Floquet Scalar Dynamics in Global AdS
Anxo Biasi, Pablo Carracedo, Javier Mas, Daniele Musso, Alexandre Serantes
FFloquet Scalar Dynamics in Global AdS
Anxo Biasi ∗ , Pablo Carracedo, † Javier Mas, ‡ Daniele Musso § and Alexandre Serantes ¶ Departamento de F´ısica de Part´ıculasUniversidade de Santiago de CompostelaandInstituto Galego de F´ısica de Altas Enerx´ıas (IGFAE)E-15782 Santiago de Compostela, Spain Meteo-Galicia, Santiago de Compostela, Spain International Centre for Theoretical Sciences-TIFR,Survey No. 151, Shivakote, Hesaraghatta Hobli,Bengaluru North, India 560 089
Abstract
We study periodically driven scalar fields and the resulting geometries with global AdSasymptotics. These solutions describe the strongly coupled dynamics of dual finite-sizequantum systems under a periodic driving which we interpret as Floquet condensates.They span a continuous two-parameter space that extends the linearized solutions onAdS. We map the regions of stability in the solution space. In a significant portionof the unstable subspace, two very different endpoints are reached depending uponthe sign of the perturbation. Collapse into a black hole occurs for one sign. For theopposite sign instead one attains a regular solution with periodic modulation. Wealso construct quenches where the driving frequency and amplitude are continuouslyvaried. Quasistatic quenches can interpolate between pure AdS and sourced solutionswith time periodic vev. By suitably choosing the quasistatic path one can obtain bosonstars dual to Floquet condensates at zero driving field. We characterize the adiabaticityof the quenching processes. Besides, we speculate on the possible connections of thisframework with time crystals. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] a r X i v : . [ h e p - t h ] M a y ontents . 52 Introduction and main results
The physics of periodically driven many-body systems is a fascinating chapter in the studyof out-of-equilibrium dynamics [1, 2]. Their behavior can differ substantially from that oftheir static counterparts. Floquet systems have been investigated at criticality with the toolsof conformal field theory (CFT) (see [3] and references therein). The naive expectation thatthey can only increase indefinitely their energy because of the driving can be avoided in someregions of the driving parameter space. This was confirmed by explicit examples that attaina steady state at finite temperature [3, 4]. Even though the actual saturation mechanism isnot completely understood, the fact that periodic driving can radically alter the stability ofequilibrium points is well known from nonlinear dynamical systems, the Kapitza pendulumbeing a prominent example [5].The AdS/CFT correspondence maps equilibration processes onto the dynamical evolutionof a dual gravitational system. The holographic dictionary defines the sources of the fieldtheory in terms of asymptotic modes of the bulk fields. It is then possible to study a drivenholographic system coupled to an external source. In the cases of interest here, the sourcecorresponds to the leading asymptotic mode of a bulk scalar field in an asymptotically globalAdS spacetime.Periodic driving has been scarcely studied in the context of AdS/CFT. The analysis per-formed in [6] comes closest in spirit to the present study (see also [7–11], all in the probeapproximation). The main difference is that their initial states are already mixed, while herewe consider the driving of pure states. Their geometrical ansatz is embedded in the Poincar´epatch, with an arbitrarily small planar horizon to cap the IR. They find a monotonousgrowth of the black hole horizon as a general outcome of the driving. Such growth leadsto an infinite temperature final state. The result is, on the one side, compatible with thealready mentioned natural expectation and, on the other side, it comes out as a consequenceof the ingoing boundary conditions at the horizon. The only remaining freedom may be inthe rate of growth of the energy density and entropy.The authors in [6] find three different phases depending upon the range of values acquiredby the dimensionless combination χ = φ b ω b , namely the product of the amplitude and thefrequency of the driving at the boundary. Small values of χ correspond to a dissipationdominated regime, where the horizon absorbs all the energy entering the system from theboundary. Here the scalar field synchronizes with the driving but does not change its ampli-tude. Raising χ they find two other phases in which the scalar field stops being “transparent”,and it may absorb part of the energy input and increase its amplitude.The picture of [6, 10] should be compared with the results we find here. The fact that ourdriving does not necessarily imply decoherence (i.e. a collapse into a black hole) is by nomeans trivial and it only occurs in certain regions of parameter space. Charting this region,3haracterizing its features, and studying its stability is the main target in the present paper.We have examined both complex and real massless scalar fields. While a priori their dynamicsis quite different, remarkably, the final results of the real and complex cases exhibit strongsimilarities, both in the structure of the phase space and in the order of magnitude of specificnumerical values like stability thresholds.Finding the manifold of driven and periodic solutions is the first step. Since such solutionscould in principle be all unstable, a thorough inspection of their stability is in order both atthe linear and nonlinear level. Linearly stable solutions proved to exist and be robust againstrather large fluctuations. For unstable solutions, we follow numerically the evolution in thesearch for the final endpoint at large times. The picture that emerges is rather interesting:in significant portions of the unstable subspace, the solutions lie at the boundary of tworadically different long time behaviors. On one side, fluctuations drive the geometry to acollapse into a black hole, the dual theory loses coherence and thermalizes as expected. Onthe other side, the geometry is regular forever. It however exhibits a periodic modulation thathas the form of a relaxation oscillation where the solution periodically bounces back andstays for a long time close to the initial unstable solution. From the dual field theory pointof view, the quantum state remains pure but exhibits an emergent pulsating modulation.The framework at hand allows us to address the important question of the adiabatic prepa-ration of Floquet condensates [12–14]. By slowly varying the amplitude and/or frequency ofthe driving, one can study if the system keeps up with the source and follows a trajectory onthe manifold of periodic solutions. This is naively expected in the limit of slow quenches, asa natural dual counterpart of the quantum adiabatic theorem. However, a careful analysisreveals subtleties whenever the driving frequency approaches that of a normal mode of AdS.Similarly, we study the quenching of the system from AdS through a cyclic protocol startingand ending with a vanishing driving. The possibility that such cycle ends on a periodicsolution with vanishing source (but not just AdS) could perhaps relate to the open questionof spontaneous breaking of continuous time translations [15, 16].The structure of the paper is as follows. In Section 2 we study the case of the driven complexfield. We characterize in detail the space of stationary solutions and focus on their linear andnonlinear stability properties. In Section 3 we analyze the response of the system upon acontinuously varying driving. This unravels interesting structures like the presence of criticalvalues of the quench parameters separating classes of qualitatively different time evolutions(ending or not in a black hole, for instance). In Section 4 we show that the unstable solutionsare, in fact, attractor solutions in the sense of Type I gravitational phase transitions studiedin [17–19]. In Section 5 we describe two alternative quenching protocols to construct a bosonstar starting from AdS. One is quasistatic, while the other involves an unstable solution as an We are adopting the usual terminology of dynamical systems like, for example, the inverted pendulum.
Our case study involves the simplest possible setup, namely a complex scalar field in globalAdS , S = 12 κ (cid:90) d x √− g ( R − − (cid:90) d x √− g (cid:0) ∂ µ φ∂ µ φ ∗ − m φφ ∗ (cid:1) , (2.1)with Λ = − /l for AdS . We will set κ = 8 πG = 1, l = 1. The action is invariant underglobal U (1) transformations φ → e iα φ . Our ansatz for the metric is ds = 1cos x (cid:0) − f e − δ dt + f − dx + sin x d Ω (cid:1) , (2.2)where x ∈ [0 , π/
2) is the radial coordinate. We will examine the space of solutions adaptedto the following time-periodic ansatz φ ( t, x ) = ρ ( x ) e iω b t , (2.3)with ρ ( x ) a real scalar function. The space of solutions contains just three functions f, δ Figure 1:
Set of profiles for a range of values of ω b holding ρ b = | φ ( π/ | fixed. ρ ; they are determined by two parameters, namely, the frequency ω b and a boundaryvalue for ρ .Introducing the ansatz (2.3) in the Einstein-Klein Gordon equations of motion, gives a set of ω -dependent static equations for ρ ( x ) which can be solved numerically by standard methods(see (A.11)). Each solution of this system gives rise to a static geometry where the scalar fieldrotates in the complex plane with angular velocity ω b while keeping constant its modulus ρ ( x ). The boundary values of ρ ( x ) are respectively ρ o = ρ (0) at the origin and ρ b at theboundary; ρ b is defined through the asymptotic Taylor expansion ρ ( x ) = ρ b ( x − π/ − ∆ + ... , with ∆ = 3 / (cid:112) / m being the conformal dimension of the dual scalar operator O .For non-vanishing ρ b , the dual state is interpreted as being driven by a time-periodic source ρ b e iω b t . We abbreviate these sourced periodic solutions as SPS. In this paper we analyzethe case of a massless scalar, m = 0, in detail. We shall also comment on partial resultsobtained in the case m = −
2, where everything seems to follow the same pattern so far. Inthe massless case, the unsourced solutions with ρ b = 0 are termed boson stars -BS- in (atleast part of) the literature [20, 21] and we shall adhere in what follows to this name.Figure 2: The surface of static geometries corresponding to a periodically driven complex scalarfield. The surface intersects the boson star plane ρ b = 0 at a set of curves as shown in the plot.In making this plot, we have adopted the phase convention that makes ρ o a positive number (seeFig. 3). Hence the sign of ρ b is nothing but the relative sgn( ρ o /ρ b ), which is correlated with thenumber of nodes of φ ( x ). ω b , ρ o , ρ b ) (see Fig. 1 for a visualization of these three parameters). TheSPS surface cuts at ω b = 0 on a line of configurations where the scalar takes a constant radialprofile, in particular ρ o = ρ b . Boson stars lie at the intersection of the SPS surface with thezero source plane ρ b = 0. The boson star curves start from the bottom plane ρ o = 0 at thevalues ω n = 3 + 2 n , given by the spectrum of normal frequencies of the massless scalar inAdS . As the value of ρ o is increased, the nonlinear dressing of the linearized solution shiftsthe value of the frequency and gives rise to the wiggly curves depicted in Fig. 2. The lowerportion of this curve that bends towards smaller values of ω b was already constructed in [21].The different sectors indicated with I,II,III,... correspond to SPS whose radial profiles ρ ( x )have 0 , , ... nodes respectively.Figure 3: Radial profiles for higher modes in regions
I, II and
III in Fig. 2 having the same valueof the scalar field at the origin, ρ o = 0 .
65 and different ω b = 1 . , .
96 and 5 .
