#### Floquet scalar dynamics in global AdS

HJE
Floquet scalar dynamics in global AdS
Anxo Biasi 0 1 3 4
Pablo Carracedo 0 1 3 4
Javier Mas 0 1 3 4
Daniele Musso 0 1 3 4
Alexandre Serantes 0 1 2 3 4
0 Santiago de Compostela , Spain
1 E-15782 Santiago de Compostela , Spain
2 International Centre for Theoretical Sciences-TIFR
3 Bengaluru North , 560 089 India
4 Survey No. 151, Shivakote, Hesaraghatta Hobli
We study periodically driven scalar elds and the resulting geometries with global AdS asymptotics. These solutions describe the strongly coupled dynamics of dual nite-size quantum systems under a periodic driving which we interpret as Floquet condensates. They span a continuous two-parameter space that extends the linearized solutions on AdS. We map the regions of stability in the solution space. In a signi cant portion of the unstable subspace, two very di erent endpoints are reached depending upon the sign of the perturbation. Collapse into a black hole occurs for one sign. For the opposite sign instead one attains a regular solution with periodic modulation. We also construct quenches where the driving frequency and amplitude are continuously varied. Quasistatic quenches can interpolate between pure AdS and sourced solutions with time periodic vev. By suitably choosing the quasistatic path one can obtain boson stars dual to Floquet condensates at zero driving eld. We characterize the adiabaticity of the quenching processes. Besides, we speculate on the possible connections of this framework with time crystals.
AdS-CFT Correspondence; Gauge-gravity correspondence; Holography and
1 Introduction and main results
Periodically driven complex scalar eld
Quenches with periodic driving
Quasistatic quenches
Non-quasistatic quenches
Type I phase transitions in driven AdS/CFT Dynamical construction of a boson star 2 3
4
5
6
7
8
3.1
3.2
5.1
5.2
Quasistatic method
Non-quasistatic method
The post-collapse regime
Periodically driven real scalar eld
Summary and outlook
A Complex periodic solutions
B The role of the pumping solution
C Study of the normal modes
D Real periodic solutions in AdS4
Introduction and main results
The physics of periodically driven many-body systems is a fascinating chapter in the study
of out-of-equilibrium dynamics [1, 2]. Their behavior can di er substantially from that of
their static counterparts. Floquet systems have been investigated at criticality with the
tools of conformal eld theory (CFT) (see [3] and references therein). The naive expectation
that they can only increase inde nitely their energy because of the driving can be avoided
in some regions of the driving parameter space. This was con rmed by explicit examples
that attain a steady state at nite temperature [3, 4]. Even though the actual saturation
mechanism is not completely understood, the fact that periodic driving can radically alter
the stability of equilibrium points is well known from nonlinear dynamical systems, the
Kapitza pendulum being a prominent example [5].
The AdS/CFT correspondence maps equilibration processes onto the dynamical
evolution of a dual gravitational system. The holographic dictionary de nes the sources of
the eld theory in terms of asymptotic modes of the bulk
elds. It is then possible to
study a driven holographic system coupled to an external source. In the cases of interest
{ 1 {
here, the source corresponds to the leading asymptotic mode of a bulk scalar eld in an
asymptotically global AdS spacetime.
Periodic driving has been scarcely studied in the context of AdS/CFT. The analysis
performed in [6] comes closest in spirit to the present study (see also [7{11], all in the
probe approximation). The main di erence is that their initial states are already mixed,
while here we consider the driving of pure states. Their geometrical ansatz is embedded
in the Poincare patch, with an arbitrarily small planar horizon to cap the IR. They
nd
a monotonous growth of the black hole horizon as a general outcome of the driving. Such
growth leads to an in nite temperature nal state. The result is, on the one side, compatible
with the already mentioned natural expectation and, on the other side, it comes out as
HJEP04(218)37
a consequence of the ingoing boundary conditions at the horizon. The only remaining
freedom may be in the rate of growth of the energy density and entropy.
The authors in [6] nd three di erent phases depending upon the range of values
acquired by the dimensionless combination
= b!b, namely the product of the amplitude
and the frequency of the driving at the boundary. Small values of
correspond to a
dissipation dominated regime, where the horizon absorbs all the energy entering the system
from the boundary. Here the scalar eld synchronizes with the driving but does not change
its amplitude. Raising
they
nd two other phases in which the scalar eld 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 nd here. The fact
that our driving does not necessarily imply decoherence (i.e. a collapse into a black hole)
is by no means trivial and it only occurs in certain regions of parameter space. Charting
this region, characterizing its features, and studying its stability is the main target in the
present paper.
We have examined both complex and real massless scalar elds. While a priori their
dynamics is quite di erent, remarkably, the
nal results of the real and complex cases
exhibit strong similarities, both in the structure of the phase space and in the order of
magnitude of speci c numerical values like stability thresholds.
Finding the manifold of driven and periodic solutions is the rst step. Since such
solutions could in principle be all unstable, a thorough inspection of their stability is in
order both at the linear and nonlinear level. Linearly stable solutions proved to exist and
be robust against rather large
uctuations. For unstable solutions, we follow numerically
the evolution in the search for the nal endpoint at large times. The picture that emerges is
rather interesting: in signi cant portions of the unstable subspace, the solutions lie at the
boundary of two radically di erent long time behaviors. On one side, uctuations drive the
geometry to a collapse into a black hole, the dual theory loses coherence and thermalizes as
expected. On the other side, the geometry is regular forever. It however exhibits a periodic
modulation that has the form of a relaxation oscillation1 where the solution periodically
bounces back and stays for a long time close to the initial unstable solution. From the
dual eld theory point of view, the quantum state remains pure but exhibits an emergent
pulsating modulation.
1We are adopting the usual terminology of dynamical systems like, for example, the inverted pendulum.
{ 2 {
The framework at hand allows us to address the important question of the adiabatic
preparation of Floquet condensates [12{14]. By slowly varying the amplitude and/or
frequency of the driving, one can study if the system keeps up with the source and follows a
trajectory on the manifold of periodic solutions. This is naively expected in the limit of
slow quenches, as a natural dual counterpart of the quantum adiabatic theorem. However,
a careful analysis reveals 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 starting and ending with a vanishing driving. The possibility that such
cycle ends on a periodic solution with vanishing source (but not just AdS) could perhaps
relate to the open question of spontaneous breaking of continuous time translations [15, 16].
linear and nonlinear stability properties. In section 3 we analyze the response of the system
upon a continuously varying driving. This unravels interesting structures like the presence
of critical values of the quench parameters separating classes of qualitatively di erent time
evolutions (ending or not in a black hole, for instance). In section 4 we show that the
unstable solutions are, in fact, attractor solutions in the sense of Type I gravitational
phase transitions studied in [17{19]. In section 5 we describe two alternative quenching
protocols to construct a boson star starting from AdS. One is quasistatic, while the other
involves an unstable solution as an intermediate step. Section 6 is devoted to the late-time
evolution of the collapsing black hole solutions and establishes contact with the results of
the analysis done in [6]. In particular we are able to pin down the three regimes found in
that paper. In section 7 we comment the analysis for the real scalar eld, stressing the
fact that it constitutes a nontrivial extension both at the conceptual and technical level.
We summarize in section 8 accounting for and interpreting the results as well as indicating
research directions for the future. Some technical material is collected in the appendices.
2
Periodically driven complex scalar
eld
Our case study involves the simplest possible setup, namely a complex scalar eld in
global AdS4,
S =
(2.1)
with
=
3=l2 for AdS4. We will set 2 = 8 G = 1, l = 1. The action is invariant under
global U(
1
) transformations
! e
i . Our ansatz for the metric is
ds2 =
1
where x 2 [0; =2) is the radial coordinate. We will examine the space of solutions adapted
to the following time-periodic ansatz
(t; x) = (x)ei!bt ;
{ 3 {
(2.2)
(2.3)
with (x) a real scalar function. The space of solutions contains just three functions f;
and ; they are determined by two parameters, namely, the frequency !b and a boundary
value for .
Introducing the ansatz (2.3) in the Einstein-Klein Gordon equations of motion, gives a
set of !-dependent static equations for (x) which can be solved numerically by standard
methods (see (A.10)). Each solution of this system gives rise to a static geometry where the
scalar eld rotates in the complex plane with angular velocity !b while keeping constant
its modulus (x). The boundary values of (x) are respectively o = (0) at the origin and
b at the boundary; b is de ned through the asymptotic Taylor expansion
(x) = b(x
=2)3
+ : : : ;
with
= 3=2 + p9=4 + m2 being the conformal dimension of the dual scalar operator
O. For non-vanishing b, the dual state is interpreted as being driven by a time-periodic
source bei!bt. We abbreviate these sourced periodic solutions as SPS. In this paper we
analyze the case of a massless scalar, m2 = 0, in detail. We shall also comment on partial
results obtained in the case m2 =
2, where everything seems to follow the same pattern
so far. In the massless case, the unsourced solutions with b = 0 are termed boson stars
| BS | in (at least part of) the literature [20, 21] and we shall adhere in what follows to
this name.
surface represents the complete set of static geometries corresponding to SPS's in the three
dimensional space spanned by (!b; o; b) (see
gure 1 for a visualization of these three
parameters). The SPS surface cuts at !b = 0 on a line of con gurations where the scalar
takes a constant radial pro le, in particular o = b
. Boson stars lie at the intersection
of the SPS surface with the zero source plane b = 0. The boson star curves start from
the bottom plane o = 0 at the values !n = 3 + 2n, given by the spectrum of normal
frequencies of the massless scalar in AdS4. As the value of o is increased, the nonlinear
dressing of the linearized solution shifts the value of the frequency and gives rise to the
wiggly curves depicted in
gure 2. The lower portion of this curve that bends towards
smaller values of !b was already constructed in [21]. The di erent sectors indicated with
I, II, III,. . . correspond to SPS whose radial pro les (x) have 0; 1; 2 : : : nodes respectively.
