4.6 . θ ( 16 ! . O ¨ isl ¨ + )] 1 R −

. That is, we subtract off (Σ dξ dζ A

[ =0 , ) would still be infinite, as a , . More precisely, we consider . 1 .,τ 5 dζ isl ¨ , isl ¨ 4.6   4 )] . ) ∞ ∗ , ) and noting from eq. ( ζ ζ R R Z σ , ( = 3 (Σ , the second term on the l.h.s. can be = 0) for 3.25 2 B A A d + 2 − τ θ [ d ( b Σ 2) L S ξ 2 S − 2 Lλ − − = 2 d ∆ − 2 d ) d ( -dimensional topological black holes, coupled H ) to eq. ( , which is given by τ ) d ( 2 ∗ , which is a dimensionless quantity. For the horizontal – 34 – QES The kinks in the plots of figure ζ s vol S +1 4.2 ξ ) + d = 1 ∆ ζ R = is also dimensionless — see further comments below. 22 Σ . ) = s ) = τ χ (1 + (Σ bulk τ d B ∗ ( ∆ G A ζ θ 4 = 0 ( S  , at which point the minimal RT surfaces in the no-island 2

O τ ∆ ∼ ∓ + = 0 ). At leading order in 2 d > is determined by the equation τ dξ dζ − d B B , the benefit of our model is that calculating the entropy of (the 3.18 θ θ ) = min 2 3 τ 2 − ( d ) S b 2 ∗ λ ζ bulk at small shows the Page curves for G 1 + ∗ 4 (1 + ζ ∗ given by eq. ( are the regulated areas of the RT surfaces, and the subscript indicates whether ζ 16a − (Σ) 2 π A dζ/dξ Figure Note that we are actually plotting 22 with respect to thetime entropy at at which the island phase of the RTaxes, surface also begins recall that to the dominate. AdS-Rindler time The corresponding Even though the UV divergencesresult have of been the removed, infinite eq.the ( extend change of in the the entropy entangling density, surface. Hence the plots in figure dependent quantity, it is convenient tothe subtract value off of the entropyphase, at to define Here we consider the extremal surface in the island or no-island phase. Since eq. ( formulas for the areas of— a number are of sufficient special to RTentanglement surfaces, entropy. calculate which — the as full shown in time section evolution ofto the a bath RT on surfaces athe and hyperbolic entropy background, thus of for of the the region cases the defined by 4.2 The PageAs curve discussed in in section complement of) the belt-shapedcalculating subregions areas in centered an on effectively the two-dimensional geometry. conformal Further defects we reduces produced explicit to with ignored and the abovefrom equation the is belt just boundary the to statement approximately that the bifurcation the surface RT on surface the should brane. stretch we find that the horizon on the brane. Applying eq. ( JHEP12(2020)025 . , 3 1 . = 1 = 4 1200 0.14 Σ d = 0 χ 1000 B θ 0.12 , and 800 2 τ = 1 600 0.10 Σ χ 400 , B b 0.08 1 θ λ = 0 200 (top to bottom). Right: b 5 . λ 0 0 0 0.06 − 500 , 1500 1000 .

0

1

Bulk , . s G 4 Δ 0 5 (top to bottom). Right: the Page 0.04 . = 0 15 = 0 . B 0 θ b 0.02 , 1200 λ 10 . 1 0 1000 - , 0.00 0 0 07 200 800 600 400 200 800 600 400 800 . 1000 1000

. The constant parameters are set to

b

P P τ τ τ λ = 0 600 B θ 400 – 35 – 1200 1200 (left to right). The entangling surface is located at 200 5 , 0 4 . The constant parameters are set to 0 , B 1000 1000 500

θ

1500 1000

Bulk = 3 s G 4 Δ d 800 800 1200 τ τ 600 600 1000 800 400 400 τ 600 as a function of the DGP coupling . P The Page curve in various dimensions. The solid blue line indicates the physical τ 400 = 4 200 200 d 200 as a function of the brane angle 0 , and 0 0 0 P 0 0 τ 500 Left: the Page curve for selected brane angles

The Page curve for dimensions Left: the Page curve for selected values of the DGP coupling

100 500 400 300 200 600 100 600 500 400 300 200 1000 1500 = 1

Bulk Bulk Bulk s G 4 s G 4 s G 4 Δ Δ Δ Σ Page curve. The dashedsurfaces. orange At lines early correspond times, totime, the entropies the RT calculated minimal surface surface by in non-minimal transitions the to extremal no-island the phase RT is surface the in minimal the surface. island After phase. some (c) the Page time χ Figure 16. time (b) (a) and the DGP coupling is set to zero. The brane angle is chosen as JHEP12(2020)025 . 3.5 (4.9) (4.11) (4.10) agrees 16b , and in the in this case. is related to d r t − 2 B ). Accidentally, agrees with the θ r 3.58 does not change. At . We verified that the ζ Σ χ ) and numerically solving goes like can be estimated from the ).  0 . d 3.25 Σ 1 4.9 − ) to numerically integrate the τ πR → . 2 B 2 θ B 2 2 / / θ = . 3.24 2) 1) 2 − − T d d − 1 , ( ( d B time-slice, as explained in section θ 2) 1) ) into eq. ( ). The dimensionful time τ 1 πT 2 2 d becomes larger, the numerics become less − 2 − − d = 0 d d 3.1 2) Σ 3.26 ( ( = 1) τ τ 4 − − – 36 – , the calculation of the entropy of the subregion d τR ∼ ( d Σ ( = χ ) and ( t s/τ ∼ in the ≤ ∆ P ]. The divergence as  3.24 Σ . A fit to the numerical data plotted in figure τ N τ Σ Σ decreases the Page time, or in other words decreases the 33 τ χ G and is given by B = 4 4 works out in such a way that the cutoff of θ d cosh cosh , the entropy can be computed by calculating the area of an RT ζ 3.4  Σ ]. As the brane angle approaches zero, the Page time diverges. for χ can be translated to the calculation of the entropy of a belt with coordinates, it turns out that the cutoff on 60 is a dimensionless time such that the temperature of the hyperbolic and Σ , & 30 . χ τ r ζ, ξ Σ 59 1 τ ± arccosh ± , we show how the Page curve and Page time change as we vary the brane (cf. the discussion in section ). There are three different regimes for the calculation of the are in the . At early times, π = 1 2 16b 3.19 0 Σ in the previous calculation. As χ ζ is the curvature scale for the spatial sections in the bath CFT, as defined in . The result is then used together with eq. ( ), and the bath CFT is taken at temperature R QES 3.3 In figure The calculation of the RT surfaces is performed as follows: the area in the island phase ζ by For example, the numericalabove coefficient formula to which be multiplies with this value. from the CFT pointin of the view island where phase theThe [ defect reason entropy is is that giventhis in in limit this terms was limit already of thesmall-angle noted an area approximation in RT of we surface [ the find island that diverges. The absence of islands in extrapolate the last fewresulting numeric data slope points agrees to with late the times, analytic result given inangle. eq. As ( we see,number of increasing microstates available to the black hole on the brane. This can also be understood surface for a zero-belt-width entanglingthe surface relation at between a timecutoff given on in eq. ( reliable. However, for moderatelywhich sized the belt area widths of we the are already RT surface well into grows the linearly regime in in time. Therefore, we use a linear fit to with boundaries at boundary As also explained inHowever, the working same in section, weintermediate need times, choose a different cutoff on eq. ( is computed by substituting eqs.for ( area in eq. ( no-island phase black hole is τ where bath at finite temperature.been The slope determined of in the section (linearly) rising portion of the Page curve has Moreover, recall that time is, of course, the natural analog of the Page time for eternal black holes coupled to a JHEP12(2020)025 . ) Σ 3.0 and χ ( b (4.12) shows λ O ) before 0 ∼ 2.5 17a 130 Σ < τ ∼ 2 Σ 2.0 P and τ ) s/∂τ ) the entanglement Σ β ∆ ( χ / 1.5 2 Σ τ O 17a ∂ regime we are interested ∼ B a universal, linear behavior Σ 1.0 θ τ ) Σ in figure χ ) evaluates to ( , 0.5 ) the Page time goes to zero. The 16 O . 2 eff 0 = 4 − 1 ∼ G d B d ∼ θ The same plot, but with axes rescaled by Σ (

0.0 2 0 8 6 4

τ

1

. The solid lines are numerical results, while

Σ

Bulk χ Δ β 2 2 d / s G 4 − Σ d (b) χ the dashed lines are themain bounds text. explained in the and 2) 1) 1 . − − d ( d – 37 – ( = 0 4 ), we obtain a linear relationship between the Page B ∼ θ 4.6 b /λ P (i.e. 3 τ , cf. eq. ( 16c (bottom to top) b 5 . λ 2 , Σ 2 2 τ , 1 it saturates and stays constant. , 1 ), then enters a period of concave growth (i.e. . P 0 τ = 0 > . Based on this argument, we can estimate the slope of the graph to be shows the dependence of the Page curve and Page time on the DGP cou- 1 b Σ 2 Σ λ χ The initial behaviour of the Page curve in four dimensions (left) and a rescaled version 16c s/∂τ and ∆ P 2 0 τ 2 0 8 6 4

This figure shows the onset of the Page curve ∂

12 10 In the following we will explain the region of fast growth and its transition into the Generally, we can separate the time-dependence of the Page curve into four different The Page curve and Page time only depends very weakly on the belt size. In fact, the Figure