04 from top to bottom.As the number of nodes increases, the energy density tends to concentrate deeper in the bulk. Thereader should understant that this profile is rotating in the transverse plane. The relative sign ρ b and ρ o correlates with the even or odd number of nodes. In the case of a driving by a relevant scalar field with m = −
2, the surface remains qualita-tively the same. The unsourced solutions analogous to the boson stars branch instead fromthe spectrum of linearized fluctuations which is now ω n = ∆ + 2 n with ∆ = 1 , ∇ µ (cid:104) T µ (cid:105) = (cid:104)O ( t ) (cid:105)∇ J . m ( t ) = − (cid:16) ˙ φ ∗ b ( t ) (cid:104)O ( t ) (cid:105) (cid:17) . (2.4)In the undriven case ˙ φ b ( t ) = 0, the r.h.s. is identically zero, and the system is isolated. For˙ φ b ( t ) (cid:54) = 0, the magnitude and sign of the product on the right hand side is unforeseeable.The product of the two factors can be either positive or negative. This implies that, for avariable source, energy can flow both in and out. The source function φ b ( t ) is known. Incontrast, the 1-point function (cid:104)O ( t ) (cid:105) is a teleological quantity (in the radial direction), likeevent horizons are (in time). It can only be extracted from the boundary behaviour of thesolution once this has been computed down to the origin and regularity has been imposed. Inthe particular case of a SPS, the vev oscillates harmonically in phase with the source. Thisparticular case yields an exactly vanishing value for the right hand side. However, as soon asthe SPS is perturbed, ˙ m will start to fluctuate around zero. The average of this fluctuationwill signal whether the mass starts to build up and the system eventually collapses or, else,if the net balance is zero, the perturbed solution stays regular in the future.An important remark is that there is a chance to find a regular solution only if the quantumsystem we drive is in a pure state. If there was a horizon, no matter how small, then part ofthe injected energy would fall behind it, and never reach back to the boundary. Unavoidably,the mass would then grow monotonically and the black hole horizon end up reaching theboundary, i.e. reaching the dual geometry to an infinite temperature state.In summary, a thorough study of the stability properties of the SPS is compulsory. We willdo it first in a linearized approximation and then in a fully nonlinear setup. Linear stability
In this section we shall establish and chart the regions of stability within the set of SPS’sgiven in Fig. 2 by performing a linearized fluctuation analysis. We find it convenient to movealong level curves with constant source amplitude ρ b . These curves are depicted in Fig. 4.Notice that from the point of view of an observer at the boundary, namely for each pair( ρ b , ω b ), there can be more than one SPS. They have different bulk profiles and, in particular,they reach the origin at different values of ρ o . For instance, in Fig. 4 a duplicity has beenhighlighted in the case ( ρ b = 0 . , ω b = 2) by the red dots on the left. In the case of multiplesolutions corresponding to the same boundary data ( ρ b , ω b ), it is natural to suspect that oneof them is stable since it represents the “ground state” of the sector.Along the lines of constant ρ b , one finds extremal values of the frequency where two solutionscorresponding to the same ω b become degenerate. At these points the spectrum of linearized8igure 4: Level curves of the SPS surface plotted in Figure 2. Each curve corresponds to a constantvalue ρ b (sometimes reported explicitly by a small number near the curve). The black curves denotethe boson stars with ρ b = 0. The two red points both correspond to ω b = 2 and ρ b = 0 .
09. Thecyan dot represents the point where ω b reaches its maximal value along the level curve. The darkyellow dots mark the stability thresholds along the boson star lines. fluctuations contains a zero mode that connects the two degenerate solutions. One suchexample is shown in Figs. 4,5 by means of a cyan dot. ρ o λ n ( ρ o ) Figure 5:
Evolution of the linearized eigenmodes along the line ρ b = 0 .
09. The red and cyan dotscorresponds to the same solutions as in Figure 4. The stability threshold occurs at ρ o = 0 .
425 andcorresponds to the cyan dot where ω b reaches its maximal value along the ρ b = 0 .
09 curve.
Turning points of physical quantities are natural locii for the onset of linear instability.The paradigmatic example is the Chandrasekhar mass of a white dwarf. In our analysis9e also find some maxima of the mass along the level curves which are accompanied by asquared eigenfrequency transiting from positive to negative values, see for instance Fig. 39.Nevertheless, looking at the extrema of the mass (or ω b ) does not yield exhaustive informationabout the stability. There are cases where complex eigenfrequencies arise because two realeigenmodes merge. Further information about the mode structure and behavior can be foundin Appendix C.Charting the complete stability region involves a numerical scan of the spectrum of linearizedperturbations of SPS’s across the space of solutions. The shaded region in Fig. 4 is our bestapproximation as to where the region of linearly stable solutions extends. Notice that partsof the edge of the stability region are given by the boson star solutions. Here the passage fromgray (stable) to white (unstable) occurs precisely because the lowest eigenmode squared turnsnegative. This observation implies that the spectrum of linearized perturbations around aboson star will always contain a zero mode. In Appendix C we prove that the zero modegenerates the boson star line and we provide more information on the building and featuresof the stability region.Also notice that the white wedges emerge from integer values of ω b . Such values correspond tospecial eigenfrequencies of the linearized scalar field problem. Indeed, over global AdS , thescalar wave equation has a general spectrum of regular solutions given by φ ( t, x ) = e ( x ) e iω b t with e ( x ) ∝ cos( x ) F (cid:18) − ω b , ω b , , cos( x ) (cid:19) + 3 cot (cid:0) πω b (cid:1) ω b ( ω b − F (cid:18) − ω b , ω b , − , cos( x ) (cid:19) . Normal modes come in two families. Solutions with ω b = 3+2 k , k = 0 , , ... are normalizableand have vanishing source, while solutions with ω b = 2 k , k = 1 , , ... are non-normalizableand have nonzero source, but vanishing vev. The fact that white wedges descend to thevicinity of these linearized solutions supports the picture of linear instability as a resonancephenomenon. With this remark in mind we would expect also an instability wedge to comedown to ω b = 2 but actually we do not find it. Nonlinear stability
In a nonlinear theory the results of a linear stability analysis are of limited range. A per-turbed unstable solution will soon start departing largely from its original, unperturbed This is more easy to see by rewriting the normal modes for AdS d +1 in terms of generalised Legendrepolynomials e k ( x ) = cos ∆ xP ( d − , ∆ − d ) k (cos(2 x )) , w k = 2 k + ∆ , with ∆ = ∆ ± the two roots of m = ∆(∆ − d ). Solutions with ∆ = ∆ + are normalizable and have vanishingsource. Solutions with ∆ = ∆ − = d − ∆ + are non-normalizable and have vanishing vev. When m = 0,∆ + = d and ∆ − = 0. t = 0 with a linearly stable SPS, the output yields consistently the expectedperiodic geometry over as long as we have let the computer go. Notice that, since the profilenever gets particularly spiky, rather low resolutions of 2 + 1 points are sufficient. Thisallows for a considerable increase in computational speed.The next numerical experiment is to initialize the code with a linearly unstable SPS slightlyperturbed with the single unstable normal mode. In other words, if φ ( x ) is an SPS, and χ ( x ) its unstable mode with a purely imaginary eigenfrecuency λ <
0, then at t = 0 wewill insert φ ( x ) + (cid:15) χ ( x ) with (cid:15) ∼ O (10 − ). In general terms, one could reasonably expectcollapse to a thermal phase as the end result of time evolutions starting from unstable initialconditions. Nevertheless, as we shall see now, even in this case, the system provides regularcounterexamples. Re [ ϕ ( t,0 )] Re [ ϕ ( t, π / )]
20 40 60 80 100 t - Figure 6:
In this plot we appreciate that the driven scalar field keeps in phase with the driving allalong its profile. The tip at the origin and at the boundary rotate at the same pace. There is, apriori, no reason why this should be so and is not an artifact or truncation of the simulation. Infact one can find (unstable) solutions where this is not the case.
In order to keep the analysis as systematic as possible, we continue working with the twosolutions marked in red on the left hand side of Fig. 4. They both represent SPS’s with ω b = 2 and ρ b = 0 .
09. The lower one lies in the region of linear stability, and we have checkedthat it also supports fairly “strong kicks”, (cid:15) ∼ O (1), leading to oscillations about the initialsolution. 11n contrast, the upper one contains in its spectrum of linearized normal modes one unstableeigenmode, χ ( x ), with a purely imaginary frequency λ = − . (cid:15) ∼ − , thetime evolution when t → ∞ differs dramatically depending upon the sign chosen for (cid:15) .With positive sign (so that (cid:15) χ (0) has the same sign as φ (0)), the evolution collapsespromptly to a black hole geometry. This can be seen in the top line of plots in Fig. 7. Thecontinuous driving reflects itself in the rising of the total mass, while the horizon radiusincreases monotonically as does the absolute value of the vev itself. The system approachesthe expected infinite temperature state. t Min ( f ) t |< O >( t )| t M t Min ( f )
50 100 150 200 t |< O >( t )|
50 100 150 200 t M Figure 7:
Evolution of three magnitudes for an unstable SPS with ρ b = 0 . ρ o = 0 .
56 and ω b = 2after being perturbed with the first unstable mode φ (0 , x ) + (cid:15) χ ( x ). The upper (lower) threeplots correspond to (cid:15) = − − (+10 − ). The magnitudes shown are the minimum of the metricfunction min x ∈ [0 ,π/ ( f ( t, x )), the energy density m , and the vev |(cid:104)O(cid:105)| . A relaxation oscillationwith a pronounced peak and a long lived plateau is apparent on the lower plots. With the opposite sign, (cid:15) ∼ − − , the evolution reaches a radically different endpoint.Instead of a collapse, the dynamics stays regular for all times. In Fig. 6 we plot the oscillationsof the real part of the scalar field at the origin and at the boundary. Both of them proceedin phase with one another, signalling that the profile of the scalar field stays in a plane whilerotating. Moreover, and this is new, the oscillation of the scalar at the origin acquires aviolent and sudden modulation in amplitude.Since we are working with a fully backreacted solution, the modulation of the driven oscil-lation affects many other physical quantities. Fig. 7 highlights the impact of the modulationon three observables: the minimum of the metric function min x [ f ( t, x )], the absolute valueof the vev (cid:104)O(cid:105) ( t ) and, finally, the energy density M . Note that min x [ f ( t, x )] reveals the12ormation of an apparent horizon whenever it drops towards zero.The periodicity of the modulation should not obscure the fact that it is highly anharmonic.The oscillations in modulation stay on long lived plateaux followed by sudden pulsationsor beats. We will refer to these regular solutions as sourced modulated solutions , SMS. Theperiod T is orders of magnitude away from the one due to the driving frequency (2 π/ω b ) and,actually, it is (weakly) dependent on the strength of the perturbation. Hence, in the strictsense, the solution is not a limiting cycle like the one found in nonlinear systems such as thevan der Pol oscillator. In such systems the asymptotic dynamics eventually loses memoryof the initial state whereas, in our case, it bears resemblance to relaxation oscillations [22].The closest mechanical analogy for a SMS would be that of an inverted pendulum slightlykicked out of its vertical unstable position, where it will return and stay for a long time untilthe next sudden turn.In fact, and very remarkably, these regular solutions also occur with vanishing source. Indeed,unstable boson stars on the black curve slightly above the yellow dots in Fig. 4 also exhibitthis modulated dynamics for one sign of the most relevant perturbation, and collapse to ablack hole for the other. Such behavior has been observed earlier, both in asymptoticallyflat [23] and AdS spacetime [24, 25]. Also, SMS profiles like the one in Fig. 6 bear someresemblance with the long time modulation of oscillating solutions that appear in confiningtheories after a global quench that injects energy below the mass gap threshold [26, 27].At first sight, the modulations we find look substantially more anharmonic and stronglypeaked. A Fourier analysis should be carried out in order to reach a definite conclusion.Another interesting proposal could relate the SMS’s to nonlinearly dressed multi-oscillatorsolutions [28]. These still have to be constructed in the driven situation, but most likely thisis feasible.The unstable SPS’s with low enough amplitude (indicatively of the order of ρ o < .
3) undergoan exotic evolution. In fact, in these cases the pulsating modulation appears for both signsof the relevant perturbation (cid:15) . With the “potentially lethal” sign (the one that usuallywould drive the SPS towards a black hole) we now observe a pulsating modulation wherethe value of ρ o increases instead of decreasing, yet reaching a maximum value and bouncingback again. The mass and the vev also increase and decrease back and the solution staysregular forever.Figure 8 shows a case of those just mentioned where we have plotted the real parts of φ at theorigin and the boundary. This highlights an intriguing feature: the modulation pulse does As explained with care in Appendix D, the spectrum of fluctuations of a boson star contains a zeromode which generates the boson star branch (black lines in Fig. 4). The next mode is the one that becomespurely imaginary at the Chandraserkar-like mass. It is perturbing with respect to the unstable mode thatthe solution behaves as mentioned above in a highly correlated way with the sign of the coefficient of theperturbation. e [ ϕ ( t,0 )] Re [ ϕ ( t, π / )]
170 180 190 200 210 220 t - - - Re [ ϕ ( t,0 )] Re [ ϕ ( t, π / )]
170 180 190 200 210 220 t - - Figure 8:
For small values of the driving amplitude and the scalar field modulus at origin, here ρ b = 10 − and ρ o = 0 .
