Probing Hawking radiation through capacity of entanglement
Kohki Kawabata, Tatsuma Nishioka, Yoshitaka Okuyama, Kento Watanabe
PPrepared for submission to JHEP
YITP-21-08
Probing Hawking radiation through capacity ofentanglement
Kohki Kawabata, a Tatsuma Nishioka, b Yoshitaka Okuyama a and Kento Watanabe c a Department of Physics, Faculty of Science, The University of Tokyo,Bunkyo-ku, Tokyo 113-0033, Japan b Yukawa Institute for Theoretical Physics, Kyoto University,Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan c Center for Quantum Mathematics and Physics (QMAP),Department of Physics, University of California, Davis, CA 95616 USA
Abstract:
We consider the capacity of entanglement in models related with the gravitational phasetransitions. The capacity is labeled by the replica parameter which plays a similar role to theinverse temperature in thermodynamics. In the end of the world brane model of a radiatingblack hole the capacity has a peak around the Page time indicating the phase transitionbetween replica wormhole geometries of different types of topology. Similarly, in a movingmirror model describing Hawking radiation the capacity typically shows a discontinuity whenthe dominant saddle switches between two phases, which can be seen as a formation of islandregions. In either case we find the capacity can be an invaluable diagnostic for a black holeevaporation process. a r X i v : . [ h e p - t h ] F e b ontents Black holes occupy a distinguished position not only as existing constitutes of our universebut also as a theoretical testing ground for searching a theory of quantum gravity that unifiesgeneral relativity and quantum mechanics. While there is no doubt that a black hole obeysthe laws of thermodynamics and owns the entropy proportional to the area of the horizon[1, 2] the thermodynamic character of black holes poses a longstanding paradox when theyare regarded as quantum mechanical objects that evolve unitarily in time. Taking quantumeffects into account, black holes begin to emit a thermal radiation and end up evaporating tonothing in a finite time [3]. If we throw objects there, we seemingly lose the information aboutthe objects because a thermal black hole radiation does not depend on what has fallen, andit evaporates completely. Hawking raised his famous information paradox in [4] by showingthe indefinite increase of radiation entropy for an evaporating black hole, which leads to thebreaking of the unitary evolution in a quantum gravity theory. To reconcile with the unitarity,Page considered the entanglement entropy for the radiation between the inside and outsideof evaporating black holes and suggested that the linear increase of the entropy caused bythe radiation at early time must stop at some typical timescale and start to decrease to zeroas black holes evaporate [5]. Reproducing the Page curve in a model of black holes withradiation is, however, still challenging and has led to considerable efforts so far (see e.g., [6]for an excellent review and references therein).The idea of associating entropy with geometric quantities was cultivated by Ryu andTakayanagi, who proposed a holographic formula equating the entanglement entropy for aspacial region in quantum field theory to the area of a minimal surface anchored on the– 1 –oundary in the dual gravitational theory [7, 8]. The Ryu-Takayanagi formula is a mani-festation of the generalized gravitational entropy [9] which derives from a gravitational pathintegral using the replica method for entanglement entropy: S A ≡ lim n → − n log Tr ρ nA . (1.1)In a field theory, Tr ρ nA is given by the partition function Z n on a manifold with singularsurface of deficit angle 2 π (1 − n ) along the boundary ∂A of a region A at some timeslice.In the bulk spacetime, Z n represents the partition function on a replica geometry with n boundaries and the singularity along ∂A extends to the bulk to form the Ryu-Takayanagisurface. There are numerous extensions of the formula including time-dependent theories[10, 11], quantum corrections [12], and R´enyi entropies [13]. The most general formulation[14] expands on the notion of the generalized second law [15] and states that the entropy ofthe boundary region A is given by the generalized entropy consisting of the area of a minimalquantum extremal surface (QES) and the entanglement entropy of a matter across the QES.Recently, a proposal for computing the entropy of quantum systems entangled with grav-itational systems, called the island formula was presented [16] and shown to hold in a certainclass of two-dimensional dilaton gravity coupled with matter of large degrees of freedom[17, 18] (see also [19] for a review). The derivation proceeds much like the Ryu-Takayanagiformula with a gravitational path integral method but a significant difference is there be-tween the two formulas. The Ryu-Takanayagi formula picks up only the replica geometrywith disconnected replicas while the island formula has a substantial contribution from thereplica wormhole with all replicas connected in gravitational regions, which instructs us toinclude a region, named an island, behind a black hole horizon as a part of the entanglingregion in calculating the entanglement entropy of the Hawking radiation. The island formulahas been exploited to resolve the information problem by successfully reproducing the Pagecurve in various types of black holes [20–30]. For further progress in replica wormholes andisland formula see also e.g., [31–73].The goal of this paper is to probe the Hawking radiation process through another quantuminformation measure known as the capacity of entanglement: C A ≡ lim n → n ∂ n log Tr ρ nA , (1.2)which is to entanglement entropy what the heat capacity is to thermal entropy when thereplica parameter n is seen as the inverse temperature [74, 75]. Some aspects of the capacityof entanglement were investigated both in holographic systems [75] and in field theories [74,76, 77], but it remains open to what extent the capacity can be a practical measure ofentanglement in a broader context. We will examine the characteristics of the capacity of One can always find a local solution of replica wormholes near the QES, but the global solutions are notguaranteed to exist in general (see e.g., [20] for discussion). – 2 –ntanglement in toy models of radiating black holes by taking A to be a region R outsidethe black holes, especially around the Page time (see figure 1). Since the contribution fromthe replica geometry with nontrivial topology is essential in the gravitational path integral,we expect the capacity of entanglement for the radiation is also capable of capturing thereplica wormholes, or equivalently the island region as in the island formula for entanglemententropy.At early time a fully disconnected replica wormhole dominates in the gravitational pathintegral. This explains the linear growth of the entropy for the radiation region R dueto the Hawking radiation while the geometry typically has a vanishing contribution to thecapacity (see section 2.1). On the other hand, a fully connected replica wormhole starts tocompete with the fully disconnected one around the Page time and take over at late time.Then the entropy either saturates or decreases depending on whether black holes are eternalor evaporating, resulting in a continuous Page curve. As for the capacity we will see it issensitive to the phase transition at the Page time, showing a crossover or discontinuity. Notethat the capacity does not equal the derivative of the entropy with respect to the radiationtime, but it will turn out to play a similar role as the latter. To our best knowledge, this isthe first case where the capacity of entanglement probes a phase transition of any kind.time S R C R ? Figure 1 . A typical shape of the entanglement entropy (orange) and an asymptotic form of thecapacity of entanglement (blue) at early and late times for black holes with the Hawking radiation.The entropy grows linearly and saturates after the Page time while the capacity is vanishing at earlytime and approaches some value at late time.
