Phases of Holographic Interfaces
PPhases of Holographic Interfaces
Constantin Bachas and Vassilis PapadopoulosFebruary 9, 2021 Laboratoire de Physique de l’ ´Ecole Normale Sup´erieure,CNRS, PSL Research University and Sorbonne Universit´es24 rue Lhomond, 75005 Paris, France
Abstract
We compute the phase diagram of the simplest holographic bottom-up model of conformal interfaces. The model consists of a thin domainwall between three-dimensional Anti-de Sitter (AdS) vacua, anchoredon a boundary circle. We distinguish five phases depending on theexistence of a black hole, the intersection of its horizon with the wall,and the fate of inertial observers. We show that, like the Hawking-Page phase transition, the capture of the wall by the horizon is also afirst order transition and comment on its field-theory interpretation.The static solutions of the domain-wall equations include gravitationalavatars of the Faraday cage, black holes with negative specific heat,and an intriguing phenomenon of suspended vacuum bubbles corre-sponding to an exotic interface/anti-interface fusion. Part of our anal-ysis overlaps with recent work by Simidzija and Van Raamsdonk butthe interpretation is different. a r X i v : . [ h e p - t h ] F e b ontents black string . . . . . . . . . . . . . . . . 72.2 Hawking-Page transition . . . . . . . . . . . . . . . . . . . . . . . . 8 & warm 196 Equations of state 22 T phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.2 Low- T phase(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.3 Warm phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
10 Outlook 41A Renormalized on-shell action 42B Opening arcs as elliptic integrals 45C Sweeping is continuous 47D Bubbles exist 48 Introduction
Begining with the classic paper of Coleman and De Lucia [1] there have beenmany studies of thin gravitating domain walls between vacua with differentvalues of the cosmological constant. Such walls figure in models of localizedgravity [2–4], in holographic duals of conformal interfaces [5–7], in effortsto embed inflation in string theory by studying dynamical bubbles [8–11],and more recently, following the ideas in refs. [12–14], in toy models of blackhole evaporation [15–23]. Besides being a simple form of matter coupled togravity, domain walls are also a key ingredient [24] in effective descriptionsof the string-theory landscape – see [25–27] for some recent discussions ofdomain walls in this context. In this paper we study a thin static domain wall between Anti-de Sitter(AdS) vacua, anchored at the conformal boundary of spacetime. If a dualholographic setup were to exist, it would have two conformal field theories,CFT and CFT , separated by a conformal interface [5–7]. We will calculatethe phase diagram of the system as function of the AdS radii, the tension ofthe wall and the boundary data. Several parts of this analysis have appearedbefore (see below) but the complete phase diagram has not, to the best ofour knowledge, been worked out. We will be interested in phenomena thatare hard to see at weak CFT coupling. A broader motivation, as in muchof the AdS/CFT literature, is understanding how the interior geometry isencoded on the boundary and vice versa, but we will only briefly allude tothis question in the present work.Our analysis is classical in gravity. Different phases are distinguishedby the presence/absence of a black hole and by the fate of inertial observers,either those moving freely in the bulk or those bound to the wall. Inertial ob-servers are a guiding fixture of the analysis, not emphasized in earlier works.In the high-temperature or ‘hot’ phase all inertial observers eventually crossthe black-hole horizon. In intermediate or ‘warm’ phases the wall avoids thehorizon, and may also shield bulk observers from falling inside. These two-center warm solutions are gravitational avatars of the Faraday cage. Finallywhat differentiates ‘cold’ horizonless phases is whether all timelike geodesicsintersect inevitably the wall, or not.Besides the domain wall and the black hole, the third actor in the problemis the center of global AdS where an inertial observer may rest. The rich phasediagram is the result of several competing forces: The attraction of the AdStrap, with or without a black hole in its center, the tension of the wall, and The above list of references is nowwhere nearly complete. It is only meant as an entryto the vast and growing literature in these subjects.
A domain wall sweeping the center of the ‘false AdS vacuum’ where an inertialobserver could rest (left), or entering the horizon of a black hole (right).
We work in 2+1 dimensions because calculations can be performed inclosed form. We expect, however, qualitative features of the phase diagramto carry over to higher dimensions. For simplicity we consider a single typeof non-intersecting wall, and only comment briefly on extended models thatallow junctions of different types of wall.The capture of the wall by the black hole is related to a transition analyzedin a very interesting recent paper by Simidzija and Van Raamsdonk [29], seealso [30, 31]. These authors consider time-dependent spherically-symmetricwalls whose intersections with the conformal boundary describe Hamiltonianquenches in the dual field theory. In this setting the boundary is the infinitecylinder, with a stripe describing the evolution of the dual CFT between thequench and ‘unquench’ times. By contrast, we are interested in equilibriumconfigurations. This means that the domain-wall geometry is static, and thestripes on the conformal boundary point in the time direction. Furthermore,the boundary is not the cylinder but an orthogonal torus, adding an extraparameter to the problem.Although the interpretation is different, many of our formulae are nev-ertheless related to those of refs. [29–31] by swapping the roles of boundaryspace and time (thereby also swapping the BTZ geometry with thermal AdS ,see section 2). This is fortuitous to 2+1 dimensions and does not carry overto higher dimensions.The gravitational action of the thin-wall model reads I gr = − Z S d x √ g ( R + 2 ‘ ) − Z S d x √ g ( R + 2 ‘ )+ λ Z W d s q ˆ g w + GHY terms + ct . , (1.1)4here R j ( g j ) are the Ricci scalars of the spacetime slices S j on either sideof the wall, and ˆ g w is the induced metric on the wall’s worldvolume. TheGibbons-Hawking-York terms and counterterms are given in appendix A.The action I gr depends on three parameters: the two AdS radii ‘ , ‘ andthe wall tension λ . The radii are related to the central charges of the dualCFTs [32], and the tension to the entropy [29,33] and to the energy-transportcoefficient [34] of the dual interface. Static solutions exist for λ min < λ < λ max , with λ max = 1 ‘ + 1 ‘ ; λ min = (cid:12)(cid:12)(cid:12) ‘ − ‘ (cid:12)(cid:12)(cid:12) . (1.2)The classical phase diagram depends on two dimensionless ratios of the above(e.g. ‘ /‘ := b and λ‘ := κ ) and on the two parameters that determine theconformal class of the striped boundary torus, e.g. τ := T L and τ := T L ,see figure 2. Without loss of generality we assume henceforth that ‘ ≥ ‘ ,i.e. that S is the true-vacuum slice.An important question is how much of this analysis has a chance to carryover to top-down holographic models, where back-reacting domain walls arenot thin. The size of the horizon and the number of stable rest points areorder parameters that can be also defined for thick walls, but a sharp crite-rion, that decides whether a thick domain wall enters or avoids the horizonis hard to imagine. Nevertheless, the field-theory interpretation of the tran-sition suggests that such an order parameter may exist, as we will explain insection 7.1.It is worth stressing that the thin-wall model is a minimal gravity dualof I(nterface) CFT in the same way that pure Einstein theory is a minimaldual for homogeneous CFT. The model captures the two universal boundaryoperators – the energy-momentum tensors on either side of the interface,as well as their combination refered to as the displacement operator [54].Top-down models have many more operators, some of which correspond tointernal excitations of the domain wall.Note also that boundaries, or end-of-the-world branes (EWBs), can beconsidered as a limit of domain walls when one side becomes the zero-radiusAdS spacetime [55]. In this sense holographic B(oundary) CFT [56, 57] canbe recovered from holographic ICFT, though the limit is subtle and shouldbe handled with care.A last remark concerns the Ryu-Takayanagi surfaces [58, 59] that delimitthe entanglement wedges of boundary subregions [60–62]. It is clearly of Many examples of supergravity domain walls have been worked out in the literature, arepresentative sample is [35–53]. None of these solutions depends, however, on non-trivial(non-Lagrangian) boundary data, indeed all but one are scale-invariant AdS n fibrations. See e.g. [63, 64] for reviews. /CFT at finite temperature.The wall separates space in two slices that we color green (true-vacuum side)and pink (false-vacuum side). Each of these comes in one of four topologicaltypes described in section 3. In section 4 we solve the matching equationsobeyed by a thin static domain wall, which we parametrize convenientlyby the blueshift metric factor g tt . This section overlaps substantially withref. [29] via double Wick rotation – a trick specific to 2+1 dimensions asearlier noted.In section 5 we start analyzing the solutions. By studying the turningpoint of the wall we classify the possible phases, i.e. the topologically distinctsolutions. We rule out in particular centerless geometries, in which no inertialobserver can avoid the wall, and solutions with two black holes whose mergingis prevented by the wall.In section 6 we write down the equations of state that characterize thesephases. They relate the canonical variables τ , τ to microcanonical variablesthat are natural for describing the interior geometry. We also point out therelevance of a critical tension λ = √ λ max λ min below which the hot solutiondisappears from a region of parameter space.In section 7 we compute the critical lines for sweeping transitions in boththe cold and the warm phases, and we show that the warm-to-hot transitionis always first-order – the domain wall cannot be lowered continuously to theblack horizon. The proof requires a detailed analysis of the region µ ≈ λ ≤ λ , where the hot and a warm solution come arbitrarily close. Wealso point out some puzzles regarding the ICFT interpretation of these phasetransitions.Section 8 presents a striking phenomenon: bubbles of the true vacuumsuspended from a point on the conformal boundary of the false vacuum.This is surprising from the perspective of ICFT, since it implies that thefusion of an interface and anti-interface does not produce the trivial (identity)defect, as expected from free-field calculations [65], but an exotic defect thatgenerates spontaneously a new scale.In section 9 we present numerical plots of the complete phase diagramin the canonical ensemble, for different values of the Lagrangian parameters λ, ‘ j . These plots confirm our earlier conclusions. We point out a criticalthreshold b = ‘ /‘ = 3, probably an artifact of the thin-wall approximation,above which black-hole solutions on the false-vacuum side of the wall cease toexist. We also exhibit coexisting black-hole solutions, including black holes6ith negative specific heat. This parallels the discussion of black holes indeformed