Bulk s G 4 Δ . The Page transition occurs whenever the area of the RT surface in the no-island phase in four dimensions. (a) for different values ofgling the surface location of the entan- b This fast growth depends ontakes the over, belt which size. is At independent time at of the the Page belt time width. The entanglement keeps growingregion until of universal linear growth. To understand the behaviour of the Page curve, first that for wide belts, the(i.e. entanglement between the beltsentering and the baths linear starts growing regime. convexly regimes. At times of the ordergrowth of increases the until thermal it scale enters a phase of fast growth between which for the parametersagrees in with the fitted value of the slope. only significant effect can be seen at very early times of the evaporation. Figure phase approximately grows linearlyapproximately with linearly on time andtime the area in the island phase depends pling. As we decrease thelinearity DGP can coupling be (i.e. increase easily explainedin be the recalling island that sits inλ close the to small the horizonexceeds and the thus area has of a the fixed RT location surface for in varying the values island of phase. Since the area in the no-island Figure 17. of the same plot with bounds (dashed) on the onset (right). JHEP12(2020)025 23 bulk 1 ) is G 4 (4.15) (4.13) (4.14) 4.13 saturates ). are region ent = 0 3.44 v χ ]. In hyperbolic from the surface Σ 101 χ , 100 have a larger regulated Σ for an arbitrary entangling χ , ; instead we will consider their [Σ] at late times; thus we will find ). S (Σ) R Σ is defined such that eq. ( A is exponentially large compared to 3.3 turns out to be independent of the . , 0] 1 1 ent ent 2 1 ent v − , as measured by the boundary metric − v 2 2 − − ]. To this end, it will be less helpful to d v d d d Σ th L R s χ > 99 surface [ 2) 1) ≤ A − − , all belts share the same rate of entanglement and the entanglement velocity bulk 1 d d = 0 ) with the monotonicity of mutual information. [Σ] th ( ( G – 38 – dt s 4 17a acts as a bottleneck, let us formulate the more χ = dS = 4.13 = ent th = 0 ) with the zero-width belt result in eq. ( v s in the boundary metric ( Σ [Σ] dτ χ Σ 4.13 dS Rχ 1 R ) cannot be tight at late times for belts of finite width. The reason 4.13 surface acts as a bottleneck for entanglement growth even for finite width )): is the area of the entangling surface 3.7 = 0 can be similarly defined by demanding that the straight surface ] which primarily considers flat space, ], it was proposed that the growth of entropy is bound by ). The thermal entropy density χ , where the defect is located, i.e. where the bath is coupled to the black hole. ), but always evaluated for the , while, as can be seen from figure ) — we shall justify this choice further below — specifically, (Σ) 99 99 ent Σ ), points on any of our entangling surfaces are a fixed distance 3.7 A v = 0 4.13 4.13 3.7 = 0] To see why the surface at It is clear that ( In [ In [ The proper distance would be χ χ 23 [ belts. refined combined bound now,consider following the closely entanglement [ entropy of the bath intervals growth at late times.we To will more therefore tightly need bound toIt combine the will eq. late ( turn time out behavioreq. of that ( finite the width optimalthat belts, bound the obtained in this way for finite-width belts uses obtained by comparison of eq. ( is that the areaA factor on the right hand side shape of the entangling surface,space, provided it is sufficiently straighteq. [ ( saturated at times just above— the this thermal definition scale is for sufficiently well-defined straight in entangling the surfaces sense that in eq. ( independent constants. The entropytimes density is horizon given area) by divided thethe by black metric the hole ( CFT entropy (i.e. volume of the spatial slices (again, measured by surface where eq. ( at However, the extrinsic curvature ofthis the distance. entangling surfaces which Similarly,volume. we the consider entangling depends surfaces on with larger consider a few characteristics of our belt geometries. As can be seen from the metric in JHEP12(2020)025 ). . ) = 0 , the τ 4.15 3.3 τ (4.19) (4.20) (4.16) (4.17) (4.18) ]( at time since the ¯ . R 0 R/L ) R 0 ¯ R τ [ . ]( S 0 R . We can find a , see below. For ](0) ¯ ¯ . 0 Σ R R 0 R χ we assume that the ∪ ¯ 0 R ](0) 0 L τ : , or in other words, by 0 R 0 is equivalent to looking ¯ R 0 L from above ¯ [ R ¯ R ¯ ) R R S is just barely a subsystem 0 [ , ∪ ) τ 0 I , which will always give the − are subregions of 0 L τ τ ]( ) 0 − ¯ ]( 0 0 R R [ τ τ . < τ R at ¯ R S τ 0 ]( 0 ¯ R ](0) τ is the mutual information between 0 R ∪ ¯ : + R R ) ¯ 0 0 L instead of R L ¯ [ τ R R ¯ R ¯ ) S ¯ : . R [ ¯ R 0 at time R [ S 1 I L ¯ 0 R ∪ ). For times before ¯ ) + ¯ ∂ 1 R − 0 R − L [ ( τ I d ¯ R can be considered a subsystem of the original A 4.17 ]( ( . It turns out that, for sufficiently large 0 ] + 0 L ](0) = 0 Σ S τ 0 – 39 – ent ) CFT at time R χ ¯ R v τ [ − R − but ¯ ) R but v 104 0 ) , yielding the prescription stated below eq. ( Σ S th , : v is allowed to decrease as fast as possible, while still χ R ¯ R 0 L ¯ Rs ∂ R − = 0 103 ¯ ) = ¯ ) is in fact a family of inequalities, parametrized by a is chosen such that ( ( R at time R 0 [ τ ≤ 0 0 S τ A 0 Σ I ]. τ ) ¯ ]( χ 0 = . We can then use monotonicity of mutual information R 4.20 | 0 = ent τ R 0 0 ) + 102 v [ τ τ ]( ¯ L R ) we can then bound S ) and right ( 0 1 R : th ¯ − R L ∆ ). Since the regions ¯ ( R 0 L τ Rs 4.13 S with ¯ ∪ R For the hyperbolic geometries considered here and the temperature [ 4.13 ≤ ≥ | 0 Σ 0 L I ] . The time 24 χ ) = ¯ 0 0 Σ R ), it is now easy to see why the entanglement growth becomes universal R [ ≥ τ χ ¯ ], we now assume that information is only transported with the butterfly R ¯ ) ]( S − but R v τ Σ R 99 is bigger than ∪ 4.20 , i.e. belts surrounding the defects, whose entropy is the same as χ in the left ( ]( ¯ it is sufficient to focus on the case where R ) ¯ or less. L R t R : if ¯ ( R 0 Σ ¯ . Using eq. ( ¯ 2 R R L [ τ χ O : but , in the sense described below eq. ( ¯ S , this velocity is given by [ v R τ [ ](0) L at ∆ I 1 ¯ πR R at with ¯ [ 2 R From eq. ( Similar to [ 0 R/L I ) The butterfly velocity is defined as the spread of the region in which the commutator of an operator t ¯ ¯ R ( = R 24 [ 1 , their mutual information bounds the mutual information of regions tight bound on the Pageminimizing curve over by all minimizing overtightest all bound choices of is obtained for O choice of regions of mutual information of subregions compatible with eq. ( τ To find a tightestany bound fixed this has tosmaller be bound. minimized The over mutual information allthe appearing initial choices on time of the slice right and hand can side be are obtained evaluated on numericallyat by late using times. the results Note of that section eq. ( Collecting everything, we find a bound on the Page curve of the black hole, In our setup, weS have that the one-sided entropies are time-independent, This implies that a beltbelt region where the regions velocity T state of both Rindler patchesat is the pure. Page curve Considering ofthe the entropy displayed black in hole the instead Page of curve that as of the radiation. It is useful to rewrite complement JHEP12(2020)025 ) 1 d but − 1 v 3.41 , the ) ) with 1)] β ] argue ( − behaves d 4.13 ) with 105 , > O [2( ). Typically, 17b ) / , independent 99 ) 1 d Σ 4.13 growth at times β , the entangling χ − 4.17 . Thus, while all ( due to our focus on 3 ( 1 2 1 25 O 2 O / 2) < 1 ∼ ∼ − d > Σ d τ ) that this surface acts τ ) for Rindler spacetime ( χ ∆ d √ 4.20 4.15 . We see that a fast growth ). For finite-width belts, plugging 17b 3.42 , again leading to quadratic entanglement with the factor of 2 τ ]. Specifically, by applying an analysis away from the entangling surface (in This growth saturates eq. ( of entanglement growth is bounded by 1 ∼ bulk 26 G 102 but 4 v ) as describing the entanglement growth in · 1 2 dS/dt ) to thermal relative entropies, [ magnifies the early time behavior of belts with 3.44 – 40 – ) by — one on either side of the TFD. 4.17 17b 3.44 = 0 times the volume between the entangling surface and χ th , for the choice of the entanglement velocity stated in s ) shows that early time evolution is equivalent to holding the cutoff 3.12 of the entangling surface by . For the zero-width, it is easily verified, at least numerically, from eq. ( τ ξ on either side of the TFD. a posteriori , one might worry about contradictions to entanglement monotonicity 0 so one might worry that entanglement is spreading faster than the speed ) into eq. ( 3 but ] dual to AdS planar black holes. Furthermore, for so that the growth is indeed quadratic by eq. ( χ > 3.56 > v τ 93 d > ] which apply above the thermal scale. However, no immediate contradictions ∼ ]. in ent ∗ v r and shifting the ). 105 ) and ( while it reduces the early time behavior of belts of width , but times the area of the tsunami wavefront — this is essentially eq. ( 101 − ζ , 1 1 99 3.55 permitted by operator commutator growth [ 4.15 √ > v For concreteness, let us interpret eq. ( Let us end with a few observations regarding the structure of entanglement spreading The bounds found in this way are presented in figure but < Note that time-reflection symmetry demands that the Page curve have an early time expansion con- To be precise, we should multiply eq. ( 100 v 26 25 Σ ent but th just two copies of the entangling surface taining only even powers of that eqs. ( at fixed growth. that, for regions and timesby above the the thermal thermal scale, entanglement entropy growtha density must be tsunami bounded wavefront propagatingeither with direction). speed Saids differently, the rate hyperbolic space without defects, specifically, computingof the half-spaces entropy for a regionv consisting v similar to the one reviewed around eq. ( velocity for a CFT onwhenever hyperbolic space exceeds the butterflylaws velocity, eq. [ ( appear in the present case, as we now explain. in [ in our system. First,with we hyperbolic note spatial slices that differs the fromin entanglement the flat analogous velocity velocity space ( [ χ other curves show the linear entanglementbehavior spreading of for the time blue scales curve issince dominated the by belt entanglement spreading width through is thermalization, ofbelow order of the the thermal thermalization scale. scale The quadratic is reminiscent of the ‘pre-local-equilibration growth’ described at early times is allowedas by the can bounds, before be theto seen linear understand growth from phase how is the to entered.qualitatively make figure, different Further, them than the these tighter. other curves. bounds Notecurve The that are is reason the controlled is fairly that by blue the the loose. early curve thermalof convex in scale the onset of and belt figure the It thus width. lasts The would for rescaling roughly be in figure interesting as a bottle neck forMatching information this transfer behaviour and to thus controlsfurther the the late justification, late time time rate growtheq. of of ( entropy. growth of the exact Page curve provides We thus see from the first term on the right hand side of eq. ( JHEP12(2020)025 , but (5.1) ), for ) to be L > v horizon.  ent 4.13 v = 0 B ` T ) with 4.13 .  . However, we replace the 1 − +1 2 d d bath dx + = 0 ··· T + 2 1 corresponds to an extremal ) with Poincaré coordinates, dx + 3.6 2 – 41 – → ∞ dt z . In hyperbolic space however, it is possible for the − 2 , i.e. a backreacting codimension-one brane extending but 2 dz  < v 2 2 z L ent v = 2 ds and the tsunami wavefront replacing the entangling surface. In flat space, The Poincare patch models a zero temperature extremal black hole. The brane ent v illustrates the Poincaré patch in our bulk geometry. 18 of minimal area; within a few thermal times, the tsunami wavefront propagating in For the most part, we will be interested in limit of large tension (i.e. = 0 Of course, the coordinate singularityFigure at which the brane theoryboundary can be CFT. described as As Einstein we gravity describe coupled to in two a copies moment, of the the brane geometry naturally inherits a bulk geometry described in section across the spacetime whichAdS-Rindler locally coordinates has introduced the in geometry eq. of ( AdS this does not contradict the bound on entanglement spreading due to the5 butterfly velocity. Extremal horizon inHere equilibrium we with turn our attention to extremal black holes. In particular, we consider the same tsunami wavefront to growthis in is both precisely what directions happens away forχ from the hyperbolic the half-space entangling which has surface.either an direction entangling Indeed, surface grows toWe an thus area see exponentially that, large though compared the to hyperbolic the half-space entangling saturates surface. eq. ( replacing the tsunami wavefront can beentangling surface typically such chosen that to itfrom propagate shrinks a in or spherical a does entangling direction surface). notsaturated, away grow Thus, one from in for must the the time require flat (e.g., space propagating equivalent inward of eq. ( Figure 18. intersects the CFT Poincaré patch at the origin and infinity. JHEP12(2020)025 : 28 5.1 = 2 . Of (5.3) (5.2) d 7 and scenario, we 6 to determine = 0 5.3 -dimensional CFT T 27 d vacuum geometry. How- +1 . , one always finds islands for d 1 − . If we consider the coordinate = 2 2 d , here for the θ , , where the information paradox d 3 dx 18 3 cos + y ) and so the induced geometry on the ··· = 5.2 a. 1 + 2 1 19 -dimensional gravity theory arises because of dx d – 42 – + 2 θ, x -dimensional Minkowski space, dt d sin bath CFT on the asymptotic AdS boundary. We note − y ) cover a wedge of the AdS ] but in a different conformal frame). = 1 b. Note that in contrast to the finite temperature TFD = = 0 5.1 z 19 2 CFT T )-dimensional Minkowski space. The Penrose diagram for this ds for the metric in eq. ( 1 ] that for the extremal case in 2 − d = 0 1 x , we are simply studying the vacuum state of the defect CFT in flat space = 0 T , we carry out this calculation using RT surfaces corresponding to island and no- . Indeed, we will find in higher dimensions that quantum extremal islands do not 3 The Poincaré coordinates ( The remainder of this section is organized as follows. We shall begin by first explicitly We may recall from [ Of course, we may also have the boundary perspective where the 5.2 Of course, this is a pure state, asComing is from manifest integrating in the bulk length since of the Poincaré the time surface slices down constitute to complete the extremal horizon. ≥ 27 28 transformation Cauchy slices. island phases, respectively. Finally, wewhen collect each these phase results dominates. in subsection ever, in the presentwould geometry appear with on either a side backreacting of brane, the a brane portion — of see two figure such wedges paradox is avoided by the appearance of quantum extremalconstructing islands. the bulk and brane metricsthe to entanglement be entropy used calculation in which theand we extremal wish case and to by consider. introducing Then, in subsections appear in the large tension limit.black Nonetheless, holes no do information paradox not arisesThis radiate, since extremal i.e. contrasts the with black thefor hole non-extremal the and eternal case the bath black inthe are hole section continuous not in exchange exchanging of the radiation. quanta effective between the black hole and the bath. Of course, the the analogous belt regions. Thisfirstly, result is there a always consequence of exists two a featuressecondly, which the bulk hold alternative for RT no-island surface RT intersecting candidateand the surface this has brane surface an to is additionald produce IR therefore an divergence subdominant. island; However, neither of these statements hold in state (entangling two copies ofonly the have bath CFT) a in single section course, copy at of the bath(analogous CFT, to e.g., what compare was the done in above [ to figures This brane perspective is illustrated in figure in Minkowski space isinsert coupled the to latter a at codimension-onedefect conformal is defect. also flat, For i.e.perspective simplicity, ( we is shown in figure which is equilibrium with the that with Poincaré coordinates,where we the are bath CFT examining is the living system on in flat a new conformal frame Poincaré metric from the bulk geometry. Hence the brane supports an extremal black hole JHEP12(2020)025 B θ . To , i.e. (5.5) (5.4) = . The θ . CFT = 0 1 ) is inter- Σ slices each 1 dimensions.  x d 5.5 ≡ , as expected d B B θ θ R AdS ∂ ] to island phase 2 sin ), eq. ( , = ), i.e. 2  L/ V 1 , is inherited from the 2.4 ∂ − = 2 d B dx ` → ∞ ), where the + y becomes the zero temperature 2.3 , or alternatively ··· 1 . defect = 0 + t = 0 2 2  2) , the brane spans one such slice at a z 2 B dx ε/ 2 /` according to eq. ( + − 2 . According to the RT formula we should o 2 L ) (1 T dt ∞ ε − b, [ – 43 – 2 = 2 ∪ dy B ] sharing the same boundary θ b  2 − B V , θ 2 sin 2 L sin −∞ 2 y . We will say that this RT surface is in the = ( = R 20 d is interpreted as the radial Poincaré coordinate running along the y 2 AdS ds The brane and boundary perspectives of the extremal black hole setup. defect = 0 ) is transformed to the form given in eq. ( t b. ). Here, determined by the brane tension — see figure a. 5.1 horizon B 2.4 θ R = Figure 19. θ Specifically, we consider the entanglement entropy calculation for a boundary region Following the brane perspective described above (and in section that is the complement of a “belt” geometry centered on the defect at extremize their area. Assurfaces we which achieves discussed this in extremization; thethe the introduction, RT one prescription there with then are instructs thebrane, generally us forming smallest to two a choose area. sets quantum of extremalwedge island The on of first the brane class which of belongs surfaces to the are entanglement those which intersect the subregions in the bath includes islands residing on theR brane. the boundary subregion consider codimension-two surfaces determine RT surface candidates among these surfaces, we must search for surfaces which preted as an extremaland black the hole CFT solution of ofbath. the the This then gravity flat provides theory a asymptoticThe induced direct boundary question brane extension of which at at the interests extremal us scenario here in [ is then whether the entanglement wedge of certain and we may thenfrom read eq. off ( the curvaturebrane, scale and of the the brane Poincarébulk. as horizon As on usual, the we wish brane, to located work at in the regime The induced metric on the brane then becomes the metric ( inherit a Poincaré metric.fixed As described in section JHEP12(2020)025 1 V − and d from 2 (5.6) − d 0) . > − , . . . , x ( 2 = 1 length x x are marked in ± , one candidate ∆ spacetime with a 5.1 5.1 +1 d AdS 29 . } , 1 directions. 2 − 1  d 1 − d , forming an island on the brane dz 1 dx has the dimensions of R − , but the RT calculations for this  2 σ d − R z Heading out of the bulk ⊥ d , . . . , x , . . . , x 2 2 1 + vol x x (b) { , r 1 3 ) and for the corresponding dx 5.7 V Z . 2 – 44 – − , are equivalent. Integrating out the ] ⊥ d in eq. ( . Various quantities defined in section b, b vol ± 1 R − . − d is an IR cutoff in the = [ L = + introduced in section IR -dimensional Minkowski CFT with a defect (green dot) along a ` R 2 ± d − ) = no-island phase d H V ( where vol A 2 − d IR of a belt geometry in the CFT. As considered in section ` ) ∞ b, [ , shown in red, intersects the brane at a QES ∪ is the volume of transverse directions ] R The bulk dual to a Definitions for the choice of 2 b Heading into the bulk Σ − . The CFT lives on the asymptotic boundary of a Poincaré − ⊥ d , ]. That is, we are considering the complement of (a) = 0 ) on the two branches of the RT surface. vol −∞ 16 1 5.8 x Note that in contrast to = ( 29 where so is essentially given by AdS [ region and for its complement, directions in which the branebecomes is constant, the area functional of a codimension-2 surface second set of surfaces falli.e. trivially these into surfaces the are bulk in and the do not produce5.1 islands on the Island brane, phase As a starting point, let us review the calculation for RT surfaces of belt geometries in pure Figure 21. eq. ( R RT surface belonging to the entanglement wedgethis of figure. Figure 20. line brane (green line) running through it. We consider the entanglement entropy of the complement JHEP12(2020)025 , , ) ∗ ) ∗ ∗ z b z z . In − ) to 2 = , (5.9) (5.7) (5.8) = (5.10) ) with z (or 5.7 z − < π/ at ) = (0 D of the RT B 1 R . In terms of . < θ z, x = 0 σ ( ) 0 D ) for pure AdS 1 ∗ , allowing us to ) with respect to z # + R 5.8 b 1) = 5.6 ) Σ − ∗ − z dz/dx d z ] and reviewed around 2( = 1 =  1 ∗ z and the RT surface begins x z z , we will have o  direction) defined by the RT T = 0 1 1 x . We can integrate eq. ( + 1; x 1) R . − ∗ ), we see that d Σ 1) z 2( − ) or heading out of the bulk ( z d 5.7 i 1 d + 1) i , rather, the relation between − − b d − 2( d 1) d 1) z . If we evaluate this expression at ; − 1 − 6= 2( d d 1) h 2( 21 2 2( ∗ D Γ above is determined by whether we are on the as one heads away from the boundary h − z d – 45 – . Further, if we were to extend the RT surface , 1 π Γ From eq. ( d b ± x q √ − 2( 20 30 ± . , = = 1 2 = -value attained by " 21 2 , which we emphasize is in the empty AdS vacuum (see 1 -separation between a point on the RT trajectory and D 1 z 1 x D dz x F dx when heading into bulk (towards when heading out of bulk (away from 2 1 and 1 − 1 20 d d ∗ x z ∆ is obtained by extremizing the area functional ( d z 1 − R x = . Further, the sign Σ ∗ ∆ D 1 z x    is the maximal . This functional, viewed as a Lagrangian, contains no explicit dependence ∆ is the absolute ∗ + ) z 1 b — see figures 0 x along the trajectory is then given by 1 − ( 1 > z x x = 1 ), we have ), this extremization leads to a boundary condition restricting the angle at which 1 x ), and hence the corresponding Hamiltonian is a constant along x must be determined by demanding that the choice of the intersection ∆ 20 1 5.8 3.21 b In general, as illustrated in figure Now returning to the geometry with the backreacting brane, each half of the RT surface The RT surface on either side of the brane will follow the trajectory given in eq. ( As noted previously, if we restrict our attention to positive tension x 30 R eq. ( the RT surface meets the brane. Again, we may reducethis this case, to the a RT two-dimensional surfacein problem must order be for increasing the in RT surface to meet the brane. and surface with the branewhen should brane extremize action the includes RT an surface’s extra area DGP (plus term). the As area described of in the [ QES, on the asymptotic boundarypast at the brane, iteq. would ( hit the asymptotic boundary again at Σ prior to meeting the brane. We have placed the defect at heading into (out of)we the obtain bulk half — of see thesurface. figure width Denoting of this the widthfigure boundary as strip (in the Here the initial (final) endpoint on the asymptotic boundary, on the portion of the RT surface for some constant portion of the RTincreasing surface heading into theand bulk therefore ( obtain the trajectory of the RT surface: the profile on deduce JHEP12(2020)025 ) : , , R 5.8    σ 1 ) and B ), the (5.17) (5.13) (5.14) (5.15) (5.16) (5.11) (5.12) 1 − θ ), with d ), ( 5.7 5.7 5.11 sin  5.9 . coordinate B B 1) θ y θ 2 − 1 d # 2( sin ∗ sin 1) z 2 − 2 b d − is the λ d − 2( ∗ L 1 −  z 1) 2 − ) 1 − QES − d QES may then be determined ∗ 1 d ⊥ d B y z z q b QES 2( QES is given by eq. ( , θ z z b + ,  ij B 2 1 2)vol v u u t ), and using eqs. ( and h ), we read off the area of  θ d, λ determined from eq. ( ∗ 1) − ± dx ) ( B z i − + 1; 5.5 θ d R 5.10 B d cos F + 1 T ( θ σ brane 2( − 2 b 1) ( = d z ∗ cot G − λ A z 4 − dz B cos d  = θ ∗ d 2    + B z ) ) QES θ 2 2( 2 z ) . Hence, upon perturbing the intersection R cot ; − − into eq. ( ij R σ ⊥ d ⊥ d QES − ( sin bulk h 1) ) is included to account for the two components  (Σ i G ∂y vol QES ∗ − 2 ∂A 4 d – 46 – 1 i (vol A z 1) z QES d − − d 1)  1) d − d = − , z 2( 1 − L sign is the same one as introduced in eq. ( , and the tangent 2( d d B , , ) to determine the variation of the RT surface area D y h θ 2( 2 2( 1 2 QES ∂ ∂ 1 ± h QES Γ − = 2 " L z d 2 π ∓ ∂y 3.21 Γ cot = ! 1 √ F B D QES 2 1 θ 2 2 ). The relationship between 1) θ + QES 0 = d ∗ z 1 ds , the QES on the brane. Here, ∓ sin 2.7 x ⇐⇒ d L QES cos R d z z ∆ 2 σ − — and the QES . An extra factor of ± + (1 ) = ( ⊥ d y B is chosen to be 1 θ = ≡ ± , z

20 21 vol j 1 − ) b 2 1 d QES x B normalized with respect to sin X ( z − − ⊥ d d i b , θ λ L b T 2 is the angle between the RT surface and the brane, = = d, λ is defined in eq. ( ( ) to find ) =vol and ) b QES R F j R QES θ — see figure λ σ θ ( X 5.15 QES (Σ As before, we may use eq. ( R A σ ∂y cos ∂A where by substituting and ( is satisfied if The extremality condition illustrated in figure of the RT surface on either side of the brane. From eq. ( where of under perturbations of deviation vector both of the RT surface with the brane, the RT area varies as and the area becomes the length of the geodesic in this geometry. where we view the RT surface as a geodesic in an effective two-dimensional geometry JHEP12(2020)025 ) ) and and ) 5.16 5.17 ) into (5.19) (5.18) 2 2 R 8 d = 7 d = 6 d = 5 d = 4 d = 3 d = − , − QES ⊥ d d z 5.13    D vol   #) 1   1) − − from above. 1) d 1.0 d − ) and ( c   2( d θ i  2( 5.7 i 1) → ! 0.5 , the hypergeometric ∗ 1) − 1 ∗ B z , we have plotted the d QES − d z ∗ θ z d 2( z QES 2( h z  b  λ 0.0 h 22a

