Holographic quenches with a gap

JHEP06(2016)172 Published for SISSA by Springer Received:May 12, 2016 Accepted:June 15, 2016 Published:June 29, 2016 Holographic quenches with a gap Emilia da Silva,aEsperanza Lopez,aJavier Masband Alexandre Serantesb aInstituto de F´ısica Te´orica IFT UAM/CSIC, Universidad Aut´onoma de Madrid, C/ Nicol´as Cabrera 13–15, 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] Abstract: In order to holographically model quenches with a gapped final hamiltonian, we consider a gravity-scalar theory in anti-de Sitter space with an infrared hard wall. We allow a time dependent profile for the scalar field at the wall. This induces an energy exchange between bulk and wall and generates an oscillating scalar pulse. We argue that such backgrounds are the counterpart of quantum revivals in the dual field theory. We perform a qualitative comparison with the quench dynamics of the massive Schwinger model, which has been recently analyzed using tensor network techniques. Agreement is found provided the width of the oscillating scalar pulse is inversely linked to the energy density communicated by the quench. We propose this to be a general feature of holographic quenches. Keywords: AdS-CFT Correspondence, Holography and condensed matter physics (AdS/CMT) ArXiv ePrint: 1604.08765 Open Access,c The Authors. Article funded by SCOAP3.doi:10.1007/JHEP06(2016)172 JHEP06(2016)172 Contents 1 Introduction 1 2 The hard wall model 2 3 Infrared physics 4 4 Time-dependent boundary conditions at the wall 7 5 The holographic dual of a quench 9 6 Discussion 13 1 Introduction Holography has proven useful for the study quantum field theories in regimes where standard techniques fail. Besides strong coupling physics, the out of equilibrium dynamics of quantum systems has emerged in the last years as a topic where holographic techniques can provide important insight. The dictionary between classical geometries and field theory states brings into correspondence black holes in an asymptotically anti-de Sitter space with states of thermal equilibrium. It is then natural to interpret processes of gravitational collapse in terms of evolution towards thermal equilibrium in the dual field theory [1,2]. Under certain conditions matter distributions give rise to oscillating geometries, which might end in the formation of a horizon on longer time scales or keep bouncing forever [3,4]. Following the previous logic, this opens the door to describe holographically excited states in a strongly interacting theory which do not lead to fast thermalization [5,6]. Although fast thermalization is generically expected, a variety of systems and initial conditions are known where it is not realized [7]. It is exciting that holography might help to address this question, which is a a subject of active research in condensed matter and quantum information theory. We are interested in holographically modeling quantum quenches where the final hamiltonian is gapped. We will use for that a very simple geometric setup consisting in introducing an infrared hard wall in AdS. Gravitational collapse of a massless scalar field in the hard wall model has been considered before [8,9]. New ingredients are however introduced. We show that a mass can be assigned to the wall which affects the infrared physics of the dual field theory. The space of couplings is parameterized by this mass together with the value of the scalar field at the wall. A time dependent profile for the scalar field at the wall allows to move in the space of couplings while introducing a homogeneous excitation in the system. In this way a quench is holographically simulated, where initial and final hamiltonian differ. – 1 – JHEP06(2016)172 Tensor network techniques have been applied very successfully to the study of ground states properties [10]. More recently they have been used to study real time dynamics, providing examples with which to crosscheck the holographic results. The response of the massive Schwinger model to a quench was addressed in [11]. In the limit of small energy quenches the resulting evolution exhibits quantum revivals. Namely, the initial out of equilibrium states is periodically reconstructed after intermediate stages of apparent decoherence [12]. We compare the revival phenomenology of the massive Schwinger model with the oscillating geometries generated by a scalar pulse in the hard wall mode, finding consistency. The plan of the paper is the following. In section 2we introduce the hard wall model. In section 3we study the infrared physics of its dual field theory. Time dependent boundary conditions at the wall are described in section 4. In section 5we identify the criteria for the hard wall model to provide a good qualitative description of a quench, taking the Schwinger model as guideline. We end in section 6with a discussion on open issues. 