{ 4 {
eld. The surface intersects the boson star plane b = 0 at a set of curves as shown in the plot.
In making this plot, we have adopted the phase convention that makes o a positive number (see
gure 3). Hence the sign of b is nothing but the relative sgn( o= b), which is correlated with the
number of nodes of (x).
eld at the origin, o = 0:65 and di erent !b = 1:9; 2:96 and 5:04 from top
to bottom. As the number of nodes increases, the energy density tends to concentrate deeper in
the bulk. The reader should understant that this pro le is rotating in the transverse plane. The
relative sign b and o correlates with the even or odd number of nodes.
In the case of a driving by a relevant scalar eld with m2 =
2, the surface remains
qualitatively the same.
The unsourced solutions analogous to the boson stars branch
instead from the spectrum of linearized
uctuations which is now !n =
+ 2n with
= 1; 2 for the alternative/standard quantization of the scalar eld.
Given the external driving force, the existence of regular solutions constitutes by itself
a nontrivial result, the natural expectation being that the system would get increasingly
excited. The total mass should grow monotonically and, eventually, lead to a collapse,
signalling thermalization in the dual eld theory side. Before closing this section let us
{ 5 {
(2.4)
HJEP04(218)37
comment why this may be avoided here. The rate at which the total mass changes with
time is controlled by the di eomorphism Ward identity
For the case at hand, this identity boils down to the following equation
r h
T 0i = hO(t)ir0J :
m_(t) =
2Re _b (t)hO(t)i :
In the undriven case _b(t) = 0, the r.h.s. is identically zero, and the system is isolated. For
_b(t) 6= 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
a variable source, energy can ow both in and out. The source function
b(t) is known.
In contrast, the 1-point function hO(t)i is a teleological quantity (in the radial direction),
like event horizons are (in time). It can only be extracted from the boundary behaviour
of the solution once this has been computed down to the origin and regularity has been
imposed. In the particular case of a SPS, the vev oscillates harmonically in phase with
the source. This particular case yields an exactly vanishing value for the right hand side.
However, as soon as the SPS is perturbed, m_ will start to
uctuate around zero. The
average of this uctuation will 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 nd a regular solution only if the
quantum system we drive is in a pure state. If there was a horizon, no matter how small,
then part of the 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 the boundary, i.e. reaching the dual geometry to an in nite temperature state.
In summary, a thorough study of the stability properties of the SPS is compulsory. We
will do it rst 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's given in
gure 2 by performing a linearized
uctuation analysis.
We nd it convenient to move along level curves with constant source amplitude b. These
curves are depicted in gure 4.
Notice that from the point of view of an observer at the boundary, namely for each
pair ( b; !b), there can be more than one SPS. They have di erent bulk pro les and, in
particular, they reach the origin at di erent values of o. For instance, in gure 4 a duplicity
has been highlighted in the case ( b = 0:09; !b = 2) by the red dots on the left. In the
case of multiple solutions corresponding to the same boundary data ( b; !b), it is natural
to suspect that one of them is stable since it represents the \ground state" of the sector.
Along the lines of constant b, one nds extremal values of the frequency where two
solutions corresponding to the same !b become degenerate. At these points the spectrum
of linearized
uctuations contains a zero mode that connects the two degenerate solutions.
One such example is shown in gures 4, 5 by means of a cyan dot. { 6 {
value b (sometimes reported explicitly by a small number near the curve). The black curves denote
the boson stars with b = 0. The two red points both correspond to !b = 2 and b = 0:09. The
cyan dot represents the point where !b reaches its maximal value along the level curve. The dark
yellow dots mark the stability thresholds along the boson star lines.
2
80
λn (ρo)
60
40
20
2
0
0.4
0.5
0.2
0.4
0.6
0.8
1.0ρo
corresponds to the same solutions as in
gure 4. The stability threshold occurs at o = 0:425 and
corresponds to the cyan dot where !b reaches its maximal value along the 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 analysis
we also
nd some maxima of the mass along the level curves which are accompanied
by a squared eigenfrequency transiting from positive to negative values, see for instance
gure 39. Nevertheless, looking at the extrema of the mass (or !b) does not yield exhaustive
information about the stability. There are cases where complex eigenfrequencies arise
because two real eigenmodes merge. Further information about the mode structure and
behavior can be found in appendix C.
Charting the complete stability region involves a numerical scan of the spectrum of
linearized perturbations of SPS's across the space of solutions. The shaded region in gure 4
{ 7 {
is our best approximation as to where the region of linearly stable solutions extends. Notice
that parts of the edge of the stability region are given by the boson star solutions. Here
the passage from gray (stable) to white (unstable) occurs precisely because the lowest
eigenmode squared turns negative. This observation implies that the spectrum of linearized
perturbations around a boson star will always contain a zero mode. In appendix C we prove
that the zero mode generates the boson star line and we provide more information on the
building and features of the stability region.
Also notice that the white wedges emerge from integer values of !b. Such values
correspond to special eigenfrequencies of the linearized scalar eld problem. Indeed, over
global AdS4, the scalar wave equation has a general spectrum of regular solutions given by
(t; x) = e(x)ei!bt with
e(x) / cos(x)3 2F1
3
!b 3 + !b 5
2
;
2
2
; ; cos(x)2
+
3 cot
!b(!b2
!b
2
1) 2F1
2
!b ; !b
;
2
1
2
; cos(x)2 :
Normal modes come in two families. Solutions with !b = 3 + 2k, k = 0; 1; 2 : : : are
normalizable and have vanishing source, while solutions with !b = 2k, k = 1; 2; 3 : : : are
non-normalizable and have nonzero source, but vanishing vev.2 The fact that white wedges
descend to the vicinity of these linearized solutions supports the picture of linear instability
as a resonance phenomenon. With this remark in mind we would expect also an instability
wedge to come down to !b = 2 but actually we do not nd it.
Nonlinear stability. In a nonlinear theory the results of a linear stability analysis are
of limited range. A perturbed unstable solution will soon start departing largely from its
original, unperturbed state. Therefore, we need to follow it to see what the end result of
its evolution can possibly be. In
nite dimensional nonlinear systems there are two
possibilities: another stable equilibrium point or a limiting cycle. We have built a numerical
evolution code which is a minimal adaptation of the ones employed in the study of
gravitational collapse in global AdS implementing a standard RK4 nite di erence scheme. We
have checked that, when initialized at t = 0 with a linearly stable SPS, the output yields
consistently the expected periodic geometry over as long as we have let the computer go.
Notice that, since the pro le never gets particularly spiky, rather low resolutions of 210 + 1
points are su cient. This allows for a considerable increase in computational speed.
The next numerical experiment is to initialize the code with a linearly unstable SPS
slightly perturbed with the single unstable normal mode. In other words, if 0(x) is an
SPS, and
1(x) its unstable mode with a purely imaginary eigenfrecuency
21 < 0, then
at t = 0 we will insert
0(x) + 1 1(x) with 1
O(10 4). In general terms, one could
reasonably expect collapse to a thermal phase as the end result of time evolutions starting
2This is more easy to see by rewriting the normal modes for AdSd+1 in terms of generalised Legendre
polynomials
ek(x) = cos xPk( d2 1; d2 )(cos(2x));
wk = 2k + ;
with
=
the two roots of m2 = (
d). Solutions with
=
+ are normalizable and have vanishing
source. Solutions with
=
= d
+ are non-normalizable and have vanishing vev. When m = 0,
+ = d and
t Re[ϕ(t,π/2)]
HJEP04(218)37
all along its pro le. The tip at the origin and at the boundary rotate at the same pace. There is,
a priori, no reason why this should be so and is not an artifact or truncation of the simulation. In
fact one can nd (unstable) solutions where this is not the case.
from unstable initial conditions. Nevertheless, as we shall see now, even in this case, the
system provides regular counterexamples.
In order to keep the analysis as systematic as possible, we continue working with the
two solutions marked in red on the left hand side of gure 4. They both represent SPS's
with !b = 2 and b = 0:09. The lower one lies in the region of linear stability, and we have
checked that it also supports fairly \strong kicks", 1
O(
1
), leading to oscillations about
the initial solution.