JT gravity in ref. [66].Section 10 contains concluding remarks. In order not to interrupt the flowof the arguments we relegate some detailed calculations to four appendices. For completeness we recall here some standard facts about AdS/CFT at finitetemperature in three spacetime dimensions. While doing this we will be alsosetting notation and conventions. black string The metric of the static AdS black string in 2+1 dimensions is ds = ‘ dr r − M ‘ − ( r − M ‘ ) dt + r dx , (2.1)where ‘ is the radius of AdS and the horizon at r H = ‘ √ M has temperature T = √ M / π . Length units on the gravity side are such that 8 πG = 1. Thedual CFT lives on the AdS boundary, at r = 1 /(cid:15) → ∞ , with conformalcoordinates x ± ≡ x ± t ∈ R . Its central charge is c = 12 π‘ [32] .The holographic dictionary becomes transparent in Fefferman-Grahamcoordinates, in which any asymptotically-Poincar´e AdS solution takes thefollowing form [67, 68] ds = ‘ dz z + 1 z (cid:18) dx + + ‘z h − dx − (cid:19)(cid:18) dx − + ‘z h + dx + (cid:19) . (2.2)Here h ± = h T ±± i are the expectation values of the canonically-normalizedenergy-momentum tensor of the CFT. Note that it is a special feature of 2+1dimensions that the Fefferman-Graham expansion stops at order z . For thestatic black string, h + = h − = M ‘/ h T tt i = M ‘/ c/ πT . Thisis indeed the energy density of finite-temperature CFT in two dimensions.The relation between z and r is r = 1 z + M ‘ z ⇐⇒ z = 2 M ‘ (cid:16) r − √ r − M ‘ (cid:17) , (2.3)and the black-string metric in the ( z, t, x ) coordinates reads ds = ‘ dz z − (cid:18) z − M ‘ z (cid:19) dt + (cid:18) z + M ‘ z (cid:19) dx . (2.4)7ote that z covers only the region outside the horizon ( r > r H ) and thatnear the conformal boundary z ≈ r − .A last change of coordinates worth recording, even though we will notuse it in this paper, is the one that maps ( z, t, x ) to the standard Poincar´eparametrization of AdS . Such a map is guaranteed to exist because allconstant-negative-curvature Einstein manifolds in three dimensions can beobtained from AdS by identifications and excisions. For the case at hand the transformation reads w ± = ζ ± − M ‘ z M ‘ z ! , y = 4 z ( M ζ + ζ − ) / M ‘ z with ζ ± = e √ M ( x ± t ) . (2.5)The reader can check that in these coordinates the metric (2.4) becomes ds = ‘ dy + dw + dw − y , (2.6)i.e. the standard Poincar´e form of AdS as advertized. Outside the blackhorizon ( M ‘ z <
4) the coordinates x ± ≡ x ± t cover only a Rindler wedgeof the w ± plane.Since we will be refering to this later, let us verify the well-known factthat no inertial observer can avoid crossing the horizon. In the proper-timeparametrization of the trajectory a simple calculation gives ‘ ¨ rr = − − M ‘ ˙ x (2.7)where dots denote derivatives with respect to proper time. Since M is positivethere is no centrifugal acceleration QED . From the perspective of the CFT, the temperature T is the only dimensionfulparameter of the infinite-black-string solution. By a scale transformation wecan always set it to one. Things get more interesting if the black string iscompactified, x ∼ x + L , thereby converting the solution (2.1) to the non-spinning BTZ black hole [72, 73]. In addition to the central charge c , there isnow a new dimensionless parameter LT . In the Euclidean geometry τ = i LT is the complex-structure modulus of the boundary torus.At the critical temperature T HP = 1 /L the theory undergoes a Hawking-Page phase transition [28, 74]. This is seen by comparing the action of the The general transformation, for an arbitrary (conformally-flat) boundary metric andvacuum expectation value h T ab i , is given in refs. [69–71]. (i) the EuclideanBTZ black hole, and (ii) thermal AdS , whose metric is the same as (2.1)but with M replaced by ˜ M = − (2 π/L ) . The difference of free energies ofthese two saddle points reads F BTZ − F TAdS = − π ‘ (cid:16) LT − L (cid:17) . (2.8)Thus thermal AdS is the dominant solution when LT <
1, while the BTZblack hole dominates when
LT > and the Euclidean BTZ black hole differ in the choiceof boundary cycle that becomes contractible in the interior geometry. Theperiodicity conditions, respectively x ∼ x +2 π/ | ˜ M | / and t E ∼ t E +2 π/M / ,ensure regularity when this contractible cycle degenerates. Below we willencounter situations in which either the center of AdS or the BTZ horizonare excised. In such cases the regularity conditions can be relaxed.One other comment in order here concerns the difference of free energies,eq. (2.8). The renormalized gravitational action I gr (where I gr = F/T ) iscalculated for the general interface model in appendix A. In the case of ahomogeneous CFT one can, however, obtain the answer faster. Indeed, fromthe Fefferman-Graham form of the metric, eq. (2.2), one reads the energy ofthe CFT state, E = L h T tt i = 12 ‘M L . (2.9)For M = (2 πT ) this is the internal energy of the high-temperature state,as previously noted, and for M = − (2 π/L ) it is the Casimir energy of thevacuum. The corresponding free energies obey the thermodynamic identity E = − T ∂∂T (cid:18) FT (cid:19) . (2.10)Eqs. (2.9) and (2.10) determine F up to a term linear in T . This can beargued to vanish both at low T , since the ground state has no entropy, andat large L since F must be extensive. The final result is eq. (2.8).Let us finally note that since in empty AdS the mass M is negative, thereis a centrifugal contribution in eq.(2.7). An inertial observer may thus eitherrest at, or orbit around the center r = 0. But in the centerless slices that weare about to discuss, all inertial observers hit the wall. Thermal AdS and Euclidean BTZ are part of an infinite SL(2, Z ) orbit of gravitationalinstantons, [74, 75] but they are the only dominant ones for an orthogonal torus. Theirregularized Euclidean actions are I TAdS = − π ‘/ | τ | and I BTZ = − π ‘ | τ | , see below. Topology of slices
Consider now two conformal field theories, CFT and CFT , coexisting atthermal equlibrium on a circle. This is illustrated in figure 1. The horizontaland vertical axes parametrize space and Euclidean time. In addition to thecentral charges c , c , and to the properties of the interfaces between the twoCFTs, there are three more parameters in this system: the sizes L , L of theregions in which each CFT lives, and the equilibrium temperature T . Thisgives two dimensionless parameters, which we can choose for instance to be τ := T L and τ := T L . T −1 CFTCFT
CFT L L + L Figure 2:
The finite-temperature interface CFT at the AdS boundary. Both space andEuclidean time are compact, so the depicted surface is an orthogonal torus.
The gravity dual of this ICFT features domain walls, i.e. strings in 2+1dimensions, anchored at the interfaces on the conformal boundary. We willmake the simplifying assumption that the two domain walls differ only inorientation, and can join smoothly in the interior of spacetime. Extendedmodels allowing junctions of different domain walls are very interesting butthey are beyond our present scope. We will comment briefly on them in alater section.The green and pink boundary regions of fig. 2, in which CFT and CFT live, extend in the interior to slices of gravitational solutions that belong toone of several topological types. These are illustrated for the green slice infigure 3. Each slice is either part of thermal AdS with the center, markedby a grey flag, included ( E1 ) or excised ( E2 ), or part of the BTZ geometry We reserve the word “interface” for the CFT, and “domain wall” or “string” for gravity.Interfaces are anchor points of domain walls on the AdS boundary. The string of ourbottom-up model should not be confused with the black string responsible for the interiorhorizon. In top-down supergravity embeddings the two types of string may be howeverinterchangeable. E2 ), included ( H1 ) or intersecting the domain wall( H2 ). The same options are available for the pink spacetime slice. E1 E2 E2′
H1 H2
Figure 3:
The different types of space-time slice described in the main text. The actualslice is colored in green, the complementary region is excised. The letters ‘E’ and ‘H’ standfor ‘empty’ and ‘horizon’, and the grey flag denotes the rest point of an inertial observer.Note that since this is excised in E2, a conical singularity in its place is permitted. Thecenterful slice E1 can act as a gravitational Faraday cage.
As was explained in section 2, we may adopt the unified parametrization(2.1) for all types of slice, with M negative for the slices of type E1 and E2 of global AdS , and positive for the slices of type E2 , H1 and H2 of theBTZ spacetime. We are interested in static configurations which are dual toequilibrium CFT states, so time is globally defined and has fixed imaginaryperiod t E ∼ t E + 1 /T . The coordinates ( x, r ) on the other hand need not becontinuous across the wall. We therefore write the spacetime metric in termsof two coordinate charts, ds = ‘ j dr j r j − M j ‘ j − ( r j − M j ‘ j ) dt + r j dx j with ( x j , r j ) ∈ Ω j , (3.1)where Ω is the range of coordinates for the green slice and Ω the range of11oordinates for the pink slice. These ranges are delimited as follows:• by the embeddings of the static wall in the two coordinate systems, { x j ( σ ) , r j ( σ ) } , where σ parametrizes the wall ;• by the horizon whenever the slice contains one, i.e. in cases H1 and H2 ;• by the cutoff surface r j ≈ /(cid:15) → ∞ .The mass parameters of the slices, M and M , are in general different.Regularity requires however that M j = (2 πT ) for slices H1, H2 (3.2)that include a horizon, whereas M j is unconstrained for the other slice types.Furthermore, for a slice of type E1 in which the spatial circle is contractible,interior regularity fixes the periodicity of x , x j ∼ x j + 2 π/ q − M j in case E1 . (3.3)For E2, E2 and H2 the coordinate x j is not periodic, while for H1 its period,proportional to the horizon size, is unconstrained.Since the horizon is a closed surface, a green slice of type H2 can only bepaired with a pink slice of the same type. This is the topology that dominatesat very high temperature when the black hole eats up most of the bulkspacetime. As the temperature is lowered different pairs of the remainingslice types dominate. The pairs that correspond to actual solutions of thedomain-wall equations will be determined in section 5.For the time being let us comment on the differences between the hori-zonless slices in the top row of fig. 3. What distinguishes E1 from the othertwo is the existence of the AdS center (or ‘refuge’) where an inertial observermay sit at rest. By contrast, in the slices of type E2 and E2 all inertialobservers will inevitably hit the domain wall as explained in the previoussubsection. This discontinuous behavior differentiates the phases on eitherside of a sweeping transition.Note that there is no topological difference between the slices of type E2 and E2 , which is why we distinguish them only by a prime. These slicesdiffer only in the sign of M j , or equivalently the energy density per degree offreedom in the boundary theory. Together E2 and E2 describe a continuum( −∞ < M j < ∞ ) of horizonless slices with no rest point.12 Solving the wall equations
