Holographic relaxation of finite size isolated quantum systems

JHEP05(2014)126 Published for SISSA by Springer Received:April 4, 2014 Accepted:April 20, 2014 Published:May 27, 2014 Holographic relaxation of finite size isolated quantum systems Javier Abajo-Arrastia,aEmilia da Silva,aEsperanza Lopez,aJavier Masband Alexandre Serantesb aInstituto de F´ısica Te´orica IFT UAM/CSIC, C-XVI, Universidad Aut´onoma de Madrid, 28049 Cantoblanco, Madrid, Spain bDepartamento de F´ısica de Part´ıculas, Universidade de Santiago de Compostela, and Instituto Galego de F´ısica de Altas Enerx´ıas IGFAE, E-15782 Santiago de Compostela, Spain E-mail: [email protected],[email protected], [email protected],[email protected], [email protected] Abstract: We study holographically the out of equilibrium dynamics of a finite size closed quantum system in 2+1 dimensions, modelled by the collapse of a shell of a massless scalar field in AdS4. In global coordinates there exists a variety of evolutions towards final black hole formation which we relate with different patterns of relaxation in the dual field theory. For large scalar initial data rapid thermalization is achieved as a priori expected. Interesting phenomena appear for small enough amplitudes. Such shells do not generate a black hole by direct collapse, but quite generically, an apparent horizon emerges after enough bounces off the AdS boundary. We relate this bulk evolution with relaxation processes at strong coupling which delay in reaching an ergodic stage. Besides the dynamics of bulk fields, we monitor the entanglement entropy, finding that it oscillates quasi-periodically before final equilibration. The radial position of the travelling shell is brought in correspondence with the evolution of the pattern of entanglement in the dual field theory. We propose, thereafter, that the observed oscillations are the dual counterpart of the quantum revivals studied in the literature. The entanglement entropy is not only able to portrait the streaming of entangled excitations, but it is also a useful probe of interaction effects. Keywords: Gauge-gravity correspondence, Black Holes in String Theory, Holography and condensed matter physics (AdS/CMT) ArXiv ePrint: 1403.2632 Open Access,c The Authors. Article funded by SCOAP3.doi:10.1007/JHEP05(2014)126 JHEP05(2014)126 Contents 1 Introduction 1 2 Scalar collapse 3 2.1 Equations of motion 4 2.2 Collapse portrait 6 2.3 Post-horizon evolution 9 3 Dual interpretation of the bounces 12 3.1 Dephasing and self-reconstruction 14 3.2 Broadness versus time span 16 4 Entanglement entropy oscillations 18 4.1 Early time dynamics 19 4.2 Holographic evolution 21 4.3 Behavior across critical points 24 4.4 Dependence on the initial state 24 5 Conclusions 26 1 Introduction Despite its fundamental importance, the relaxation of closed quantum systems is still a subject of debate both from the theoretical and the experimental perspective. The recent availability of highly controllable quantum simulators, together with the awareness of its conceptual importance, has stimulated the interest on quantum thermalization (see [1] for a review with references). To place this topic in a historical perspective, already in the classical realm, interest in a similar question was behind the seminal work of Fermi, Pasta and Ulam (FPU) on the dynamics of a one-dimensional anharmonic chain [2,3]. There, contrary to Fermi’s expectation, the presence of nonlinearities was not enough to trigger the ergodicity required for a statistical behavior at long times. Fermi suspected this was something deep and new, and indeed, this problem marks the starting point for two branches of classical dynamics that developed rapidly: integrability and chaos. For quantum systems the situation is much less clear even at the theoretical level. From the experimental data, mounting evidence points towards a rich variety of evolutions, depending on the microscopic dynamics as well as on the initial conditions. In some cases, like for hard core atomic interactions, integrability inhibits thermalization by freezing the momentum distribution, such that memory of the initial state is not lost [4]. In others, the system thermalizes after passing through a quasi-stationary plateau at intermediate times, which has received the name of prethermalization [5,6]. Theoretical efforts have been put – 1 – JHEP05(2014)126 into trying to derive a statistical description for these quasi-stationary states by means of a generalised Gibbs ensemble [7–11]. Further work is still necessary to clarify the different routes from integrability to quantum chaos and quantum ergodicity. This paper aims to show that holographic techniques can contribute to the study of this fascinating subject. The distinction between classical and quantum receives an unexpected twist by means of the holographic duality. In short, it states that some quantum systems at strong coupling are believed to admit a description in terms of a dual picture that involves classical General Relativity in one more dimension. Moreover, some non-local observables are computable from purely geometrical constructs. This is the case of the entanglement entropy, SA, of a region Ain space. It has been conjectured in [12], and recently put on firmer grounds in [13,14], to be given by the area of a minimal surface γAwhich extends into the bulk while being homologous to A. More precisely SA=Area(γA) 4GN .