Entanglement growth during thermalization in holographic systems
MMIT-CTP 4510
November 7, 2013
Entanglement growth during thermalization in holographic systems
Hong Liu and S. Josephine Suh
Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139
We derive in detail several universal features in the time evolution of entanglement entropy andother nonlocal observables in quenched holographic systems. The quenches are such that a spatiallyuniform density of energy is injected at an instant in time, exciting a strongly coupled CFT whicheventually equilibrates. Such quench processes are described on the gravity side by the gravitationalcollapse of a thin shell that results in a black hole. Various nonlocal observables have a unifieddescription in terms of the area of extremal surfaces of different dimensions. In the large distancelimit, the evolution of an extremal surface, and thus the corresponding boundary observable, iscontrolled by the geometry around and inside the event horizon of the black hole, allowing us toidentify regimes of pre-local- equilibration quadratic growth, post-local-equilibration linear growth,a memory loss regime, and a saturation regime with behavior resembling those in phase transitions.We also discuss possible bounds on the maximal rate of entanglement growth in relativistic systems.
CONTENTS
I. Introduction 2II. General setup 3A. Vaidya metric 3B. Extremal surfaces and physical observables 5C. Vacuum and thermal equilibrium propertiesof extremal surfaces 51. Vacuum properties 52. Equilibrium properties 6D. Further comments on the Vaidya setup 6III. Equations of motion for extremal surfaces 7A. Strip 7B. Sphere 8IV. General geometric features and strategy 9V. Evolution in (1 + 1) dimensions 11A. Early growth 12B. Linear growth 12C. Saturation 12D. Memory loss regime 13VI. Pre-local-equlibrium quadratic growth 13VII. Critical extremal surfaces 15A. Strip 15B. Sphere 18C. Summary 19VIII. Linear growth: strip 20A. Linear growth 20B. Example: Schwarzschild 211. η > η = 1 213. η < v E from a Schwarzschild BH inGauss-Bonnet gravity 27XI. Saturation 28A. Strip 29B. Sphere 30C. More on the saturation time 30XII. Memory loss regime 31A. Strip 31B. Sphere 32C. Memory loss 33XIII. Conclusions and discussions 34A. More general equilibration processes 34B. Entanglement growth 35C. Tsunami picture: local propagation ofentanglement 35D. Application to black holes 36A. Equilibrium behavior of extremal surfaces 361. Strip 372. Sphere 37B. Details in the saturation regime 371. Strip 372. Sphere 383. Discussion of z when Σ is a sphere 40C. Details in the memory loss regime for Σ asphere 411. Critical extremal surface 412. Equations 423. Time 43a. Region I 43 a r X i v : . [ h e p - t h ] N ov b. Region II 43c. Region III 444. Action 45References 46 I. INTRODUCTION
Understanding whether and how quantum matter equi-librate is a question of much importance in many differentareas of physics. Yet such non-equilibrium problems arenotoriously difficult to deal with; universal characteriza-tions are scarce and far between.For a non-integrable system it is expected that ageneric (sufficiently excited) non-equilibrium state even-tually thermalizes. For strongly coupled systems with agravity dual this expectation is borne out as holographicduality maps equilibration from such a state to black holeformation from a gravitational collapse, and gravitationalcollapse of a sufficiently massive body is indeed genericin General Relativity.Questions related to equilibration then become inti-mately connected to those of black hole physics. Thison the one hand brings in powerful gravity techniquesfor studying thermalization processes, and on the othergives new perspectives on the quantum nature of blackholes.One of the simplest settings for equilibration is theevolution of a system after a global quench, which canbe divided into two types. In the first type one changessome parameter(s) of a system at t = 0 within a short in-terval δ t . The previous ground state becomes an excitedstate with respect to the new Hamiltonian and evolves toequilibrium under the evolution of the new Hamiltonian.In the second type, one turns on a uniform density ofsources for a short interval δ t at t = 0 and then turns itoff. The work done by the source takes the system to anexcited state which subsequently equilibrates (under theevolution of the same Hamiltonian before the quench).In both situations, the interval δ t is taken to be muchsmaller than any other scale in the system. For conve-nience we will take δ t to be zero in subsequent discus-sions.In (1+1)-dimension, by tuning a parameter of a gappedsystem to criticality Calabrese and Cardy found that [1]the entanglement entropy for a segment of size 2 R growswith time linearly as∆ S ( t , R ) = 2 t s eq , t < R (1.1)and saturates at the equilibrium value at a sharp satu-ration time t s = R . In the above equation ∆ S denotesdifference of the entanglement entropy from that at t = 0and s eq is the equilibrium thermal entropy density. Fur-thermore, they showed that this remarkably simple be-havior can be understood from a simple model of entan-glement propagation using free-streaming quasiparticlestraveling at the speed of light. Subsequently, the linear behavior (1.1) was found inholographic context for (1 + 1)-dimensional systems dualto a bulk Vaidya geometry [2] (see also [3, 4]). An AdSVaidya geometry, as we will review in more detail inSec. II A, describes the gravitational collapse of a thinshell of matter to form a black hole. It corresponds toa quench process of the second type in a boundary con-formal field theory, where at t = 0, a uniform density ofoperators are inserted for a very short time. The entan-glement entropy is obtained from the area of an extremalsurface in the Vaidya geometry with appropriate bound-ary conditions [5–7].The agreement of results between the very differentsetups of [1] and [2] is in some sense not surprising.Both setups involve a homogeneous excited initial stateevolving under a gapless Hamiltonian, and the powerfulboundary CFT techniques of [1] should apply in bothcontexts. Behavior similar to that of entanglement en-tropy has also been found in correlation functions in bothcontexts [3, 4, 8, 9] (see also [5, 10–20] for other studiesof two-dimensional systems).Given the simplicity and elegance of (1.1), it is nat-ural to wonder: (i) whether similar linear growth oc-curs in higher dimensions; (ii) whether other nonlocalobservables such as equal-time correlation functions andWilson loops share similar behavior; (iii) if such lineargrowth exists, whether it can still be understood fromfree-streaming quasiparticles.For entanglement entropy we recently reported the an-swers to some of these questions for a class of quenchedholographic systems [21]. Interested in long-distancephysics, we focused on entangled regions of large size,and found that the time evolution of entanglement en-tropy is characterized by four different scaling regimes:1. Pre-local-equilibration quadratic growth in time.2. Post-local-equilibration linear growth in time.3. A saturation regime in which the entanglement en-tropy saturates its equilibrium value. The satu-ration can be either continuous or discontinuousdepending on whether the time derivative of theentanglement entropy is continuous at saturation.In the continuous case saturation is characterizedby a “critical” exponent.4. When the entangled region is a sphere, there isan additional scaling regime between linear growthand saturation, which we dub “late time memory-loss”, and in which the entanglement entropy onlydepends on time remaining till saturation, and noton the size of the region and time separately.These results are generic in the sense that they are in-sensitive to the specific details of the system as well asthose of the quench.The above scaling regimes were obtained by identifyingvarious geometric regimes for the bulk extremal surface.An important observation was the existence of a family of“critical extremal surfaces” which lie behind the horizonand separate extremal surfaces that reach the boundaryfrom those which fall into the black hole singularity. Inthe large size limit, one finds that the time evolution ofentanglement entropy is controlled by these critical ex-tremal surfaces. In this paper we give a detailed deriva-tion of these results and provide generalizations to othernon-local observables such as equal-time correlation func-tions and Wilson loops.Also, with M´ark Mezei [22], we generalized the free-streaming model of [1] to higher dimensions. It turns outthat such a model also exhibits post-local-equilibrationlinear growth of entanglement entropy, but that intrigu-ingly, the rate of growth of entanglement entropy result-ing from free-streaming particles moving at the speed oflight is less than what we find here for strongly coupledholographic systems.In [21], we argued that the evolution of entanglemententropy can be captured by the picture of an entangle-ment wave propagating inward from the boundary ofthe entangled region, which we called an “entanglementtsunami” (see also [20]). There we also suggested a pos-sible upper bound on the rate of entanglement growthin relativistic systems. The results of [21] and the cur-rent paper also have potential applications for variousissues associated with black hole physics. The fact thatthe growth of entanglement is controlled by some criti-cal extremal surfaces inside the horizon of a collapsingblack hole also suggests new avenues for probing physicsbeyond horizons in holography. Similar processes as weconsider here were also considered in [23] to obtain in-sights into the “scrambling time” of a black hole. Wewill elaborate more on these issues in the discussion sec-tion.To conclude this introduction, we note that earlierwork on quenches in higher dimensional holographic sys-tems include [3, 4, 24, 25] (see also [12–14, 17, 26–28]).In particular, for d = 3, a linear growth toward satu-ration was mentioned in [24], although it appears thatthe linear regime mentioned in [24] is different from thatof [21] and the current paper. Ref. [24] was also the firstto observe discontinuous saturation in various examples.In [3, 4] non-analyticity near saturation was emphasized.In a different gravity setup, linear growth of entangle-ment entropy was also observed [25], whose connectionto that in [21] will be discussed in detail in the main text.In [13] it was pointed out that the presence of a nonzerochemical potential in the final equilibrium state tends toslow the growth of entanglement. II. GENERAL SETUP
In this paper we consider the evolution of various non-local observables, including entanglement entropy, equal-time correlation functions, and Wilson loops, after asharp quench of a strongly coupled gapless system witha gravity dual. More explicitly, at t = 0 in the boundary system we turn on a spatially uniform density of exter-nal sources for an interval δ t , creating a spatially ho-mogeneous and isotropic excited state with nonzero en-ergy density, which subsequently equilibrates. The pre-cise manner (e.g. what kind of sources are turned on andhow) through which the excited state is generated andits microscopic details will not concern us. We are inter-ested in the macroscopic behavior of the system at largedistances and in extracting “universal” behavior in theevolution of these observables that are insensitive to thespecific nature of final equilibrium states.On the gravity side such a quench process is describedby a thin shell of matter starting from the boundary andcollapsing to form a black hole, which can in turn bedescribed by a Vaidya metric, see Fig. 1. The matterfields making up the shell and their configuration aredetermined by the sourcing process in the boundary the-ory and are again not important for our purposes. Seee.g. [29–35] for more explicit discussions. In the classicalgravity regime we are working with, which translates tothe large N and strongly coupled limit of the boundarytheory, all of our observables are only sensitive to themetric of the collapsing geometry.In this section we give a detailed description of oursetup and review the vacuum and equilibrium propertiesof the class of systems under consideration. ← AdS black hole h o r i z o n ( v <
0) ( v > nu ll s h e ll ( v = ) b o und a r y ( z = ) t = 0 FIG. 1. Vaidya geometry: One patches pure AdS with a blackhole along an in-falling collapsing null shell located at v = 0.We take the width of the shell to be zero which corresponds tothe δ t = 0 limit of the boundary quench process. The spatialdirections along the boundary are suppressed in the figure. A. Vaidya metric
We consider a metric of the form ds = L z (cid:0) − f ( v, z ) dv − dvdz + d(cid:126)x (cid:1) . (2.1)In the limit the sourcing interval δ t goes to zero, thewidth of the collapsing shell goes to zero and f ( v, z ) canbe expressed in terms of a step function f ( v, z ) = 1 − θ ( v ) g ( z ) . (2.2)For v <
0, the metric is given by that of pure AdS, ds = L z (cid:0) − dt + dz + d(cid:126)x (cid:1) (2.3)where v = t − z , t = v + z . (2.4)For v >
0, (2.1) is given by that of a black hole inEddington-Finkelstein coordinates, ds = L z (cid:0) − h ( z ) dv − dvdz + d(cid:126)x (cid:1) , (2.5)which in terms of the usual Schwarzschild time t can bewritten as ds = L z (cid:18) − h ( z ) dt + 1 h ( z ) dz + d(cid:126)x (cid:19) (2.6)with h ( z ) ≡ − g ( z ) , v = t − σ ( z ) , σ ( z ) = (cid:90) z dz (cid:48) h ( z (cid:48) ) . (2.7)The functions h ( z ) in the black hole metric (2.5)–(2.6)may be interpreted as “parameterizing” different typesof equilibration processes with different final equilibriumstates. We assume that (2.1) with some g ( z ) can alwaysbe achieved by choosing an appropriate configuration ofmatter fields. In following discussions we will not needthe explicit form of h ( z ), and only that it gives rise to ablack hole metric. We will work with a general boundaryspacetime dimension d .More explicitly, we assume h ( z ) has a simple zero atthe horizon z = z h >
0, and that for z < z h , it is pos-itive and monotonically decreasing as a function of z asrequired by the IR/UV connection. As we approach theboundary, i.e. as z → h ( z ) approaches zero with theleading behavior h ( z ) = 1 − M z d + · · · (2.8)where M is some constant. From (2.8), one obtains thatthe energy density of the equilibrium state is E = L d − πG N d − M , (2.9)while its temperature and entropy density are given by T = | h (cid:48) ( z h ) | π , s eq = L d − z d − h G N . (2.10)Representative examples of (2.5) include the AdSSchwarzschild black hole with h ( z ) = 1 − z d z dh (2.11) which describes a neutral final equilibrium state, and theAdS Reissner-Nordstrom (RN) black hole with h ( z ) = 1 − M z d + Q z d − , (2.12)which describes a final equilibrium state with a nonzerochemical potential for some conserved charge.A characteristic scale of the black hole geometry (2.5)–(2.6) is the horizon size z h which from (2.10) can beexpressed in terms of the entropy density s eq as z h = (cid:18) L d − G N s eq (cid:19) d − . (2.13)Were we considering a gas of quasiparticles, the prefac-tor L d − G N in (2.13) could be interpreted as the numberof internal degrees of freedom of a quasiparticle, and z h would then be the average distance between quasiparti-cles, or mean free path. Here of course we are consideringstrongly coupled systems which do not have a quasiparti-cle description. Nevertheless, z h provides a characteristicscale of of the equilibrium state. For example, as we willsee below it controls the correlation length of equal-timecorrelation functions and Wilson loops in equilibrium.For the collapsing process described by (2.1) we canalso identify z h as a “local equilibrium scale” (cid:96) eq , whichcan be defined as the time scale when the system hasceased production of thermodynamic entropy, or in otherwords, has achieved local equilibrium at distance scalesof order the “mean free path” of the equilibrium state.We will discuss further support for this identification atthe end of Sec. IV.We note that in the AdS Schwarzschild case (2.11), thetemperature T is the only scale and controls both thelocal equilibrium scale z h and energy density E (givenby (2.9)), T = d πz h , M = 1 z dh = (cid:18) πTd (cid:19) d , (2.14)but that in a system with more than one scale as in theReissner-Nordstrom case, z h and E (or M ) do not dependonly on T . In the Reissner-Norstrom case, it is convenientto introduce a quantity u ≡ πz h Td (2.15)which decreases monotonically from its Schwarzschildvalue of unity to 0, as the chemical potential is increasedfrom zero to infinity at fixed T . Thus with a large chem-ical potential (compared to temperature), the local equi-librium scale (cid:96) eq ∼ z h can be much smaller than the Note that while the horizon location is a coordinate dependentquantity, in the particular radial coordinate used in (2.5)–(2.6) z h corresponds to a meaningful boundary scale as for exampleindicated by (2.13). thermal wave length 1 /T . In this regime, the system iscontrolled by finite density physics which gives rise to thescale z h . For recent related discussions, see [36].Finally, we note that the metric (2.1) is not of themost general form describing a spatially homogenous andisotropic equilibration process. If the equilibrium statehas a nontrivial expectation value for (or sourced by)some scalar operators, the metric has the form ds = L z (cid:0) − f ( v, z ) dv − q ( v, z ) dvdz + d(cid:126)x (cid:1) (2.16)with f ( v, z ) = 1 − θ ( v ) g ( z ) and q ( v, z ) = 1 − θ ( v ) m ( z ).The black hole part of the spacetime now has a metric ofthe form ds = L z (cid:0) − h ( z ) dv − k ( z ) dvdz + d(cid:126)x (cid:1) (2.17)with h ( z ) ≡ − g ( z ) and k ( z ) ≡ − m ( z ), and can alsobe written as ds = L z (cid:18) − h ( z ) dt + dz l ( z ) + d(cid:126)x (cid:19) , k ( z ) = h ( z ) l ( z ) . (2.18)We will restrict our discussion mostly to (2.1), but it isstraightforward to generalize our results to (2.16) as willbe done in various places below. B. Extremal surfaces and physical observables
We are interested in finding the area A Σ of an n -dimensional extremal surface Γ Σ in the Vaidya geome-try (2.1) which ends at an ( n − spatial surface Σ lying at some time t in the boundary theory.We will use A Σ to denote the area of Σ. Since (2.1) isnot invariant under time translation, Γ Σ and therefore A Σ will depend on t . A Σ can be used to compute various observables in theboundary theory:1. For n = 1, we take Σ to be two points separatedby some distance 2 R . Γ Σ is then the geodesic con-necting the two points, and its length A ( R, t ) givesthe equal-time two-point correlation function of anoperator with large dimension, G (2 R, t ) ∝ e − m A ( R, t ) , (2.19)where m is the mass of the bulk field dual to theoperator.2. For n = 2, we take Σ to be a closed line, whichdefines the contour of a spacelike Wilson loop. Thearea A Σ ( t ) then gives the expectation value of theWilson loop operator [37, 38], (cid:104) W Σ ( t ) (cid:105) ∝ e −A Σ ( t ) / πα (cid:48) , (2.20)where (2 πα (cid:48) ) − is the bulk string tension. 3. For n = d −
1, we take Σ to be a closed surfacewhich separates space into two regions. The area A Σ ( t ) then gives the entanglement entropy associ-ated with the region bounded by Σ [5, 7], S Σ ( t ) = A Σ ( t )4 G N , (2.21)where G N is Newton’s constant in the bulk.When there are multiple extremal surfaces correspond-ing to the same boundary data, we will choose the sur-face with the smallest area. For entanglement entropy,this allows the holographic prescription to satisfy strongsub-additivity conditions [39, 40], while for correlationfunctions and Wilson loops, the smallest area gives themost dominant saddle point.We will often consider as examples the following twoshapes for Σ, which are the most symmetric representa-tives of two types of topologies for the boundary surface: • a sphere of radius R : with d(cid:126)x in (2.1) written inpolar coordinates for the first n directions, d(cid:126)x = dρ + ρ d Ω n − + dx n +1 + · · · + dx d − , (2.22)Σ is specified by ρ = R , x a = 0 , a = n + 1 , · · · , d − . (2.23) • boundary of a strip of half-width R : Σ consists oftwo ( n − x = ± R , x a = 0 , a = n + 1 , · · · , d − x , · · · , x n . For n = 1, Σ consists of two points separated by 2 R .For n = 2, it defines a rectangular Wilson loop,and for n = d −
1, it encloses the strip region x ∈ ( − R, R ). For brevity, we will refer to a Σ with thissecond shape as a “strip”.