3, perturbations along the unstable mode (cid:15) χ with both signs (cid:15) = ± − destabilize the initial unstable SPS reaching pulsating solutions with regular behavior. not only perturb the absolute value of the scalar field in the bulk, but also its phase. Moreprecisely, during the pulsation, the two values φ o ( t ) at the origin and φ b ( t ) at the boundarystart de-phasing in one sense or the other depending upon the sign of (cid:15) , and the scalarprofile becomes increasingly helical. After the pulsation has ended they again re-phase andthe profile recovers its planar shape.The study of nonlinear stability is always a long and very much resource dependent task.What we have done so far is to characterize the endpoint of linearly unstable SPS’s and wehave found two possibilities, a dynamical black hole and a SMS. Concerning this last one, wehave indeed checked for their robustness by adding a perturbation to them. Perturbed SMS’sdevelop wiggling plateaux but remain regular all along the simulation, suggesting that theirnormal mode spectrum contains only real eigenfrequencies. The very late time behaviourof the solution is another delicate point of the nonlinear stability analysis. As far as ourcodes have run, up to t = O (10 ), we have not found the slightest evidence of a nonlinearinstability setting in at late times. After the discovery of the AdS instability [29] this isan issue one should be concerned about. However, the essential ingredient found in thatcontext, namely full resonance, is very unlikely to be present in the spectrum of linearizedperturbations of a SMS. In the absence of any potential mechanism for destabilization atlong times, any runtime is disputable in what concerns any claim for stability.Finally, let us stress again that all the analysis has been performed in region I of the phasespace shown in Fig. 4. It would be interesting to do an accurate scan to see if further exoticevolutions can be obtained along the wedge of unstable solutions. In particular, we havenot analyzed the fate of unstable solutions in regions II, III... This remains for a laterinvestigation. 14 Quenches with periodic driving
The sourced periodic solutions we have found are eternal, extending from t = −∞ to t = + ∞ .Therefore, it makes sense to try to study if they can be constructed by means of a slowlygrowing source starting from AdS. In this section we perform this investigation by consideringquenches in a generalized sense. The word quench usually refers to a change in some couplingconstant, which can be either sudden or slow. In the present context, it will refer to a certainprocess that interpolates between two different periodically driven Hamiltonians. In short, weshall explore how the system responds when the boundary data that determine the periodicdriving of the dual QFT become, themselves, functions of time ( ρ b ( t ) , ω b ( t )). We analyze the system response to very slow changes of the driving parameters, whose vari-ation occur on a typical time scale β (cid:29) ω − b . In the context of static quantum systems theadiabatic theorem states that, for sufficiently slow quenches, the ground state of the systemfollows the change in the Hamiltonian. Even if a non-vanishing transition amplitude to anexcited state is generated, there is a well defined way to bring it to arbitrarily small values(by making β large enough). Here we find that a similar phenomenon occurs, albeit the un-derlying unquenched Hamiltonian is time-periodic. The system follows the slow modulationof the parameters of the periodic driving, moving from one ground state to another.As we already know, SPS exist on a codimension-one submanifold in ( ω b , ρ o , ρ b ) space. Inthis subsection, we demonstrate that linearly stable SPS’s can be reached from the globalAdS vacuum by means of a sufficiently slow quench. Conversely, we also show that theselinearly stable SPS determine which quench processes cannot be regarded as adiabatic. Weemploy the following ansatz for the scalar field source φ ( t, π/
2) = 12 (cid:15) (cid:18) − tanh (cid:18) βt + βt − β (cid:19)(cid:19) e iω b t , ≤ t < β,φ ( t, π/
2) = (cid:15)e iω b t , t ≥ β (3.1)i.e., the source starts being zero at t = 0, and reaches its final amplitude (cid:15) after a timespan β . Afterward, it oscillates harmonically. During the whole process, the frequency ω b is kept constant. We refer to the regime taking place at t < β as the build-up phase,while the driving phase corresponds to t ≥ β . The quench profile is plotted in Fig. 9, where ρ b ( t ) ≡ | φ ( t, π/ | . 15 .2 0.4 0.6 0.8 1.0 t / β | ϕ ( t, π / )| Figure 9: ρ b ( t ) for the quench profile (3.1) at (cid:15) = 0 . Quasistatic quench to a SPS
Imagine that the final values of (cid:15) and ω b correspond to a SPS in the region of stability, asgiven in Fig. 4. For concreteness, let us focus on region I, and consider the lower red dotdepicted at ρ b = 0 .
09 and ω b = 2. In this case, if β is sufficiently large, the state obtainedafter the build-up process is, to an excellent approximation, the SPS corresponding to thefinal harmonic driving. Furthermore, during the driving phase the numerical solution alsoremains stationary (up to the largest times we have simulated and with high accuracy).Let us choose β = 2500. We have determined numerically that, after the quench (i.e., for t > β ), the energy density of the geometry corresponds to m = 0 . ρ and f during the drivingphase (solid curves) and their profiles in the corresponding SPS (dashed curves).Having demonstrated that the linearly stable SPS can be reached from the vacuum by asufficiently slow quench process, it remains to analyze the adiabaticity of the whole procedure.Specifically, if the system’s response to the time-dependent source is perfectly adiabatic, weexpect that Ψ( t, x ) = Ψ SP S ( ρ b ( t ) , ω b ; x ) , (3.2)where, with no loss of generality, Ψ denotes any field of the geometry. Ψ SP S is the valueof this field in the SPS at the given instantaneous ρ b and ω b . The equality (3.2) thusentails that the dynamics does not depend explicitly on time, but only implicitly throughthe instantaneous value of the source. In other words, in the quasistatic large β limit, onecan draw the evolution as a path on the surface of SPS’s (in this case, the vertical dashedline displayed in Fig. 4).Note that these observations should also hold for one-point functions such as m and (cid:104)O(cid:105) : if16 .5 1.0 1.5 x f x ρ Figure 10:
In dotted black f ( x ) and ρ ( x ) at the end of the driving phase for a quench processof the form (3.1) with time span β = 2500 and final harmonic driving data ( ω b = 2 , (cid:15) = 0 . f ( x ) and ρ ( x ) of the SPS corresponding to the same data. The curves lie with highaccuracy on top of each other. the response of the system to the quasistatic quench process is perfectly adiabatic, it mustbe the case that m ( t ) = m SP S ( ρ b ( t ) , ω b ) , (3.3)and similarly for the scalar vev.
500 1000 1500 2000 2500 t m t / β m Figure 11:
Left: m ( t ) for quenches of the form (3.1) to the harmonic driving (cid:15) = 0 . , ω b = 2.From left to right, β = 500 , , , m ( t/β ) for the quenches of the right figure.The curves clearly collapse into a universal profile (dotted black). We also plot m SP S ( ρ b ( t/β ) , ω b ) in(solid) yellow. The agreement between both curves implies that the system responds adiabatically. As we illustrate in Fig. 11, these expectations are fulfilled. Let us look first at Fig. 11a. There,we compare the time evolution of m for four quench processes, with β = 500 , , , ρ b ( t ) only depends on the dimensionless ratio t/β , relation173.3) implies that for adiabatic response the energy density can only be a function of t/β ,but not of t and β separately. Therefore, when plotted in terms of t/β , the four instances ofthe time evolution of m shown in Fig. 11a must collapse to a universal curve. This is indeedwhat happens, as we illustrate in Fig. 11b (dotted black). Finally, in Fig. 11b we also depict m SP S ( ρ b ( t/β ) , ω b ) (solid yellow). The evident agreement between both curves shows thatrelation (3.3) is satisfied. Quasistatic quench to a black hole
Even in the β → ∞ limit, there exists a bound to the amplitude or the frequency of thedriving that one can reach. Consider a quench at constant ω b from global AdS to a finalsource amplitude, (cid:15) , such that there is no SPS associated to ω b and (cid:15) . As the source amplitudebuilds up, and in the quasistatic limit, we expect that the system evolves through a successionof SPS’s, at most up to the time t ∗ when the SPS associated to ρ ∗ b ≡ ρ b ( t ∗ ) ceases to exist. , Past this point, the system exits the surface of SPS’s and adiabatic evolution cannot proceedfurther, no matter how slowly the source amplitude increases afterwards.Let us analyze these questions in a specific example. We consider the quench describedby equation (3.1), again at driving frequency ω b = 2, but increasing now the final sourceamplitude to (cid:15) = 0 .
1. There does not exist a SPS associated to this particular driving:the highest driving amplitude compatible with the existence of a SPS with ω b = 2 is ρ ∗ b ∈ [0 . , . t ∗ ∈ [0 . , . β ≡ ηβ . Here η is critical value of the scaling variable t/β after whichwe expect adiabatic evolution to break down.In Fig. 12a, we plot the time evolution of m for quench profiles with time spans β =500 , , , , , t < β . In order to understand adiabaticity and its breakdown, in Fig. 12b we plot m ( t/β ). Two facts are manifest.The first one is that, for t/β < η (signaled by the vertical dashed red line), all the energydensity curves merge into a universal profile. This observation immediately leads to theconclusion that m ( t ) is set solely by the instantaneous value of ρ b ( t/β ) and, as a consequence,any nontrivial dynamics that depends explicitly on t is highly suppressed in the quasistaticlimit. The system evolves adiabatically, as expected.The second one is that, for t/β > η , the system undergoes gravitational collapse (as can This point corresponds to the tip of the ρ b = ρ ∗ b curve in the ( ω b , ρ o ) plane Note that, strictly speaking, adiabatic evolution can break down for t c ≤ t ∗ , t ∗ − t c (cid:28)
1; for instance, t c could be the moment when the intrinsic response time of the system is above the rate of change of thesource amplitude. In this sense, the time t ∗ just sets an upper bound on t c , and should be understood inthis sense. On the other hand, in the quasistatic β → ∞ limit, we must have that t c → t ∗ .
500 1000 1500 2000 t m t / β m Figure 12:
Left: time evolution of the energy density for the quench processes de-scribed in the main text. From left to right, the quench time span corresponds to β =500 , , , , , t/β . We clearly observe that, prior to the adiabaticity breaking, m ( t/β ) ap-proaches a limiting curve as β increases. The vertical line marks the time t ∗ /β = η = 0 . be seen in the unbounded increase of its energy density). Given that, for t/β > η , we havethat ρ b ( t ) ≥ ρ ∗ b (i.e., there is no SPS associated to the driving), we conclude that adiabaticevolution breaks immediately after crossing the ρ b ( t ) = ρ ∗ b threshold. Let us address now the non-quasistatic regime, namely that of quenches with finite time span β < ∞ . From the QFT side the standard lore is, according to the adiabatic theorem, thatthe system will not keep up with the variation of the source, and will generically transit toan excited state. If this state is highly excited, it is natural to expect a decohering evolutiontowards an (effective) thermal mixed state.In order to inspect the nature of this excited state in the dual gravitational theory we considera one-parameter family of quench profiles (3.1) with varying β ∈ (0 , ∞ ). For definiteness,we focus on the same ω b = 2 , (cid:15) = 0 .
09 case as before. Recall that our findings from theprevious section established that, in the β → ∞ limit, the constructed state was the stableground state SPS at ρ o = 0 .
33 given by the lowest red dot in Fig. 4. Instead, for β < ∞ ,the results of our simulations vindicate the existence of a critical value β c separating tworadically different regimes. • For β ≥ β c , the system always remains regular in the driving regime, and settles downto a time-periodic geometry. For finite β , in addition to the harmonic response given19igure 13: Evolution of m for different build-up processes for a driving characterized by ω b = 2, (cid:15) = 0 .