(1.1) For non-static backgrounds the condition of minimal surface should be replaced by that of an extremal one [15]. Entanglement entropy, which provides a measure of quantum entanglement in extended systems, is a notion receiving increasing attention. It has been studied in dynamical situations as a probe on the evolution towards equilibration. The seminal work of Cardy and Calabrese [16,17] focuses on quantum quenches from gapped to critical 1+1 field theories. The authors prove that the entanglement entropy grows with time until it saturates at a constant value. The saturation time increases proportionally to the length of the interval, and the emerging picture is that of propagation of entanglement at the speed of light. Consistently, the same results have been recovered within a class of holographic models where the gravity dual involves a shell of null dust infalling from the AdS boundary and forming a black hole deep inside [18,19]. This set up has been extended to higher dimensions [19–21] and to local quenches [22]. We want to push this venue further, and construct holographic models whose dynamics out of equilibrium departs from a fast approach to ergodic behavior. We will focus on isolated quantum systems of finite size. A simple example where neither equilibration nor thermalization takes place involves free fields with a linear dispersion relation on a circle. For such system any initial state will reconstruct periodically in time. In [23] a dual counterpart of this behavior was proposed to be given by a succession of quantum black holes that form and evaporate in an asymptotically global AdS space. Instead, a calculation involving only classical general relativity would be addressing the problem of a strongly coupled quantum field theory living in the boundary. Our central result is that an evolution pattern where the initial state is partially reconstructed several times before reaching equilibration, is also possible in strongly coupled theories. The simple gravitational system we will consider involves a massless scalar field coupled to Einstein gravity with negative cosmological constant. In [24], working on AdS3, collapse to a black hole was seen for amplitudes above a threshold in a similar fashion as found some years ago by Choptuik for asymptotically flat spacetimes [25]. A radical difference appears below threshold. Now the compactness of global AdS, together with the fact that the scalar field has pressure (unlike the case of null dust), implies there should – 2 – JHEP05(2014)126 appear a periodic regime where the scalar shell bounces back and forth between the origin and the boundary. Althought this was indeed seen in [24], a step further came out of the work by Piotr Bizo´n and Andrzej Rostworowski on AdS4[26]. They pushed the simulations far enough in time and resolution so as to establish that, even for subcritical pulses, after a large enough number of bounces, the solution ended up forming an apparent horizon. We will propose to related this type of bulk solutions with a field theory dynamics able to retain quantum coherence on a long time scale. The results of [26] triggered the interest on this topic [31]–[40], and the subsequent analysis gave rise to a richer landscape. Despite the initial suspect that horizon formation was the unavoidable end from evolving generic initial data, using perturbation theory, the authors of [36] gave evidence for the existence of fully stable periodic non-linear solutions, and conjectured the existence of “islands of stability” around them. Some regular solutions were explicitly constructed in [37] confirming the perturbative analysis. Also long lived, presumably stable, solutions were obtained in [38] (see also [39]). In summary, there is a rich landscape of initial conditions, and it is very tempting to encompass this fact with the variety of relaxation processes that are being observed in real closed quantum systems. It would be extremely interesting to setup a concrete dictionary. This paper intends to be a step in that direction. The nonlinear nature of the problem calls for the use of numerical simulations. We have setup a calculational program that allows us to generate collapses of the above type and compute extremal surfaces on them. We have checked stability and convergence of our code, and reproduced most of the results in the literature. We have focused on the AdS4case. Qualitatively, higher dimensions share many relevant features [31]. In fact the physics is essentially the same as the one for the spherical scalar collapse on Minkowski space-time with reflective boundary conditions at some finite radius [41]. The paper is organised in the following way. Section 2 includes background material as well as a portrait of the landscape of collapse stories. We have been able to prolong our simulations past the moment of horizon formation and, in some cases, until a black hole of almost the total scalar shell mass is stablished. Section 3 is devoted to a detailed dual interpretation of the bouncing solutions. The space of initial conditions contains cases whose evolution looks either as a sharp packet propagation or as a radially delocalized wave. We relate these features with the entanglement pattern in the dual field theory. In section 4 we analyze the evolution of the entanglement entropy. The bouncing geometries induce an oscillatory pattern on it, which follows from the quasi-periodic behavior of the metric. We show that the entanglement entropy can not only portrait kinematical effects, such as the streaming away of entangled excitations, but also interaction phenomena. In section 5 we present our conclusions and comment on the calculations that are underway. 2 Scalar collapse We consider Einstein gravity with negative cosmological constant coupled to a real massless scalar field in four dimensions, S=Zd4x√g1 16πG4 R−2Λ −1 2∂µφ∂µφ,(2.1) – 3 – JHEP05(2014)126 with Λ = −3/l2. We set 8πG4= 2 as well as l= 1. This system has been examined recently for the numerical study of gravitational collapse in asymptotically global AdS spaces, and we summarize some of the known results. Our emphasis, however, is set in pushing the simulation beyond the apparent horizon formation. With the aim at motivating our interpretation in the dual theory, we pursue the evolution as far as the numerical code permits us to approach the final stationary black hole. 2.1 Equations of motion We will follow the ansatz and conventions in [26] for a spherically symmetric collapse. For the line element this gives ds2=1 cos2x−A(t, x)e−2δ(t,x)dt2+dx2 A(t, x)+ sin2x dΩ2 2,(2.2) where x∈[0, π/2] is a compact radial coordinate, and dΩ2 2stands for the metric of the unit sphere. The equations of motion can be casted in a first order form ˙ Φ = Ae−δΠ0 ,˙ Π = 1 tan2xtan2x Ae−δΦ0 ,(2.3) A0=1 + 2 sin2x sin xcos x(1 −A)−sin xcos x A (Φ2+ Π2),(2.4) δ0=−sin xcos x(Φ2+ Π2).