Γ . ; 2) Γ 1 bulk π QES . 1) G − 2 z √ brane -0.5 2 ∞ 2 " − − d ; in particular, note in eq. ( d G − ⊥ d ( ∗ + d d ∗ , we shall discuss the fact that, for O z to z   2( vol → + 2 -1.0 ). For 5.3 i ). In figure 1; 0.5 0.4 0.3 0.2 0.1 0.0 Critical brane angle as a function of the ratio bulk 2 − eff i 1) QES d G 1 − d − G y π 1) − c 5.17 d 5.15 4 2 θ (b) of − 1 ) — keeping in mind that we have not shown 1) 2( − d ( h 2( is given by 2 2 − 1.2 d 2 h Γ − − 10 1 – 47 – d ⊥ d − 2 d 5 π R 1 2 d , giving ) and ( . In section 2( ] ! δ 1 2 √ 1)Γ . We have noted in the second equality of eq. ( 1 2 vol , 1) ∗ 1) 1 2 + -1 = - 1 = = 0 = - = 2 z − − b b b b b = 3 L 5.15 , the intersection of the RT surface with the brane, and the d QES λ λ λ λ λ   2 ∓ d − z d = R 2( ( ) σ d

z (1 ∗ F 2 between the RT surface and the brane as a function of the 2 1 ( , the extremal surfaces discussed here fail to exist. That is,    /z − and c 2 ⊥ d 1 R θ 2 b 1 − 0.5 − σ by eqs. ( bulk ⊥ d − λ d − QES bulk 3 = d QES b d G z d L z G 4 is determined by eq. ( L vol + vol [( 0.4 4 ∗ O = ∓ + /z 2 ) 0.3 2 2 ) − 1 + 2 R as a function of brane angle. π − ⊥ d − B R ⊥ d − d σ brane θ QES R d σ ( QES brane δ for various z 0.2 ( σ G vol z A vol G B 4 A   4 θ defines the UV cutoff surface near the asymptotic AdS boundary, and Plots of the position of 2 + δ 2 0.1 + ) − 2 − brane ) d = R d L R G bulk z 0.0 bulk 8 6 4 2 0 4 + (Σ Position of 14 12 10 , the position of the QES on the brane, runs off to infinity as G Having determined the profile of the RT surfaces, we may proceed to evaluate their cor- (Σ G 4 A below some critical angle are linearly related to = 4 A b (a) QES B ∗ QES where z function becomes that these surfaces minimize thethe entropy functional generalized yet. entropy Inserting functional, eqs.hence ( we of find the that complementary the bath entropy region of the belt geometry brane angle θ y responding entropies using the RT formula ( where the top (bottom)left signs (right) of chosen the above extremal ifthat point the all terms RT surface ofthat intersects the the the previous brane expression ratio are toposition linear the of in the intersection Figure 22. critical brane angle at which this surface runs off to y JHEP12(2020)025 , ) 32 at eff D 1 , G 4 ) 5.15 (5.20) (5.21) 2 CFT 2 , which − − Σ ⊥ d T d c D vol ∼ 1 . − ), we have the ]. d with entangling  bulk 1 , which corresponds   85 B − D i 5.19 θ 2 d /G i 1) 1 dx 1) − geometry of the brane 1 − tan d d − d from the bulk metric. d + d 2( L ]. However, by a simply Weyl 2 2( h ) > δ/ h ··· 106 1 Γ x 2) Γ + L/δ π ( 2 1 − √ 2 ) for d dx ( ) but rather we are considering the . +   δ 5.21 2 5.5 in the first line. Note from eq. ( 2 = dt z 1 − D − . Imagine we begin with a single copy of d  0 ]. Further, we note that both contributions B − θ < with 17 2 2 , 2 2 1 − ∗ – 48 – ). Beginning with the leading term of the second δ ⊥ d − x z sin 16 d 2 1 δ vol x . For this geometry, the holographic entanglement 5.19 b 2 = 1 − − so the correction is indeed smaller than the other terms d 1) D 2 CFT − + d = ds ( 2 ) to replace / ) contain a prefactor proportional to 2 B 1 1 − ] θ ⊥ d − x ), and evaluate the entropy of a belt of width B 5.9 d 2 θ δ 2 ] 5.18 vol δ 5.2 sin 2 and 17 2 1 , Turning to the first term in the first line of eq. ( + 1) . It is interesting to note that there are no higher curvature corrections 1 b x − b R 16 − d λ σ = 31 [( ( ), we find that it corresponds to Bekenstein-Hawking of the QES, i.e. domain: 2 = 1 1 ∼ 1 ds x − bulk ∗ x 5.19 d G . However the latter can be produced by making a local Weyl transformation in L /z 4 0 . This leaves us with the second term in the first line. Upon closer examination can b but the corresponding copy of the CFT resides onto the AdS > = QES 0 ± z 1 We can see that these results correspond approximately to the expected entropy from Using the brane perspective, let us examine the various contribution to the generalized EE x Further, we are only performing the Weyl transformation ( One can argue that all of the higher curvature corrections to the Wald-Dong entropy must cancel against vanish. Hence in this flat conformal frame, the higher curvature corrections to the Wald-Dong entropy < = S 32 31 1 1 R on the gravitating brane andtransformation, so the should brane be metric conformally becomesσ invariant, flat e.g., and see further [ bothindividually the vanish. intrinsic and Hence the while extrinsic thesecurvature curvatures entropy curvatures of do corrections not must vanish all in cancel the against original one conformal another. to frame, the the intersection higher of the brane with the UV cutoff surface Note that this is geometrystandard is not conformal the frame induced where metric one ( strips off the factor of one another as follows: in the present case, these terms would arise from integrating out the boundary CFT x for the positive where we have separated the area lawsurface. contributions of Now from the the two components brane of perspective the in entangling our system, the bath CFT reside in flat space for to one side ofthe the CFT conformal in flat defect, spacesurfaces say ( at entropy becomes [ up to an additional factoron of the 2, first e.g., line seemeasures of [ the eq. number ( of degrees of freedom in thethe boundary brane CFT, perspective e.g., as [ follows: we begin by considering the contribution from the CFT times the area of to the generalized entropyentropy of formula. the QES asarea might law have divergence been associated expected withx the from two the components Wald-Dong of thebe entangling recognized surface as the finite contribution to the entanglement entropy for a belt of width that shown here in high tension limit. entropy on the right-hand side ofline eq. in ( eq. ( where we have used eq. ( JHEP12(2020)025 . ) , i.e. 5.20 → ∞ b (5.23) (5.24) (5.22) ) is to Unlike z − 33 D ) and the . 5.20 0 b ) = ± 2 2 > 1 − − x = ⊥ d planes anchored d 1 1 x 1 ) with that from D x vol x 1 − 5.22 d   i . Hence the entropy ( i 1) 1 1) − 1 d , . − d  . d 2( 2 2 2 1 B 2( h 2 − − θ ⊥ d − h − ⊥ d − d d dz d Γ δ z δ 2) Γ vol vol π V 2 − √ Z − 2 d 2 d bulk ( − ! 1 G ⊥ d   , the integral produces an IR divergence at − 2) d 2 L vol QES L 1 = 2 1 − − z . − – 49 – d d d ). These surfaces are constant d ), which then yields 3

L 2( ≥ − , these planes trivially extremize the area functional, 1.2 d b , these planes do not intersect the brane and thus no 5.23 2 bulk = 2 − ± ) G ⊥ d − ) = 2 d L 5.1 R 2 = 2) δ V bulk vol on the asymptotic boundary and fall straight into the bulk. ( 1 (Σ − corresponding to the no-island phase. To determine which G x 2 A 4 A d d > ( 1 − CFT 5.2 ). Hence this simple CFT argument allows us to match the 1 4 d Σ , where the latter assumes that for ( + 5.19 1 QES z − = 0 bulk d ), the term on the second line can be recognized as the contribution of G ' L T 4 B 2.7 θ ' ), there exists another set of simple extremal surfaces which must also be has been included above to account for the two planes at 2 2 0 EE ) sin S b is again the UV cutoff in the boundary CFT. − < π/ which intersect the brane to form a quantum extremal island, and the extremal δ B D ( θ 5.1 Further, let us note that for the special case 33 → Altogether, we have twotion candidate RT surfaces:planes the described extremal in surfaces section described in sec- However, there is no such IR divergence for where 5.3 Islands at A factor of the surfaces considered in section islands are formed onfrom the evaluating the brane. area functional The ( entropy in this no-island phase is easily obtained considered under the RT prescription ( on the entangling surface By reflection symmetry about which becomes leading contributions in the holographic result with the5.2 expected entanglement entropy. No-island phase Above, we studied the(for set of candidate RT surfaces which intersect the brane. In fact one of the boundarysurface CFTs on to the the Bekenstein-Hawkingthe brane. entropy other of copy the Hence ofDGP quantum combining extremal the contribution the boundary to above the CFTcontributions contribution Bekenstein-Hawking (which in ( entropy, extends eq. we to ( precisely the recover bath the for leading Now using eq. ( modify the cutoff appearing inδ the area law contributionbecomes for the surface at Now the net effect of this Weyl transformation on the entanglement entropy ( JHEP12(2020)025 , . ; c 2 ). 2 2 1 θ at 1 . = 2 − 2 < = 1 ∞ − d > π/ < π/ < (5.25) (5.26) d < π/ B b ). Then ) θ b B B = c . Now as are that b 1 θ θ λ θ ( y λ ) − , we have < λ 20 ) indicates B 1 1 1 = . As increasing . > , θ 2 − b < b . b (1 + ∞ 5.15 λ for which there } λ ( π/ ] b = 2 b d, λ ) ≤ ( λ B B = 1 θ QES < λ F θ when z — see figure b ∗ 2 − 1 D λ z function introduced in ∞ scales as − π ). Below, we first observe c [( F = < π/ θ or to the horizon )] + The physical significance of O ∗ B 1.2 2 B z ) are unphysical, i.e. have the RT , this implies that when θ . This differs from the θ produce an IR divergence and ( 36 . For 5 = 0 1 + = 1 5.15 O { , there is a critical angle y b 2 1 b = 2 λ − − λ d d d [1 + ) = 0 − c 2 1 ) − − B ). For , θ d d θ 2 b − B θ − π/ 1 , respectively. d, λ π + 1 − 1 ( ( in the regime of interest with d B , 1 ) θ F 1 1 1 − – 50 – b . Moreover, d = B > λ ) θ b b . As we shall find that , the QES is stuck to the defect, i.e. , we see that this is the range of λ c → − λ 2 , the island phase exists and is dominant. At QES c θ θ b (1 + − ( , an island can only possibly exist when λ 22a 2 2 < π/ − > θ for which B as π/ B It is slightly more involved to determine when the extremal θ θ c − 2 θ ) = (1 < π/ that supports the island phase — recall that this critical angle 34 B , some (numerically deduced) facts about 0) = . π θ B B 1 2 , π/ ) means the DGP entropy contribution exerts a greater force pushing the θ θ = 0 1 . For < π/ − ( b From figure such that the DGP gradient has not overpowered the bulk term of → QES and increasing in ≤ 1 θ 2 c 22b B b 35 close to θ close to . < λ θ λ 1 B , sensible extremal surfaces intersecting the brane can only possibly exist B 1 2 , θ < π/ < − , θ b with b beyond b B θ b 2 < π/ λ d, λ , one can argue that for d, λ ( < λ ( 1 B from above, the QES runs off towards the horizon and consequently no QES exists for ) to push the QES to the asymptotic boundary ). For θ − F 1 F 2 ) indicates that < π/ has an IR-finite area, the analogous surfaces in − = c π/ b 5.14 5.17 To be more precise, we must consider properties of the Let us begin our analysis by constraining the parameter space in which each type of RT 3 5.15 In particular then, no islands form with Of course, this was our regime of interest,Specifically, as this this was can the be regime where seen a as (nearly) follows: massless graviton let is us take the extreme case of λ → < θ 36 34 35 B θ For (decreasing) QES towards the horizon (theIn defect), these it parameter follows ranges, thatsurface the no anchored naive in QES ‘solutions’ the exists obtained unphysical for region from behind eq. the ( brane. induced on the brane. eq. ( the RT surface falls straight into the bulk until it hits the brane, i.e. Since the former divergesthere negatively exists while a the critical0 latter angle diverges positively, it follows that eq. ( it is decreasing in surfaces in the islandthat phase for exists.when For a start, theexists first some equality ofeq. eq. ( ( d > thus are never dominant. candidate surface exists. Itexist is if easy and only to if see that the extremal planes of the no-island phase was plotted in figure the entropies computed by the islandsition and to no-island RT the surfaces no-island equalize,its leading phase smallest, to below this a precludes tran- thedescribed possibility by of QFT islands on incase, semiclassical the where gravity regime the — where island see the phase footnote brane always is exists; well- furthermore, while the no-island RT surface in exists. When both typesthe surfaces one exist which has simultaneously, the thethat correct smallest on generalized RT a surface entropy, as brane is inmore at given eq. specifically, by angle for ( this rangewhich of gives the the DGP minimum parameter is the correct RT surface, we must first study the parameter space for which each surface JHEP12(2020)025 . > 37 as |  b w ∞ λ (5.27) (5.28) | + + 1; 1) ) B − d , θ d run off to b . , the RT formula , extremal planes 2( d 2 2 ; d, λ QES , there exist extremal ( π/ 1) c θ F , y < π/ − d ≤ 2 d − B B d . Altogether, we find that, ∗ QES 2( θ 2 z , , z − above < θ for various ∗ ⊥ d 1 2 z bulk B c  b ) = 0 θ θ c λ vol G from above, the coefficient of the 1 F , θ 2) − 1 2 1 b d ]. for which no extremal surfaces of any −  L ): for 2 d d 107 w d, λ then corresponds leads to an unphysical ( → − 2( π/  1 F from the brane perspective, we note from 5.16 b − from above, + − & λ c c ), and the hypergeometric function identity θ B < w = θ θ . b – 51 – − ¨ λ isl → where ¨ as a function of , we transition to an island phase where the RT 5.24 1  c c B c ) √ θ θ θ ), ( R > θ bulk = (Σ G  B 5.18 , no extremal surfaces exist which intersect the brane. In 4 A θ w c  ; ), ( , extremal surfaces which intersect the brane, corresponding − 1) c < θ . B − d isl 5.17 θ . > θ  d 1 . Further taking ) B 2( θ 5 R < σ 1; brane b ( , there exists a range of G − 1 A 4 < λ > 1) + ), we see that whenever the island- and no-island-type surfaces coexist, the 1 b ; finally, for ) − − d 1 λ R d − , we are in the no-island phase where the RT surface is given by planes falling , we plot the critical angle 5.27 bulk ) c c 2( (Σ G θ , 4 A 1 2 < θ 22b −   B B To gain intuition for the critical angle Now we have two competing possible RT surfaces: for Before continuing, let us briefly note a number of peculiarities which arise when θ This can be proven using eq. (15.1.8) and (15.2.25) of [ θ can be seen from the second equality of eq. ( ( F 37 . First, for c 2 1 straight into the bulk,surface and, is for given by extremal surfaces which intersect the brane and form an island. From eq. ( island-type surface always gives aformula. lower area Moreover, and we is seeno-island thus that phases the surface at entropy picked transitions the outfor critical continuously by between angle the the RT island and where we have used eqs. ( island phase; and, for to an island phase.instructs us As to choose both theconsider types surface the with of area the surfaces smallest difference: exist area in for this parameter space. Thus, we mentioned in footnote ghost-like gravity action in theour focus brane to theory. At any rate, from here on,anchored we on shall the restrict entangling surfaces to either side of the brane, which correspond to a no- kind exists, i.e. the RT prescription failsplus completely. defect This theory may which indicate that canthe there be is brane no dual CFT has to a atheory bulk negative on with tension this the in range brane. thisEinstein-Hilbert of action regime parameters vanishes, Second, leading — and to recall of a so breakdown course, that of there semiclassical as is Einstein gravity, as no effective gravitational surfaces which intersect the brane;∼ as Figure 1 θ JHEP12(2020)025 , our (5.29) 0 due to → and brane QFT B S 3 θ ), we interpret #) ], the addition of 1) ), we see that this 1 − 5.19 d .). We remark that, 2.7 2( 5  . To be more precise, , there is no immediate 3 ∗ z 3 QES ≥ z  d 1 bulk ) gives the change in ], we have found that in the small . It is still possible to stay in the G 2 b " λ 5.29 2 which, as mentioned in the main text, O 2 − − ⊥ d . + d ∗ 1 + case [ ¨ . However, from eq. ( isl ¨ z 2  vol = 2 eff − ) ∼ i 1 d , leading to a breakdown of the semiclassical B d G R i θ 2 = 2 1) 4 bulk ∞ 1) − – 52 – − ] (as mentioned in footnote 1 (Σ d , the QES ceases to exist and only the no-island ∼ d ( c d G 1 − + d c θ 4 d 2 A 2( −  2( h → d h < θ − ! Γ . B eff 2) Γ π θ isl G L  − √ QES z 4 ) d to scale as (