C. Vacuum and thermal equilibrium properties ofextremal surfaces
1. Vacuum properties
Before the quench, our system is in the vacuum stateof a strongly coupled CFT with a gravity dual. Consideran extremal surface Γ Σ (with boundary Σ) in pure AdS,whose area gives the vacuum value of the correspondingphysical observable. When Σ is a sphere, A sphere = local divergences+ L n ω n − (cid:40) ( − n b n n even( − n − b n log R n odd (2.25) The following expressions for Σ a sphere or strip have appearedin many places in the literature. For the case of entanglemententropy with n = d −
1, they were first obtained in [5]. where ω n − is the area of unit ( n − b n = ( n − n − . (2.26)When Σ is a strip, A strip = local divergences + (cid:40) L log R n = 1 − L n ( a n ) n n − A strip R n − n > ,a n ≡ √ π Γ( + n )Γ( n ) (2.27)where A strip is the area of the strip Σ with both sidesincluded. The local divergences in (2.25) and (2.27) canbe interpreted as coming from short-range correlationsnear Σ and its leading contributions are proportional to A Σ .The number of degrees of freedom in a CFT can becharacterized by a central charge s d , defined in all dimen-sions in terms of the universal part of the entanglemententropy of a spherical region in the vacuum [41], S (vac)sphere = local divergences+ (cid:40) ( − d − s d d odd( − d − s d log R d even , (2.28)where from (2.25), s d = L d − G N ω d − b d − = π d Γ( d ) L d − G N × (cid:40) d odd π d even . (2.29)Note that for d = 2 the above central charge is related tothe standard central charge c as s = c . (2.30)From the standard AdS/CFT dictionary, s d ∝ N where N is the rank of the gauge group(s) of the boundarytheory. If we put such a holographic CFT on a lattice, s d is heuristically the number of degrees of freedom on asingle lattice site.From (2.20) and (2.25)–(2.27), a Wilson loop of circu-lar and rectangular shape respectively have the vacuumbehavior W Σ ∼ (cid:40) e − √ λ circle e − √ λ (cid:96)R rectangle , √ λ = L α (cid:48) (2.31)where (cid:96) denote the length of the long side of a rectangu-lar Wilson loop. Similarly one finds that the two-pointcorrelation function of an operator with large dimension∆ ≈ mL (cid:29) G (2 R ) ∼ R . (2.32)
2. Equilibrium properties
After the quench, our system eventually evolves to afinal equilibrium state dual to a black hole in the bulk.Here we briefly review properties of an extremal surfaceΓ Σ (with boundary Σ) in the black hole geometry (2.6),whose area gives the equilibrium value of the correspond-ing physical observable.To leading order in large size limit, one can show thatfor Σ of any shape [42] (see also Appendix A) A (eq)Σ = L n V Σ z nh ≡ a eq V Σ , a eq = L n z nh , (2.33)where V Σ denotes the volume of the boundary regionbounded by surface Σ, and a eq can be interpreted as anequilibrium “density.” This result has a simple geometricinterpretation in the bulk – in the large size limit, mostof the extremal surface simply runs along the horizon. Inparticular, for entanglement entropy, S (eq)Σ = L d − G N V Σ z d − h = s eq V Σ (2.34)where we have used the entropy density s eq from (2.10).For a Wilson loop we have W eq ∼ e − √ λ V Σ z h (2.35)where V Σ is now the area of the region enclosed by theloop. The two-point correlation function of an operatorwith dimension ∆ ≈ mL (cid:29) G eq (2 R ) ∼ e − ∆ Rzh . (2.36) D. Further comments on the Vaidya setup
To conclude this section we make some further com-ments on the Vaidya setup:1. It should be kept in mind that while the final equi-librium state has a temperature and coarse grainedthermal entropy density, the Vaidya geometry de-scribes the evolution of a pure state. As a con-sistency check, one can show that for such a pro-cess the entanglement entropy for a region A is thesame as that of its complement [2, 10, 24]. Thusthe equilibrium entanglement entropy (2.34), de-spite having a thermal form, reflects genuine long-range quantum entanglement. The reason (2.34)has exactly the form of a thermal entropy is as fol-lows. We are considering a finite region in a systemof infinite size. Thus the number of degrees of free-dom outside the region is always infinitely largerthan that inside. As a result in a typical excitedpure state the reduced density matrix for the finiteregion appears thermal [43].2. Before the quench, our system is in a vacuum stateof a CFT and thus already has long range correla-tions, whereas the initial state of [1] only has short-range correlations. However, this difference is likelynot important for the questions we are interestedin, which concern the build-up of the finite den-sity of entanglement entropy in (2.34). The long-range entanglement in the vacuum, quantified bythe universal part in (2.28), is measure zero com-pared to (2.34). Heuristically, for odd d , the long-range entanglement entropy in the vacuum, being a R -independent constant, amounts to that of a fewsites inside the region that are fully entangled withthe outside, while in equilibrium, almost all pointsinside the region become entangled. For even d ,there is a logarithmic enhancement of the long-range entanglement in the vacuum, but it is stillmeasure zero compared to the final entanglementin the large region limit.3. From the perspective of entanglement entropy, theequilibration process triggered by the quench buildsup long-range entanglement, as can be seen by com-paring (2.34) and (2.28), whereas from the per-spective of correlation functions (2.19) and Wil-son loops (2.20) in which A appears in the expo-nential with a minus sign, the same process cor-responds to the destruction of correlations (com-pare (2.35)–(2.36) with (2.31)–(2.32)). More specif-ically, long range correlations in the latter observ-ables which were present in the vacuum are re-placed by short-range correlations with correlationlength controlled by z h . However, there is no con-tradiction, as the process of building up entangle-ment also involves redistribution of those in the vac-uum – pre-existing correlations between local oper-ators and over the Wilson loop get diluted by theredistribution process. III. EQUATIONS OF MOTION FOREXTREMAL SURFACES
Here we describe equations of motion for Γ Σ and itsgeneral characteristics when Σ is a strip or a sphere.In such cases Γ Σ can be described by two functions, z ( ρ ) , v ( ρ ) for a sphere, or z ( x ) , v ( x ) for a strip. Forboth shapes the functions satisfy the following boundaryconditions at the boundary as well as regularity condi-tions at the tip of the surface, z ( R ) = 0 , v ( R ) = t , z (cid:48) (0) = v (cid:48) (0) = 0 . (3.1)For a strip we will write x simply as x . It is convenientto introduce the location ( z t , v t ) of the tip of Γ Σ , z (0) = z t , v (0) = v t . (3.2)The sphere and strip being highly symmetric, specifying( z t , v t ) completely fixes Γ Σ . The relations between ( R, t ) and ( z t , v t ) are in general rather complicated and re-quire solving the full equations for z ( ρ ) , v ( ρ ) or z ( x ) , v ( x ).Also, it is possible that a given ( R, t ) corresponds to mul-tiple ( z t , v t )’s, i.e. multiple extremal surfaces have thesame boundary data. Then as mentioned earlier we willchoose the extremal surface with smallest area.For Σ a sphere or strip we will simply denote A Σ ( t ) as A ( R, t ). A. Strip
The area of an n -dimensional surface in (2.1) endingon the strip Σ given by (2.24) can be written as A = 12 ˜ K (cid:90) R − R dx √ Qz n , Q ≡ − v (cid:48) z (cid:48) − f ( z, v ) v (cid:48) (3.3)where ˜ K = L n A strip , (3.4)with A strip being the area of Σ (both sides of Σ are in-cluded which gives the factor in (3.3)). z ( x ) , v ( x ) thensatisfy the equations of motion z n (cid:112) Q∂ x (cid:18) z (cid:48) + f v (cid:48) z n √ Q (cid:19) = 12 ∂f∂v v (cid:48) , (3.5) z n (cid:112) Q∂ x (cid:18) v (cid:48) z n √ Q (cid:19) = n Qz + 12 ∂f∂z v (cid:48) . (3.6)Since the integrand of A does not depend explicitly on x , there is a first integral z n (cid:112) Q = J = const . (3.7)Furthermore, when ∂ v f = 0, equation (3.5) can be inte-grated to give another first integral, z (cid:48) + f v (cid:48) = E = const . (3.8)We are mainly interested in Γ Σ which go through bothAdS and black hole regions. With reflection symmetryabout x = 0, we only need to consider the x > Σ . We now discuss equations in each regionseparately:1. AdS region: From (3.1) and (3.8) we have E = z (cid:48) + v (cid:48) = 0 (3.9)and from (3.7) z (cid:48) = − z n (cid:112) J − z n , J = z nt , (3.10)which give x ( z ) = (cid:90) z t z dy y n (cid:112) z nt − y n , v ( z ) = v t + z t − z . (3.11)2. Matching conditions at the shell: Denoting the val-ues of z and x at the intersection of Γ Σ and the nullshell v = 0 as z c and x c , respectively, we have z c = z t + v t (3.12)and derivatives on the AdS side of the null shell are z (cid:48)− = − v (cid:48)− = − z nc (cid:113) z nt − z nc . (3.13)To find derivatives on the other side, we integratethe equations of motion (3.5)–(3.6) across the nullshell to find the matching conditions v (cid:48) + = v (cid:48)− , Q + = Q − ,z (cid:48) + = z (cid:48)− + 12 g ( z c ) v (cid:48) = (cid:18) − g ( z c ) (cid:19) z (cid:48)− . (3.14)Note we have used the subscript − (+) to refer toquantities on the AdS (black hole) side of the nullshell.3. Black hole region: From matching condi-tions (3.14), J is the same as in the AdS region,i.e. given by (3.10), while E is given by E = 12 g ( z c ) z (cid:48)− < t is no longer constant. From (3.8), v (cid:48) = E − z (cid:48) h (3.16)which can be substituted into (3.7) to obtain z (cid:48) = h ( z ) (cid:18) z nt z n − (cid:19) + E ≡ H ( z ) . (3.17)Substituting (3.17) back in (3.16) we also have dvdz = − h (cid:18) E √ H + 1 (cid:19) . (3.18)Collecting equations in the two regions we find from(3.10) and (3.17) R = (cid:90) z t z c dz (cid:113) z nt z n − (cid:90) z c dz (cid:112) H ( z ) , (3.19)where we have assumed that z ( x ) monotonically de-creases as x increases (recall we let x > z ( x ) can be non-monotonic in which case the aboveequation should be suitably modified. Similar caveatsshould be kept in mind for other equations below. Fromintegrating (3.18), t = (cid:90) z c dzh ( z ) (cid:32) E (cid:112) H ( z ) + 1 (cid:33) . (3.20) Note that at z = z h , h ( z ) − has a pole but the inte-grand in (3.20) remains finite as the second factor van-ishes at z = z h , due to H ( z h ) = E and E <
0. Finally,from (3.10) and (3.17) we have that the area of Γ Σ isgiven by A = A AdS + A BH (3.21)where 1˜ K A AdS = z − nt (cid:90) zczt dy y n (cid:112) − y n (3.22)and 1˜ K A BH = z nt (cid:90) z c dz z n (cid:112) H ( z ) . (3.23)For a given R and t , we can use (3.19) and (3.20) to solvefor z t ( R, t ) , z c ( R, t ) after which (3.21) can be expressedin terms of R and t . B. Sphere
The area of an n -dimensional surface in (2.1) endingon a sphere Σ given by (2.23) can be written as A = K (cid:90) R dρ ρ n − z n (cid:112) Q , Q = 1 − v (cid:48) z (cid:48) − f ( z, v ) v (cid:48) (3.24)where K = L n A sphere R n − . (3.25)It follows that z ( ρ ) , v ( ρ ) satisfy the equations of motion z n √ Qρ n − ∂ ρ (cid:20) ρ n − z n √ Q v (cid:48) (cid:21) = nQz + 12 ∂f∂z v (cid:48) , (3.26) z n √ Qρ n − ∂ ρ (cid:20) ρ n − z n √ Q ( z (cid:48) + f v (cid:48) ) (cid:21) = 12 ∂f∂v v (cid:48) , (3.27)and boundary conditions (3.1). When ∂ v f = 0, equa-tion (3.27) can be integrated to give ρ n − z n √ Q ( z (cid:48) + f v (cid:48) ) = E = const (3.28)which can also be expressed as ρ n − z n f √ Q dtdρ = E (3.29)where t is the Schwarzschild time.Again, we are interested in Γ Σ which go through bothAdS and black hole regions:1. AdS region: Given (3.1), we again have E = 0,which implies that the solution in the AdS regionis the same as that in pure AdS, i.e. is given by [6] z ( ρ ) = (cid:113) z t − ρ , v ( ρ ) = z t + v t − z ( ρ ) . (3.30)2. Matching conditions at the shell: Denoting valuesof z and ρ at the intersection of Γ Σ and the nullshell v = 0 as z c and ρ c , respectively, we have z c = z t + v t , ρ c = (cid:113) z t − z c (3.31)and derivatives on the AdS side of the null shell are z (cid:48)− = − v (cid:48)− = − ρ c z c . (3.32)To find the corresponding derivatives on the otherside, we integrate (3.26) and (3.27) across the shell,which again leads to the matching conditions (3.14)but with z (cid:48)− , v (cid:48)− now as in (3.32).3. Black hole region: The matching implies E = − (cid:18) ρ c z c (cid:19) n g ( z c ) z t < t is no longer constant. Solving for v (cid:48) and Q in terms of z (cid:48) using (3.28), we obtain v (cid:48) = 1 h ( z ) − z (cid:48) + EB (cid:113) z (cid:48) h (cid:113) E B h , B ≡ z n ρ n − (3.34)which, when substituted in (3.26), gives the equa-tion for z (cid:0) h + E B (cid:1) z (cid:48)(cid:48) + (cid:0) h + z (cid:48) (cid:1) (cid:18) n − ρ z (cid:48) + nhz (cid:19) + (cid:0) E B − z (cid:48) (cid:1) ∂ z h . (3.35)From integrating (3.34), the boundary time is t = (cid:90) Rρ c dρh − z (cid:48) + EB (cid:113) z (cid:48) h (cid:113) E B h = (cid:90) Rρ c dρh + E B E B − z (cid:48) EB (cid:113) h + z (cid:48) h + E B + z (cid:48) (3.36)where the second expression is manifestly well-defined atthe horizon, and the integral is evaluated on shell, with z ( ρ ) satisfying equation (3.35) and boundary conditions(3.14) at ρ = ρ c and z ( R ) = 0. Finally, from (3.30) and(3.34), the area of Γ Σ can be written as A = A AdS + A BH (3.37)where1 K A AdS = (cid:90) ρ c dρ ρ n − z n (cid:112) z (cid:48) = (cid:90) ρczt dx x n − (1 − x ) n +12 (3.38) and 1 K A BH = (cid:90) Rρ c dρ ρ n − z n (cid:113) z (cid:48) h (cid:113) E B h . (3.39)Note the story here is significantly more complicatedthan for a strip. One needs to first solve the differen-tial equation (3.35) with initial condition given by thelast equation of (3.14). Imposing the boundary condi-tion z ( R ) = 0 gives a relation between ρ c and z c . Onethen needs to evaluate (3.36) to find z c ( R, t ) , ρ c ( R, t ) andfinally use (3.37) to obtain A ( R, t ). IV. GENERAL GEOMETRIC FEATURES ANDSTRATEGY
We now describe geometric features of Γ Σ during itstime evolution, using as examples the case of Σ being asphere or a strip. For the two shapes the equations ofmotion (given in Sec. III) can be readily solved numer-ically. We are interested in long-distance behavior, i.e.we take R (cid:29) z h . (4.1) A B C D · ··· (a) · ·· C A C B · (b) FIG. 2. Cartoon of the curve ( z t ( R, t ) , v t ( R, t )) for (a) contin-uous and (b) discontinuous saturation. Cartoons of variousextremal surfaces whose tip are labelled above are shown inFig. 3. (a): For continuous saturation the whole curve has aone-to-one correspondence to ( R, t ), and saturation happensat point C continuously. (b): Discontinuous saturation hap-pens via a jump of the extremal surface from one with tipat C (cid:48) to one with tip at C . Along the dashed portion of thecurve, different points can correspond to the same ( R, t ). At fixed R , as t is varied, the tip (3.2) of Γ Σ traces outa curve ( z t ( R, t ) , v t ( R, t )) in the Penrose diagram. Thisprovides a nice way to visualize the evolution of Γ Σ with t . See Fig. 2.0 · A (a) · B (b) · D (c) FIG. 3. Cartoons of extremal surfaces with tip at variouspoints labelled in Fig. 2. Spatial directions are suppressed.(a): At t = 0 + , the extremal surface starts intersecting thenull shell, with z c very small. (b) When t (cid:38) z h , the extremalsurface starts intersecting the null shell behind the horizon.(c) The extremal surface close to continuous saturation forwhich z t − z c is small. Instead of ( z t , v t ) it is sometimes convenient to use( z t , z c ) or ( z t , ρ c ) to specify Γ Σ , where z c and ρ c are thevalues of z and ρ at which the Γ Σ intersects the null shell.For both sphere and strip z c = z t + v t . For a sphere ρ c isgiven by (3.31), while for a strip x c can be obtained bysetting z = z c in (3.11).We now elaborate on various stages of the time evolu-tion of Γ Σ , and strategies for obtaining A ( R, t ) in eachof them.For t <
0, Γ Σ lies entirely in AdS, and z t ( R, t <
0) = (cid:40) R sphere Ra n strip , v t = t − z t (4.2)where a n was introduced in (2.27). A ( R, t ) is indepen-dent of t and is given by its vacuum value. In Fig. 2 thiscorresponds to the part of curve below point A . Notethat as R → ∞ , z t → ∞ .At t = 0 + , or point A , Γ Σ starts intersecting the nullshell (see Fig. 3(a)). For t (cid:28) z h , the point of intersectionis close to the boundary, i.e. z c (cid:28) z h . This defines thepre-local-equilibrium stage mentioned in the Introduc-tion. In this regime, one can extract A Σ ( t ) by expanding both t and A in small z c , which we will do for arbitraryΣ in Sec. VI.When t becomes of order z h , at some point Γ Σ startsintersecting the shell behind the horizon, i.e. z c > z h .An example is point B in Fig. 2, whose correspondingΓ Σ is shown in Fig. 3(b).There exists a sharp time t s after which Γ Σ lies en-tirely in the black hole region. Γ Σ then reduces to thatin a static black hole geometry. It lies on a constantSchwarzschild time t = t outside the horizon and is timeindependent. That is, for t > t s z t ( R, t ) = z b ( R ) < z h , v t = t − σ ( z t ) (4.3)where z b denotes the location of the tip of Γ Σ in thestatic black hole geometry, and in the second equationwe have used (2.7). This corresponds to the part of thecurve above point C in Fig. 2. For t > t s , A ( R, t ) is timeindependent and given by its equilibrium value.The saturation at the equilibrium value at t s can pro-ceed as a continuous or discontinuous transition, as illus-trated in Fig. 2. For a continuous transition, depictedon the left, the entire curve ( z t , v t ) as a function of t has one-to-one correspondence with ( R, t ) and saturationhappens at point C , with t s given by v t ( t s ) = 0 , t s ( R ) = σ ( z b ( R )) = (cid:90) z b dzh ( z ) . (4.4)In contrast, for a discontinuous saturation, depicted onthe right plot of Fig. 2, in the dashed portion of the curve,there are multiple ( z t , v t ) associated with a given ( R, t ).As a result, the minimal area condition requires that theextremal surface jump from point C (cid:48) to C at some t s .In this case there does not exist a general formula for t s .For a discontinuous saturation, A Σ ( t ) is continuous at t s ,but its first time derivative becomes discontinuous.In the case of a continuous saturation, for which thefirst time derivative of A Σ ( t ) is continuous, one can thendefine a critical exponent γ (by definition γ > A Σ ( t ) − A (eq)Σ ∝ − ( t s − t ) γ . (4.5)The “critical” behavior around saturation can be ob-tained as follows. As t → t s , the tip of Γ Σ approachesthe null shell, i.e. z t − z c → z t , z c → z b (this isdepicted by point D in Fig. 2 and Fig. 3(c)). Thus onecan expand both t − t s and A − A eq in small z t − z c , aswe discuss in detail in Sec. XI.So far we have based our discussion on generic featuresof bulk extremal surfaces without referring to explicit so-lutions. To understand what happens during intermedi-ate stages of time evolution, i.e. between B and C in thefigures of Fig. 2, it is useful to work out specific exam-ples of the evolution of ( z t ( R, t ) , v t ( R, t )). In Fig. 4, wegive the parametric plots of ( z t ( R, t ) , z c ( R, t )) for variousvalues of R , for Σ a strip and a sphere, for Schwarzschild h ( z ) with d = 3. From these plots we see a remarkablephenomenon: curves of varying R , after a brief period of1 z t z c (a) z t z c (b) FIG. 4. Parametric curves ( z t ( R, t ) , z c ( R, t )) at fixed R andvarying t for Schwarzschild h ( z ) in d = 3. Different curvescorrespond to R = 2 , , · · · ,
10. In both plots, we choose unitsso that the horizon is at z h = 1. (a): For a strip. Note thesaturation is discontinuous with z c lying behind the horizonat the saturation point where each curve stops. (b): For asphere. The saturation is continuous and z c lies outside thehorizon at the saturation point (in the plot it is too close tothe horizon to be discerned). order O ( z h ), all collapse into a single curve z ∗ c ( z t ) high-lighted by the dashed line in each plot.In Sec. VII, we will show that the universal curve z ∗ c ( z t )corresponds to a critical line in ( z t , z c ) space: for a given z t , Γ Σ reaches the boundary only for z c < z ∗ c . In par-ticular, for a Γ Σ with z c = z ∗ c ( z t ), to which we will referas a “critical extremal surface,” the surface stretches to ρ, v = ∞ . As a consequence, for sufficiently large R and t , ( z t , z c ) lies very close to the critical line, and theevolution of A ( R, t ) is largely governed by properties ofthe critical extremal surfaces. We will show in Sec. IXand XII that this is responsible for the linear growth andmemory loss regimes discussed in [21].To conclude this section we comment on the role of z h in the evolution. As can be seen from the above discus-sion, z h plays the characteristic scale for the evolution ofΓ Σ . There is an important geometric distinction betweenthe time evolution of surfaces with R (cid:46) z h and of thosewith R (cid:29) z h . In the former case, Γ Σ ( t ) stays outsidethe horizon during ts entire evolution, while in the lat-ter case important parts of its evolution are controlled bythe geometry near and behind the horizon. This supports the identification of z h as a “local equilibrium scale” asonly after such time scale does an extremal surface startprobing the geometry around the black hole horizon. V. EVOLUTION IN (1 + 1)
DIMENSIONS
Before going to general dimensions, let us first considerthe case where d = 2 and the final equilibrium state isgiven by the BTZ black hole, i.e. g ( z ) = z /z h . Then n = 1, and Γ Σ is a geodesic whose length can be expressedanalytically in closed form [3, 4], which enables us to di-rectly extract its scaling behavior in various regimes. Re-lated boundary observables are the entanglement entropyof a segment of length 2 R , and equal-time two-point cor-relation functions of operators with large dimension, atseparation 2 R . For definiteness, we consider the entan-glement entropy, and show that its evolution exhibits thefour regimes discussed in the introduction.It is convenient to introduce the dimensionless vari-ables τ ≡ πT t , (cid:96) ≡ πT R , (5.1)where T is the equilibrium temperature. First, recallthe result for entanglement entropy in a CFT at thermalequilibrium [44, 45], S eq ( (cid:96) ) = c (cid:18) sinh (cid:96)(cid:96) (cid:19) + c Rδ = ∆ S eq + S vac . (5.2)Here, the second term S vac is the vacuum value (with δ a UV cutoff), c is the central charge, and ∆ S eq denotesthe difference between thermal and vacuum values. Note∆ S eq is free of any UV ambiguities, and that for (cid:96) (cid:29) S eq = c (cid:96) − c πT δ ) + O ( e − (cid:96) ) . (5.3)Here we see that the log R piece in S vac has been replacedby a log T term, signaling a redistribution of long-rangeentanglement. Also note that the equilibrium entropyand energy densities are given by s eq = πcT , E = πcT . (5.4)Now, the evolution of entanglement entropy in theVaidya geometry (2.1) with g ( z ) = z /z h is given by S ( R, t ) = ∆ S ( R, t ) + S vac , (5.5)where (following expressions are obtained from Eqs. (3)-(5) of [3] with a slight rewriting)∆ S = c (cid:18) sinh τ(cid:96)s ( (cid:96), τ ) (cid:19) , (5.6)and the function s ( (cid:96), τ ) is given implicitly by (cid:96) = 1 ρ cs + 12 log (cid:18) c ) ρ + 2 sρ − c c ) ρ − sρ − c (cid:19) (5.7)2with ρ ≡
12 coth τ + 12 (cid:114) τ + 1 − c c , c = (cid:112) − s . (5.8)At a given (cid:96) , the above expressions only apply for τ < τ s ( (cid:96) ) ≡ (cid:96) . (5.9)At τ = τ s , one finds that c = 0 (i.e. s = 1), ρ = coth τ s ,and ∆ S = ∆ S eq . (5.10)For τ > τ s , ∆ S remains ∆ S eq .To make connections to the discussion in Sec. IV, notethat ρ and s can be related to z t and z c , locations of thetip of Γ Σ and its intersection with the null shell, respec-tively, as ρ = z h z c , s = z c z t . (5.11)Thus equations (5.7)–(5.8) provide an explicit mappingbetween boundary data ( τ, (cid:96) ) and bulk data ( z t , z c ). Inthe discussions that follow, it is convenient to introducean angle φ ∈ [0 , π/
2] with c = cos φ , s = sin φ . (5.12)Then saturation happens at φ = π/
2, when z c = z t ,while φ → z t /z c → ∞ . At fixed τ ,as we vary φ from π/ (cid:96) increases monotonicallyfrom τ to + ∞ . At fixed (cid:96) , as we increase φ from 0 to π/ τ increases monotonically from 0 to τ s . Note we willmostly consider the limit (cid:96) (cid:29)
1, as we are interested inlong-distance physics.
