Comments on Entanglement Propagation
PPrepared for submission to JHEP
Comments on Entanglement Propagation
Moshe Rozali , Alexandre Vincart-Emard
Department of Physics and Astronomy, University of British Columbia,Vancouver, BC V6T 1Z1, Canada
E-mail: [email protected] , [email protected] Abstract:
We extend our work on entanglement propagation following a local quenchin 2+1 dimensional holographic conformal field theories. We find that entanglementpropagates along an emergent lightcone, whose speed of propagation v E seems distinctfrom other measures of quantum information spreading. We compare the relations wefind to information and hydrodynamic velocities in strongly coupled 2+1 dimensionaltheories. While early-time entanglement velocities corresponding to small entanglingregions are numerically close to the butterfly velocity, late-time entanglement velocitiesfor large regions show less regularity. We also generalize and extend our previous resultsregarding the late-time decay of the entanglement entropy back to its equilibrium value. a r X i v : . [ h e p - t h ] J un ontents The generation and propagation of quantum information is a fascinating subject, bring-ing together insights from quantum information theory, many-body physics and perhapsmost surprisingly, studies of the quantum mechanics of black holes. Here we focus onentanglement as a measure of quantum information.One way to generate entanglement is by quenching the system, i.e. starting theevolution from an atypical excited state of the Hamiltonian, usually generated as theground state of another, closely related Hamiltonian. The quenching process generatesshort range entanglement which then evolves and propagates as the system reaches thetypical, thermal state . We note in passing that much of the work on holographic quenches has been done at finite tem-perature, for example quenching past thermal critical points. Such studies mix quantum entanglementand classical correlations. To directly probe the quantum entanglement of the ground state one needsto work at zero temperature, for example quenching past quantum critical points. While some workin that direction has been done, much more remains to be explored. On the holographic side, the bulkgeometry at zero temperature does not involve a regular horizon, which makes both the mechanicsand physics quite different from the thermal case. – 1 –n the holographic context, quenching the system can be achieved by starting atequilibrium and turning on external sources (non-normalizable modes) for marginal orrelevant operators, which drive the system out of equilibrium for a finite duration oftime. Much attention has been given to global, i.e. spatially homogeneous, quenches. Inthis case the time-dependence of the entanglement entropy is the observable of interest,and many insights have been gained both in the holographic context, as well as in moretraditional approaches to many body physics. Models of entanglement evolution, basedon those results, are put forward in [1–5]. It would be interesting to incorporate thespatially-resolved holographic results, discussed here and in [6], into such models.Indeed, the setup of local quenches, whereby the system is excited locally in thespatial domain, provides a spatially-resolved probe of the generation and propagationof entanglement. In [6] we initiated the study of such quenches, and here we continuethat study in a more general set of holographic theories involving a charged black holehorizon, corresponding to strongly coupled conformal field theories in 2+1 dimensions atfinite charge density. We focus on testing our previous results concerning entanglementpropagation in this more general, yet quite similar, context. We are thus able togeneralize and improve our original discussion, to test which of our previous results arerobust, and to investigate which of our conjectures hold in a more general context .Similarly to our previous work, we find that entanglement propagation defines anemergent lightcone structure for the theory. The maximal value of entanglement definesa lightcone, except for narrow transition regimes. We typically find two associatedlightcone velocities, one to do with short times, and one with longer times . Theassociated lightcone velocity v E in those regimes depends on various parameters, andwe have previously found some regularities in the quenches for neutral spacetimes.Here we extend that analysis: we find that the early-time velocity seems to berelated to the butterfly velocity, while late-time velocities have more complicated phe-nomenology. We discuss the phenomenology of v E in this more general setup, andcompare our results to other measures of entanglement propagation in that regime. Wealso discuss the return of the entanglement entropy to its equilibrium value, where weare able to give more precise results than previously due to improved numerics.The outline of this paper goes as follows: In Section 2 we discuss our setup for localquenches in charged spacetimes, our numerical integration strategy using the character-istic formulation of general relativity, and our holographic calculation of the extremalsurfaces encoding the entanglement entropy of regions on the boundary. Section 3contains analysis of the dynamics of holographic entanglement entropy. We continue This is similar in spirit to [7, 8], where it was found that breaking conformal invariance has onlya limited effect on holographic results. Due to numerical limitations, these are not asymptotically long times. – 2 –ur investigation of the emergent lightcone structure that encodes the spatial propaga-tion of entanglement entropy, by including the effects of charge and discussing variousmechanisms that may underlie the phenomenology of entanglement dynamics. We alsoextend our description of entanglement thermalization, for which an improved numeri-cal implementation of the quenches’ evolution at late times reveals a logarithmic returnto equilibrium rather than an exponential damping. We provide a brief summary ofour results in Section 4 as well as further details on the numerical aspects of this workin Appendix A.