In this section we find the general solution of the domain wall equations interms of the mass parameters M , M , the AdS radii ‘ , ‘ , and the tension ofthe wall λ . That a solution always exists for any bulk geometries is a specialfeature of 2+1 dimensions, as is the double Wick rotation that relates thispart of our analysis to ref. [29]. The matching conditions at a thin domain wall have appeared in numerousstudies of cosmology and AdS/CFT. They are especially simple in the case athand, where the wall/string is static and is characterized only by its tension.Matching the induced worldsheet metric of the two charts (3.1) gives onealgebraic and one first-order differential equation for the embedding functions x ( σ ) , r ( σ ) , x ( σ ) and r ( σ ): r − M ‘ = r − M ‘ ≡ f ( σ ) (4.1)and f − ‘ r + r x = f − ‘ r + r x ≡ g ( σ ) , (4.2)where the prime denotes a derivative with respect to σ . We have defined theauxiliary functions f and g in terms of which the induced worldsheet metricreads d ˆ s | W = − f ( σ ) dt + g ( σ ) dσ . A third matching equation expressesthe discontinuity of the extrinsic curvature in terms of the tension, λ , of thewall [76, 77]. It can be written as follows : r x ‘ + r x ‘ = λ q f g . (4.3)Our convention is that σ increases as one circles Ω j in the ( x j , r j ) planeclockwise. Other conventions introduce signs in front of the two terms onthe left-hand side of this equation.Eqs. (4.1)–(4.3) are three equations for four unknown functions, but oneof these functions can be specified at will using the string-reparametrization The Israel-Lanczos matching conditions are matrix equations, [ K αβ ] − [tr K ]ˆ g αβ = λ ˆ g αβ , where K αβ is the extrinsic curvature, ˆ g αβ the induced metric, and brackets denotethe discontinuity across the wall. Only the trace part of this equation is non-trivial. Thetraceless part of K is automatically continuous by virtue of the momentum constraints D α K αβ − D β K = 0 , where D α is the covariant derivative with respect to the inducedmetric. Equation (4.3) is the tt component of the matrix equation. x j , so theintegration constants are irrelevant choices of the origin of the x j axes. Forgiven ‘ , ‘ and λ , the string embedding functions x j ( r j ), are thus uniquelydetermined by the parameteres M and M . Different choices of ( M , M )may correspond, however, to the same boundary data ( L , L , T ). These arethe competing phases of the system. Near the conformal boundary, r j → ∞ , the parameters M j can be neglectedand the worldsheet metric asymptotes to AdS by virtue of scale invariance.Explicitly the solution reads [78] r ≈ r , x j ≈ − ‘ j (tan θ j ) r − j , (4.4)where θ j is the angle in the ( x j , ‘ j /r j ) plane between the normal to theboundary and the interface, see figure 4. The matching eqs. (4.2) and (4.3)relate these angles to the bulk radii ‘ j and to the string tension λ : ‘ cos θ = ‘ cos θ ≡ ‘ w and tan θ + tan θ = λ ‘ w , (4.5)where ‘ w is the radius of the AdS worldsheet, and − π/ < θ j < π/ ℓ / r θ x Figure 4:
Near the AdS boundary in the ( x j , ‘ j /r j ) plane the string is a straight linesubtending an angle θ j with the normal. Without loss of generality we assume that ‘ ≤ ‘ , so that CFT has thesmaller of the two central charges. Its gravity dual has the lower vacuumenergy, i.e. the green slice is the ‘true vacuum’ side of the domain wall andthe pink slice is the ‘false-vacuum’ side. The first eq. (4.5) then implies that14 tan θ | ≥ | tan θ | and, provided that the tension is positive, the secondeq. (4.5) implies that θ >
0. The sign of θ , on the other hand, depends onthe precise value of λ . Expressing the tangents in terms of cosines bringsindeed this equation to the form( 1 ‘ − ‘ w ) / + ε s ‘ − ‘ w = λ with ε = sign( θ ) . (4.6)Since λ is real we must have ‘ < ‘ w < ∞ . Furthermore to each value of theworldsheet radius ‘ w there correspond two values of the tension λ , dependingon sign( θ ). Explicitly, λ min < λ < λ for ε = − and λ < λ < λ max for ε = + , (4.7)where the three critical tensions read λ min = 1 ‘ − ‘ , λ max = 1 ‘ + 1 ‘ , λ = q λ max λ min . (4.8)Let us pause here to discuss the significance of these critical tensions. The meaning of the critical tensions λ min and λ max has been understood inthe work of Coleman-De Lucia [1] and Randall-Sundrum [2]. Below λ min the false vacuum is unstable to nucleation of true-vacuum bubbles, so thetwo phases cannot coexist in equilibrium. The holographic description ofsuch nucleating bubbles raises fascinating questions in its own right, see e.g.refs. [8–10]. It has been also advocated that expanding true-vacuum bubblescould realize accelerating cosmologies in string theory [11]. Since our focushere is on equilibrium configurations, we will not discuss these interestingissues any further.The maximal tension λ max is a stability bound of a different kind. For λ > λ max the two phases can coexist, but the large tension of the wall forces It was argued in ref. [79] that the walls in the λ < λ range are unstable. But the radiusinstability in this reference reduces the action by an amount proportional to the infinitevolume of AdS and does not correspond to a normalizable mode. The only normalizablemode of the wall in the thin-brane model corresponds to the displacement operator whichis an irrelevant (dimension = 2) operator [34]. Ref. [1] actually computes the critical tension for a domain wall separating Minkowskifrom AdS spacetime. Their result can be compared to λ min in the limit ‘ → ∞ . Both the λ = λ max and the λ = λ min walls can arise as flat BPS walls in supergravitytheories coupled to scalars [80, 81]. These two extreme types of flat wall, called type IIand type III in [81], differ by the fact that the superpotential avoids, respectively passesthrough zero as fields extrapolate between the AdS vacua [82]. ‘ , ‘ → ∞ .The meaning of λ is less clear, its role will emerge later. For now notethat it is the turning point at which the worldsheet radius ‘ w ( λ ) reaches itsminimal value ‘ . Note also that the range λ min < λ < λ only exists fornon-degenerate AdS vacua, that is when ‘ is strictly smaller than ‘ .Since the wall in this minimal model is described by a single parameter,its tension λ , all properties of the dual interface depend on it. These includethe interface entropy, and the energy-transport coefficients. The entropy or g -factor, computed in [29, 33], reads log g I = 2 π‘ ‘ " λ max tanh − (cid:18) λλ max (cid:19) − λ min tanh − (cid:18) λ min λ (cid:19) . (4.9)It varies monotonically between −∞ and ∞ as λ varies inside its allowedrange (4.7).The fraction of transmitted energy for waves incident on the interfacefrom the CFT side, respectively CFT side, was computed in [34] with theresult (reexpressed here in terms of critical tensions) T → = λ max + λ min λ max + λ , T → = λ max − λ min λ max + λ . (4.10)Note that using λ max + λ min = 2 /‘ and λ max − λ min = 2 /‘ , one can check thatthese coefficients obey the detailed-balance condition c T → = c T → . Thelarger of the two transmission coefficients reaches the unitarity bound when λ = λ min , and both coefficients attain their minimum when λ = λ max . Totalreflection (from the false-vacuum to the true-vacuum side) is only possible if ‘ /‘ →
0, i.e. when the “true-vacuum” CFT is almost entirely depleted ofdegrees of freedom relative to CFT .Using eqs. (4.8) and the Brown-Henneaux formula one can express thecentral charges c , in terms of the critical tensions λ min and λ max . As wejust saw, λ parametrizes two key properties of the interface. The triplet( λ min , λ max , λ ) of parameters in the gravitational action defines thereforethe basic data of the putative dual ICFT. We will now derive the general solution of the equations (4.1) - (4.3), andthen relate the geometric parameters M j to the data ( T, L j ) of the boundary There are many calculations of the boundary, defect, and interface entropy in a varietyof holographic models – a partial list is [56, 57, 84–87]. The formula for arbitrary left andright central charges, which we rederive below, was found in ref. [29]. f ( σ ) = | σ | = ⇒ r j = q | σ | + M j ‘ j . (4.11)In this parametrization d ˆ s | W = −| σ | dt + g ( σ ) dσ . Let σ + be the minimalvalue of | σ | , which is either zero or positive. If σ + = 0 the string enters thehorizon. If on the other hand σ + > r j ( σ ) where both x and x diverge.A static string has (at most) one turning point, and is symmetric underreflection in the axis that passes through the centers of the boundary arcs, as illustrated in figure 5. It follows that the parametrization is one-to-one ifwe let σ take values in ( −∞ , − σ + ] ∪ [ σ + , ∞ ) where the reflection symmetryinverts the sign of σ . Henceforth we focus on the half string parametrized bypositive σ . [E1,E2] [H2,H2] L L L L Figure 5:
Schematic drawing of a low-temperature and a high-temperature solution,corresponding to pairs of type [E1,E2] and [H2,H2]. The broken line is the axis of reflectionsymmetry. The blueshift parameter | σ | decreases monotonically until the string reacheseither the turning point or the black-hole horizon. Eqs. (4.11) imply that 2 r j r j = 1. Inserting in eq. (4.2) gives( x j ) = r − j (cid:18) g ( σ ) − ‘ j σr j (cid:19) . (4.12)Squaring now twice eq. (4.3) and replacing ( x j ) from the above expressionsleads to a quadratic equation for g ( σ ), the σσ component of the worldsheet In ref. [29] this corresponds to the time-reflection symmetry of the instanton solutions. g = 0, and a non-trivial one g ( σ ) = λ (cid:18) r r ‘ ‘ (cid:19) − (cid:18) r ‘ + r ‘ − λ σ (cid:19) − = λ Aσ + 2 Bσ + C , (4.13)where in the second equality we used eqs. (4.11), and A = ( λ − λ )( λ − λ ) ; B = λ ( M + M ) − λ ( M − M ) ; C = − ( M − M ) . (4.14)We expressed the quadratic polynomial appearing in the denominatorof (4.13) in terms of M j , λ and the critical tensions, eqs. (4.8), in order torender manifest the fact that for λ in the allowed range, λ min < λ < λ max ,the coefficient A is positive. This is required for g ( σ ) to be positive near theboundary where σ → ∞ . In addition, AC ≤ σ ± = − B ± ( B − AC ) / A (4.15)are real, and that the larger root σ + is non-negative. Inserting eq. (4.13) ineq. (4.12) and fixing the sign of the square root near the conformal boundarygives after a little algebra x ‘ = − σ ( λ + λ ) + M − M σ + M ‘ ) q Aσ ( σ − σ + )( σ − σ − ) , (4.16a) x ‘ = − σ ( λ − λ ) + M − M σ + M ‘ ) q Aσ ( σ − σ + )( σ − σ − ) . (4.16b)We may now confirm our earlier claim that if σ + > x ∝ dx /dr and x ∝ dx /dr diverge at this point. Furthermore, since σ + M j ‘ j = r j is positive, the x j are finite at all σ > σ + . Thus σ + is the unique turningpoint of the string, as advertized.Eqs. (4.11) and (4.16) give the general solution of the string equations forarbitrary mass parameters M , M of the green and pink slices. These mustbe determined by interior regularity, and by the Dirichlet conditions at the Except for the measure-zero set of solutions in which the string passes through thecenter of global AdS . L j = 2 Z ∞ σ + dσ x j for E2, E2 ; (4.17a) L j = nP j + 2 Z ∞ σ + dσ x j for E1, H1 ; (4.17b) L j = ∆ x j (cid:12)(cid:12)(cid:12) Hor + 2 Z ∞ σ + dσ x j for H2 . (4.17c)The integrals in these equations are the opening arcs, ∆ x j , between thetwo endpoints of a half string. They can be expressed as complete ellipticintegrals of the first, second and third kind, see appendix B. For the slices E1,H1 where x j is a periodic coordinate, we have denoted by P j > n the string winding number. Finally for strings entering the horizonwe denote by ∆ x j | Hor the opening arc between the two horizon-entry pointsin the j th coordinate chart.Possible phases of the ICFT for given torus parameters T, L j must besolutions to one pair of conditions (4.17). Apart from interior regularity, wewill also require that the string does not self-intersect. In principle, two stringbits intersecting at an angle = π could join into another string. Such stringjunctions would be the gravitational counterparts of interface fusion [65], andallowing them would make the holographic model much richer. To keep,however, our discussion simple we will only allow a single type of domainwall in this work.The reader can easily convince herself that to avoid string intersectionswe must have P j > L j and n = 1 in (4.17b), and ∆ x j (cid:12)(cid:12)(cid:12) Hor > & warm Among the five slice types of figure 3, H2 stands apart because it can onlypair with itself. This is because a horizon is a closed surface, so it cannotend on the domain wall. We will now show that the matching equationsactually rule out several other pairs among the remaining slice types.One pair that is easy to exclude is [