09. From top-left to right bottom, the build-up time β decreases as shown in the plots. by e iω b t , it develops an additional periodic modulation. This can be seen in severalone-point functions such as m and |(cid:104)O(cid:105)| , as illustrated in Fig. 13. This additionalmodulation is tantamount to the fact that our system is not in the ground state any-more, in agreement with expectations. As β is lowered two phenomena can be seenfrom the plots. On one side, both the amplitude and the periodicity, T , of the modu-lation grow. At the same time, the injected mass also increases and ends up oscillatingaround larger mean values. This reflects the fact that the quench has injected moreenergy and, consequently, the state is more excited. On the other, periodic modula-tions become less and less harmonic, and start developing plateaux where the systemstays for progressively longer times as β → β c . This can be observed in Fig. 13 aswe move to plots on the lower right part. Interestingly, the mass curve in the lastplot matches very well with the corresponding one in Fig. 7. As we will show in thenext section, this is more than a coincidence, and indeed, in the limit β → β c the endresult of the non-quasistatic quench is the SMS to which the unstable SPS with theprescribed values of ( ω b , (cid:15) ) will decay when perturbed with the appropriate sign of themost relevant fluctuation. To be most transparent, and referring to Fig. 4, the impliedunstable SPS is the top red dot sitting at ρ o = 0 .
56 along the vertical dashed line. Insummary, a quench to the same final boundary data ( (cid:15), ω b ) yields the stable solution(lower red dot) if performed in a quasistatic way β → ∞ , and the SMS associated to20he unstable SPS (higher red dot) in the critical limit β → β c . However, the dashedline should not cause confusion in the later case. The process, not being quasistatic,has not proceeded through a sequence of SPS’s. It cannot be drawn as a path in thesolution space.
30 40 50 60 β < m > T
30 40 50 60 β T Figure 14:
Average mass (cid:104) m (cid:105) T (left) and period T (right) of the time-periodic solutions generatedby the build-up phase when β > β c . In Fig. 14, we plot the period T of the periodic modulation of the final solutions aswell as the average mass (cid:104) m (cid:105) T ≡ T (cid:90) t + Tt dt m ( T ) (3.4)as a function of β . Both functions increase monotonically with decreasing β . For β → ∞ , (cid:104) m (cid:105) T tends to its value on the stable SPS, while T is compatible with theperiod of the fundamental normal mode of this SPS. • For β < β c , the quench results in an energy injection process strong enough to triggergravitational collapse. After a transitory regime both m and |(cid:104)O(cid:105)| increase withoutbound, while the value min x [ f ( t, x )] drops to zero, signaling the approach to an appar-ent horizon. The duration of this transitory regime grows with smaller | β − β c | . Afterthe collapse, the harmonic driving keeps injecting energy continuously into the system,which increases its mass monotonically. Representative examples of the behavior justdescribed can be found in Fig. 15.A pertinent question that cannot be definitely answered with our numerical methodsis whether the system reaches an infinite mass state in finite or infinite field theorytime. A careful analysis of this post-collapse regime should connect with the findings We cannot discard that the growing black hole finally equilibrates into a steady state of finite entropy,although it seems intuitively unlikely, giving that we are considering a system with an infinite number ofd.o.f. It would be nice to have some sharp physical arguments regarding this point.
21n [6]. We have indeed pinned down different regimes for the growth of the one-pointfunctions and refer the interested reader to Section 6.Figure 15:
Energy density (left) and absolute value of the vev (right) for a family of build-upprocesses of the harmonic driving with ω b = 2, (cid:15) = 0 . β increases from β = 5 to β = 21 . β = 21 . > β c . The unstable attractor
We have clearly demonstrated that, for non-quasistatic build-up processes, the major prop-erty is the existence of the time scale β c , which separates two different phases during thedriving regime. These phases are distinguished by the final fate of the solution. The newtime scale seems to be related to the existence of an intermediate time attractor, and wementioned that this attractor was nothing but the unstable SPS associated to ω b and (cid:15) . Inthe following discussion we provide evidence in favor of this statement.For concreteness, let us consider the β = 21 . < β c case. In Fig. 16a, we plot ρ ( t , x ), f ( t , x ), as obtained from the numerical simulation at t = 26 . ρ SP S ( x ), f SP S ( x ), where we plotthe differences( δρ )( t, x ) = ρ ( t, x ) − ρ SP S ( x ) , ( δf )( t, x ) = f ( t, x ) − f SP S ( x ) , (3.5)at t . Clearly, these differences are small.In order to gain further understanding, let us compute the time evolution of the norms of22 .5 1.0 1.5 x ρ ,f x - - - - δρ , δ f Figure 16:
Left: ρ ( t , x ) (blue) and f ( t , x ) (red), as obtained from a numerical simulation with ω = 2, (cid:15) = 0 . β = 21 . t = 26 . δρ )( t , x ) (blue) and ( δf )( t , x ) (red). δρ and δf , defined by ∆( δρ ) = (cid:32)(cid:90) π tan ( x )( δρ − δρ | x = π/ ) (cid:33) , (3.6)∆( δf ) = (cid:32)(cid:90) π tan ( x )( δf ) (cid:33) . (3.7)We plot these quantities in Fig. 17. It is clearly seen that there is a time window in whichthe distance between the fields in the driving phase and the unstable SPS is negligible.In Section 4, we employ the identification of the intermediate attractor and the unstableSPS to argue that the results presented in this section can be naturally understood as a dynamical type I phase transition. Our results bear some resemblance to critical phenomena in asymptotically flat gravitationalcollapse. In both cases, there exists an intermediate attractor with one unstable modethat separates two qualitatively different dynamical regimes. In the flat case, they aregravitational collapse versus dispersion to asymptotic infinity; in our case, they are the flowto the infinite energy phase or the establishment of a regime of persistent, exactly periodicoscillations in the system. ρ ( t, x ) is not normalizable when the source is nonzero; we define its norm as the norm of its normalizablepart. t - - - - Δ ( δρ ) , Δ ( δ f ) Figure 17:
Time evolution of ∆( δρ ) (blue) and ∆( δf ) (red) for ω = 2, (cid:15) = 0 .
09 and β = 21 . There are also apparent differences between our findings and the type II phase transitionuncovered by Choptuik for a massless scalar field [30]; actually, our results are akin tothe type I phase transitions studied in asymptotically flat space [17–19]. In the first ofthese works, the authors considered a SU (2) non-Abelian gauge field minimally coupledto gravity. There, for certain one-parameter families of initial data, a static soliton - theBartnik-Mckinnon soliton- played the role of an intermediate time unstable attractor. Theexistence of this attractor entailed that, for supercritical data and as the critical surface wasapproached, the mass of the black hole formed did not display Choptuik scaling: there wasa mass gap, set precisely by the soliton mass.As summarized in [18], there are two model-independent assumptions that must be satisfiedin order to have a first-order phase transition in spherically symmetric gravitational collapse,which we state here for completeness: • Assumption a . There exists a static, regular solution with only one unstable eigenmode .Let Ψ u ( x ) denote a generic field of this unstable geometry. Having only one unstableeigenmode, the evolution of any small spherically symmetric fluctuation around Ψ u ( x )can be decomposed as δ Ψ u ( t, x ) = Ψ( t, x ) − Ψ u ( x ) = αe λt χ λ ( x ) + (cid:88) n> α n e λ n t χ n ( x ) , (4.1)where λ ∈ R + is the eigenvalue associated to the unstable eigenmode. Modes with n > λ n ≤ There is actually an infinite countable family of them, we are referring to the one of lowest mass. Assumption b . The final fate of the perturbation δ Ψ u ( t, x ) depends solely on the signof α . For one sign, a black hole forms; for the other, gravitational collapse does nottake place. As pointed out in [18], assumption a means that the stable manifold W S of the solution Ψ u has codimension one. Assumption b entails that the stable manifold is a critical surface thatdivides the phase space into collapsing and non-collapsing initial data.Consider now a one-parameter family of initial data Ψ ( x ; p ) that crosses the critical surface W S at p = p ∗ . Ψ ( x ; p ∗ ) flows to Ψ u ( x ) along W S . By continuity, initial data Ψ ( x ; p ) withsufficiently small | p − p ∗ | have a time development Ψ( t, x ; p ) that remains sufficiently closeto W S so as that, after some time t , the linearization (4.1) holds. The precise numericalvalues of the coefficients α, α n are set by the initial data: α = α ( p ) , α n = α n ( p ).Given that by definition α ( p ∗ ) = 0, we must have that α ( p ) = a ( p − p ∗ ) + . . . (4.2)and, as a consequenceΨ( t, x ; p ) = Ψ u ( x ) + a ( p − p ∗ ) e λ ( t − t ) χ λ ( x ) + . . . (4.3)The above linearization is valid up to some time t > t at which the unstable eigenmodebecomes dominant and the solution is repelled away from the critical surface. Therefore, at t = t we have that ( p − p ∗ ) e λ ∆ t = O (1) , (4.4)where we have defined ∆ t = t − t . This last condition forces ∆ t to scale as∆ t ∼ − λ log | p − p ∗ | . (4.5)In Section 3, we have analyzed a one-parameter family of build-up processes with differenttime spans β , but the same final harmonic driving, characterized by ρ b = 0 .
09 and ω b = 2.We found out that there exists a critical β c that divides the post-quench response of thesystem into two different regimes, which correspond to gravitational collapse ( β < β c ) or theestablishment of persistent and exactly periodic oscillations in the system ( β > β c ). Fur-thermore, we demonstrated that, as β → β c , at intermediate times, the system is attractedto a SPS with just one unstable eigenmode. This observation, together with the fact thatthe nonlinear evolution of a perturbed, unstable SPS falls generically into two different dy-namical regimes depending solely on the sign of the perturbation (as shown in Section 2),implies that assumptions a and b apply naturally to the problem we are considering. As a The final state depends on the details of the Einstein-matter action. p , p ∗ correspond to β, β c . Inparticular, the lowest eigenfrequency of the unstable SPS, λ , has to be extractable from ournumerical simulations.In Fig. 18 we plot ∆ t versus | log( β − β c ) | for a series of quenches with different values of β > β c . The linear fit yields a value of λ = 0 . λ we have obtainedindependently by means of the numerical solution to the eigenmode equations.