(2.5) where Φ = φ0and Π = A−1eδ˙ φ, with φ0and ˙ φthe space and time derivatives of the scalar field respectively. Equations (2.3) are evolution equations for the scalar field, and (2.4) is the Hamiltonian constraint. Due to isotropy, there are no evolution equations for the metric components. We want to solve the previous system of differential equations given smooth initial data for the scalar field. Regularity at the origin implies A(t, 0) = 1, which fixes the integration constant from (2.4). Equation (2.5) is invariant under time dependent shifts of the function δ(t, x). This is equivalent to reparameterizing the time direction, a gauge freedom left over by the ansatz (2.2). Since we are interested in the holographic dictionary, a natural way to fix this freedom is by requiring tto be the proper time at the boundary, namely δ(t, π/2) = 0. Further imposing that the total mass remains constant along the evolution sets to zero the non-normalizable mode of the scalar, associated with a source term in the dual field theory lagrangian.1Under these conditions, the expansion of the fields close to the origin is φ=φ0(t) + O(x2), A = 1 + O(x2), δ =δ0(t) + O(x2),(2.6) whereas close to the boundary one finds φ=φ∞(t)y3+O(y5), A = 1 −2My3+O(y6), δ =O(y6),(2.7) 1The scalar field can also take a constant value at infinity of no physical relevance. – 4 – JHEP05(2014)126 with y=π/2−x. The total mass is M=1 2Zπ/2 0 dx ρ(t, x), ρ(t, x) = tan2x(Φ2+ Π2)e−δ,(2.8) where ρprovides a description of the radial energy distribution of the scalar pulse.2 Concerning initial conditions, they will be set as values for Π(0, x) and Φ(0, x). The numerical integration of equations (2.3)–(2.5) subject to the above described initial and boundary conditions is accomplished using a finite difference method with a fourth order Runge-Kutta algorithm. In addition to the equations (2.3), (2.4) and (2.5) there is an additional “momentum constraint” ˙ A+ 2 sin xcos xA2e−δΦΠ = 0. Fulfillment of this equation and constancy of the mass will be used as quality check of our numerical simulations. The choice of the slice t=0 as initial data surface is suited to the observable we want to study, namely the entanglement entropy. Ideally, we would require that its holographic derivation at any positive boundary time does not require information from across our initial data surface. As recalled in (1.1), the entanglement entropy is captured by the area of extremal surfaces in the bulk that anchor on the boundary of AdS to the boundary of a chosen region. In a static spacetime, these extremal surfaces are minimal surfaces and can be shown to live on constant tslices. Although this does not hold for a dynamical background, deviations are small in the cases we consider here as we will see below. For comparison, an alternative choice is to set initial data on the null infalling surface v= 0, where vis an Eddington-Finkelstein coordinate (see for example [42,43]). This would be inconvenient in our case, as extremal surfaces ending at a boundary time t&0 would pierce that surface towards negative values of v. Instead of performing a scan over the full space of initial conditions for the scalar field, we have restricted to gaussian type profiles localized either close to the origin of AdS Φc(x)=0,Πc(x) = 2 πexp −4 tan2x π2σ2.(2.9) as in [26], or close to the boundary Φb(x) = 0 ,Πb(x) = 12 πexp −4 tan2(π/2−x) π2σ2cos3x . (2.10) These second type of boundary conditions, although exhibiting a very similar phenomenology to the first type, looks more akin to the Vaidya setup that has been used to model an analog to a quench in the dual field theory. Altogether, our initial conditions are therefore parameterised by two variables, the amplitude and the width σ. Of course, the space of initial conditions is infinite dimensional, and may hide surprises that deserve to be unveiled. Some of them involve stationary regular solutions with and without rotation [36] and, from them, at least one has been constructed numerically [37]. 2The integrand of (2.8) is not uniquely defined. The choice ˜ρ= tan2x(Φ2+ Π2)Areconstructs equally M[26]. Both functions give quite similar results. – 5 – JHEP05(2014)126 2.2 Collapse portrait Black holes formed in the collapse of a spherical shell of a massless real scalar field are of Schwarzschild type and can have either positive or negative specific heat. With our conventions, the threshold mass separating both cases is3 Mth =2 3√3∼0.385 .(2.11) Big (small) black holes refer to those with masses above (below) this threshold, which correspondingly have positive (negative) specific heat. Due to the negative specific heat, small black holes can not be in thermal equilibrium with their own Hawking radiation, and they will evaporate very much as they do in flat space. For small GNhowever, Hawking radiation is suppressed and the process of evaporation becomes very slow as compared with the time scales we will be interested in. Moreover this type of configurations has been shown to have higher entropy than thermal AdS with the same energy [44]. Hence in the microcanonical ensemble we are considering, both small and large AdS black are valid collapse end products. Regardless of which one of the profiles (2.9) or (2.10) are used as initial data, for masses above or comparable to Mth, the direct formation of a black hole of the total mass is observed. Decreasing the mass neatly below Mth, the emerging apparent horizon starts trapping only a fraction of the scalar pulse, while the rest scatters towards the boundary. Upon lowering the amplitude further, a critical black hole (actually a naked singularity) of vanishing radius (hence mass) can form. This is in total agreement with the findings by Choptuik [25] in asymptotically flat space. The presence of an apparent horizon is signalled by a zero of the function A(t, x). However this function does not strictly vanish at any finite value of t, since as it drops the relative redshift factor with respect to the boundary grows very large and the dynamics gets frozen around the emergent horizon. Hence in the coordinate system we are using, it only makes sense to define the time of horizon formation as that when the minimum of A(t, x) drops below a sufficiently small value. In this sense it should be understood below. New effects appear below the threshold amplitude for critical collapse. In flat space, all the mass would just be reflected back to infinity and that would be the end of the story. Here instead, the massless scalar pulse that escapes towards the boundary of AdS reflects in finite proper time, and falls in again. In [26,31] support for a remarkable phenomenon was provided for AdSd+1 when d≥3: well localized scalar