2 The hard wall model We want to model a 1+1 dimensional QFT on an infinite line, hence we will work in AdS3 with Poincare slicing. A crude holographic representation of a mass gap is achieved by imposing an infrared cutoff in the bulk, modeled as an infinitely hard wall. Such simple set up will yet allow us to get interesting insight into the holographic dictionary. In the hard wall framework, we will consider gravity coupled to a massless scalar field [8,9]. Configurations with translational invariance along the boundary spatial direction can be described using the diagonal ansatz for the metric ds2=1 z2−A(t, z)e−2δ(t,z)dt2+dz2 A(t, z)+dx2.(2.1) The AdS boundary corresponds to z= 0, while the location of the infrared wall will be denoted by z0. The equations of motion are ˙ Φ = Ae−δΠ0 ,˙ Π = zAe−δΦ z0 ,(2.2) δ0=z 2(Φ2+ Π2), A0=z 2A(Φ2+ Π2) + 2 z(A−1) ,(2.3) where Φ = φ0and Π = A−1eδ˙ φ, with φ0and ˙ φthe space and time derivatives of the scalar field respectively. The equations of motion allow an inflow of energy from the AdS boundary into the bulk, triggered by the variation of the non-normalizable component of the scalar field. We set this component to zero along the paper by requiring the scalar field to vanish asymptotically. This insures the conservation of the total mass M=z−2 0(1 −A0) + 1 2Zz0 0 y−1(Φ2+ Π2)A dy . (2.4) – 2 – JHEP06(2016)172 The value of the metric function Aat the wall, A0, is determined by the momentum constrain ˙ A=z A2e−δΦΠ .(2.5) We assume that the position of the infrared wall is a parameter of the model not susceptible of any variation. Under this condition, A0can only be constant if the scalar field satisfies Dirichlet or Neumann boundary conditions at the wall [8,9]. In the following we introduce a freedom in the hard wall model which has not been addressed before. We will allow for a time dependent profile of the scalar field at the wall. In particular, our seed for the evolution will be a non-trivial function Π0(t) = Π(t, z =z0).(2.6) The momentum constraint ensures the conservation of the total mass also under these general boundary conditions. The first term on the r.h.s. of (2.4) can be considered a property of the wall, which we will refer to as M0, the mass of the wall, while the second is the contribution of the scalar profile to the mass. A time dependent scalar profile at the wall induces an energy exchange between the gravity sector and the scalar field, represented respectively by the first and second terms in (2.4). A vanishing value of the wall mass corresponds to introducing an infrared cutoff in empty AdS space. Positive values M0z2 0<1 relate to black hole geometries whose horizon lies behind the wall. Negative wall masses are linked to geometries which in the absence of the wall contain a naked singularity at the origin. The wall excises the singularity, leaving open their consistency as holographic backgrounds. A trivial case of (2.6) are Dirichlet boundary conditions. The equations (2.2)–(2.3) with Dirichlet boundary conditions at the wall admit static solutions with a non-trivial scalar profile [9]. The time independent Klein-Gordon equation simplifies to z−1Ae−δΦ = α , (2.7) where αis an integration constant. The metric (2.1) allows the gauge freedom of reparameterizations along the time coordinate. Using this, tcan be fixed to be the proper time at the boundary, which is the natural choice for holographic applications. The integration constant αis then identified with the expectation value of the field theory operator dual to the scalar field, α= 2hOi. As an example, the set of static solutions without horizons with vanishing wall mass is plotted in figure 1a; other values of M0behave similarly. Static solutions exists up to a threshold for φ0, the scalar field at wall. There are two branches of solutions joining at it. Those with higher mass are linearly unstable. They approximate a black hole solution with horizon at the position of the wall as φ0decreases. Consequently Mz2 0tends to unity and hOi to zero along the unstable branch. The lower mass branch is stable. Remarkably it includes regular solutions whose Schwarzchild radius is neatly above the wall position. We will use the integrand in (2.4) ρ=1 2z(Φ2+ Π2)A , (2.8) to describe the radial energy distribution of the scalar configuration. Figure 1b shows this function for several values of φ0along the stable branch. – 3 – JHEP06(2016)172 Figure 1. Left: properties of static scalar configurations with vanishing wall mass and Dirichlet boundary conditions φ(z0) = φ0. Right: radial energy profile of solutions along the stable branch for φ0=.5, .6,...,1.3 and the threshold value φ0= 1.32 from bottom to top. 3 Infrared physics We analyze the dual field theory infrared physics associated to the static solutions by focussing on two quantities: i) the correlation length, defined as the size of the interval that saturates the large N contribution to the entanglement entropy; and ii) the mass gap, defined as the lowest normal frequency over a static solution without horizon. The entanglement entropy of a region is defined as the von Neumann entropy of the density matrix traced over the degrees of freedom in its complementary. The entanglement entropy