10 15 20 25 log ( β - β c ) Δ t Figure 18:
Scaling of the permanence time ∆ t along the unstable SPS with ω b = 2. We are starting to get operational control over the phase diagram of periodic solutions.Armed with a quench protocol like (3.1) we are able to target both at SPS’s as well as atSMS’s by appropriately tuning the parameters ω b , (cid:15) and β . In the cases we have analyzedso far, the end result is some sourced periodic solution, whether modulated or not. Anintriguing legitimate question is if we can engineer a protocol such that the end solution isunsourced, yet not AdS but, instead, a boson star.Boson stars are dual to excited QFT states that, despite being unsourced, exhibit a periodicone-point function. It is more than tempting to try to establish a link between this type ofsolutions to the so called time crystals. Strictly speaking, these last systems are postulated tobe true ground states, yet to break time translation invariance spontaneously [15]. However, The boson star can be envisaged as circular polarized solution for a pair of real scalar fields rather thana complex field. The one-point function of each of these real scalar fields varies harmonically in time. Anatural extension of the model would include a gauge field with which one could gauge away the phaserotation by a gauge transformation linear in time. This is a different scenario which also deserves a thoroughstudy. However, the single real scalar field will also exhibit time periodic unsourced solutions. In this caseno gauge transformation is available to gauge the periodic motion away. The no-go theorem leaves openthe possibility of finding spontaneous time shift symmetry breaking in excited states (see [16]for a review with references).In parallel to what just described, there is another possible connection to time crystals. Dueto interactions, upon a small periodic driving certain systems can respond sub-harmonically;they are dubbed discrete time crystals or Floquet time crystal. Such proposal has beenrealized experimentally in spin systems in which the response beats at twice the drivingperiodicity [4, 32, 33].We only take these as inspiring considerations trying to highlight similarities and differences.As a major target, we try to devise a quench protocol that connects continuously vacuum-AdS with a boson star solution. The most naive guess immediately faces a seeminglyunavoidable obstruction. Imagine, for concreteness, drawing a straight vertical line at ω b =2 . ρ b ( t ) >
0, at a constant frequency ω b . If performed quasistatically, β → ∞ , the quench protocolof the previous section moves the state along the vertical segment drawn in the figure. Asexplained before, this can only proceed up to the point where the line exits the gray (stability)region. If once there one continues increasing the source amplitude, unavoidably will exitthe surface and face collapse and thermalization. If one starts decreasing quasistatically thesource amplitude, the system will proceed backward through the same states until AdS isagain recovered. It is impossible to slide back along the vertical line in the white (unstable)sector down to zero ρ b . Bosons stars look unreachable from this side of the phase diagramby means of a quasistatic process. We will provide two ways to get around this caveat.Needless to say, one of the assumptions made in the previous argument will have to berelaxed. For example, if one insists in using a quasistatic quench, one may still reach the BScurve from region II without crossing any instability curve. A remarkable possibility arisesby employing, instead, non-quasistatic processes. This takes us out of the surface of SPS’s,but an unexpected attractor will bring us back to it. In this section we explicitly construct a boson star, starting from AdS, via a quasistaticquench. As advertised above, the clue to this procedure comes from drawing a quasistaticpath on the space of SPS’s across region II, always staying within the region of linear stability.It is fortunate that boson stars lie at the boundary of such region. The advantage of thismethod is that we control the frequency of the final boson star. We must start by extend The no-go theorem [15] refers to an infinite-volume thermodynamic limit. As we comment later, globalAdS holography relies on a different large N thermodynamic limit φ ( t, π/
2) = φ b ( t ) = (cid:15) ( t ) e iω b ( t ) t , (5.1)where (cid:15) ( t ) = 12 (cid:15) m (cid:18) − tanh (cid:18) βt + βt − β (cid:19)(cid:19) , ≤ t < β,(cid:15) ( t ) = 12 (cid:15) m (cid:18) (cid:18) βt − β + βt − β (cid:19)(cid:19) , β ≤ t < β (5.2) ω b ( t ) = ω i + 12 ( ω f − ω i ) (cid:18) − tanh (cid:18) βt + 2 βt − β (cid:19)(cid:19) , ≤ t < β . (5.3)Note that, although ω b ( t ) has the quench shape of Fig. 9 (interpolating between ω i and ω f ),the actual instantaneous frequency of the source is given rather by the time derivative ofits phase, ω ins ( t ) = d/dt ( ω b ( t ) t ) = ω (cid:48) b ( t ) t + ω b ( t ), which has the same asymptotic values as ω b ( t ). This quench process starts from the global AdS vacuum at t = 0; we expect it toland on the stable boson star with ω b = ω f at t = 2 β when β is sufficiently large (modulosmall corrections that vanish in the β → ∞ limit).As an example, we consider (5.2), (5.3) with ω i = 3 . ω f = 2 . (cid:15) m = 0 .
01 and β = 1000.See Fig. 19 for a plot of (cid:15) ( t ) and ω ins ( t ) in this case.
500 1000 1500 2000 t ϵ ( t )
500 1000 1500 2000 t ω ins ( t ) Figure 19:
The functions (cid:15) ( t ) and ω ins ( t ) employed to build a boson star starting from AdS. In Fig. 20, we plot the time evolution of m and |(cid:104)O(cid:105)| . It is clearly seen that they relaxto (almost) time-independent values after the quench. In particular, from the numericalsimulation, we have that m ( t > β ) = 0 . , |(cid:104)O ( t > β ) (cid:105)| = 1 . . (5.4)28igure 20: m and |(cid:104)O(cid:105)| for the quench process described in the main text. After the quench,both functions (approximately) relax to their values in the boson star solution corresponding to ω f = 2 . On the other hand, by solving the ODE system that determines the boson star solution with ω f = 2 .
9, we obtain m BS = 0 . , |(cid:104)O(cid:105)| BS = 1 . . (5.5)The agreement between them is rather good, in particular the relative error between thevalues obtained from the numerical simulation and the solution of the ODE system is below1%. Another useful quantities to monitor the difference between the end product of thequench and the stable boson star are ∆ ρ , ∆ f -cf. equations (3.6),(3.7)-. We plot them inFig. 21a.Figure 21: Time evolution of ∆ ρ (blue) and ∆ f (red) for the quench process described in the maintext. Left: we compare the simulation results to the boson star at the target frequency ω f = 2 . ω BS = 2 .
29y looking at Fig. 21a, we clearly see that, although small, the discrepancy between theend state of the quasistatic quench process and the target boson star is not negligible. Itis natural to wonder about the origin of this difference. Is the boson star oscillating at afrequency different than the target frequency? Does this difference vanish when β → ∞ ?Let us address the first issue. In our example we have that, for t > β , ρ ( t,
0) = 0 . m BS = 0 . |(cid:104)O(cid:105)| BS = 1 . ω BS = 2 . < ω f = 2 . ρ and ∆ f are compatible with the numericalnoise.The discussion in the previous paragraph shows that the frequency of the boson star obtainedafter the slow quench does not need to agree with the target frequency. We expect that thedifference between both frequencies vanishes in the strict β = ∞ limit, i.e., for a quasistaticquench. Equivalently, we expect that, when we employ the target boson star to perform thecomparison, ∆ ρ, ∆ f → β → ∞ .
700 900 1100 1300 1500 1700 1900 β × - × - × - × - Δρ , Δ f Figure 22: ∆ ρ (upper curve) and ∆ f (lower curve) vs β for the boson star building processesdescribed in the main text. The blue lines correspond to the fits ∆ ρ , ∆ f ∼ β − . . In Fig. 22, we plot ∆ ρ and ∆ f for the quench profile (5.2),(5.3) with ω i = 3 . ω f = 2 . (cid:15) m = 0 .
01 and varying β . We consider β = 700 , , , , ρ , ∆ f decrease with increasing β . In particular, a log-log fit of these data gives∆ ρ, ∆ f ∼ β − . . (5.6)This behavior makes plausible to assume that in the strict quasistatic limit we will recoverthe target boson star. 30 .2 Non-quasistatic method As advertised before, we can also reach BS’s with a quench that starts from region I but,in turn, we must give up quasistaticity. The crucial observation is that, if instead of usinga large value of β , we let β → β c , we reach a SMS associated to an unstable SPS. Once inthe SMS, which is stable by definition, we may slowly turn off the source amplitude. If theend result is a stationary solution that corresponds neither to a BH nor to AdS, the onlyremaining option is a BS. We shall provide numerical evidence that this is indeed the case.We construct the SMS by the procedure described in section 3.2. We work with the sourceprofile (3.1) with β = β in (cid:38) β c , and wait until a time t out ≥ β in after build up. Once theSMS has formed, we start to turn off the source amplitude, reaching zero at t = t out + β out .The extinction quench follows the time reversed profile with, now, a larger time scale β out φ ( t, π/
2) = 12 (cid:15) (cid:18) (cid:18) β out t − t out + β out t − t out − β out (cid:19)(cid:19) e iω b t , t ≥ t out , t < t out + β out ,φ ( t, π/
2) = 0 . t ≥ t out + β out . (5.7)As an example, in Fig. 23 we have considered a build-up phase characterized by ω b = 2 . (cid:15) = 0 .
09 and a quench-in time β in = 23 . β c that we havemanaged to approach from above. As shown in Fig. 23, the system first “jumps” on top ofthe unstable SPS attractor, where is stays for more than 100 seconds, until it falls onto theside of the modulated SMS, as expected. After the first beating, we begin to turn off thesource amplitude slowly, with a time scale β out = 1000. The final, sourceless solution has anon-vanishing mass and frequency ω consistent with some BS.A very important difference between the setup at hand and the quasistatic method describedin the previous subsection is that, in the present case, even for large β out , the frequency ofthe reached BS, ω BS , has no obvious relation with the driving frequency, ω b , that has beenused throughout the quench.A second very important point is that, presumably due to the violence of the build-up phase(i.e. the smallness of β in ), the system does not return to a strict BS state after the sourcehas been turned off: the final geometry can be understood as the linear superposition of aBS and its first nontrivial eigenmode. Our numerical experiments point to the fact that thisconclusion holds irrespectively of the value of β out . In fact, a stronger statement seems to betrue: the final BS is actually independent of β out and there is a univocal relation betweenthis BS and the SMS generating it; β out only controls the amplitude of a final oscillationabout the final BS, which occurs with the frequency of its first nontrivial normal mode. Aninteresting open question is determining which property of the original SMS sets ω BS .In order to strengthen these statements above, we have considered a set of non-quasistaticquenches leading to a BS, now with ω b = 2, β in = 22, and quench-out time scales β out =31
00 400 600 800 1000 t m
200 400 600 800 1000 t |< O >( t )| Figure 23:
A close-to-critical quench with build up time scale β in = 23 . β out = 1000 and leaves a very slightly perturbed BS at theend. , , , , ρ o ( t ) = ρ ( t,
0) = | φ ( t, | oscillates around ρ o,BS = 0 . ω BS =2 . ω b = 2. For normal mode fluctuationsaround this BS, the first nontrivial eigenfrequency is λ = 1 . ρ ( t, x )drawn from the simulation is given by ρ ( t, x ) = ρ BS ( x ) + αχ ( x ) cos( λ t + δ ) + e ( t, x ) , (5.8)where χ ( x ) is the spatial profile of the BS first nontrivial eigenmode, and e ( t, x ) an errorterm that takes into account both linear contributions from higher eigenmodes as well aspossible nonlinear corrections. The parameters α and δ are fixed by demanding that (5.8)and its first time derivative (both without the error term) hold exactly at the origin at t = t out + β out . We define ( δ ρ )( t, x ) ≡ ρ ( t, x ) − ρ BS ( x ) , (5.9)( δ ρ )( t, x ) ≡ ρ ( t, x ) − ρ BS ( x ) − αχ ( x ) cos( λ t + δ ) = e ( t, x ) . (5.10)After the source has been turned off, we expect that ∆( δ ρ ) (cid:28) δ ρ ) (cid:28) ∆( δ ρ ) (i.e., the leading order deviation withrespect to the BS solution is controlled by its first nontrivial eigenmode). Both of theseexpectations hold, as the reader can convince herself by looking at Fig. 24. Of course, this choice is not unique. Alternative ones do not modify the conclusions of the main text. .5 1.0 1.5 x - t - t out - β out × - × - × - × - × - Δ ( δ ρ ) Figure 24:
Left: ρ (black), 10 δ ρ (brown) and 10 δ ρ (orange) for β out = 5000 at a time t = 600after the quench-out. The three magnitudes clearly display the hierarchy described in the maintext. Right: ∆( δ ρ ) (brown) and ∆( δ ρ ) (orange) during the whole post-quench-out regime for β out = 5000. We clearly see that the inequality ∆( δ ρ ) (cid:28) ∆( δ ρ ) holds. Our numerical techniques allow us to follow the system after gravitational collapse hastaken place, at least during some time. In the examples we have analyzed, we have notfound traces of equilibration to a stationary regime: the harmonic driving always led to amonotonic increase of the energy density. This increase is responsible for a steady growthof the apparent horizon, which gets progressively closer to the boundary. The way inwhich the energy density increases is not universal, and depends on the specific form of theharmonic driving.Let us discuss some examples in detail. The first one involves a perturbation of the unstableSPS with ρ b = 0 . , ω b = 2 .
375 by its first, unstable eigenmode. In Fig. 4 this point is tobe located to the right of the upper red dot in region I, on the isocurve with ρ b = 0 .
01. Aswe have illustrated in Section 2, choosing the right sign of the perturbation leads right awayto gravitational collapse. Following the post-collapse evolution of the system, we find theresults depicted in Fig. 29: |(cid:104)O(cid:105)| stays approximately constant, while the energy densityincreases in an approximately linear fashion, as prescribed by the differomorphism Wardidentity given in eq. 2.4.As explained in [6], it is useful to portrait the system trajectory in the ( φ b , (cid:104)O(cid:105) ) plane inorder to understand its response to the harmonic driving. We plot the real section of thiscomplex curve in Fig. 26a. It is clearly observed that the trajectory remains bounded. In Our results indicate, but not demonstrate, that the boundary is not hit in finite time. Our results show a small oscillation around a nonzero mean. t m
50 100 150 t |< O >| Figure 25:
Time evolution of the energy density (left) and vev modulus (right) of the SPS with ρ b = 0 . , ω b = 2 .