profiles of arbitrary small initial amplitude always generate a horizon after a sufficient number of bouncing cycles. The mechanism behind this phenomenon is the transfer of energy from long to small wavelengths along the evolution [26,32]. Analogous effects are well known from fluid dynamics [45], where they are referred to as weak turbulence. This fact is responsible for the change of shape of the traveling wave and the sharpening of at least one of its fronts. Its action is most effective at the origin, where the bouncing produces extremely high values of the 3Schwarzschild-AdS4black holes have a temperature T=(3 tan2xh+1)/tan xh, where xhis the largest root of 2Mcos2x=tan x. The threshold mass is set by requiring dT/dM =0. – 6 – JHEP05(2014)126 2 4 6 8 10 t 0.2 0.4 0.6 0.8 1.0 Min!A" 0.0 0.1 0.2 0.3 0.4 0.5 0.6 x 0.0 0.1 0.2 0.3 0.4 Ρ Figure 1. Evolution of a narrow pulse (2.9) with σ=1/16 and M=0.015. Left: the dashed line shows the initial mass distribution function. The curves in color denote the mass density profile at the times when the pulse bounces against the origin producing a minimum of A(t, x) (see inset). Field values grow wild with time at the origin, enhancing the non-linear effects. Right: scalar profiles from the first (blue), second (red) and third (brown) bouncing cycles; the arrows indicate the direction of movement. The scalar profile changes its shape when scattering at the origin, while it travels almost unaltered along the full radial coordinate. fields, therefore enhancing the effect of the nonlinearities. After enough number of bounces a sufficiently sharp subpulse freezes, and the radial minimum of A(t, x) drops abruptly (see inset in figure 1a). This is the defining time for an apparent horizon formation, although as said before, in these coordinates the exact formation occurs in infinite time. We would like to point out an effect accompanying the transfer of energy towards small wavelengths. Each time the signal scatters at the origin and a part of it sharpens, the rest instead tends to increase its radial dispersion. This behavior, which will prove to have important consequences for the dual field theory, is illustrated in figure 1a using a narrow initial profile which requires three bounces for collapse. There we have plotted the mass distribution function ρ(t, x) (2.8) at the times of closest approach to the origin. On figure 1b we illustrate how the scalar profile changes shape when it scatters at the origin, while its shape remains largely unaffected upon propagating and reflecting against the boundary. The profiles in color blue, red and brown correspond respectively to configurations along the first, second and third bouncing cycles. The apparent horizon is generated from the small spiky front in the brown profile. Besides the mass, a parameter which has strong influence on the evolution of a scalar pulse is its broadness. The relevance of this parameter was made manifest in [38]. It was shown that the turbulent mechanism characteristic of the narrow pulse evolution becomes less efficient as the broadness of the profile grows. Broad pulses quickly develop a subpulse structure with infalling and outgoing components which scatter among themselves. In order to better understand the interaction among subpulses, let us consider a different initial profile: a linear combination of (2.9) and (2.10) producing two well localized pulses close to the origin and the boundary Φ(x)=0,Π(x)=Πc(x)+Πb(x).(2.12) – 7 – JHEP05(2014)126 0.5 1.0 1.5 x 0.1 0.2 Ρ 0 5 10 15 20 25 30 t 0.2 0.4 0.6 0.8 1.0 M in!A" 0.5 1.0 1.5 x 0.1 0.2 Ρ 0 2 4 6 8 10 t 0.2 0.4 0.6 0.8 1.0 M in!A" Figure 2. Evolution of the initial data (2.9) (blue), (2.10) (red) and the combined profile (2.12) (green) for the following subpulse data. Left: σ= 0.25 and M= 0.012. Right: σ= 1/16 and M= 0.029. In the inset, the initial mass density function. The left initial data have more overlap than the right ones. The horizon formation time for the fast collapsing pulse (red curve) gets affected and delayed (green curve), whereas it remains unaltered in the non-overlapping case on the right. We choose them to have the same broadness and the same mass. When the tails of the initial pulses have some overlap, the creation of an apparent horizon is delayed with respect to the independent evolution of the pulse that would collapse first, Πb. We observe in figure 2a that the number of bounces necessary for the creation of a horizon increases. Hence scattering tends to work against weak turbulence, which in this example finally wins. Interestingly, pulses with a negligible initial superposition are practically transparent to each other, as shown in figure 2b. Collapse processes with very broad initial profiles present distinguished characteristics. They are delocalized along the complete radial direction in the major part of the evolution. The oscillation periodicities of these solutions are determined, besides radial displacement, by their internal subpulse structure. As a result, the bouncing cycles are not neatly defined, see figure 3a. Moreover, the horizon emerges supported by a finite fraction of the pulse mass, in contrast to narrow pulses where it can be vanishingly small. Delocalized pulses require masses around 40% the threshold mass (2.11) to generate an apparent horizon. When the total mass is decreased, a point is reached where the time elapsed until horizon formation abruptly increases [38,39]. For masses below this threshold our results join previous analysis supporting the establishment of a regular quasi-standing wave [36]–[39]. For initial conditions of the form (2.9) the threshold broadness for the existence of regular solutions appear to be around σ∼0.35 [38]. In [39] the numerical analysis at σ= 0.6 confirmed the regularity of the evolution. However, for large values of σ, again collapse to black hole formation is observed. Their interpretation thereafter is that, for some intermediate values of σ, the evolution lies within the domain of attraction of stable periodic solution that bifurcate from the fundamental mode of the linearized scalar wave equation in AdS, which the authors in [35] termed “oscillons”. For the lowest frequency in AdS4they are given by φosc j=0(t, x) = acos(3t+α) cos3x , (2.13) – 8 – JHEP05(2014)126 Its spectrum is given by non-interacting modes of momentum p=2πn/N with n=0, . . . , N− 1 and frequency ωp= 2νsin p 2.