In contrast, the upper one contains in its spectrum of linearized normal modes one
unstable eigenmode,
1(x), with a purely imaginary frequency
fundamental eigenmode has no nodes and its shape resembles the SPS itself. With 1
10 4, the time evolution when t ! 1 di ers dramatically depending upon the sign chosen
for 1
.
With positive sign (so that 1 1(0) has the same sign as
0(0)), the evolution
collapses promptly to a black hole geometry. This can be seen in the top line of plots in
gure 7. The continuous driving re ects itself in the rising of the total mass, while the
horizon radius increases monotonically as does the absolute value of the vev itself. The
system approaches the expected in nite temperature state.
With the opposite sign, 1
10 4, the evolution reaches a radically di erent
endpoint. Instead of a collapse, the dynamics stays regular for all times. In
gure 6 we plot
the oscillations of the real part of the scalar eld at the origin and at the boundary. Both
of them proceed in phase with one another, signalling that the pro le of the scalar eld
stays in a plane while rotating. Moreover, and this is new, the oscillation of the scalar at
the origin acquires a violent and sudden modulation in amplitude.
Since we are working with a fully backreacted solution, the modulation of the driven
oscillation a ects many other physical quantities. Figure 7 highlights the impact of the
modulation on three observables: the minimum of the metric function minx[f (t; x)], the
absolute value of the vev hOi(t) and, nally, the energy density M . Note that minx[f (t; x)]
reveals the formation of an apparent horizon whenever it drops towards zero.
{ 9 {
30
t
t
|<O>(t)|
0
|<O>(t)|
3.0
2.5
10 4 (+10 4). The magnitudes shown are the minimum of the metric
function minx2[0; =2)(f (t; x)), the energy density m, and the vev jhOij. A relaxation oscillation
with a pronounced peak and a long lived plateau is apparent on the lower plots.
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
pulsations or beats. We will refer to these regular solutions as sourced modulated solutions,
SMS. The period T is orders of magnitude away from the one due to the driving frequency
(2 =!b) and, actually, it is (weakly) dependent on the strength of the perturbation. Hence,
in the strict sense, the solution is not a limiting cycle like the one found in nonlinear systems
such as the van der Pol oscillator. In such systems the asymptotic dynamics eventually
loses memory of 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 slightly kicked out of its vertical unstable position, where it will return and stay
for a long time until the 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
also exhibit this modulated dynamics for one sign of the most relevant perturbation, and
collapse to a black hole for the other.3 Such behavior has been observed earlier, both
in asymptotically
at [23] and AdS spacetime [24, 25]. Also, SMS pro les like the one
in
gure 6 bear some resemblance with the long time modulation of oscillating solutions
that appear in con ning theories after a global quench that injects energy below the mass
gap threshold [26, 27]. At rst sight, the modulations we nd look substantially more
3As explained with care in appendix D, the spectrum of uctuations of a boson star contains a zero mode
which generates the boson star branch (black lines in
gure 4). The next mode is the one that becomes
purely imaginary at the Chandraserkar-like mass. It is perturbing with respect to the unstable mode that
the solution behaves as mentioned above in a highly correlated way with the sign of the coe cient of
the perturbation.
0.2
0.1
-0.1
-0.2
-0.3
t
Re[ϕ(t,π/2)]
0.2
-0.2
-0.4
170
180
190
200
210
220
170
180
190
200
210
220
Re[ϕ(t,0)]
t
Re[ϕ(t,π/2)]
o = 0:3, perturbations along the unstable mode 1 1 with both signs 1 =
destabilize the initial unstable SPS reaching pulsating solutions with regular behavior.
anharmonic and strongly peaked. A Fourier analysis should be carried out in order to reach
a de nite conclusion. Another interesting proposal could relate the SMS's to nonlinearly
dressed multi-oscillator solutions [28]. These still have to be constructed in the driven
situation, but most likely this is feasible.
The unstable SPS's with low enough amplitude (indicatively of the order of o < 0:3)
undergo an exotic evolution. In fact, in these cases the pulsating modulation appears for
both signs of the relevant perturbation 1
. With the \potentially lethal" sign (the one that
usually would drive the SPS towards a black hole) we now observe a pulsating modulation
where the value of o increases instead of decreasing, yet reaching a maximum value and
bouncing back again. The mass and the vev also increase and decrease back and the
solution stays regular forever.
at the origin and the boundary. This highlights an intriguing feature: the modulation
pulse does not only perturb the absolute value of the scalar eld in the bulk, but also its
phase. More precisely, during the pulsation, the two values o(t) at the origin and b(t) at
the boundary start de-phasing in one sense or the other depending upon the sign of 1, and
the scalar pro le becomes increasingly helical. After the pulsation has ended they again
re-phase and the pro le 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 we have found two possibilities, a dynamical black hole and a SMS. Concerning this
last one, we have indeed checked for their robustness by adding a perturbation to them.
Perturbed SMS's develop wiggling plateaux but remain regular all along the simulation,
suggesting that their normal mode spectrum contains only real eigenfrequencies. The very
late time behaviour of the solution is another delicate point of the nonlinear stability
analysis. As far as our codes have run, up to t = O(104), we have not found the slightest
evidence of a nonlinear instability setting in at late times. After the discovery of the
AdS instability [29] this is an issue one should be concerned about. However, the essential
ingredient found in that context, namely full resonance, is very unlikely to be present in the
spectrum of linearized perturbations of a SMS. In the absence of any potential mechanism
for destabilization at long 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
phase space shown in gure 4. It would be interesting to do an accurate scan to see if further
exotic evolutions can be obtained along the wedge of unstable solutions. In particular, we
have not analyzed the fate of unstable solutions in regions II, III. . . This remains for a
later investigation.
3
Quenches with periodic driving
The sourced periodic solutions we have found are eternal, extending from t =
t = +1. Therefore, it makes sense to try to study if they can be constructed by means of
a slowly growing source starting from AdS. In this section we perform this investigation by
considering quenches in a generalized sense. The word quench usually refers to a change
in some coupling constant, which can be either sudden or slow. In the present context, it
will refer to a certain process that interpolates between two di erent periodically driven
Hamiltonians. In short, we shall explore how the system responds when the boundary
data that determine the periodic driving of the dual QFT become, themselves, functions
1 to
of time ( b(t); !b(t)).
3.1
Quasistatic quenches
We analyze the system response to very slow changes of the driving parameters, whose
variation occur on a typical time scale
!b 1. In the context of static quantum systems
the adiabatic theorem states that, for su ciently slow quenches, the ground state of the
system follows the change in the Hamiltonian. Even if a non-vanishing transition amplitude
to an excited state is generated, there is a well de ned way to bring it to arbitrarily small
values (by making
large enough).
Here we
nd that a similar phenomenon occurs,
albeit the underlying unquenched Hamiltonian is time-periodic. The system follows the
slow modulation of 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 (!b; o; b) space.
In this subsection, we demonstrate that linearly stable SPS's can be reached from the global
AdS4 vacuum by means of a su ciently slow quench. Conversely, we also show that these
linearly stable SPS determine which quench processes cannot be regarded as adiabatic. We
employ the following ansatz for the scalar eld source
1
2
(t; =2) =
1
tanh
(t; =2) = ei!bt;
t
+
t
ei!bt;
0
t
t < ;
(3.1)
i.e., the source starts being zero at t = 0, and reaches its nal amplitude
after a time
span . Afterward, it oscillates harmonically. During the whole process, the frequency !b
is kept constant. We refer to the regime taking place at t <
as the build-up phase, while
the driving phase corresponds to t
. The quench pro le is plotted in gure 9, where
b(t)
j (t; =2)j.
1.00
0.99
0.98
0.97
0.96
f
solid yellow, f (x) and (x) of the SPS corresponding to the same data. The curves lie with high
accuracy on top of each other.
Quasistatic quench to a SPS. Imagine that the nal values of
and !b correspond
to a SPS in the region of stability, as given in
gure 4. For concreteness, let us focus on
region I, and consider the lower red dot depicted at b = 0:09 and !b = 2. In this case,
if
is su ciently large, the state obtained after the build-up process is, to an excellent
approximation, the SPS corresponding to the nal harmonic driving. Furthermore, during
the driving phase the numerical solution also remains stationary (up to the largest times
we have simulated and with high accuracy).
Let us choose
= 2500. We have determined numerically that, after the quench (i.e.,
for t >
), the energy density of the geometry corresponds to m = 0:2121, which agrees
with the energy density of the SPS we expect to land on up to the fth signi cant digit. A
more detailed check is provided in gure 10, where we compare the elds
and f during the
driving phase (solid curves) and their pro les in the corresponding SPS (dashed curves).
Having demonstrated that the linearly stable SPS can be reached from the vacuum
by a su ciently slow quench process, it remains to analyze the adiabaticity of the whole
procedure. Speci cally, if the system's response to the time-dependent source is perfectly
|ϕ(t,π /2)|
m
0.20
The curves clearly collapse into a universal pro le (dotted black). We also plot mSPS( b(t= ); !b) in
(solid) yellow. The agreement between both curves implies that the system responds adiabatically.
adiabatic, we expect that
this eld in the SPS at the given instantaneous b and !b. The equality (3.2) thus entails
that the dynamics does not depend explicitly on time, but only implicitly through the
instantaneous value of the source. In other words, in the quasistatic large
limit, one can
draw the evolution as a path on the surface of SPS's (in this case, the vertical dashed line
displayed in gure 4).