R b , the island phase is favoured as the introduction of the island σ 2 brane c 1 λ ( − − G suggest that bulk ⊥ d A d 4 ) can may be approximated as > θ G L 1 + 4 B + 22b θ ) − + vol 5.27 R = bulk corresponds to limit. Building upon the discussion given below eq. ( (Σ G ), the first term on the r.h.s. of eq. ( + 4 A B 1 θ  1.1 → − ) and figure ) that eq. ( b geometries. Finally, we study extremal surfaces serving as candidate RT surfaces to λ 5.25 5.19 In closing, we note that, unlike the We briefly comment that, unlike for the CFT region considered in [ 2 limit, where an effective theory of gravity plus quantum matter emerges on the brane, B requires a slightly differenton treatment. the We brane, begin supplementedAdS with with a JT discussion gravity. ofdetermine Next, the the we induced entropy review in action the thecurve. two bulk phases, At AdS with leading and order without in an an island, expansion leading in to the terms Page of small brane angles, i.e. unlike for non-extremal blackinformation paradox holes that to arises as be a discussed result in of the section lack6 of islands in the Two extremal dimensions case revisited here. In this section, we specialize to the case of islands typically doeq. not ( exist forisland extremal phase black by tuning holeslimit in description of the effective brane theory [ RT surface of the islandthe phase no-island has the phase topology consists ofnot of an favour two infinite one half-planes. strip phase while over Thus, the the the RT other. topological surface of contributionθ would topological terms to the bulkisland gravity and theory no-island does not phases change ofonly the the effect favourability belt a between the topological geometry.geometry, contribution This the to is RT the because surfaces in Wald-Dong such entropy both a formula phases modification and, have can vanishing for Euler the characteristic. belt Namely, the the effective theory ofisland the rule asymptotic ( boundary andthe brane. introduction of Namely, comparing thethe with island, QES. the and Hence, the for secondreduces term generalized gives entropy. Bekenstein-Hawking For entropyphase of is possible. in the small the r.h.s. as giving a change in generalized entropy due to the introduction of the island in eq. ( JHEP12(2020)025 , . , ) ), 2 and 2.4 ) as 109 = 2 2.1 (6.5) (6.3) (6.4) (6.1) (6.2) → d induced L 6.2 d I  . B # ` ··· + /L . 2 brane with radius of ˜ bulk R 2 2 G ]. In the absence of any 8 L 1 = AdS . + eff ˜ ˜ g , R ! ]. − 2 2 B + 1 ` L p )). The unusual logarithmic term ˜ ! g. x ˜ − 2 R − 2.7 d 1 2 ], which appears from integrating out 2 p leads to an L s Z x − 2 108 L − d

1 bulk Z

1 ,G (cf. eq. ( induced log o )), which prevent a naive substitution 2 I ) T πG ˜ – 53 – R o ≡ 4 − 2.6 = 2 − ! LT − = d 2 ). In two dimensions, an Einstein-Hilbert action is 2 B 2 eff 2 ]. However, we can also retain the subleading terms, ` L JT ` bulk 2 I ). In this limit, the brane moves towards the would- " 6.1

], we then choose the brane action as brane ˜ g 1 f = πG I 2.4 4 (see eq. ( − = − p , giving rise to a logarithmic UV divergence in eq. ( 2 2) brane 2 eff x I L ) and ( ` 2 in the same way as in higher dimensions (i.e. through eqs. ( − d = 0 in eq. ( d eff 2.2 ( Z ` θ , the induced action evaluated for higher dimensions contains coef- = 2 (1 B θ 2 2 eff  ]). Following [ 1 πG eff L ` 16  111 , related to = B 15 , ` . boundary at 0 )): 3 13 induced , I 2 → 2.7 Throughout the main text, we considered supplementing the brane action with a DGP Let us start in the absence of JT gravity, considering only the brane action B ] in the brane theory (e.g., see recent discussions of quantum extremal islands in AdS topological and so it110 is common to insteade.g., consider [ Jackiw-Teitelboim (JT) gravity [ a small brane angle be L/` term — compare eqs. ( curvature and ( Thus, as in the higher dimensional case, the large tension limit leads to Notice that while thethe second first one equality must followsabove be the arises redefined same from for the definitionthe nonlocal as two-dimensional Polyakov in CFT action on [ higher theDGP dimensions, brane terms — in see the the brane, discussion extremization in of [ where the two effective scales are As we saw in section ficients with factors of Instead, redoing the calculation specificallyfound in to two be dimensions, the induced brane action is induced by the bulk Einstein-Hilbertits action (with corresponding cosmological Gibbons-Hawking constant) action given in on eq. the ( brane, and the brane tension term which produce corrections due to the finite UV6.1 cutoff on the Brane brane. action We begin by brieflymensions reviewing — the a more modifications complete for discussion the can induced be brane found in action [ in two di- results precisely agree to those of [ JHEP12(2020)025 ) for (6.9) (6.6) (6.7) (6.8) (6.10) (6.11) 6.7 , # #  ) similar to the JT L ··· ··· ` ), we arrive at the 6.7 ) now reads , +  . + , ) and instead chosen 6.5 2 2 6.7 ˜ ˜ R log R  6.5  brane Φ 2 2 8 eff 2 JT 2 8 ]. However, we would like 2 JT 2 L G L ` 1 1 ` G 4 , we are omitting boundary 2 + + + 2 + + in eq. ( ˜ ˜ R − ! ˜ R 0 R o ˜  φ φ R ), is therefore obtained by taking . The limit of small brane angle +  T 2 . . 2 JT ! L = = 2.4 `  ˜ eff + Φ R ) yields an equation of motion simply + Φ − ˜ φ 0 = ˜ ˜

2 JT ˜ R 2 φ /G R Φ ` B 0 0 ` − − Φ ˜ log Φ ˜ brane  φ

 ( ˜ ˜ g R G ˜ g − 2 JT 2 − log − + ` " – 54 – ). Similarly, the source-free equations of motion p 0 ˜ p ˜ g R x + , x 2 6.4 − − 2 ˜ d R " d  = Φ p ˜ ˜ φ g 0 Z x JT Z L  2 ` ˜ − Φ ˜ g d  p − brane Z brane x 1 p 2 1 log eff ) may be interpreted as the renormalized effective action pro- d πG x πG eff , φ 2 1 1 16 Z d πG 16 G 6.11 2 eff ), it is clear that varying 16 Z = 0 + brane − Φ π 1 1 = JT 6.7 πG G 0 4 I 4 φ 16 ]. + still through the first equality of eq. ( 1 = = . With the addition of JT gravity on the brane in eq. ( B = induced ` 0 0 0 I ]. To facilitate this comparison, we make the following field redefinitions ˜ Φ φ φ 2 . Note that this leads to a logarithmic UV divergence in eq. ( induced I L The above reviews our discussion of the induced action in [  , related to B JT Here, the first lineduced eq. ( by integating out the brane CFT, and the second line contains the renormalized giving the bare andthis renormalized renormalization values shortly. of In the terms dilaton of — the we latter, shall induced clarify action the ( meaning of to compare our resultsderived in for [ the quantum extremal surfaces and the Page curve to those ` non-JT case, as mentionedfor below the eq. dilaton ( can thenas be discussed obtained in by varying [ the metric and further shifting the dilaton, Note that we have discarded thethe usual tension tension coefficient such thatsimplicity. no In cosmological eq. constantsetting ( appears the radius in of theθ curvature first on line the of brane eq. to ( where we have redefined themultiplying topological an part Einstein-Hilbert of terms, the i.e. dilaton upon collecting the coefficients following induced action on the brane, terms), The Einstein-Hilbert term, though topological,with weight still contributes to the generalized entropy with the JT action taking the usual form (again as in section JHEP12(2020)025 ]) , we 3 112 (6.14) (6.15) (6.12) (6.13) , 13 using the . ) and then tune CFT ij e ! T 6.5 2 and one exterior 2 JT L ) `

∞ , f in eq. ( , o eff T ! −∞ G ij ( 2 2 i ˜ g 2 L ∈ dχ 4 2 r − . ˜ r dϕ τ, χ 2 + = ! . The equation of motion for the 1 2 2 JT 2 JT L ` − CFT ` ij + sin ]. Here, ‘renormalized’ means that we dr e

2 T 2 ]. Recall that the central charge of the boundary 2 , the AdS-Rindler coordinates are useful r f = ) introduces a new cosmological constant term. 1 ˜ π r 2 . But the latter also receives additional corrections d eff + B geometry becomes and hence this expression yields the expected trace =2. Here, ` JT 6.11 2 G = 2 + 3 d ) 2 JT 2 ` 2 ). dτ 4 JT ! L L/` ˜ τ – 55 – 0 4 d /` 6.5 . 1) ˜ ] 4 φ ) for − + L 2 JT h − ( − ` 0 2 3.6 ˜ r O π, π ˜ ˜ φ φ r 2 2 ( + − The standard discussions of JT gravity (e.g., [ [ L = − − 2 JT cos 0 ˜ ∈ φ

ˆ 39 φ 2 /` 2 ϕ 2 = ). ∇ acts as a Lagrange multiplier which fixes the brane geometry L L 2 and the extra factor of two in the trace anomaly arises because the brane

to leading order in ˜ φ 40 6.4 = ds metric as ˜ and ij R ) = 2 g ). 3 bulk π . As described in section 2 JT c 2] into the JT action, which was achieved by the renormalization of ds 1 G 24 + ˜ /` 2 0 6.12 2 ˜ ). × with radius of curvature φ AdS , π/ j L L/ ( , on the other hand, yields the dilaton equation of motion r > 2 → [0 ∇ f = 2 ij i 6.10 = 3 i ˜ g B ], ∈ i 1 i c AdS ˜ ) r − ∇ , L/` defined in eq. ( R CFT e T f in eq. ( as ∈ ( h 0 ˜ τ ˜ 38 φ In the AdS-Rindler coordinates, the AdS As before, the dilaton Note that implementing this shift in the action ( Recall that we also removed the power lawAs divergence corresponding noted to in the [ induced cosmological con- → 40 38 39 0 due to the finite UV cutoffCFT on the is brane given — by see eq.supports (2.45) two in [ copies of this CFT. Hence an alternative approach wouldthe be latter to to introduce absorb a bothof general the the brane corresponding dilaton tension (power equation law) ( UV divergence in the induced action and the r.h.s. region is given by stant term by introducing a counterterm in eq.anomaly ( which is just the special case of eq. ( may write the bulk where 6.2 Bulk andLet brane us geometries now reviewwill the geometry find for it our convenientwill current to be setup. describe considering Due RT Rindler to surfaces time the using evolution, simplicity as global of in coordinates, AdS the even main though text. we In global coordinates, we function refer to the source-freemodated dilaton by equation, a i.e. further the shift r.h.s. vanishes, but this is easily accom- induced metric In the final expression, we evaluated the renormalized CFT stress tensor have absorbed the logarithmic UVaction divergence that would otherwiseφ appear in the induced to be locally JT action, which can be compared to eq. (2) in [ JHEP12(2020)025 , π 41 2 . : with ± is (6.21) (6.18) (6.19) (6.20) (6.16) (6.17) 2 /β y loses its τ or , we have = 1 . AdS π R τ ρ ˜ r T . 2 ) with respect = 2  . ) β # sin − 6.7 metric written as ) y ϕ − − − 3 β ˜ r y 2 + , 2 y dy ( ], we take the induced # − + π β sin 2 ) cos +  − dy − 2 y y ( in the second line element ˜ τ π 2 − ) β " sinh − + , cos y , y 2 JT ) ( ` 2 2 + π = 2 y slice of the AdS β " ), though the relation 2 ( dχ π = coth B are related to the global coordinates 4 3.6 θ + ) , − coth 2 = , r = JT r ), we introduce a surface of large constant L θ dτ ` r β ! − πφ τ, r, χ ( where the temperature becomes = 1 ( 2 τ 6.15 sin ˜ 2 B with respect to dimensionless time − 2 − UV + – 56 – θ y ϕ r . dρ 2 cos ˜ 2 π 0 2 + 1 ρ 2 ˆ + φ ] (see eq. (18) and discussion below (2) there). L sin = 2 sin y 2 , the boundary geometry is flat, i.e. it is simply two , ρ + ) fixes the intrinsic brane geometry to be d = , extremizing the brane action in eq. ( = ˜ 2 = φ = ρ πt β t = 2 χ 2 dτ r 6.1 2 CFT is simply the special case of AdS-Rindler coordinates given d β 1) ) = πφ ds 2 ) tanh − . This becomes the ) is easily solved for the dilaton profile in terms of τ, ρ − necessary for a smooth Euclidean continuation — we shall also 2 + ( y JT ρ 0 ( ` iτ . The light-cone coordinates ˆ , φ + 6.12 β − = r is determined by +

= B = 2 y B ˜ ( ` ], whose results we wish to compare against. The relationship between ) by φ τ is given by 2 JT sin ˜ θ d 2 π ` ) with respect to ) ϕ . Note that in sin ˜ − = = 6.14 2 , y τ 6.11 cos = 1 ). We write the induced metric on the brane as + . The AdS-Rindler coordinates ) with 2 y r 2 AdS = ( ), where 3.8 is a constant introduced in [ R τ ds which will serve as the UV cutoff surface. Then following [ in eq. ( r 3.10 2.3 ) 2 JT φ and ` (or eq. ( UV In the AdS-Rindler metric given in eq. ( On the brane, eq. ( As described above in section ) r tanh This is the same time coordinate introduced below eq. ( ˜ r, ϕ Φ 41 = τ, ρ τ, r metric on this surface as the background metric for the bath CFT, i.e. meaning as there is no spatial curvature in where Given that the TFDintroduced has the temperature dimensionful time in eq. ( are those used by [ ( as in eq. ( The first line element with radius of curvature in eq. ( (˜ to is in a thermofieldgiving double the periodicity state. of define The a inverse dimensionful temperature time withhorizon and respect at temperature to shortly. time Indeed,copies these of coordinates describe a for the description of vacuum AdS as a topological black hole, such that the boundary CFT JHEP12(2020)025 ). ), 2.2 (6.24) (6.25) (6.22) (6.23) → ∞ r . , by taking Σ B θ and ϕ . 2 = as − describes the brane. in the two baths θ π/ Σ 0 dy can be extended to Σ χ + < → χ ± − ) at y − ˜ r ). Therefore, the extension dy y = 2 − | ≥ 6.18  χ , 6.21 , which we define as χ + . | 0 Σ , y . UV χ ¨ χ τ < isl ¨ β π  β Σ 2 ) πLr orbifold of our setup (see section χ 2 R sinh 2 cosh = bulk brane with JT gravity. Specifically, −  Z (Σ b 2 G = = − 4 A τ  χ = complementary to belts centered on the ], the sign of the spatial coordinate on the brane . In the island and no-island phases the 2 , 42 tan ˜ R 2 CFT . 6.2 isl , with  – 57 – ) 0 with τ χ correspond respectively to the bath and brane. QES B τ, χ φ ] can be seen as a ( b > 2 ). Indeed, on the asymptotic boundary, with metric sinh cosh + π, θ . Now the light-cone coordinates = ) 2.3 = = of the flat boundary geometry in eq. ( R − θ bulk, and in particular the bath region on the asymptotic are simply geodesics. Although we are primarily concerned bulk = 0 y 0 ϕ 3 (Σ 3 G , ds 2 χ − 4 A ) tan χ < describes the asymptotic boundary while  + ). Focusing on a single Rindler wedge and on one side of the χ comprised of all of the points with AdS y 0 π ∓ . Similar assignments apply for the patches covering the other black hole coupled to the AdS > 23 are related to 2 R τ τ 3 − as well as the brane geometry given in eq. ( ( ± y β y π − ), = = + y θ ± y 6.21 is always positive and − y metric in the form eq. ( ) yields the relation of the global and Rindler coordinates on the boundary: − 3 + y 6.16 Finally, we note that going to the asymptotic boundary (with As in higher dimensions, we are interested in computing the entanglement entropy in the bulk, which in AdS We should note that the geometry in [ 42 R Hence they would only consider of the null coordinatesconformal defect. that As we a are technicalis point, reversed discussing let so us here that add hasHere, that in to [ be considered separately for each side of the RT formula equates the entropy to: The RT variational problemΣ instructs us to consider extremal co-dimension two surfaces Now we turn to the problem ofof computing the entropies hyperbolic using the AdS RT formulawe in the wish background to computedefects, the as entropy of described the at region the end of subsection which allow us to simplifydependent) global some coordinate calculations angle below. of It the will entangling be6.3 surface useful at to denote Entropies: the island (time- and no-island phases for all Rindlerportions times of the boundary. eq. ( two boundaries (which correspondsboundary to — the see intersection figure brane, of the the entangling brane surface with is the located at asymptotic a fixed of a boundary region (associated with the tworegion copies is the of complement the of two CFT intervals (‘belts’) entangled centered in on the the conformal TFD defects state). in the That is, this describe the geometry of boundary near an AdS given in eq. ( with the conformal defect at JHEP12(2020)025 ) = 2 is by (6.27) (6.28) (6.29) (6.30) (6.26) in the d ˜ τ UV r , ) 4 2 =  ( r ) O . } # 2 , ) +  23 1 i 4 1 ˜ r  ( . . ∈ { − O  ) i ( r ϕ 1 2 (˜ ϕ . The area (length in 2 ∆ ) + 2 2 2 2  −  ϕ i 2 ˜ + r ϕ cos 1) cos 2 1 −   and  ( − i 1 2 π r  ϕ 2 UV + ˜ ϕ = ] (but allowing now for two different UV r 2 = cos i ( at ∆ ϕ 2  2 2 − ∆ 2 113 2 ,  π/ ϕ ) 17 ϕ – 58 – 2 + ) corresponds to the standard entanglement en- → cos 1 ˜ r cos − ϕ 1 + 3 cot r 6.28 s  −  ϕ L , where geodesics are given by 12 ϕ 2  2 ,  − ∆ | , ϕ 2 # ) = (sin ˜ 1  ϕ τ ) cos ϕ (˜ 2) r 2 − csc " ϕ/ 1 ). This length stretches with Rindler time and leads to a growing 2 sin 1 sin(˜ ϕ  | − (∆ 1 as seen below.  2 = 6.25 43 ϕ tanh ). As in the rest of the paper, we restrict to the regime of small brane } 4 sin ∆ 2 , " 1 6.25 X ∈{ i log L L . B = = θ A Now as usual, one must appropriately regularize the areas of the RT surfaces. As The leading order term in eq. ( We begin by considering geodesics and their lengths in global coordinates. As is well Now just as in higher dimensions, the minimization procedure yields two competing The fact that the endpoints reside at constant global time, together with the conservation of the charge 43 Rindler radial coordinate. In terms of global coordinates, this describes the surface associated with the globalRT time surfaces Killing themselves vector must (obtained reside by on dotting constant with global the time RT slices. tangent) implies that the tropy formula of an intervalcutoffs). on We the have circle also [ includedfor the computing next-to-leading corrections order to terms entropy as formulas these on will the be brane. explained important above, we place the cutoff surface at a large holographic radius are respectively the openingradial coordinate) angle at of which the the area RT integral surface is and terminated, the see figure UV cutoffs (in the global where such that the curves hitof the an boundary RT surface with this trajectory is given by known, a convenient way tousing parametrize two anchoring the points RT surfaces on constant global time and the entropy isof given the purely expressions ( byentropy. the At bulk late length timeswhere of the the the RT RT surfaces contribution surface, goexpressions of across as ( the the in dilaton brane theangle becomes instead, second leading important, to as an shown island in the first of the points; these can alwaysThis property, be not simultaneously present in placed higherglobal on dimensions, coordinates allows a us surface to simplify of the constant analysis by global using phases. time. At early times, the minimal surfaces cross the Rindler horizon avoiding the brane with evolution in Rindler time, the boundaries of the entangling surface are simply four JHEP12(2020)025 Σ ϕ 2 below (6.31) − Σ π ϕ = relate to the ϕ 2 ∆ , ϕ 1 ϕ UV r ) with ϕ b to denote the cutoff at = . ϕ , on the other hand, will UV cutoff 1 6.28 r ) 2  − 3  2 QES 1  − UV  = ϕ r ) (due to the two pieces) with ( 1 ϕ O 2 (recall the definition of 6.28 ϕ + Σ ) ϕ χ + , showing the two phases of the generalized π 3 − = 2 2 ) + cosh(2 – 59 – ]. ϕ τ 2 and ). Substituting this into eq. ( is the angle at which the RT surface intersects the brane and Σ cosh(2 , one finds that the UV cutoff is associated with a length ϕ s − 6.31 UV QES r ). Here and below, we shall use 1 ϕ UV = r involved in the computation are associated to the UV cutoff at the 1 ϕ 2 6.24 = , 1 1   We begin with the no-island phase. Here once again due to the given by eq. ( , the minimal surfaces lie on constant global time slices. The RT surface ) where 3 2 ,  6.26 , these contributions will cancel once we consider the difference between the 1  A slice of constant global time in AdS ) given by UV r )). The total RT length is given by double eq. ( 6.29 6.24 Equipped with this, we can now compute the generalized entropy in the two phases simplicity of AdS consists of two pieces,given by one eq. ( connectedeq. piece ( on eitherboth side cutoffs of the brane with trajectory island and no island phases, as seen below. and reproduce the Page curve found in [ No-island phase. where we have usedthe eq. end-point ( of theeither RT be surface a at cutoff thewe at asymptotic are the boundary; in asymptotic the boundary no-islandregulator or or due island to phase. the Note brane, that depending although on the whether entropies diverge with the corresponds to the boundaryby of gluing the it island. to Recall another the copy. geometry is cut at the brane andExpanding continued to leading order in in eq. ( Figure 23. entropy. The two cutoffs asymptotic boundary and the brane,RT surface respectively. opening The angle, global while coordinate angles JHEP12(2020)025 , ± y (6.35) (6.36) (6.37) in (6.38) (6.32) (6.33) (6.34) , since ) is then δ 2 . The UV as measured 6.32 δ π/β , ) and short distance − L − w , − L dy y  ∂ + + L dy rather than w πt/β − . We may equivalently take + L 4 y , ± − ∂ β/π y sides respectively. We have used − R ce ) L  ), with coordinate cutoff w  τ  − R πt y β O ± L 6.22 ∂ and 2 β  + R + πy  ) R cosh w with proper distance cutoff −  L  + R ) matches the entropy from eq. (29) w 2 y UV πt ∂ β r cosh − ds 2 coth . , in eq. ( )) and the short-distance cutoff in the − R 6.32 − log( β − + πδ w UV in a CFT with metric 2 3.6 bulk dy L c  β )( = δ  12 log (2 3 + R + L π r G  Eq. ( − 2 2 w πδ dy β log ] ± R bulk 2 2 L – 60 – − c β
= = , δ 3 44 G 2  πy δ + L R − c δ  w = = ( For times much larger than the thermal scale, as measured by the induced metric. dw log − . +  − L ¨  isl 45 ¨ c  3 dw 2 ) = tanh − log [ ]: inside the logarithm, it should be R c = 6 ± 2 S bulk ), we have expressed the answer in terms of the dimensionful . w (Σ G ¨ ] = ] = isl ¨ are the entangling surfaces on the 4 A  , δ , δ b 6.32 )  − − ∓ R ) (see also below eq. ( t ) computes the entropy of bulk dy dw + (Σ = + G ). 4 A 6.32 6.19 dy ± ), the associated entanglement entropy in the no-island phase is L to denote entropy in a CFT living in dw y  − ] Let us next consider the island phase. As explained in section slice. Notice that this is also a symmetry of the dilaton profile as is clear [ is both the proper distance cutoff and the cutoff in − 6.37 [ S , δ δ τ S 6.24 2 ) gives the length of the piece of the RT surface to one side of the brane; eq. ( and ). → b ds [ ± S 6.37 coordinates on the boundary. t 6.20 = 0 = = ± — here, is also hidden. The full answer is obtained by applying the conformal transformation y ˜ τ ± R , as in eq. ( δ t y δ . Eq. ( 2 ], accounting for the fact that here the central charge is doubled since we include the To be precise, eq. ( There is a typo in eq. (29) of [ 2 ds 44 45 where the notation by exactly double eq. ( (mapping the vacuum to a TFD) to the entropy formula cutoff the CFT metric tocorresponding be to a the proper induced distance metric cutoff cutoff which corresponds to the linear growth predictedIsland by phase. Hawking. translations in Rindler time are anto isometry, we the can use this symmetryfrom to eq. bring the ( problem of [ regions on both sides of the brane. boundary CFT in the In the second line oftime eq. ( where we used the Brown-Henneaux central charge and using eq. ( JHEP12(2020)025 , ) 0 and ! → 6.18 1 (6.41) (6.42) (6.43) (6.39) (6.40) 4 B B ϕ )) with and the θ bulk Lθ b G of the QES 6.12 =