In this section we introduce our setup for local quenches in charged spacetimes. Thelocal quench is generated by an inhomogeneous scalar source which is turned on fora finite duration, disrupting the initially uniform energy and charge densities in theprocess. The resulting bulk solution is found numerically, and the extremal surfaces inthat geometry encode the dynamics of the entanglement entropy. Here we describe thatsetup, before turning to the results in the next section. We focus mostly on differencesfrom [6], and the reader may wish to consult that reference for additional details.
We choose our metric to be a generalization of the infalling Eddington-Finkelsteincoordinates for black holes in an asymptotically AdS spacetime [9, 10] ds = − A e χ dt + 2 e χ dt dr − F x dtdx + Σ (cid:0) e B dx + e − B dy (cid:1) , (2.1)and we introduce a gauge field V in the radial gauge V = V dt + V x dx. (2.2)The coordinate r denotes the radial bulk coordinate, with the boundary located at r = ∞ , and t is a null coordinate that coincides with time on the boundary. Ourquench, controlled by a relevant scalar on the boundary, will have local support in x while being translationally invariant in the y direction. Hence all the fields underconsideration depend only on the coordinates { r, t, x } with ∂ y being an isometry.This null slicing of spacetime, known as the characteristic formulation, is welladapted to treat gravitational infall problems since the coordinates remain regular– 3 –verywhere as the quench propagates through the bulk. Our ansatz also provides uswith a residual radial diffeomorphism r → r = r + λ ( x µ ) , (2.3)which we use to fix the coordinate location of the apparent horizon and thus keep thecomputational domain rectangular.The Einstein-Maxwell equations in the presence of a scalar field are given by R MN − R G MN − (cid:96) G MN = T Φ MN + T VMN , (2.4) ∇ M F MN = 0 (2.5)where the matter stress tensors are given by T Φ MN = ∇ M Φ ∇ N Φ + G MN L Φ , L Φ = − (cid:0) G MN ∇ M Φ ∇ N Φ + m Φ (cid:1) , (2.6) T VMN = G AB F MA F NB − F G MN , F = dV. (2.7)Before the quench, the spacetime geometry obeys the vacuum Maxwell-Einstein equa-tions and is described by the RNAdS black hole of mass M and charge Qds = − r f ( r ) dt + 2 dt dr + r (cid:0) dx + dy (cid:1) , f ( r ) = 1 − Mr + Q r , (2.8)and the time-component of the gauge field is V = µ − Qr , µ ≡ Qr + . (2.9)The chemical potential µ is chosen so that V vanishes at the event horizon. In fact,RN black holes typically possess two horizons r ± , which correspond to the two realsolutions of f ( r ) = 0. The black hole’s Hawking temperature is given by T = r f (cid:48) ( r + )4 π , (2.10)and extremality occurs when T = 0, i.e. when Q = (cid:112) M r + / .2 Asymptotic Analysis We now turn our attention to the asymptotic behaviour of our system. We first makea simplifying choice and take m (cid:96) = − /r Φ( r, t, x ) = φ ( t, x ) r + φ ( t, x ) r + · · · . (2.11)Requiring that the Einstein-Maxwell equations in the presence of Φ are satisfied as r → ∞ informs us that the gauge field behaves like V ( r, t, x ) = µ ( t, x ) − ρ ( t, x ) r + · · · (2.12) V x ( r, t, x ) = µ x ( t, x ) + j x ( t, x ) r + V (2) x ( t, x ) r · · · (2.13)whereas the metric components have the asymptotic expansion A ( r, t, x ) = ( r + λ ( t, x )) − ∂ t λ ( t, x ) − φ ( t, x ) + a (3) ( t, x ) r + · · · (2.14) χ ( r, t, x ) = c (3) ( t, x ) r + · · · (2.15) F x ( r, t, x ) = − ∂ x λ ( t, x ) + f (3) ( t, x ) r + · · · (2.16)Σ( r, t, x ) = r + λ ( t, x ) − φ ( t, x ) + · · · (2.17) B ( r, t, x ) = b (3) ( t, x ) r + · · · . (2.18)The functions G (3) µν are undetermined by the equations of motion and require the inputof boundary data via the stress tensor T µν [11], defined in its Brown-York form as [12] T µν = K µν − Kγ µν + 2 γ µν − (cid:18) γ R µν − γ R γ µν (cid:19) + 12 γ µν φ , (2.19)where γ µν is the induced metric on the boundary, K µν , K ≡ γ µν K µν its extrinsic cur-vatures, and γ R µν , γ R its intrinsic curvatures. It is straightforward to show that T = 2 a (3) + 4 c (3) + φ φ response , (2.20) T tx = 32 f (3) − φ ∂ x φ , (2.21)– 5 –nd that these components obey the conservation equations ∂ t T = ∂ x T tx + ∂ t φ φ response − ( ∂ t µ x − ∂ x µ ) − j x ( ∂ x µ − ∂ t