A. Early growth
For any (cid:96) , in the limit τ (cid:28) ρ is large, and in orderfor (5.7) to be satisfied we need s to be small (i.e. φ small). We find that ρ = 1 τ + τ
12 + · · · , s = 1 (cid:96) (cid:18) τ − τ
12 + · · · (cid:19) (5.13)and 3 c ∆ S = τ − (cid:18)
196 + 116 (cid:96) (cid:19) τ + O ( τ ) . (5.14)Note that for z t and z c , (5.13) translates to z c = t (cid:0) O ( t ) (cid:1) , z t = R (cid:0) O ( t ) (cid:1) (5.15)which is consistent with the regime of early growth out-lined in Sec. IV.Thus at early times, the entanglement entropy growsquadratically as∆ S = c τ O ( τ ) = 2 π E t + O ( t ) , (5.16)where we have used (5.4). This result was also obtainedrecently in [46]. B. Linear growth
We now consider the regime (cid:96) (cid:29) τ (cid:29)
1, which corre-sponds in (5.7)–(5.8) to e − τ (cid:28) φ (cid:28) e − τ/ , τ (cid:28) ρ = 12 + φ O (cid:18) e − τ φ (cid:19) , (cid:96) = 2 φ + τ + log φ + O (1) . (5.18)Then from (5.6) we find that∆ S = c τ − c O (cid:18) τ(cid:96) , log (cid:96)(cid:96) , e − τ (cid:19) = 2 s eq t − c · · · . (5.19)The leading term agrees with (1.1). Also note that thesubleading term is negative which is important for themaximal rate conjecture of [21], which we will furtherelaborate in the conclusion section.Note that for z t and z c , equations (5.17)–(5.18) trans-late to z c = 2 z h + · · · , z t z c = 1 φ (cid:29) . (5.20)In Sec. VIII A and Sec. IX we will see that the lineargrowth of entanglement entropy in (5.19) is generic forall dimensions and collapsing geometries, being a conse-quence of the critical surface referred to at the end ofSec. IV. C. Saturation
Let us now examine the behavior of entanglement en-tropy as τ → τ s . For this purpose, consider φ = π − (cid:15) with (cid:15) (cid:28)
1. Then from (5.6)–(5.8), ρ = coth τ −
12 tanh τ (cid:15) − (cid:0) tanh τ (tanh τ − (cid:1) (cid:15) + O ( (cid:15) ) , (5.21) (cid:96) = τ + 12 tanh τ (cid:15) + O ( (cid:15) ) , (5.22)and3 c ∆ S = log sinh ττ + 12 (cid:18) − tanh ττ (cid:19) (cid:15) + O ( (cid:15) ) . (5.23)Now fix (cid:96) and expand τ near τ s , i.e. let τ = τ s − δ , δ (cid:28) δ = 12 tanh τ s (cid:15) + 16 tanh τ s (cid:15) + ( (cid:15) ) (5.24)and3 c δS = 3 c ∆ S eq − √ (cid:112) tanh τ s δ −
16 tanh τ s δ + O ( δ / ) . (5.25)3In particular, in the limit (cid:96) (cid:29) c ∆ S = 3 c ∆ S eq − √ δ − δ + O ( δ / , e − τ s δ ) . (5.26)We see that the approach to saturation has a nontrivialexponent ,∆ S − ∆ S eq ∝ ( t s − t ) + · · · , t → t s . (5.27)This result was also recently obtained in [46].To make connections to the discussion in Sec. IV, notethat for z t and z c , equations (5.22) and (5.23) translateto z c = z t (cid:18) − (cid:15) · · · (cid:19) , z c = z h tanh τ s + · · · (5.28)which is consistent with the picture of continuous satu-ration presented there. D. Memory loss regime
We now show that for τ, (cid:96) (cid:29) τ < τ s , S − S eq de-pends on a single combination of τ and (cid:96) and interpolatesbetween the linear growth of Sec. V B and the saturationregime of Sec. V C. Thus in this regime the “memory” ofthe size (cid:96) of the region is lost.First notice from (5.7) and (5.8) that for any φ , ρ > ρ ∗ ≡ (cid:18) φ (cid:19) , (5.29)and that τ, (cid:96) → ∞ as ρ → ρ ∗ . (5.30)Thus to explore the regime τ, (cid:96) (cid:29)
1, take ρ = ρ ∗ + (cid:15) with (cid:15) (cid:28)
1. Then τ = −
12 log (cid:15) + 12 log (cid:18) φ (cid:19) + O ( (cid:15) ) , (5.31) (cid:96) = −
12 log (cid:15) + (cid:18) cot φ − (cid:19) + 12 log (cid:18) − cos φ + sin φ φ (cid:19) + O ( (cid:15) ) , (5.32)and the entropy (5.6) can be written as3 c ∆ S − c ∆ S eq = τ − (cid:96) − log (sin φ ) + O (cid:0) e − τ , e − (cid:96) (cid:1) . (5.33)Equations (5.31) and (5.32) imply that (cid:96) − τ = χ ( φ ) + O ( (cid:15) ) , χ ( φ ) ≡ (cid:18) cot φ − (cid:19) + log tan φ , (5.34)i.e. as (cid:15) → τ, (cid:96) → ∞ but (cid:96) − τ remains finite. Inverting(5.34) to express φ in terms of (cid:96) − τ , we can write (5.33)in the scaling form∆ S − ∆ S eq = c λ ( (cid:96) − τ ) + O ( e − τ ) (5.35) where the scaling function λ is given by λ ( y ) = − y − log (cid:0) sin h − ( y ) (cid:1) . (5.36)Note that χ ( φ ) monotonically decreases from + ∞ to 0as φ increases from 0 to π . More explicitly, as δ → φ = δ : χ ( φ ) = 2 δ + log δ − O ( δ ) ,φ = π − δ : χ ( φ ) = δ O ( δ ) , (5.37)from which λ has the asymptotic behavior λ ( y ) = − y − log (cid:16) y (cid:17) + O (cid:16) log yy , y − (cid:17) y (cid:29) − √ y − y + O ( y / ) y (cid:28) . (5.38)Then using the expression for large y , we find from (5.35)and (5.3) that for (cid:96) (cid:29) τ (cid:29) c ∆ S = τ − log 4 + O (cid:18) e − τ , τ(cid:96) , log (cid:96)(cid:96) (cid:19) , (5.39)which recovers (5.19), and that for δ ≡ (cid:96) − τ (cid:28) c ∆ S − c ∆ S eq = − √ δ − δ O ( δ ) , (5.40)which recovers (5.26).In Sec. VII A, we will show that (5.29) is precisely thecritical line z ∗ c ( z t ) alluded to near the end of Sec. IV, andthat the scaling behavior discussed above is controlledby properties of critical extremal surfaces associated withthe critical line.Finally, we remark that in higher dimensions, theredoes not exist a closed expression like (5.6), and we haveto rely on geometric features of bulk extremal surfaces toaccess the above regimes of evolution, as was outlined inSec. IV. VI. PRE-LOCAL-EQULIBRIUM QUADRATICGROWTH
In this section, we consider the growth of A Σ ( t ) relativeto the area of a minimal surface in AdS with the sameboundary Σ for t (cid:28) z h . (6.1)Recall our earlier discussion in which we identified z h as alocal equilibrium scale – at the stage of (6.1) the systemhas not yet achieved local equilibrium. Except for theenergy density which is conserved in time, equilibriumquantities such as temperature, entropy, or chemical po-tential are not yet relevant at this stage.We work in general dimensions, and only assume that g ( z ) has the asymptotic expansion (2.8). We will derivea universal result that applies to Σ of arbitrary shape.4At early times, the null shell lies in the UV part of thegeometry, i.e. near the boundary, and the bulk extremalsurface crosses the shell near the boundary, i.e. z c → t → + (see Fig. 3(a)). This implies that: (i) the partof the surface lying in the black hole region is very small,and (ii) the black hole region can be approximated byperturbing pure AdS. Thus our strategy in finding thesmall t behavior of A is to expand t and A in small z c .A general ( n − x a = x a ( ξ α ) , a = 1 , , · · · , d − , α = 1 , , · · · , n − x a are spatial coordinates along the boundary and ξ α are coordinates parameterizing the surface. The area A Σ of Σ is given by A Σ = (cid:90) d n − ξ (cid:112) det h αβ , h αβ = ∂x a ∂ξ α ∂x a ∂ξ β . (6.3)The n -dimensional bulk extremal surface Γ Σ ending onΣ can be parametrized by v ( ξ α , z ) , x a = X a ( ξ α , z ) (6.4)which satisfy the z = 0 boundary conditions v ( ξ α , z = 0) = t , X a ( ξ α , z = 0) = x a ( ξ α ) . (6.5)We also require Γ Σ to be smooth at the tip z t . The area A Σ of Γ Σ can be written as A Σ ( t ) = L n (cid:90) z t dz (cid:90) d n − ξ z − n (cid:112) det γ = (cid:90) z t dz (cid:90) d n − ξ L ( X a , v ) (6.6)where z γ is the induced metric on Γ Σ , γ αβ = ∂X a ∂ξ α ∂X a ∂ξ β − f ( v, z ) ∂v∂ξ α ∂v∂ξ β , (6.7) γ αz = ∂X a ∂ξ α ∂X a ∂z − f ( v, z ) ∂v∂ξ α ∂v∂z − ∂v∂ξ α , (6.8) γ zz = ∂X a ∂z ∂X a ∂z − f ( v, z ) (cid:18) ∂v∂z (cid:19) − ∂v∂z . (6.9) Near the boundary of an asymptotic AdS spacetime, i.e.as z → z/z h (cid:28) X a ( z, ξ α ) = x a ( ξ α ) + O ( z ) , v ( z, ξ α ) = t − z + O ( z ) . (6.10)Now, we denote the solution in pure AdS ( f = 1) withthe same boundary conditions as Γ Σ by X (0) a , v (0) , andas having tip z (0) t and area A (0)Σ . Recall that our goal isto work out the difference∆ A Σ ( t ) = A Σ ( t ) − A (0)Σ (6.11)to leading order in small t . First, note that the pure AdSsolution lies at constant t , i.e. from (2.4) v (0) ( ξ α , z ) = t − z , (6.12)and that as discussed earlier, X a ( ξ, z ) , v ( ξ, z ) deviate bya small amount from corresponding quantities in pureAdS, i.e. X a ( ξ, z ) = X (0) a + δX a , v ( ξ, z ) = v (0) + δv (6.13)where from (6.10), lowest order terms in δX a and δv in z should start at O ( z ). Solving v ( ξz c ) = 0, we then find t = z c + O ( z c ) (6.14)which in turn implies that expanding δX a and δv in small t , the lowest order terms should start at O ( t ).Next, to leading order in small t , (6.11) can be foundby varying the action (6.6),∆ A Σ ( t ) = (cid:90) z (0) t dzd n − ξ δ L δf (cid:12)(cid:12)(cid:12)(cid:12) δf + (cid:90) d n − ξ L ( X (0) , v (0) ; z (0) t ) δz t + (cid:90) d n − ξ (cid:0) Π zA (cid:12)(cid:12) δX A (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) z (0) t , (6.15)where | denotes that a quantity should be evaluated onthe pure AdS solution, X A = ( X a , v ), andΠ zA = ∂ L ∂∂ z X A , δX A = X A − X (0) A . (6.16)In deriving (6.15) we have assumed that the boundary terms associated with integration by part over ξ α vanish.This is true when Σ is compact and there is no boundaryin the ξ α directions, and also when Σ has no dependenceon ξ α , as in the case when Σ is a strip. We proceed toobserve that L ( X (0) , v (0) ; z (0) t ) = 0 (6.17)5as the area element vanishes at the tip of the bulk surface,and that similarly, regularity conditions at the tip forΓ (0)Σ and boundary conditions at infinity imply that thelast term in (6.15) vanishes. Thus only the first termin (6.15) is non-zero. Now note δ L δf = L n z n (cid:112) det γ tr (cid:18) γ − δγδf (cid:19) , (6.18)and from (6.12) δγ αβ δf (cid:12)(cid:12)(cid:12)(cid:12) = 0 , δγ αz δf (cid:12)(cid:12)(cid:12)(cid:12) = 0 , δγ zz δf (cid:12)(cid:12)(cid:12)(cid:12) = − . (6.19)Given that for small z , X (0) a ( ξ α , z ) = x a ( ξ α ) + O ( z ) , (6.20)we find γ αβ = h αβ + O ( z ) , γ αz = O ( z ) , γ zz = 1 + O ( z ) . (6.21)Thus to leading order δ L δf (cid:12)(cid:12)(cid:12)(cid:12) = − L n z n √ det h (6.22)and since δf = − M z d + · · · (6.23)is nonvanishing only for z ∈ (0 , z c ), we find (recall (6.14))∆ A Σ = L n A Σ M (cid:90) z c z d − n dz = L n A Σ M d − n + 1) t d − n +1 + · · · . (6.24)For entanglement entropy, we have n = d − S = ∆ A Σ G N = L d − M G N A Σ t + · · · = πd − E A Σ t + · · · (6.25)where E given in (2.9) is the energy density of the sys-tem. This expression is free of any UV ambiguities andis universal for any Σ and bulk geometry g ( z ), dependingonly on the energy density of the state.More general metrics (2.16)–(2.18) typically involvescalar fields and the asymptotic behavior of the metriccomponents h ( z ) and l ( z ) in the black hole region in gen-eral depend on the falloff of the scalar fields. Further-more the energy density can also receive contributionsfrom scalar fields. Thus it appears likely that (6.25) maynot generalize to such a case. It would be interesting tounderstand this further. This term has to vanish to ensure X (0) A is a proper solution toequations of motion. VII. CRITICAL EXTREMAL SURFACES
In this section, using as examples cases of Σ being astrip or sphere, we show that the universal curve z ∗ c ( z t )for different R ’s observed in Fig. 4 corresponds to a crit-ical line in ( z t , z c ) space: for a given z t , Γ Σ reaches theboundary only if z c < z ∗ c . In particular, when Γ Σ liesprecisely on the critical line z c = z ∗ c ( z t ), in which casewe refer to it as a critical extremal surface , it asymptotesto ρ, v = ∞ along some constant z = z m ≥ z h . A. Strip
With Σ a strip, the black hole portion of Γ Σ is givenby z ( ρ ) satisfying the equation of motion (3.17), z (cid:48) = h ( z ) (cid:18) z nt z n − (cid:19) + E ( z t , z c ) ≡ H ( z ) ,E = g c (cid:18) z nt z nc − (cid:19) (7.1)and the boundary condition at z c (3.14), z (cid:48) + = − (cid:16) − g c (cid:17) (cid:115) z nt z nc − , (7.2)where we have denoted g c ≡ g ( z c ) (7.3)and E has been obtained from (3.15). As discussed inSec. IV, for t (cid:29) z h , the extremal surface intersects theshell behind the horizon, i.e. z c > z h , except possiblynear saturation.Equation (7.1) specifies a one-dimensional classical me-chanics problem, with the qualitative behavior of z ( ρ )readily deduced from properties of H ( z ). To acquiresome intuition on such behavior, we proceed to work con-cretely with the Schwarzschild (or Reissner-Norstrom) g ( z ). Since our discussion clearly applies to more thanthe examples of g ( z ) being examined, we maintain thegeneral notation g ( z ) and h ( z ) = 1 − g ( z ) in all expres-sions. However, we do not attempt to characterize themost general class of g ( z ) for which H ( z ) exhibits prop-erties discussed below, nor do we attempt to classify al-ternative possibilities.To begin, note that from (7.2), when g c > z (cid:48) + > Σ initiallymoves away from the boundary to larger values of z . Weintroduce z s as g ( z s ) = 2 , z s > z h . (7.4) z (cid:48) + changes sign when z c crosses z s . Next, note thatfor Schwarzschild g ( z ), the first term in (7.1) is zero at z = z h and z = z t , and negative in between. Thus H ( z )6has a minimum between z h and z t which we denote z m .Setting H (cid:48) ( z m ) = 0, we find z m satisfies the equation z nt = h (cid:48) ( z m ) z n +1 m z m h (cid:48) ( z m ) − nh ( z m ) . (7.5)It is easy to see that such a minimum also exists forReissner-Norstrom g ( z ). The following discussion onlydepends on the existence of such a minimum. We nowintroduce z ∗ c given by H ( z m ) (cid:12)(cid:12) z c = z ∗ c = 0 . (7.6)Note z ∗ c and z m are functions of z t only. Also note thatthere is a special value of z t , which we call z ( s ) t , where z m ( z ( s ) t ) = z s . Evaluating (7.5) at z m = z s , we find that z ( s ) t = (cid:18) z s h (cid:48) ( z s ) z s h (cid:48) ( z s ) + 2 n (cid:19) n z s . (7.7)In fact, there are two additional occurences at z t = z ( s ) t .First, one can check z ∗ c = z s = z m . (7.8)Second, by taking the derivative of E in (7.1) with re-spect to z c and plugging in the values z ( s ) t and z ∗ c = z s ,we find dE dz c (cid:12)(cid:12)(cid:12)(cid:12) z ∗ c = 0 . (7.9)In the limit z t → ∞ , assuming that z m remains finite(which is not always true, see e.g. (7.24)–(7.25) below),equations (7.5) can be simplified to z m h (cid:48) ( z m ) h ( z m ) = 2 n . (7.10)Similarly in the z t → ∞ limit, assuming that z t z m , z t z ∗ c →∞ , equation (7.6) can be simplified to g ( z ∗ c )4 z ∗ nc = − h ( z m ) z nm . (7.11)In general, for a given z t there are multiple positiveroots to the equation (7.6). In fact, equation (7.9) sug-gests that two branches of roots of (7.6) are convergingat z ( s ) t . However, for any g ( z ) which satisfies g ( z h ) = 1and g (cid:48) ( z h ) >
0, it can be checked that as z t → z h so that z t = z h (1 + (cid:15) ), (cid:15) (cid:28)
1, we have z m = z t (cid:18) − (cid:15) + · · · (cid:19) = z h (cid:18) (cid:15) + · · · (cid:19) (7.12)and there is a unique z ∗ c satisfying z ∗ c = z t (cid:0) − O ( (cid:15) ) (cid:1) . (7.13)Now, increasing z t and following this root, we note that: 1. In region I given by z h < z t < z ( s ) t , z s > z ∗ c > z m > z h , dE dz c (cid:12)(cid:12)(cid:12)(cid:12) z ∗ c < , (7.14)and thus for z c < z ∗ c , z (cid:48) + < . (7.15)2. In region II given by z t > z ( s ) t , z s < z ∗ c < z m , dE dz c (cid:12)(cid:12)(cid:12)(cid:12) z ∗ c > , (7.16)and for z s < z c < z ∗ c , z (cid:48) + > . (7.17)See Fig. 5 for plots of z ∗ c and z m as functions of z t forSchwarzschild g ( z ) and one instance of RN g ( z ). (a) z t (cid:72) s (cid:76) z t z s (b) z t FIG. 5. Examples of z m (blue) and z ∗ c (red) as functions of z t for (a): Schwarzschild g ( z ) with d = 4 and n = 3, (b): RN g ( z ) with d = 4, u = 0 .