) (as well as , . ), the area of − O (together with ! y ! QES 2 2 6.17 4 − + ϕ ˜ 4 JT δ ϕ 6.28 + 4 B ` c , y    bulk = ) 1

, Lθ 5 B G π ϕ θ − O (

i = ); we have also used the + i O ) . First, O b ) θ ), such that the trajectory, b  − β + + 23 a β ) + 6.34 ( a πb β   ]. Using eq. ( ( 2 π . From eq. ( ], reviewed around eqs. ( i is now provided by the brane 6.26 π 1  h ) 1  2   ) is defined in terms of h b QES 2 2  2 +  ϕ β a  QES ( ( ) ]), we find 6.28 ϕ 2 sinh 2 ) and ( π QES B 2 + QES h θ sinh  ϕ 2 3 sinh Σ 2 ϕ + QES ϕ β πa of [ 6.33 ) cos Σ 2    ϕ 2  ϕ b  2  sinh sin QES )] 2 B 2 sin( bulk ϕ and G sinh Lθ 4 sin sin coordinates of [ by QES a – 61 – 12 B 2 sin( 2 JT ϕ + ± ˜ θ δ ` 4 . ), the regulator y 1 3 c B + 1 3 6 3 2 QES  θ ϕ ϕ −   ) — see discussion in [    − appearing in eq. ( ) sin( 6.31   i 6= on the brane (hinted at earlier below eq. ( B ) ) +    ϕ θ b log 2 B i 6.18 and L + ∆ β ) , as in the no-island phase. But, the RT surface will now ). Placing the belt boundary at bulk a β πa b ( QES QES 2 Σ Lθ Σ + G = π L bulk β ϕ ϕ ϕ [tan( a β ϕ πa 42 h ( 2 ˜ , G δ 1 (matching the + 2 π − B − h sinh sin( θ a = 2 = B . sinh = sinh θ 1 isl ) in the bath, given in eqs. ( ϕ  − = = tan sinh B y ) ± UV θ 2 2 ). Second, there is the global angular coordinate r y − R  ˜ JT δ 4 + bulk y (Σ β`    πδ G (in 6.29 , . 2 is still given by eq. ( 4 A B δ isl  θ    1 log but, at finite   ) . The opening angle 2 = 2 ) (see also footnote log R ϕ θ L ϕ bulk bulk c 3 (Σ G 2 G → 6.22 4 A While We will leave point 1 anchored on the cutoff surface near the asymptotic boundary = = and as per eq. (  ) characterize the trajectory of the RT surface, as in eq. ( and cutoff 1 2 1 QES where, in the secondc line, we have written theproper answer distance in UV terms of cutoff the the induced CFT metric central given charge in eq. ( and ( QES at We can also write this in terms of the the RT surface (includinghorizon) the is pieces given to in either terms of side of the brane and to either side of the which we use below perturbatively in the regime of ϕ ϕ where the RT surfaceϕ intersects the brane. In the limit of vanishingposition brane and angle is given by at global coordinate intersect the brane at itsdifferent angles other appearing endpoint. in the Here islandϕ it calculation — is see important figure towhen distinguish maximally between extended two (even behind the brane), reaches the asymptotic boundary at JHEP12(2020)025 ) 0 a ), — ). < ∓ , ) t 0 (6.45) (6.46) 6.43 − − Σ 6.20 (6.44) (6.47) y w 6.43 = are the ) which → − − Σ  y ) + ∂ JT y JT ± QES + . Σ    6.43 y 0 i w − QES ], accounting ˆ ˜ δ/` φ ) and ( ) 2 + ˜ Σ δ/` w b and y ) to + ∂ and − β 0 a β πa ( 6.10 2 b − Σ π 6.35 > log( w h ± ) is precisely four times − )( 2 t c δ y 12 sinh ) precisely recovers the = − , 6.43 − + QES sinh + eq. (19) of [ w !   ± Σ y y 4 6.43 β − ] is exact even after keeping all 47 ˜ ,L 4 JT δ πδ 2 2 . ` c + Σ       completely contains the baths. w

( written in eq. ( , and − !! O R R log may be interpreted as corrections bath ) y 4 brane 45 , given by eqs. ( 4 JT c + ˜ δ 3 ` φ 2 JT − QES 

and − w 2 + ) πb β ˜ δ/` − w O dw 2 − JT − i  ` ) and (19) in [ + ) w  dw 2 + b 2 + − dw δ β πa + , i.e. when + QES + β 6.47 a dw 2 ) matches exactly 2 JT 2 w 2 JT w ( 2 ( ` ` ˜ ( δ sinh π  L 4 6 L = 0 h  – 62 – − −  b 6.47 2 β πa    conveniently absorbs the part of eq. ( 2 1 + coth log    0

c 6 sinh S r ˆ φ introduced in footnote ), as well as the topological dilaton contribution = = ˜ β JT δ ] πφ ` • sinh 4     , 6.46 • 2 [ = 2 JT ,L ,L + ˜ δ S ` B c 0 6       θ ˆ φ − ). This is unsurprising given that the renormalized entropy is brane relative to the main text, so that here bath bath while the higher orders in = 2 2 brane brane ) to the bare dilaton profile . 46 6.13 isl AdS − −  2 2 6.43 ) ) − dw dw − − − + + w w dw dy QES 2 + − − 2 + dw dw δ φ is precisely designed to eliminate the UV divergence of the matter effective δ + + on the ) and ( dw dy w w 2 JT 2 JT ] considers only one side of the TFD and they work with an end of the world 0 2 2 ( ( + ` ` − ˆ 2 L 4 L 4 φ y ) − − − − 6.10 R ) is to be interpreted as the von Neumann entropy of the effective CFT spanning − ∼ ), we can write bulk       0 + ). (Σ     y ˜ G JT φ ` 4 6.43 S S A 6.46  In fact, the match between the first line if eq. ( To see this, we may apply the transformation between → = → 47 46 0 B correspond to the entanglingsign surface of and thecorrespond QES respectively respectively. to theeq. (In bath ( this and footnote, brane.) we Then, have the swapped first the terms term collected in of their eq. “constant”.entropy ( This calculation, can described be in checked eq. by keeping ( all constant terms in the von Neumann where we have used the notation corrections due to the UV cutoff, inherited from the von Neumann entropy in eq. ( φ action on the brane. Thefor first the line of doubling eq. and ( eq. quadrupling (19) of of [ thebrane dilaton with and bulk von spacetime Neumann only entropies to here one (since side). The terms of higher order in where we have includedthe dilaton right contributions of from the thebecomes QES TFD. logarithmically points Recall divergent on that onsee both the eqs. the brane ( left as and derivable we from take the the renormalized matter UV effective limit action, and that the renormalization of due to the finite UV cutoffvanishes on for the brane. the case Curiously, the of leadingWe a order zero-width may correction belt in add eq. ( eq.evaluated at ( the QES, to obtain the generalized entropy Eq. ( the asymptotic boundary and theexpected brane. CFT The result, first term of eq. ( ` JHEP12(2020)025 . 0 ˆ φ (6.50) (6.51) (6.48) (6.49)  πb ) in the β 4 . − ! )), and the e 6.47 4 , 4 JT − ˜ δ ` # 1 6.32 . 

 as the anchoring r O φ i cβ 2 and corresponds to is to push the QES − 2 JT 2 r . In particular, we ˜ δ y `  + c c 2 6 + JT to zero, we obtain the ∞ /φ O +    2 + y  a −  β ˜ δ/` + 2 = πb c β = β ! → BH 2  i πcb 3 ) evaluate the von Neumann − t S 4 )  4 4 JT a y b ˜ δ O ` 2 ), we have a correction due to − + ∝ β + a