(3.6) For pπthe dispersion relation becomes linear, ωp≃νp. An initial wave packet constructed out of low momenta will reconstruct itself with period t0=N/2ν, as in (3.3). No sign of relaxation will appear until enough time has passed to render important the non-linearity of the dispersion relation. If the wave packet is centered around frequency ¯ω, this time is [1] t≃N2 ¯ω.(3.7) Afterwards the system dephases and tends to a stationary state.6The dephasing time can be much larger than the propagation time t0if ¯ωis chosen sufficiently small. It is important to emphasize here that this dependence of the relaxation process on the initial state is indeed seen in experimental setups [4–6,9]. We can now state the main proposal of this work: collapses which require bouncing on the AdS boundary before forming a horizon are holographically dual to field theory evolutions where the initial state is partially reconstructed several times before achieving equilibration. In other words, the dephasing time is much larger than the typical propagation time, in analogy with the case of the harmonic chain above. We have argued that in an interacting field theory this is not the generic behavior. However, that reasoning can fail for states with small enough energy density. The finite size of the system introduces an intrinsic scale and, hence, the dynamical process can also depend on the energy density of the initial state. This is precisely what is found in the holographic models based on the collapse of a massless scalar profile [26]. When the mass of the scalar shell is above the threshold (2.11) for the formation of a large black hole, the shell is completely trapped behind a horizon by direct collapse. Bouncing with the AdS boundary is only required when the final black hole to be formed is small. In the same way that an infalling scalar pulse is to be holographically interpreted as a growing separation between entangled excitations, the stages of the evolution when it moves towards the AdS boundary should represent entangled excitations joining again. This can be neatly seen using a thin shell of null dust that travels outwards and reaches the boundary at t=0. The same reasoning that for an infalling shell sets the bound (3.1), leads now to 0≤l≤ −2t , (3.8) for the size of intervals producing a thermal result for the entanglement entropy. Hence their size decreases as the system evolves towards t= 0, as can be predicted from the qualitative picture in figure 7. A very nontrivial support for this picture comes from the periodicity of the scalar pulse in the bulk. From the numerical simulations, one can see that its evolution from the boundary to the center and back completes a full roundtrip with a period of approximately 6The stationary state generically differs from thermal equilibrium since the occupation numbers of noninteracting modes are conserved along the evolution. – 15 – JHEP05(2014)126 π(see for example the inset in figure 1a). Now, recalling that in the gravitational system we have fixed the radius of the boundary sphere to unity, the expected reconstruction time (3.3) is t0=L 2=π(3.9) where L= 2πis the length of an equator. As we have mentioned in section 2, the exact periodicity is slightly bigger than πand this delay increases with the amplitude of the pulse. We will relate this fact to the presence of interactions in section 4.1. Besides dephasing, another process which is crucial in order to achieve thermal equilibrium is the equipartition of energy among degrees of freedom. If the radial position of the traveling shell would also describe this process, we should expect an oscillatory pattern in the evolution of the occupation numbers. However periodicities in occupation numbers are more naturally associated with Poincare recurrences, and in a large Nsystem these take a much longer time than L/2. Hence we are led to conclude that the radial displacement of the scalar shell is not directly related with the redistribution of energy, and its periodic behaviour is tantamount to the reconstruction, or revival, of the initial dual quantum state. Evidence for revivals has been observed experimentally in cold atoms systems [27]. At a theoretical level, it has been seen in quantum spin chains [28,29] and also in CFT in 1+1 dimensions [30]. 3.2 Broadness versus time span In our set up no reference is made as to how the initial state that triggers the evolution was created. In order to understand the implications of broad scalar profiles, it is however useful to discuss rough characteristics of field theory perturbations that can holographically relate to them. To gain insight we can resort again to the familiar case of the Vaidya metric ds2=1 cos2x−1−m(v)cos2x tan xd2v+ 2dvdx + sin2x dΩ2 2(3.10) with van Eddington-Finkelstein infalling coordinate. We choose a mass function satisfying m(v) = (0v < −∆t , M v > 0.(3.11) It represents the building up of an energy density =Min a finite time span ∆tstarting from the CFT vacuum. The null dust shell sourcing the metric (3.10) has support in the region v∈[−∆t, 0]. Upon transforming to the Schwarzschild coordinates (t, x), the shell exhibits a finite broadness in the radial direction. This is shown in figure 8a, where the intersection of a succession of slices of constant twith the location of the shell, highlighted in yellow, determines its radial localization. Its profile at several tslices is plotted in figure 8b. The mass distribution function in figure 8b corresponding to t=0 provides the analogue of the initial data we are dealing with in this paper. Its value at a given xreflects the – 16 – JHEP05(2014)126 Figure 8. Left: intersection a null dust shell with ∆ = 1/2, signaled in yellow, with the lines of constant t=−0.5,0,...,2 in the (v, x) plane. Right: mass distribution function at the same tslices. 0.5 1.0 1.5 x 0.5 Ρ 0 5 10 15 20 t 0.2 0.4 0.6 0.8 1.0 MinHAL Figure 9. Left: excitations sourced during small time intervals around t1∼x1−π/2 and t2∼ x2−π/2. Right: evolution of minxA(t, x) for several pulses: σ= 0.1 and M=0.035 (blue), σ=0.2 and M=0.066 (black) and σ=0.2 and M= 0.035(red). In the inset we plot the initial mass density distribution for each pulse. density of excitations created at a prior time t∼x−π/2≤0.