2, and n = 3. We have fixed z h = 1.Note in (b), z s does not exist and there is only region I (7.14). With the above properties established, the behavior of z ( ρ ) can be read off from Fig. 6–Fig. 7. In particular, fora given z t , Γ Σ only reaches the boundary for z c < z ∗ c ( z t ),and at z c = z ∗ c ( z t ), it asymptotes to a critical extremalsurface z = z m . Note that this conclusion holds in thepresence of other roots to (7.6) as long as the followingare satisfied:1. In region I there is no other root lying between z m and z ∗ c .2. In region II there is no other root lying between z s and z ∗ c .7 z m z c z H z ← (a) z m z c z H z ← ← (b) z m z c z H z A ← ← (c) FIG. 6. H ( z ) for z t < z ( s ) t . In this case z ∗ c > z m and z (cid:48) + < z c (cid:46) z ∗ c . (a): z c = z ∗ c . z ( x ) decreases then asymptotesto z = z m . (b): z c = z ∗ c − (cid:15) for (cid:15) >
0. Since dE dz c (cid:12)(cid:12) z ∗ c < H ( z m ) > z (cid:48) remains negative throughout and Γ Σ can reachthe boundary. If (cid:15) is small, then H ( z m ) is small (positive) andΓ Σ hangs near the critical extremal surface z = z m for a longinterval in x before eventually reaching the boundary. (c): z c = z ∗ c + (cid:15) . Since dE dz c (cid:12)(cid:12) z ∗ c >
0, now H ( z m ) < z ( x ) firstdecreases to point A, then turns around and never reaches theboundary. It can be readily checked that these conditions are sat-isfied by Schwarzschild and Reissner-Nordstrom g ( z ) forgeneral d . In Fig. 8–9 we plot some examples of near-critical surfaces with z c ≈ z ∗ c .Now let us mention some explicit results. ForSchwarzschild g ( z ) (2.11) and d = 2, the case discussedin Sec. V, one finds z s = √ z h , z ( s ) t = 2 z h , z m = √ z t z h , (7.18)and z ∗ c = 12 (cid:18)(cid:113) z t + 4 z t z h − z h − z t (cid:19) + z h , (7.19)where z ∗ c → z h , as z t → ∞ . (7.20) Note that in this case there are two positive roots to equa-tion (7.6). The root below is the branch chosen by (7.13). z m z c z H z ← (a) z m z c z H z ← ← B (b) z m z c z H z ← ← (c) FIG. 7. H ( z ) for z t > z ( s ) t . In this case z ∗ c < z m . If z c < z s , z (cid:48) c < z ( ρ ) monotonically decreases to zero.These plots show what happens when z c > z s so that z (cid:48) + > z c = z ∗ c . z ( ρ ) increases and asymptotes to z = z m .(b): z c = z ∗ c − (cid:15) for a positive (cid:15) >
0. With dE dz c (cid:12)(cid:12) z ∗ c > H ( z m ) < z ( x ) first increases, then turns around at point B and monotonically decreases to zero. If (cid:15) is small, then H ( z m )is small (negative), and Γ Σ hangs near the critical surface z = z m (i.e. near B ) for a long interval in x before eventuallyreaching the boundary. (c): z c = z ∗ c + (cid:15) . With H ( z m ) > z ( x ) only increases and never reaches the boundary. Using (5.11) and (5.12), one finds that the criticalline (7.19) is precisely equivalent to (5.29). Simi-larly, (7.20) maps to (5.20).For Schwarzschild g ( z ) (2.11) in general d , one has z s = 2 d z h , z ( s ) t = (cid:18) dd − n (cid:19) n d z h . (7.21)but the expressions for z m and z ∗ c get complicated. Inthe following discussion we will mostly be interested inthe z t → ∞ limit, for which introducing η ≡ nd , (7.22)we find:1. For η > z m = (cid:18) ηη − (cid:19) d z h , z ∗ c = (cid:18) η − η − η η (cid:19) d − n ) z h . (7.23)8 xz c (cid:42) z m z h z (cid:72) x (cid:76) FIG. 8. Left: Behavior of near-critical surfaces with (cid:15) = ± − for z t = 1 . z h < z ( s ) t , for Schwarzschild with d = 3, n = 2, and Σ a strip. The critical surface runs to infinite x along z = z m . For small (cid:15) , the solution runs along the criticalsurface for a while before reaching the boundary or black holesingularity, depending on the sign of (cid:15) . Right: Cartoon of thenear-critical surfaces on the Penrose diagram. Dashed curveis constant z = z m slice. xz c (cid:42) z m z h z (cid:72) x (cid:76) FIG. 9. Left: Behavior of near-critical surfaces with (cid:15) = ± − for z t = 3 z h > z ( s ) t , for Schwarzschild g ( z ) with d = 3, n = 2, and Σ a strip. Right: Cartoon of the near-criticalsurfaces on the Penrose diagram. Note that both z m and z ∗ c remain finite as z t → ∞ and z ∗ c z m = (cid:16) η − η (cid:17) d − n ) < η < z m = (1 − η ) n z t , z ∗ c ∼ z d − n d − n ) t (cid:28) z m . (7.24)Note that both z m and z ∗ c approach infinity as z t →∞ .3. For η = 1, i.e. n = d , z m = √ z t z h , z s < z ∗ c = 2 d − n z h (cid:28) z m . (7.25)In this case z m approaches infinity, but z ∗ c remainsfinite as z t → ∞ .For Reissner-Nordstrom g ( z ) (2.12), we find that for n = d − z t → ∞ , z m = (cid:32) d − d − Q z d − h (cid:33) d z h (7.26) and z ∗ c is also finite but is given by a complicated expres-sion which is not particularly illuminating. Also notethat in the extremal limit, z m → z h , z ∗ c → (cid:18) − d (cid:19) d − z h . (7.27)and that for sufficiently large Q , z m never reaches z s forall z t . B. Sphere
We now examine the case of Σ being a sphere with n ≥ d ≥ z ( ρ ) giving the black holeportion of Γ Σ is now a second order nonlinear differentialequation, (3.35). We copy it here for convenience, (cid:0) h + E B (cid:1) z (cid:48)(cid:48) + (cid:0) h + z (cid:48) (cid:1) (cid:18) n − ρ z (cid:48) + nhz (cid:19) + (cid:0) E B − z (cid:48) (cid:1) ∂ z h E = − (cid:18) ρ c z c (cid:19) n g ( z c ) z t , B ≡ z n ρ n − . (7.29)We again expect that for a given z t , there is a critical z ∗ c beyond which Γ Σ never reaches the boundary. For agiven h ( z ), z ∗ c ( z t ) can be readily found by numericallysolving (7.28). From the strip analysis (7.23)–(7.25), anatural guess for Schwarzschild h ( z ) is that for η = nd ≥ z ∗ c remains finite as z t → ∞ . This appears to besupported by numerical results. In Fig. 10 we show someexamples. z t z c (cid:42) FIG. 10. Plot of z ∗ c as a function of z t for d = 3 Schwarzschildwith n = 2 (blue), d = 4 with n = 3 (red), and d = 6 with n = 2 (yellow). We plot in the unit z h = 1. For the lastcase z ∗ c appears to grow with z t as z t . This should also becompared with the strip case (7.24) where z ∗ c grows with z t as z t . At z c = z ∗ c the critical solution z ∗ ( ρ ) should reach ρ = ∞ along some constant z surface. Now, solving (7.28)9for a constant z in the limit ρ → ∞ , one finds the uniquesolution z = z h . (7.30)In other words, independent of the choice of z t and thefunction g ( z ), the critical extremal surface approachesand runs along at the horizon to ρ = ∞ . Expand-ing about the solution (7.30) in the equation (7.28), onefinds a perturbation which grows exponentially in ρ (inSec. XII we work this out explicitly). By tuning z c to z ∗ c ,one ensures that this exponentially growing perturbationis absent and z → z h as ρ → ∞ . For z c = z ∗ c (1 − (cid:15) ), (cid:15) (cid:28)
1, the perturbation acquires a small coefficient, and z ( ρ ) runs along the horizon for a while before eventu-ally breaking away. Depending on the sign of (cid:15) , it eitherapproaches the boundary ( (cid:15) >
0) or turns away from it( (cid:15) < Ρ z c (cid:42) z h z (cid:72) Ρ (cid:76) FIG. 11. Left: Behavior of near-critical surfaces with (cid:15) = ± − for z t = 1 . z h ∼ z ∗ c , for Schwarzschild with d = 4, n = 3, and Σ a sphere. Note the surfaces now run along thehorizon (c.f. Figs. 8, 9). Right: Cartoon of the near-criticalsurfaces on the Penrose diagram. When z t is large and z ∗ c remains finite in the large z t limit, the critical extremal solution z ∗ ( ρ ) has another in-teresting feature which will play an important role in ourdiscussion of the linear growth of entanglement entropyin Sec. IX. From (3.31), for a finite z c ≈ z ∗ c , ρ c = z t + O (1 /z t ) , z t → ∞ . (7.31)Then for the range of ρ satisfying ρ ≥ ρ c and ρρ c ≈ z ∗ ( ρ ) ≈ z m with z m given by nh ( z m ) z m + (cid:18) z m z ∗ c (cid:19) n g ( z ∗ c ) h (cid:48) ( z m )8 = 0 . (7.32)The above equation is obtained from (7.28) by setting z ( ρ ) = z m , z c = z ∗ c and ρ nc z t ρ n − = 1. This results ina plateau at z = z m for a range of ρ ∼ ρ c as indicatedin Fig. 12. Note equation (7.32) agrees prescisely withequations (7.10)–(7.11) for a strip. That is, provided the z ∗ c in (7.32) agrees with that of the strip, the z m deter-mined from (7.32) agrees precisely with the location ofthe critical surface for a strip. We will show in Sec. IX Bthis is indeed the case. Ρ z h z c z m z Ρ FIG. 12. Cartoon of z ∗ ( ρ ) for z t (cid:29) z ∗ c , with z ∗ c ∼ O (1)as z t → ∞ : There is an intermediate plateau at z = z m for ρ ∼ ρ c . The critical surface eventually approaches the horizonfor ρ (cid:29) ρ c . C. Summary
FIG. 13. The dotted line denotes a curve at constant z ,along which v increases from −∞ to + ∞ from bottom (notshown) to top. The dashed purple line corresponds to Γ ∗ Σ , thecritical extremal surface, while the green lines correspond toΓ Σ with v t just above and below v ∗ t . In this section, we showed explicitly for cases of Σ beinga strip or sphere that in the Penrose diagram there ex-ists a critical line v ∗ t ( z t ): Γ Σ reaches the boundary onlyfor v t < v ∗ t , with the critical extremal surface Γ ∗ Σ corre-sponding to v ∗ t ( z t ) stretching to R, t = ∞ . See Fig. 13.The same phenomenon should apply to general shapes.In the numerical plots presented in Sec. IV (see Fig. 4),we saw that for t (cid:38) O ( z h ), constant R trajectories of Γ Σ in the ( z t , z c ) plane collapse onto a single curve. Fromthe above discussion, we now understand that this is a Recall v t = z c − z t . Thus statements regarding z ∗ c can immedi-ately be translated to those about v ∗ t . z ∗ c ( z t ) exists on which Γ Σ asymptotes to a critical extremal surface that extendsto infinite R and t , and (ii) z ∗ c remains finite (of order O ( z h )) as z t → ∞ . Thus at large fixed R , when t becomessufficiently large, i.e. of order O ( z h ), ( z t , z c ) quickly ap-proaches the critical line z ∗ c ( z t ). This is clearly exempli-fied in the (1 + 1)-dimensional story in Sec. V D. There, (cid:15) , parameterizing the distance to the critical line (5.29)(or (7.19)), gave the leading large (cid:96) and τ behavior, while φ in (5.29) (or z t in (7.19)), parametrizing the locationon the critical line, mapped to (cid:96) − τ or τ s − τ .In short, for large R, t (cid:29) z h , with corresponding( z t , v t ) lying very close to the critical line v ∗ t ( z t ), Γ Σ closely follows Γ ∗ Σ before deviating away to reach theboundary. The evolution of A Σ can then be largely de-termined from that of Γ ∗ Σ . Again, this is seen in thediscussion in (1 + 1)-dimensions of Sec. V. In higher di-mensions, with much less analytic control, this featureprovides a powerful tool for extracting the evolution of A Σ ( t ).For Σ a strip, the critical extremal surfaces asymptotesto a constant- z hypersurface z = z m lying inside thehorizon, i.e. z m > z h with z m depending on the function h ( z ) in (2.5). It is important to keep in mind that z t changes during the time evolution, and so does z m .For Σ a sphere, the critical extremal surface for largeenough z t forms an intermediate plateau at some z = z m before running along the horizon z = z h all the way to ρ, v = ∞ , see Fig. 12. For moderate z t > z h , the criticalextremal surface runs along the horizon z = z h to ρ, v = ∞ with no plateau at z = z m , see Fig. 11.We will see below that for a sphere, the plateau at z = z m governs a linear growth in A at early times, whilethe plateau at the horizon governs a memory loss effectat late times. VIII. LINEAR GROWTH: STRIP
In this section, we show that with Σ given by a strip A ( R, t ) grows linearly with t for R (cid:29) t (cid:29) z h . The evolu-tion can be straightforwardly worked out from the discus-sion of Sec. VII A and as we will see is largely controlledby the critical extremal surface discussed in the last sec-tion. The same growth also applies to a sphere and othershapes as will be discussed in the next section. A. Linear growth
To obtain the behavior for R (cid:29) t (cid:29) z h , we consider z c close to z ∗ c for some z t , z c = z ∗ c (1 − (cid:15) ) , (cid:15) (cid:28) z ∗ c z t , z m z t (cid:28) , z ∗ c | log (cid:15) | (cid:28) . (8.2) In this regime we can expand t , R , and A in a doubleexpansion of 1 /z t and (cid:15) .We now proceed to evaluate the boundary quantities t , R , and A using (3.19)–(3.23). Note that these equationsshould be modified when z ( x ) is not monotonic, whichhappens, for example, for z t > z ( s ) t . Then from (7.16), z c ≈ z ∗ c < z m , i.e. after intersecting the shell, z ( x ) firstmoves to larger values of z before turning around as illus-trated in Fig. 7(b) and Fig. 9. In this case equation (3.19)should be modified to R = (cid:90) z t z c dz (cid:113) z nt z n − (cid:18)(cid:90) z r z c dz + (cid:90) z r dz (cid:19) (cid:112) H ( z )(8.3)and similarly for others. In the above equation z r is theroot of H ( z ) which is slightly smaller than z m (i.e. point B of the second plot of Fig. 7), and z r = z m for (cid:15) = 0.It is useful to separate z ( x ) into four regions (seeFig. 9): (i) AdS region from z t to z c , (ii) from z c tonear z m , (iii) running along z m , and (iv) from near z m toboundary z = 0. One can then check that contributionsto t , R , and A − A vac from regions (ii) and (iv) are atmost O ( z ∗ c ). Now let is look at region (iii). Near z = z m , with z c = z ∗ c (1 − (cid:15) ), we have H ( z ) = H ( z − z m ) + b(cid:15) , (8.4)where H = 12 H (cid:48)(cid:48) ( z m ) , b = − z ∗ c dE dz c (cid:12)(cid:12)(cid:12)(cid:12) z ∗ c . (8.5)Note H > b < b >
0) for z t > z ( s ) t ( z t 1, we have the entanglemententropy ∆ S = ∆ A G N = s eq A strip v E t + O (1) (8.16)where s eq is the equilibrium entropy density in (2.10),and v E ≡ v d − = (cid:18) z h z m (cid:19) d − (cid:112) − h ( z m ) . (8.17)In the regime of (8.2) we can approximate the valueof z m in various equations above by that at z t = ∞ . Soto leading order in large R limit, the evolution is linear.Note in order for (8.2) to be satisfied we need t to be largeenough so that z c is sufficiently close to z ∗ c , but not toolarge such that z t becomes comparable to z ∗ c (see (8.8))to invalidate (8.2). B. Example: Schwarzschild Let us now consider the Schwarzschild case for explicitillustration. From (7.23)–(7.25), depending on the valueof η = nd , z ∗ c and z m behave differently in the limit of alarge z t . Below we consider these situations separately.While we are considering Schwarzschild, the discussiononly depends whether z ∗ c and z m have a finite limit as z t → ∞ . So we will still keep h ( z ) general in our discus-sion. η > For η > 1, which covers the case of entanglement en-tropy n = d − d > 2, both z ∗ c and z m remain finiteof order O ( z h ) in the limit of large z t . The assump-tions (8.2) then apply when R (cid:29) t (cid:29) O ( z h ).In this case we can show that the linear growth (8.14)in fact persists all the way to saturation, which happensvia a discontinuous transition. We do this by assumingthe conclusion, strongly suggested by Fig. 4, and checkingself-consistency.With the linear growth (8.14), A will reach its equilib-rium value (2.33) at time t s = Rv n = R (cid:16) z h z m (cid:17) n (cid:112) − h ( z m ) , (8.18)when, from (8.8) and (8.12), z t = Ra n (cid:18) − (cid:18) z m z t z h (cid:19) n (cid:19) + · · · . (8.19)From (7.23), for η > z t , so we find that when the systemreaches the equilibrium value, z t is still very large.When t is greater than (8.18), equation (8.14) ex-ceeds its equilibrium value, and the extremal surface withsmallest area is no longer a near-critical extremal surfaceto which (8.14) applies, but one that lies solely in theblack hole region. Thus the extremal surface jumps at t s , and the saturation is discontinuous. Note that forentanglement entropy, the saturation time is t s = Rv E (8.20)where v E was given in (8.17). η = 1 For η = 1, which covers the case of entanglement en-tropy in d = 2 examined earlier in Sec. V and that of aspacelike Wilson loop in d = 4, z ∗ c remains finite but z m increases with z t in the large z t limit. In this case, thereis still a linear regime, with v n = 1 . (8.21)2Furthermore, due to (7.25), the expression inside paren-theses in (8.19) becomes zero at the time (8.18), i.e. z t becomes comparable to z c before (8.18) is reached. Thusthe system exits the linear growth regime before satura-tion. This is consistent with what we saw in Sec. V forthe d = 2 case. In Sec. XI and Sec. XII we discuss thebehavior of the system after exiting the linear regime inhigher dimensions. η < For η < 1, from (7.24) both z ∗ c ∼ z αt (with α < 1) and z m ∝ z t grow with z t in the limit z t → ∞ . Then since z ∗ c is also very large for large z t , it may take a long time for z c to reach z ∗ c . If z t is still O ( R ) as z c first approaches z ∗ c , the linear regime could still exist. Supposing sucha regime exists, equation (8.15) gives for Schwarzschild h ( z ) v (S) n = (cid:18) z m z h (cid:19) d − n → ∞ , (8.22)which is physically unreasonable and suggests that a lin-ear regime does not exist. Explicit numerical calculationappears to be consistent with this expectation [47]. IX. LINEAR GROWTH: GENERAL SHAPES In this section we generalize the linear growth foundin the last section for a strip to general shapes. We showthat for t in the range R (cid:29) t (cid:29) z h , A Σ ( t ) genericallyexhibits linear growth in t with a slope independent of theshape of Σ. Again the technical requirement is that z ∗ c should remain finite as z t → ∞ , which for Schwarzschild g ( z ) amounts to 2 n ≥ d .We first revisit the strip story and rederive the lin-ear growth from a scaling limit, which we can extendstraightforwardly to general shapes. We will also extendresults to the wider class of metrics (2.16). A. Revisiting strip: a scaling limit The linear growth of the last section occurs when z t islarge but z ∗ c remains finite in the limit z t → ∞ . In thisregime, with z c ≈ z ∗ c we have (from (3.11)) x c = x ( z c ) = a n z t − z n +1 c nz nt + · · · . (9.1)Also from (8.8) and (8.12) a n z t = R − O ( z − nt ) . (9.2)The above equations suggest that in the black hole regionwe should consider a scaling coordinate y = ( R − x ) z nt . (9.3) Indeed, in terms of y equation (7.1) (or (3.17)) has ascaling form independent of z t to leading order as z t →∞ , (cid:18) dzdy (cid:19) = h ( z ) z n + a , a = g c z nc . (9.4)Similarly, to leading order in 1 /z t , equation (3.18) be-comes dvdz = 1 h a (cid:113) h ( z ) z n + a − . (9.5) z c z FIG. 14. In the limit of a large z t and a finite z c ≈ z ∗ c , theevolution in the black hole region is essentially solely in thetime direction, with two sides of the strip evolving indepen-dently. From (9.4) and (9.5), we conclude dxdz ∼ z nt , dvdz ∼ O (1) . (9.6)Then using z as the independent variable, the action (3.3)in the black hole region is A BH = L n A strip (cid:90) z c dz z n (cid:115)(cid:18) dxdz (cid:19) − dvdz − h (cid:18) dvdz (cid:19) = L n A strip (cid:90) z c dz z n (cid:115) − dvdz − h (cid:18) dvdz (cid:19) (9.7)where in the second equality we have dropped the term (cid:0) dxdz (cid:1) ∼ O ( z − nt ). It may look odd that in (9.7) x ( z )completely drops out. This in fact has a simple geometricinterpretation: from (9.1)–(9.2), by the time the extremalsurface reaches z c , x ( z c ) = R − O ( z − nt ) has essentiallyreached its boundary value R , while v ( z c ) is zero and stillfar away from its boundary value v ( z = 0) = t . Thus theevolution of the extremal surface in the black hole region(for z < z c ) is almost completely in the time direction.See Fig. 14 for an illustration. For purposes of calculatingthe area A to leading order in 1 /z t , we can simply ignore3the evolution in x -direction. As a consistency check, weindeed recover (9.5) by variation of (9.7).Integrating (9.5) we find that t = (cid:90) z c dzh a (cid:113) h ( z ) z n + a − (9.8)and further substituting (9.5) into (9.7) we have A BH = L n A strip (cid:90) z c dz z n (cid:113) h ( z ) z n + a (9.9)The linear growth of A ( t ) can now be immediately un-derstood from (9.8) and (9.9). As before, for z c = z ∗ c with z ∗ c given by (7.11), h ( z ) z n + a has a double zero atits minimum z m which precisely coincides with (7.10) .For z c = z ∗ c (1 − (cid:15) ) with (cid:15) → 0, both the integrals for t and A BH are then dominated by region around z m , andwe precisely recover (8.14).Note that the action (9.7) as well as the linear growthof A is in fact identical to that of [25], where entangle-ment entropy between half spaces lying on two asymp-totic boundaries of an eternal AdS black hole was con-sidered. The agreement can be easily understood fromFig. 