µ x ) , (2.22) ∂ t T tx = 12 (cid:0) ∂ x T − ∂ x b (3) + ∂ x φ φ response − φ ∂ x φ response (cid:1) + ρ ( ∂ x µ − ∂ t µ x ) . (2.23)In addition to energy and momentum, the electric charge and current are also conserved ∂ t ρ = − j x − ∂ x µ + ∂ t ∂ x µ x , (2.24) ∂ t j x = V (2) x + j x λ − ∂ x ρ. (2.25) The characteristic formulation of the Maxwell-Einstein and Klein-Gordon equationsconveniently reorganizes the coupled PDEs in two simpler categories: equations forauxiliary fields that are local in time and that obey nested radial ODEs, and equationsfor dynamical fields that propagate the geometry from one null slice to the next [9, 10].Here we outline our numerical integration strategy, and refer the reader to AppendixA for a discussion on the more technical aspects of our implementation.We modelled the quench source function as φ ( t, x ) = f ( x ) g ( t ), with f ( x ) = α (cid:20) tanh (cid:18) x + σ s (cid:19) − tanh (cid:18) x − σ s (cid:19)(cid:21) , g ( t ) = sech (cid:18) t − t q ∆ t q (cid:19) . (2.26)We let the scalar field profile reach a maximum value α at time t = t q ∆. We set t q = 0 .
25 and ∆ = 8, and chose the steepness s according to the width σ of theperturbation in order to have a smooth profile. By t = 3, φ ≈
0, and the quench hasconcluded.We performed domain decomposition in the radial direction, using 4 domains eachdiscretized by a Chebyshev collocation grid containing 11 points. In doing so, errorslocated near the boundary or near the apparent horizon remain localized within theirrespective subdomain [13], thus improving the solutions for auxiliary fields over theentire radial domain. We discretized the spatial direction using a uniformly-spacedFourier grid over the interval [ − ,
30] and used 121 points for σ = 2, and 173 pointsfor σ = 0 . t = 20, the approximate time at which the fields per-turbations reach the spatial boundaries. We also got rid of high-frequency modes that– 6 –ontaminated our solutions by applying a smooth low-pass filter that discarded the topthird of the Fourier modes. However, we remark that it is important not to filter thebulk scalar field Φ if we want its RKF-propagated boundary profile to agree with thesource φ at all times. The next step after obtaining numerical solutions for our local quench is to studythe evolution of the holographic entanglement entropy (HEE) of a region A on theboundary. For simplicity, we consider a strip that extends infinitely in the y direction A = { ( x, y ) | x ∈ ( − L, L ) , y ∈ R } , ∂ A = { ( x, y ) | x = ± L, y ∈ R } . (2.27)The covariant HEE prescription [14] tells us that the entanglement entropy is deter-mined by the area of extremal surfaces anchored on ∂ A . It is natural to use the quench’stranslational invariance to parametrize the extremal surfaces by the coordinates τ and y . The extremal surfaces we are looking for will also be translationally invariant in y , and the problem of calculating their area reduces to that of calculating the properlength of the geodesics X M ( τ ) = { t ( τ ) , r ( τ ) , x ( τ ) } arising from the Lagrangian L = G yy G MN ˙ X M ˙ X N . (2.28)The resulting system of 3 second order ODEs can be transformed into a system of 6first order ODEs in the variables (cid:8) t, P t ≡ Σ ˙ t, r, P + ≡ e χ (cid:0) ˙ r − A ˙ t (cid:1) , x, P x ≡ Σ ˙ x − e − B F x ˙ t (cid:9) , (2.29)for which L = 2 P + P t + P x .Keeping in mind that the length of a geodesic in an asymptotically AdS spacetimeis formally infinite, we introduce a UV cutoff r = (cid:15) − and use a regularization schemein which we subtract the entanglement entropy of a RNAdS geometry expressed withthe radial coordinate ¯ r = r + λ ( t, x ), thus effectively matching asymptotic coordinatecharts in both setups and setting ∆ S A = 0 prior to the quench .To solve the Euler-Lagrange equations derived from (2.28), we adopt an initialvalue problem point of view in which the initial conditions at the turning point are { t = t ∗ , P t = 0 , r = r ∗ , P + = 0 , x = 0 , P x = ± } , (2.30)and we use a shooting method in r ∗ so that x = L when r = (cid:15) − . Note that the toleranceparameters of the ODE solver must be chosen so that L = 1 along the geodesic, whichin turn provides us with a safety check for our solutions. This regularization procedure is equivalent to subtracting the vacuum entanglement entropy forthe region A with a dynamical cutoff (cid:15) vac ( t, x ) related to the radial shift λ ( t, x ). – 7 – Results