+ + . ( 6.50 y ), obtaining sinh π τ   O  h  6.50 numerically or analytically with an β 4 JT r 2 πδ + ). Symmetry has already allowed us β ] accounting for the fact that we have πa 6.47 b ) in /` 2 β  2 .  πφ , again accounting for the doubling of 4  2 ˜ δ sinh πb β JT 6.47 ∞  4 log 6.47 + ) in the limit − c O 3 ˜ 0 δ/` sinh e 2 h ˆ = + φ as these are inherently smaller than either the c − 2 JT 2 6.50  + from the bifurcation point. Here, the dilaton − 2 ` ˜ δ – 63 – 1 ) into eq. ( ˜ + y δ 2 JT 4  ` β 2 a  − r r of the QES, the RT prescription instructs us to 2 πb β + c φ 4 = 2 y 2 JT 2 6.49 β a πφ 1 + − ` ˜ δ 2 e 4    BH r = S i + − β ) 2 b 0 πφ 2 : −  ˆ c y φ − 1 β r a 18 2 −  ( i + π  − ) ), after re-absorbing the UV divergence on the brane into πφ y cβ h b r , this matches eq. (20) in [ + = 2 12 in terms of the belt width cβ φ β a ] at leading order in ( . JT 6.43  2 a π sinh isl h   corrections.) The first line simply evaluates the generalized entropy, ˜ δ/` log ), at the bifurcation surface, i.e. taking " 2 r β πa 2 QES sinh π β /φ  φ 2 2 6.47 ]. The other terms on the first line of eq. ( β + + 3 2 ) c sinh b R bulk = = or correction. Moving to the second line in eq. ( (Σ G a r 4 A 4 JT Having found the location of the QES, we may re-evaluate the generalized entropy of To find the location 2 JT  cβ πφ /` 6 /` 4 2 ˜ δ ˜ Specifically, the second termthe gives asymptotic the boundary UV (alsothird contribution appearing and from fourth in terms the theδ give no-island entangling finite surface phase contributions in on the to eq. displacement the ( of renormalized entropy the including QES a location entropy of two black holes This classical contribution dominateseq. eq. (30) ( in [ entropy, given in eq. ( (We have also dropped terms ofc order given in eq.recognize ( the first term as giving the Bekenstein-Hawking result for the course-grained matching eq. (21) inthe [ CFT. We seefurther that from the the leading bifurcation order point correction at due to finite the island phase by substituting eq. ( for the QES position additional expansion in At leading order in two copies of the CFT versus a single copy of JT gravity. This equation can be solved extremize the generalized entropyto given restrict in the eq. QES ( point to on the the same asymptoticspacial slice direction. boundary. of Setting It Rindler theextremization thus derivative time condition: of remains eq. only ( to extremize eq. ( JHEP12(2020)025 + ). ). 2 of 2 JT ˜ δ 0 ` c r ˆ φ cβ φ 6.51 6.38 (6.52) ). · = ) which r ). Over ˜ φ β r πb β φ , resembles 4 ): correction 6.50 ] to higher − πφ 6.50 2 6.32 = 0 2 JT , leading to the and 12 6.50 % cβe /` ) reads 2 r 2 2 JT ˜ δ cβ φ ` ˜ δ − r cβ · φ 6.20  r and the number β φ ) and ( πb β 4 and /β − 1 r e 6.38 cβ φ ]. From the boundary per- − The order 15 1 , near the horizon  48  ], which uses Randall-Sundrum 2 1 2 JT 2 +1 at orders ` ˜ the usual radial coordinate. The dilaton δ 2 d% % 4 % + = 0 − -dimensional boundary CFT coupled 2 d , this setup precisely realizes the three . , in terms of which eq. ( dτ πb β QES 2 2 1 2 %  % − 2 r − 2  ρ /φ 2 2 JT p – 64 – ` β marking the transition between the two phases, 2 = c =  /β 2 % ) is not visible at the order shown in eq. ( + log(2) + P O ) should then have an expansion in terms of non-negative integer 2 AdS  πt + 6.49 r ds in polar coordinates with  6.47 β 2 JT 2 πφ = 2 ` 4 JT 2 d% P /` τ + + 4 ˜ 2 δ 0  ˆ φ , we plot the Page curve after subtracting off the initial entropy dτ 2  O % ) gives the first corrections to c 3 24 + − 6.49 = ) as the course-grained entropy for the two sides of the black hole, this is P . Note that there are no dilaton corrections at orders β πt 6.51 β r . Eq. ( 2 2 and the brane metric φ c 2 QES = % ∼ % P 2 τ 2 r 1 + β φ 2 c To find the Page time p It is helpful to consider the coordinate · r 48 β r πφ β 2 φ the standard flat metric and the von Neumann entropypowers in of eq. ( corrections mentioned in the main text. dimensional black holes.different As perspectives reviewed of in the section holographicspective, system described the in system [ isto a described conformal in defect. terms The of usual the holographic dictionary then yields the bulk perspective, (which includes the UV divergences from the asymptotic boundary). 7 Discussion In this paper, weplus applied DGP the gravity, framework to introduced extend in the [ discussion of quantum extremal islands in [ Overall, we recover ato Page the curve, Page with time, and entropyPage saturating growing time. to linearly a In in constant figure a maximal value no-island in phase an up island phase after the we equate the corresponding generalized entropies given in eqs. ( on the brane justof outside the the black black hole holelatter horizon interior, phase, surround now generalized an belonging entropy island, is tois containing given the dominated by a entanglement by the portion double wedge constant the of valueViewing written Bekenstein-Hawking the eq. black in ( bath. hole eq. entropy, ( In asprecisely given this the in expected eq. maximal ( entropy of the system. Collecting together the resultstimes, of we the have previous thetime, subsection, no-island this we entropy phase, grows have at two with amatter phases. rate generalized degrees proportional of entropy to At freedom given the early participating temperature This by in growth, Hawking eq. however radiation, ( is as emphasized capped in off eq. by ( an island phase, where quantum extremal surfaces because the bifurcation point extremizesin the the dilaton QES profile. location given in eq. ( 6.4 Page curve and von Neumann components of generalized entropy both receive contributions at order JHEP12(2020)025 , 4 ], the and 30 3 . We plot the = 2 d runs into the bulk R . As noted in [ R ], our approach provides σ marks the boundary of an 30 t R σ -dimensional AdS bulk space- , viewing our setup in Poincaré 5 + 1) d ( -dimensional effective theory induced by the d – 65 – . and intersects the brane at = 0 t R ∂ = bath CFT on the asymptotic boundary. CFT stretches through the bulk and is manifestly connected to the S -dimensional brane. The brane perspective is an intermediate Σ d R ∆ of the subregion on the CFT which is associated to the radiation, where BH σ = 0 S . Similarly, as explained in section 2 T (0) . From the bulk perspective, the RT surface of S 1 πR R -dimensional brane perspective, when computing the entropy of a bound- 2 − d -dimensional theory. To determine the RT surface in an island phase, we ) t d = ( S in the island phase, a quantum extremal surface T Page curve for the equilibration of our topological black hole in = R S ∆ While islands have been numerically studied previously in [ We have considered the vacuum state of the system with respect to global time, which island on the brane inin this the higher-dimensional effective picture, despitemust the not apparent only disconnection extremizewith the respect area to functional the(IR) locally and intersection within near-brane of the (UV) the bulk, contributions RT (further but surface modified also and by extremize the DGP contributions) brane. to the Since the deep bulk higher. From the ary region island on the brane stretching toof the the horizon; this bath island region belongsfrom to its the anchoring entanglement surface wedge entanglement wedge of a relatively simple settingdoubly-holographic in nature which of our analytic modelin calculations reduces are the the possible. entropy presence calculationsto of involving In holographic islands massless particular, entanglement the hyperbolic, entropy calculations or in extremal (locally) pure black AdS holes in of one arbitrary dimension dimension brane geometry as athen black play hole the of role one oftemperature lower bath dimension. CFTs in The equilibrium ‘two’ withcoordinates, asymptotic the we boundaries black have hole an oncoupled the extremal to brane horizon a at in (single) a finite the bulk and on the brane. The latter was which supports a gravitational theory by the usual Randall-Sundrumsimplifies mechanism. the bulk geometryby to viewing be this setup pure in AdS.as AdS-Rindler in coordinates, However, the terms as global of discussed vacuum a can in massless be sections re-interpreted hyperbolic black hole. This induces a similar description of the where the dual descriptiontime is bi-partitioned Einstein by gravity a characterization in of a this system givenbulk by theory the on theboundary asymptotic CFT boundary spans and the the asymptotic brane. boundary, which That is is, non-gravitational, and in the this brane description, the Figure 24. entropy we subtract the value of the entropy at JHEP12(2020)025 , ] 2 6 3 49 . At shorter L ) computes the ' 1.1 ] in deriving their ˜ δ 2 ] at leading order in an 2 imposes a finite UV cutoff ). Of course, with a finite B θ 6.50 , in analogy to the DGP terms As noted above, in section , the interpretation of the brane 2 = 2 d . A finite ), and therefore subleading corrections 1 , JT gravity does not appear automatically  6.44 6 B θ bulk and including the DGP entropy, as in the – 66 – 3 AdS ], our braneworld construction clarifies further concep- 1 ). Our model puts the explanation of this fact given in [ 1.1 ]. One particularly confusing feature of the island rule is the ), correctly reproduces the results of [ ], this bulk calculation is equivalent to the island prescription of 1.2 111 1 , 15 , 2 ] is that, as detailed in section -dimensional Randall-Sundrum gravity coupled to a CFT only holds for 2 d on both sides of eq. ( ]. These corrections have the effect of pushing the QES slightly further from the 1 R analogue of eq. ( As discussed extensively in [ The most striking difference between our holographic construction and the two-dimen- By the standard rules of the AdS/CFT correspondence, the boundary and bulk perspectives give an complementary to belts centered around the defects in the two Rindler wedges. This = 2 49 geometry is induced on theasymptotic brane, boundary coupled in to both andR Rindler in wedges. equilibrium with We considered bath regions the on entropy of the bath regions equivalent descriptions of the physical phenomena. perspective or the bulk perspective gives a complete descriptionNon-extremal of quantum black state. holeswe in considered AdS-Rindler higher coordinatesAdS dimensions. in spacetime the as bulk, a providing two-sided a massless description non-extremal of black the hole. pure A similar black hole the entropy on the right(our hand brane side perspective). is In toperspective partiular, be as as interpreted noted in in an section the effective, semiclassical low theory energy physicsdistance at scales, scales longer gravity than is the no short longer distance localized cutoff to the brane. In contrast, the boundary framework, e.g., [ (implicit) appearance of theregion entanglement entropy of theon QFT solid degrees footing. of The freedomfull entanglement entropy in entanglement in the entropy the in left the hand UV side of complete eq. picture ( (the boundary perspective), while results [ horizon, lowering the entropyearlier of time. the island phase, and shifting the Pagetual transition puzzles that to appeared an early discussions of quantum extremal islands in a holographic expansion in terms of smallin brane angles, the i.e. effective braneto theory, as entropy formulas shown appear inUV in eq. cutoff, the ( we island would phase not,the — for CFT see instance, transformation eq. expect ( rules the of holographic the entropy to entanglement precisely entropy satisfy used by [ in higher dimensions.branes However, this in may higher be dimensions,model, contrasted where but with is adding the not aCFT induced strictly coupled DGP gravity necessary to on term gravity. for the Having providesthat interpreting added finer applying the JT the control brane gravity RT as perspective over formulad a the as in DGP the an term, effective we showed in section Wald-Dong entropies [ extremizing generalized entropy over candidate quantum extremal surfaces. sional model of [ but has to be added by hand to the brane theory for RT area, respectively, can be interpreted as renormalized von Neumann and gravitational JHEP12(2020)025 R . Without R ]. 2 on either side B ] and our setup, is and 93 A . This gives a useful alternative view R consists of a belt region centered on the R – 67 – ). In particular, at late times, the minimal RT en- 1.2 ). Our holographic construction translates this competition . In fact, for the holographic system, the late time entropy ], the entropy grows at early times but then quickly thermal- ) 1.1 B 93 ( S . From the brane perspective, the intersection of this RT surface with ) + ], we find that the island extends outside the event horizon, i.e. the ] makes an expected re-appearance in higher dimensions, as reviewed 2 2 A 2 ] can be understood as a consequence of the large number of degrees of ( S 93 ≤ ], except that here the spatial sections of the bath geometry are hyperbolic is identical to that of its complement . Again, this information paradox is resolved by the appearance of a quan- ) 93 R B 3.6 ∪ A ( Further as in [ Recall that the global state is pure, i.e. from the boundary perspective, it is a ther- We find, in particular, that the information paradox for eternal black holes and its res- S quantum extremal surfaces appear outsideas of above, the this horizon. feature Ifof again we entanglement focus has wedge on a nesting. thethe simple entropy Recall RT of explanation in surface in the yield our islandRindler the holographic phase, entropy wedge. setup, of the Since the in individual these individual terms components belts belt of are regions on subregions the of boundary the of full either hyperbolic slice on which the the addition of asurfaces backreacting to brane traverse which in createswhere extra the this entropy spacetime late-time is geometry saturated. islandtime for From phase relative the the to boundary and RT [ perspective, so thisfreedom delays longer introduced thermalization the by the onset conformal of defect. this phase in the present case.izes. As in In [ this case,i.e. the growth of the entropysaturates stops, this because inequality it which is erasesregions. bounded the by mutual The subadditivity, information primary between difference two between boundary the sub- framework studied in [ of the evolution ofconformal the defect entropy. in the Theering two region the bath entanglement regions. entropyof of Hence two the from isolated corresponding this boundary eternal pointstudied regions black in of hole [ view, in we the are bulk. consid- This is essentially the same system connected pieces of thewedges RT and surface are in thus invariant the under island time translation phase (i.e. are forwardmofield isolated boosts double to in state individual both of wedges). Rindler twoentropy copies of of the boundary CFT plus conformal defect. Hence the background. The island belongs tothe the appearance entanglement wedge of of islands, thethe the bath islands entropy region of however, the bath ever-growingoff subregions entropy would by of grow the the ad-infinitum. constant no-islanddimensional finite With phase discussion entropy is of provides eventually capped this a island simple phase explanation at for late the times. saturation Further, of our entropy: higher- the entropy in the island rulebetween quantum ( extremal surfaces to thesurfaces usual in competition the between holographic different possible formulatropy RT ( is provided by aillustrated second in extremal figure surface withthe components which brane cross becomes the the brane, quantum as extremal surfaces bounding the island in the black hole analogous to the two-dimensional setup at finite temperature consideredolution in studied [ in [ in section tum extremal island when a second quantum extremal surface minimizes the generalized setup, from the perspective of the effective theory on the brane and asymptotic baths, is JHEP12(2020)025 ], 2 from ] that c on the 2 θ 1 − ], and the d H 92 approaches from the conformal , their candidacy for B 2 b θ d > . As 0 , by taking a Poincaré patch 5 > ]. In contrast, we have found in c 2 , we proportionately reduce the θ b far away from the defect, the RT R In section – 68 – below some critical ], we calculated the entanglement entropy for a bath B , we find that regions of the brane arbitrarily close to 2 θ 0 → , no quantum extremal surface exists on the brane and the c b < θ B θ ) that the horizon on the brane is precisely the intersection of the . Recall that, in the two-dimensional JT model, the island phase is is precisely the angle at which the entropies of the no-island-type and θ c θ 3.11 ], the RT surface and entanglement wedge for any subregion of (or brane tension) and the DGP coupling. B 115 θ , which corresponded to points greater than some distance 114 R that islands cease to exist for Interestingly, a further qualitative deviation from the two-dimensional case is seen at Due to the scale invariance of Poincaré coordinates, it is clear that as we push the 3 bulk RT surface is simplyinto given the by bulk. two Since planes the onRT area surfaces either of these side must latter of be surfaces the considered isfact, brane IR we even finite running find when in straight that theisland-type alternative surfaces island-phase match surfaces — exist. above this In angle, the island-type surfaces remain favourable small brane angles always dominant for belt geometriesd in > the extremal case [ above, the quantum extremal surfaceextremal of horizon). the For island phase runs off infinity (i.e. towards the the spatial infinity of theframe. Poincaré patch In which the corresponds otherthe to asymptotic extreme a boundary single can point bedefect. in recovered the This by global is portions in of contrastexists. the to bath the sufficiently two close dimensional to JT the model, where a maximum island size entangling surface out insize the of bath the region, island.the i.e. region increase Again, near thisbe the behaviour contained extremal reproduces within horizon the thepicture deep shows intuition far-away in that portion suggested these the of regions in gravitating are the [ not region bath. far (our from each Actually, brane) other our at is all higher-dimensional can — they are both close to