14; in the large z t limit, each half of the strip evolvesindependently in the black hole region solely in the timedirection, which coincides with the set-up of [25]. B. General shapes z c z FIG. 15. A cartoon of an extremal surface for Σ with somearbitrary shape, in the large size limit and t in the linearregime. Upon entering the black hole region, the extremalsurface has essentially attained its boundary shape Σ. Theevolution in the black hole region is essentially solely in thetime direction and is the same as that for a strip. h ( z ) z n + a differs from H ( z ) of (7.1) only by an overall scalingand thus has the same minimum and zero. The intuition obtained from the above discussion fora strip and Fig. 14 can now be generalized to arbitraryshapes. For arbitrary Σ, we again expect that in the limit R (cid:29) t (cid:29) z h , the evolution of the extremal surface afterentering the shell will be essentially solely in the timedirection, as indicated in Fig. 15. In other words, in thelarge size limit, when z c is much smaller than the size ofΣ, the curvature of Σ should not matter in the black holeand each point of the extremal surface essentially evolveslike one on a strip. Below we present arguments that thisis indeed the case.Consider a smooth entangling surface Σ which can beparameterized in terms of polar coordinates (2.22) as ρ = Rr (Ω) , x a = 0 (9.10)where Ω denotes collectively the angular coordinates pa-rameterizing Σ, R is the size of Σ, and the function r (Ω)specifies the shape of Σ . The bulk extremal surface canthen be parameterized in terms of ρ ( z, Ω) , v ( z, Ω) withboundary conditions ρ ( z = 0 , Ω) = R r (Ω) , v ( z = 0 , Ω) = t (9.11)and regularity at the tip of the surface.Writing (see (2.22)) d Ω n − = (cid:88) i g i (Ω) dθ i , d n − Ω = (cid:89) i √ g i dθ i (9.12)the area of Σ can be written as A Σ = R n − (cid:90) d n − Ω r n − (Ω) (cid:115) r (cid:88) i r i g i (9.13)where r i ≡ ∂ θ i r (Ω) . (9.14)Meanwhile, in the Vaidya geometry, the action for an n -dimensional extremal surface ending on the above Σ canbe written as A Σ = L n (cid:90) z t δ dz (cid:90) d n − Ω ρ n − z n (cid:112) Q (9.15)with Q = ρ (cid:48) − v (cid:48) − f ( v, z ) v (cid:48) + 1 ρ (cid:88) i g i G i − ρ (cid:88) i,j ( ρ i v j − ρ j v i ) g i g j (9.16)where we have used the notation ρ (cid:48) ≡ ∂ z ρ, ρ i ≡ ∂ i ρ, v (cid:48) ≡ ∂ z v, v i ≡ ∂ i v (9.17)and G i = − f ( v, z )( ρ (cid:48) v i − ρ i v (cid:48) ) +2 ρ i ( ρ (cid:48) v i − ρ i v (cid:48) ) − v i . (9.18)4In (9.15) δ is a short-distance cutoff. It is readily foundthat in the black hole region ρ and v have the followingsmall z expansion (for z (cid:28) z h ) ρ ( z, Ω) = Rr (Ω) − z R ˜ r (Ω) + · · · (9.19) v ( z, Ω) = t − z + O ( z n +1 ) (9.20)where ˜ r (Ω) is a function which can be determined from r (Ω).For R (cid:29) t , to leading order in 1 /R , the part of theextremal surface in the AdS region can be approximatedby that in pure AdS, which we denote ρ (0) ( z, Ω) (andfor which t constant). For z/R (cid:28) ρ (0) has the theexpansion ρ (0) ( z, Ω) = Rr (Ω) + O ( R − ) (9.21)Note that in contrast to (9.19) which applies only to z (cid:28) z h , due to the scaling symmetry of pure AdS and that Σas defined in (9.10) has a scalable form, equation (9.21) infact applies to any z/R (cid:28) z ∼ z c ≈ z ∗ c . Thus we conclude that when the extremal surfaceenters the shell at z c , ρ ( z c , Ω) = Rr (Ω) − O ( R − ) . (9.22)From (9.19)–(9.20) and (9.22), the extremal surfacein the black hole region should then have the followingscaling ρ (cid:48) ∼ O ( R − ) , ρ i ∼ O ( R ) , v i ∼ O ( R − ) , v (cid:48) ∼ O (1) . (9.23)Plugging in the above scaling into the action (9.15) wefind that to leading order in 1 /R , A Σ , BH = L n R n − (cid:90) z c δ dz (cid:90) d n − Ω r n − (Ω)1 z n (cid:112) − v (cid:48) − hv (cid:48) (cid:115) r (cid:88) i r i g i = L n A Σ (cid:90) z c δ dzz n (cid:112) − v (cid:48) − hv (cid:48) (9.24)which reduces to (9.7). In particular, all evolution in ρ and Ω directions have dropped out. Thus we concludethat (8.14) in fact applies to all shapes with A strip re-placed by A Σ .The above discussion encompasses the case of Σ beinga sphere for which r (Ω) = 1. In that case one can derivethe above scaling limit explicitly from equations (3.26)–(3.27). In particular, the linear growth regime is con-trolled by the first plateau of the critical extremal surfaceas indicated in Fig. 16. C. More general metrics The above discussion can be readily extended to moregeneral metrics of the form (2.16)–(2.18). The ac-tion (9.24) is replaced by A BH = L n A Σ (cid:90) z c z n (cid:112) − h ( z ) v (cid:48) − k ( z ) v (cid:48) , (9.25) Ρ z h z c z m z Ρ FIG. 16. Cartoon: For a sphere, in the linear regime theextremal surface follows the critical extremal surface for awhile but exits near the first plateau. The dashed curve isthe critical extremal surface. from which v ( z ) satisfies the equation1 z n hv (cid:48) + k √− hv (cid:48) − kv (cid:48) = const (9.26)which can be solved as ( b is a positive constant) v (cid:48) = k ( z ) h ( z ) b (cid:113) h ( z ) z n + b − (9.27)with d A dz = L n A Σ k ( z ) z n (cid:113) h ( z ) z n + b . (9.28)Other than a prefactor k ( z ) appearing in both equations,equations (9.27)–(9.28) are identical to (9.8)–(9.9). Theconstant b should be determined by matching conditionsat the null shell, i.e. be expressible in terms of z c alonein the limit z t → ∞ . Its precise form is not important.As far as a z ∗ c exists such that h ( z ) z n + b is zero at itsminimum z m , A will have a linear growth regime for z c close to z ∗ c .Since in the linear regime the leading behavior is givenby the behavior of the RHS of (9.27)–(9.28) near z m ,the factor k ( z m ) cancels when we relate A to t and weconclude A is still given by (8.14) with the same v n , i.e.the additional function k ( z ) in (9.25) cannot be seen inthe linear regime. X. LINEAR GROWTH: AN UPPER BOUND? In previous sections we found that for any metric ofthe form (2.16) and for Σ of any shape, provided that z ∗ c remains finite in the limit z t → ∞ , there is a lineargrowth regime ∆ A ( t ) = a eq A Σ v n t + O (1) (10.1)5for R (cid:29) t (cid:29) z h . In the above equation a eq is the equi-librium density introduced in (2.33), A Σ is the area of Σ,and the velocity v n is given by v n = (cid:18) z h z m (cid:19) n (cid:112) − h ( z m ) . (10.2)Here z m is the minimum of h ( z ) z n and lies inside the blackhole event horizon. In particular, for entanglement en-tropy we have n = d − S Σ ( t ) = s eq A Σ v E t + O (1) , v E = v d − (10.3)where s eq is the equlibrium entropy density.Now let us specialize to the evolution of entanglemententropy which has the cleanest physical interpretation.The linear growth regime (10.3) sets in for t (cid:38) z h ∼ O ( (cid:96) eq ), i.e. after local equilibration has been achieved.This explains the appearance of the equilibrium entropydensity s eq in the prefactor. In contrast, the pre-local-equilibration quadratic growth (6.25) is proportional tothe energy density E . Indeed, at very early times beforethe system has equilibrated locally, the only macroscopicdata characterizing the state is the energy density.It is natural that in both regimes ∆ S Σ is proportionalto A Σ , as the time evolution in our system is generatedby a local Hamiltonian which couples directly only to thedegrees of freedom near Σ, and the entanglement has tobuild up from Σ. When R is large, the curvature of Σis negligible at early times, which explains the area lawand shape-independence of (6.25) and (10.3).Note that if we stipulate that before local equilibration S Σ ( t ) should be proportional to A Σ and E , the quadratictime dependence in (6.25) follows from dimensional anal-ysis. Similarly, if we require that after local equilibration, S Σ ( t ) is proportional to A Σ and s eq , linearity in time fol-lows.As discussed in [21], equations (6.25) and (10.3) sug-gest a simple geometric picture: entanglement entropyincreases as if there was a wave with a sharp wave-frontpropagating inward from Σ, with the region that has beencovered by the wave entangled with the region outside Σ,and the region yet to be covered not yet entangled. SeeFig. 17. This was dubbed an “entanglement tsuanmi”in [21]. In the linear regime, the tsunami has a constantvelocity given by v E , while in the quadratic regime thefront velocity increases linearly with time. The tsunamipicture highlights the local nature of the evolution of en-tanglement. For quadratic and linear growth regimes,when the curvature of Σ can be neglected, different partsof the tsunami do not interact with one another. But asthe tsunami advances inward, curvature effects will be-come important, and the propagation will become morecomplicated.In a relativistic system, v E should be constrained bycausality, although in a general interacting quantum sys-tem relating it directly to the speed of light appears diffi-cult. In the rest of this section we examine v E for known ΣΣ − v E t FIG. 17. The growth in entanglement entropy can be visual-ized as occuring via an “entanglement tsunami” with a sharpwave-front carrying entanglement inward from Σ. The regionthat has been covered by the wave (i.e. yellow region in theplot) is entangled with the region outside Σ, while the whiteregion is not yet entangled. black hole solutions and also various h ( z ) satisfying nullenergy conditions. We find support that v E ≤ v (S) E = ( η − ( η − η η = d = 2 √ = 0 . d = 3 √ = 0 . d = 4 d = ∞ (10.4)where v (S) E is the value for a Schwarzschild black hole with η = d − d .There are reasons to suspect that the Schwarzschildvalue in (10.4) may be special. The gravity limit corre-sponds to the infinite coupling limit of the gapless bound-ary Hamiltonian, in which generation of entanglementshould be most efficient. From the bulk perspective, itis natural to expect that turning on additional matterfields (satisfying the null energy condition) will slow downthermalization. From the boundary perspective, the cor-responding expectation is that when there are conservedquantities such as charge density, the equilibration pro-cess becomes less efficient.With M´ark Mezei, we generalized the free-streamingmodel of [1] to higher dimensions and find that at earlytimes there is linear growth as in (10.3) with s eq inter-preted as giving a measure for quasiparticle density. For d ≥ 3, quasiparticles can travel in different directions,and as a result although they travel at the speed of lightthe speed of the entanglement tsunami turns out to besmaller than 1 [22], v (streaming) E = Γ( d − ) √ π Γ( d ) < v ( S ) E < . (10.5)Comparing with the Schwarzschild value (10.4), we con-clude that in strongly coupled systems, the propagationof entanglement entropy is faster than that from free-streaming particles moving at the speed of light!6It is important to examine whether (10.4) could beviolated from higher derivative corrections to Einsteingravity. As a preliminary investigation, at the end ofthis section we consider the example of a Schwarzschildblack hole in Gauss-Bonnet gravity in d = 4, but as weexplain there one cannot draw an immediate conclusionfrom it. A. Schwarzschild, RN and other black holes Let us now consider some examples. For Schwarzschild h ( z ) (2.11), plugging (7.23) into (10.2) we find v (S) n = ( η − ( η − η η , η = 2 nd . (10.6)Recall that our current discussion only applies to η ≥ v (S) n < η > , v (S) d = 1 . (10.7) v (S) n is a monotonically decreasing function of η . Themaximal value of η is for entanglement entropy, for which η = d − d and v (S) E = d ( d − − d (2( d − − d . (10.8)The above expression and (8.17) were also obtained ear-lier in [25] in a different set up.For Reissner-Nordstrom h ( z ), from (7.26) the velocityfor entanglement entropy is given by v (RN) E = (cid:114) dd − (cid:18) − d u d − (cid:19) d − d − (1 − u ) (10.9)where u was defined in (2.15)–recall that 1 ≥ u ≥ u = 1 , v E decreases with increasingchemical potential. For the extremal black hole, one finds v E = 0 which implies that the linear growth regime nolonger exists.We now consider the behavior of v E for more generalblack holes. Other than Schwarzschild and RN blackholes there are no known examples of explicit supergrav-ity solutions of the form (2.5). Given that (10.2) dependson some location z = z m behind the horizon, which couldbe shifted around by modifying h ( z ), one may naively ex-pect that v E could easily be increased by changing h ( z )arbitrarily. However, in the examples we studied, the nullenergy condition z h (cid:48)(cid:48) − ( d − zh (cid:48) ≥ v E ≤ v (S) E . Here are some examples: • Consider h ( z ) = 1 − M z d + qz d + p , p > . (10.11)The null energy condition (10.10) requires q ≥ q ≤ dp . (Here and belowwe set z h = 1). This constrains v E ≤ v ( S ) E , anexample of which we show in Fig. 18. Note that for q < v E does exceed v ( S ) E . • A three-parameter example with h ( z ) = 1 − M z d + q z d +1 + q z d +2 . (10.12)The null energy condition (10.10) requires both q and q to be non-negative, and the existence of ahorizon requires q + 2 q ≤ d . Then again v E ≤ v ( S ) E , an example of which is shown in Fig. 18.We have also looked at some non-polynomial examplesand found v E ≤ v (S) E . The phase space we have exploredis not big, nor do we expect that the null energy condi-tion is the only consistency condition. Nevertheless, theexamples seem suggestive. v E /v ( S ) E q q q v E v ( S ) E FIG. 18. Plots of v E /v ( S ) E in examples of h ( z ) with parameterspace restricted by the NEC and the existence of a horizon. Upper : For (10.11) with d = 3 and p = 2. Lower : For (10.12)with d = 4. B. Other supergravity geometries 1. Charged black holes in N = 2 gauged supergravity in AdS [48]: ds = L H ( y ) y (cid:18) − h ( y ) dt + d(cid:126)x + dy f ( y ) (cid:19) (10.13)where h ( y ) = f ( y ) H ( y ) , f ( y ) = H ( y ) − µy , H ( y ) = (cid:89) i =1 (cid:0) q i y (cid:1) . (10.14)We normalize y so that the horizon is at y h = 1, then µ = (cid:81) i =1 (1 + q i ). From (10.2) we find v E = 2 + κ y m − κ y m κ + κ + κ y − m (10.15)with κ = q + q + q , κ = q q + q q + q q , κ = q q q , (10.16)and y m = κ + (cid:112) κ + 3(1 + κ + κ )1 + κ + κ . (10.17)Note for the temperature to be non-negative requires κ ≤ κ + 2 . (10.18)It can be readily checked analytically that for one- andtwo-charge cases with q = κ = 0, the bound is satisfiedfor any ( q , q ), including regions which are thermody-namically unstable. After numerical scanning we findthat (10.15) satisfies v E ≤ v ( S ) E in the full three-chargeparameter space. 2. Charged black holes in N = 8 gauged supergravityin AdS [49]: ds = L H ( y ) y (cid:18) − h ( y ) dt + d(cid:126)x + dy f ( y ) (cid:19) , (10.19)where h ( y ) = f ( y ) H ( y ) , f ( y ) = H ( y ) − µy , H ( y ) = (cid:89) i =1 (1 + q i y ) . (10.20)We again set y h = 1. Then µ = (cid:81) i =1 (1 + q i ) and re-quiring non-negative temperature gives κ ≤ κ + κ + 3 (10.21)where κ i are defined analogously to (10.16), with e.g. κ = q q q q . We then find that v E = 3 + 2 κ y m + κ y m − κ y m κ + κ + κ + κ y − m (10.22)where y m is the smallest positive root of the equation(1 + κ + κ + κ ) y − κ y − κ y − . (10.23) It can again be readily checked that for a single charge q (cid:54) = 0 v E ≤ v ( S ) E is satisfied for any q . One finds afternumerical scanning that the bound is in fact satisfied inthe full four-parameter space. 3. Metrics with hyperscaling violation: Now let us con-sider metrics with hyperscaling violation [50, 51]. Sincewe are interested in theories which have a Lorentz in-variant vacuum, we restrict to examples with dynamicalexponent unity, ds = L y (cid:18) yy F (cid:19) θd − (cid:18) − f ( y ) dt + dy f ( y ) + d(cid:126)x (cid:19) (10.24)where f ( y ) = 1 − (cid:16) yy h (cid:17) ˜ d and ˜ d ≡ d − θ . y F is somescale and θ is a constant. Example of (10.24) includedimensionally reduced near-horizon Dp-brane spacetimesfor which d = p + 1 and θ = − ( d − − d . With boundaryat y = 0, such metrics are no longer asymptotically AdS,but our discussion can still be applied. We find in thiscase v E = (˜ η − ˜ η − ˜ η ˜ η , ˜ η = 2( ˜ d − d . (10.25)The null energy condition now reads [51]˜ dθ ≤ θ ≤ d ≤ 0. The former leadsto ˜ d ≥ d and thus v E ≤ v (S) E , while the latter is inconsis-tent with small y describing UV physics. For examplescoming from Dp-branes, θ is clearly negative with d ≤ d the metric no longer describes a non-gravitational field theory. C. v E from a Schwarzschild BH in Gauss-Bonnetgravity In this subsection as a preliminary investigation ofthe effect of higher derivative gravity terms, we computethe v E from a Schwarzschild black hole in Gauss-Bonnetgravity [52], I = 116 πG N (cid:90) d x √− g [ R + 12 L + λ L ( R − R µν R µν + R µνρσ R µνρσ )] . (10.27)We consider the following the Vaidya metric ds = ˜ L z (cid:0) − f ( v, z ) dv − dvdz + d(cid:126)x (cid:1) (10.28)with f ( v < , z ) = 1, f ( v > , z ) = h ( z ), and [53, 54]˜ L = a L , a ≡ (cid:16) √ − λ (cid:17) ,h ( z ) = a λ (cid:32) − (cid:115) − λ (cid:18) − z z h (cid:19)(cid:33) . (10.29)8Various thermodynamical quantities are given by T = a πz h , s = 14 G N ˜ L z h , E = 34 T s . (10.30)The entanglement entropy is obtained by extremizing theaction [55, 56] A = (cid:90) d σ √ γ (cid:0) λL R (cid:1) (10.31)where γ is the induced metric on the extremal surface and R is the intrinsic scalar curvature of the extremal surface.We have also suppressed a boundary term which will notbe relevant for our discussion below.As v E is shape-independent, it is enough to examinethe extremal surface for a strip, whose induced metriccan be written as ds = ˜ L z (cid:0) Qdx + d(cid:126)y (cid:1) (10.32)with Q = 1 − f v (cid:48) − v (cid:48) z (cid:48) , √ γ = ˜ L z (cid:112) Q , R = − Q ˜ L (cid:0) Qz (cid:48) + zQ (cid:48) z (cid:48) − Qzz (cid:48)(cid:48) (cid:1) , (10.33)where primes denote differentiation with respect to x .We need to extremize the action A = K (cid:90) R dx √ Qz (cid:0) λL R (cid:1) (10.34)with K = ˜ L A strip . (10.35)It is convenient to split the Lagrangian as L = L + L , L = √ Qz , L = λL √ Qz R . (10.36)Note that L depends on λ through h ( z ). We focus onthe black hole region where equations of motion can bewritten as z (cid:48) + hv (cid:48) z √ Q + O v = const , (10.37) ∂ x (cid:18) v (cid:48) z √ Q (cid:19) = 1 z √ Q (cid:18) Qz + 12 h (cid:48) ( z ) v (cid:48) (cid:19) + O z , (10.38)with O v = − ∂ L ∂v (cid:48) + ∂ x (cid:18) ∂ L ∂v (cid:48)(cid:48) (cid:19) (10.39) O z = − ∂ L ∂z + ∂ x (cid:18) ∂ L ∂z (cid:48) (cid:19) − ∂ x (cid:18) ∂ L ∂z (cid:48)(cid:48) (cid:19) (10.40)To identify the linear regime, we look for a solutionwith z = z m = const , v (cid:48) = const , Q = const (10.41)One can check explicitly that 1. Every term in O v contains at least a factor of z (cid:48) or z (cid:48)(cid:48) . It will thus contribute zero.2. Every term in O z contains at least a factor of z (cid:48) or z (cid:48)(cid:48) or Q (cid:48) . It will thus contribute zero.So to find the value of z m and v (cid:48) we can simply ignore L , and the story is exactly the same as before exceptthat h ( z ) is now given by (10.29). That is, z m is deter-mined by z m h (cid:48) ( z m ) − h ( z m ) = 0 (10.42)and Q = − h ( z m ) v (cid:48) . (10.43)We