Having described our setup and methods of calculation, we now turn to summarizingthe patterns observed in our extended framework. In each case, we provide contextby starting our discussion with a brief reminder of our observations for local quenchesin neutral spacetimes before broadening the scope of our analysis to account for theeffects of charge.
The local nature of the quenches (having finite energy at infinite volume, i.e. zeroenergy density) implies that the entanglement entropy of any region A initially growswith time, reaching a maximum, before inevitably decaying to its pre-quench valueas the perturbation dissipates away. Much of our analysis has to do with the spatialstructure of that maximum, as a function of the spatial extent L of A and the time t .We find that, except for narrow transition ranges, the curve traced by the maximum inthe L − t plane is linear: the spatial propagation of entanglement defines a new lightconestructure, distinct from the causal structure of both bulk and boundary theories.We note that a similar observation was made in [15], in which local quenches areimplemented as a perturbative approximation to the backreaction caused by a massiveinfalling particle in pure AdS. In that context, the trajectory traced by ∆ S A ( t max , L )in the L − t plane always follows a slope of v E = 1 (additionally, the amplitude of thatmaximum remains constant throughout).It turns out that the structure of our results is much richer since our numericalscheme accounts for the full backreaction of the quench on the geometry. While ourdata reveals the appearance of an emergent lightcone, this result emerges from theanalysis rather than being one of the assumptions put in by hand. Indeed, as we willdetail below, we typically find two linear regimes separated by a narrow transition,with distinct velocities at early and late times.The slope of the curve traced by the maximum, v E , is a natural measure of howfast entanglement propagates spatially. Much of our analysis has to do with analyzingthis velocity v E . We find a rich structure in the dependence of the emergent lightconevelocity on parameters. In particular, while it is conceptually similar to other measuresof quantum information spreading such as the butterfly or tsunami velocities, we findthat it is numerically distinct from them under most circumstances.– 8 –et us now turn to describing the regularities found in the entanglement velocity v E . Entanglement velocity
As is expected from a relativistic theory, we found that v E was bounded from above bythe speed of light, with the bound being saturated universally in the high temperatureregime.Perhaps more interesting was the discovery of a lower bound on v E different fromthe speed of sound of a three-dimensional CFT, v sound = 1 / √ . v sound , and in fact was consistently very close to v ∗ E (3), the tsunami velocity of aSchwarzschild-AdS black hole [1] v ∗ E ( d = 3) = ( η − ( η − η η (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d =3 = √ = 0 . , with η = 2( d − d . (3.1)The tsunami velocity is a holographic measure of how fast entanglement propagatesspatially when spacetime is globally quenched and depends uniquely on a black hole’sconserved charges. Given the naturalness of this velocity in matters related to en-tanglement entropy propagation, we conjectured that v E should be found within thebounds v ∗ E (3) = 0 . ≤ v E ≤ . (3.2)This situation is in a way reminiscent of quantum spin systems, which admit an upper Lieb-Robinson bound on the speed at which information can travel despite the absenceof relativistic constraints [16]. However our holographic calculation also provides uswith an unexpected lower bound on information processing based on the properties ofspacetime itself.We now extend our analysis of the structure of the entanglement lightcone and thevelocity v E by including the effects of charge. Our main result persists: in all the caseswe examined, the entanglement traces a lightcone structure. We can therefore lookmore closely at the relation between the entanglement propagation velocity v E definedby our emergent lightcone structure, and other closely related velocities. We note thatwhile those velocities are conceptually similar, and numerically close to each other forneutral black holes, their dependence on charge is distinct. We can therefore hope tomake better distinction between them by examining our results for different parameterranges, in particular by focusing on the charge dependence.