375 when perturbed by its fundamental, unstable eigenmode with amplitude (cid:15) = 0 . - - Re ( ϕ b ) - - - Re (< O >)
50 100 150 t | σ in / σ | Figure 26:
Left: Phase portrait of the time evolution of the SPS with ρ b = 0 . , ω b = 2 . (cid:15) = 0 . t = 30. Each period of the source has a different color, whose wavelength increases the later itstarts. Right: Relative value of the imaginary part of nonlinear conductivity σ . A nonzero valueindicates that the system response is partially in-phase with the driving. order to understand the phase alignment of the source and the response, it is convenient tointroduce the nonlinear conductivity σ ( t ) ≡ iω b (cid:104)O ( t ) (cid:105) φ b ( t ) ≡ σ out ( t ) + iσ in ( t ) . (6.1)We plot the relative contribution of σ in to the total conductivity in Fig. 26b. It is clearlyobserved that, while small, it typically has a non-negligible value. The boundedness of theresponse, together with the fact that it is not in complete phase opposition with respect tothe source, leads to identify the post-collapse evolution of the system as belonging to the34 ynamical crossover tilted regime obtained in [6].
20 40 60 80 100 t m
20 40 60 80 100 t |< O >| Figure 27:
Time evolution of the energy density (left) and vev modulus (right) for a quenchprocess of the form (3.1) with (cid:15) = 0 . , ω b = 2 and β = 20. - - Re ( ϕ b ) - - - Re (< O >)
20 40 60 80 100 t | σ in / σ | Figure 28:
Left: Phase portrait of the late-time dynamics of a quench process of the form (3.1)with (cid:15) = 0 . , ω b = 2 and β = 20. We start at t = β . Each period of the source has a differentcolor, whose wavelength increases the later it starts. Right: Relative value of the imaginary partof the nonlinear conductivity σ . A nonzero value indicates that the system response is partiallyin-phase with the driving. The second example of post-collapse evolution we would like to discuss involves a SPS with ρ b = 0 . , ω b = 2, perturbed by its fundamental, unstable eigenmode. In Fig. 4 this isprecisely the upper red dot in sector I. Since this solution controls the late-time dynamicsof the non-quasistatic quench processes we discussed in depth in Section 3, we can insteadfocus on those. For definiteness, we take β = 20. In Fig. 27, we plot the time evolution of m and | (cid:104)O(cid:105) | , while in Fig. 28a we depict the phase diagram (along the lines of Fig. 26a).It is clear that this late-time regime is qualitatively different from the previous one: | (cid:104)O(cid:105) | does not remain bounded and, as a consequence, m increases faster than linearly. On the35hase portrait, we observe a clear precession of the system’s trajectory, which is accompaniedby an increase in its amplitude. This precession is due to the fact that the response is notentirely out-of-phase with the source, as illustrated by Fig. 26b. This new late-time phase isbest identified with the unbounded amplification regime found in [6], as the trademark ofthis regime is the unbounded and partially in-phase response to the harmonic source. t m t |< O >| Figure 29:
Time evolution of the energy density (left) and vev modulus (right) for a quenchprocess of the form (3.1) with (cid:15) = 10 − , ω b = 20 π and β = 1. - - Re ( ϕ b ) - - Re (< O >) t | σ in / σ | Figure 30:
Left: Phase portrait of the late-time dynamics of a quench process of the form(3.1) with (cid:15) = 10 − , ω b = 20 π and β = 1. We start at t = β . The different periods of timeevolution collapse into a single curve. Right: Relative value of the imaginary part of the nonlinearconductivity σ . Its vanishing after the quench indicates that the response is in perfect phaseopposition with the source. We close this section by providing one last example. It corresponds again to a quench profileof the form (3.1), this time with frequency ω b = 20 π , amplitude (cid:15) = 10 − and time span In particular, compare our Fig. 26 with with Fig.10a in [6]. = 1. As illustrated in Fig. 29, the energy density grows linearly after the quench, while theabsolute value of the scalar vev remains constant. In accordance with these facts, the phasediagram of the system corresponds to a closed, non-precessing trajectory of fixed amplitude(see Fig. 30a). Consistently, the imaginary part of the nonlinear conductivity vanishes, i.e.,the system response is in complete phase opposition with respect to the harmonic source(see Fig. 30b). The features discussed so far imply right away that the system finds itself inthe linear response regime. We now turn the attention to the case of a real scalar field. Now, unlike the complex case,the metric of a time-periodic real solution will not be static. This implies a dramatic changein the formalism and the methodology. Naturally, what makes the problem tractable is time-periodicity itself which, for instance, allows us to replace the linear analysis of normal modesby the equivalent Floquet analysis. Qualitatively, the final results we obtain are surprisinglyclose to those of the complex case.Another important difference with respect to the complex case is the fact that the solution,albeit being periodic, is not harmonic. For the complex scalar the time dependence can befactorized as a complex exponential -cf. eqn. (2.3)-, but for the real one the time periodicityrequires an infinite spectrum of Fourier modes given by the following ansatz φ ( t, x ) = ∞ (cid:88) k =0 φ k ( x ) cos((2 k + 1) ω b t ) . (7.1)Only the boundary source remains harmonic, φ ( t, π/
2) = ρ b cos ω b t . As before, ω b = 2 π/T is the driving frequency in the boundary gauge δ ( π/
2) = 0 (the techniques needed to obtainthe fully nonlinear solutions are reviewed in appendix D). Although we cannot factorize theradial dependence into a function ρ ( x ), we will keep the same notation as for the complexcase, i.e., ρ o ≡ φ (0 ,
0) and ρ b ≡ φ (0 , π/ sourced periodic solutions -SPS- is a surface embeddedin the three dimensional space spanned by coordinates ( ω b , ρ b , ρ o ). In the ρ o → , while the limiting,unsourced cases with ρ b = 0 are sometimes termed nonlinear oscillon solutions in the lit-erature, and were first constructed in [34] (see also [24] for an extensive account of resultsand methods). Figure 32 is the counterpart of Fig. 2 in the real case. The similarity is re-markable. The quality of the plot is substantially smaller, as oscillatory solutions are muchmore demanding in terms of computational resources than stationary ones. This implies37
10 15 20 25 k10 - - - | ϕ k ( )| Figure 31: Fourier spectrum at the origin of the periodic solution given by ω b = 2, ρ b = 0 . ρ = 0 . ρ o is very difficult. Although not being apparent from Fig. 32,the oscillon curves tilt and wiggle in a similar way as the BS lines do.Figure 32: The surface of time-periodic geometries responding to a periodically driven real scalarfield. The surface traverses the nonlinear oscillon plane ρ b = 0 at a set of curves. This plot corre-sponds to the one on Fig. 2, to which the similarity is evident. The range exhibited is considerablysmaller in this plot and this is due to the higher technical difficulty in computing each solutionindividually. inear stability The study of stability is more involved now than in the complex case. There, a set ofstatic equations was perturbed, and the linearized spectrum of normal modes was easy toestablish. In the present situation, the linearized fluctuations obey a time-dependent systemof equations. The pertinent tool to employ now is provided by the Floquet analysis. In oursituation we consider linearized fluctuations for the fields of the form x ( t, x ) = x p ( t, x ) + ˜ x ( t, x ) (7.2)where the subindex p stands for “periodic solution” and ˜ x denotes the perturbations. Atfirst order in the perturbations amplitude we obtain a system of the form˙˜ x = L ( t )˜ x (7.3)where L ( t ) = L ( t + T ) is a time periodic linear integro-differential operator (in x ) built outof the solution x p . The Floquet theorem establishes the existence of a solution of the form x ( t ) = e λt P ( t ) with P ( t ) = P ( t + T ) . (7.4)The information about the stability resides in λ i , with i = 1 , , . . . In the general case, both λ and P will be complex. From equation (7.3), we immediately see that λ ∗ and P ∗ are alsosolutions. The real solution thus involves the appropriate linear combination of both. Also,though not evident, one can show that solutions come in pairs, with eigenvalues of oppositesigns, ± λ . This is ultimately a reflection of the Hamiltonian character of the dynamicalsystem (7.3). Hence the stability analysis collapses to one of two possible cases, dependingon whether Re( λ ) is zero or not. The first case yields a stable (cycle) solution, and thesecond one an unstable (saddle point) one.Computing λ i requires an exact integration of (7.3) over a full period T . The same techniquesused to obtain the exact nonlinear SPS’s can be applied here; the reader can find furtherdetails in appendix D. The numerical analysis gives the region of stability that can beobserved in Fig. 33, whose similarity with Fig.4 is manifest. The linear analysis says thatthis similarity goes beyond the shape, also at boundary of the stability region we find thesame algebraic structure, either the lowest eigenvalue λ becoming purely real or two realeigenvalues fusing and developing imaginary components. Nonlinear stability
As for the complex case, analyzing nonlinear stability requires to study the full numericalevolution of the system departing from the perturbed unstable solution. The main conclu-sions remain similar to the ones obtained in the case of a complex scalar. The generic end39igure 33:
Level curves of the SPS surface plotted in Figure 32. Each curve corresponds to aconstant value ρ b (reported explicitly by a small number near the curve). Solutions belonging toshaded (white) regions are linearly stable (unstable). result of an initial unstable SPS is a black hole. However, in the lower part of the whitewedge in sector I unstable solutions decay either to a black hole or to a sourced modulatedsolution , SMS, depending on the sign of the single unstable linear perturbation. As for thecomplex case, the pulsation is a modulation of the driven oscillation, with sudden beatsseparated by a substantially larger plateaux.Like the Boson Stars, nonlinear oscillons become unstable beyond the turning point for themass [24]. At such high values of the field at the origin, the SMS’s are far more fragile thantheir complex counterparts and usually decay after a couple of modulating beats. This maycome from the fact that the whole geometry now shakes. Or maybe from the fact that theharmonic ansatz for the source starts conflicting with the anharmonic response of the fieldin the interior and calls itself for a generalized periodic ansatz.Similarly to the complex case, for sufficiently small values of the source, ρ b ∼ − , we havealso found exotic real SMS’s where the bounce occurs in both directions (i.e, towards loweror higher masses) when the initial perturbation is very weak. We plot an example in thesecond line of Fig. 34. Type I phase transition in the real case
In the real case, the ingredients necessary for the assumptions a and b to hold are present(see Section 4). Therefore, we expect the same type I phase transition we have uncovered in40igure 34: Sourced modulated solutions (SMS) with ρ b = 10 − . In the first line we depict astandard SMS with ω b = 2 .
944 and ρ o = 0 .
15. The blue filling corresponds to the fast oscillationsof the scalar field (as compared to the pulsating modulation). In second line, we provide an exampleof an exotic SMS, where pulsating modulations of both signs alternate. The unstable initial SPShas, in this case, ω b = 2 .
722 and ρ o = 0 . the complex case. In order to confirm this expectation, we consider a time-dependent sourcegiven by the real part of eq. (3.1) (i.e., we replace exp( iω b t ) → cos( ω b t )).In Fig. 35 we plot m and (cid:104)O(cid:105) for the one-parameter family of build-up processes of theharmonic driving ω b = 2 . (cid:15) = 0 .
09. We clearly see that there exists a β c that separatestwo different late-time phases, as for the complex scalar field. Furthermore, as β → β c , thesystem seems to spend progressively larger time around an intermediate attractor. Dynamical construction of a nonlinear oscillon
Nonlinear undriven real SPS’s ( nonlinear oscillons ) are the equivalent to BS’s in the realcase. In this section we will show how nonlinear oscillons can be dynamically constructed,in analogy to the results we presented in section 5.1 for the BS case. Same as there, herewe will describe a quasistatic quench connecting the AdS vacuum with a nonlinear oscillonacross the linearly stable part of region II. The protocol requires the appropriate tuning ofthe source parameters, frequency and amplitude, as a function of time φ ( t, π/
2) = φ b ( t ) = (cid:15) ( t ) cos( ω b ( t ) t ) . (7.5)The specific profiles for (cid:15) ( t ) and ω b ( t ) we shall employ are again given by (5.2)-(5.3), andplotted in Fig. 19. 41igure 35: Left: Energy density for a family of build-up processes of the harmonic driving ω b = 2 . (cid:15) = 0 .