Note that these observations should also hold for one-point functions such as m and
hOi: if the response of the system to the quasistatic quench process is perfectly adiabatic,
it must be the case that
m(t) = mSPS( b(t); !b);
and similarly for the scalar vev.
As we illustrate in
gure 11, these expectations are ful lled. Let us look rst at
gure 11a. There, we compare the time evolution of m for four quench processes, with
= 500; 1000; 1500; 2500. Since for the quench pro le (3.1) b(t) only depends on the
dimensionless ratio t= , relation (3.3) implies that for adiabatic response the energy density
can only be a function of t= , but not of t and
separately. Therefore, when plotted in
terms of t= , the four instances of the time evolution of m shown in gure 11a must collapse
to a universal curve. This is indeed what happens, as we illustrate in
gure 11b (dotted
black). Finally, in
gure 11b we also depict mSPS( b(t= ); !b) (solid yellow). The evident
agreement between both curves shows that relation (3.3) is satis ed.
Quasistatic quench to a black hole.
Even in the
! 1 limit, there exists a bound
to the amplitude or the frequency of the driving that one can reach. Consider a quench
at constant !b from global AdS4 to a
nal source amplitude, , such that there is no SPS
associated to !b and . As the source amplitude builds up, and in the quasistatic limit, we
expect that the system evolves through a succession of SPS's, at most up to the time t
(3.2)
(3.3)
3.5
3.0
0.6
0.5
text. From left to right, the quench time span corresponds to
Right: time evolution of the energy density, now plotted against the scaling variable t= . We clearly
observe that, prior to the adiabaticity breaking, m(t= ) approaches a limiting curve as increases.
The vertical line marks the time t = =
when the SPS associated to b
b(t ) ceases to exist.4;5 Past this point, the system exits
the surface of SPS's and adiabatic evolution cannot proceed further, no matter how slowly
the source amplitude increases afterwards.
Let us analyze these questions in a speci c example. We consider the quench described
by equation (3.1), again at driving frequency !b = 2, but increasing now the
nal source
amplitude to
= 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 !b = 2 is
b 2 [0:099200; 0:099225]. According to the quench pro le (3.1), this value is reached at a
time t 2 [0:73482; 0:73581]
which we expect adiabatic evolution to break down.
. Here
is critical value of the scaling variable t= after
In
gure 12a, we plot the time evolution of m for quench pro les with time spans
= 500; 1000; 1500; 2000; 2500; 3000. We clearly see that gravitational collapse takes place
at times t <
. In order to understand adiabaticity and its breakdown, in gure 12b we
plot m(t= ). Two facts are manifest.
The rst one is that, for t=
<
(signaled by the vertical dashed red line), all the
energy density curves merge into a universal pro le. This observation immediately leads
to the conclusion that m(t) is set solely by the instantaneous value of b(t= ) and, as a
consequence, any nontrivial dynamics that depends explicitly on t is highly suppressed in
the quasistatic limit. The system evolves adiabatically, as expected.
The second one is that, for t=
> , the system undergoes gravitational collapse (as
can be seen in the unbounded increase of its energy density). Given that, for t=
> ,
we have that b(t)
b (i.e., there is no SPS associated to the driving), we conclude that
adiabatic evolution breaks immediately after crossing the b(t) = b threshold.
4This point corresponds to the tip of the b = b curve in the (!b; o) plane.
5Note that, strictly speaking, adiabatic evolution can break down for tc
t , t
tc
1; for instance,
tc could be the moment when the intrinsic response time of the system is above the rate of change of the
source amplitude. In this sense, the time t just sets an upper bound on tc, and should be understood in
this sense. On the other hand, in the quasistatic
! 1 limit, we must have that tc ! t .
3.2
Let us address now the non-quasistatic regime, namely that of quenches with
nite time
span
< 1. From the QFT side the standard lore is, according to the adiabatic theorem,
that the system will not keep up with the variation of the source, and will generically
transit to an excited state. If this state is highly excited, it is natural to expect a decohering
evolution towards an (e ective) thermal mixed state.
In order to inspect the nature of this excited state in the dual gravitational theory
we consider a one-parameter family of quench pro les (3.1) with varying
de niteness, we focus on the same !b = 2; = 0:09 case as before. Recall that our ndings
from the previous section established that, in the
! 1 limit, the constructed state was
the stable ground state SPS at o = 0:33 given by the lowest red dot in
gure 4. Instead,
for
< 1, the results of our simulations vindicate the existence of a critical value c
separating two radically di erent regimes.
2 (0; 1). For
For
c, the system always remains regular in the driving regime, and settles
down to a time-periodic geometry. For nite , in addition to the harmonic response
given by ei!bt, it develops an additional periodic modulation. This can be seen in
several one-point functions such as m and jhOij, as illustrated in
gure 13. This
additional modulation is tantamount to the fact that our system is not in the ground
state anymore, in agreement with expectations. As
is lowered two phenomena can
be seen from the plots. On one side, both the amplitude and the periodicity, T , of
the modulation grow. At the same time, the injected mass also increases and ends
up oscillating around larger mean values. This re ects the fact that the quench has
injected more energy and, consequently, the state is more excited. On the other,
periodic modulations become less and less harmonic, and start developing plateaux
where the system stays for progressively longer times as
!
c. This can be observed
!
c
in gure 13 as we move to plots on the lower right part. Interestingly, the mass curve
in the last plot matches very well with the corresponding one in
gure 7. As we
will show in the next section, this is more than a coincidence, and indeed, in the
limit
!
c the end result of the non-quasistatic quench is the SMS to which the
unstable SPS with the prescribed values of (!b; ) will decay when perturbed with
the appropriate sign of the most relevant uctuation. To be most transparent, and
referring to gure 4, the implied unstable SPS is the top red dot sitting at o = 0:56
along the vertical dashed line. In summary, a quench to the same nal boundary
data ( ; !b) yields the stable solution (lower red dot) if performed in a quasistatic
way
critical limit
! 1, and the SMS associated to the unstable SPS (higher red dot) in the
. However, the dashed line 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 the solution space.
In gure 14, we plot the period T of the periodic modulation of the nal solutions as
well as the average mass
hmiT
1 Z t+T
T t
dt m(T )
(3.4)
= 0:09. From top-left to right bottom, the build-up time
decreases as shown in the plots.
T
35
30
25
20
15
10
5
0.4
0.3
0.2
0.1
0.0
as a function of . Both functions increase monotonically with decreasing . For
! 1, hmiT tends to its value on the stable SPS, while T is compatible with the
period of the fundamental normal mode of this SPS.
c, the quench results in an energy injection process strong enough to
trigger gravitational collapse. After a transitory regime both m and jhOij increase
without bound, while the value minx[f (t; x)] drops to zero, signaling the approach
to an apparent horizon. The duration of this transitory regime grows with smaller
j
cj. After the collapse, the harmonic driving keeps injecting energy continuously
into the system, which increases its mass monotonically.n Representative examples
of the behavior just described can be found in gure 15.
processes of the harmonic driving with !b = 2,
= 0:09.
increases from
= 5 to
= 21:222683
from left to right. The non-collapsing simulation corresponds to
= 21:222690 > c
.
A pertinent question that cannot be de nitely answered with our numerical methods
is whether the system reaches an in nite mass state in
nite or in nite eld theory
time.6 A careful analysis of this post-collapse regime should connect with the ndings
in [6]. We have indeed pinned down di erent regimes for the growth of the one-point
functions and refer the interested reader to section 6.
The unstable attractor.
We have clearly demonstrated that, for non-quasistatic
buildup processes, the major property is the existence of the time scale
c, which separates
two di erent phases during the driving regime. These phases are distinguished by the
nal fate of the solution. The new time scale seems to be related to the existence of an
intermediate time attractor, and we mentioned that this attractor was nothing but the
unstable SPS associated to !b and . In the following discussion we provide evidence in
favor of this statement.
For concreteness, let us consider the
= 21:222683 < c case. In gure 16a, we plot
(t0; x), f (t0; x), as obtained from the numerical simulation at t0 = 26:075. In
gure 16b,
these functions are compared with their values on the unstable SPS, SPS(x), fSPS(x),
where we plot the di erences
( )(t; x) = (t; x)
SPS(x);
( f )(t; x) = f (t; x)
fSPS(x);
(3.5)
at t0. Clearly, these di erences are small.
of
and f , de ned by7
In order to gain further understanding, let us compute the time evolution of the norms
( ) =
( f ) =
Z 2
0
Z 2
0
tan2(x)(
tan2(x)( f )2
1
! 2
:
jx= =2)
2
1
! 2
;
(3.6)
(3.7)
6We cannot discard that the growing black hole nally equilibrates into a steady state of nite entropy,
although it seems intuitively unlikely, giving that we are considering a system with an in nite number of
d.o.f. It would be nice to have some sharp physical arguments regarding this point.