– 9 – elation to other velocities In our simulations for wide quenches we find two stages for entanglement propagation,both exhibiting a lightcone structure, and a narrow transition regime between them.For the early-time results, governing the evolution of small entangling surfaces, it isnatural to suspect some relation to the butterfly velocity, quantifying the spatial spreadof chaos [17–19]. We note that the presence of charge does not affect its value: v butterfly = √ / . v E ∈ [0 . , . L < σ regime under consideration approximates a global quench forwhich t max can be thought of as the saturation time’s counterpart, v butterfly seems alikely candidate to quantify the initial spread of quantum information that we observe.Analytical solutions for global quenches in the limit of very narrow regions L (cid:28) u h feature two additional characteristic velocities [21]. One of them, dubbed the maxi-mum velocity v max = 0 . d = 2, and we donot observe v E in that vicinity in our setup. The other one, called the time-averagedvelocity v avg = 0 . S A as a function of strip width. Given thatwe are in an intermediate scaling regime where L ∼ u h and thus receive additionalcontributions from the bulk, it is not surprising that we did not observe this velocityin our lightcone analysis either.For the late-time velocities, governing the evolution of larger entangling surfaces,we had previously found a relation to the tsunami velocity, which appears as a lowerbound of entanglement propagation in the neutral case. It turns out that the tsunamivelocity of RNAdS black hole decreases as its charge increases, ultimately vanishing atextremality. If the tsunami velocity serves as a lower bound for all values of the charge,then the addition of charge should change the measured slopes v E in a predictable way.In particular, we should find that the spatial propagation of the entropy significantlyslows down near extremality.Note however a subtle order of limits issue. Our numerics, performed outsidethe apparent horizon, are restricted to sufficiently narrow entangling surfaces. This issufficient for discovering the emergent lightcone structure, which we investigate here.However, the asymptotic IR limit L → ∞ is a priori distinct and may exhibit differentregularities. In particular, even in the extremal limit, the entangling surfaces relevant– 10 – .0 3.5 4.0 4.5 5.00.51.01.52.0 t max L (a) M = 0 . , σ = 2 t max L (b) M = 0 . , σ = 2 Fig. 1 : The curves traced by the maximum of ∆ S A ( t ) in the L − t plane. Note that all chargedconfigurations have been included in the same figure to illustrate the weak dependence of the lightconebehaviour with respect to charge. In both cases, the early-time velocities are found in close proximity tothe butterfly velocity ( v E ∈ [0 . , . ), whereas the late-time velocities are found within v E ∈ [0 . , . ,an interval containing various velocities of interest. for the emerging lightcone are not deformed much in the near-horizon region. It may bethe case that infinitely wide surfaces are more sensitive to the near-horizon geometry,and thus exhibit a more dramatic behaviour in the near-extremal limit.As it turns out, the inclusion of charge does not affect our results in a dramaticway in this regime. Figure 1 shows the small effect charge has on the lower bound forentanglement propagation speeds; the slopes v E all fall within the same range for allcharged configurations. In the case where the minimal surfaces can penetrate deeper inthe bulk, we still observe two linear regimes (as in Fig. 1b) corresponding approximatelyto L < σ and
L > σ . The tsunami velocity originally appeared in the large L limit,and we observe that charge only marginally decreases the slope v E .The range of the lightcone velocities found at large L in our simulations, v E ∈ [0 . , . v sound = 0 .