The structure of this paper is as follows. In section 2, we treat a quantum mechanicalmodel of evaporating black holes known as the end of the world (EOW) brane model intro-duced in [18] that is simple enough for us to perform the exact gravitational path integralwith all possible replica wormholes taken into account. We show that the capacity of en-tanglement shows a peak around the Page time due to the change of the dominant saddlefrom the replica geometries in the model. In section 3 we then move onto two-dimensionalholographic conformal field theories (CFTs) in moving mirrors which mimic radiating blackholes and reproduce the Page curves [78]. We find the capacity becomes discontinuous whenthe dominant saddle to the entropy switches between two phases, corresponding to black holewith and without an island region. Finally we devote section 4 to discussions and futuredirections. – 3 –
End of the world brane model
In this section, we will consider a toy 2 d gravity model of an evaporating black hole in theJackiw-Teitelboim (JT) gravity [79, 80] with an end of the world (EOW) brane entangledwith an auxiliary system [18]. In this simple model, island contributions to entanglemententropy in the auxiliary system are obtained from replica wormhole contributions to theR´enyi entropies. For large dimension of the systems, the replica wormholes contributions canbe calculated by summing up all planar topologies in the full gravitational path-integral asperformed in [18]. It can be done analytically in the microcanonical ensemble and numericallyin the canonical ensemble. For both cases, we will find smooth curves of the capacity. First of all, we will briefly review the simple gravity model [18] and discuss about asymptoticbehaviors of the entanglement entropy and the capacity of entanglement at early and latetime (sufficiently away from the Page time), where only the topology of replica wormholesmatters. More detailed analysis of replica geometries around Page time will be deferred tothe following subsections.In this section, we consider a planar approximation of the replica wormhole calculationin a simple 2 d gravity model consisting of a black hole in the JT gravity, a brane and anauxiliary system, discussed in [18] (dual to a pure state in the Sachdev-Ye-Kitaev model[81]). To model the early radiation of an evaporating black hole, we consider a quantum statedescribing a black hole system B of dimension dim H B = e S with a brane entangled with anauxiliary system R of dimension dim H R = k : | Ψ (cid:105) = 1 √ k k (cid:88) i =1 | ψ i (cid:105) B | i (cid:105) R . (2.1)Here we take an orthogonal basis | i (cid:105) R of the auxiliary system R and an unnormalized basis | ψ i (cid:105) B of the black hole system. The 2 d gravity system has two asymptotic 1 d boundaries on B and the brane (the left panel of figure 2). The boundary conditions are given by the inversetemperature β or the renormalized length on B and the brane tension µ . As the black hole B evaporates the auxiliary system R will collect more radiated particles. In that sense, wecan regard log( k e − S ) as time in the model.To discuss the island contributions in this model, we work on the replica method forthe R´enyi entropies S ( n ) R in the auxiliary system R . The basic object we calculate is the n th moment of the reduced density matrix ρ R = Tr B | Ψ (cid:105)(cid:104) Ψ | for R ,Tr ρ nR = 1( k e S ) n k (cid:88) i , ··· ,i n =1 (cid:104) ψ i | ψ i (cid:105) B · (cid:104) ψ i | ψ i (cid:105) B · · · (cid:104) ψ i n | ψ i (cid:105) B . (2.2)– 4 – i hyperbolicdiskEOWbrane Black holesystem B Auxiliarysystem R k ( Z ) k Z Z k Z Figure 2 . [Left] The geometry of (cid:104) ψ i | ψ j (cid:105) B in the EOW brane model. The hyperbolic disk (blue) hasthe asymptotic boundary and terminates on the EOW brane (orange). [Right] The replica geometriesfor the moment (2.3) of ρ R with n = 3. There are three ways to connect the EOW branes in thebulk region. The planar replica wormholes with l -dotted loops and m b -disconnected disk regions with b -asymptotic boundaries provide the factors k l (cid:81) m b ,b ( Z b ) m b respectively. This is divided by e nS so that Tr ρ R = 1. The products of the amplitudes (cid:81) k (cid:104) ψ i k | ψ i k +1 (cid:105) B isevaluated by the gravitational path integral on replicated manifolds with boundaries connect-ing some of the n -copies of the auxiliary system R . By taking the replica wormhole saddlesin the gravitational path integral, and considering large k, e S with a fixed ratio α ≡ k e − S ,we can pick up the replica wormholes with disk topology as dominant contributions (the rightpanel of figure 2). This is nothing but the planar approximation:Tr ρ nR ≈ k Z ) n (cid:20) k ( Z ) n + (cid:18) n (cid:19) · k Z ( Z ) n − + · · · + k n Z n (cid:21) = 1 k n − (cid:20) (cid:18) n (cid:19) · k Z ( Z ) + · · · + k n − Z n ( Z ) n (cid:21) . (2.3)where Z n is a partition function for a replica wormhole connecting n -copies of the auxiliarysystem R . Since the wormhole has disk topology, Z n ∝ e S and then the moment is a finiteseries of α . The n th term ∝ α n − corresponds to the fully connected replica wormhole withthe coarse-grained or black hole R´enyi entropy S ( n )BH ≡ − n log Z n ( Z ) n .Although many types of the replica wormhole configurations contribute to the momentat intermediate order of α , the entanglement entropy has two kinds of dominant phases inthe asymptotic limits, the fully disconnected ( α (cid:28)