of an interval is measured holographically by the length of a geodesic that anchors at the AdS boundary on the interval endpoints [13]–[15], and is homologous to it [16]. We will introduce a modification of the second condition in the hard wall model. Besides geodesics which satisfy the homology constraint, we also allow geodesics which together with their associated AdS boundary interval are homologous to an interval at the wall. In mathematical terms this means that the geodesic should be homologous to its associated interval in the relative homology defined with respect to the infrared boundary. The relaxation of the homology constraint is necessary because geodesics which intersect the AdS boundary at the endpoints of a sufficiently large interval unavoidably intersect the infrared wall. A question arises on the possible contribution of the length of the interval at the wall to the entanglement entropy. Including it would lead to an extensive behavior of the entanglement entropy, which is characteristic of a thermal state or a highly excited state which violates the area law. The eigenfrequencies of the laplacian in the dual geometries should have an imaginary component, reflecting the finite lifetime of excitations in such field theory states. This is verified in the black hole backgrounds, dual to thermal equilibrium. However, the frequencies of the harmonic modes in the hard wall setup with Dirichlet or Neumann boundary conditions at the wall are purely real. Hence in order to obtain a consistent infrared physics we must assume that the interval at the wall does not play a role in the computation of the holographic entanglement entropy. Since any point in the AdS boundary can be joined by a geodesic with an arbitrary point in the wall, there is an infinite set of geodesic configurations associated to a given – 4 – JHEP06(2016)172 Figure 2. Connected and disconnected geodesic configurations (black) anchoring at the endpoints of a boundary interval (pink). The interval, the geodesic and possibly an interval at the wall (purple), determine a closed area in the bulk (blue). field theory interval which reach the wall. Among them we should select that extremizing the length functional with respect to the endpoints at the wall. When there is more than one geodesic configuration satisfying all the stated requirements, that with smallest length will determine the entanglement entropy [13,14]. This is the case for sufficiently small intervals. There is a connected geodesic joining its endpoints without reaching the wall, along with the disconnected configuration composed of two geodesic segments extending from the interval endpoints to the wall, see figure 2. Let us consider static solutions of the hard wall model with vanishing scalar profile for simplicity. A geodesic anchoring at the endpoints of an interval of size l, as far as it does not reach the wall, is not affected by its presence. The length of such geodesic in a black hole background of mass M0is 2 log 2 sinh pM0l/2−2 log pM0 , (3.1) where the bulk has been cut at a UV scale z=in order to regularize the otherwise divergent length. The disconnected choice which extremizes the length is given by geodesics starting on the interval endpoints and extending radially to the wall. Its length is 2 log 2z0/−2 log 1 + q1−M0z2 0.(3.2) For small intervals the connected choice is preferred, but for large ones it is the disconnected configuration which has smaller length. Once the latter dominates, the entanglement entropy saturates to a value independent of the length of the interval. This is the behavior expected in a gapped theory, whose holographic derivation was sketched in [14]. The length for which (3.1) and (3.2) coincide gives the correlation length as defined above ξ=2 √M0 arcsinh √M0z0 1 + p1−M0z2 0!.(3.3) For vanishing wall mass we obtain, ξ=z0. The correlation length increases with the wall mass up to a finite value attained at the horizon threshold M0z2 0= 1. Expression (3.3) can – 5 – JHEP06(2016)172 (a) (b) (c) Figure 3. Correlation length (a), mass gap (b) and their product (c) as a function of φ0for M0= 1/2,0,−1,−4,−8 from red to blue. be smoothly continued to negative M0, causing a monotonic decrease of the correlation length. For very large negative masses the correlation length tends to zero as ξ≈π p|M0|.