707 and v shear = 0 . v shear = (cid:114) D η τ Π ≈ . , (3.3)where D η is the effective shear “diffusion” constant obtained from an analysis of thesound pole, and the hydrodynamic parameter τ Π is the shear relaxation time, which– 11 –an be calculated from AdS/CFT [23] D η = 14 πT , and τ Π = 34 πT (cid:20) − (cid:18) log 3 − π √ (cid:19)(cid:21) . (3.4)As this velocity has to do with entropy production, it can naturally affect the evolutionof holographic entanglement entropy in our setup.In summary, it remains unclear exactly what phenomena come into play to influenceentanglement propagation in the late-time regime, where we find an emergent lightcone.On one hand, we have seen that the slope traced by ∆ S A ( t max , L ) does not decreaseas we approach extremality, which suggests that the charged tsunami velocity doesnot provide an appropriate description of the lower bound for v E . Additionally, ouranalysis remains inconclusive as to the relevance of the neutral tsunami velocity v ∗ E (3).We also see that the entanglement velocity is fairly close to hydrodynamical velocitiesrelated to entropy production. As such we are unable to disentangle the various effectswhich may influence entanglement propagation, and it is entirely possible that differentmechanisms may compete to influence the shape of the entanglement lightcone in thelate-time regime, resulting in the variations observed in v E . In the neutral case, we found that the value of the entanglement entropy maximum∆ S A ( t max ) increased linearly with the size L of the entangling region for small L . Thisincrease was also quantified as a function of the scalar source’s maximal amplitude α∂∂L ∆ S A ( t max , L ; α ) ∼ α . (3.5)For fixed amplitudes, we observe that the maximum ∆ S A ( t max ) was not affected by theaddition of charge for small L , and increased marginally when changing Q , even as weapproach extremality (see Figure 2). Thus, our previously discovered regularities seemrobust to the addition of charge. – 12 – .5 1.0 1.5 2.00.000.020.040.060.08 L (cid:68) S (cid:65) (cid:72) t max (cid:76) (a) M = 0 . , σ = 2 (cid:68) S (cid:65) (cid:72) t max (cid:76) (b) M = 0 . , σ = 2 Fig. 2 : The maximum of ∆ S A ( t ) as a function of strip width L for α = 0 . . Note that all chargedconfigurations have been included in the same figure to illustrate the weak dependence of theentanglement entropy with respect to charge. We now turn our attention to the late-time behaviour of holographic entanglemententropy. Our earlier work on local quenches showed evidence that the process of returnto equilibrium was best described by an exponential damping∆ S A ( t ) = a e − a ( t − a ) + a . (3.6)The parameters a i depended on the particular features of the quench but did not seemto follow any discernible pattern. However, our analysis was limited by the quality ofour numerical quench solutions. In particular, the bulk fields could not be propagatedpast t = 9 without loss of accuracy at large x and large memory requirements. Wemanaged to evolve the quenches up until t = 20 in a reasonable time by making a fewmodifications, including increasing the spatial resolution by discretizing the x directionwith a uniform Fourier grid and by solving the radial ODEs for the auxiliary fieldsindependently for each discretized x j .These improvements allowed us to investigate the late-time behaviour of the entan-glement entropy over much larger time intervals. This additional information revealedthat the exponential decay we observed previously was due to fitting the late-time dataover too short of a time interval. In fact, the new data instead suggests that∆ S A ( t ) ∼ a log t + a t δ , (3.7)is a much better fit, as illustrated in Figure 3. This result is more in line with thosederived from spin chain models [24]. – 13 – (cid:68) S (cid:65) (cid:72) t (cid:76) Fig. 3 : The late-time behaviour of the entanglement entropy closely follows the logarithmic decay (3.7)for L = { , , } , from top to bottom, for α = 0 . , M = 0 . , σ = 2 and Q = 0 . . In this particularcase, the best fit exponents are, respectively, δ = { . , . , . } . (cid:68) S (cid:65) (cid:72) t (cid:76) (a) M = 0 . , σ = 2 , L = 3 . (cid:68) S (cid:65) (cid:72) t (cid:76) (b) M = 0 . , σ = 2 , L = 3 . Fig. 4 : The decay of ∆ S A ( t ) and its best logarithmic fit for Q = 0 . Q ext and α = 0 . . The sizes L have been chosen such that the extremal surfaces probe the near-horizon geometry at one point duringthe quench’s evolution, i.e. L is taken as large as the quench allows it to be. We find δ = 1 . for thefigure on the right. Interestingly, the best-fit exponents δ , obtained by a least-square fit, are generallyclustered around either δ = 1 or δ = 1 .