1) and fully connected replica wormholephases ( α (cid:29) S R ≈ log k ( α (cid:28) ,S BH ≡ lim n → S ( n )BH ( α (cid:29) . (2.4)Here we take the limit n → α . Theentropy starts with a linear growth in log α and ends up with the saturation to the largevalue of S BH as the island formula suggests. Note that, as we will see later in this section,the intermediate parts of the Page curves are smoothed out in this model.– 5 –imilarly, we can find the capacity of entanglement (1.2) in the asymptotic limits: C R ≈ k Z ( Z ) ∝ α ( α (cid:28) ,C BH ≡ − ∂ n S ( n )BH (cid:12)(cid:12)(cid:12) n =1 ( α (cid:29) . (2.5)At early time, the capacity grows exponentially in log α from zero (or, if plotted in log k ,exponentially small initial value e − S for large S ). But, at late time, it decreases to the finalvalue C BH which can be nonzero.Note that, although we cannot draw any implication about the behavior of the capacityaround the Page time ( α ∼
1) from this simple discussion, as shown in the following sub-sections, the capacity takes the maximum around the Page time after the initial growth andthen decreases exponentially to the late time value C BH . In semiclassical regime the capacitycan be discontinuous due to the phase transition between the fully connected and fully dis-connected replica wormholes, but the curve should be smoothed out once the other replicageometries are taken into account. Even if the capacity does not exhibit a discontinuity itmay still be a good indicator for the phase transition between the disconnected (no island)and connected replica wormhole (island) phases.In the rest of this section, we will perform more detailed calculations of the entropyand capacity in the microcanonical and canonical ensembles. They incorporate the contri-butions from planer replica wormholes which smooth out the sharp transition from the fullydisconnected to the fully connected phases around the Page time. To probe the entropy and capacity around the Page time, we need the explicit forms ofthe partition functions Z i ( i = 1 , · · · , n ) for the replica wormholes. To this end, it will beconvenient to use the density of states D ( λ ) of the eigenvalue λ for the reduced density matrix ρ R , D ( λ ) = lim (cid:15) → +0 π Im[ R ( λ − i (cid:15) )] , (2.6)where R ( λ ) = (cid:80) i R ii ( λ ) is the trace of the resolvent matrix R ij ( λ ), R ij ( λ ) = (cid:18) λ − ρ R (cid:19) ij = 1 λ δ ij + ∞ (cid:88) n =1 λ n +1 ( ρ nR ) ij , (2.7)which can be determined via the Schwinger-Dyson equation for large k and S with fixedratio α or in the planar limit. By using the density of states D ( λ ), the entanglement entropy S R and capacity C R is expressed as integrals of the eigenvalues λ : S R = − (cid:90) ∞ d λ D ( λ ) λ log λ , (2.8) C R = (cid:90) ∞ d λ D ( λ ) λ (log λ ) − ( S R ) . (2.9)– 6 –n the microcanonical ensemble, we fix the energy E in the asymptotic region rather thanthe temperature or the renormalized length β . In a small energy band around E with theentropy S ∼ log(number of states in the energy band), the Schwinger-Dyson equation for R ( λ ) is simplified to a quadratic equation and the solution gives the density of states D ( λ )in the microcanonical ensemble [18]: D ( λ ) = k e S πλ (cid:115)(cid:18) λ − (cid:16) k − − e − S (cid:17) (cid:19) · (cid:18)(cid:16) k − + e − S (cid:17) − λ (cid:19) + δ ( λ ) ( k − e S ) θ ( k − e S )= k · (cid:20) πα ˜ λ (cid:113) (˜ λ − (1 − √ α ) ) · ((1 + √ α ) − ˜ λ ) + δ (˜ λ ) (cid:18) − α (cid:19) θ ( α − (cid:21) ≡ k · ˜ D (˜ λ ) . (2.10)Here we rescaled the variables as ˜ λ = k λ and denote the ratio of the dimensions of thesystems in the energy band as α = k e − S . For (1 − √ α ) ≤ ˜ λ ≤ (1 + √ α ) , the first term in˜ D (˜ λ ) takes real values. And the second term localized at ˜ λ = 0 gives us the normalization ofthe density of states as follows, (cid:90) ∞ d λ D ( λ ) = k (cid:90) ∞ d˜ λ ˜ D (˜ λ ) = k , (cid:90) ∞ d λ λ D ( λ ) = (cid:90) ∞ d˜ λ ˜ λ ˜ D (˜ λ ) = 1 . (2.11)For α = k e − S <