find d A dv = K √ Qz m v (cid:48) = K (cid:112) − h ( z m ) z m (10.44)and v E = z h (cid:112) − h ( z m ) z m . (10.45)Expanding in small λ , we thus have v E = √ − √ λ + O ( λ ) . (10.46)Entanglement entropy in Gauss-Bonnet gravity was stud-ied numerically in [57] and their results are consistentwith the above.While in principle λ can take both signs, in all knownexamples λ appears to be positive [58]. We shouldalso note that in all known examples where the Gauss-Bonnet term arises, there are probe branes and orien-tifolds which back-react on the metric and give rise toadditional contributions at the same (or a more domi-nant) order. Thus it seems one cannot draw a conclu-sion based on (10.46) alone. XI. SATURATION In this section we consider the saturation time and crit-ical behavior in the case of continuous saturation. Thebasic strategy was outlined in Sec. IV near (4.4) – forcontinuous saturation, z t − z c → R, t and A in termsof small z t − z c . Such an expansion also provides a sim-ple diagnostic of whether saturation is discontinuous. Forcontinuous saturation, t − t s must be negative in the limit z t − z c goes to zero. If it is positive, then saturation isdiscontinuous, and equation (4.4) does not give the sat-uration time. See [59–61] for recent progress in computing contributions toentanglement entropy from probe branes. A. Strip We already saw in Sec. VIII B 1 that for Schwarzschild g ( z ) and η = 2 n/d > d ≥ 3) saturation is discontinuous– at saturation time given by (8.18), Γ Σ jumps directlyfrom a near-critical extremal surface whose area growslinearly in time, to one residing entirely in the black holeand corresponding to equilibrium. Here we consider gen-eral g ( z ) and n .Let us start by supposing that saturation is continuouswith saturation time given by (4.4). In the large R limit, z b is close to the horizon z h , and (4.4) has the leadingbehavior t s = 1 h (cid:48) ( z b ) log( z h − z b ) + · · · . (11.1)In this limit z b can be found as in Appendix A (see (A1)and (A4)), from which t s = 1 c n R + O ( R ) , c n = (cid:114) z h | h (cid:48) ( z h ) | n = (cid:114) πz h Tn . (11.2)Next, introducing the expansion parameter (cid:15)z c = z t (cid:18) − (cid:15) n (cid:19) , (11.3)we find that t given by (3.20) has the expansion (seeAppendix B 1 for details) t − t s = u (cid:15) + O ( (cid:15) ) + · · · (11.4)where u = 12 g ( z b ) (cid:18) z b nh ( z b ) F (cid:48) ( z b ) − H ( z b ) (cid:19) (11.5)with F ( z b ) ≡ (cid:90) dy (cid:112) y − n − z b (cid:112) h ( z b y ) ,H ( z b ) ≡ (cid:90) dy (cid:112) h ( z b y )( y − n − z b h ( z b y ) . (11.6)Note that u < t < t s as z c → z t , as one expectsfor continuous saturation, while u > t > t s as z c → z t , indicates that the saturation is discontinuous.The sign of u as given in (11.5) is not universal anddepends on d , n , and g ( z ). In the case of Schwarzschild g ( z ), for d = 2 and n = 1, u = 0, which agrees with theresult of Sec. V C. For d = 3 , 4, we find that u < n = 1, but u > n > 1. Thus for Schwarzschild g ( z ),correlation functions in d = 3 , g ( z ) and d = 3 , u can have either sign for n = 1 but again u > n > 1, implying discontinuous saturation for Wilson linesand entanglement entropy.Meanwhile, for A given by (3.21)–(3.23), one finds thesmall (cid:15) expansion (see Appendix B 1)∆ A − ∆ A eq ∝ (cid:15) (11.7)which for a generic continuous transition (i.e. one with u < 0) gives ∆ A eq − ∆ A ∝ ( t s − t ) . (11.8)In the language of phase transitions, such a quadraticapproach corresponds to mean-field behavior.Note that for a given R , a solution which lies fully inthe back hole region exists only for t > t s ( R ), so for adiscontinuous saturation the “genuine” saturation time t (true) s is always larger than that given by (8.18). SeeFig. 19 for an explicit example.To summarize, for Σ a strip the saturation leading toequilibrium is non-universal, with possibilities of bothdiscontinuous and continuous saturation. When the sat-uration is continuous one finds that ∆ A approaches itsequilibrium value quadratically in t s − t irrespective of n .In contrast, we will see below that for Σ a sphere, satu-ration is almost always continuous (except when n = 2)and there is a nontrivial n -dependent critical exponent. t − t s A − A s FIG. 19. Plots of t − t s and A − A s as functions of (cid:15) in(11.3), with d = 4 Schwarzschild, n = 3, and z b = 0 . t s is the time when continuous saturation would have occurred,but true saturation t (true) s occurs at the dashed line, for which t (true) s > t s . B. Sphere Again let us first assume that saturation is continuous.Then from (4.4) and (A6), we find that in the large R limit t s = 1 c n R − n − πT log R + O ( R ) (11.9)where c n was given earlier in (11.2). For entanglemententropy we then have t s ( R ) = 1 c E R − d − πT log R + O ( R ) (11.10)where c E is the dimensionless number c E = (cid:115) z h | h (cid:48) ( z h ) | d − 1) = (cid:114) πz h Td − . (11.11)To find the critical behavior during saturation we needto solve for z ( ρ ), which we accomplish by expandingabout the solution at equilibrium, z ( ρ ). After a some-what long calculation (outlined in Appendix B 2), we findthat using the expansion parameter (cid:15) defined by ρ c = z c (cid:15) , (11.12) t given by (3.36) has the expansion t − t s = − (cid:18) z b + g ( z b ) z b h ( z b ) (cid:18) b b + I (cid:19)(cid:19) (cid:15) + · · · n = 2 − z b (cid:15) + · · · n > b , b and I are some constants which are de-fined in Appendix B 2. Thus for n > 2, saturation isalways continuous, while for n = 2 it is model depen-dent. Computing b , b , I in (11.13) explicitly, one findsthat the coefficient before (cid:15) is positive for Schwarzschild g ( z ) (saturation is continuous), but becomes negative forReissner-Norstrom g ( z ) at sufficiently large chemical po-tential and for sufficiently large R (saturation is discon-tinuous). Meanwhile, A given by (3.37) has the expan-sion∆ A − ∆ A eq = K g ( z b )8 h ( z b ) (cid:15) log (cid:15) + O ( (cid:15) ) n = 2 − K g ( z b )2( n − (cid:18) n − n + 2 + g ( z b )4 h ( z b ) (cid:19) (cid:15) n +2 + · · · n > . We thus find∆ A eq − ∆ A ∝ (cid:40) − ( t s − t ) log( t s − t ) + · · · n = 2( t s − t ) n +1 + · · · n > . (11.14)Characterizing continuous saturation with a nontrivialscaling exponent S ( R, t ) − S (eq) ( R ) ∝ − ( t s − t ) γ , t s − t (cid:28) (cid:96) eq , (11.15) we thus find that for an n -dimensional extremal surface γ n = n + 22 . (11.16)Note that the above exponent depends only on n andis independent of the boundary spacetime dimension d .Also note that in (11.14), the n = 2 expression appliesto cases of continuous saturation. There is a logarithmicprefactor by which the scaling barely avoids the “mean-field” exponent γ = 2. For d = 2, only n = 1 is possibleand γ = which was previously found in [46]. For en-tanglement entropy, n = d − 1, giving γ E = d + 12 . (11.17) C. More on the saturation time Let us now collect the results we have obtained so faron saturation time. For a strip we showed in Sec. VIII B(see (8.19)) that for z t (cid:29) z m , the linear regime persistsall the way to discontinuous saturation, with saturationtime in the large R limit given by t s = Rv n + · · · , v n = (cid:18) z h z m (cid:19) n (cid:112) − h ( z m ) . (11.18)This happens, for example, for Schwarzschild with η = nd > t s = Rc n + · · · , c n = (cid:114) z h | h (cid:48) ( z h ) | n . (11.19)It is tempting to speculate that the above result appliesto continuous saturation for all shapes.For Schwarzschild and RN black holes c n is given by c (S) n = 1 / √ η , c (RN) n = (cid:112) u/η ≤ c ( S ) n . (11.20)In particular for entanglement entropy we have c ( S ) E = (cid:115) d d − . (11.21)It can be readily checked that for η > v (S) n < c (S) n < η = 1 v (S) d = c (S) d = 1 . (11.23)1As discussed earlier for n = 1 in d = 2, the saturationis continuous, but is discontinuous for n = 2 in d = 4.In the latter case the “true” saturation time should begreater than (8.18) which at leading order in the large R expansion gives (11.19). Numerical results suggest thatthe difference is O (1) in the large R limit and thus atleading order the “true” saturation time is still given by t (true) s = R .For η < 1, as in the case of equal-time correlationfunctions in d = 3 , 4, the saturation is continuous and c (S) n > . (11.24)That t s < R has been observed before numerically ine.g. [4, 13]. Recall that in this case A appears in anexponential with a minus sign. Since c n does not corre-spond directly to any physical propagation, there is noobvious constraint on it from causality. XII. MEMORY LOSS REGIME In this section, we examine implications of the criticalextremal surface for the evolution of A ( R, t ) for a stripand sphere in the regime t s (cid:29) t s − t (cid:29) z h . In (1 +1)-dimensions, we saw in Sec. V D that in this regimethe difference between A ( R, t ) and the equilibrium value A eq ( R ) is a function of t s ( R ) − t = R − t only and not of R and t separately. In other words, at late times in theevolution, the size R has been “forgotten”. We emphasizethat since t s ∝ R → ∞ in the large R limit, such memoryloss can happen long before saturation.We will generalize this result to higher dimensions. Ata heuristic level the existence of such a scaling regime isexpected, as for large R and t ( z t , z c ) very closely followsthe critical line z ∗ c ( z t ) as time evolves. Thus in the limit R, t → ∞ the system is controlled by a single parameteralong the line z ∗ c ( z t ) rather than two separate variables R and t . Recall that in the (1 + 1)-dimensional story inSec. V D, (cid:15) , parameterizing the distance to the criticalline (5.29) (or (7.19)), gave the leading large (cid:96) and τ behavior, while φ in (5.29) (or z t in (7.19)), parametrizingthe location on the critical line, mapped to (cid:96) − τ or τ s − τ .In the limit (cid:96), τ → ∞ with their difference finite, (cid:15) dropsout to leading order and A−A eq is determined by a singleparameter φ only.In general dimensions, the story becomes technicallymuch more involved. For example, for Σ a sphere,even determining the scaling variable (the analogue of (cid:96) − τ in (5.35)) is a nontrivial challenge. We willleave the explicit scaling functions (the analogue of λ in (5.36)), which requires working out the O (1) counter-parts of (5.31)–(5.33), for future investigation. A. Strip For definiteness we will restrict our discussion toSchwarzschild. With a given R , as t increases, z t de- creases. For η > 1, as discussed in Sec. VIII B z t remainslarge compared to log (cid:15) term in (8.7) all the way to sat-uration, in which case the linear regime persists to thesaturation. But this is no longer so for η ≤ 1. For η = 1,in Sec. VIII B 2 we showed that before saturation z t willbecome comparable to z h and the system will eventu-ally exit the linear growth regime. For η < 1, for whichthe linear regime appears not to exist, from discussionof Sec. XI A, we saw at least for d = 3 , 4, the saturationis continuous which implies that z t again has to becomecomparable to z h before saturation.We will now focus on η ≤ 1. We show below thatfor η = 1 there is another scaling regime prior to thesaturation when z t is O (1) (i.e. no longer scales with R ).We again consider z c = z ∗ c (1 − (cid:15) ) , (cid:15) → 0. Following adiscussion similar to that of Sec. VIII A we find that t = − E ( z ∗ c ) h ( z m ) √ H log (cid:15) + O (1) , (12.1) R = − √ H log (cid:15) + O (1) , (12.2)1˜ K ∆ A = − z nt z nm √ H log (cid:15) + O (1) . (12.3)Note that z t is now considered to be O (1), which varieswith R, t , and both z ∗ c , z m are functions of z t .For Schwarzschild with h ( z ) = 1 − z d z dh , we findfrom (7.5) z nt = dz d +2 nm nz dh + ( d − n ) z dm (12.4)and from (7.6) E ( z ∗ c ) = − (cid:115) − h ( z m ) (cid:18) z nt z nm − (cid:19) = h ( z m ) (cid:113) η − − z dm z dh . (12.5)For η = 1, we then have z nt = z nm z nh , E ( z ∗ c ) = h ( z m ) . (12.6)Using these equations in (12.1)–(12.3) we find that t = − √ H log (cid:15) + O (1) , (12.7) R = − √ H log (cid:15) + O (1) , (12.8)1˜ K ∆ A = Rz nh + O (1) . (12.9)Note that O (1) terms are evaluated in the (cid:15) → z c → z ∗ c ( z t ) and therefore are functions only of z t . Inother words, R − t = χ ( z t ) , ∆ A = ∆ A eq + α ( z t ) as (cid:15) → χ and α are some functions whose explicit formwe have not determined for general n , and in the secondequation we have used (2.33). We thus conclude that for t , R (cid:29) R − t (cid:29) z h , A ( R, t ) has the scaling behavior A ( R, t ) − A eq ( R ) = λ ( R − t ) + · · · (12.11)where λ ( x ) = α ( χ − ( x )) and · · · are terms suppressed inthe large R, t limit. Here we will not attempt to find thesefunction explicitly for general d . For d = 2, functions χ , α , and λ are given in (5.34)–(5.36). The above discussiondoes not apply near saturation when R − t (cid:46) O ( z h ).Recall from Sec. V C that in d = 2 ( n = 1) saturationis continuous. But in d = 4 with n = 2, the results inSec. XI A show that saturation is discontinuous. In bothcases the saturation time is given by t s = R for large R and thus (12.11) can also be written as A ( R, t ) − A eq ( R ) = λ ( t s − t ) + · · · . (12.12)For η < 1, from (12.1)–(12.3) we find that R − h ( z m ) E ( z ∗ c ) t = O (1) , ∆ A ˜ K = z nt z nm R + O (1) (12.13)but in this case from (12.4)–(12.5) the prefactor h ( z m ) E ( z ∗ c ) before t as well as the prefactor before R on the right sideof the second equations depends on z t . Thus a scalingregime does not appear to exist. B. Sphere We now consider Σ being a sphere. Since the discussionis rather involved, here we only outline the basic stepsand final results, leaving details to Appendix C.The basic strategy is the same as in previous sections;we consider z c close to the critical line, z c = z ∗ c (1 − (cid:15) ) , (cid:15) (cid:28) , (12.14)and expand the quantities t , R and A in (cid:15) . In contrastto the linear regime, where R (cid:29) t ∼ z h | log (cid:15) | (cid:29) z h andwe expressed all quantities in a double expansion of 1 /R and (cid:15) , here we have instead R → ∞ , − log (cid:15) ∼ O ( R ) → ∞ , z t , ρ c , z ∗ c ∼ O (1) . (12.15)That is, evolution of the extremal surface happens largelyafter the surface has entered the black hole region.We denote the critical extremal surface for z c = z ∗ c as z ∗ ( ρ ). As discussed earlier in Sec. VII B, z ∗ asymptotesto the horizon z h for sufficiently large ρ . In the regimeof z t ∼ z ∗ c , an example of z ∗ was given in Fig. 11. Moreexplicitly, for large ρ (cid:29) z t we can write z ∗ as (see Ap-pendix C for more details) z ∗ ( ρ ) = z h + χ ∗ ( ρ ) (12.16) where χ ∗ has the asymptotic behavior χ ∗ ( ρ ) = αρ n − + O ( ρ − n ) , ρ (cid:29) ρ c (12.17)with α some constant.With (12.14), we can expand solution z about z ∗ , z ( ρ ) = z ∗ ( ρ ) − (cid:15)z ( ρ ) + O ( (cid:15) ) . (12.18)At the shell z satisfies the boundary conditions z ( ρ c ) = z ∗ c , z (cid:48) ( ρ c ) = ρ c z ∗ c (cid:18) − g ( z ∗ c ) + 12 z ∗ c g (cid:48) ( z ∗ c ) (cid:19) (12.19)which can be obtained from the matching conditions dis-cussed in Sec. III B. Focusing on large ρ for which z ∗ asymptotes to the horizon, we have z ( ρ ) = z h + χ ∗ ( ρ ) − (cid:15)z ( ρ ) + O ( (cid:15) ) . (12.20)The equation for z can be obtained by insert-ing (12.20) into (3.35) and expanding in (cid:15) . Due to h ( z h ) = 0, this expansion differs depending on the rela-tive magnitudes of χ ∗ and (cid:15)z , and as a result, the near-horizon region for z can be further subdivided into threeregions in which z can have distinct behavior (see Ap-pendix C for details):1. Region I: χ ∗ (cid:29) (cid:15)z . In this region, z is well ap-proximated by z ∗ and approaches the horizon fromthe inside. Solving for z , we find it has the leadinglarge ρ behavior z ( ρ ) = A e γ n ρ ρ − β n (cid:0) O (cid:0) ρ − (cid:1)(cid:1) (12.21)where β n = n − b γ n , b = δ n, | E | ( h − h ) √ h . (12.22)Here A ( ρ c ) is a positive O (1) constant determinedby the boundary conditions (12.19), and γ n , h , are some constants given in (C8) and (C4). Equa-tion (12.21) applies in the region αρ n − (cid:29) (cid:15)A e γ n ρ ρ − β n (12.23)which translates into ρ c (cid:28) ρ (cid:28) − γ n log (cid:15) + b γ n log log 1 (cid:15) + O (1) + · · · (12.24)which, when written using R (see (12.30) below),is ρ c (cid:28) ρ (cid:28) R − γ n (cid:18) n − − b γ n (cid:19) log R + O (1) . (12.25)2. Region II: χ ∗ ∼ (cid:15)z . Since z grows exponentiallywith ρ , at a certain point (cid:15)z surpasses χ ∗ and z crosses the horizon. Close to this crossing χ ∗ and (cid:15)z are comparable and need to be treated on equalground, making the equation for z complicated.33. Region III: χ ∗ (cid:28) (cid:15)z (cid:28) 1. In this region, z hasgrown sufficiently large that it dominates over χ ∗ ,and has leading large ρ behavior z ( ρ ) = A ρ − ( n − e γ n ρ (cid:0) O ( ρ − ) (cid:1) (12.26)with A ( ρ c ) a positive O (1) constant. The domainof the region is 1 ρ n − (cid:28) (cid:15)z (cid:28) − γ n log (cid:15) (cid:28) ρ (cid:28) − γ n log (cid:15) + n − γ n log log 1 (cid:15) . (12.28)Note that (cid:15)z should become O (1) when ρ ≈ R ,and z ( ρ ) then quickly deviates from the horizon toreach the boundary, i.e. (cid:15)z ( R ) ∼ O (1) (12.29)which leads to − log (cid:15) = γ n R − ( n − 1) log R + O (1) . (12.30)This relation can be established rigorously by care-fully matching (12.26) with an expansion of z near the boundary following techniques developedin [42]. Using (12.30), we can rewrite (12.27) as R − n − γ n log R (cid:28) ρ (cid:28) R . (12.31)Note that for n > b = 0 in (12.22), and the leadingbehavior (12.21) and (12.26) in regions I and III match upto an overall constant factor. Consistently, the domainof the regions (12.25) and (12.31) are adjacent to eachother, i.e. the width of region II is O (1) as (cid:15) → R → ∞ . In contrast, for n = 2, b (cid:54) = 0 sothat the power of ρ in (12.21) and (12.26) do not match,and region II should be of width O (log R ). One can proceed to use z obtained as above in thethree regions to calculate the boundary quantities t (3.36)and A (3.37) (see Appendix C for details). We find thatfor n > t = t s ( R ) + O (1) , (12.32)where t s ( R ) is the saturation time and was given beforein (11.9), and ∆ A − ∆ A eq = O (1) . (12.33) This is evidently the case when b < 0, for example forSchwarzschild h ( z ). However, b can also be positive, for ex-ample for Reissner-Norstrom h ( z ) at sufficiently large chemicalpotential. When b is positive, even though naively it appearsthat (12.25) and (12.31) overlap with each other, it is likely thatthe width of region II is still O (log R ) in order for the exponentof ρ to evolve from that of (12.21) to that of (12.26). Working in the (cid:15) → O (1) terms in (12.32)and (12.33) can be functions of z t only. Eliminating z t -dependence between (12.32) and (12.33), we find the scal-ing behavior A ( t , R ) − A eq = − a eq λ ( t s ( R ) − t ) (12.34)for some function λ . In (12.34) we have included a pref-actor a eq as A eq ( R ) ∝ a eq and a minus sign, so that λ is positive and has the dimension of volume enclosed byΣ. Finding the explicit form of λ requires computing the O (1) terms in (12.32)–(12.33), which is a rather intricatetask and will not be attempted here.For n = 2 (which gives the entanglement entropy in d = 3), we cannot rule out a possible additional log R term in (12.32), due to complications in region II men-tioned earlier. Thus we do not yet have a clean answerin that case. C. Memory loss Let us again specialize to the case of entanglement en-tropy with n = d − 1. Given that S eq ( R ) = V Σ s eq , onecan interpret λ in (12.34) as the volume which has notyet been entangled. Equation (12.34) then implies thatthe “left-over” volume only depends on the difference t s − t and not on R and t separately. In other words,at late times of evolution, the size R has been “forgot-ten”. We