H1,H1 ], i.e. solutions that describe twoblack holes sitting on either side of the wall. Interior regularity would requirein this case M = M = (2 πT ) . But eqs. (4.14) and (4.15) then imply that σ + = 0, so the wall cannot avoid the horizon leading to a contradiction. Generically the intersection point in one slice will correspond to two points that mustbe identified in the other slice; this may impose further conditions. Except possibly in the limiting case where the wall is the boundary of space. λ > λ max ) inflate and couldthus prevent the black holes from coalescing. A second class of pairs one can exclude are the ‘centerless geometries’[
E2,E2 ], [
E2,E2 ], [ E2 ,E2 ] and [ E2 ,E2 ]. We use the word ‘centerless’ forgeometries that contain neither a center of global AdS, nor a black hole inits place (see fig. 3). If such solutions existed, all inertial observers wouldnecessarily hit the domain wall since there would be neither a center whereto rest, nor a horizon where to escape. The argument excluding such solutions is based on a simple observation:What distinguishes the centerless slices E2 and E2 from those with a AdScenter ( E1 ) or a black hole ( H1 ) is the sign of x j at the turning point,sign (cid:16) x j (cid:12)(cid:12)(cid:12) σ ≈ σ + (cid:17) = + for E2 , E2 , − for E1 , H1 . (5.1)Now from eqs. (4.16) one has( σ + M ‘ ) x ‘ + ( σ + M ‘ ) x ‘ < , (5.2)so both x j cannot be simultaneously positive. This holds for all σ , and hencealso near the turning point QED . This is our second no-go lemma:‘Centerless’ static spacetimes in which all inertial observers wouldinevitably hit the domain wall are ruled out.We can actually exploit this argument further. As is clear from eq. (4.16a),if M > M then x is manifestly negative, i.e. the green slice is of type E1 or H1 . The pairs [ E2 ,E1 ] and [ E2 ,E2 ] for which the above inequality isautomatic are thus ruled out. One can also show that x | σ ≈ σ + is negativeif M > > M . This is obvious from eq. (4.16b) in the range λ > λ ,and less obvious but also true as can be checked by explicit calculation for Asymptotically-flat domain walls, which have been studied a lot in the context ofGrand Unification [83], are automatically in this range. In the double-Wick rotated context of Simidzija and Van Raamsdonk the [E2,E2]geometries give traversible wormholes [29]. H1H2 E1 E2 E2′ H1 hotcoldwarmexcluded2 centers Color code
Figure 6:
Phases of the domain-wall spacetime. The type of the green slice labels therows of the table, and that of the pink slice the columns. In the hot (red) phase the wallenters the black-hole horizon, while in the warm (yellow-orange) phases it avoids it. Thecold (blue) phases have no black hole. Geometries in which an inertial observer is attractedto two different centers are indicated by a different shade (yellow or darker blue). λ < λ . The pairs [ E1,E2 ] and [ E2,E2 ] for which the above mass inequalityis automatic, are thus also excluded.Recall that the energy density of the j th CFT reads h T tt i = ‘ j M j .Ruling out all pairs of E2 with E1 or E2 implies therefore that in the groundstate the energy density must be everywhere negative. When one L j is muchsmaller than the other, the Casimir energy scales like E ∼ /L j . The factthat the coefficient cannot be paired with a horizonless BTZ slice.This implies that in the ground state of the putative dual ICFT theenergy density is everywhere negative.We have collected for convenience all these conclusions in fig. 6. The tableshows the eligible slice pairs, or the allowed topologies of static-domain-wallspacetimes. It also defines a color code for phase diagrams.The light yellow phases that feature a wall between the black hole andan AdS restpoint are the gravitational avatars of the Faraday cage. Suchsolutions are easier to construct for larger λ . Domain walls lighter than λ ,in particular, can never shield from a black hole in the ‘true-vacuum’ side.Indeed, as follows easily from eq. (4.16b), for λ < λ and M > > M , thesign of x | σ ≈ σ + is positive, so geometries of type [ H1,E1 ] are excluded.21
Equations of state
The different colors in figure 6 describe different phases of the system, sincethe corresponding geometries are topologically distinct. They differ in howthe wall, the horizon (if one exists) and inertial observers intersect or avoideach other.Let us now think thermodynamics. For fixed Lagrangian parameters λ, ‘ j , the canonical variables that determine the state of the system are thetemperature T and the volumes L , L . Because of scale invariance only twodimensionless ratios matter: τ := T L , τ := T L or γ := L L = τ τ . (6.1)The microcanonical variables, the energy density and the entropy of eachsubsystem, read (see section 2, and recall that 8 πG = 1) E j L j = ‘ j M j and S j = r H j ∆ x j | Hor G = 2 π‘ j q M j ∆ x j | Hor . (6.2)These are the natural parameters of the interior geometry. The entropies arescale invariant. The other key dimensionless variable is the mass ratio, viz.the ratio of energy densities per degree of freedom in the two CFTs µ := M M . (6.3)When several phases coexist the dominant one is the one with the lowestfree energy F = P j ( E j − T S j ). As a sanity check, we rederive F from therenormalized on-shell gravitational action in appendix A.The Dirichlet conditions, eqs. (4.17), give for each type of geometry tworelations among the above variables that play the role of equations of state. They relate the natural interior parameters S j and µ to the variables τ j and γ of the boundary torus. Note that in each phase of the system only twoamong the variables S j , µ are non-trivial, since for horizonless slices S j = 0and for slices with horizon M j = (2 πT ) . In computing the phase diagramwe will have to invert these equations of state. T phase For fixed L j and very high temperature the black hole grows so large that iteats away a piece of the domain wall and the AdS rest points. The dominant In homogeneous systems there is a single equation of state. Here we have one equationfor each subsystem. [H2,H2] and regularity fixes the mass parameters inboth slices, M = M = (2 πT ) . The boundary conditions (4.17c) reduce inthis case to simple equations for the opening horizon arcs ∆ x j | Hor . Perform-ing explicitly the integrals (see appendix B) gives L − ∆ x (cid:12)(cid:12)(cid:12) Hor = − πT tanh − ‘ ( λ + λ )2 λ ! , (6.4a) L − ∆ x (cid:12)(cid:12)(cid:12) Hor = − πT tanh − ‘ ( λ − λ )2 λ ! . (6.4b)For consistency we must have ∆ x j | Hor >
0, which is automatic if λ > λ . If λ < λ , on the other hand, positivity of ∆ x | Hor puts a lower bound on τ , τ ≥ π tanh − ‘ ( λ − λ )2 λ ! := τ ∗ . (6.5)We see here a first interpretation of the critical tension λ encountered insection 4.3. For walls lighter than λ there is a region of parameter spacewhere the hot solution ceases to exist, even as a metastable phase.The total energy and entropy in the high- T phase read E [hot] = 12 ( ‘ L M + ‘ L M ) = 2 π T ( ‘ L + ‘ L ) , (6.6) S [hot] = 4 π T (cid:16) ‘ ∆ x (cid:12)(cid:12)(cid:12) Hor + ‘ ∆ x (cid:12)(cid:12)(cid:12) Hor (cid:17) = 4 π T ( ‘ L + ‘ L ) + 2 log g I , (6.7)where log g I is given by eq. (4.9) and the rightmost expression of the entropyfollows from eqs. (6.4) and a straightforward reshuffling of the arctangentfunctions. This is a satisfying result. Indeed, the first term in the right-handside of (6.7) is the thermal entropy of the two CFTs (being extensive theseentropies cannot depend on the ratio L /L ), while the second term is theentropy of the two interfaces on the circle. The Bekenstein-Hawking formulacaptures nicely both contributions.Eqs.(6.4) and (6.7) show that shifting the L j at fixed T does not changethe entropy if and only if ‘ δL = − ‘ δL . Moving in particular a defect(for which ‘ = ‘ ) without changing the volume L + L is an adiabaticprocess, while moving a more general interface generates/absorbs entropy bymodifying the density of degrees of freedom.23 .2 Low- T phase(s) Consider next the ground state of the system, at T = 0. The dual geometrybelongs to one of the three horizonless types: the double-center geometry [E1, E1] , or the single-center ones [E1, E2] and [E2, E1] (see fig. 6). Herethe entropies S j = 0, and the only relevant dimensionless variables are thevolume and energy-density ratios, γ and µ . Note that they are both positivesince L j > M j < j .The Dirichlet conditions (4.16) for horizonless geometries read q | M | L = 2 π δ S , E1 − f ( µ ) , q | M | L = 2 π √ µ δ S , E1 − f ( µ ) , (6.8)where δ S j , E1 = 1 if the j th slice is of type E1 and δ S j , E1 = 0 otherwise, and f ( µ ) = ‘ √ A Z ∞ s + ds s ( λ + λ ) − µ ( s − ‘ ) q s ( s − s + )( s − s − ) , (6.9a) f ( µ ) = ‘ √ A Z ∞ s + ds s ( λ − λ ) + 1 − µ ( s − µ‘ ) q s ( s − s + )( s − s − ) , (6.9b)with A s ± = λ (1 + µ ) − λ (1 − µ ) ± λ s − µ‘ + µ − µ‘ + µλ . (6.10)The dummy integration variable s is the appropriately rescaled blueshiftfactor of the string worldsheet, s = σ/ | M | .Dividing the two sides of eqs. (6.8) gives γ as a function of µ for each of thethree possible topologies. If the ground state of the putative dual quantum-mechanical system was unique, we should find a single slice-pair type andvalue of µ for each value of γ . Numerical plots show that this is indeed thecase. Specifically, we found that γ ( µ ) is a monotonically-increasing functionof µ for any given slice pair, and that it changes continuously from onetype of pair to another. We will return to these branch-changing ‘sweepingtransitions’ in section 7. Let us stress that the uniqueness of the cold solutiondid not have to be automatic in classical gravity, nor in the dual large- N quantum mechanics. The functions f j ( µ ) are combinations of complete elliptic integrals of the first, secondand third kind, see appendix B. The value µ = 1 gives γ = 1, corresponding to the scale-invariant AdS string worldsheet. The known supersymmetric top-down solutions live atthis special point in phase space. ‘ j , λ ) parameter space, as γ ranges in (0 , ∞ ) the massratio µ covers also the entire range (0 , ∞ ). However, if ‘ < ‘ (strict inequal-ity) and for sufficiently light domain walls, we found that γ vanishes at somepositive µ = µ ( λ, ‘ j ). Below this critical value γ becomes negative signalingthat the wall self-intersects and the solution must be discarded. This leadsto a striking phenomenon that we discuss in section 8. The last set of solutions