5, which marks a departure from the prediction– 14 – ig. 5 : This figure illustrates the sharp transition between δ = 1 and δ = 1 . in the logarithmic decay ofthe HEE as a function of Q and L for M = 0 . and σ = 2 . ∆ S A ( t ) ∼ t − made in the perturbative analysis of [15]. Our findings show that there isa complex interplay between the size L , the initial energy density M , the initial chargedensity Q , and the amount of injected energy in the characterization of entanglemententropy’s return to equilibrium. When M = 0 .
1, the logarithmic decay fits the datawith δ = 1 . Q and small L for both σ = 0 . σ = 2. However, (3.7) becomesa bad fit as either the charge and/or the size of A are increased, as showcased in Figure4a. We find that the breakdown occurs around Q ∼ Q ext / Q and L when M = 0 .
01. When σ = 0 .
5, thermalization is dominated by δ = 1 exceptfor near-extremal black holes Q = 0 . Q ext , for which δ = 1 . σ = 2 reveals an even richer picture in which we observe asharp transition between decays characterized by δ = 1 and δ = 1 .
5. As illustrated inFigure 5, the late-time evolution of holographic entanglement entropy in the neutral,large size limit is fitted best with δ = 1. The exponent δ = 1 . σ = 0 . L limit, as in the M = 0 . S A ( t ) isinfluenced not only by the parameters characterizing the geodesics and the geometry ofthe unquenched spacetime, but also by the amount of energy injected by the scalar. Assuch it is hard to disentangle and generalize our findings when the underlying competing– 15 –rocesses arbor inherently different length scales. We have studied the spatial propagation of entanglement entropy following a localexcitation of the system. We find that the entanglement generically propagates alongan emergent lightcone, whose velocity may change over a narrow transition regime. Inour simulation we find early and late-time velocities, and look at their dependence onparameters and relation to other interesting information and hydrodynamical velocities.The early-time entanglement velocity for small strips seems similar to the butterflyvelocity. As both have to do with the initial propagation of quantum information, wefind that relation plausible, especially as it mirrors an analytical result derived in ananalogous global quench scenario. We are however unable to disentangle the variouseffects that could influence the late-time entanglement velocity: the propagation inthat regime seems likely to be controlled by a combination of many mechanisms.We are also able to exhibit some universality in the logarithmic return of theentanglement to its equilibrium value. In particular, the relation to known results forspin chains in 1+1 dimensional CFTs is intriguing.There are very few avenues to investigate the propagation of quantum informationin higher-dimensional strongly coupled conformal field theories. We hope that thephenomenology we present can illuminate that difficult subject: in particular it wouldbe instructive to have a simple model incorporating the regularities we find in theholographic results. We hope to return to these issues in the future.