7 (t; x) is not normalizable when the source is nonzero; we de ne its norm as the norm of its
normalizable part.
ρ,f
1.0
= 21:222683 at t0 = 26:075. Right: di erence of the functions on the left plot
with respect to their values on the unstable SPS, ( )(t0; x) (blue) and ( f )(t0; x) (red).
0.0002
0.0001
10-2
10-3
10-4
10-5
10
20
30
40
50
60 t
We plot these quantities in gure 17. It is clearly seen that there is a time window in which
the distance between the elds in the driving phase and the unstable SPS is negligible.
In section 4, we employ the identi cation of the intermediate attractor and the unstable
SPS to argue that the results presented in this section can be naturally understood as a
dynamical type I phase transition.
4
Type I phase transitions in driven AdS/CFT
Our results bear some resemblance to critical phenomena in asymptotically at
gravitational collapse. In both cases, there exists an intermediate attractor with one unstable
mode that separates two qualitatively di erent dynamical regimes. In the at case, they
are gravitational collapse versus dispersion to asymptotic in nity; in our case, they are the
ow to the in nite energy phase or the establishment of a regime of persistent, exactly
periodic oscillations in the system.
5.×10-6
2.×10-6
(black), 103 1 (brown) and 105 2 (orange) for out = 5000 at a time t = 600
after the quench-out. The three magnitudes clearly display the hierarchy described in the main
text. Right:
(
1
) (brown) and
( 2 ) (orange) during the whole post-quench-out regime for
out = 5000. We clearly see that the inequality
( 2 )
(
1
) holds.
HJEP04(218)37
scales out = 1000; 2000; 3000; 4000; 5000. In each one of them, the post quench value of
o(t) =
(t; 0) = j (t; 0)j oscillates around o;BS = 0:092828, which corresponds to a BS
frequency !BS = 2:983133, very di erent from the driving frequency, !b = 2. For normal
mode
uctuations around this BS, the
rst nontrivial eigenfrequency is
According to our discussion so far, we expect that, after the quench out, the time evolution
of the (t; x) drawn from the simulation is given by
(t; x) = BS(x) +
1(x) cos( 1t + ) + e(t; x);
(5.8)
where 1(x) is the spatial pro le of the BS rst nontrivial eigenmode, and e(t; x) an error
term that takes into account both linear contributions from higher eigenmodes as well as
possible nonlinear corrections. The parameters
and
are xed by demanding that (5.8)
and its rst time derivative (both without the error term) hold exactly at the origin at
t = tout + out.12
After the source has been turned o , we expect that
(
1
)
1 (i.e., we are close to the
BS solution) and, furthermore, that
( 2 )
(
1
) (i.e., the leading order deviation
with respect to the BS solution is controlled by its rst nontrivial eigenmode). Both of
these expectations hold, as the reader can convince herself by looking at gure 24.
6
The post-collapse regime
Our numerical techniques allow us to follow the system after gravitational collapse has
taken place, at least during some time. In the examples we have analyzed, we have not
12Of course, this choice is not unique. Alternative ones do not modify the conclusions of the main text.
1.5
1.0
t
b = 0:01; !b = 2:375 when perturbed by its fundamental, unstable eigenmode with amplitude
found traces of equilibration to a stationary regime: the harmonic driving always led to a
monotonic increase of the energy density. This increase is responsible for a steady growth
of the apparent horizon, which gets progressively closer to the boundary.13 The way in
which the energy density increases is not universal, and depends on the speci c form of the
harmonic driving.
Let us discuss some examples in detail. The rst one involves a perturbation of the
unstable SPS with b = 0:01; !b = 2:375 by its rst, unstable eigenmode. In
gure 4 this
point is to be located to the right of the upper red dot in region I, on the isocurve with
b = 0:01. As we have illustrated in section 2, choosing the right sign of the perturbation
leads right away to gravitational collapse. Following the post-collapse evolution of the
system, we
nd the results depicted in
gure 25: jhOij stays approximately constant,14
while the energy density increases in an approximately linear fashion, as prescribed by the
di eromorphism Ward identity given in eq. (2.4).
As explained in [6], it is useful to portrait the system trajectory in the ( b; hOi) plane
in order to understand its response to the harmonic driving. We plot the real section of this
complex curve in gure 26a. It is clearly observed that the trajectory remains bounded. In
order to understand the phase alignment of the source and the response, it is convenient
to introduce the nonlinear conductivity
(t)
1 hO(t)i
i!b b(t)
out(t) + i in(t):
(6.1)
We plot the relative contribution of in to the total conductivity in gure 26b. It is clearly
observed that, while small, it typically has a non-negligible value. The boundedness of the
response, together with the fact that it is not in complete phase opposition with respect to
the source, leads to identify the post-collapse evolution of the system as belonging to the
dynamical crossover tilted regime obtained in [6].
The second example of post-collapse evolution we would like to discuss involves a SPS
with b = 0:09; !b = 2, perturbed by its fundamental, unstable eigenmode. In
13Our results indicate, but not demonstrate, that the boundary is not hit in nite time.
14Our results show a small oscillation around a nonzero mean.
1.0
0.8
t
100
|<O>|
5
4
3
2
1
Re(<O>)
t
perturbed by its fundamental, unstable eigenmode with amplitude
= 0:001. We start at t = 30.
Each period of the source has a di erent color, whose wavelength increases the later it starts. Right:
relative value of the imaginary part of nonlinear conductivity . A nonzero value indicates that the
system response is partially in-phase with the driving.
20
40
60
80
20
40
60
80
t
100
process of the form (3.1) with
this is precisely the upper red dot in sector I. Since this solution controls the late-time
dynamics of the non-quasistatic quench processes we discussed in depth in section 3, we
can instead focus on those. For de niteness, we take
= 20. In gure 27, we plot the time
evolution of m and j hOi j, while in gure 28a we depict the phase diagram (along the lines
of gure 26a).
It is clear that this late-time regime is qualitatively di erent from the previous one:
j hOi j does not remain bounded and, as a consequence, m increases faster than linearly.
On the phase portrait, we observe a clear precession of the system's trajectory, which is
accompanied by an increase in its amplitude. This precession is due to the fact that the
response is not entirely out-of-phase with the source, as illustrated by gure 26b. This new
late-time phase is best identi ed with the unbounded ampli cation regime found in [6],15
as the trademark of this regime is the unbounded and partially in-phase response to the
harmonic source.
15In particular, compare our gure 26 with gure 10a in [6].
1.0
0.8
t
100
. Each period of the source has a di erent
color, whose wavelength increases the later it starts. Right: relative value of the imaginary part
of the nonlinear conductivity . A nonzero value indicates that the system response is partially
in-phase with the driving.
1
2
3
4
5
6
7
1
2
3
4
5
6
7
t
process of the form (3.1) with
= 10 4; !b = 20 and
= 1.
We close this section by providing one last example. It corresponds again to a quench
pro le of the form (3.1), this time with frequency !b = 20 , amplitude
= 10 4 and time
span
= 1. As illustrated in gure 29, the energy density grows linearly after the quench,
while the absolute value of the scalar vev remains constant. In accordance with these facts,
the phase diagram of the system corresponds to a closed, non-precessing trajectory of xed
amplitude (see gure 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 gure 30b). The features discussed so far imply right away that the
system
nds itself in the linear response regime.
7
Periodically driven real scalar eld
We now turn the attention to the case of a real scalar eld. Now, unlike the complex case,
the metric of a time-periodic real solution will not be static. This implies a dramatic change
in the formalism and the methodology. Naturally, what makes the problem tractable is
20
-20
-40
1.0
0.8
t
. The di erent periods of time evolution collapse
into a single curve. Right: relative value of the imaginary part of the nonlinear conductivity
. Its
vanishing after the quench indicates that the response is in perfect phase opposition with the source.
time-periodicity itself which, for instance, allows us to replace the linear analysis of normal
modes by the equivalent Floquet analysis. Qualitatively, the nal results we obtain are
surprisingly close to those of the complex case.
Another important di erence 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 be factorized as a complex exponential | cf. eq. (2.3) |, but for the real one the time
periodicity requires an in nite spectrum of Fourier modes given by the following ansatz
1
X
k=0
k(x) cos((2k + 1)!bt):
(7.1)
Only the boundary source remains harmonic, (t; =2) = b cos !bt. As before, !b = 2 =T
is the driving frequency in the boundary gauge ( =2) = 0 (the techniques needed to obtain
the fully nonlinear solutions are reviewed in appendix D). Although we cannot factorize the
radial dependence into a function (x), we will keep the same notation as for the complex
case, i.e., o
(0; 0) and b
spectrum of these solutions.