(3.4) In figure 3a we plot ξas a function of φ0for several fixed values of the wall mass. At the topological level, all values M0z2 0<1 give equivalent geometries. In particular, the entanglement entropy of an interval is equivalent to that of its complementary region for all of them. It follows immediately from requiring that a region at the AdS boundary and its associated geodesic are homologous in the relative homology defined with respect to the infrared wall. This suggests that all geometries with M0z2 0<1 describe pure states [17,18]. An important information on the static backgrounds is their spectrum of normal modes. The holographic dictionary relates it with the non-analyticities in the two-point function of the dual field theory operator O. Hence we will use the lowest normal frequency as definition of the mass gap, which we plot in figure 3b. Both the correlation length and the mass gap give a measure of the characteristic infrared scale of the theory. In an interacting theory we do not expect both quantities to coincide, but they should be of the same order. Figure 3c shows that this is verified, except close to the stability threshold, where ωvanishes while ξremains finite. The correlation length has been calculated using the geodesic length, which gives the leading N contribution. However the mode becoming unstable close to the threshold for horizon formation reflects an order 1 effect. For this reason ξand ωmeasure different aspects of the physics in that limit, and thus give independent answers. For very large negative wall mass and vanishing scalar profile, where the correlation length is given by (3.4), we have obtained numerically ξω ≈6.5,(3.5) in accord as well with the field theory reasoning. A necessary consistency check on the backgrounds with negative wall mass is that they do not induce superluminal propagation at the boundary. Indeed, we have evaluated the dispersion relation for several normal modes on static backgrounds with negative M0, – 6 – JHEP06(2016)172 and always found group velocities smaller than one. A related check requires that, after reflection on the hard wall, light rays leaving the AdS boundary return to it within the light-cone of the starting point. For static solutions with a vanishing scalar profile, the distance travelled by such light ray verifies1 tanh √M0∆t 2= cosh γtanh √M0∆x 2,(3.6) where γdetermines its incidence angle with the AdS boundary: dz dx z=0 = sinh γ. Hence ∆t≥∆x. Relation (3.6) extends with the same conclusion to backgrounds with negative wall mass as far as sinh γ≥p|M0|z0. Smaller incidence angles give rise to null geodesics that get repelled towards the AdS boundary without reaching the wall. Independently of γthey verify ∆t= ∆x=π p|M0|.(3.7) They return to the boundary on the light cone of the starting point, respecting thus causality. 4 Time-dependent boundary conditions at the wall We have seen that the infrared physics is not only determined by the position of the wall. The correlation length and the mass gap depend strongly on M0and φ0. Since boundary conditions can not be changed by bulk dynamics these parameters should be interpreted as couplings, in accordance with the usual holographic dictionary. Parameters in the yellow area of figure 4admit stable static solitonic configurations. These configurations provide the ground states of the dual field theory for different couplings. Pure AdS with a cutoff, corresponding to M0=φ0= 0, respects Lorentz invariance, while non-zero values of M0and φ0in general break it. These backgrounds represent pure states with order N2 expectation values of the stress tensor and the operator Odual to the scalar field. Ground states of these characteristics describe condensates. Therefore generic couplings {M0, φ0} create a non-trivial medium in the dual field theory. We can simulate an action on the dual field theory in which both {M0, φ0}vary, by imposing time dependent boundary conditions at the wall. An static state is assumed at early times. We consider boundary conditions of the form (2.6) with the particular profile Π0(t) =  α 1 cosh2t α ,(4.1) which forces the value of φ0to change. In the process a scalar pulse is generated which enters from the wall into the bulk. As we explained in section 2, the total mass (2.4) is conserved under generic boundary conditions at the wall. Hence the scalar pulse draws its energy from the wall, determining in this way the final value of M0. In consequence M0can be rendered negative as a result of (4.1). This is always the case when the initial background is pure AdS with an infrared cutoff. 1Null geodesics are described by z√M0= sinh γsinh(√M0x), t√M0= arctanhcosh γtanh(√M0x), which ca be analytically continued to negative masses. – 7 – JHEP06(2016)172 Figure 4. Left: in yellow, couplings which admit stable solitonic solutions. Final couplings after an action (4.1) which starts at M0=φ0= 0: as a function of αfor = 0.3 (blue), and as a function of negative for α=.8 (red). Right: relation (4.4) as a function of φ0for fixed M0, with the same color code as in figure 3. Geometries with M0<0 lead to regular evolutions of the homogeneous Einstein equations much as non-negative values do, providing one more check of their consistency in the context of the hard wall model. The blue line in figure 4shows the final couplings reached after an action (4.1) starting from pure AdS, as a function of αat fixed = 0.3. Slow actions, namely α→ ∞, drive the system to the point signaled with a dot. Decreasing α produces scalar pulses sharply localized on the radial direction of increasingly large mass, which in turn implies large negative M0. The value of φ0keeps practically constant, showing that it is mainly determined by . For completeness, we plot in red the