region defect. In the casetime of evolved, extremal but black instead holes, foundbrane we that did angle the not appearance find of a an transition island as is the linked system to was the choice of of the bath which are furthest from theExtremal black black hole. holes in higherof dimensions. the bulk, we consideredCFT an in extremal a black flat hole background. on the As brane in coupled [ to a (single) bath to see from eq.Rindler ( horizon in thebrane, bulk i.e. with the the intersection of brane.horizon. RT surface Hence This with the also the quantum meanssurface brane, extremal that must will surface if lie pass on outside we close of the considerinformation to the regions about the black the horizon. hole horizon Thus, seems analogously to to be the contained situation in discussed the in entanglement of [ CFT regions wedge. Thatsurface is, corresponding the to bifurcation eitherRindler surface of wedge the of is copies the thenesting of corresponding [ Rindler the entanglement horizon CFT wedge. in inboundary Hence, the must the by lie TFD entanglement bulk within state wedge the is [ corresponding the Rindler wedge. RT Finally it was straightforward corresponding CFT resides, the RT surface must remain within the corresponding Rindler JHEP12(2020)025 . For ], or has = 0 T 119 – 116 ) is that in this c < θ B θ ] in the bulk perspective, 122 50 , 121 ]. This becomes particularly clear in our 1 (and in particular ]. B θ – 69 – 120 ], the existence of wormholes follows from the fact that ]. One might then wonder why — despite the absence of 1 [ 41 ], one might be tempted to turn the logic around and, given , 1 52 39 → n In order to derive the island formula, a crucial ingredient was the is strikingly different from the two-dimensional case, we remark that, in c < θ B θ ] formulates a point of view where integrating out the bath CFT generates an averaging over 56 Following the logic of [ In fact, this is not a problem, since the different effective gravity theories in the brane On the contrary, our construction relies only on the standard holographic rules of the Of course, an interesting question may be to examine how varying the geometry of Ref. [ 50 the appearance of wormholesis in some the form brane of description ensemble of averaging our in model, the concludecouplings in dual that the boundary there theory of theory. the conformal However, defect. this line of theories, i.e. replica wormholessetup on where the the brane braneWe [ lives emphasize in that here the thisRT bulk prescription discussion and implicitly for does relies holographic notwhere on entanglement serve again the entropy we as standard [ assume derivation a that of boundary there the of is no spacetime. ensemble averaging. picture are UV completedconsider by a geometries single connecting theory theIn of branes, gravity fact in i.e. considering the replica Renyi bulkcorresponding wormholes entropy bulk and in calculations geometry so the induces in it connections effective the is between theory. boundary natural the different to theory, copies one of sees the brane that the lack of averaging characterizes the UV-complete descriptionperspective. of the Nonetheless, system, i.e. quantum the extremal boundarythe islands brane appear perspective in and the oncewormholes effective again in description one the likes of limit toensemble understand averaging — them replica as wormholes remnants shouldin of appear different replica and copies connect of the replica gravitating trick region calculations. gravity emerges as the lowa energy definition effective description in of terms the of SYK a model matrix [ model [ AdS/CFT correspondence where there is notheory. such averaging This of is the in couplings line in with the the boundary general expectations for higher dimensional holography. This Not an ensemble. appearance of wormholes inJT gravity the studied replica so trick.JT far gravity [ In is the defined two-dimensional by models averaging over involving an ensemble of Hamiltonians. For example, JT non-extremal case, where islands areunbounded growth necessary, at of all black hole brane entropy angles, at to late tame times thethe and otherwise entangling avoid the surface information affects paradox. example, the rather appearance than of belt quantum geometries,conformal extremal one defect. islands might consider at spherical regions bisected by the limit, the effective theorycurvature on corrections, the which brane is isislands described the for by most Einstein interesting gravity parameterthe with regime. extremal small case, While higher islands thetheir are lack absence of not should required perhaps from not an be information-theoretic terribly standpoint surprising. and This is to be contrasted with the as RT surfaces. The relevance of small JHEP12(2020)025 ], 2 ] is 2 ], correlation ]. Apart from ] that in non- 50 123 , , 125 , 41 41 120 ], that this shift would 127 , 126 ], the spectral form factor [ – 70 – 50 , ] do not extend to this situation. Instead, in our 41 52 Having produced a setup in which quantum extremal islands can be ], and overlap of black hole microstate wavefunctions [ 124 , 89 ]. There, the authors investigated whether a protocol can be implemented to retrieve Recall that above, we described how in the present discussion the appearance of the For example, one may consider information-theoretic questions similar to those raised Nonetheless, this issue is certainly worth further examination since in two dimensions, 2 have the final event horizon swallow the island, preventing post-quench communication the defects. Naively, thiseven appears if the to coupling allow between communication thesefrom systems between noting is the that severed. baths a The and splittingcreate resolution quench defects a of between this positive the paradox energy comes defect shockusing causing and an bath a outward systems shift JT would of inevitably version the horizon. of It the was quantum argued in focusing [ conjecture [ quantum extremal surfaces outsideentanglement wedges of from the the bulk horizon perspective.whether this was However, protrusion another a question of simple raised islandsportion by outside result of [ the the of horizon island the violates of causality. nesting the In of baths particular, outside the the horizon appears to be causally connected to to pull information from thisand region bath. into the One left could exterior,insertions try to to be of reproduce picked operators up this byenergy on protocol the produced in the left would our defect left then higher-dimensionalbrane. and shift setup the using right bulk asymptotic horizon boundaries. and hence the The induced negative horizon null on the in [ information from the island.system plus In an particular, interval the ofdisconnected from the entanglement the right wedge left bath and of the containsthe the right left an bath complete and interval. island right left However, that by baths, acting naively it with appears was operators argued causally in that sufficient negative null energy can be generated hope that studying our system will giveFuture directions. an idea of howstudied this with suggestion relative might ease, beabove, realized. some but possible a number avenues of of other further possible investigation extensions were to suggested the present work also come to mind. higher dimensions more generally.averaged Furthermore, theories it wormholes was might suggested appear inobtain as [ a a full result quantum of gravitational answer, someGiven additional diagonal off-diagonal that approximation. terms we To need to havewhile be a added. at system, the where same wormholes time appear having some in control an over approximate a formulation, UV complete description, one might the partition function of the boundary CFT with areplica wormholes conformal have defect now been will shown stille.g., to factorize. calculations play an of important Renyi role entropiesfunctions in [ [ a variety of situations, Renyi entropies, it is not clear how to reproduce these effects in our construction, or in ‘bulk’ gravity theory (containinghold wormholes). in our Wetheory construction. stress and that Rather this sosituation the equivalence the does replica gravitational arguments wormholes not theory of appear,the on [ but boundary the theory wormholes brane do connecting is not independent play an instances a effective of role. For example, this implies that higher powers of argument implicitly assumes a precise equivalence between the boundary theory and the JHEP12(2020)025 . d ], ], ]. 1 πR 93 2 138 136 ). This , = 52 T 3.11 135 ) otherwise the T c , the splitting surface on = 2 d ]. For instance, as noted in [ The latter is particular interesting 139 ]. In 51 – 128 on the horizon. While a direct boundary 137 , i ij T 132 , however, the calculations will become more h 3 – 71 – d > ] allows us to recover information about the island with 134 – in the vicinity of the horizon. 129 i , ij T h 114 , 24 on the horizon and throughout the black hole solution on the brane. = 0 i ij T h Yet another direction to take would be to consider our setup from the perspective of This is related to the fact that in the present paper, for the sake of simplicity, we have Returning to the issue of extracting information from the island, entanglement wedge We thank Ahmed AlmheiriThe for vanishing raising of this the question. stress tensor on the brane is an essential feature of our construction as the AdS 52 51 brane geometry must bethe brane a cannot solution providegeometry of a source would the in deviate corresponding these frombath gravitational equations CFT AdS (at equations. is least space. coupled to That toacceleration leading Recall the allows is, order the brane that the for along bath while large an CFT CFT the accelerating to on trajectory achieve brane equilibrium — CFT at see a is discussion finite under in temperature. eq. its ( vacuum state, the tensor networks and errorthe correction MERA-like codes tensor network [ constructing theon time-evolved CFT the thermofield double asymptotic state boundaryslices stretching shares through a the similar bulk geometry wormhole. to codimension-one One might bulk then spatial be motivated, as in [ must be solved simultaneously withthe the equilibrium former configuration back-reacting will onthe now the effective involve latter. Einstein excitations In equationsboundary of particular, on CFT the the residing CFT there. brane on will the be brane, sourced i.e. by the stress tensor of the chosen to work with aThe bulk Rindler that horizon is in pureblack this AdS, hole. geometry i.e. An the consequently temperature obvious correspondscalculations was extension will to tuned be would a to made then massless difficult be hyperbolic by to the fact consider that massive the black brane holes. and bulk Again, equations of motion Page curve can be addressedreconstruction by of evaluating the latter remainsin to our be framework. done, The we reason areare is simply confident that examining that in this no the state singularities bulk,that from arise the a system Rindler is frame in of the reference. vacuum Hence state in and fact, we we expect question would be to evaluate thee.g., expectation reconstructing value of various CFTbecause operators in while the the island, appearanceof of the quantum information extremal paradox, thisHere islands does pointed asking not out if directly a address the new the black resolution issue hole of firewalls horizon [ develops a firewall in the late time phase of the from a planar branecomplicated, in e.g., pure the AdS. end-of-the-worldsuch In brane that will, the in bulk is general, no backreact longer on locally the pure geometry reconstruction AdS. [ data from the boundary CFT in the corresponding boundary subregion. One interesting setup to probeperspective, the a quantum splitting focusing quenchanchored conjecture would asymptotically in on be higher the implementedthe splitting dimensions. by asymptotic surface boundary a [ can bulk From bein the end-of-the-world obtained by the brane bulk a bulk, conformal transformation the from a end-of-the-world full brane plane; can similarly be obtained by a diffeomorphism between the bath and defect. It would be interesting to re-create this problem in our JHEP12(2020)025 grows. It Σ χ ]. ]) on the brane and 147 , 1 141 , 140 geometry with the bulk descrip- +1 d – 72 – ]. ), which permits any use, distribution and reproduction in 146 we computed bounds on entanglement growth in hyperbolic space. CC-BY 4.0 ], e.g., see [ 4.2 This article is distributed under the terms of the Creative Commons 145 – ], the brane perspective also provides an effective description of the coupling 1 142 Lastly, in order to explain the fast growth of entanglement at early times for large Above, we emphasized the effective character of the gravitational theory on the brane where part of this work was done. Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. Scholarship. RCM andthe DN “It also from received Qubit”Fields funding and collaboration. from Information group the The atHumboldt work Simons AEI, Foundation which of Foundation and is through IR generously the supported isSofja Federal by Ministry funded Kovalevskaja the for by Alexander Award. Education von the and IR Gravity, Research also Quantum through acknowledges the the hospitality of Perimeter Institute, of Canada throughCanada the and Department by the of ProvinceRCM Innovation, of is Science Ontario supported through in and the part Ministry by EconomicResearch a of Council Development Discovery Colleges Grant of and from the Universities. Canada,the Natural and Sciences Province and by Engineering of the Ontario BMO and Financial the Group. University HZC of is Waterloo supported through by an Ontario Graduate Acknowledgments We would like toNetta thank Engelhardt, Ahmed Zach Fisher, Almheiri, Greg Gabadadze, RaphaelRuan, Juan Bousso, Hernandez, Edgar Don Xi Shaghoulian, Marolf, Dong, Shan-Ming Antonyand Roberto Speranza discussions. Emparan, and Research at Raman Perimeter Sundrum Institute for is supported useful in comments part by the Government regions, in section While they display thestead, expected the qualitative difference behavior, between theywould bounds are be and not interesting numerical to particulary data improve tight. becomes these bigger bounds. In- as couplings, which can seen astion. coming Given through the the simplicity AdS ofto our further understand construction, these it nonlocal may couplings,between provide which the a implicitly island useful provide degrees framework subtle in correlations of which freedom and those in the bath CFT [ conjectures [ with the appearance ofdiscussed a in short [ distance cutoffof in the bath Randall-Sundrum CFT gravity. tolocalized However, the at as conformal the defect. defect, In which particular, dominate it at only low accounts energies, for the but couplings ignores the subtle ‘nonlocal’ correction codes between the bulka and network boundary. would It tell wouldhow us be information about interesting to the on see effective thetheory. what theory brane such (see On is e.g., a ultimately [ rations, related encoded for note, in example, one the using might asymptotic the also CFT higher-dimensional study bulk and the to defect complexity probe of holographic these complexity brane configu- to view these spatial slices as supporting tensor networks implementing quantum error JHEP12(2020)025 , ] 09 26 ]. (2006) 14 JHEP SPIRE . 08 , ]. IN ]. [ (1975) Rev. Mod. Phys. JHEP SPIRE , , SPIRE ]. IN 43 , IN arXiv:2006.04851 , vol. 931. Springer, ][ Phys. Rev. D [ [ ]. (1972) 737. , , pp. 353–397, 2017, Class. Quant. Grav. (1976) 191 (1993) 3743 (2019) 063 ]. 4 , SPIRE Quantum Extremal Islands 13 IN 71 12 ]. SPIRE ][ ]. SPIRE IN IN [ (2020) 166 (1973) 2333 JHEP ][ SPIRE The entropy of bulk quantum fields , 7 SPIRE 10 Theoretical Advanced Study Institute IN The Page curve of Hawking radiation IN ][ ][ arXiv:1908.10996 , in Phys. Rev. D (1974) 30 A Covariant holographic entanglement [ , Nuovo Cim. Lett. JHEP Commun. Math. Phys. Phys. Rev. Lett. , , , Quantum corrections to holographic ]. , 248 arXiv:1307.2892 Islands outside the horizon [ ]. Phys. Rev. D ]. , SPIRE arXiv:0705.0016 (2020) 149 – 73 – IN [ Nature SPIRE , ][ 03 Holographic Entanglement Entropy hep-th/0603001 IN ]. SPIRE [ arXiv:1609.01287 ][ IN (2013) 074 [ ]. ][ ]. 11 JHEP SPIRE ]. (2007) 062 Aspects of Holographic Entanglement Entropy , Holographic derivation of entanglement entropy from AdS/CFT ]. IN SPIRE ][ 07 SPIRE IN JHEP SPIRE IN ][ , SPIRE ]. IN (2006) 181602 [ ][ IN arXiv:1409.1231 JHEP ][ [ Black holes and the second law Black holes and entropy 96 Particle creation by black holes , Black Holes and Thermodynamics Black hole explosions Breakdown of Predictability in Gravitational Collapse arXiv:0909.1038 SPIRE Entanglement Wedge Reconstruction and the Information Paradox The Black Hole Information Problem [ The Information paradox: A Pedagogical introduction IN arXiv:1905.08255 Information in black hole radiation [ Jerusalem Lectures on Black Holes and Quantum Information [ ]. arXiv:1609.04036 10.1007/978-3-319-52573-0 hep-th/0605073 [ [ (2016) 015002 SPIRE arXiv:1905.08762 hep-th/9306083 IN entropy proposal 2017, entanglement entropy 045 Phys. Rev. Lett. [ (2020) 002 from semiclassical geometry [ and the entanglement wedge of an evaporating black hole (2009) 224001 in Elementary Particle Physics:DOI New Frontiers in Fields and Strings 88 (1976) 2460 Made Easy, Part I:[ Entanglement on the Brane arXiv:1910.11077 M. Rangamani and T. Takayanagi, T. Faulkner, A. Lewkowycz and J. Maldacena, S. Ryu and T. Takayanagi, S. Ryu and T. Takayanagi, V.E. Hubeny, M. Rangamani and T. Takayanagi, G. Penington, A. Almheiri, R. Mahajan, J. Maldacena and Y. Zhao, S.W. Hawking, D.N. Page, A. Almheiri, N. Engelhardt, D. Marolf and H. Maxfield, J.D. Bekenstein, S.D. Mathur, J.D. Bekenstein, J. Polchinski, D. Harlow, S.W. Hawking, S. W. Hawking, S.W. Hawking, H.Z. Chen, R.C. Myers, D. Neuenfeld, I.A. Reyes and J. Sandor, A. Almheiri, R. Mahajan and J. Maldacena, [8] [9] [6] [7] [3] [4] [5] [1] [2] [19] [20] [16] [17] [18] [14] [15] [11] [12] [13] [10] References JHEP12(2020)025 , ] ]. , , 06 ] 11 Gravity ] ] SPIRE ] D IN 2 JHEP ][ JHEP , , ]. The entropy of Replica Wormholes Information ]. arXiv:2007.04877 ]. SPIRE [ Geometric secret IN Page Curve for an ][ arXiv:2006.10754 SPIRE [ arXiv:1910.12836 SPIRE ]. (2020) 033 arXiv:2006.02438 IN [ IN [ arXiv:1911.12333 [ [ [ arXiv:1408.3203 03 [ ]. SPIRE IN (2020) 475401 Hawking Radiation Correlations of ][ JHEP (2020) 004 53 ]. ]. , SPIRE (2020) 121 (2020) 086009 (2020) 013 IN Islands in Asymptotically Flat Island in the Presence of Higher 05 ]. ]. (2015) 073 BCFT entanglement entropy at large 09 ][ ]. arXiv:2004.00598 05 102 [ SPIRE SPIRE 01 Including contributions from entanglement IN IN Entanglement islands in higher dimensions arXiv:2004.13088 SPIRE SPIRE JHEP arXiv:2003.05448 Islands in Schwarzschild black holes ][ ][ ]. ]. SPIRE JHEP , , J. Phys. A , IN IN JHEP [ [ , Unitarity of Entanglement and Islands in Two-Sided – 74 – IN , [ , JHEP , (2020) 091 SPIRE SPIRE arXiv:2004.14944 Islands and Page Curves for Evaporating Black Holes in [ IN IN Phys. Rev. D 05 ][ ][ Quantum Extremal Surfaces: Holographic Entanglement , ]. arXiv:1911.09666 Reflected Entropy for an Evaporating Black Hole Notes on islands in asymptotically flat 2d dilaton black holes [ JHEP Massive islands SPIRE , (2020) 094 arXiv:2004.01601 arXiv:2004.13857 IN [ [ ][ arXiv:2006.11717 arXiv:2006.06872 08 arXiv:2005.08715 , , , (2020) 001 9 JHEP arXiv:2004.05863 arXiv:2006.10846 , [ [ (2020) 036 (2020) 022 Pulling Out the Island with Modular Flow ]. ]. ]. ]. ]. 