(3.12) The earlier some excitations have been created, the further entangled components are able to fly apart and the deeper its holographic representation reaches in the xdirection. We will adopt this point of view in order to interpret the scalar profiles, as sketched in figure 9. Hence the dual field theory state associated to a broad pulse should describe a configuration with entanglement over many length scales. The field theory dual to the collapse processes we are considering is in a pure state along the complete evolution (see below) [18,23]. One pertinent question is: how much correlation exists between the components x1and x2that build up the profile in figure 9a? Let us recall the evolution of the initial profile (2.12), which is composed of two well localized subpulses. When the overlap between subpulses is negligible, the gravitational dynamics renders them transparent to each other, see figure 2b in section 2. This is however not the case for the profile in figure 2a, where the pulses have a small overlap. Both effects – 17 – JHEP05(2014)126 point to the existence of entanglement among the adjacent components of the profile in a way that grows with their proximity. This point deserves a deeper investigation. On general grounds, the time that a quantum system takes to dephase should depend on the amount of strongly correlated components it contains rather than on the total energy density. According to our interpretation, the height in ρ(0, x) provides a qualitative measure of the number of initially strongly correlated excitations. Hence the time for horizon formation, to be related with the dephasing time in the dual field theory, must be influenced by this value. This is what we find in figure 9b, where we plot the evolution of three different pulses whose initial distributions can be seen in the inset. One of them, blue, has half the broadness of the other two. It coincides with the black one in the value of the initial amplitude, while with the red one in the total mass. The time of horizon formation is very similar for the pulses with the same amplitude (blue and black). However when we compare pulses of the same mass, it is much longer for the broader one (red).7 4 Entanglement entropy oscillations All the information necessary to describe the evolution of the dual field theory can be derived from knowledge of the metric and the scalar field. However, local observables on the quantum field theory only require knowledge about the asymptotic behavior of the bulk fields. The deep interior geometry can only be accessed through the imprint it leaves on non-local observables. Following much of the recent literature, we will use to this aim the entanglement entropy. It is defined as the von Neumann entropy of the reduced density matrix for a certain subregion Aof the system SA=−TrA(ρAln ρA).(4.1) We have already mentioned that the entanglement entropy has a simple holographic representation in terms of the area of the extremal surface γAthat anchors on the AdS boundary at the boundary of the region A[12,15]. For global AdS4, the boundary is R×S2. In the present paper, the regions Awe will be interested in are circular caps. Exploiting the symmetry of the problem, γAwill be a surface of revolution which only depends on a polar angle of the boundary S2. Hence the task is to find functions x(θ) and t(θ) that extremize the area functional subject to the boundary conditions x(θ0) = π/2, t(θ0) = t0, x0(0) = t0(0) = 0 ,(4.2) where θ0is the angular aperture of the cap. This is a boundary value problem and to solve it we have used a relaxation algorithm [49]. The entanglement entropy is a UV divergent quantity and, correspondingly, the area of any surface satisfying (4.2) is also divergent. The area of the extremal surface anchoring on a cap on empty AdS4has been obtained explicitly [50] Area(γA)=2πsin θ0 −1.(4.3) 7See [38] for a systematic study of the collapse time upon varying the amplitude and the broadness. – 18 – JHEP05(2014)126 The factor 2πis due to the rotation symmetry of the configuration. In order to get a finite result the radial direction has been cut at xM.π/2. The quantity = cot xMhas the interpretation of a UV field theory cutoff. Hence the first term in parenthesis gives the area law characteristic of the entanglement entropy [51]. It is then natural to define the finite contribution to the EE by [19] S(t, θ0) = π 2G4Area(γA) 2π−sin θ0 (4.4) The entanglement entropy measured in a pure state satisfies the following important property SA=S¯ A(4.5) where ¯ Ais the region complementary to A. The holographic prescription for the calculation of the EE requires that γAand Abe homologous to one another [52]. This leads to the failure of the previous equality in a static black hole background, as it can be expected from the fact that it represents a field theory thermal state. Although the final product of the gravitational collapses we are analyzing is a static black hole, the smoothness conditions (2.6) that we imposed at x= 0 imply that γAis homologous to both Aand ¯ A. Thus (4.5) is satisfied, reflecting the fact that we are holographically modelling the unitary evolution of a pure state [18,23]. The above equality implies that it suffices to study the EE of regions not larger than a hemisphere, namely θ0∈[0, π/2]. The portion of spacetime covered by the coordinates (t, x) does not reach behind the apparent horizon. Using a lightlike coordinate vinstead of t, it has been shown that the extremal surfaces calculating the EE in a Vaidya collapse can cross both the event and the apparent horizons [18]. They do so for boundary times and regions whose size is larger than the scale set by the collapsing shell, which is proportional to M−1 3. However we are focussing in scalar configurations for which this scale is larger than the size of the boundary sphere, since these are the only ones that require several bounces before forming a horizon. Therefore the extremal surfaces we need to calculate will not reach the apparent horizon, and the coordinates (t, x) suffice to describe them. The bouncing geometries induce an oscillating pattern in the entanglement entropy. An example is shown figure 10 for a pulse with a regular evolution, namely no horizon seems to emerge, and whose dynamics is somewhat between a quasi-standing wave and a localized pulse. The holographically associated entanglement entropy exhibits oscillations with two clear periodicities, close to π/3 and π. These are respectively the periodicities characteristic of the internal dynamics of the pulse and of its radial displacement. In the next subsections we will analyze the most relevant features of the holographic entanglement entropy evolution. Our aim is twofold: we want to learn about the nonequilibrium dynamics of finite size closed systems at strong coupling and, at the same time, explore the holographic dictionary in dynamical situations. 