again emphasize that with (12.34) valid for t s (cid:29) t s − t (cid:29) (cid:96) eq , such memory loss can happen longbefore saturation.Note that the existence of the memory loss regime it-self is not related to the tsunami picture discussed ear-lier. However, the tsunami picture does lend a naturalgeometric interpretation to the regime, as the memoryloss of the wave front of the entanglement tsunami. Itis tempting to speculate that due to interactions amongdifferent parts of the tsunami wavefront, for a genericsurface Σ in the limit of large R , memory of both the sizeand shape of Σ could be lost during late times in evolu-tion. See Fig. 20 for a cartoon. It would be interestingto understand whether this indeed happens.If such “memory loss” as indicated in Fig. 20 indeed oc-curs, we expect that in the infinite size limit, the space ofall possible Σ separates into different basins of attraction,defined by various attractors (or “fixed points”) such asthe sphere and strip. For example, for a smooth compactΣ, at late times the wave front of the tsunami may ap-proach that of the sphere, while for an elongated surfaceΣ with topology that of a strip, it may approach thatof the strip. This would also imply that the saturationbehavior for generic Σ could be classified using those ofthe “fixed points.”4 FIG. 20. A cartoon picture for late-time memory loss. The(hypothetical) tsunami picture discussed in Sec. X can beused to visualize the memory loss regime–for a wide class ofcompact Σ, in the limit of large size, at late times the wavefront may approach that of a spherical Σ. XIII. CONCLUSIONS AND DISCUSSIONS In this paper we considered the evolution of entangle-ment entropy and various other nonlocal observables dur-ing equilibration, in a class of quenched holographic sys-tems. In the bulk the equilibration process is describedby a Vaidya geometry, with different observables havinga unified description as functions of the area of extremalsurfaces of different dimension n . We were able to derivegeneral scaling results for these observables without usingthe explicit bulk metric. Some of these lead to universalbehavior in the boundary theory.It is important to keep in mind that while the entan-glement entropy is proportional to the area, for other ob-servables the area appears in an exponential with a minussign. So the boundary interpretation of the evolution of A could be very different. We also see interesting differ-ences in the evolution of A for different n . For example,there appears to be no linear evolution for n < d , whichincludes correlation functions in d > 2. See tables I–IIfor a list of the time-dependence of various observablesin d = 3 and d = 4.In the rest of this section we discuss some future direc- tions, using language for entanglement entropy. A. More general equilibration processes In this paper we restricted our discussion to the equi-libration following a global quench. It is interesting toconsider more general equilibration processes, in particu-lar those with inhomogeneous or anisotropic initial states(see [62–64] for recent related work).There are reasons to believe some of our results mayapply to these more general situations. In particular, animportant feature of the linear growth (10.3) is that thespeed v E characterizes properties of the equilibrium state ,as it is solely determined by the metric of the black hole.This highlights the local nature of entanglement propa-gation. At corresponding times, locally, the system hasalready achieved equilibrium, although for large regionsnon-local observables such as entanglement entropy re-main far from their equilibrium values. Thus v E shouldbe independent of the nature of the initial state, includ-ing whether it was isotropic or homogeneous. Similarly,the memory loss regime occurs long after a system hasachieved local equilibration, and we again expect that itshould survive more general initial states.The pre-local-equilibration stage is likely sensitiveto the nature of initial states, including the value ofthe sourcing interval δ t . Nevertheless, that the earlygrowth (6.25) is proportional to the energy density isconsistent with other recent studies of the entanglemententropy of excited states [65–68].Finally with a nonzero sourcing interval δ t , we expectthe wave front of “entanglement tsunami” to develop a fi-nite spread, but the picture of an entanglement wave thatpropagates may still apply as long as δ t is much smallerthan the size of the region one is exploring. If δ t is com-parable to or larger than the local equilibration scale (cid:96) eq ,the pre-local-equilibration and saturation regimes likelycan no longer be sharply defined. t (cid:28) z h z h (cid:28) t (cid:28) R z h (cid:28) t s − t (cid:28) t s saturationEqual-time two-point function G vac exp (cid:0) − t (cid:1) no linear regime no scaling G eq exp( t s − t ) )Wilson loop (rectangular) W vac exp( − t ) W vac exp( − t ) linear regime persists discontinuousWilson loop (circular) W vac exp( − t ) W vac exp( − t ) undetermined W eq exp (cid:0) − t s − t ) log( t s − t ) (cid:1) EE (strip) S vac + t S vac + t linear regime persists discontinuousEE (sphere) S vac + t S vac + t undetermined S eq + t s − t ) log( t s − t )TABLE I. Time-dependence of non-local variables in d = 3 for Schwarzschild. R limit, t s ∝ R , with coefficients as follows: for the equal-time two-point function, t s /R = (cid:112) / 3, for the rectangular Wilson loop and strip EE, t s /R = 2 / / / , and for the circular Wilson loop and sphereEE, t s /R = 2 / √ t (cid:28) z h z h (cid:28) t (cid:28) R z h (cid:28) t s − t (cid:28) t s saturationEqual-time two-point function G vac exp (cid:0) − t (cid:1) no linear regime no scaling G eq exp( t s − t ) )Wilson loop (rectangular) W vac exp( − t ) W vac exp( − t ) W eq exp ( λ ( t s − t )) discontinuousWilson loop (circular) W vac exp( − t ) W vac exp( − t ) undetermined W eq exp (cid:0) − t s − t ) log( t s − t ) (cid:1) EE (strip) S vac + t S vac + t linear regime persists discontinuousEE (sphere) S vac + t S vac + t S eq − λ ( t s − t ) S eq − t s − t ) / TABLE II. Time-dependence of non-local variables in d = 4 for Schwarzschild. λ and˜ λ are those from (12.11) and (12.34). The saturation times are: for the equal-time two-point function, t s /R = 1 / √ 2, for therectangular and circular Wilson loops, t s /R = 1, for strip EE t s /R = 3 / / √ 2, and for sphere EE, t s /R = (cid:112) / B. Entanglement growth It is interesting to compare the growth of entangle-ment entropy among different systems. For this purposewe need a dimensionless quantity in which the system sizeor total number of degrees of freedom has been factoredout, since clearly for a subsystem with more degrees offreedom the entanglement entropy should increase faster.In [21], motivated by the linear growth (10.3) we intro-duced a dimensionless rate of growth R Σ ( t ) ≡ s eq A Σ dS Σ d t . (13.1)In the linear regime, R Σ is a constant given by v E , whilein the pre-local-equilibration regime t (cid:28) (cid:96) eq , from (6.25), R Σ ( t ) = 2 πd − E t s eq (13.2)grows linearly with time. In Fig. 21 we give numericalplots of R Σ for some examples.In all explicit examples we studied, it appears that af-ter local equilibration (i.e. after the linear growth regimehas set in), R Σ monotonically decreases with time. Giventhat we also found earlier that v E appears to have an up-per bound at the Schwarzschild value (10.4), it is tempt-ing to speculate that after local equilibration R Σ ( t ) ≤ v (S) E . (13.3)Before local equilibration, the behavior of R Σ appearsto be sensitive to the initial state. In particular for aRN black hole with Σ a sphere or strip, we find R Σ canexceed v (S) E near (cid:96) eq (see Fig. 21). Also, for a highlyanisotropic initial state, R Σ could for a certain period oftime resemble that of a (1 + 1)-dimensional system. Asin (1 + 1)-dimensions v ( S ) E = 1, it then appears at bestone can have R Σ ( t ) ≤ . (13.4)It is clearly of great interest to explore more systemsto see whether the inequalities (10.4), (13.3) and (5.34)are valid, or to find a proof. If true, the inequalities (10.4), (13.3) and (5.34) maybe considered as field theory generalizations of the smallincremental entangling conjecture [69] for ancilla-assistedentanglement rates in a spin system, which was re-cently proved in [70]. The conjecture states that dSdt ≤ c || H || log D where S is the entanglement entropy betweensubsystems aA and bB , || H || is the norm of the Hamil-tonian H that generates entanglement between A and B ( a , b are ancillas), D = min( D A , D B ) where D A is thedimension of the Hilbert space of A , and c is a constantindependent of D . In our case, the Hamiltonian is lo-cal and thus couples directly only the degrees of freedomnear Σ–the analogue of log D is proportional to A Σ , andthe entropy density s eq in (13.1) can be seen as giving ameasure of the density of excited degrees of freedom. C. Tsunami picture: local propagation ofentanglement In [21] and Sec. X we discussed that the time evolutionof S Σ ( t ) suggests a picture of an entanglement wave frontpropagating inward from the boundary of the entangledregion. See Fig. 17. We stress that at the level of our dis-cussion so far this is merely a hypothetical picture to ex-plain the time dependence of S Σ ( t ). As mentioned earlier,from the field theory perspective, the existence of such anentanglement wave front may be understood heuristicallyas resulting from evolution under a local Hamiltonian. Itwould be very interesting to see whether is possible to“detect” such local propagation using other observables.In the free streaming quasiparticle model of [22], the pic-ture of an entanglement tsunami does emerge at earlystages of time evolution in terms of propagating quasi-particles. But as the system evolves, in particular towardthe late stage, the picture becomes more murky.On the gravity side it should be possible to make thetsunami picture more precise. It is tempting to interpretthe black hole and pure AdS regions of the extremal sur-face as respectively corresponding to parts covered andnot yet covered by the tsunami wave. The two bulk re-gions of the extremal surface are separated sharply at thecollapsing shell and their respective sizes are controlled6 R t tR R t FIG. 21. R Σ for Σ a sphere or strip, for Schwarzschild and RNblack holes. We use units in which the horizon is at z h = 1. Upper : d = 3 and Σ a sphere. The dot-dashed curves arefor the Schwarzschild black hole with R = 7, 13, and 50, re-spectively (larger values of t for the R = 13 , 50 curves are notshown due to insufficient numerics), with the top horizontaldashed line marking v (S) E . Red, green, and blue curves are forthe RN black hole with ( u = 0 . , R = 20), ( u = 0 . , R = 50),and ( u = 0 , R = 50) respectively, and the two lower dashedhorizontal lines mark v E for u = 0 . . Middle : For d = 3 and Σ a strip. The dot-dashed curves are for theSchwarzschild black hole with R = 7 , , 15. It is interestingto note their evolution is essentially identical with the ex-ception of different saturation times. The visible end of thedot-dashed curves coincides with discontinuous saturation for R = 7. For R = 12 and 15 the curves have not been extendedto saturation due to insufficient numerics. The red, green, andblue curves are for the RN black hole with ( u = 0 . , R = 5),( u = 0 . , R = 6), and ( u = 0 , R = 6), respectively. The u = 0 . u = 0 . t than shown. Lower :For d = 4 and Σ a sphere. The color and pattern scheme isidentical to the upper plot, but the Schwarzschild curves areat R = 7, 12, and 50, respectively, and u = 0 . 5, 0 . 2, 0 curvesare all at R = 20. by the tip of the surface z t ( t ) and its intersection with theshell z c ( t ). It should be possible to describe the motionof the tsunami wave front in terms of these data. D. Application to black holes One striking feature of our results, which was alsoemphasized in [25, 71] in different contexts, is that thegrowth of entanglement entropy as well as the evolutionof other nonlocal observables, such as correlation func-tions and Wilson loops, is largely controlled by geome-tries inside the horizon of the collapsing black hole. Inparticular, the linear growth (10.1)–(10.3) is controlledby a constant- z hypersurface inside the horizon while thememory loss regime discussed in Sec. XII B is controlledby an extremal surface which asymptotes to the horizonfrom the inside. In contrast, for a static eternal blackhole an extremal surface whose boundary is at fixed timealways lies outside the horizon [72]. The relation between entanglement growth and certainspatial hypersurfaces inside the horizon is tantalizing. Inparticular, possible bounds on v E (10.2) and the entan-glement growth rate (13.1) impose nontrivial constraintson the geometry inside the horizon. Acknowledgements We thank M´ark Mezei for many discussions, and thankE. Berrigan, J. Maldacena, V. Hubeny, M. Rangamani,B. Swingle, T. Takayanagi, J. Zaanen for conversations.Work supported in part by funds provided by the U.S.Department of Energy (D.O.E.) under cooperative re-search agreement DE-FG0205ER41360. Appendix A: Equilibrium behavior of extremalsurfaces Here we briefly review the behavior of Γ Σ in a blackhole geometry, corresponding to the equilibrium behaviorof various boundary observables. In a black hole geom-etry, an extremal surface always lies outside the hori-zon [72], i.e. denoting the location of the tip of Γ Σ by z b , z b < z h . In our regime of interest R (cid:29) z h , z b is veryclose to the horizon, and we will write z b = z h (1 − (cid:15) ) , (cid:15) (cid:28) . (A1) While for correlation functions separated in the time direction itis possible to relate the geometry inside the horizon to certainfeatures of boundary correlation functions via analytic continu-ation [73–78], the relation is less direct. 1. Strip With Σ a strip, R and A in the black hole geometrycan be obtained from (3.19) and (3.23) by setting E = 0( z c = z t ) and z t = z b , R = (cid:90) z b dz (cid:114) h (cid:16) z nb z n − (cid:17) , (A2) A eq = z nb ˜ K (cid:90) z b dz z n (cid:114) h (cid:16) z nb z n − (cid:17) . (A3)Thus we find that in the large R limit, with z b givenby (A1), R = − γ n log (cid:15) + O (1) , γ n ≡ z h (cid:112) nh , h ≡ − z h h (cid:48) ( z h )(A4)and A eq = − ˜ Kz nh γ n log (cid:15) + O (1) = L n V strip z nh + O ( R ) (A5)where V strip = A strip R is the volume enclosed by the stripΣ. 2. Sphere For Σ a sphere, the story is more complicated. Oneneeds to solve the differential equation (3.35) with E = 0to find the relation between z b and R . In the large R limit, this can be done by matching an expansion nearthe horizon with an expansion near the boundary [42].With z b given by (A1) one finds [42] − log (cid:15) = γ n R − ( n − 1) log R + O ( R ) (A6)and z ( ρ ) can be written near the horizon as z ( ρ ) = z h − (cid:15)z ( ρ ) + O (cid:0) (cid:15) (cid:1) (A7)with z ( ρ ) = Ae γ n ρ ρ − ( n − (cid:0) O ( ρ − ) (cid:1) (A8)where A is some constant. Meanwhile, one finds that theleading contribution to the area of Γ Σ , given by (3.39)with E = ρ c = 0, comes from near the horizon, and thus A eq = K (cid:90) R dρ ρ n − z n (cid:114) z (cid:48) h = KR n nz nh + · · · = V sphere L n z nh + · · · (A9)where · · · denotes terms lower in the large R expansion.This behavior for a general shape Σ has been provedin [42]. Appendix B: Details in the saturation regime1. Strip Near saturation we expect both z c and z t of Γ Σ to beclose to z b , where z b is the tip of the equilibrium Γ Σ withthe same boundary Σ, i.e. same R . We thus write z c = z t (cid:18) − (cid:15) n (cid:19) , z t = z b (cid:18) δ n (cid:19) (B1)where both (cid:15) and δ are small parameters. Thenfrom (3.13) and (3.15) we have E = − g ( z t ) (cid:15) + O ( (cid:15) ) . (B2)First, we determine the relation between δ and (cid:15) byequating (3.19) with (A2). For this purpose it is conve-nient to write (A2) as R = F ( z b ) , F ( z b ) ≡ (cid:90) dy (cid:112) y − n − z b (cid:112) h ( z b y ) . (B3)To expand (3.19) in terms of z t − z b and E , we write itas R = A − A + A + F ( z t ) (B4)where A = (cid:90) z t z c dz (cid:113) z nt z n − , A = (cid:90) z t z c dz (cid:114) h (cid:16) z nt z n − (cid:17) + E , (B5)and A = (cid:90) z t dz (cid:114) h (cid:16) z nt z n − (cid:17) + E − (cid:114) h (cid:16) z nt z n − (cid:17) . (B6)For small (cid:15) we find that A , A , and A have the expan-sions A = z t (cid:15)n (cid:0) O ( (cid:15) ) (cid:1) , A = z t (cid:15)n (cid:0) O ( (cid:15) ) (cid:1) , (B7)and A = − n z t g ( z t ) h ( z t ) (cid:15) + O ( (cid:15) ) , (B8)where in (B7) we used h ( z ) = 1 − g ( z ). Then equat-ing (B3) and (B4), we have δ = g ( z b ) h ( z b ) F (cid:48) ( z b ) (cid:15) + O ( (cid:15) ) . (B9)Next, let us look at (3.20) which can be written as t − t s = B + B − B (B10)8where B = (cid:90) z c z b dzh ( z ) , B = (cid:90) z t dzh ( z ) E (cid:114) h (cid:16) z nt z n − (cid:17) + E ,B = (cid:90) z t z c dzh ( z ) E (cid:114) h (cid:16) z nt z n − (cid:17) + E . (B11)The integrals can be expanded in small (cid:15) as B = 1 h ( z b ) z b n δ + O ( (cid:15) ) , B = H ( z t ) E + O ( (cid:15) ) ,B = Eh ( z b ) z t (cid:15)n + O ( (cid:15) ) , (B12)with H ( z t ) ≡ z t (cid:90) dyh ( z t y ) 1 (cid:112) h ( z t y )( y − n − . (B13)Since B ∼ O ( (cid:15) ), we find t − t s = u (cid:15) + O (cid:0) (cid:15) (cid:1) (B14)where u = 12 g ( z b ) (cid:18) z b nh ( z b ) F (cid:48) ( z b ) − H ( z b ) (cid:19) . (B15)Now let us look at the area of Γ Σ . The area of theequilibrium Γ Σ (A3) can be written as1˜ K A eq = G ( z b ) , G ( z b ) ≡ z nb (cid:90) z b dz z n (cid:114) h (cid:16) z nb z n − (cid:17) . (B16) The area of Γ Σ itself (3.21) can be written as1˜ K A = C − C + C + G ( z t ) (B17)where C = z nt (cid:90) z t z c dz z n (cid:113) z nt z n − , (B18) C = z nt (cid:90) z t z c dz z n (cid:114) h ( z ) (cid:16) z nt z n − (cid:17) + E (B19) C = z nt (cid:90) z t dzz n (cid:114) h ( z ) (cid:16) z nt z n − (cid:17) + E − (cid:114) h ( z ) (cid:16) z nt z n − (cid:17) . (B20)To leading order the expansion of the above quantities isthe same as that for (B5)-(B6), C = z − nt (cid:15)n + O ( (cid:15) ) , C = z − nt (cid:15)n + O ( (cid:15) ) ,C = − n z − nt g ( z t ) h ( z t ) (cid:15) + O ( (cid:15) ) . (B21)Thus we find that1˜ K ( A − A eq ) = z − nb g ( z b )2 nh ( z b ) (cid:18) z nb G (cid:48) ( z b ) F (cid:48) ( z b ) − (cid:19) (cid:15) + O ( (cid:15) ) . (B22)Note that while G ( z b ) is a divergent integral (i.e. dependson a cutoff at small z ), G (cid:48) ( z b ) should have a well definedlimit when the cutoff is taken to zero. In fact, in (B22)the coefficient of the O ( (cid:15) ) term is identically zero, whichcan be seen by writing G (cid:48) ( a ) and F (cid:48) ( a ) as a n G (cid:48) ( a ) = lim δ → (cid:32) − n (cid:90) − δ y n (1 − y n ) (cid:112) h ( ay ) + y − n (cid:112) h ( ay )(1 − y n ) (cid:12)(cid:12)(cid:12)(cid:12) − δ (cid:33) ,F (cid:48) ( a ) = lim δ → (cid:32) − n (cid:90) − δ dy y n (1 − y n ) (cid:112) h ( ay ) + y n (cid:112) h ( ay )(1 − y n ) (cid:12)(cid:12)(cid:12)(cid:12) − δ (cid:33) , (B23)from which we confirm that a n G (cid:48) ( a ) = F (cid:48) ( a ) (B24)for any h ( z ). However, one can check that the O ( (cid:15) ) termin (B22) (whose coefficient is rather long and which wewill not give here) is generically nonzero. 