1, we have k -states distributed over (1 − √ α ) ≤ ˜ λ ≤ (1 + √ α ) . For α > e S -states in the same region and ( k − e S )-states at λ = 0. The density of statesis nothing but the entanglement spectrum of a subsystem of dimension k in a random stateof total dimension k e S in the planar approximation [82]. The capacity of entanglement forrandom pure states was also examined in [76].The density of states (2.10) gives us express the entropy (2.8) and the capacity (2.9), S R = S + log α + (cid:90) d˜ λ ˜ D (˜ λ ) ( − ˜ λ log ˜ λ ) , (2.12) C R = (cid:90) d˜ λ ˜ D (˜ λ ) ˜ λ (log ˜ λ ) − (cid:18)(cid:90) d˜ λ ˜ D (˜ λ ) ˜ λ log ˜ λ (cid:19) , (2.13)as integrals of ˜ λ . As we will see soon, the entropy and capacity give us smooth curves inlog α = log k − S with the expected asymptotics.In the case of the microcanonical ensemble, we can compute analytically the R´enyi en-tropy S ( n ) R , which is now given by S ( n ) R − S = log α + 11 − n log (cid:34)(cid:90) (1+ √ α ) (1 −√ α ) d˜ λ ˜ D (˜ λ ) ˜ λ n (cid:35) . (2.14)– 7 –y changing the variable as λ (cid:48) = (˜ λ − (1 −√ α ) ) / (4 √ α ), which takes real values for 0 ≤ λ (cid:48) ≤ n th moment by the hypergeometric functions: (cid:90) (1+ √ α ) (1 −√ α ) d˜ λ ˜ D (˜ λ ) ˜ λ n = (1 − √ α ) n − π (cid:90) d λ (cid:48) (cid:112) λ (cid:48) (1 − λ (cid:48) ) (cid:18) √ α (1 − √ α ) λ (cid:48) (cid:19) n − = (1 − √ α ) n −
1) 2 F (cid:18) − n,
32 ; 3; − √ α (1 − √ α ) (cid:19) = F (1 − n, − n ; 2; α )= α n − F (cid:18) − n, − n ; 2; 1 α (cid:19) . (2.15)Here we used some formulas for the linear and quadratic transformations of the hypergeo-metric functions. Note that the n th moment (2.15) is the special case of the result for therandom pure states obtained in [76]. This is controlled by only one parameter α as expectedin the planar limit. For 0 ≤ α ≤
1, the third expression in (2.15) is a convergent power seriesof α and it can be expanded in ( n −
1) as follows: F (1 − n, − n ; 2; α )= 1 + ( n − α − ( n − (cid:20) α − α α log(1 − α ) − ( α ) (cid:21) + O (( n − ) . (2.16)For 1 ≤ α , the last line of (2.15) gives us a similar expansion by replacing α to 1 /α in theabove expansion. Then the R´enyi entropy (2.14) is expanded as S ( n ) R − S = log α − α − ( n − · (cid:20) − − α − α − − α α log(1 − α ) + 2 Li ( α ) (cid:21) + · · · (0 < α ≤ , − α − ( n − · (cid:20) − − α − α − α − α log (cid:18) α − α (cid:19) + 2 Li (cid:18) α (cid:19)(cid:21) + · · · (1 ≤ α ) . (2.17)Taking the limit n → S R , S R − S = lim n → ( S ( n ) R − S ) = log α − α < α ≤ , − α (1 ≤ α ) , (2.18)which is smooth around the Page time ( α ∼
1) and has the desired asymptotic behaviors, S R ≈ { log k ( α (cid:28) , S ( α (cid:29) } (see the left plot in figure 3).– 8 – Figure 3 . The entanglement entropy S R (2.18) [Left] and capacity of entanglement C R (2.19) [Right]for the EOW brane model with S = 2 in the microcanonical ensemble. The capacity has a peak at thePage time log k = S , which clearly shows the crossover from the fully disconnected to fully connectedreplica wormhole solutions describing the Hawking radiation in an evaporating black hole. Finally we can find a smooth curve of the capacity of entanglement C R from the firstorder coefficient in (2.17): C R = − ∂ n S ( n ) R (cid:12)(cid:12)(cid:12) n =1 = − − α − α − − α α log(1 − α ) + 2 Li ( α ) (0 < α ≤ , − − α − α − α − α log (cid:18) α − α (cid:19) + 2 Li (cid:18) α (cid:19) (1 ≤ α ) . (2.19)As shown in the right plot of figure 3, it has a peak around the Page time ( α ∼ k ∼ S ) signaling the phase transition. In the asymptotic regions, the capacity decays tozero polynomially in α or exponentially in log α : C R ∼ α α (cid:28) , α (1 (cid:28) α ) . (2.20)The capacity becomes maximum at the Page time ( C R, max = π − ≈ .
54) and decaysto zero at the late time. Note that, since the capacity C R is defined via replica parameter n not temperature, the late time value C BH introduced in (2.5) is well-defined even in themicrocanonical ensemble, and does not necessarily equal to thermal capacity which we willsee in the canonical ensemble.From these observations we conclude that the sharp peak at the Page time is a gooddiagnostics of the phase transition between the fully connected and fully disconnected replicawormholes. We consider the canonical ensemble by fixing the inverse temperature β of the system ratherthan the energy. In the present model, β corresponds to the renormalized length at the– 9 –symptotic boundary. Since β appears with G N as G N β and we consider small G N or thesemi-classical regime, we will take small β .In the canonical ensemble, the density of state may not be simplified analytically. So wewill try to analyze the resolvent equation (2.21) numerically with some approximation. (Forthe detail analysis, see [18].)To calculate the resolvent R ( λ ), we use the Schwinger-Dyson equation. Taking the planarlimit (large k and S with α = k e − S fixed) and rewriting the replica partition functions Z n as integrals of the energy s = √ E , the equation ends up with a geometric series: λ R ( λ ) = k + e S (cid:90) ∞ d s ρ ( s ) w ( s ) R ( λ ) k − w ( s ) R ( λ ) , (2.21)where ρ ( s ) ≡ s π sinh(2 πs ) , y ( s ) ≡ e − βs − µ (cid:12)(cid:12)(cid:12)(cid:12) Γ (cid:18) µ −