(0; =2). Figure 31 shows an example of the Fourier
As in the complex case, the space of sourced periodic solutions | SPS | is a surface
embedded in the three dimensional space spanned by coordinates (!b; b; o). In the o ! 0
limit the periodic solutions go to (sourced and unsourced) linear modes of AdS4, while the
limiting, unsourced cases with b = 0 are sometimes termed nonlinear oscillon solutions
in the literature, and were rst constructed in [34] (see also [24] for an extensive account
of results and methods). Figure 32 is the counterpart of gure 2 in the real case. The
similarity is remarkable. The quality of the plot is substantially smaller, as oscillatory
solutions are much more demanding in terms of computational resources than stationary
ones. This implies that going to high values of o is very di cult. Although not being
apparent from
gure 32, the oscillon curves tilt and wiggle in a similar way as the BS
lines do.
1|0ϕ4k(0)|
10-16
10-36
10-56
5
10
15
20
25
k
0 = 0:1.
eld. The surface traverses the nonlinear oscillon plane b = 0 at a set of curves.
This plot
corresponds to the one on
gure 2, to which the similarity is evident. The range exhibited is
considerably smaller in this plot and this is due to the higher technical di culty in computing each
solution individually.
Linear stability.
The study of stability is more involved now than in the complex case.
There, a set of static equations was perturbed, and the linearized spectrum of normal
modes was easy to establish. In the present situation, the linearized
uctuations obey a
time-dependent system of equations. The pertinent tool to employ now is provided by
the Floquet analysis. In our situation we consider linearized
uctuations for the elds of
the form
x(t; x) = xp(t; x) + x~(t; x)
(7.2)
gure 32. Each curve corresponds to a
constant value b (reported explicitly by a small number near the curve). Solutions belonging to
shaded (white) regions are linearly stable (unstable).
where the subindex p stands for \periodic solution" and x~ denotes the perturbations. At
rst order in the perturbations amplitude we obtain a system of the form
x~_ = L(t)x~
(7.3)
(7.4)
where L(t) = L(t + T ) is a time periodic linear integro-di erential operator (in x) built out
of the solution xp. The Floquet theorem establishes the existence of a solution of the form
x(t) = e tP(t)
with
P(t) = P(t + T ) :
The information about the stability resides in i, with i = 1; 2; : : : In the general case,
both
and P will be complex. From equation (7.3), we immediately see that
and P
are also solutions. 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 opposite signs,
. This is ultimately a re ection of the Hamiltonian character of the
dynamical system (7.3). Hence the stability analysis collapses to one of two possible cases,
depending on whether Re( ) is zero or not. The rst case yields a stable (cycle) solution,
and the second one an unstable (saddle point) one.
Computing i requires an exact integration of (7.3) over a full period T . The same
techniques used to obtain the exact nonlinear SPS's can be applied here; the reader can
nd further details in appendix D. The numerical analysis gives the region of stability that
can be observed in gure 33, whose similarity with gure 4 is manifest. The linear analysis
says that this similarity goes beyond the shape, also at boundary of the stability region we
nd the same algebraic structure, either the lowest eigenvalue
1 becoming purely real or
two real eigenvalues fusing and developing imaginary components.
Nonlinear stability.
As for the complex case, analyzing nonlinear stability requires to study the full numerical evolution of the system departing from the perturbed unstable { 33 {
standard SMS with !b = 2:944 and o = 0:15. The blue lling corresponds to the fast oscillations
of the scalar eld (as compared to the pulsating modulation). In second line, we provide an example
of an exotic SMS, where pulsating modulations of both signs alternate. The unstable initial SPS
has, in this case, !b = 2:722 and o = 0:54.
solution. The main conclusions remain similar to the ones obtained in the case of a complex
scalar. The generic end result of an initial unstable SPS is a black hole. However, in the
lower part of the white wedge in sector I unstable solutions decay either to a black hole
or to a sourced modulated solution, SMS, depending on the sign of the single unstable
linear perturbation. As for the complex case, the pulsation is a modulation of the driven
oscillation, with sudden beats separated by a substantially larger plateaux.
Like the Boson Stars, nonlinear oscillons become unstable beyond the turning point
for the mass [24]. At such high values of the eld at the origin, the SMS's are far more
fragile than their complex counterparts and usually decay after a couple of modulating
beats. This may come from the fact that the whole geometry now shakes. Or maybe from
the fact that the harmonic ansatz for the source starts con icting with the anharmonic
response of the eld in the interior and calls itself for a generalized periodic ansatz.
Similarly to the complex case, for su ciently small values of the source, b
10 3, we
have also found exotic real SMS's where the bounce occurs in both directions (i.e, towards
lower or higher masses) when the initial perturbation is very weak. We plot an example in
the second line of gure 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 in the complex case. In order to con rm
this expectation, we consider a time-dependent source given by the real part of eq. (3.1)
(i.e., we replace exp(i!bt) ! cos(!bt)).
In gure 35 we plot m and hOi for the one-parameter family of build-up processes of the
harmonic driving !b = 2:15,
= 0:09. We clearly see that there exists a c that separates
= 0:09. The
of the last build-up process ending in the runaway phase has been marked by
the vertical dashed red line. Right: hOi for the last simulation ending in the runaway phase (solid
orange) and the rst one ending in the non-collapsing phase (solid black).
region corresponds to a highly dense number of fast oscillations compared with the duration of the
process. In the rst plot this region is not appreciated but as the inset shows it is also present.
two di erent late-time phases, as for the complex scalar eld. Furthermore, as
system seems to spend progressively larger time around an intermediate attractor.
! c, the
Dynamical construction of a nonlinear oscillon.
Nonlinear undriven real SPS's
(nonlinear oscillons ) are the equivalent to BS's in the real case. 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, here we will describe a quasistatic
quench connecting the AdS4 vacuum with a nonlinear oscillon across the linearly stable
part of region II. The protocol requires the appropriate tuning of the source parameters,
frequency and amplitude, as a function of time
(t; =2) = b(t) = (t) cos(!b(t)t) :
(7.5)
The speci c pro les for (t) and !b(t) we shall employ are again given by (5.2){(5.3), and
plotted in gure 19.
As a particular example, we consider a quench protocol given by !i = 3:1, !f = 2:9,
m = 10 3 and
= 1500. Figure 36 contains the time evolution of m and hOi along
the process. The main di erence with respect to gure 20 is that now we are plotting
(blue) and
f (red). We compare the end eld con gurations
and f after the quench with two nonlinear oscillons: one with !NLO = 2:9 (left), and another one
with !NLO = 2:89992 (right). The agreement is better in the second case.
the oscillation of real quantitities. The fast oscillations pile up and, due to their high
density, form the blue shaded area visible in the
gure. Now, instead of (3.6){(3.7), we
de ne two new quantities, (
, f ), which measure the distance to the nonlinear oscillon
( NLO; fNLO) built with the target data,
(t) =
f (t) =
Z 2
0
0
Z 2
tan2(x) ( (t; x)
NLO(t; x))2
tan2(x) (f (t; x)
fNLO(t; x))2
1
! 2
1
! 2
(7.6)
(7.7)
In gure 37 we have plotted the time evolution of these two quantities for our particular
example. On the left plot we observe that, after the quench, the geometry is very close to
the oscillon with frequency !NLO = 2:9 but, as also happened in the complex case, even
more to the one with !NLO = 2:89992 (see right plot).
8
Summary and outlook
In this paper we analyzed periodically driven scalar elds on global AdS4. This framework
allows one to study di erent aspects of holographic Floquet dynamics, such as dynamical
phase diagrams and late-time regimes alternative to thermalization.
We constructed zero-temperature solutions subjected to a constant-amplitude periodic
driving of the scalar and dubbed them Sourced Periodic Solutions (SPS). They are dual
to so called Floquet condensates. We have characterized the SPS solution space in detail
and studied throughly both linear and nonlinear stability properties. SPS extend beyond
linearity known perturbative solutions about AdS.
The unstable SPS's feature a rich phenomenology upon time evolution. In
particular, there exist horizonless stable late-time solutions which evade gravitational collapse
and develop instead a pulsating behavior. We named them Sourced Modulated Solutions
(SMS). SMS's are themselves stable solutions where the modulation pulsation impacts in
the vev pro le and the total mass while, remarkably, its frequency is not imposed by that
of the driving.
quench protocols.
behaviours found elsewere [6].
We addressed various types of quench processes concerning both the amplitude and
the frequency of the scalar source.
We focused on both slow and fast quenches. The
rst allow the study of quasistatic processes and showed that they can be used to prepare
SPS's, dual to Floquet condensates, starting from the AdS vacuum. The study of
nonquasistatic quenches uncovered a nice surprise: by suitably
ne-tuning the quench rate,
the system stays for an arbitrary 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
starting from AdS. From the gauge/gravity duality, this is motivated by the exciting possibility
of preparing experimentally sourceless Floquet condensates in strongly coupled systems. In
fact, we nd explicit solutions to this problem using both quasistatic and non-quasistatic
Finally we have studied the post-collapse evolution. We have unravelled the three basic
The analysis has been performed both for the case of complex and real scalars. We
showed that the phenomenology is qualitatively similar despite the technical treatment is
di erent (and considerably more involved in the real case). We mainly focused on massless
scalars, but preliminary results for the massive situation with m2 =
2 point to the
persistence of the same picture.