09. The β of the last build-up process ending in the runaway phase has beenmarked by the vertical dashed red line. Right: (cid:104)O(cid:105) for the last simulation ending in the runawayphase (solid orange) and the first one ending in the non-collapsing phase (solid black). Figure 36:
Time evolution of m and (cid:104)O(cid:105) along the quench described in the main text. The blueregion corresponds to a highly dense number of fast oscillations compared with the duration of theprocess. In the first plot this region is not appreciated but as the inset shows it is also present. As a particular example, we consider a quench protocol given by ω i = 3 . ω f = 2 . (cid:15) m = 10 − and β = 1500. Figure 36 contains the time evolution of m and (cid:104)O(cid:105) alongthe process. The main difference with respect to Fig. 20 is that now we are plotting theoscillation of real quantitities. The fast oscillations pile up and, due to their high density,form the blue shaded area visible in the figure. Now, instead of (3.6)-(3.7), we define twonew quantities, (∆ φ ,∆ f ), which measure the distance to the nonlinear oscillon ( φ NLO , f
NLO )42uilt with the target data,∆ φ ( t ) = (cid:32)(cid:90) π tan ( x ) ( φ ( t, x ) − φ NLO ( t, x )) (cid:33) , (7.6)∆ f ( t ) = (cid:32)(cid:90) π tan ( x ) ( f ( t, x ) − f NLO ( t, x )) (cid:33) . (7.7)In Fig. 37 we have plotted the time evolution of these two quantities for our particularexample. On the left plot we observe that, after the quench, the geometry is very close tothe oscillon with frequency ω NLO = 2 . ω NLO = 2 . Time evolution of ∆ φ (blue) and ∆ f (red). We compare the end field configurations φ and f after the quench with two nonlinear oscillons: one with ω NLO = 2 . ω NLO = 2 . In this paper we analyzed periodically driven scalar fields on global AdS . This frameworkallows one to study different aspects of holographic Floquet dynamics, such as dynamicalphase diagrams and late-time regimes alternative to thermalization.We constructed zero-temperature solutions subjected to a constant-amplitude periodic driv-ing of the scalar and dubbed them Sourced Periodic Solutions (SPS). They are dual to socalled Floquet condensates. We have characterized the SPS solution space in detail and stud-ied throughly both linear and nonlinear stability properties. SPS extend beyond linearityknown perturbative solutions about AdS. 43he unstable SPS’s feature a rich phenomenology upon time evolution. In particular, thereexist horizonless stable late-time solutions which evade gravitational collapse and developinstead a pulsating behavior. We named them Sourced Modulated Solutions (SMS). SMS’sare themselves stable solutions where the modulation pulsation impacts in the vev profileand the total mass while, remarkably, its frequency is not imposed by that of the driving.We addressed various types of quench processes concerning both the amplitude and thefrequency of the scalar source. We focused on both slow and fast quenches. The first allowthe study of quasistatic processes and showed that they can be used to prepare SPS’s, dual toFloquet condensates, starting from the AdS vacuum. The study of non-quasistatic quenchesuncovered a nice surprise: by suitably fine-tuning the quench rate, the system stays for anarbitrary long time on an unstable SPS and then decays either to a SMS or a black hole.These SPS’s act therefore as attractors in a Type I gravitational phase transition.Next, we examined the possibility of using such quenches to prepare boson stars startingfrom AdS. From the gauge/gravity duality, this is motivated by the exciting possibility ofpreparing experimentally sourceless Floquet condensates in strongly coupled systems. Infact, we find explicit solutions to this problem using both quasistatic and non-quasistaticquench protocols.Finally we have studied the post-collapse evolution. We have unravelled the three basicbehaviours found elsewere [6].The analysis has been performed both for the case of complex and real scalars. We showedthat the phenomenology is qualitatively similar despite the technical treatment is different(and considerably more involved in the real case). We mainly focused on massless scalars,but preliminary results for the massive situation with m = − Acknowledgements
We would like to thank Andrea Amoretti, Daniel Are´an, Riccardo Argurio, An´ıbal Sierra-Garc´ıa, Blaise Gout´eraux, Carlos Hoyos, Piotr Bizo´n, Keiju Murata and Alfonso Ramallofor pleasant and insightful discussions.This work of was supported by grants FPA2014-52218-P from Ministerio de Economia yCompetitividad, by Xunta de Galicia ED431C 2017/07, by FEDER and by Grant Mar´ıa deMaeztu Unit of Excellence MDM-2016-0692. A.S. is happy to acknowledge support fromthe International Centre for Theoretical Sciences (ICTS-TIFR), and wants to express hisgratitude to the ICTS community, and especially to the String Theory Group, for theirwarm welcome. D.M. thanks the FRont Of pro-Galician Scientists (FROGS) for uncondi-tional support. A.B. thanks the support of the Spanish program ”Ayudas para contratospredoctorales para la formaci´on de doctores 2015” associated to FPA2014-52218-P. This re-search has benefited from the use computational resources/services provided by the GalicianSupercomputing Centre (CESGA).
A Complex periodic solutions
This is by now standard material but we include it here for completeness and in order to fixthe notation. The general gravitational action for a complex scalar field is S = 12 κ (cid:90) d d +1 x √− g ( R − − (cid:90) d d +1 x √− g ( ∂ µ φ∂ µ φ ∗ + V ( | φ | )) , (A.1)with κ = 8 πG , Λ = − d ( d − / l for AdS d +1 . Note that it is assumed V (0) = 0 otherwisethe constant term of the scalar potential would contribute to the cosmological constant. The45pecific action (2.1) corresponds to taking d = 3 and V ( | φ | ) = − m φ ∗ φ . As the scalar fieldis complex, the action is invariant under the global U (1) transformations φ → e iα φ . Theequations of motion are R µν − g µν R + Λ g µν = κ (cid:2) ∂ µ φ ∗ ∂ ν φ + ∂ ν φ ∗ ∂ µ φ − g µν (cid:0) | ∂φ | + V ( | φ | ) (cid:1)(cid:3) , (A.2)1 √− g ∂ µ (cid:0) √− gg µν ∂ ν φ (cid:1) = ∂V ( φ, φ ∗ ) ∂φ ∗ . (A.3)(A.4)The isotropic ansatz for the metric of AdS d +1 can be expressed as follows ds = l cos x (cid:0) − f e − δ dt + f − dx + sin x d Ω d − (cid:1) , (A.5)with x ∈ [0 , π/ t, x ) = φ (cid:48) ( t, x ) , Π( t, x ) = e δ ( t, x ) f ( t, x ) ˙ φ ( t, x ) . (A.6)Upon this redefinition, the scalar field equations of motion can be cast in the form˙Φ = (cid:0) f e − δ Π (cid:1) (cid:48) , (A.7)˙Π = 1tan d − x (cid:0) tan d − xf e − δ Φ (cid:1) (cid:48) − l cos x e − δ ∂ φ c V ( | φ | ) . (A.8)From the Einstein equations we obtain, after setting κ = ( d − / l = 1, f (cid:48) = d − x sin x cos x (1 − f ) − sin x cos x f (cid:0) | Φ | + | Π | (cid:1) − tan x V ( | φ | ) , (A.9) δ (cid:48) = − sin x cos x (cid:0) | Φ | + | Π | (cid:1) . (A.10)In the main text we consider the time-periodic ansatz (2.3) where ω is an angular frequencyand ρ ( x ) is a real function. We are adhering here to the boundary gauge δ ( π/
2) = 0 thatmakes t equal to the proper time of an observer at the boundary. For any given ω there arestatic solutions for ρ ( x ), f ( x ) and δ ( x ) to be obtained from the equations of motion ρ (cid:48)(cid:48) + (cid:18) d − x cos x + f (cid:48) f − δ (cid:48) (cid:19) ρ (cid:48) + (cid:18) ω e δ f − m f cos x (cid:19) ρ = 0 ,f (cid:48) − x sin x cos x (1 − f ) − f δ (cid:48) + m ρ tan x = 0 , (A.11) δ (cid:48) + sin x cos x (cid:18) ρ (cid:48) + ω e δ f ρ (cid:19) = 0 , that result from the above-mentioned ansatz.46 The role of the pumping solution
In a previous work [40], some of the authors analyzed a massless, real scalar field in AdS subjected to a linearly rising pumping source, φ ( t, π/
2) = α b t . One of the major findingswas that there exist extremely simple pumping solutions , in which the scalar field profile isflat in the radial direction, φ ( t, x ) = α b t , while the metric functions f ( x ) , δ ( x ) are nontrivial,but static.Our objective in this appendix is to comment on the interplay between these pumpingsolutions and the four-dimensional SPS’s we have found in this paper. From Fig. 2 it isapparent that the surface of SPS’s extends to values ω b = 0 on a line ρ b = ρ o . Naively thesewould be related to the mentioned pumping solutions. The correct answer involves a carefulscaling limit. The scaling limit
Consider the SPS equations of motion (A.11) (for d = 3 , m = 0) and set ρ ( x ) = ˆ ρ b /ω b . (B.1)By taking the ω b → ρ b , these equations reduce to f (cid:48) ( x ) − x sin x cos x (1 − f ( x )) − f ( x ) δ (cid:48) ( x ) = 0 , (B.2) δ (cid:48) ( x ) + cos x sin x e δ ( x ) ˆ ρ b f ( x ) = 0 , (B.3) e δ ( x ) f ( x ) ˆ ρ b ω b = 0 . (B.4)The equations are nothing but the equations of motion for the pumping ansatz (as reportedin [40]), provided we identify ˆ ρ b = α b and fix ω b = 0. This simple observation establishesthat the pumping solution controls the SPS’s in the limit of infinite source amplitude andzero source frequency (i.e., on the upper left part of the diagram displayed in Fig. 4).As demonstrated in [40], there is a critical pumping rate α ∗ b above which no pumping solu-tion exist. Numerically, α ∗ b ≈ . ω b (cid:28)
1, the upper boundary of the region containing the linearly stable SPS’s in the ω b − ρ b plane must approach the curve ω b ρ b = α ∗ b . (B.5)Note in particular that, if this expectation is correct, ρ b must diverge as ω b →
0. Theanalogous statement ω b ρ o = α ∗ b also has to hold in this limit, since the difference between ρ b ρ o is also expected to vanish as ω b →
0: the pumping solution is radially flat and hasno vev.In order to confirm these expectations, we have determined numerically the upper boundaryof the stability region in the ( ω b , ρ b ) phase diagram (at small frequency). The results areshown in Fig. 38a. It is clearly seen that, as ω b →
0, this upper boundary lies progressivelycloser to the predicted curve ρ b,c = α ∗ b /ω b .As a further consistency check, in Fig. 38b we establish that |(cid:104)O c (cid:105)| → ω b → |(cid:104)O c (cid:105)| is the vev of the last linearly stable SPS-. The numerical data are compatible with alinear decrease in driving frequency, |(cid:104)O c (cid:105)| ∼ ω b . ●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●● - - - ω b - ρ b , c ●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●● - - - ω b - - - |< O >| Figure 38:
Left: Critical values ρ b,c as a function of ω b (blue dots); the red line corresponds tothe analytical prediction ρ b,c = α ∗ b /ω b . Right: |(cid:104)O c (cid:105)| of the critical SPS’s (blue dots), together withthe linear fit |(cid:104)O c (cid:105)| = 1 . ...ω b (red line). C Study of the normal modes
This appendix provides details on the spectrum of linearized normal modes of the complexSPS’s and BS’s presented in the main text.Linear stability is the first important piece of information one gets from the normal modespectrum. A point-wise study of several individual solutions in the plane ( ω b , φ o ) leads toestablish the stability diagram reported in Fig. 4 whose details are highlighted below.In Fig. 4, the solid lines departing from the φ o = 0 axis at ω b = 3 + 2 n with n ∈ N + representing the BS branches that dress nonlinearly the AdS oscillons. Moving along the BSbranches by raising φ , one encounters BS’s with increasing mass which are stable until themass reaches a maximum. This points provides a Chandrasekar-like bound beyond whichthe BS’s become linearly unstable. This is seen in Fig. 39 were we plot the evolution of the48quared frequencies of the first two normal eigenmodes λ , . Precisely at the Chandrasekarmass, the second eigenmode λ becomes purely imaginary. ϕ o - - ( λ ) ϕ o ϕ o - - ( λ ) ϕ o Figure 39:
Top: Real part of the first two squared eigenfrequencies of the lower Boson Starbranch. The red point signals the onset of linear instability which corresponds to the maximumvalue attained by the mass along the same branch. Bottom: Analogous plots for the φ s = − . The second set of plots shows the evolution of the same modes along a line of SPS’s with asmall source in sector II (profiles with one node), running closely parallel to the BS curve. Thesituation here changes qualitatively, though continuously. The instability is now triggeredby the first two modes fusing at positive values and developing imaginary parts of oppositesign. This now happens before the maximum of the mass is reached. The continuation ofthis point away from the line of BS deep into sector II draws the upper limit of the stabilityregion displayed on Fig. 4 (in gray) on the segment ω b ∈ (3 , ω b = 4 and very small49 .5 1 1.5 x f (cid:72) x (cid:76) A.U.