12 + i s (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ,w ( s ) ≡ y ( s ) Z , Z n ≡ e S (cid:90) ∞ d s ρ ( s ) y ( s ) n , (2.22)and µ is the brane tension (or mass of a particle behind the horizon) and β is the renormalizedlength at the asymptotic boundary (or the temperature of the black hole) in the dual JTgravity. To gain the physical intuition for the spectrum, we divide the spectrum into threeregimes: pre-Page, near-Page and post-Page phases. The density of state D ( λ ) is well-localized near λ = 1 /k at early time (the pre-Page phase). But, as the subsystem evolves,the distribution decays into the thermal spectrum which has a long tail in high eigenvalues(the near-Page and post-Page phases). In order to truncate the thermal tail, we introduce anenergy cutoff s k such that k e − S = (cid:90) s k d s ρ ( s ) = cosh(2 πs k )8 π · (2 πs k − tanh(2 πs k )) . (2.23)The spectrum has a minimal eigenvalue λ which is determined by d λ/ d R = 0 and theresolvent equation (2.21): λ ≈ e S k (cid:90) ∞ s k d s ρ ( s ) w ( s ) = 1 k (cid:20) − e S (cid:90) s k d s ρ ( s ) w ( s ) (cid:21) . (2.24)Under the condition k (cid:28) e S (cid:82) s k d s ρ ( s ) w ( s ) / ( λ − λ − w ( s )) or λ > λ + w ( s k − δ ) with asmall control parameter δ , the density of state is approximated as D ( λ ) = e S (cid:90) s k d s ρ ( s ) δ ( λ − λ − w ( s )) . (2.25)Using this density of state, we find the entanglement entropy S R and the capacity C R : S R = − e S (cid:90) s k d s ρ ( s ) ( λ + w ( s )) log( λ + w ( s )) , (2.26) C R = e S (cid:90) s k d s ρ ( s ) ( λ + w ( s )) (log( λ + w ( s ))) − ( S R ) . (2.27)– 10 –
10 20 30 40 501020304050
Figure 4 . The entropy (blue) and capacity (orange) of the EOW brane model in the canonicalensembles. Both approach the asymptotic values S BH and C BH shown in the dashed lines in the large k limit. While the analytic forms of S R and C R are out of our reach, it is straightforward to performthese integrals numerically. We plot them as functions of log k in figure 4.We can compare the numerical results with the analytic results in limits. In the small k region where D ( λ ) is localized near λ = 1 /k , we find [18] S R = log k + O ( k e − S ) ,C R = O ( k e − S ) . (2.28)This explains the linear and exponential growths of S R and C R at the early time respectively.In the large k region, by iterative uses of the resolvent equation (2.21) for λ (cid:29) w ( s k ), wecan approximate the resolvent R ( λ ) by the first order perturbation of 1 /k : R ( λ ) ≈ e S (cid:90) s k d s ρ ( s ) 1 λ − w ( s ) . (2.29)Here we also used the definition of k (2.23). Then the density of state has the form of thethermal spectrum (consistent with λ ≈ k ): D ( λ ) = e S (cid:90) s k d s ρ ( s ) δ ( λ − w ( s )) . (2.30)Indeed it turns out that D ( λ ) = 0 for λ (cid:29) w ( s k ), and there is no analytic control for theintermediate values. The entanglement entropy approaches to the coarse-grained or blackhole entropy S BH in the canonical ensemble: S BH = − e S Z (cid:90) ∞ d s ρ ( s ) y ( s ) log y ( s ) + log Z . (2.31)– 11 –or µ (cid:29) /β , y ( s ) ∼ y (0) e − βs / since | Γ( µ − / s ) | / (Γ( µ − / = (cid:81) ∞ m =0 (1 + s / ( m + µ − / ) − ∼
1. Then the disk partition function and the entropy can be approximated as Z ≈ e S +2 π /β √ π β / ,S BH ≈ S + 4 π β + 32 log 2 πβ + 32 − π ) . (2.32)Similarly, the capacity of entanglement approaches to the black hole capacity C BH in thecanonical ensemble: C BH = e S Z (cid:90) ∞ d s ρ ( s ) y ( s ) (log y ( s )) − (cid:18) e S Z (cid:90) ∞ d s ρ ( s ) y ( s ) log y ( s ) (cid:19) . (2.33)For µ (cid:29) /β , the capacity approaches to C BH ≈ π β + 32 , (2.34)which is nonzero in contrast to the capacity (2.19) in the microcanonical ensemble.It might be worthwhile to mention that, although the capacity of entanglement C R isdefined by using the derivatives of the replica parameter n not the temperature β , interestinglythe capacity in the canonical ensemble coincides with thermodynamic one C BH at late time. We consider another model of Hawking radiation known as a moving mirror model in two-dimensional CFT, where the mirror plays a role of a black hole horizon and a thermal energyflux can be measured at the infinity [83–87]. The moving mirror may also be regarded as theEOW brane and this model has a similarity to the EOW brane model. The entanglemententropy of a particular class of the moving mirror model has been investigated [88–92] andshown to reproduce the Page curves for eternal and evaporating black holes recently in [78].In this section we will extend the analysis of [78] to the capacity of entanglement.