The complex
eld setup in the boson star con guration can be regarded as a
timelike version of the spatial Q-lattice model [35]. These models have been recently shown
to provide a framework where to study spontaneous symmetry breaking of space
translations [36{38]. This observation gives support to the speculation that relates boson stars
to time crystals. The statement as such could be a ected by subtleties which need to be
carefully studied. It is however interesting to note that the holographic setup could avoid
the known no-go theorems [31, 39]. Indeed, the no-go theorems rely on standard arguments
about the large-volume thermodynamic limit, while holography in global AdS is related to
an alternative thermodynamic limit at nite volume.
The precise origin of the frequency of the pulsating solutions is an interesting open
problem to be addressed. It resembles the possible spontaneous generation of a time scale
observed for similar late-time solutions in the pumping setup [40], although in the present
case the pulsating frequency does depend on the amplitude of the initial perturbation of the
unstable harmonic solution. There is a lower bound to this amplitude set by the numerical
noise, hence we cannot establish whether in the limit of vanishing amplitude the period of
the modulating pulsation diverges or not.
While entering the
nal stages of this work, paper [11] appeared. Albeit in a totally
di erent context, their results bear intriguing resemblance with ours. In fact their gure 3
and our
gure 2 are very similar. Our boson stars and nonlinear oscillons are, in their
language, vector meson Floquet condensates.
However, the physics behind looks very
di erent. In their case the instability is related to a Schwinger pair dielectric breakdown.
Also their process of building a sourceless Floquet condensate di ers from ours. We intend
to investigate further these similarities by going to richer models for a scalar eld in AdS
involving potentials which have a phase transition.
Acknowledgments
We would like to thank Andrea Amoretti, Daniel Arean, Riccardo Argurio, An bal
SierraGarc a, Blaise Gouteraux, Carlos Hoyos, Piotr Bizon, Keiju Murata and Alfonso Ramallo
for pleasant and insightful discussions.
This work of was supported by grants FPA2014-52218-P from Ministerio de
Economia y Competitividad, by Xunta de Galicia ED431C 2017/07, by FEDER and by Grant
Mar a de Maeztu Unit of Excellence MDM-2016-0692. A.S. is happy to acknowledge
support from the International Centre for Theoretical Sciences (ICTS-TIFR), and wants to
express his gratitude to the ICTS community, and especially to the String Theory Group,
for their warm welcome. D.M. thanks the FRont Of pro-Galician Scientists (FROGS) for
unconditional support. A.B. thanks the support of the Spanish program \Ayudas para
contratos predoctorales para la formacion de doctores 2015" associated to FPA2014-52218-P.
This research has bene ted from the use computational resources/services provided by the
Galician Supercomputing Centre (CESGA).
A
Complex periodic solutions
This is by now standard material but we include it here for completeness and in order to
x the notation. The general gravitational action for a complex scalar eld is
S =
2 2
g (R
2 )
+ V (j j)) ;
(A.1)
with 2 = 8 G,
d(d 1)=2l2 for AdSd+1. Note that it is assumed V (0) = 0 otherwise
the constant term of the scalar potential would contribute to the cosmological constant.
The speci c action (2.1) corresponds to taking d = 3 and V (j j) =
eld is complex, the action is invariant under the global U(
1
) transformations
m2
. As the scalar
! e
i .
The equations of motion are
R
1
p
g
R +
g
tand 1 xf e
0
l
2
{ 38 {
The isotropic ansatz for the metric of AdSd+1 can be expressed as follows
with x 2 [0; =2). A useful eld rede nition is given by
f e 2 dt2 + f 1dx2 + sin2 x d 2d 1 ;
ds2 =
l
2
_ =
1
tand 1 x
0
;
(t; x) = 0(t; x) ;
f (t; x)
e (t; x) _(t; x) :
Upon this rede nition, the scalar eld equations of motion can be cast in the form
(A.2)
(A.3)
(A.4)
(A.5)
(A.6)
(A.7)
From the Einstein equations we obtain, after setting 2 = (d
1)=2 as well as l = 1,
f 0 =
0 =
d
2 + 2 sin2 x
sin x cos x j j2 + j j
f )
sin x cos x f j j2 + j j
2
tan x V (j j) ;
In the main text we consider the time-periodic ansatz (2.3) where ! is an angular
frequency and (x) is a real function. We are adhering here to the boundary gauge ( =2) =
0 that makes t equal to the proper time of an observer at the boundary. For any given
! there are static solutions for (x), f (x) and (x) to be obtained from the equations
d
sin x cos x
0
0 +
f cos2 x
= 0 ;
0 + sin x cos x
02 + !
2
= 0 ;
2 e
2
f 2
f )
f 0 + m2 2 tan x = 0 ;
(A.10)
that result from the above-mentioned ansatz.
B
The role of the pumping solution
In a previous work [40], some of the authors analyzed a massless, real scalar eld in AdS4
subjected to a linearly rising pumping source, (t; =2) =
bt. One of the major ndings
was that there exist extremely simple pumping solutions, in which the scalar eld pro le is
at in the radial direction, (t; x) =
bt, while the metric functions f (x); (x) are nontrivial,
but static.
Our objective in this appendix is to comment on the interplay between these pumping
solutions and the four-dimensional SPS's we have found in this paper. From
apparent that the surface of SPS's extends to values !b = 0 on a line b =
gure 2 it is
o
. Naively
these would be related to the mentioned pumping solutions. The correct answer involves
a careful scaling limit.
and set
The scaling limit.
Consider the SPS equations of motion (A.10) (for d = 3; m = 0)
By taking the !b ! 0 limit at xed ^b, these equations reduce to
f 0(x)
(x) = ^b=!b:
(1
f (x))
f (x) 0(x) = 0;
0(x) + cos x sin x
e2 (x) ^2
f (x)2
b = 0;
e (x)
f (x) ^b!b = 0:
(A.8)
(A.9)
(B.1)
(B.2)
(B.3)
(B.4)
10-3
10-2
● ● ●●●●●●●●
0.1
1
ωb
HJEP04(218)37
104 ●
1000
100
10
1
10-2
●
●
1
0.1
10-2
10-3
ωb
10-4 ●
10-4
10-4
10-3
10-2
analytical prediction b;c =
b =!b. Right: jhOcij of the critical SPS's (blue dots), together with the
linear t jhOcij = 1:1543165 : : : !b (red line).
b plane must approach the curve
when !b
The equations are nothing but the equations of motion for the pumping ansatz (as reported
in [40]), provided we identify ^b =
b and
x !b = 0. This simple observation establishes
that the pumping solution controls the SPS's in the limit of in nite source amplitude and
zero source frequency (i.e., on the upper left part of the diagram displayed in
gure 4).
As demonstrated in [40], there is a critical pumping rate b above which no pumping
solution exist. Numerically, b
0:785187. This fact, together with the relation between
the pumping solutions and the SPS's in the scaling limit, leads to the prediction that,
1, the upper boundary of the region containing the linearly stable SPS's in the
!b b =
b
:
(B.5)
Note in particular that, if this expectation is correct, b must diverge as !b ! 0. The
analogous statement !b o =
b also has to hold in this limit, since the di erence between
b and o is also expected to vanish as !b ! 0: the pumping solution is radially at and
has no vev.
In order to con rm these expectations, we have determined numerically the upper
boundary of the stability region in the (!b; b) phase diagram (at small frequency). The
results are shown in
gure 38a. It is clearly seen that, as !b ! 0, this upper boundary lies
progressively closer to the predicted curve b;c =
b =!b.
As a further consistency check, in
gure 38b we establish that jhOcij ! 0 as !b ! 0
| here jhOcij is the vev of the last linearly stable SPS. The numerical data are compatible
with a linear decrease in driving frequency, jhOcij
!b.
C
Study of the normal modes
This appendix provides details on the spectrum of linearized normal modes of the complex
SPS's and BS's presented in the main text.
Linear stability is the
rst important piece of information one gets from the
normal mode spectrum.
A point-wise study of several individual solutions in the plane
2
1.5
1.0
0.5
0.0
-0.5
-1.0
m
0.30
0.25
The red point signals the onset of linear instability which corresponds to the maximum value
attained by the mass along the same branch. Bottom: analogous plots for the s =
:001 sourced
solution. The red point represents the onset of linear instability. Here the two lower eigenfrequencies
meet and develop opposite complex parts. The instability (red dot) occurs before the maximum
value of the mass along the branch is attained (light-blue dot).
(!b; o) leads to establish the stability diagram reported in
gure 4 whose details are
highlighted below.
In gure 4, the solid lines departing from the o = 0 axis at !b = 3 + 2n with n 2 N +
representing the BS branches that dress nonlinearly the AdS oscillons. Moving along the BS
branches by raising 0, one encounters BS's with increasing mass which are stable until the
mass reaches a maximum. This points provides a Chandrasekar-like bound beyond which
the BS's become linearly unstable. This is seen in gure 39 were we plot the evolution of the
squared frequencies of the rst two normal eigenmodes 21;2. Precisely at the Chandrasekar
mass, the second eigenmode 2 becomes purely imaginary.