Figure 40:
Zero-mode of a Boson Star compared with the “profile increment” of the scalar fieldalong the Boson Stars branch. The plots are rescaled in order to overlap. φ o : the solutions are unstable, even though surrounded by stable regions. Such qualitativebehavior is repeated for ω b = 2 n with n ≥ ω b = 2. A direct look to themodes shows that the instability here is again triggered by the fusion of the first two modesdeveloping opposite imaginary parts (see Fig. 41). ω b - - ( λ ) ω b ( λ ) Figure 41:
First two normal modes squared developing an imaginary part (dashed) upon fusing.The plots correspond to φ b = − .
001 (left) and φ b = − .
01 (right). In the left plot the imaginaryparts have been rescaled by a factor 500 to be visible.
It should be noted that the “encounter” of two normal modes does not lead always to afusion and an instability. For instance, at ω b ∼ φ b . Let us showthis by means of an explicit example. Consider two SPS’s corresponding to φ b = 0 . φ b = 0 . ω b ∼ ϕ ( λ ) ϕ ( λ ) Figure 42: Second and third modes repelling at φ b = . φ b = . ω b (cid:28) λ n behave as λ n = λ ∗ n ± ω b , where λ ∗ n are the eigenfrequencies of AdS. This behavior is main-tained also at higher values of ω b as the numerics shows, see Fig. 43. Nevertheless, the linearbehavior of the modes has to break down as two modes approach in order to be compatiblewith a repulsion. ω b λ n Figure 43:
Normal frequencies plotted against the boundary frequency of the source whose ampli-tude is φ b = . Real periodic solutions in AdS . The action is S = 12 κ (cid:90) d x √− g ( R − − (cid:90) d x √− g (cid:18) ∂ µ φ∂ µ φ + V ( φ ) (cid:19) (D.1)where we will take as before κ = 8 πG = 1, Λ = − /l . The isotropic ansatz for the metricof AdS d +1 can be expressed as follows ds = l cos x (cid:0) − f e − δ dt + f − dx + sin x d Ω d − (cid:1) . (D.2)We are interested in time-periodic solutions with harmonic boundary conditions such that φ ( t, π/
2) = ρ b cos( ω b t ). Continuing with the same notation as in the complex case we will use ρ o = φ (0 ,
0) and ρ b = φ (0 , π/ F = f e − δ Φ ≡ φ (cid:48) ( t, x )Π ≡ ˙ φ/F + ρ b ω b sin ω b t. Working in the boundary gauge, we have that Π( t, π/
2) = 0, while Φ( t, π/
2) = 0 due tothe asymptotic near-boundary expansion. With these definitions, the equations of motionbecome ˙Φ = ( F Π) (cid:48) − ρ b ω b sin ω b t F (cid:48) (D.3)˙Π = 1tan x (tan xF Φ) (cid:48) + ρ b ω b cos ω b t (D.4) δ (cid:48) = − sin x cos x (cid:0) Φ + Π − ρ b ω b sin ω b t Π + ρ b ω b sin ω b t (cid:1) (D.5) F (cid:48) = 1 + 2 sin x sin x cos x ( e − δ − F ) (D.6)The periodicity of the solution (7.1) instructs us to consider the most general Fourier seriesexpansion. However, the structure of the equations of motion allows for a truncation to oddmodes, which correspond to bifurcations emanating from single normal modes of AdS,Φ = ∞ (cid:88) k =0 Φ k ( x ) cos[(2 k + 1) ω b t ] Π = ∞ (cid:88) k =0 Π k ( x ) sin[(2 k + 1) ω b t ] . (D.7)While δ and F are obtained from Φ and Π through (D.5) and (D.6). Following the methodsintroduced in [24], we rescale time as τ = ω b t and use Chebyshev polynomials to expand52he spatial dependence. The collocation grids are associated to these two basis of functions,Fourier and Chebyshev, τ n = π n − K + 1 with n = 0 , , ..., K − x i = π πi/ (2 N + 1)] with i = 0 , , ..., N . (D.8)Let us define f ki ≡ f k ( x i ). Then, the discretized values areΦ( τ n , x i ) = ∞ (cid:88) k =0 Φ ki cos[(2 k + 1) τ n ] Π( τ n , x i ) = ∞ (cid:88) k =0 Π ki sin[(2 k + 1) τ n ] . (D.9)Using the boundary conditions discussed previously (Φ( t, π/
2) = Π( t, π/
2) = 0), the valuesof the fields at x are restricted to Φ k = Π k = 0. Hence, the unknowns to be fixed are ω b , ρ b , Φ ki and Π ki for k = 0 , ..., K − i = 1 , ..., N , while the equations are (D.3) and(D.4) evaluated at each of these collocation points ( τ n , x i ). This gives 2 KN equations for2 KN + 2 unknowns. We fix a numerical value for one of them, being it either ω b or ρ b , andadd another equation of motion that sets the value of ρ o . Finally, the number of equationsand variables match, 2 KN + 1, and the discretized system yields an algebraic nonlinearsystem of equations that can be solved using a Newton-Raphson algorithm.After obtaining time-periodic solutions we are interested in their linear stability. Considerlinearized fluctuations of the form φ ( t, x ) = φ p ( t, x ) + ˜ φ ( t, x ) , Π( t, x ) = Π p ( t, x ) + ˜Π( t, x ) ,δ ( t, x ) = δ p ( t, x ) + ˜ δ ( t, x ) ,F ( t, x ) = F p ( t, x ) + ˜ F ( t, x ) , where the subindex p stands for “periodic solution” and fields with a tilde are the perturba-tions. Inserting these ansatz into the equations of motion, and at first order in the amplitude,we obtain the equations for the perturbations˜ δ (cid:48) = − x cos x (cid:16) Φ p ∂ x ˜ φ + (Π p − ρ b ω b sin ω b t ) ˜Π (cid:17) , (D.10)˜ F (cid:48) = − x sin x cos x (cid:16) ˜ δe − δ p + ˜ F (cid:17) , (D.11)˙˜ φ = F p ˜Π + (Π p − ρ b ω b sin ω b t ) ˜ F , (D.12)˙˜Π = 1tan x ∂ x (cid:104) tan x (cid:16) F p ∂ x ˜ φ + Φ p ˜ F (cid:17)(cid:105) . (D.13) hereafter we will find it convenient to work with the φ rather than Φ δ and ˜ F can be seen as linear operators acting on ˜ φ and ˜Π, with this point of view, equations(D.12) and (D.13) are expressed in the following form (cid:32) ˙˜ φ ˙˜Π (cid:33) = L ( t ) (cid:18) ˜ φ ˜Π (cid:19) , (D.14)where L ( t ) is a linear integro-differential operator (in x ) constructed with the periodic fields.The periodicity of the background is inherited by the operator, L ( t ) = L ( t + T ), T being theperiod of φ p . For x ( t ) = ( ˜ φ, ˜Π), the Floquet theorem establishes the existence of a solutionof the form x ( t ) = e λt P ( t ) with P ( t ) = P ( t + T ) . (D.15)The structure of (D.10)-(D.13) allows a truncated version of the Fourier expansion for P ( t )in odd modes which rule the transitions of stability that we have found. It is convenient toexpress P ( t ) in two parts P ( t, x ) = p (1) ( t, x ) + p (2) ( t, x ) = (cid:18) ˜ φ (1) ˜Π (1) (cid:19) + (cid:18) ˜ φ (2) ˜Π (2) (cid:19) , (D.16)where the time dependence is distributed as follows˜ φ (1) = ∞ (cid:88) k =0 ˜ φ (1) k ( x ) cos [(2 k + 1) ω b t ] , ˜Π (1) = ∞ (cid:88) k =0 ˜Π (1) k ( x ) sin [(2 k + 1) ω b t ] , (D.17)˜ φ (2) = ∞ (cid:88) k =0 ˜ φ (2) k ( x ) sin [(2 k + 1) ω b t ] , ˜Π (2) = ∞ (cid:88) k =0 ˜Π (2) k ( x ) cos [(2 k + 1) ω b t ] . (D.18)Inserting this ansatz into (D.17)-(D.18) and taking advantage of the linear independence ofthe trigonometric functions, equations split in the following form λ ˜ φ (2) = ˜ F (1) (Π p − ρ b ω sin ωt ) + F p ˜Π (1) − ˙˜ φ (1) , (D.19) λ ˜ φ (1) = ˜ F (2) (Π p − ρ b ω sin ωt ) + F p ˜Π (2) − ˙˜ φ (2) , (D.20) λ ˜Π (2) = 1tan x (cid:20) tan x ˜ F (1) Φ p + tan xF p (cid:16) ˜ φ (1) (cid:17) (cid:48) (cid:21) (cid:48) − ˙˜Π (1) , (D.21) λ ˜Π (1) = 1tan x (cid:20) tan x ˜ F (2) Φ p + tan xF p (cid:16) ˜ φ (2) (cid:17) (cid:48) (cid:21) (cid:48) − ˙˜Π (2) , (D.22)where we have defined ˜ F ( i ) ≡ ˜ F ( ˜ φ ( i ) , ˜Π ( i ) ) and used the time structure of the periodic fields(D.7). This system of equations has the property that given a solution (cid:0) λ, p (1) , p (2) (cid:1) anothersolution exists with (cid:0) − λ, p (1) , − p (2) (cid:1) , implying that an unstable mode exists if Re ( λ ) (cid:54) = 0.On the other hand, if all λ ∈ i R , the periodic solution is linearly stable.54olving (D.19) through (D.22) is very similar to obtaining the time-periodic solutions φ p andΠ p themselves. So the same techniques explained in the previous section are in order here.The differences lie in the fact that, in this case, the ansatz (D.16) has two more unknownfunctions ( p (2) ) and the role of ω is played by λ . Namely, we discretize the problem asin (D.8) and set boundary conditions so as to study perturbations which don’t modify thesource ( ˜ φ ( i ) ( t, π/
2) = ˜Π ( i ) ( t, π/
2) = 0). Finally, by adding an additional equation to fix theamplitude of ˜ φ (1) (0 ,
0) = 1, we obtain a system of 4
N K + 1 nonlinear equations for our4
N K + 1 unknowns, which can be solved using a Newton-Raphson algorithm.
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