Suppose the trajectory of a mirror profile is given by x = z ( t ). We consider a CFT living inthe region x ≥ z ( t ) and map it by a conformal transformation:˜ u = p ( u ) , ˜ v = v , (3.1)where u = t − x and v = t + x . In addition, we define a new coordinate after the conformalmap ˜ x and ˜ t through the relation ˜ u = ˜ t − ˜ x and ˜ v = ˜ t + ˜ x . We choose the function p ( u ) suchthat the mirror trajectory is mapped into a static one ˜ u − ˜ v = 0 or equivalently v = p ( u ),which can be written in the original coordinates as t + z ( t ) = p ( t − z ( t )) , (3.2)– 12 – tu v insidethe mirror conformalmap ˜ x ˜ t ˜ u ˜ v insidethe mirror B H h o r i z o n trajectory of v = − p ( u )mirror trajectory v = p ( u ) Figure 5 . A moving mirror model in the ( t, x )-coordinate [Left] and in the tilde coordinates [Right].A CFT lives outside the mirror v ≥ p ( u ) ( x ≥ z ( t )), which can be mapped to the RHP by a conformaltransformation. The red region (˜ x ≥ , ˜ t ≥ ˜ x ) in the right panel has no counterpart in the originalcoordinate system and can be seen as a black hole interior. as in figure 5. More explicitly p ( u ) can be written as p ( u ) = 2 τ u − u , (3.3)where τ u is subject to the condition τ u − z ( τ u ) = u . Then, after the conformal map p ( u ),CFT lives in the right half plane (RHP) ˜ x ≥
0. If CFT is in the vacuum state in the tildecoordinates, the stress tensor in the original coordinates can be given by the Schwarzian termfrom the conformal anomaly: (cid:104) T uu (cid:105) = − c π { p ( u ) , u } , (3.4)which can be matched to the energy flux of the Hawking radiation by an appropriate choiceof the mirror trajectory p ( u ) [83].Typically the mirror describing a (non-evaporating) black hole extends into the x < x = 0 at early time and asymptoting to the lightlike curve v = 0 at latetime. This means that the function p ( u ) is subject to the following conditions: p ( u ) < , p ( u ) = (cid:40) u ( u → −∞ ) , − βe − u/β ( u → ∞ ) . (3.5)Here β corresponds to the inverse temperature of the black hole. Also it follows from (3.3)that p (cid:48) ( u ) = 1 + z (cid:48) ( t )1 − z (cid:48) ( t ) ≥ , (3.6)unless the mirror travels faster than light. We probe a CFT with this mirror trajectory by asemi-infinite line R = [ x , ∞ ) with x > x ≥ (cid:104) σ n ( t, x ) (cid:105) RHP = e (1 − n ) S bdy (cid:12)(cid:12)(cid:12)(cid:12) x ˜ (cid:15) (cid:12)(cid:12)(cid:12)(cid:12) − c ( n − n )= e (1 − n ) S bdy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t + x − p ( t − x ) (cid:15) (cid:112) p (cid:48) ( t − x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − c ( n − n ) , (3.7)where the UV cutoff ˜ (cid:15) is introduced in the half space ˜ x ≥ (cid:15) = (cid:112) p (cid:48) ( u ) (cid:15) to the cutoff (cid:15) in the original half space x ≥ z ( t ). S bdy = log (cid:104) | B (cid:105) is the boundary entropydetermined by the overlap between the vacuum | (cid:105) and a conformal boundary state | B (cid:105) characterizing the vacuum degeneracy of the CFT with a boundary [94]. It can be interpretedas the entropy of a black hole the moving mirror describes.Through the above one-point function, the R´enyi entropy for the subsystem R at time t is given by S ( n ) R = c (cid:18) n (cid:19) log (cid:34) t + x − p ( t − x ) (cid:15) (cid:112) p (cid:48) ( t − x ) (cid:35) + S bdy . (3.8)We then immediately obtain the entanglement entropy for R at time t : S R = c (cid:34) t + x − p ( t − x ) (cid:15) (cid:112) p (cid:48) ( t − x ) (cid:35) + S bdy , (3.9)and the capacity of entanglement C R = − ∂ n S ( n ) R (cid:12)(cid:12) n =1 = S R − S bdy . (3.10)For the mirror trajectory satisfying (3.5) it follows that the capacity is constant at earlytime and grows linearly in t at late time: C R = c (cid:18) x (cid:15) (cid:19) ( t → −∞ ) ,c β t + c (cid:18) t(cid:15) (cid:19) ( t → + ∞ ) . (3.11)The entropy takes the same form as C R with a shift by S bdy , so it does not reproduce thePage curve. Hence, there are no phase transitions to be detected by the capacity in thismodel. Next we consider a moving mirror in a class of two-dimensional CFTs known as holographicCFTs with a large central charge. We take the radiation region to be a finite subsystem– 14 – x ˜ t (cid:101) σ n (cid:101) ¯ σ n , bulk-bulkOPE channel = max ˜ x ˜ t (cid:101) σ n (cid:101) ¯ σ n bulk-boundaryOPE channel (cid:104) ˜ σ n (˜ t , ˜ x ) ˜¯ σ n (˜ t , ˜ x ) (cid:105) RHP
Figure 6 . The bulk-bulk OPE channel and the bulk-boundary OPE channel contributing to thetwo-point function of twist operators in two-dimensional holographic BCFT. A channel with largervalue is favored, and there can be a phase transition between the two. R = [ x , x ] at time t . To obtain the R´enyi entropy, we need to compute the two-pointfunction of twist operators: (cid:104) σ n ( t , x ) ¯ σ n ( t , x ) (cid:105) = (cid:0) p (cid:48) ( u ) p (cid:48) ( u ) (cid:1) h n · (cid:104) ˜ σ n (˜ t , ˜ x ) ˜¯ σ n (˜ t , ˜ x ) (cid:105) RHP . (3.12)In this case, the correlation function on the RHP can be expressed by two kinds of the identityblocks in different operator product expansion (OPE) channels of the two-point functions on R , in the large central charge limit c → ∞ [95] (see figure 6): (cid:104) ˜ σ n (˜ t , ˜ x ) ˜¯ σ n (˜ t , ˜ x ) (cid:105) RHP = max (cid:104) ˜ σ n (˜ t , ˜ x ) ˜¯ σ n (˜ t , ˜ x ) (cid:105) R , ,e − n ) S bdy · (cid:89) i ∈{ , } (cid:104) ˜ σ n (˜ t i , ˜ x i ) ˜¯ σ n (˜ t i , ˜ x i ) (cid:105) R , , (3.13)where the correlator of the twist operators on flat space is given by (cid:104) ˜ σ n (˜ t, ˜ x ) ˜¯ σ n (˜ t (cid:48) , ˜ x (cid:48) ) (cid:105) R , = (cid:12)(cid:12) (˜ t − ˜ t (cid:48) ) − (˜ x − ˜ x (cid:48) ) (cid:12)(cid:12) − c ( n − n ) . (3.14)Whether a phase transition between the