of the model are the yellow- or orange-coloured onesin fig. 6. Here the string avoids the horizon, so the slice pair is of type [H1,X] or [X,H1] with X one of the horizonless types: E1, E2 or E2 .Assume first that the black hole is on the green side of the wall, so that M = (2 πT ) . In terms of µ the Dirichlet conditions (4.17a, 4.17b) read:2 πT ∆ x (cid:12)(cid:12)(cid:12) Hor − πτ = ˜ f ( µ ) , πτ = 2 π √− µ δ S , E1 − ˜ f ( µ ) , (6.11)where ˜ f ( µ ) = ‘ √ A Z ∞ ˜ s + ds s ( λ + λ ) + 1 − µ ( s + ‘ ) q s ( s − ˜ s + )( s − ˜ s − ) , (6.12a)˜ f ( µ ) = ‘ √ A Z ∞ ˜ s + ds s ( λ − λ ) − µ ( s + µ‘ ) q s ( s − ˜ s + )( s − ˜ s − ) , (6.12b)and the roots ˜ s ± = σ ± /M inside the square root are given by A ˜ s ± = − λ (1 + µ ) + λ (1 − µ ) ± λ s − µ‘ + µ − µ‘ + µλ . (6.13)In the first condition (6.11) we have used the fact that the period of the greenslice that contains the horizon is P = ∆ x | Hor .If the black hole is on the pink side of the wall, the conditions take asimilar form in terms of the inverse mass ratio ˆ µ = µ − = M /M ,2 πT ∆ x (cid:12)(cid:12)(cid:12) Hor − πτ = ˆ f (ˆ µ ) , πτ = 2 π √− ˆ µ δ S , E1 − ˆ f (ˆ µ ) , (6.14)where hereˆ f (ˆ µ ) = ‘ √ A Z ∞ ˆ s + ds s ( λ + λ ) + ˆ µ − s + ˆ µ‘ ) q s ( s − ˆ s + )( s − ˆ s − ) , (6.15a)25 f ( µ ) = ‘ √ A Z ∞ ˆ s + ds s ( λ − λ ) − ˆ µ + 1( s + ‘ ) q s ( s − ˆ s + )( s − ˆ s − ) . (6.15b)and the roots ˆ s ± = σ ± /M inside the square root are given by A ˆ s ± = − λ (ˆ µ + 1) + λ (ˆ µ − ± λ s ˆ µ − ˆ µ‘ + 1 − ˆ µ‘ + ˆ µλ . (6.16)The functions ˜ f j and ˆ f j , as well as the f j of the cold phase, derive fromthe same basic formulae (4.16a, 4.16b) and differ only by a few signs. Wechose to write them out separately because these signs are important. Notealso that while in cold solutions µ is always positive, here µ and its inverseˆ µ can have either sign.All the values of µ and ˆ µ do not, however, correspond to admissiblesolutions. For a pair of type [H1,X] we must demand (i) that the right-handsides in (6.11) be positive – the non-intersection requirement, and (ii) that x | σ ≈ σ + be negative – the turning point condition (5.1). Likewise for solutionsof type [X, H1] we must demand that the right-hand sides in (6.14) be positiveand that x | σ ≈ σ + be negative.The turning-point requirement is easy to implement. In the [H1,X] case, x | σ ≈ σ + is negative when the numerator of the integrand in (6.12a), evaluatedat at s = ˜ s + , is positive. Likewise for the [X,H1] pairs, x | σ ≈ σ + is negativewhen the numerator of the integrand in (6.15b), evaluated at at s = ˆ s + , ispositive. After a little algebra these conditions take a simple formfor [H1 , X] µ ∈ ( −∞ ,
1] ; for [X , H1] ˆ µ = µ − ∈ ( −∞ , . (6.17)Recalling that µ = ˆ µ − = M /M , we conclude that in all the cases theenergy density per degree of freedom in the horizonless slice is lower thanthe corresponding density in the black hole slice.This agrees with physical intuition: the energy density per degree offreedom in the cooler CFT is less than the thermal density πT / µ → µ →
1, the wallenters the horizon and the energy is equipartitioned.This completes our discussion of the equations of state. To summarize,these equations relate the parameters of the interior geometry ( µ, S j ) to thoseof the conformal boundary ( γ, τ j ). The relation involves elementary functionsin the hot phase, and was reduced to a single function γ ( µ ), that can be read-ily plotted, in the cold phases. Furthermore at any given point in parameterspace the hot and cold solutions, when they exist, are unique. The excluded26egions are τ < τ ∗ ( λ, ‘ j ) for the hot solutions, and µ > µ ( λ, ‘ j ) for the coldsolution with µ the point where γ = 0.In warm phases the story is richer since more than one solutions typicallycoexist at any given value of ( γ, τ j ). Some solutions have negative specificheat, as we will discuss later. To find the parameter regions where differentsolutions exist requires inverting the relation between ( γ, τ j ) and ( µ, S j ). Wewill do this analytically in some limiting cases, and numerically to computethe full phase diagram in section 9. The transitions between different phases are of three kinds:• Hawking-Page transitions describing the formation of a black hole.These transitions from the cold to the hot or warm phases of fig. 6are always first order;• Warm-to-hot transitions during which part of the wall is captured bythe horizon. We will show that these transitions are also first-order;• Sweeping transitions where the wall sweeps away a center of global AdS,i.e. a rest point for inertial observers. These are continuous transitionsbetween the one- and two-center phases of fig. 6.It is instructive to picture these transitions by plotting the metric factor g tt while traversing space along the axis of reflection symmetry. The curvechanges qualitatively as shown in figure 9, illustrating the topological natureof the transitions on the gravity side.Before embarking in numerical plots, we will first do the following things:(i) Comment on the ICFT interpretation of these transitions; (ii) Computethe sweeping transitions analytically; and (iii) Prove that the warm-to-hottransitions are first order, i.e. that one cannot lower the wall to the horizoncontinuously by varying the boundary data. When a holographic dual exists, Witten has argued that the appearance ofa black hole at the Hawking-Page (HP) transition signals deconfinement inthe gauge theory [90]. Assuming this interpretation leads to the conclusion There is an extensive literature on the subject including [91, 92], studies specific totwo dimensions [93, 94], and recent discussions in relation with the superconformal indexin N = 4 super Yang Mills [95–98]. For an introductory review see [99]. + H-P H - P sweep w a r m - h o t Figure 7:
Curves of the blueshift factor g tt as one traverses space along the Z symmetryaxis. The color code is the same as in fig. 6. The wall is located at the turning point g tt = σ + where the curve is discontinuous. The grey arrows indicate possible transitions.The blackened parts of the curves are regions behind the horizon. that in warm phases a confined theory coexists with a deconfined one. Wewill see below that such coexistence is easier when the confined theory isCFT , i.e. the theory with the larger central charge. This is natural fromthe gravitational perspective. Solutions of type [H1, X] are more likely thansolutions of type [X, H1] because a black hole forms more readily on the ‘true-vacuum’ side of the wall. We will actually provide some evidence later thatif c > c there are no equilibrium phases at all in which CFT is deconfinedwhile CFT stays confined.The question that jumps to one’s mind is what happens for thick walls,where one expects a warm-to-hot crossover rather than a sharp transition.One possibility is that the coexistence of confined and deconfined phases isimpossible in microscopic holographic models. Alternatively, an appropri-ately defined Polyakov loop [90] could provide a sharp order parameter forthis transition.For sweeping transitions the puzzle is the other way around. Here a sharporder parameter exists in classical gravity – it is the number of rest pointsfor inertial observers. This can be defined both for thin- and for thick-wallgeometriess. The interpretation on the field theory side is however unclear.The transitions could be related to properties of the low-lying spectrum atinfinite N , or to the entanglement structure of the ground state.We leave these questions open for future work. Even though for homogeneous 2-dimensional CFTs the critical temperature, τ HP = 1,does not depend on the central charge by virtue of modular invariance. .2 Sweeping transitions Sweeping transitions are continuous transitions that happen at fixed valuesof the mass ratio µ . We will prove these statements here.Assume for now continuity, and let the j th slice go from type E1 to type E2 . The transition occurs when the string turning point and the center ofthe j th AdS slice coincide, i.e. when r j ( σ + ) = q σ + + M j ‘ j = 0 . (7.1)Clearly this has a solution only if M j <
0. Inserting in (7.1) the expressions(4.15) - (4.14) for σ + gives two equations for the critical values of µ with thefollowing solutions µ ∗ = 1 − ‘ λ ‘ /‘ and µ ∗ = ‘ /‘ − ‘ λ . (7.2)In the low- T phases both M j are negative and µ is positive. Furthermore,a little algebra shows that for all λ ∈ ( λ min , λ max ) the following is true x (cid:12)(cid:12)(cid:12) σ ≈ σ + < µ (cid:29) x (cid:12)(cid:12)(cid:12) σ ≈ σ + < µ (cid:28) . (7.3)This means that for µ (cid:29) E1 , and for µ (cid:28) E1 . A sweeping transition can occur if the critical massratios (7.2) are in the allowed range. We distinguish three regimes of λ :• Heavy ( λ > /‘ ): None of the µ ∗ j is positive, so the solution is oftype [E1,E1] for all µ , i.e. cold solutions are always double-center ;• Intermediate (1 /‘ > λ > /‘ ): Only µ ∗ is positive. If this is insidethe range of non-intersecting walls, the solution goes from [E1,E2] atlarge µ , to [E1,E1] at small µ . Otherwise the geometry is always of thesingle-center type [E1,E2] ;• Light ( λ < /‘ ): Both µ ∗ and µ ∗ are positive, so there is the pos-sibility of two sweeping transitions: from [E2,E1] at small µ to [E1,E2] at large µ passing through the double-center type [E1,E1] . Note thatsince λ min = 1 /‘ − /‘ , this range of λ only exists if ‘ < ‘ , i.e. whenCFT has no more than twice the number of degrees of freedom of themore depleted CFT .We can now confirm that sweeping transitions are continuous, not only interms of the mass ratio µ but also in terms of the ratio of volumes γ . To thisend we expand the relations (6.8) around the above critical points and show29hat the L j vary indeed continuously across the transition. The calculationscan be found in appendix C .For the warm phases we proceed along similar lines. One of the two M j isnow equal to (2 πT ) >
0, so sweeping transitions may only occur for negative µ . Consider first warm solutions of type [H1,X] with the black hole in the‘true vacuum’ side. A little calculation shows that x | σ ≈ σ + is negative, i.e. X=E1 , if and only if λ > ‘ and µ < µ ∗ < . (7.4)Recall that when X=E1 some inertial observers can be shielded from theblack hole by taking refuge at the restpoint of the pink slice. We see thatthis is only possible for heavy walls ( λ > /‘ ) and for µ < µ ∗ . A sweepingtransition [H1,E1] → [H1,E2] takes place at µ = µ ∗ .Consider finally a black hole in the ‘false vacuum’ side, namely warmsolutions of type [X,H1] . Here x | σ ≈ σ + is negative, i.e. X has a rest point, ifand only if the following conditions are satisfied λ > ‘ and ˆ µ := µ − < ( µ ∗ ) − < . (7.5)Shielding from the black hole looks here easier, both heavy and intermediate-tension walls can do it. In reality, however, we have found that solutionswith the black hole in the ‘false vacuum’ side are rare, and that the aboveinequality pushes ˆ µ outside the admissible range.The general trend emerging from the analysis is that the heavier thewall the more likely are the two-center geometries. A suggestive calculationactually shows that ∂σ + ∂λ (cid:12)(cid:12)(cid:12)(cid:12) M j fixed is +ve for two-center solutions − ve for single-center solutions (7.6)where the word ‘center’ here includes both an AdS restpoint and a black hole.At fixed energy densities a single center is therefore pulled closer to a heavierwall, while two centers are instead pushed away. It might be interesting toalso compute ∂σ + /∂λ and ∂V /∂λ at L j fixed, where V is the regularizedvolume of the interior space. In the special case of the vacuum solution withan AdS wall, the volume (and the associated complexity [100]) can be seento grow with the tension λ . In warm-to-hot transitions the thin domain wall enters the black-hole horizon.One may have expected this to happen continuously, i.e. to be able to lower30he wall to the horizon smoothly, by slowly varying the boundary data L j , T .We will now show that, if the tension λ is fixed, the transition is actuallyalways first order.Note first that in warm solutions the slice that contains the black holehas M j = (2 πT ) . If the string turning point approaches continuously thehorizon, then σ + →