4.1 Early time dynamics We shall start our analysis by focusing on the growth of the entanglement entropy as the scalar pulse first falls towards the interior geometry. To that purpose we consider narrow initial profiles localized close to the boundary as described by (2.10). – 19 – JHEP05(2014)126 Figure 10. EE oscillation for an initial profile (2.9) with σ= 0.4 and M= 0.09. Different colors correspond to caps with angle of aperture θ=.9,1.2,1.5. Figure 11. EE evolution for several pulses with σ= 1/16. Different colors correspond to caps with θ=.5, .6,...,1.4. In each graph, the EE values for the lower mass pulse have been rescaled to coincide with those of the larger mass one at their maxima for the sake of comparison. The red line on the left figure gives, for the pulse with larger mass, the EE at t=θ. In figure 11a we compare two well localized pulses of the same broadness but different amplitudes. One of them gives rise to a black hole of the total mass by direct collapse (M=0.3), while the other requires three bounces for the emergence of an apparent horizon (M= 0.012). For the sake of comparison, we have rescaled the entanglement entropies of the later case such that they coincide with the former one at their maxima. We find no significant difference between the EE growth to its first maxima for the small mass process and to its final values for the direct collapse one. Pursuing this line, in figure 11b we compare a one-bounce (M= 0.014) with a many-bounce pulse (M= 0.008) using the same rescaling of entropies as before. In this case there is a perfect match in the growth of the EE for both pulses. Moreover, also the decrease to the subsequent minimum is very similar. The only important difference is in the time that S(t, θ) spends at its maximum, which grows with the mass. These results allow us to sharpen the dual interpretation. The early time dynamics proceeds very much as in non-compact space. Namely, the evolution of the entanglement entropy is qualitatively well described by the propagation model in figure 7, where the – 20 – JHEP05(2014)126 entangled components of the plasma separate at the speed of light. This is illustrated by the red curve in figure 11a. This curve gives the value of the EE at t=θ, and very approximately signals the moment at which S(t, θ) saturates to its maxima. Moreover, we have compared the EE growth for narrow shells which form a black hole by direct collapse and Vaidya configurations of approximately the same broadness and mass, observing again no relevant difference. The effective propagation of entanglement at the speed of light implies that the bulk of entangled excitations have reached their maximal separation on the two-sphere at t∼π/2. However the period needed by the scalar shell to complete a bouncing cycle is always above, although close to π. It is practically πfor pulses of very small mass and increases for more massive ones, as can be seen in figure 11b. A dual heuristic picture for this effect could be as follows. Strong interactions might have generated a phase shift on the field theory wavefunction that effectively induces a larger radius for the two-sphere. Since the entanglement entropy depends on the actual size of the region considered, the only natural imprint of the phase shift on the EE would be to prolong the time interval that it keeps at its maximum values. Being an effect due to interaction, it should increase with the energy density of the state, =M. This pattern is precisely what we observe in figure 11b. 4.2 Holographic evolution In this subsection we study the evolution of the entanglement entropy based on radially localized pulses. The collapse of narrow pulses is led by a transference of the energy towards high momentum modes, such that a fraction of the pulse develops a peak sufficiently sharp to become trapped by an emerging horizon. The remaining pulse is swallowed stepwise by a growing horizon, until a final black hole of the total scalar shell mass sets up. Let us analyze first the evolution of the entanglement entropy before a horizon emerges. Remarkably we find that the EE of large regions not only oscillates, but its maxima in each bouncing cycle slightly decrease. We illustrate this effect in figure 12a with a narrow pulse which starts close to the origin and requires three bounces to generate a horizon. We showed in section 2 that when the pulse reaches the origin two opposite effects take place. Namely, together with the sharpening of a fraction of the pulse, the rest tends to increase its radial dispersion, see figure 1a. As a result the extremal surfaces associated to the EE maxima of large boundary regions intersect, at each successive bounce, a growing and more spread fraction of the scalar pulse. This causes the decrease in area and, hence on S(t, θ), visible in figure 12a. Although this is a small effect, we find it relevant. It has been suggested that the entanglement entropy of half the space could provide a definition of coarse grained entropy [23]. In spite of the oscillations, we might have expected that the maxima of the entanglement entropy monotonically increase along the evolution, and their value still serves as a notion of coarse grained entropy. We have seen that not even this is true in general. Recently it has been proposed a different holographic definition of coarse grained entropy [53], related to holographic causal information [54]. It would be very interesting to study its evolution in the collapse processes we are considering. – 21 – JHEP05(2014)126 Figure 12. Scalar profile (2.9) with σ=0.1 and M=0.012. Left: evolution of the EE for caps with θ=.5,...,1.5. We have superposed in orange the minimal radial value of A(t, x). Right: projection on the (t, x) plane of the surfaces responsible for the EE maxima of