Acknowledgements
We thank Mukund Rangamani for initial collaboration on the paper and many insightfulcomments on related subjects. We thank Hong Liu and Mark Mezei for interestingconversations. The research is supported by a Discovery grant from NSERC.– 16 –
Numerical Details
The characteristic formulation of Einstein’s equations in the presence of matter reor-ganizes all the fields into two categories: auxiliary fields obeying radial ODEs thatcan be solved sequentially, and dynamical fields which are used to evolve the geometryfrom one null slice to the next. This separation of fields can be achieved by expressingtime derivatives in terms of the directional derivative along outgoing null geodesics, d + = ∂ t + A ∂ r , thereby completely eliminating the presence of A from our scheme.Changing to a compact variable u = 1 /r , we rewrite the fields appearing in our equa-tions as Φ( u, t, x ) ≡ φ ( u, t, x ) u,E r ( u, t, x ) ≡ e r ( u, t, x ) u , Σ( u, t, x ) ≡ λ ( t, x ) uu − φ ( u, t, x ) u,B ( u, t, x ) ≡ b ( u, t, x ) u ,χ ( u, t, x ) ≡ c ( u, t, x ) u ,F x ( u, t, x ) ≡ − ∂ x λ ( t, x ) + f x ( u, t, x ) ,d + Σ( u, t, x ) ≡ (1 + λ ( t, x ) u ) u + ˜Σ( u, t, x ) ,d + Φ( u, t, x ) ≡ − φ ( u, t, x ) + (cid:18) ˜Φ( u, t, x ) + 12 ∂ u φ ( u, t, x ) (cid:19) ,d + B ( u, t, x ) ≡ ˜ B ( u, t, x ) u ,A ( u, t, x ) ≡ (1 + λ ( t, x ) u ) u + a ( u, t, x ) , (A.1)in order to subtract the divergent parts. The field E r ( r, t, x ) above is defined as E r = ∂ r V + e − B Σ F x ∂ r V x ∼ F tr (A.2)in order to decouple the radial equations satisfied by V and F x . However we note thatthe equations for d + B and d + V x form a linear system of radial ODEs that cannot bedecoupled.Given initial conditions specified by φ , λ , b and V x all being 0, as well as the CFTdata T and T tx , we can solve the radial ODEs for the auxiliary fields c , e r , f x , V ,˜Σ, ˜Φ, and for the coupled system ˜ B and d + V x , in that order. These fields obey the– 17 –oundary conditions ∂ u c ( u = 0) = − λφ + 16 φ φ , (A.3) e r ( u = 0) = ρ, (A.4) f x ( u = 0) = 0 and ∂ u f x ( u = 0) = f (3) = 23 T tx + 13 φ ∂ x φ , (A.5) V ( u = 0) = µ, (A.6)˜Φ( u = 0) = − φ − λφ + ∂ t φ , (A.7)˜ B ( u = 0) = 16 (cid:18) ( ∂ x φ ) − φ ∂ x φ − ∂ x T tx (cid:19) − j x (cid:18) ∂ t µ x − ∂ x µ − j x (cid:19) − ∂ u b (cid:12)(cid:12)(cid:12) u =0 , (A.8) d + V x ( u = 0) = ∂ t µ x − j x . (A.9)There are two options when treating the field ˜Σ, one of which is to impose the condition (cid:20) d + Σ − e − B (cid:18) F x ∂ x B − ∂ x F x − e − χ F x ∂ r ΣΣ (cid:19)(cid:21) r = r h = 0 , (A.10)which determines the location of the apparent horizon as the boundary of trappedsurfaces [6]. Our second option is to set ∂ u ˜Σ( u = 0) = 12 T − φ φ − λφ (A.11)on the boundary, as required by self-consistency of the equations of motion. Eitherconditions imply the other; imposing the latter should yield the former and vice-versa,and we can use this as a safety check for our numerics.Now that we have solved for the necessary auxiliary fields, we have to propagate thesolutions along null slices. In order to propagate λ , we require a horizon stationaritycondition, obtained by differentiating (A.10) with respect to time and thus ensuringthat the location of the apparent horizon remains fixed at all times. This procedureyields a boundary value problem in x for the field A ( u h , t, x ). We can then extract ∂ t λ from the relation d + Σ = ∂ t λ + A − d + (cid:18) φ (cid:19) (A.12)evaluated at the horizon. The same equation in turn enables us to solve for A every-where in the bulk since λ does not depend on the radial coordinate. With A in hand,it now becomes straightforward to extract the time derivatives for b , φ and V x from thesolutions for d + B , d + Φ and d + V x , and from the definition of d + = ∂ t + A ∂ r . At this– 18 –oint, all that is left to do is propagate these fields, along with T , T tx and ρ usingthe conservation equations (2.22), (2.23), and (2.24), and to repeat the process on newslices until satisfied with the time evolution of the quench. References [1] H. Liu and S. J. Suh,
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