final couplings as a function of at fixed α=.8. The Schwarzchild radius for most solitonic solutions is below the wall position, see figure 1a. Conservation of energy forbids that actions (4.1) starting on such states lead to the formation of a horizon. Instead they will result in a scalar pulse that bounces forever back and forth between AdS boundary and infrared wall [8]. Gravitational collapse ending in the formation of a black hole holographically models quantum field theory thermalization. On the contrary, oscillating geometries where interpreted in [5,6] as geometrical counterparts of quantum revivals in the unitary evolution of the dual field theory. Quantum revivals denote periodic reconstructions of an initial out of equilibrium state [12]. Interacting field theories typically exhibit a fast approach to a thermal state, which prevents revivals. Hence oscillating geometries should be disfavored. A natural exception to this rule can be found in out of equilibrium states where the scale set by the energy density is smaller than the infrared gap. Namely, revivals in d-dimensional field theory might happen only when E/n)1 d< mgap ,(4.2) where mgap is a measure of the gap, Ethe energy density and nthe number of elementary fields. The dynamics of infalling matter shells in global AdS was found to be consistent with this relation [5,6]. In that case mgap is set by the inverse of the boundary sphere radius and n∼c, the central charge of the dual CFT. – 8 – JHEP06(2016)172 an irreversible decay of the electric field. Virtual pairs can not propagate further than the correlation length before recombining, what is consistent with the picture of a thin pulse traveling back and forth between wall and AdS boundary. It is unlikely that such a field theory state, which combines characteristics proper of high and low energies, can be produced without fine tuning. In spite of these issues, thin pulses of small mass were important for interpreting oscillating geometries in global AdS in terms of quantum revivals [5,6]. Their periodicity is τ≈πR , (6.2) with Rthe radius of the global AdS boundary sphere, where the dual field theory lives. Revivals of a free theory on a finite size space have the same periodicity, which also agrees with the simple quasiparticle picture of [27]. Broad pulses have a different period, inherited from the lowest normal mode and difficult to interpret in purely field theory terms. An additional hint for the relation of thin pulses with field theory states of high energy is the following. The evolution of rational CFT’s on a circle after a quench with energy much larger than the inverse radius was studied in [28]. As it happens for holographic models based in thin pulses, the overlap of the initial state with the state describing the system at time texperiences a very fast decay. Subsequently the initial state is partially reconstructed with periodicity (6.2). It is the very constrained dynamics of RCFT’s what allows to overcome the energy bound for revivals (4.2), which holographic theories respect [29]. Finally we want to stress some properties of the very simple hard wall setup considered in this paper. By introducing time dependent boundary conditions at the wall, we move in coupling space between theories with different infrared physics. The wall acts as an infinite reservoir from which to drag energy at the price of increasing the mass gap of the dual theory. Hence we come closer to a quench scenario than just simulating an energy injection without real change in the hamiltonian. Moreover, the energy density resulting from a wall action (4.1) is mainly determined by one of its two parameters, the time span. This renders again more direct the map with a quench. These characteristics make the wall driven holographic quenches worth further studying. Acknowledgments We want to thank J. Ignacio Cirac, Roberto Emparan, Karl Landsteiner, Fernando Marchesano, Erik Tonni and Tadashi Takayanagi for discussions. We very especially thank Boye Buyens, Jutho Haegeman, Karel Van Acoleyen and Frank Verstraete for sharing with us some of their results prior to publication. The work of E.daS. is financed by the spanish grant BES-2013-063972. E.L. has been supported by the spanish grant FPA2012-32828 and SEV-2012-0249 of the Centro de Excelencia Severo Ochoa Programme. The work of J.M. is supported in part by the spanish grant FPA2011-22594, by Xunta de Galicia (GRC2013- 024), by the Consolider-CPAN (CSD2007-00042), and by FEDER. A.S. is supported by the European Research Council grant HotLHC ERC-2011-StG-279579 and by Xunta de Galicia (Conselleria de Educaci´on). Part of the numerical calculations were performed at the Centro de Supercomputaci´on de Galicia (CESGA). – 15 – JHEP06(2016)172 Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. References [1] T. Banks, M.R. Douglas, G.T. Horowitz and E.J. Martinec, AdS dynamics from conformal field theory,hep-th/9808016 [INSPIRE]. [2] U.H. Danielsson, E. Keski-Vakkuri and M. 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