large caps in the last bouncing cycle before the horizon forms. The radial minimum of the metric function A(t, x) is a useful indicator of how far from horizon formation the gravitational system is at a given time slice. The example plotted in figure 12a suggests that the maxima of the EE do not necessarily relate to the minima of A(t, x). Since extremal surfaces do not lie in a constant tslice in our dynamical geometries, we have to analyze what region they explore deep in the bulk before reaching the previous conclusion. For the same example, figure 12b shows a projection on the (t, x) plane of the surfaces whose area gives the EE maxima of large caps in the last pre-horizon cycle. They stay indeed well before the time slice where the value of Adrops to zero, th∼11. The area of a surface seems to be maximized by a competition between reaching deep in the bulk and keeping outside the traveling shell. After the weak turbulence mechanism has acted on the scalar pulse, the area maximizing configuration arises slightly before Adrops to its minimal value. Indeed, the minima along the time evolution of Adescribe a different situation: the moments at which a well localized peak of the scalar profile is at its closest approach to the origin. Hence the entanglement entropy turns out not to be precisely correlated with the moment at which a horizon emerges. In particular, it decreases the instants before an apparent horizon first forms. It is interesting to describe the behavior of the extremal surfaces with respect to the tslicing. As long as they do not reach the scalar shell, they live on slices of constant t. If an extremal surface intersects a fraction of the falling pulse, the part involved deviates from constant ttowards smaller values of the time coordinate. On the contrary, it deviates towards bigger values when it intersects a fraction of the pulse travelling towards the AdS boundary. This is the case in figure 12b where the projections in the (t, x) plane show that the extremal surfaces tilt towards larger values of tat their inner portion. Indeed, they reach the part of the scalar profile not trapped by the emerging horizon, and which has started to move away from the origin before the horizon neatly forms. – 22 – JHEP05(2014)126 Figure 13. Post-horizon evolution of the EE for the same case plotted in figure 12. The time for the first collapse is th∼11, and at around t= 28 the horizon radius is 83% of its final value. The green line gives S(θ=1.5) for a static black hole of the total mass, where we did have into account the numerical mass loss along the evolution. We have plotted in figure 13 the post-horizon evolution of the entanglement entropy. Using a grid of 7×104points we could complete five oscillations beyond horizon formation with a mass loss below 3%. The oscillations of entanglement entropy neatly reflect the impact of the horizon, decreasing their amplitude at a pace correlated with the approach of the horizon to its final value. The maxima of the EE, whose value dropped along the pre-horizon phase, should rise to the result prescribed by a black hole of the total mass. Indeed, we observe that the maxima slowly but monotonically increase along the posthorizon cycles. The green dashed line in figure 13 signals the value that the EE of a θ=1.5 cap should reach. Its slight decrease just reflects that we did have into account the small mass loss of the numerically simulation. The traveling pulse keeps radial localization along the first five post-horizon cycles.8 Following the argumentation in section 3, this suggests that while part of the system dephases in correspondence with the appearance of an apparent horizon, part of it still retains quantum coherence. Moreover, a typical separation could be associated to the remaining entangled degrees of freedom, linked to the radial position of the pulse. Thus a pattern emerges in which the system undergoes a stepwise loss of quantum coherence, triggered by the dephasing of a subset of the degrees of freedom. Before closing this section, we would like to draw attention to the striking similarity between the oscillations in EE shown in figure 13, and those of a different, albeit also non-local, operator of the XY quantum spin chain studied in [28] (see figure 1). 8Radial localization is manifest in the time span of the oscillations, which is close to πbefore and after a horizon first forms. In spite of this, the radial spread of the profile progressively increases (recall figure 4a for a similar example). This can be detected in the emergence of a small modulation in the EE with a shorter period, consistent with π/3. – 23 – JHEP05(2014)126 Figure 14. Two pulses with σ= 0.1 and masses slightly above (blue) and below (orange) critical collapse. Left: minimal radial value of A(t, x). Right: EE evolution for caps with θ= 0.9,1.2,1.4,1.56. The green line marks horizon formation time for the above critical pulse. 4.3 Behavior across critical points A very relevant characteristic in the collapse of narrow pulses is that the fraction of energy in the sharp front which generates the horizon can become arbitrarily small. The transition between processes with nand n+1 bounces happens indeed as the energy of the trapped front vanishes. Therefore, it becomes relevant to investigate the behavior of the entanglement entropy across these critical collapses. The question we want to answer is whether the appearance of a horizon, no matter how small, leaves an imprint in the posterior evolution of the entanglement entropy. On line with the results in the previous subsection, the answer we find is negative. In figure 14a we plot the evolution of the radial minimum of A(t, x) for two initial profiles (2.9) with broadness σ= 0.1 and masses M= 17.324 and M= 17.32. The former generates a horizon after two bounces with radius xh= 0.0025, which is 4.4% of the Schwarzschild radius associated to its total mass. A trapped horizon emerges for the latter after three bounces with a radius one order of magnitude larger, xh= 0.02, which is 35% of its corresponding Schwarzschild radius. Hence the masses of the two profiles are close to the critical value for the transition between two and three bouncing pre-horizon cycles, being the former slightly above and the latter slightly below. 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