two channels occurs depends on both S bdy and thepositions of the twist operators.It is then straightforward to calculate the R´enyi entropy: S ( n ) R = 11 − n log (cid:104) σ n ( t, x ) ¯ σ n ( t, x ) (cid:105) = min c (cid:18) n (cid:19) log (cid:34) ( x − x ) ( p ( t − x ) − p ( t − x )) (cid:15) (cid:112) p (cid:48) ( t − x ) p (cid:48) ( t − x ) (cid:35) ,c (cid:18) n (cid:19) log (cid:34) t + x − p ( t − x ) (cid:15) (cid:112) p (cid:48) ( t − x ) (cid:35) + c (cid:18) n (cid:19) log (cid:34) t + x − p ( t − x ) (cid:15) (cid:112) p (cid:48) ( t − x ) (cid:35) + 2 S bdy . (3.15)– 15 –n the n → S R = S ( n ) R (cid:12)(cid:12) n =1 = min (cid:104) S dis R , S con R (cid:105) , (3.16)where S dis R = c (cid:34) t + x − p ( t − x ) (cid:15) (cid:112) p (cid:48) ( t − x ) (cid:35) + c (cid:34) t + x − p ( t − x ) (cid:15) (cid:112) p (cid:48) ( t − x ) (cid:35) + 2 S bdy ,S con R = c (cid:34) ( x − x ) ( p ( t − x ) − p ( t − x )) (cid:15) (cid:112) p (cid:48) ( t − x ) p (cid:48) ( t − x ) (cid:35) . (3.17)When the conditions (3.5) and (3.6) are met, the connected channel entropy S con R is alwaysfavored at late time as the disconnected channel entropy grows as S dis R → c log ( t/(cid:15) ) in t → ∞ while S con R is bounded from above. On the other hand, the dominant channel at earlytime depends on the value of the boundary entropy S bdy . When the disconnected channelis dominant at early time, S dis R < S con R , there must be a transition at the Page time where S con R = S dis R and the entropy S R continuously changes its behavior from the linear growth inlog t to plateau, reproducing the Page curve [78].The capacity of entanglement also depends on the dominant channel as given by C R = − ∂ n S ( n ) R (cid:12)(cid:12) n =1 = (cid:40) S con R (connected channel) ,S dis R − S bdy (disconnected channel) . (3.18)When the phase transition occurs for the entropy the capacity shows a discontinuity by 2 S bdy at the Page time. This is a universal behavior independent of the profile of the moving mirroras long as the conditions (3.5) are met. Eternal black hole
As a concrete example, we consider the following moving mirror tra-jectory satisfying the conditions (3.5) and (3.6): p ( u ) = − β log (cid:16) e − u/β (cid:17) . (3.19)The stress tensor (3.4) is given by (cid:104) T uu (cid:105) = c πβ e − u/β (1 + e − u/β ) , (3.20)which is zero at early time ( u → −∞ ), but approaches to the energy flux of the Hawkingradiation from a black hole at inverse temperature β at late time ( u → ∞ ). Thus this modelis seen to describe an eternal black hole with radiation. The original mirror trajectory z ( t )can be read off from p ( u ) through the relation (3.2). For small β , the mirror trajectory z ( t )corresponding to the equation (3.19) can be approximated as follows: at the early period– 16 – tu vR Figure 7 . A mirror trajectory describing an eternal black hole (3.19). It starts from x = 0 andasymptotes to the light-ray t = − x at late time. The entropy and capacity of entanglement areprobed by the interval R = [ z ( t ) + 0 . , z ( t ) + 10] at a fixed distance from the mirror. t <
0, the mirror stays around the origin z ( t ) = 0 and approaches to the light-ray z ( t ) = − t at the late time t > R = [ z ( t ) + 0 . , z ( t ) + 10] at time t . The interval moves along the mirror trajectory with the distance kept fixed. Using theequation (3.17) with the boundary entropy S bdy /c = 0 .
1, we plot the entanglement entropyin the left panel of figure 8. The disconnected channel is dominant at early time while theconnected channel becomes dominant at late time. The capacity of entanglement is obtainedthrough the equation (3.18) as depicted in the right panel of figure 8. The discontinuity ofthe capacity at the phase transition time t phase , C con − C dis (cid:12)(cid:12) t phase = 2 S bdy , (3.21)is seen in the figure. Evaporating black hole
Next we consider another moving mirror trajectory [78] p ( u ) = − β log (cid:16) e − u/β (cid:17) + β log (cid:16) e ( u − u ) /β (cid:17) . (3.22)In terms of the original coordinates z ( t ) the mirror starts from the origin at early time,linearly decreases around the period 0 < t < u / u >
0, then asymptotes to a constant z ( t ) = − u / (cid:104) T uu (cid:105) = c πβ (cid:34) e − u/β (1 + e − u/β ) + 1 + 2 e ( u − u ) /β (1 + e ( u − u ) /β ) (cid:35) (3.23)– 17 – onnecteddisconnected - - Figure 8 . The entropy [Left] and capacity [Right] in the moving mirror model whose trajectory mimicsan eternal black hole with radiation. We set the subsystem to be an interval R = [ z ( t ) + 0 . , z ( t ) + 10]with β = 0 . (cid:15) = 0 . S bdy /c = 0 .
1. The phase transition from the disconnected channel toconnected one occurs around t phase = 5. The capacity shows a discontinuity by 2 S bdy = 0 . c at thattime. xt x = − u u vR Figure 9 . The mirror trajectory describing an evaporating black hole (3.22). It starts from x =0, approaches to the light-ray t = − x around the time interval t = [0 , u / x = − u / R = [ z ( t ) + 0 . , z ( t ) + 10]. localized around finite time interval 0 < t < u / R = [ z ( t )+0 . , z ( t )+10]. In this case the entropy reproduces the Page curve of an evaporatingblack hole, but with two peaks as in the left panel of figure 10. While there are both connectedand disconnected solutions, the disconnected one is always favored and no phase transitionoccurs between the channels in this model. Correspondingly the capacity of entanglement– 18 – onnecteddisconnected0 2 4 6 8 10 12 141.01.52.02.53.03.54.0 Figure 10 . The entropy [Left] and capacity [Right] in the moving mirror model whose trajectorymimics an evaporating black hole. We set the subsystem to be R = [ z ( t ) + 0 . , z ( t ) + 10] with β = 0 . (cid:15) = 0 . S bdy /c = 0 .