0. From eqs. (4.14, 4.15) we see that this can happen ifand only if ( M − M ) →
0, which implies in passing that the solution mustnecessarily be of type [H1,E2 ] or [E2 ,H1] . Expanding around this putativepoint where the wall touches the horizon we set M − M M + M := δ with | δ | (cid:28) ⇒ σ + ≈ πTλ ! δ . (7.7)Recalling that the horizonless slice has the smaller M j we see that for positive δ the black hole must be in the green slice and µ = 1 − δ + O ( δ ), while fornegative δ the black hole is in the pink slice and ˆ µ = 1 + 2 δ + O ( δ ).The second option can be immediately ruled out since it is impossibleto satisfy the boundary conditions (6.14). Indeed, ˆ f (ˆ µ ≈
1) is manifetslypositive, as is clear from eq. (6.15a), and we have assumed that S is oftype E2 . Thus the second condition (6.14) cannot be obeyed. By the samereasonning we see that for δ positive, and since now S is of type E2 , weneed that ˜ f ( µ ≈
1) be negative. As is clear from the expression (6.12b) thisimplies that λ < λ .The upshot of the discussion is that a warm solution arbitrarily close tothe hot solution may exist only if λ < λ and if the black hole is on thetrue-vacuum side.It is easy to see that under these conditions the two branches of solutionindeed meet at µ = 1, ∆ x | Hor = 0 and hence from (6.4b) τ = 1 π tanh − ‘ ( λ − λ )2 λ ! := τ ∗ . (7.8)Recall from section 6.1 that this is the limiting value for the existence ofthe hot solution – the solution ceases to exist at τ < τ ∗ . The nearby warmsolution could in principle take over in this forbidden range, provided that τ ( δ ) decreases as δ moves away from zero. It actually turns out that τ ( δ )initially increases for small δ , so this last possibility for a continuous warm-to-hot transition is also ruled out.To see why this is so, expand (6.11) and (6.12b) around µ = 1,˜ s + = δ λ + O ( δ ) , ˜ s − = − λ A (cid:18) − δ (1 + λ λ ) (cid:19) + O ( δ ) , s := y + ˜ s + so that (6.12b) reads2 π τ ( δ ) = ‘ √ A Z ∞ dy y ( λ − λ ) + 2 δ ( y + µ‘ ) q y ( y + ˜ s + )( y − ˜ s − ) + O ( δ ) . (7.9)We neglected in the integrand all contributions of O ( δ ) except for the ˜ s + inthe denominator that regulates the logarithmic divergence of the O ( δ log δ )correction. Now use the inequalities y ( λ − λ ) + 2 δ q ( y + ˜ s + )( y − ˜ s − ) > y ( λ − λ ) + 2 δ q ( y + δ /λ )( y + 4 λ /A ) > √ y ( λ − λ ) q ( y + 4 λ /A ) , where the second one is equivalent to 2 δ > ( λ /λ − δ , which is true forsmall enough δ . Plugging in (7.9) shows that τ ( δ ) > τ (0) at the leadingorder in δ , proving our claim. - - - - - μ - - τ Figure 8:
The function τ ( µ ) in the [H1,E2] and [H1,E2 ] branches of solutions, for2 ‘ = 3 ‘ and λ = 3 / ‘ < λ = √ / ‘ . The red line indicates the bound τ ∗ belowwhich the hot solution ceases to exist. A typical τ ( µ ) in the [H1,E2] and [H1,E2 ] branch of solutions, and for λ < λ , is plotted in figure 8. The function grows initially as µ moves awayfrom 1, reaches a maximum value and then turns around and goes to zero as µ → −∞ . The red line indicates the limiting value τ ∗ below which there is nohot solution. For τ slightly above τ ∗ we see that there are three coexistingblack holes, the hot and two warm ones. For τ < τ ∗ , on the other hand, onlyone warm solution survives, but it describes a wall at a finite distance fromthe horizon. Whether this is the dominant solution or not, the transition istherefore necessarily first order. 32 Exotic fusion and bubbles
Before proceeding to the phase diagram, we pause here to discuss the peculiarphenomenon announced earlier, in section 6.2. This arises in the limits γ = L /L → γ → ∞ , with L + L and T kept fixed. In these limits theconformal boundary of one slice shrinks to a point.Consider for definiteness the limit L →
0. In the language of the dualfield theory the interface and anti-interface fuse in this limit into a defect ofCFT . The naive expectation, based on free-field calculations [65, 101, 102],is that this is the trivial (or identity) defect. Accordingly, the green interiorslice should recede to the conformal boundary, leaving as the only remnanta (divergent) Casimir energy.We have found that this expectation is not always borne out as we willnow explain.Suppose first that the surviving CFT is in its ground state, and thatthe result of the interface-antiinterface fusion is the expected trivial defect.The geometry should in this case approach global AdS of radius ‘ , with M tending to − (2 π/L ) , see section 2. Furthermore, σ + should go to infinity inorder for the green slice to shrink towards the ultraviolet region. As seen fromeqs. (4.15, 4.14) this requires M → −∞ , so that µ should vanish togetherwith γ . This is indeed what happens in much of the ( λ, ‘ , ‘ ) parameterspace. One finds µ ∼ γ →
0, a scaling compatible with the expectedCasimir energy ∼ /L .Nevertheless, sometimes γ vanishes at finite µ . In such cases, as µ → µ the green slice does not disappear even though its conformal boundary hasshrunk to a point. This is illustrated by the left figure 9, which shows a staticbubble of ‘true vacuum’ suspended from a point on the boundary of the ‘falsevacuum’. To convince ourselves that the phenomenon is real, we give ananalytic proof in appendix D of the existence of such suspended bubbles inat least one region of parameters ( ‘ > ‘ and λ ≈ λ min > γ is unique, there is no other competingsolution. In the example of appendix D, in particular, γ is finite and negativeat µ = 0.In the language of field theory this is a striking phenomenon. It impliesthat interface and anti-interface do not annihilate, but fuse into an exotic de-fect generating spontaneously a new scale in the process. This is the blueshiftat the tip of the bubble, σ + ( µ , L ), or better the corresponding frequency These are static solutions, not to be confused with ‘bags of gold’ which are cosmologiesglued onto the backside of a Schwarzschild-AdS spacetime, see e.g. [30, 103]. The phe-nomenon is reminiscent of spacetimes that realize ‘wedge’ or codimension-2 holography,like those in refs. [104, 105]. L ≈ 0 L L = 0 Figure 9:
Left: A bubble of true vacuum that survives inside the false vacuum despitethe fact that its conformal boundary shrinks to a point. Right: A bubble of false vacuumwith λ = λ inscribed between the boundary and the horizon of a black hole. scale r ( σ + ) in the D(efect)CFT.The phenomenon is not symmetric under the exchange 1 ↔
2. Staticbubbles of the false vacuum (pink) spacetime inside the true (green) vacuumdo not seem to exist. We proved this analytically for λ < λ , and numericallyfor all other values of the tension. We have also found that the suspendedgreen bubble can be of type E1 , i.e. have a center. The redshift factor g tt inside the bubble can even be lower than in the surrounding space, so thatthe bubble hosts the excitations of lowest energy. We did not show thisanalytically, but the numerical evidence is compeling.Do suspended bubbles also exist when the surrounding spacetime containsa black hole ? The answer is affirmative as one can show semi-analyticallyby focussing on the region λ ≈ λ . We have seen in the previous section thatnear this critical tension there exist warm solutions of type [H1,E2 ] with thewall arbitrarily close the horizon. Let us consider the function τ ( µ, λ ) givenin this branch of solutions by eqs. (6.12b) and (6.11) (with S = E1 ). Thisis a continuous function in both arguments, so as λ increases past λ , τ (1)goes from positive to negative with the overall shape of the function varyingsmoothly. This is illustrated in figure 10, where we plot τ ( µ ) for λ slightlybelow and slightly above λ . It should be clear from these plots that for λ > λ (the plot on the right) τ vanishes at a finite µ ≈
1. This is a warmbubble solution, as advertized.We have found more generally that warm bubbles can be also of type E1 , thus acting as a suspended Faraday cage that protects inertial observersfrom falling towards the horizon of the black hole. Contrary, however, towhat happened for the ground state, warm bubble solutions are not unique.34 - - μ - - τ - - - μ - - - - - τ Figure 10:
Plots of the function τ ( µ ) in the [H1,E2] or [H1,E2 ] branch of solutions for ‘ /‘ = 1 .
5. The critical tension is λ ‘ ≈ .
12. The curve on the left is for λ ‘ = 1 . λ ‘ = 1 . There is always a competing solution at µ → −∞ , and it is the dominantone by virtue of its divergent negative Casimir energy. A stability analysiswould show if warm bubble solutions can be metastable and long-lived, butthis is beyond our present scope.As for warm bubbles of type [X, H1] , that is with the black hole in thefalse-vacuum slice, these also exist but only if ‘ < ‘ . Indeed, as we willsee in a moment, when ‘ > ‘ the wall cannot avoid a horizon located onthe false-vacuum side.Finally simple inspection of fig. 10 shows that by varying the tension, thebubble solutions for λ > λ go over smoothly to the hot solution at λ = λ .At this critical tension the bubble is inscribed between the horizon and theconformal boundary, as in figure 9. This gives another meaning to λ : Onlywalls with this tension may touch the horizon without falling inside. In this last section of the paper we present numerical plots of the phasediagram of the model. We work in the canonical ensemble, so the variables arethe temperature and volumes, or by scale invariance two of the dimensionlessratios defined in (6.1). We choose these to be τ = τ + τ = T ( L + L ) and γ = L /L . The color code is as in fig. 6.We plot the phase diagram for different values of the action parameters ‘ , ‘ , λ . Since our analysis is classical in gravity, Newton’s constant G playsno role. Only two dimensionless ratios matter, for instance b := ‘ ‘ = c c ≥ κ := λ‘ ∈ ( b − , b + 1) . (9.1) Dimensionless in gravity, not in the dual ICFT. b = 1 corresponds to a defect CFT, while b (cid:29) ovewhelm thoseof CFT . The true vacuum approaches in this limit the infinite-radius AdS,and/or the false vacuum approaches flat spacetime. The critical tension λ corresponds to κ = √ b − µ in terms of theboundary data ( γ, τ ) and for all types of slice pair, and compared their freeenergies when solutions of different type coexist. As explained in the intro-duction, although the interpretation is different, our diagrams are related tothe ones of Simidzija and Van Raamsdonk [29] by double-Wick rotation (spe-cial to 2+1 dimensions). Since time in this reference is non-compact, onlythe boundaries of our phase diagrams, at γ = 0 or γ = ∞ , can be compared.The roles of thermal AdS and BTZ are also exchanged Consider first b = 1. By symmetry, we may restrict in this case to γ ≥ γ, τ ) plane for a very light( κ = 0 .