2. Sphere Let z ( ρ ) correspond to the equilibrium Γ Σ and denotethe location of its tip as z b . Then near saturation, z ( ρ ),corresponding to the black hole portion of the actual Γ Σ ,9can be obtained by perturbing z ( ρ ), z ( ρ ) = z ( ρ ) + δz ( ρ ) + δ z ( ρ ) + · · · (B25)where δ is a small parameter which we will obtain pre-cisely later on. Note that near the boundary, z n shouldsatisfy the boundary condition z n ( R ) = 0 , n = 1 , , · · · (B26)They should also satisfy the boundary condition (3.14)at the shell, order by order. For small δ , z c and z t areclose to z b , and ρ c = (cid:112) z t − z c and E are all small. It isconvenient to introduce another small parameter (cid:15) by ρ c = z c (cid:15) (B27)after which from (3.31)–(3.33), z t = z c (cid:18) (cid:15) O ( (cid:15) ) + · · · (cid:19) ,E = − (cid:15) n g ( z c ) z c (cid:0) O ( (cid:15) ) + · · · (cid:1) . (B28)Note that specifying R and (cid:15) fixes Γ Σ entirely. Thuswe can expand t − t s and A − A eq in terms of (cid:15) , then A − A eq in terms of t − t s . In order to do so we firstneed to relate z c − z b and δ to R and (cid:15) . This requiressolving for z near ρ c by expanding it as a power series insmall ρ , but only after imposing the boundary condition(B26) at z = 0. We leave the detailed analysis of z toAppendix B 3, and for now merely list the results. Wefind that for n = 2, δ = − g ( z b ) z b r (cid:15) + O ( (cid:15) log (cid:15) ) + · · · ,z c = z b (1 + c (cid:15) log (cid:15) + c (cid:15) + · · · ) , (B29)with c = − g ( z b )2 , c = − g ( z b )2 r ( r − r + r log z b ) − , (B30)and for n > δ = g ( z b ) z n − b n − r (cid:15) n + O ( (cid:15) n +2 )+ · · · , z c = z b (1+ d (cid:15) + · · · ) , (B31)with d = n − n − g ( z b ) − . (B32)In the above equations r and r are numerical constantsthat we define in (B57). Note that z t > z b while z c doesnot have to be greater than z b .Now let us look at the boundary time (3.36), writingit as t = t + t (B33) with t = − (cid:90) Rρ c dρ z (cid:48) h , t = (cid:90) Rρ c dρ EB (cid:113) z (cid:48) h ( z ) h (cid:113) B E h . (B34)Note that t can be written as t = (cid:90) z c dzh ( z ) = t s + (cid:90) z c z b dzh = t s + z c − z b h ( z b ) + · · · (B35)where in the second equality we have used that z c − z b issmall. Meanwhile, from (B28) E ∼ O ( (cid:15) n ), and to leadingorder in small (cid:15) t can be evaluated by replacing z in itsintegrand by the equilibrium solution z . The resultingintegral receives the dominant contribution from its lowerend, and we have t = Ez nb h ( z b ) − log( z b (cid:15) ) + I + · · · n = 2( z b (cid:15) ) − n n − · · · n > I = lim ρ c → h ( z b ) z b (cid:90) Rρ c dρ z ρ (cid:113) z (cid:48) h ( z ) h ( z ) + log ρ c (B37)and we have replaced z c by z b wherever it appears. Col-lecting (B35), (B36) and using (B29), (B31), we find that t − t s = − (cid:18) z b + g ( z b ) z b h ( z b ) (cid:18) b b + I (cid:19)(cid:19) (cid:15) + · · · n = 2 − z b (cid:15) + · · · n > . (B38)Next, we proceed to compute the area of Γ Σ givenby (3.38) and (3.39). The AdS portion can be easilyexpanded as 1 K A AdS = (cid:15) n n − (cid:15) n +2 n + 2) + · · · , (B39)while the black hole portion can be written as A BH = A + A + O ( E ) + · · · (B40)with1 K A = (cid:90) Rρ c dρ L ( z, z (cid:48) ) ≡ (cid:90) Rρ c dρ ρ n − z n (cid:114) z (cid:48) h , K A = − E (cid:90) Rρ c dρ z n ρ n − h (cid:114) z (cid:48) h . (B41)Since A is multiplied by E ∼ O ( (cid:15) n ), it can be com-puted by replacing z with z in its integrand, and we find0the leading order results1 K A = − E z nb h ( z b ) − log( z b (cid:15) ) + I + · · · n = 2( z b (cid:15) ) − n n − · · · n > . (B42)To compute A , we consider the variation of L undera variation about the equilibrium solution z = z + ∆ z ,which gives1 K A = (cid:90) Rρ c dρ L ( z , z (cid:48) ) − Π( z )∆ z (cid:12)(cid:12) ρ c + · · · , Π = ∂ L ∂z (cid:48) . (B43)Note that in (B43) there is also a potential boundaryterm at ρ = R , but that it is zero due to z and z bothending at ρ = R . Meanwhile, (cid:90) Rρ c dρ L ( z , z (cid:48) ) = 1 K A eq − n (cid:18) (cid:15) √ (cid:15) (cid:19) n (cid:18) z t z b (cid:19) n − ( n + 1) h ( z b ) (cid:15) n +2 n + 2) + · · · (B44)andΠ( z ) (cid:12)(cid:12) ρ c = − ρ nc z n +1 b = − (cid:15) n z b , ∆ z c = z c − z ( ρ c ) = z c − z b − z (cid:48) (0) ρ c + · · · (B45)where the small ρ expansion of z is given in equa-tion (B50). Finally, collecting all the results above wehave A − A eq = K g ( z b )8 h ( z b ) (cid:15) log (cid:15) + O ( (cid:15) ) + · · · n = 2 − K g ( z b )2( n − (cid:18) n − n + 2 + g ( z b )4 h ( z b ) (cid:19) (cid:15) n +2 + · · · n > . 3. Discussion of z when Σ is a sphere Here we give the derivation of (B29)–(B32). To firstorder in δ , z satisfies the equation of motion z (cid:48)(cid:48) + p ( ρ ) z (cid:48) + q ( ρ ) z = s ( ρ ) (B46) where p ( ρ ) = z (cid:48) (cid:18) nz − h (cid:48) h (cid:19) + ( n − ρ (cid:18) z (cid:48) h (cid:19) , (B47) q ( ρ ) = h (cid:48) (cid:18) nz + ( n − z (cid:48) h ρ (cid:19) + nz (cid:18) z (cid:48) h (cid:19) (cid:18) h (cid:48) − h z (cid:19) + 1 h (cid:18) h (cid:48) z (cid:48)(cid:48) − h (cid:48)(cid:48) z (cid:48) (cid:19) , (B48)and s ( ρ ) = − E δ z n h ( z ) ρ n − (cid:18) z (cid:48)(cid:48) + ∂ z h ( z )2 (cid:19) . (B49)While the full analytic solution z is not known, its be-havior near ρ = R and ρ = 0 can be obtained by seriesexpansions and the same applies to functions p and q - this is sufficient for our purposes. Now, near the tip ρ = 0, z ( ρ ) = z b − h ( z b )2 z b ρ + h ( z b )(( n + 1) z b h (cid:48) ( z b ) − ( n + 2) h ( z b ))8( n + 2) z b ρ + O ( ρ )(B50)while near the boundary ρ = R with σ ≡ R − ρ (cid:28) z ( ρ ) = √ Rσ + (cid:40) O ( σ ) n = 2 O ( σ ) n > . (B52)Then we find that the leading terms in p and q are givenby:1. Near ρ = 0, p ( ρ ) = n − ρ + (( n − h ( z b ) + z b h (cid:48) ( z b )) z b ρ + O (cid:0) ρ (cid:1) ,q ( ρ ) = n ( − h ( z b ) + z b h (cid:48) ( z b )) z b + O (cid:0) ρ (cid:1) . (B53)2. Near ρ = R , p ( ρ ) = n − σ + (cid:40) O ( σ − ) n = 2 O (1) n > ,q ( ρ ) = − n σ + (cid:40) O ( σ − ) n = 2 O ( σ − ) n > . (B54)Let us first look at the homogenous part of equa-tion (B46). Near ρ = R , it is convenient to work with abasis of solutions given by the expansions k ( ρ ) = ( R − ρ ) n (cid:16) O (( R − ρ ) ) (cid:17) , ˜ k ( ρ ) = ( R − ρ ) − (cid:16) O (( R − ρ ) ) (cid:17) , (B55)while near ρ = 0, it is more convenient to work with abasis of solutions given by the expansions1 g ( ρ ) = 1 + ∞ (cid:88) m =1 g m ρ m , g ( ρ ) = (cid:40) g log ρ (cid:0) (cid:80) ∞ m =1 g m ρ m (cid:1) + ρ − ( n − (cid:80) (cid:48) ∞ m =0 ˜ g m ρ m n even ρ − ( n − (cid:0) (cid:80) ∞ m =1 g m ρ m (cid:1) n odd , (B56)where in g for even n the prime in (cid:80) (cid:48) indicates thatthe sum does not include m = ( n − / 2. Since we aredealing with a linear equation, the two bases are related by linear superposition, k ( ρ ) = r g ( ρ ) + r g ( ρ ) , ˜ k ( ρ ) = ˜ r g ( ρ ) + ˜ r g ( ρ ) , (B57)where r , r , ˜ r , ˜ r are constants which can be evaluatednumerically.Now we consider a particular solution to the full inho-mogeneous equation (B46), z s ( ρ ) = k ( ρ ) (cid:90) Rρ dρ (cid:48) s ( ρ (cid:48) )˜ k ( ρ (cid:48) ) W k ( ρ (cid:48) ) − ˜ k ( ρ ) (cid:90) Rρ dρ (cid:48) s ( ρ (cid:48) ) k ( ρ (cid:48) ) W k ( ρ (cid:48) ) = g ( ρ ) (cid:90) Rρ dρ (cid:48) s ( ρ (cid:48) ) g ( ρ (cid:48) ) W g ( ρ (cid:48) ) − g ( ρ ) (cid:90) Rρ dρ (cid:48) s ( ρ (cid:48) ) g ( ρ (cid:48) ) W g ( ρ (cid:48) )(B58)where W k ( ρ ) = k ˜ k (cid:48) − ˜ kk (cid:48) ∼ ( R − ρ ) n − as ρ → R (B59)and W g ( ρ ) = g (cid:48) g − g (cid:48) g ∼ ρ − ( n − as ρ → . (B60)Noting that s ( ρ c ) = − E δ z n − n ( z b h (cid:48) ( z b ) − h ( z b ))2 h ( z b ) ρ n − c ,s ( ρ → R ) = E δ R (cid:18) σR (cid:19) n − + · · · (B61)we find z s ( ρ ) ∼ E δ σ n + as ρ → R . (B62)As is shown after the matching in (B28), (B29), and(B31), E δ ∼ δ and thus it is consistent to ignore z s near ρ = R . However, it cannot be ignored near ρ = 0 be-cause the source term s becomes singular. We find thatas ρ → z s has the behavior z s ( ρ ) ∼ E δ (log ρ ) + · · · n = 2 E δ ρ − n − + · · · n > . (B63) This boundary term at z = 0 should be treated with some careas Π is divergent there. In terms of ρ ( z ) we have the expansion ρ ( z ) = R − z R + · · · (B51)where the expansion is identical to that in AdS until the O ( z n )term whose coefficient is undetermined. The actual matching that results in (B29)–(B32) isperformed as follows. The boundary condition (B26) re-quires us to choose k ( ρ ) near ρ = R . Then near ρ = 0, z can be written as z = r g ( ρ ) + r g ( ρ ) + z s . (B64)Plugging in (B27) and (B28) into (B64) and (B56), weobtain z b and δ in terms of z t and (cid:15) as in (B29)–(B32).We note that at leading orders z s does not contribute. Appendix C: Details in the memory loss regime for Σ a sphere Here we give the equations underlying (12.21)and (12.26), and the derivation of (12.32) and (12.33).Recall the expansion parameter and expansion given in(12.14) and (12.18). 1. Critical extremal surface Let us first examine in some detail the asymptotic be-havior of z ∗ ( ρ ) for ρ (cid:29) ρ c where it approaches the hori-zon. Letting z ∗ ( ρ ) = z h + χ ∗ ( ρ ) (C1)with χ ∗ small and requiring it to decrease with increasing ρ , we find that z ∗ has the asymptotic behavior χ ∗ ( ρ ) = z ∗ ( ρ ) − z h = αρ n − + α ρ n + α ρ n +1 + · · · , ρ (cid:29) ρ c (C2)where α = | E | z n +1 h √ nh , α = δ n, E z h h . (C3)2Here we have used the notation h ≡ − z h h (cid:48) ( z ) , h ≡ z h h (cid:48)(cid:48) ( z h ) , · · · (C4)Note that α is positive, i.e. z ∗ approaches the horizonfrom above, or inside. The leading two terms in (C2) canbe obtained by equating the two most dominant termsin (3.35) as ρ → ∞ , i.e. (note B is defined in (3.34)) nh z + 12 E B ∂ z h = 0 → χ ∗ = α ρ n − z n +1 z n +1 h , (C5)while in order to obtain terms of O ( ρ − ( n +1) ) and higherin (C2), one needs to take into higher-order termsin (3.35). Note the leading term in (C2) can also bewritten as χ ∗ = 1 γ n | E | B + O (cid:0) ρ − n (cid:1) (C6)or h ( z ∗ ( ρ )) = − c n | E | B + O (cid:0) ρ − n (cid:1) (C7)where γ n = 1 z h (cid:112) nh , c n = (cid:114) h n = h z h γ n . (C8)Let us now calculate v ∗ ( ρ ; ρ c ) and A ∗ ( ρ ; ρ c ) corre-sponding to z ∗ ( ρ ; ρ c ), where we have traded z t for ρ c and made explicit in our notation that ρ c is the only pa-rameter. Evaluating (3.34) on z ∗ , for large ρ we find that v ∗ (cid:48) = 1 c n − (cid:18) n − πT − δ n, | E | (3 h + h ) z h h (cid:19) ρ + O ( ρ − ) , (C9)from which v ∗ ( ρ ; ρ c )= ρc n − (cid:18) n − πT − δ n, | E | (3 h + h ) z h h (cid:19) log ρ + O (1) . (C10)Note that the leading term, and for n > ρ c . Similarly,evaluating the integrand of (3.24) on z ∗ , we find L ∗ = ρ n − z nh + δ n, E (3 h + h ) z h h ρ + O ( ρ − ) (C11)from which1 K A ∗ ( ρ ; ρ c ) = ρ n nz nh + δ n, E (3 h + h ) z h h log ρ + O (1) . (C12)The leading coefficients are again independent of ρ c andthere is a logarithmic term only for n = 2. 2. Equations We now examine the equation for z ( ρ ) as introducedin (12.18). Let us first look at the region in which χ ∗ (cid:29) (cid:15)z . Plugging (12.18) into (3.35) we find that z satisfiesa linear differential equation z (cid:48)(cid:48) + p ( ρ ) z (cid:48) + p ( ρ ) z = 0 (C13)where p and p are some complicated functions of ρ ,expressed via χ ∗ ( ρ ) and h ( z h + χ ∗ ( ρ )). They have thelarge ρ expansions p ( ρ ) = a ρ + a ρ + · · · a = 2( n − , a = δ n, | E | z h h (13 h − h ) (C14)and p ( ρ ) = − γ n + b ρ + b ρ + · · · , b = δ n, | E | ( h − h ) √ h . (C15)Equation (C13) can then be solved in terms of an expan-sion z ( ρ ) = A e γ n ρ ρ − β n (cid:18) c ρ + O (cid:0) ρ − (cid:1)(cid:19) + · · · (C16)with β n = n − b γ ,c = 18 γ (cid:0) b + 2 γb + (2 a − a + 4 b ) γ + 4 a γ (cid:1) , (C17)where A ( ρ c ) is a positive O (1) constant determined byboundary conditions (12.19) at ρ c , and in (C16) we havesuppressed terms that are exponentially small, i.e. thoseproportional to e − γρ .In the region in which χ ∗ (cid:28) (cid:15)z (cid:28) 1, we canplug in (12.18) into (3.35) while ignoring χ ∗ and termsin (3.35) proportional to E . We then find a nonlinearequation for z , z (cid:48)(cid:48) z − z (cid:48) z + n − ρ z (cid:48) z − γ , (C18)which has the solution z ( ρ ) = ρ − ( n − (cid:18) I n − (cid:18) γρ (cid:19) + K n − (cid:18) γρ (cid:19)(cid:19) = A e γρ ρ − ( n − (cid:0) O (cid:0) ρ − (cid:1)(cid:1) + · · · (C19)where we have again suppressed exponentially smallterms. For Schwarzschild h ( z ), b < h ( z ), b > 3. Time In this subsection and the next, for purposes of clarity,will use a new symbol to denote the polynomial part ofthe large ρ limit of z ∗ , P ( ρ ) ≡ χ ∗ ( ρ ) , z ∗ ( ρ ) = z h + P ( ρ ) . (C20)Recall the labeling of regions I, II, and III given near(12.25) and (12.31). Delineating the regions more explic-itly, the boundary time can be divided as t = v ( R ) = t I + t II + t III ≡ (cid:32)(cid:90) R − k k + (cid:90) R − k R − k + (cid:90) R − k R − k (cid:33) dρ v (cid:48) (C21) where the first equality holds up to O (1) terms and k = 1 γ n (cid:18) n − − b γ n (cid:19) log R + C ,k = 1 γ n ( n − 1) log R − C . (C22)Here C , C , k , k are all positive O (1) constants and k must be chosen sufficiently large that large ρ expansionsapply in region I. We now proceed to calculate (C21),recalling (3.34) v (cid:48) = 1 h (cid:16) − z (cid:48) + EB (cid:112) Q (cid:17) , Q = 1 + z (cid:48) h E B h . (C23) a. Region I Here z = z h + P + O ( (cid:15)z ) + · · · , z (cid:48) = O (cid:18) Pρ (cid:19) + O ( (cid:15)z ) + · · · (C24)1 + z (cid:48) h = 1 + O (cid:18) Pρ (cid:19) + O (cid:18) (cid:15)z ρ (cid:19) + · · · , E B h = 1 − γ n c n P (cid:18) − α α ρ + (cid:18) n + h h (cid:19) Pz h (cid:19) + O ( (cid:15)z ) + · · · (C25)Then v (cid:48) = 1 c n (cid:18) − (cid:18) n − γ n + α α (cid:19) ρ + (cid:18) n + h h + γ n z h c n (cid:19) Pz h (cid:19) + O (cid:18) ρ (cid:19) + (cid:16) (cid:15)z P (cid:17) + · · · (C26)from which t I = R − k c n − (cid:18) n − πT − δ n, | E | (3 h + h ) z h h (cid:19) log R + O (1) . (C27)Comparing with (C10), we see that the two leading termscome from the solution on the critical line, z ∗ = z h + P . b. Region II Here the expansions require more care than in regionsI and III. Let us assume that n = 2 , z inter-polates between the leading behavior in regions I and IIIgiven in (C16) and (C19).First, define D and X by z = z h + D , αρ n − = P (1 − X ) (C28) and note D = P − (cid:15)z + O (cid:0) (cid:15) (cid:1) + · · · (cid:46) O ( P ) ,X = α α ρ + O (cid:18) ρ (cid:19) + · · · ∼ O ( P ) . (C29)Using D and X we can expand h = − c n γ n D + O (cid:0) D (cid:1) + · · · ,EB = − γ n P (1 − X ) (1 + O ( D ) + · · · ) . (C30)Also define Y by z (cid:48) = − γ n ( P − D ) (1 + Y ) , (C31)noting Y = O (cid:18) b ρ (cid:19) + O (cid:18) ρ (cid:19) + · · · + O (cid:18) D ( P − D ) ρ (cid:19) + · · · ,Y ( D = 0) ∼ O ( P ) . (C32)4Now we divide region II into three subregions II : | D | (cid:28) O (cid:0) P (cid:1) , II : | D | ∼ O (cid:0) P (cid:1) , II : | D | (cid:29) O (cid:0) P (cid:1) , (C33)and focus on calculating Q s ≡ Q − z (cid:48) − E B f + E B (C34)in subregions II and II . Then f + E B = γ n P (cid:18) O (cid:18) DP (cid:19) + O ( X ) + · · · (cid:19) (II ) − c n γ n D (cid:18) O ( D ) + O (cid:18) P D (cid:19) + · · · (cid:19) (II ) , (C35)and z (cid:48) − E B = γ n P (cid:18) X + Y + · · · − DP + · · · (cid:19) (II ) Dγ n ( − P + D + · · · ) (II ) , (C36)from which Q s = (cid:18) X + Y + · · · + O (cid:18) DP (cid:19) + · · · (cid:19) (II )2 γ n c n (cid:18) P − D · · · (cid:19) (II ) . (C37)Using expansions (C30), (C31), and (C37), we have − z (cid:48) + EB (cid:112) Q = (cid:26) O ( D ) + · · · (II ) − γ n D + · · · (II ) , (C38) from which v (cid:48) = O (1) + · · · (II )1 c n + · · · (II ) (C39)in subregions II and II . But since the differential equa-tion (3.35) does not contain any scales other than z h , v (cid:48) should smoothly interpolate between subregions II andII , i.e. it should also be O (1) in subregion II . Thus weconclude t II = k − k c n + O (log R ) + O (1) n = 2 k − k c n + O (1) n = 3 . (C40) c. Region III Here z = z h − (cid:15)z + P + O (cid:0) (cid:15) (cid:1) + · · · ,z (cid:48) = − γ n (cid:15)z (cid:18) O (cid:18) ρ (cid:19) + O ( (cid:15) ) + · · · (cid:19) , (C41)and1 + z (cid:48) h = 1 + 2 nz h ( (cid:15)z + P ) + O (cid:18) (cid:15)z ρ (cid:19) + O (cid:0) (cid:15) (cid:1) + · · · , E B h = 1 + O (cid:18) P (cid:15)z (cid:19) + · · · . (C42)Then v (cid:48) = 1 c n (cid:18) O ( (cid:15)z ) + O (cid:18) P(cid:15)z (cid:19) + · · · (cid:19) (C43)and t III = 1 c n k + O (1) . (C44)Finally, collecting (C27), (C40), and (C44), we have t = Rc n − (cid:18) n − πT − | E | (3 h + h ) z h h (cid:19) log R + O (log R ) + O (1) n = 2 Rc n − n − πT log R + O (1) n = 3 . (C45) Note subregions II and II each have two connected pieces. Here we have only made potential leading terms explicit, with n = 2 there is an O (log R ) piece that we werenot able to determine. 4. Action To calculate the action, we proceed in similar fashion.The action with its equilibrium value subtracted can bedivided as A − A eq = A I + A II + A III ≡ (cid:32)(cid:90) R − k k + (cid:90) R − k R − k + (cid:90) R − k R − k (cid:33) dρ (cid:0) A (cid:48) − A (cid:48) eq (cid:1) (C46)where the first equality holds up to O (1) terms includ-ing the contribution from the AdS portion of extremalsurfaces, and from (3.39), A (cid:48) = ρ n − z n (cid:112) Q , A (cid:48) eq = ρ n − z eq (cid:115) z (cid:48) h ( z eq ) . (C47) Here A (cid:48) is evaluated on the near-horizon expansion(12.18) of the near-critical solution, and A eq is evaluatedon the near-horizon expansion (A7) of the equilibriumsolution, where the (cid:15) ’s in the two expansions can be setequal. Note that from (A7),1 + z (cid:48) h ( z eq ) = 1 + 2 nz h (cid:15)z , eq + · · · (C48)and A (cid:48) eq = ρ n − z nh (cid:18) nz h (cid:15)z , eq + · · · (cid:19) . (C49)Then in region I, from (C24) and (C25), A (cid:48) − A (cid:48) eq = ρ n − z nh (cid:18) O (cid:18) Pρ (cid:19) + O ( (cid:15)z ) + · · · (cid:19) − ρ n − z nh (1 + O ( (cid:15)z , eq ) + · · · ) (C50)and one can check (cid:90) R − k k dρ (cid:15)z P ∼ O (1) , (cid:90) R − k k dρ (cid:15)z , eq P ∼ O (cid:16) R b / γ n (cid:17) , (C51)so assuming b < 0, we have A I = O (1) . (C52)In region II, from (C28) and (C37), A (cid:48) − A (cid:48) eq = ρ n − z nh (cid:18) − n Dz h + · · · (cid:19) (cid:18) Q s + · · · (cid:19) − ρ n − z nh (cid:18) − n (cid:15)z , eq z h + · · · (cid:19) = ρ n − z nh ( X + Y ) + · · · (II ) ρ n − z nh (cid:18) γ n c n P − (cid:18) nz h − γ n c n (cid:19) D + · · · (cid:19) (II ) (C53) the exception that in the expression for subregion II , the leadingterm proportional to D have been noted, although it is sublead-ing to terms without factors of D . This is used in calculating v (cid:48) in subregion II . Although the definition of the two (cid:15) ’s in (12.14) and (A1) aredifferent, their expansions in large R (12.30) and (A4) show that fixing R , they agree up to an O (1) factor. This factor than canbe absorbed into z , eq in (A8). D , X , and Y in(C29) and (C32), A (cid:48) − A (cid:48) eq is O (1) in subregions II andII . But as was the case with v (cid:48) , A (cid:48) −A (cid:48) eq must interpolatesmoothly between subregions II and II , so we conclude A (cid:48) − A (cid:48) eq is O (1) throughout region II and that A II = (cid:26) O (log R ) + O (1) n = 2 O (1) n = 3 . (C54) Lastly, in region III, from (C41) and (C42), A (cid:48) − A (cid:48) eq = ρ n − z nh (cid:18) nz h ( (cid:15)z − P ) + O (cid:0) (cid:15) (cid:1) + · · · (cid:19) (cid:18) nz h ( (cid:15)z + P ) + O (cid:18) (cid:15)z ρ (cid:19) + O (cid:0) (cid:15) (cid:1) + · · · (cid:19)(cid:18) O (cid:18) P (cid:15)z (cid:19) + · · · (cid:19) − ρ n − z nh (cid:18) nz h (cid:15)z , eq + O (cid:18) (cid:15)z , eq ρ (cid:19) + O (cid:0) (cid:15) (cid:1) + · · · (cid:19) (C55)where from (A8) and (C19), z , eq ∼ z . (C56)One can check the leading terms in (C55) contribute at (cid:90) R − k R − k dρ (cid:15)z P ∼ O (cid:16) e − γ n R R n − (cid:17) ρ ∼ O (cid:18) log RR (cid:19) so we have A III = O (1) . (C57)Collecting (C52), (C54), and (C57), we arrive at A − A eq = (cid:40) O (log R ) + O (1) n = 2 O (1) n = 3 , (C58)where for n = 2 we have an undetermined O (log R )piece. [1] P. Calabrese and J. L. Cardy, J.Stat.Mech. , P04010(2005), arXiv:cond-mat/0503393 [cond-mat].[2] J. Abajo-Arrastia, J. 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