1. In this model, there is no phase transition between the connected anddisconnected channels, hence the capacity does not have any discontinuity. xtu vR
Figure 11 . The static moving mirror trajectory (3.24) probed by an expanding interval R = [0 . , . t ]. does not show any discontinuity as seen in the right panel of figure 10. Static mirror probed by an expanding interval
Finally we consider a different setupfrom the previous ones, which has a static mirror but a radiation region expands in time.While it does not describe a radiating black hole, it turns out that this model has a phasetransition and may serve as a good testing ground for the capacity as an order parameter.For a static mirror z ( t ) = 0 we have p ( u ) = u . (3.24)The energy flux (3.4) vanishes, so there are no Hawking radiations in this model. Insteadwe expand the interval R = [0 . , . t ] in time as in figure 11, which mimics the Rindlerobserver who feels thermal radiation. In this case, there is a phase transition from theconnected channel to the disconnected one as shown in the left panel of figure 12. The– 19 – isconnectedconnected0 1 2 3 40.00.20.40.60.81.01.2 Figure 12 . The entropy [Left] and capacity [Right] in the static mirror model probed by an expandinginterval. We set the subsystem to be R = [0 . , . t ] with (cid:15) = 0 . S bdy /c = 0 .
1. There is aphase transition from the connected channel to disconnected one, and the capacity drops down by2 S bdy = 0 . c at the transition time. capacity of entanglement shows a discontinuity at the transition point t phase as in the rightpanel of figure 12. In contrast to the previous case for an eternal black hole, the discontinuityis negative: C dis − C con | t phase = − S bdy , (3.25)as the direction of the phase transition is opposite to the previous case. In this paper, we found that the capacity of entanglement has a peak or discontinuity atthe Page time in two simple models of evaporating black holes, the EOW brane model [18](section 2) and the moving mirror model [78] (section 3). These observations lead us toconclude that the capacity can be a useful probe of the Hawking radiation in evaporatingblack holes. Comparison with other quantum information measures which can capture similarsignals, e.g., the reflected entropy or the entanglement wedge cross section [43, 44] and therelative entropy in charged states [67], would also be interesting to examine the similarityor difference between the capacity and the other measures.The simple model calculations elaborated in section 2 revealed the behavior of the ca-pacity highly depends on the choice of ensembles while the entropy takes almost the sameform in both cases. In the microcanonical ensemble, the capacity has a peak at the Page time(log k = S ) and decays rapidly to zero in the late time. On the other hand, we find a similarpeak at the Page time (log k = S ) in the canonical ensemble, but the capacity does not decayand approaches to a constant value fixed by the temperature β and the brane tension µ at late For the topic discussed in [67] or a global symmetry violation in the Hawking radiation, see also [66, 68, 98]. – 20 –ime. It would be worthwhile to further study the dependence of the capacity on the choiceof ensembles by introducing chemical potentials to the two-dimensional dilaton gravity. The capacity in the canonical ensemble of the EOW brane model clearly exhibits a peakaround the Page time, but there is a bump after the Page time peak when S is small as inthe left plot of figure 4. Note that we use an approximation in the plot which may give rise toan O (( β G N ) ) error to the entropy around the Page time in comparing with the exact result[18]. The capacity should also suffer from a similar type of error around the Page time andthe exact numerical calculation will reveal whether the bump is caused by an error or not.If it would not originate from an error, but still remain, the bump could be a signature of amore refined phase transition in the radiation process [18]. The phase transition between the replica wormholes are caused by non-perturbative in-stanton corrections to the replica partition functions. We believe the capacity of entanglementis a invaluable probe not only for the topology change of replica wormholes but also for othernon-perturbative phenomena in non-gravitating theories where replica instanton effect doesmatter, e.g., confinement in a simple gauge theory with an axion-like coupling to a scalarfield [98, 103]. Other cases where instanton corrections matter are the fixed-area states inAdS/CFT [104–106] and the random tensor networks [107] (see Appendix E of [18]). A simi-lar calculation as in [18] may be applied to the capacity of entanglement while it will requiremore effort due to multiple instanton corrections than the entanglement entropy which onlyhas a one instanton correction.In the moving mirror models, the differences of the capacities of entanglement betweenbefore and after the phase transitions took negative values, except for the case for staticmirror probed by an expanding interval. The sign of the discontinuity is positive when thedisconnected channel is favored at early time and taken over to the connected channel at latetime, while it is negative when the transition goes in the other way. It is not clear whetherthe sign of the discontinuity is related to some nature of the Hawking radiations such as thestability of evaporating black holes. It would deserve further studies to see if the sign can beconstrained by some physical conditions such as unitarity and causality in a broader class ofmodels than those in this paper.We have been focused on the exploration of the capacity only in the simple toy models ofblack holes, but it should be feasible to extend our work to more realistic black holes wherewe believe the capacity will exhibit a peak or discontinuity when there happens a phasetransition between replica wormhole geometries. We hope to report further investigations ofthe capacity in a 2 d CFT coupled to the JT gravity in future work [108]. For the grand canonical ensembles associated to asymptotic boundaries or baby universes, see [56, 99],and to defects in the bulk see [100, 101]. See also [68] for charged states in the EOW brane model. Random pure states in a Floquet model describe thermal process similar to the Hawking radiation, whichhave prethermalized and chaos phases in addition to the early and late time phases [102]. The peak and bumpin the capacity might be associated with a development of these phases. – 21 – cknowledgments
We are grateful to K. Goto, T. Ugajin, T. Takayanagi and K. Tamaoka for valuable discussions.The work of T. N. was supported in part by the JSPS Grant-in-Aid for Scientific Research(C) No.19K03863 and the JSPS Grant-in-Aid for Scientific Research (A) No.16H02182. Thework of K. W. was supported by the Grant-in-Aid for JSPS Fellows No.18J00322 and U.S.Department of Energy grant DE-SC0019480 under the HEP-QIS QuantISED program, andby funds from the University of California. The works of K. K. and Y. O. were supportedby Forefront Physics and Mathematics Program to Drive Transformation (FoPM), a World-leading Innovative Graduate Study (WINGS) Program, the University of Tokyo.
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