03) and a very heavy ( κ = 1 .
8) domain wall. For the light, nearlytensionless, wall the phase diagram approaches that of a homogeneous CFT.The low- T solution is single-center, and the Hawking-Page (HP) transitionoccurs at τ ≈
1. Light domain walls follow closely geodesic curves, and avoidthe horizon in a large region of parameter space. κ = κ = Figure 11:
Phase diagrams of a very light (left) and a very heavy (right) domain wallbetween degenerate vacua ( b = 1). The horizontal and vertical axes are γ and τ . Thebroken line in the left diagram separates solutions of type [H1, E2 ] and [H1,E2] that onlydiffer in the sign of the energy of the horizonless slice. The color code is as in fig. 6. One can compute this phase diagram analytically by expanding in powers of λ . T , and the warmphase recedes to L (cid:29) L . Furthermore, both the cold and the warm solu-tions have now an additional AdS restpoint. This confirms the intuition thatheavier walls repel probe masses more strongly, and can shield them fromfalling inside the black hole.The transition that sweeps away this AdS restpoint is shown explicitlyin the phase diagrams of figure 12. Recall from the analysis of section 7.2that in the low- T phase such transitions happen for λ < /‘ = ⇒ κ < b = 1.Furthermore, the transitions take place at the critical mass ratios µ ∗ j , givenby eq. (7.2). Since in cold solutions the relation between µ and γ is one-to-one, the dark-light blue critical lines are lines of constant γ . These statementsare in perfect agreement with the findings of fig. 12. κ = κ = Figure 12:
Phase diagrams for intermediate-tension walls exhibiting sweeping transi-tions. On the left a restpoint of the vacuum solution is swept away as γ increases beyonda critical value. On the right the same happens in the warm solution but for decreasing γ . Warm solutions of type [H1,E1] , respectively [E1,H1] , exist for tensions λ > /‘ = ⇒ κ > b , respectively λ > /‘ = ⇒ κ >
1. In the case of a defectthese two ranges coincide. The stable black hole forms in the larger of thetwo slices, i.e. for γ > j = 1 slice. The sweeping transition occursat the critical mass ratio µ ∗ = ( b − κ ) − , which through eqs. (6.11) and(6.12a) corresponds to a fixed value of τ . Since τ = τ (1 + γ ), the criticalorange-yellow line is a straight line in the ( γ, τ ) plane, in accordance againwith the findings of fig. 12.A noteworthy fact is the rapidity of these transitions as function of κ . For κ a little below or above the critical value the single-center cold, respectively37arm phases almost disappear. Note also the cold-to-warm transitions arealways near τ ≈
1. This is the critical value for Hawking-Page transitionsin the homogeneous case, as expected at large γ when the j = 1 slice coversmost of space.The critical curves for the cold-to-hot and warm-to-hot transitions alsolook linear in the above figures, but this is an illusion. Since the transitionsare first order we must compare free energies. Equating for example the hotand cold free energies gives after some rearrangements (and with ‘ = ‘ := ‘ )2 π τ + 2 ‘ log g I = 12 τ | M | L (1 + µγ ) . (9.2)Now | M | L can be expressed in terms of µ through eq. (6.8, 6.9a), and µ inthe cold phase is a function of γ . Furthermore log g I /‘ = 4 π tanh − ( κ/
2) isconstant, see eq. (4.9), and τ = τ / (1 + γ − ). Thus (9.2) can be written as arelation τ = τ hc ( γ ), and we have verified numerically that τ hc is not a linearfunction of γ . Figure 13 presents the phase diagram in the case of non-degenerate AdSvacua, b = ‘ /‘ = c /c = 3, and for different values of the tension inthe allowed range, κ ∈ (2 , γ → γ − symmetry, γ herevaries between 0 to ∞ . To avoid squeezing the γ ∈ (0 ,
1) region, we use forhorizontal axis α := γ − γ − . This is almost linear in the larger of γ or γ − ,when either of these is large, but the region γ ≈ γ <
1. This shows that it is impossible to keepthe wall outside the black hole when the latter forms on the false-vacuumside. From the perspective of the dual ICFT, see section 7.1, the absenceof [X,H1] -type solutions means that no interfaces, however heavy, can keepCFT in the confined phase if CFT (the theory with larger central charge)has already deconfined.We suspect that this is a feature of the thin-brane model which does notallow interfaces to be perfectly-reflecting [34].Warm solutions with the horizon in the pink slice appear to altogetherdisappear above the critical ratio of central charges b c = 3. The boundary This critical value was also noticed in ref. [29], who also note that multiple branes canevade the bound confirming the intuition that it is a feaure specific to thin branes. As amatter of fact, although [X,H1] solutions do exist for b < γ , outside the range of our numerical plots, unless b is very close to 1. - - - - κ = - - - - - κ = - - - - - κ = - - - - - κ = Figure 13:
Phase diagrams for b = 3, and values of the tension that increase from thetop-left figure clockwise. The horizontal and vertical axes are α := γ − γ − and τ . Thebroken red cruve is the bound τ = τ ∗ (1 + γ ) below which there is no hot solution. Notethe absence of a warm phase in the left ( γ <
1) region of the diagrams. For the heaviestwall all non-hot solutions are two-center. conditions corresponding to topologies of type [X,H1] are given by eqs. (6.14).We plotted the right-hand side of the second condition (6.14) for differentvalues of λ and µ in their allowed range, and found no solution with positive τ for b >
3. Analytic evidence for the existence of a strict b c = 3 bound canbe found by considering the limit of a maximally isolating wall, λ ≈ λ max ,and of a shrinking green slice ˆ µ → −∞ . In this limit, the right-hand side of(6.14) can be computed in closed form with the result τ (ˆ µ ) = π √− ˆ µ − s ‘ ‘ ! + subleading . (9.3)We took X=E1 as dictated by the analysis of sweeping transitions, see section7.2 and in particular eq. (7.5). This limiting τ (ˆ µ ) is negative for b >
3, andpositive for b < [E1,H1] solutions do exist, as claimed.An interesting corollary is that end-of-the-world branes cannot avoid thehorizon of a black hole, since the near-void limit, ‘ (cid:28) ‘ , is in the rangethat has no [X,H1] solutions. 39 .3 Unstable black holes The phase diagrams in figs. 11, 12, 13 show the solution with the lowest freeenergy in various regions of parameter space. Typically, this dominant phasecoexists with solutions that describe unstable or metastable black-holes whichare ubiquitous in the thin-wall model. Figure 14 shows the number of black hole solutions in the degenerate case, b = 1, for small, intermediate and large wall tension, and in different regionsof the ( τ, γ ) parameter space. The axes are the same as in figs. 11 and 12but the range of γ is halved. At sufficiently high temperature the growinghorizon captures the wall, and the only solution is the hot solution. We see (cid:1) = Out[ (cid:1) ]= (cid:1) = (cid:1) = Figure 14:
The number of independent black hole solutions in the ( γ, τ ) parameter spacefor b = 1, and three values of the tension ( κ = 0 .
2; 1 .
1; and 1 . however that in a large region of intermediate temperatures the hot solutioncoexists with two warm solutions. Finally at very low temperature the hotsolution coexists with four other black-hole solutions, two on either side ofthe wall. The dominant phase in this region is vacuum, so the black holesplay no role in the canonical ensemble.The hot solution exists almost everywhere, except when λ < λ and τ = τ (1 + γ ) < τ ∗ (1 + γ ) with τ ∗ given by eq. (6.5). It has positive specificheat even when it is not the dominant phase. For warm black holes, on For a similar discussion of deformed JT gravity see ref. [66]. Note that in the absenceof a domain wall, the only static black hole solution of pure Einstein gravity in 2+1dimensions is the non-spinning BTZ black hole. λ ≈ λ . Simple inspection of fig. 8 shows that in some range τ ∗ < τ < τ max2 the hot solution coexists with two nearby warm solutions. At the maximum τ max2 , where dτ /dµ = 0, the warm solutions merge and then disappear. Sincethe black hole is in the j = 1 slice, M = (2 πT ) and their energy reads E [warm] = 12 ( ‘ M L + ‘ M L ) = 2 π T L (cid:16) ‘ γ + ‘ µ (cid:17) . (9.4)Taking a derivative with respect to T with L , L kept fixed we obtain ddT E [warm] = 2 T E [warm] + 2 π T L ‘ dµdτ . (9.5)Near τ max2 the dominant contribution to this expression comes from thederivative dµ/dτ which jumps from −∞ to + ∞ . It follows that the warmblack hole with the higher mass has negative specific heat, and should decayto its companion black hole either classically or in the quantum theory. It would be very interesting to calculate this decay process, but we leavethis for future work.One last comment concerns transitions from the double-center vacuumgeometries, of type [E1,E1] , to warm solutions where the wall avoids thehorizon. One can ask what side of the wall does the black hole choose. Anatural guess is that it forms in the deepest of the two AdS wells. Therelative depth is the ratio of blueshift factors at the two rest points, R := vuut g tt | r =0 g tt | r =0 = ‘ ‘ q µ ( γ ) . (9.6)One expects the black hole to form in the j = 1 (green) slice if R < j = 2 (red) slice if R >
1. Our numerical plots confirmed in all casesthis expectation.
10 Outlook
One urgent question, already noted in the introduction, is how much of thisanalysis will survive in top-down interface models, where gravitating domain We have verified numerically that the black holes with negative specific heat are neverthe ones with lowest free energy, a conclusion similar to the one reached in deformed JTgravity in ref. [66].
Aknowledgements
We are grateful to Mark Van Raamsdonk for his critical reading of apreliminary draft of this paper and for many useful comments. Many thanksalso to Panos Betzios, Shira Chapman, Dongsheng Ge, Elias Kiritsis, IoannisLavdas, Bruno Le Floch, Emil Martinec, Olga Papadoulaki and GiuseppePolicastro for discussions during the course of this work.