The second set of plots shows the evolution of the same modes along a line of SPS's
with a small source in sector II (pro les with one node), running closely parallel to the
BS curve. The situation here changes qualitatively, though continuously. The instability
is now triggered by the rst two modes fusing at positive values and developing imaginary
parts of opposite sign. This now happens before the maximum of the mass is reached. The
continuation of this point away from the line of BS deep into sector II draws the upper
limit of the stability region displayed on gure 4 (in gray) on the segment !b 2 (3; 4).
Figure 39 shows also that the lowest normal eigenmode in the uctuation spectrum of
BS's is a zero mode (in blue). This eigenmode corresponds to the zero-frequency
deformation generating the BS branch itself. To verify this statement, one can compare the pro le
0.5
1
1.5
x
along the Boson Stars branch. The plots are rescaled in order to overlap.
ωb
The plots correspond to b =
:001 (left) and
b =
parts have been rescaled by a factor 500 to be visible.
:01 (right). In the left plot the imaginary
of the zero-frequency eigenfunction with the increment of the scalar eld pro le along the
branch (see gure 40).
The stability diagram displayed in gure 4 presents a peculiarity at !b = 4 and very
small
o
: the solutions are unstable, even though surrounded by stable regions. Such
qualitative behavior is repeated for !b = 2n with n
2 but not for !b = 2. A direct look
to the modes shows that the instability here is again triggered by the fusion of the rst
two modes developing opposite imaginary parts (see gure 41).
It should be noted that the \encounter" of two normal modes does not lead always to
a fusion and an instability. For instance, at !b
2 for the SPS's with very small source
the modes do not fuse and do not spoil linear stability.
When two modes encounter they can either cross or repel. Deciding which is the
case could be however numerically demanding, especially at small values of the source b
Let us show this by means of an explicit example. Consider two SPS's corresponding to
b = 0:2 and b = 0:1 respectively. In a region where !b
1 the second and third modes
approach and repel each other. The two modes however get closer as the source is lowered.
Speci cally, the minimal distance between two modes decreases more than linearly with
the source, as illustrated in gure 42.
8
5
0.0
0.2
The repulsion phenomenon departs from the perturbative expectation about the mode
behavior. In fact a perturbative analysis at !b
1 suggests that the eigenfrequencies
n behave as n =
n
!b, where
n are the eigenfrequencies of AdS. This behavior is
maintained also at higher values of !b as the numerics shows, see gure 43. Nevertheless,
the linear behavior of the modes has to break down as two modes approach in order to be
compatible with a repulsion.
D
Real periodic solutions in AdS4
The action is
S =
d x
4 p g (R
2 )
d x
4 p g
+ V ( )
(D.1)
1
2
15.4
15.35
(λ2,3)2
ϕ0 0.1273
Z
where we will take as before 2 = 8 G = 1,
of AdSd+1 can be expressed as follows
ds2 =
l
2
We are interested in time-periodic solutions with harmonic boundary conditions such
that (t; =2) = b cos(!bt). Continuing with the same notation as in the complex case we
6=l2. The isotropic ansatz for the metric
f e 2 dt2 + f 1dx2 + sin2 x d 2d 1 :
(D.2)
will use o = (0; 0) and b = (0; =2). The e ective system of equations is conveniently
expressed in terms of the following variables
F = f e
0(t; x)
_=F + b!b sin !bt:
Working in the boundary gauge, we have that (t; =2) = 0, while (t; =2) = 0 due to
the asymptotic near-boundary expansion. With these de nitions, the equations of
motion become
HJEP04(218)37
_ = (F )0
1
_ =
0 =
F 0 =
sin x cos x
1 + 2 sin x
sin x cos x
b!b sin !bt F 0
tan2 x (tan2 xF )0 + b!b2 cos !bt
2 +
2
F )
2 b!b sin !bt
+ b2!b2 sin2 !bt
(D.3)
(D.4)
(D.6)
The periodicity of the solution (7.1) instructs us to consider the most general Fourier
series expansion. However, the structure of the equations of motion allows for a
truncation to odd modes, which correspond to bifurcations emanating from single normal modes
of AdS,
While
and F are obtained from
and
through (D.5) and (D.6). Following the methods
introduced in [24], we rescale time as
= !bt and use Chebyshev polynomials to expand
the spatial dependence. The collocation grids are associated to these two basis of functions,
Fourier and Chebyshev, =
k(x) cos[(2k + 1)!bt]
k(x) sin[(2k + 1)!bt]:
with
n = 0; 1; : : : ; K
1
cos[ i=(2N + 1)] with i = 0; 1; : : : ; N :
(D.8)
Let us de ne fki
fk(xi). Then, the discretized values are
1
X
k=0
( n; xi) =
ki cos[(2k + 1) n]
( n; xi) =
ki sin[(2k + 1) n]:
Using the boundary conditions discussed previously ( (t; =2) =
(t; =2) = 0), the
values of the elds at x0 are restricted to
k0 =
k0 = 0. Hence, the unknowns to be xed
are !b; b; ki and
ki for k = 0; : : : ; K
1 and i = 1; : : : ; N , while the equations are (D.3)
and (D.4) evaluated at each of these collocation points ( n; xi). This gives 2KN equations
for 2KN + 2 unknowns. We x a numerical value for one of them, being it either !b or
b, and add another equation of motion that sets the value of o
. Finally, the number of
equations and variables match, 2KN + 1, and the discretized system yields an algebraic
nonlinear system of equations that can be solved using a Newton-Raphson algorithm.
n =
xi =
2n
1
2 2K + 1
2
1
X
k=0
F~0 =
2 sin x cos x
~_ = Fp ~ + ( p
b!b sin !bt) ~ ;
e p + F~ ;
b!b sin !bt) F~;
~_ =
1
tan2 x @x htan2 x Fp@x ~ +
~ i
~_ !
= L(t)
~ and F~ can be seen as linear operators acting on ~ and ~ , with this point of view,
equations (D.12) and (D.13) are expressed in the following form
After obtaining time-periodic solutions we are interested in their linear stability.
Consider linearized
uctuations of the form16
(t; x) = p(t; x) + ~(t; x);
p(t; x) + ~ (t; x);
(t; x) = p(t; x) + ~(t; x);
F (t; x) = Fp(t; x) + F~(t; x);
where the subindex p stands for \periodic solution" and elds with a tilde are the
perturbations. Inserting these ansatz into the equations of motion, and at rst order in the
amplitude, we obtain the equations for the perturbations
(D.10)
(D.11)
(D.12)
(D.13)
(D.14)
(D.15)
(D.16)
(D.17)
(D.18)
where L(t) is a linear integro-di erential operator (in x) constructed with the periodic
elds. The periodicity of the background is inherited by the operator, L(t) = L(t + T ), T
being the period of p. For x(t) = ( ~; ~ ), the Floquet theorem establishes the existence of
a solution of the form
x(t) = e tP(t)
with
P(t) = P(t + T ) n:
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
to express P(t) in two parts
P(t; x) = p(
1
)(t; x) + p(2)(t; x) =
~(
1
) !
~ (
1
)
~(2) !
~ (2)
where the time dependence is distributed as follows
~(
1
) =
~(2) =
1
k=0
1
k=0
X ~(k1)(x) cos [(2k + 1)!bt] ;
~ (
1
) =
X ~ (k1)(x) sin [(2k + 1)!bt] ;
X ~(k2)(x) sin [(2k + 1)!bt] ;
~ (2) =
X ~ (k2)(x) cos [(2k + 1)!bt] :
1
k=0
1
k=0
16hereafter we will nd it convenient to work with the rather than .
Inserting this ansatz into (D.17){(D.18) and taking advantage of the linear independence
of the trigonometric functions, equations split in the following form
~(2) = F~(
1
) ( p
~(
1
) = F~(2) ( p
~ (2) =
~ (
1
) =
1
1
tan2 x
tan2 x
b! sin !t) + Fp ~ (
1
)
b! sin !t) + Fp ~ (2)
tan2 xF~(
1
) p + tan2 xFp
tan2 xF~(2) p + tan2 xFp
~_ (
1
);
(D.19)
(D.20)
(D.22)
HJEP04(218)37
where we have de ned F~(i)
F~( ~(i); ~ (i)) and used the time structure of the periodic
elds (D.7). This system of equations has the property that given a solution
; p(
1
); p(2)
another solution exists with
; p(
1
); p(2) , implying that an unstable mode exists if
Re( ) 6= 0. On the other hand, if all
2 iR, the periodic solution is linearly stable.
Solving (D.19) through (D.22) is very similar to obtaining the time-periodic solutions
p and
p themselves. So the same techniques explained in the previous section are in order
here. The di erences lie in the fact that, in this case, the ansatz (D.16) has two more
unknown functions (p(2)) and the role of ! is played by . Namely, we discretize the problem
as in (D.8) and set boundary conditions so as to study perturbations which don't modify
the source ( ~(i)(t; =2) = ~ (i)(t; =2) = 0). Finally, by adding an additional equation to
x the amplitude of ~(
1
)(0; 0) = 1, we obtain a system of 4N K + 1 nonlinear equations for
our 4N K + 1 unknowns, which can be solved using a Newton-Raphson algorithm.
Open Access.
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