07 07 SPIRE SPIRE SPIRE SPIRE SPIRE IN IN IN IN IN arXiv:1912.02210 Evaporating Black Holes in[ JT Gravity and the Entropy of[ Hawking Radiation islands to the reflected entropy [ Hawking radiation [ Janus Black Holes (2020) 155 radiation in BCFT models[ of black holes (2020) 085 Derivative Terms SciPost Phys. sharing in a model of Hawking radiation JHEP JT Gravity [ JHEP Entropy beyond the Classical Regime Evaporating Black Hole central charge and the black hole interior T.J. Hollowood, S. Prem Kumar and A. Legramandi, A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, V. Chandrasekaran, M. Miyaji and P. Rath, A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, D. Bak, C. Kim, S.-H. Yi and J. Yoon, T. Li, J. Chu and Y. Zhou, K. Hashimoto, N. Iizuka and Y. Matsuo, H. Geng and A. Karch, M. Alishahiha, A. Faraji Astaneh and A. Naseh, A. Almheiri, R. Mahajan and J.E. Santos, M. Rozali, J. Sully, M. Van Raamsdonk, C. Waddell and D. Wakeham, T. Hartman, E. Shaghoulian and A. Strominger, T.J. Hollowood and S.P. Kumar, Y. Chen, T. Anegawa and N. Iizuka, V. Balasubramanian, A. Kar, O. Parrikar, G. Sárosi and T. Ugajin, F.F. Gautason, L. Schneiderbauer, W. Sybesma and L. Thorlacius, J. Sully, M. Van Raamsdonk and D. Wakeham, N. Engelhardt and A.C. Wall, [38] [39] [36] [37] [34] [35] [32] [33] [29] [30] [31] [27] [28] [24] [25] [26] [22] [23] [21] JHEP12(2020)025 , , ]. Black (2020) 09 102 SPIRE ]. ]. , IN ][ JHEP (2020) 066005 (2020) 044 (2020) 111 , SPIRE SPIRE , 08 11 IN IN arXiv:2008.05275 [ 102 arXiv:2008.07994 , ][ , Phys. Rev. D Evaporating Black Holes JHEP (2011) 101602 , ]. JHEP ]. , , 107 SPIRE SPIRE arXiv:2003.13147 ]. Phys. Rev. D IN IN [ , ][ Replica wormholes and the black hole ][ arXiv:2008.08570 SPIRE arXiv:1810.10601 , [ IN [ (2020) 121 Islands in de Sitter space Entanglement between two disjoint universes Phys. Rev. Lett. Islands in cosmology , Bra-ket wormholes in gravitationally prepared 06 ]. Simple holographic models of black hole – 75 – (2019) 065 ]. ]. arXiv:2003.05451 ]. 07 JHEP arXiv:1910.00972 SPIRE [ , [ Replica wormhole and information retrieval in the SYK IN The ghost in the radiation: Robust encodings of the black Wormhole calculus, replicas, and entropies SPIRE SPIRE ][ Gravity/ensemble duality Unitarity From a Smooth Horizon? ]. ]. ]. IN IN SPIRE [ ]. ]. ]. ]. ]. arXiv:2007.11658 JHEP Transcending the ensemble: baby universes, spacetime IN ][ , , [ A dynamical mechanism for the Page curve from quantum chaos (2020) 031 SPIRE SPIRE SPIRE (2020) 032 SPIRE SPIRE SPIRE SPIRE SPIRE IN IN IN Comments on wormholes, ensembles, and cosmology 06 IN IN IN IN IN [ [ [ [ ][ ][ ][ 08 ][ Holographic Dual of BCFT Pure de Sitter space and the island moving back in time JHEP More quantum noise from wormholes JHEP ER = EPR revisited: On the Entropy of an Einstein-Rosen Bridge , arXiv:2004.02900 , [ arXiv:1911.11977 ]. ]. arXiv:1911.06305 , arXiv:2007.16091 [ , SPIRE SPIRE IN IN arXiv:1105.5165 arXiv:2002.08950 arXiv:2008.01022 arXiv:2006.16289 [ [ [ states arXiv:2008.02259 [ [ (2020) 194 Hole Microstate Cosmology wormholes, and the order and disorder of black hole information arXiv:2008.05274 Coupled to a Thermal Bath arXiv:2003.13117 arXiv:2002.05734 [ model coupled to Majorana chains hole interior 106019 interior evaporation V. Balasubramanian, A. Kar and T. Ugajin, T. Takayanagi, Y. Chen, V. Gorbenko and J. Maldacena, M. Van Raamsdonk, W. Sybesma, T. Hartman, Y. Jiang and E. Shaghoulian, S.B. Giddings and G.J. Turiaci, D. Stanford, S. Cooper, M. Rozali, B. Swingle, M. Van Raamsdonk, C.D. Waddell and Marolf D. and Wakeham, H. Maxfield, V. Balasubramanian, A. Kar and T. Ugajin, H.Z. Chen, Z. Fisher, J. Hernandez, R.C. Myers and S.-M. Ruan, H. Liu and S. Vardhan, R. Bousso and E. Wildenhain, Y. Chen, X.-L. Qi and P. Zhang, I. Kim, E. Tang and J. Preskill, H. Verlinde, G. Penington, S.H. Shenker, D. Stanford and Z. Yang, C. Akers, N. Engelhardt and D. Harlow, R. Bousso and M. Tomašević, [58] [59] [55] [56] [57] [53] [54] [50] [51] [52] [48] [49] [46] [47] [43] [44] [45] [41] [42] [40] JHEP12(2020)025 ] ] ]. (2010) (1999) JHEP Phys. Phys. JHEP , SPIRE , (2002) , ]. , , 82 ] 83 IN [ (1994) 6587 Fortsch. 04 Minkowski , 49 SPIRE D IN (2011) 043 ]. − hep-th/0011156 ]. JHEP ][ arXiv:0910.0466 5 [ (1993) 3427 , 11 [ 48 Phys. Rev. D SPIRE SPIRE , IN IN Phys. Rev. Lett. ]. JHEP ][ arXiv:1107.1722 , ][ Phys. Rev. D , [ arXiv:1303.6955 , (2001) 008 , (2010) 066 SPIRE 05 02 Defect conformal field theory and IN arXiv:0705.0024 Exact half-BPS string-junction Phys. Rev. D ][ ]. ]. [ , (2011) 086 JHEP gravity on a brane in JHEP , 12 gr-qc/9403028 , ]. SPIRE D [ SPIRE hep-th/0303249 IN IN − [ ][ 4 ]. ][ Half-BPS Solutions locally asymptotic to (2007) 022 ]. SPIRE gravity in anti-de Sitter space JHEP Aspects of AdS/BCFT , IN On black hole entropy – 76 – Exact half-BPS Type IIB interface solutions. II. 06 D ][ hep-th/0005016 SPIRE − (1994) 846 ]. SPIRE [ Absence of a VVDZ discontinuity in AdS(AdS) IN 4 (2003) 030 IN 50 ][ ]. JHEP ][ ]. , 07 SPIRE An Alternative to compactification A Large mass hierarchy from a small extra dimension ]. IN ]. hep-ph/9905221 SPIRE hep-th/0109017 ]. ]. Locally localized gravity ][ [ SPIRE (2000) 208 [ IN Some properties of Noether charge and a proposal for dynamical JHEP IN ][ , SPIRE ][ hep-th/0011160 SPIRE 485 IN [ SPIRE SPIRE IN Phys. Rev. D [ ][ IN IN , ][ ][ hep-th/0106261 arXiv:1310.5713 [ [ (1999) 3370 Entanglement Renyi entropies in holographic theories Entanglement Entropy at Large Central Charge The Entanglement Renyi Entropies of Disjoint Intervals in AdS/CFT A Power law for the lowest eigenvalue in localized massive gravity Black hole entropy is the Noether charge Higgs phenomenon for Mass and gauge invariance 4. Holography for the Karch-Randall model and interface conformal field theories (2002) 044015 83 Holographic Entanglement Entropy for General Higher Derivative Gravity 3 (2001) 747 ]. ]. ]. arXiv:1006.0047 S 65 [ Phys. Lett. B 49 hep-th/9906064 × , hep-th/0112166 [ 3 [ (2001) 016 (2014) 044 SPIRE SPIRE SPIRE gr-qc/9312023 IN gr-qc/9307038 IN IN arXiv:1108.5152 058 locally localized gravity Phys. Rev. D 12 126010 arXiv:1303.7221 black hole entropy [ 01 solutions in six-dimensional supergravity [ [ Flux solutions and multi-Janus AdS [ [ space [ Rev. Lett. 4690 O. Aharony, O. DeWolfe, D.Z. Freedman and A. Karch, A. Miemiec, M. Porrati, A. Karch, E. Katz and L. Randall, M. Porrati, M. Headrick, T. Faulkner, T. Hartman, T. Jacobson, G. Kang and R.C. Myers, X. Dong, R.M. Wald, V. Iyer and R.M. Wald, M. Chiodaroli, M. Gutperle and D. Krym, M. Chiodaroli, E. D’Hoker, Y. Guo and M. Gutperle, A. Karch and L. Randall, G.R. Dvali, G. Gabadadze and M. Porrati, E. D’Hoker, J. Estes and M. Gutperle, L. Randall and R. Sundrum, L. Randall and R. Sundrum, M. Fujita, T. Takayanagi and E. Tonni, [78] [79] [75] [76] [77] [72] [73] [74] [70] [71] [68] [69] [66] [67] [63] [64] [65] [61] [62] [60] JHEP12(2020)025 , ] 14 ]. ]. , ] JHEP (2017) 44 16 , ]. Phys. 05 ]. , SPIRE SPIRE IN IN Holographic SPIRE Phys. Rev. ][ ][ IN , SPIRE [ JHEP , IN arXiv:1211.7370 ][ (2001) 223 [ Nuovo Cim. B hep-th/9912001 Class. Quant. Grav. , [ , 502 (2003) 021 Class. Quant. Grav. ]. , ]. 04 ]. (2018) 129 hep-th/9903238 ]. arXiv:0911.4257 [ [ ]. SPIRE 10 arXiv:1402.6378 SPIRE IN , JHEP SPIRE (2001) 084017 IN ]. arXiv:1305.7244 , ][ SPIRE IN Phys. Lett. B [ ][ IN SPIRE , 63 ][ JHEP ][ IN , ]. SPIRE (2010) 111 ][ IN Surface terms as counterterms in the Complexity of Formation in Holography ]. (1999) 104001 03 ][ SPIRE 60 ]. – 77 – IN ]. Towards a derivation of holographic entanglement SPIRE (2014) 011601 [ gr-qc/9706018 IN JHEP Phys. Rev. D [ gr-qc/9709048 Action Integrals and Partition Functions in Quantum , , [ ][ ]. SPIRE 112 SPIRE arXiv:1303.1080 Time Evolution of Entanglement Entropy from Black Hole IN arXiv:1102.0440 IN [ Rindler-AdS/CFT ]. ][ [ ][ arXiv:1406.7659 SPIRE Minimal surfaces and entanglement entropy in anti-de Sitter ]. ]. On entanglement spreading in chaotic systems [ Phys. Rev. D hep-th/9903203 (1977) 2752 IN , [ (1997) L163 (1998) 251 ][ SPIRE Accelerated detectors and temperature in (anti)-de Sitter spaces 15 IN Entanglement Tsunami: Universal Scaling in Holographic SPIRE SPIRE 14 15 (2013) 014 [ IN IN (2011) 036 ][ ][ Eternal black holes in anti-de Sitter 05 Topological black holes in Anti-de Sitter space (2014) 077 Phys. Rev. Lett. The Emergence of localized gravity 05 arXiv:1610.08063 , What is the dual of two entangled CFTs? [ Comment on ‘Accelerated detectors and temperature in anti-de Sitter spaces’ 10 gr-qc/9607071 AdS/CFT and gravity hep-th/9808032 Stress tensors and Casimir energies in the AdS/CFT correspondence Pair production of topological anti-de Sitter black holes [ Role of conformal three geometry in the dynamics of gravitation [ (1999) 046002 Singular hypersurfaces and thin shells in general relativity JHEP ]. ]. Phys. Rev. D JHEP (1972) 1082 , . 60 , , JHEP 28 , arXiv:1608.05101 [ (2017) 062 SPIRE SPIRE IN hep-th/0106112 IN hep-th/0011177 065 Thermalization Class. Quant. Grav. [ space Interiors Class. Quant. Grav. Rev. D 01 entropy (1999) 1197 [ GB gravity in arbitrary dimensions [ (1997) L109 AdS/CFT correspondence (1966) 1 [ Lett. Gravity H. Liu and S.J. Suh, S.D. Mathur, M. Mezei and D. Stanford, M. Parikh and P. Samantray, P. Krtous and A. Zelnikov, T. Hartman and J. Maldacena, S. Deser and O. Levin, T. Jacobson, S. Chapman, H. Marrochio and R.C. Myers, H. Casini, M. Huerta and R.C. Myers, D. Birmingham, J.M. Maldacena, R.C. Myers, S.S. Gubser, R.B. Mann, R. Emparan, C.V. Johnson and R.C. Myers, W. Israel, A. Buchel, J. Escobedo, R.C. Myers, M.F. Paulos, A. Sinha and M. Smolkin, J.W. York, G.W. Gibbons and S.W. Hawking, M.D. Schwartz, [98] [99] [96] [97] [93] [94] [95] [91] [92] [88] [89] [90] [86] [87] [83] [84] [85] [81] [82] [80] [100] JHEP12(2020)025 ] 94 , , , 105 ]. ] (2014) 085 ]. (2016) 069 J. Stat. , 03 ]. SPIRE 10 IN Phys. Rev. D ]. SPIRE , ][ (2015) 051 IN ]. [ SPIRE JHEP arXiv:1606.01857 JHEP , [ 03 arXiv:1512.02695 IN , SPIRE Phys. Rev. Lett. , , ][ , Information Flow in IN SPIRE ][ arXiv:1211.3494 IN [ JHEP ][ ]. , (1985) 343 ]. . ]. 252 SPIRE (2016) 12C104 arXiv:1907.08030 IN SPIRE [ IN ][ , talks at KITP on 7 April 2015 and 27 SPIRE Scrambling in Hyperbolic Black Holes: ][ 2016 (2014) 225007 IN arXiv:1911.03402 hep-th/9910023 [ [ ][ ]. Localized shocks 31 ]. cond-mat/9212030 (2019) 257 PTEP Conformal symmetry and its breaking in two [ Nucl. Phys. B , – 78 – , SPIRE 10 ]. SPIRE IN Speed Limits for Entanglement IN Quantum effective action from the AdS /CFT (2020) 152 (2000) 316 ][ Handbook of Mathematical Functions with Formulas, , Dover, New York City, 10th printing ed. (1964). ][ arXiv:1311.1200 03 Remarks on the Sachdev-Ye-Kitaev model SPIRE [ JHEP arXiv:0802.3117 472 ]. , IN Entanglement entropy and quantum field theory [ ]. ]. (1993) 3339 hep-th/0405152 [ [ 70 JHEP Class. Quant. Grav. SPIRE , Gapless spin fluid ground state in a random, quantum Heisenberg SPIRE SPIRE , IN Entanglement growth during thermalization in holographic systems IN IN ][ (1983) 41 (2008) 305 ][ ][ Entanglement entropy, conformal invariance and extrinsic geometry (2014) 066012 Phys. Lett. B arXiv:1604.07818 arXiv:1006.3794 Bounding the Space of Holographic CFTs with Chaos Gravitation and Hamiltonian Structure in Two Space-Time Dimensions , 126 665 89 [ [ General properties of holographic entanglement entropy (2004) P06002 Holographic metals and the fractionalized Fermi liquid Maximin Surfaces, and the Strong Subadditivity of the Covariant Holographic Lower Dimensional Gravity A simple model of quantum holography https://online.kitp.ucsb.edu/online/entangled15/kitaev/ ]. ]. ]. Phys. Rev. Lett. , 0406 SPIRE SPIRE SPIRE arXiv:1312.6717 IN IN IN arXiv:1409.8180 arXiv:1602.08272 (2010) 151602 May 2015, https://online.kitp.ucsb.edu/online/entangled15/kitaev2/ [ (2016) 106002 magnet Mech. Entanglement Entropy [ Black Hole Evaporation dimensional Nearly Anti-de-Sitter space [ Phys. Lett. B Phys. Lett. B Graphs, and Mathematical Tables correspondence shock waves and pole-skipping [ Phys. Rev. D [ [ S. Sachdev, A. Kitaev, J. Maldacena and D. Stanford, S. Sachdev and J. Ye, A.C. Wall, M. Headrick, J. Maldacena, D. Stanford and Z. Yang, P. Calabrese and J.L. Cardy, R. Jackiw, C. Teitelboim, H.Z. Chen, Z. Fisher, J. Hernandez, R.C. Myers and S.-M. Ruan, M. Abramowitz and I.A. Stegun, K. Skenderis and S.N. Solodukhin, Y. Ahn, V. Jahnke, H.-S. Jeong and K.-Y. Kim, T. Hartman and N. Afkhami-Jeddi, S.N. Solodukhin, D.A. Roberts, D. Stanford and L. Susskind, E. Perlmutter, H. Liu and S.J. Suh, [118] [119] [116] [117] [114] [115] [112] [113] [109] [110] [111] [107] [108] [104] [105] [106] [102] [103] [101] JHEP12(2020)025 , ] , , ]. , ]. Phys. , Black Holes SPIRE (2013) 090 (2012) 065007 IN SPIRE IN ][ (2017) 151 08 Entanglement 86 ][ ]. hep-th/0401024 (2019) 031011 arXiv:1903.11115 07 [ , 9 (2016) 021601 JHEP ]. (2015) 149 , SPIRE The Gravity Dual of a JHEP 117 IN 06 An Apologia for Firewalls , (2018) 002] ][ ]. SPIRE ]. Causality \& holographic (2004) 053 Phys. Rev. D IN 09 , ][ 02 Relative entropy equals bulk JHEP Phys. Rev. X arXiv:1812.01176 , arXiv:1204.1330 ]. SPIRE SPIRE , [ [ IN IN Holographic quantum error-correcting Quantum focusing conjecture ][ ][ ]. Black Holes: Complementarity or JHEP ]. SPIRE , Phys. Rev. Lett. IN , ][ Erratum ibid. Deriving covariant holographic entanglement SPIRE arXiv:1408.6300 (2019) 165 [ SPIRE [ IN IN ][ 03 (2012) 155009 ][ JT gravity as a matrix integral Holographic Quantum Circuits from – 79 – arXiv:1512.06431 [ 29 Reconstruction of Bulk Operators within the JHEP Generalized gravitational entropy arXiv:1207.3123 (2017) 118 (2014) 162 , arXiv:1512.06109 Bulk locality from modular flow [ [ Holographic Proof of the Quantum Null Energy Condition Wormholes in AdS 05 ]. ]. ]. ]. ]. 12 ]. ]. ]. (2016) 004 arXiv:1506.02669 [ 06 SPIRE SPIRE SPIRE SPIRE SPIRE JHEP JHEP SPIRE SPIRE SPIRE arXiv:1304.6483 arXiv:1607.07506 , (2013) 062 IN IN IN IN IN , [ [ IN IN IN [ ][ ][ ][ ][ ][ ][ ][ 02 (2016) 024026 JHEP Class. Quant. Grav. , , 94 Entanglement Renormalization and Holography JHEP (2016) 064044 (2013) 018 (2016) 028 , Late Time Correlation Functions, Baby Universes, and ETH in JT Gravity ]. ]. 93 09 11 SPIRE SPIRE arXiv:0905.1317 arXiv:1503.06237 arXiv:1704.05839 arXiv:1601.05416 arXiv:1704.05464 IN arXiv:1611.04650 IN arXiv:1304.4926 [ codes: Toy models for the[ bulk/boundary correspondence [ Firewalls? JHEP Entanglement Wedge in Gauge-Gravity Duality [ [ Wedge Reconstruction via Universal Recovery Channels entanglement entropy relative entropy Phys. Rev. D Splitting/Joining Local Quenches Density Matrix [ Rev. D and Random Matrices [ arXiv:1910.10311 [ [ JHEP B. Swingle, F. Pastawski, B. Yoshida, D. Harlow and J. Preskill, A. Almheiri, D. Marolf, J. Polchinski and J. Sully, A. Almheiri, D. Marolf, J. Polchinski, D. Stanford and J. Sully, T. Faulkner and A. Lewkowycz, J. Cotler, P. Hayden, G. Penington, G. Salton, B. Swingle and M. Walter, M. Headrick, V.E. Hubeny, A. Lawrence and M. Rangamani, D.L. Jafferis, A. Lewkowycz, J. Maldacena and S.J. Suh, X. Dong, D. Harlow and A.C. Wall, T. Shimaji, T. Takayanagi and Z. Wei, B. Czech, J.L. Karczmarek, F. Nogueira and M. Van Raamsdonk, J.M. Maldacena and L. Maoz, R. Bousso, Z. Fisher, S. Leichenauer and A.C. Wall, J. Koeller and S. Leichenauer, J.S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski, P. Saad, S.H. Shenker et al., P. Saad, A. Lewkowycz and J. Maldacena, X. Dong, A. Lewkowycz and M. Rangamani, P. Saad, S.H. Shenker and D. Stanford, [137] [138] [135] [136] [133] [134] [130] [131] [132] [128] [129] [125] [126] [127] [123] [124] [121] [122] [120] JHEP12(2020)025 ] ]. 90 ]. SPIRE IN ][ SPIRE IN ]. ][ arXiv:1411.0690 Phys. Rev. D [ Holographic duality , ]. SPIRE Holographic Complexity Complexity, action, and IN ]. ][ SPIRE , in preparation. (2016) 49 IN arXiv:1509.07876 ][ SPIRE 64 [ IN ][ arXiv:1601.01694 [ Effective entropy of quantum fields coupled A defect in holographic interpretations of arXiv:1512.04993 Island complexity ]. [ (2016) 191301 Fortsch. Phys. (2016) 009 , – 80 – arXiv:1612.05698 116 [ 11 SPIRE IN arXiv:2007.02987 [ ][ JHEP Complexity and Shock Wave Geometries (2016) 086006 , (2017) 090 93 03 (2020) 052 Phys. Rev. Lett. , 10 JHEP , arXiv:1406.2678 [ JHEP Entanglement is not enough Phys. Rev. D , , ]. SPIRE IN Equals Bulk Action? black holes (2014) 126007 [ from random tensor networks tensor networks with gravity A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y.J. Zhao, Hernandez, R.C. Myers, and S.-M.D. Ruan, Neuenfeld et al., in preparation. D. Stanford and L. Susskind, L. Susskind, A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, B. Czech, P.H. Nguyen and S. Swaminathan, X. Dong, X.-L. Qi, Z. Shangnan and Z. Yang, P. Hayden, S. Nezami, X.-L. Qi, N. Thomas, M. Walter and Z. Yang, [145] [146] [147] [142] [143] [144] [140] [141] [139]