A Renormalized on-shell action
The Euclidean action of the holographic-interface model, in units 8 πG = 1,is the sum of bulk, brane, boundary and corner contributions, see e.g. [107] I gr = − R S d x √ g ( R + ‘ ) − R S d x √ g ( R + ‘ ) + λ R W d s √ ˆ g w + R ∂ S d s √ ˆ g K + R ∂ S d s √ ˆ g K + R C ( θ − π ) √ ˆ g c + c . t . (A.1)where the counterterms, abbreviated above by c.t., read [108]c . t . = 1 ‘ Z B q ˆ g + 1 ‘ Z B q ˆ g − Z B ∩ B ( θ + θ ) q ˆ g c . (A.2)42ere S j are the spacetime slices whose boundary is the sum of the cutoffsurface B j and of the string worldsheet W , i.e. ∂ S j =B j ∪ W . The inducedmetrics are denoted by hats. The K j are traces of the extrinsic curvatureson each slice computed with the inward-pointing normal vector. Finally, inaddition to the standard Gibbons-Hawking-York boundary terms, one mustadd the Hayward term [107, 109] at corners of ∂ S j denoted by C. There isat least one such corner at the cutoff surface, B ∩ B , where θ − π is the sumof the angles θ j defined in figure 4.Let us break the action into an interior and a conformal boundary term, I gr = I int + I B , with the former including contributions from the worldsheetW. Using the field equations R j = − /‘ j and K | W + K | W = − λ , and thevolume elements that follow from eqs. (3.1) and (4.1, 4.2), √ g j d x = ‘ j r i dr j dx j dt and q ˆ g w d s = q f g dσdt , we can write the interior on-shell action as follows : I int = 2 ‘ Z Ω r dr dx dt + 2 ‘ Z Ω r dr dx dt − λ Z W q f g dσdt . (A.3)We have been careful to distinguish the spacetime slice S j from the coordinatechart Ω j , because we will now use Stoke’s theorem treating Ω j as part of flatEuclidean space, X j =1 , ‘ j Z Ω j r j dr j dx j dt = X j =1 , ‘ j I ∂ Ω j r j (ˆ r j · d ˆ n j ) dt , (A.4)with d ˆ n j dt the surface element on the boundary ∂ Ω j . Crucially, the boundaryof Ω j may include a horizon which is a regular interior submanifold of theEuclidean spacetime and is not therefore part of ∂ S j . In particular, there isno Gibbons-Hawking-York contribution there.The boundary integral in eq. (A.4) receives contributions from the threepieces of ∂ Ω , : the cutoff surface B ∪ B , the horizon if there is one, and theworldsheet W . Conveniently, this last term precisely cancels the third term in(A.3) by virtue of the Israel-Lanczos equation (4.3). Thus, after all the dusthas settled, the action can be written as the sum of terms evaluated either atthe black-hole horizon or at the cutoff. After integrating over periodic timethe interior part of the action, eq. (A.3) , reads I int = 1 ‘ T (cid:20) r ∆ x (cid:21) B Hor + 1 ‘ T (cid:20) r ∆ x (cid:21) B Hor (A.5) These play no role here, but they can be important in the case of string junctions. X ] ba = X | b − X | a , and X | a for X evaluated at a . If the slice S j does not contain a horizon the correspondingcontribution is absent.We now turn to the conformal-boundary contributions from the lower linein the action (A.1). For a fixed- r j surface, the inward-pointing unit normalexpressed as a 1-form is n j = − dr j / q r j − M j ‘ j . One finds after a littlealgebra (we here drop the index j for simplicity) K xx = K tt = − r‘ √ r − M ‘ = ⇒ q ˆ g K = − ‘ (2 r − M ‘ ) . (A.6)Combining the Gibbons-Hawking-York terms and the counterterms gives I B = 1 ‘ T ( r q r − M ‘ − r + M ‘ ) ∆ x (cid:12)(cid:12)(cid:12)(cid:12) B + (1 → . (A.7)Expanding for large cutoff radius, r j | B j → ∞ , and dropping the terms thatvanish in the limit we obtain I B = 1 ‘ T ( − r + 12 M ‘ ) ∆ x (cid:12)(cid:12)(cid:12)(cid:12) B + (1 → . (A.8)Upon adding up (B.6) and (A.8) the leading divergent term cancels, givingthe following result for the renormalized on-shell action : I gr = M ‘ T (cid:16) L − x (cid:12)(cid:12)(cid:12) Hor (cid:17) + M ‘ T (cid:16) L − x (cid:12)(cid:12)(cid:12) Hor (cid:17) . (A.9)We used here the fact that ∆ x j | B j = L j , and that r j = M j ‘ j at the horizonwhen one exists. We also used implicitly the fact that for smooth strings theHayward term receives no contribution from the interior and is removed bythe counterterm at the boundary.As a check of this on-shell action let us compute the entropy. Using ourformula for the internal energy h E i = ( M ‘ L + M ‘ L ), see section 2,and I gr = h E i /T − S we find S = 1 T (cid:16) M ‘ ∆ x (cid:12)(cid:12)(cid:12) Hor + M ‘ ∆ x (cid:12)(cid:12)(cid:12) Hor (cid:17) = 4 π T (cid:16) ‘ ∆ x (cid:12)(cid:12)(cid:12) Hor + ‘ ∆ x (cid:12)(cid:12)(cid:12) Hor (cid:17) = A (horizon)4 G . (A.10)In the lower line we used the fact that M j = (2 πT ) and r H j = 2 πT ‘ j forslices with horizon, plus our choice of units 8 πG = 1. The calculation thusreproduces correctly the Bekenstein-Hawking entropy.44 Opening arcs as elliptic integrals
In this appendix we express the opening arcs, eqs. (4.17), in terms of completeelliptic integrals of the first, second and third kind, K ( ν ) = Z dy q (1 − y )(1 − νy ) (B.1) E ( ν ) = Z √ − νy dy √ − y . (B.2) Π ( u, ν ) = Z dy (1 − uy ) q (1 − y )(1 − νy ) . (B.3)Consider the boundary conditions (4.17a). The other conditions (4.17b, 4.17c)differ only by the constant periods or horizon arcs, P j or ∆ x j | hor . Insertingthe expression (4.16) for x gives L = − Z ∞ σ + ‘ dσ ( σ + M ‘ ) ( λ + λ ) σ + M − M q Aσ ( σ − σ + )( σ − σ − ) , (B.4)and likewise for L . The roots σ ± are given by eqs. (4.14, 4.15). We assumethat we are not in the case [H2, H2] where M = M >
0, nor in the fringecase σ + = − M j ‘ j when the string goes through an AdS center. These caseswill be treated separately.Separating the integral in two parts, and trading the integration variable σ for y , with y := σ + /σ , we obtain L = − ‘ √ A σ + " M − M M ‘ Z dy q (1 − y )(1 − νy )+ (cid:18) ( λ + λ ) − M − M M ‘ (cid:19) Z y dy (1 − u y ) q (1 − y )(1 − νy ) (B.5)where ν = σ − /σ + and u = − M ‘ /σ + . Identifying the elliptic integralsfinally gives L = − ‘ √ A σ + " M − M M ‘ (cid:18) K ( ν ) − Π ( u , ν ) (cid:19) + ( λ + λ ) Π ( u , ν ) , (B.6)45nd a corresponding expression for L L = − ‘ √ A σ + " M − M M ‘ (cid:18) K ( ν ) − Π ( u , ν ) (cid:19) + ( λ − λ ) Π ( u , ν ) (B.7)with u = − M ‘ /σ + . The prefactors in (B.6) diverge when M → Π ( u , ν ) around u = 0. In this limit L ( M = 0) = − ‘ √ A σ + " M σ − ( E ( ν ) − K ( ν )) + ( λ + λ ) K ( ν ) (B.8a)and similarly L ( M = 0) = − ‘ √ A σ + " M σ − ( E ( ν ) − K ( ν )) + ( λ − λ ) K ( ν ) (B.8b)with E ( ν ) the complete elliptic integral of the second kind.The M = M > M j = (2 πT ) , σ + = 0 and σ − = − (4 πT λ ) /A . The integrals (4.17c)simplify to elementary functions in this case: L − ∆ Hor1 = − ‘ ( λ + λ ) q A | σ − | Z ∞ ds ( s + a ) √ s + 1 | {z } = √ − a arctanh( √ − a )with a = A‘ / λ . Using the expression (4.14) for A , and going through thesame steps for j = 2, gives after a little algebra L − ∆ Hor1 = − πT tanh − ‘ ( λ + λ )2 λ ! , (B.9a) L − ∆ Hor2 = − πT tanh − ‘ ( λ − λ )2 λ ! . (B.9b)Interestingly, since ∆ Hor2 must be positive,
T L is bounded from below in therange λ < λ as discussed in section 6.2.In the high-temperature phase the on-shell action, eq. (A.9), reads I (high − T)gr = 4 π T (cid:20) −
12 ( ‘ L + ‘ L ) + ‘ ( L − ∆ Hor1 ) + ‘ ( L − ∆ Hor2 ) (cid:21) . (B.10)Using the expressions (B.9) and rearranging the arc-tangent functions gives I (high − T)gr := ET − S = − π T ( ‘ L + ‘ L ) − log g I (B.11)where the interface entropy S = log g I is given by eq. (4.9).46 Sweeping is continuous
In this appendix we show that sweeping transitions are continuous.We focus for definiteness on the sweeping of the j = 2 AdS center at zerotemperature (all other cases work out the same). The transition takes placewhen µ crosses the critical value µ ∗ given by eq. (7.2). Setting µ = µ ∗ (1 − δ )in expression (6.9b) gives f ( µ ) = ‘ √ A Z ∞ s + ds ( λ − λ )( s − µ‘ ) + δ ( s − µ‘ ) q As ( s − s + )( s − s − )= 2 ‘ ( λ − λ ) √ As + K (cid:16) s − s + (cid:17) + ‘ δ √ A Z ∞ s + ds ( s − µ‘ ) q s ( s − s + )( s − s − ) | {z } J . (C.1)The first term is continuous at δ = 0, but the second requires some carebecause the integral J diverges. This is because for small δs + − µ‘ = δ λ µ ∗ + O ( δ ) , (C.2)as one finds by explicit computation of the expression (6.16). If we set δ = 0, J diverges near the lower integration limit. To bring the singular behaviorto 0 we perform the change of variable u = s − s + , so that J = Z ∞ du ( u + δ / λ µ ∗ ) q ( u + s ∗ + )( u + s ∗ + − s ∗− ) , (C.3)where we kept only the leading order in δ , and s ∗± are the roots at µ = µ ∗ .Since s ∗ + and s ∗ + − s ∗− are positive and finite, the small- δ behavior of theintegral is (after rescaling appropriately u ) J = 4 λ | µ ∗ || δ | q s ∗ + ( s ∗ + − s ∗− ) Z ∞ duu + 1 | {z } π/ + finite . (C.4)Inserting in expression (C.1) and doing some tedious algebra leads finally toa discontinuity of the function f ( µ ) equal to sign( δ ) π/ √ µ ∗ . This is preciselywhat is required for L , eq. (6.8), to be continuous when the red ( j = 2) slicegoes from type E1 at negative δ to type E2 at positive δ .47 Bubbles exist
We show here that the bubble phenomenon of section 8 is indeed realized ina region of the parameter space of the holographic model.This is the region of non-degenerate gravitational vacua ( ‘ strictly biggerthan ‘ ) and a sufficiently light domain wall. Specifically, we will show thatfor λ close to its minimal value, λ min , the arc L ( µ = 0) is negative, so thewall self-intersects and µ is necessarily finite.Let λ = λ min (1 + δ ) with δ (cid:28)
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