\section{Introduction}

The AdS/CFT correspondence \cite{Maldacena:1997re,Gubser:1998bc,Witten:1998qj}
provides a controlled environment to study the physics of a strongly-coupled conformal field theory (CFT)
 on the boundary by dealing with a weakly-coupled gravitational system in the Anti-de Sitter (AdS) bulk.
 In particular,   it gives a very useful semiclassical picture of entanglement
 in strongly-coupled systems \cite{Ryu:2006bv,Hubeny:2007xt,Rangamani:2016dms},
 by relating the entanglement entropy of a subsystem in a CFT
  to the area of an extremal surface   anchored at the boundary
of the  subregion in the AdS dual.  Holographic entanglement entropy is also important to understand black holes: for example,
at finite temperature and in the limit of large subregions,  the entanglement entropy is dominated
 by the thermodynamical contribution,  and the Bekenstein-Hawking entropy \cite{Bekenstein:1973ur,Hawking:1974sw}
is recovered.

 The AdS/CFT correspondence provides several setups  to investigate the thermalisation of
 out of equilibrium systems. Among these, quenches are conceptually simple theoretical settings, describing the evolution of a system triggered
 by a sudden injection of energy or a change of coupling constants.
 In the case of global quenches, the perturbation is spatially homogeneous
 and, in the bulk, it corresponds to the formation of a black hole.
The evolution of the entanglement entropy has been studied
in several examples of quench protocols, both on the CFT
  \cite{Calabrese:2005in} and on the gravity side
   \cite{Hubeny:2007xt,AbajoArrastia:2010yt,Albash:2010mv,Balasubramanian:2010ce,Balasubramanian:2011ur,
   Buchel:2012gw,Buchel:2013lla,Auzzi:2013pca,Buchel:2014gta,Liu:2013iza,Liu:2013qca}.
While the simplest features of the spread of entanglement can be described by a model
with free quasiparticles   \cite{Calabrese:2005in},  a more detailed understanding
needs to take into account  the role of interactions \cite{Casini:2015zua}.

 A common feature of global quenches is that the energy is
 injected in the whole space. Once some perturbation
 in the entanglement entropy is detected, there is no clear way
 to discriminate the point where it originated. Local quenches allow for a more transparent
 analysis of the spread of quantum entanglement, because the
 initial perturbation is localised in a finite region of space.
 Moreover, local quenches can be realised in condensed matter systems
 such as cold atoms \cite{Langen:2013,Meinert:2013}  and they may provide a setup in which
 entanglement entropy might be  measured experimentally  \cite{Klich:2008un,Cardy:2011zz}.

  In CFT, local quenches can be modelled by joining
 two initially decoupled field theories \cite{Calabrese:2007mtj,Calabrese:2016xau}
 and then evolving with a time-translation invariant Hamiltonian.
 Another approach in field theory is to consider excited states
 obtained by acting with local operators on the vacuum \cite{Nozaki:2014hna,Caputa:2014vaa,Nozaki:2014uaa,Asplund:2014coa}.
In holography, the latter kind of  local quenches  can be described
by a free falling particle-like object in AdS \cite{Nozaki:2013wia}.
The topic of local quenches was  studied by many authors
both on the CFT and on the gravity side: for example, mutual information
was considered in \cite{Asplund:2013zba} and finite temperature aspects
were investigated in \cite{Caputa:2014eta,Rangamani:2015agy,David:2016pzn}.
Bulk quantum corrections were recently studied in \cite{Agon:2020fqs}.
Local quenches obtained  by splitting an initial CFT into two disconnected pieces
were considered in \cite{Shimaji:2018czt}.

In principle, the arbitrariness of choosing the falling particle on the  gravity side corresponds to
several choices of local quench protocols on the field theory side.
  In a local quench, it is interesting to understand which aspects of the physics are universal
 and which one depend on the details of the quenching protocol.
 This raises the question of how the choice of the falling particle in AdS
 affects the physics of the local quench of the boundary theory.
 Two natural candidates which are free of singularities
 are black holes (BH) or solitons.   The  black hole case was studied by several authors,
 see  e.g. \cite{Nozaki:2013wia,Jahn:2017xsg,Ageev:2020acl}.
In this paper we will explore the possibility in which the falling particle
is a soliton in AdS$_4$, focusing on the case of the 't Hooft-Polyakov monopole \cite{tHooft:1974kcl}.

 Monopole solutions in global AdS$_4$ have been considered by several authors,
 starting from \cite{Lugo:1999fm,Lugo:1999ai}.
 AdS monopole wall configurations have been studied in  \cite{Bolognesi:2010nb,Sutcliffe:2011sr,Bolognesi:2012pi,Kumar:2020hif}.
Holographic phase transitions for AdS monopoles have
 been investigated in \cite{Lugo:2010qq,Lugo:2010ch,Giordano:2015vsa,Miyashita:2016aqq}.
In this paper,  we will consider the theoretical setting introduced in  \cite{Esposito:2017qpj}.
In this situation, specialising to a multitrace deformation,
 the monopole in global AdS is dual to a theory with spontaneous symmetry breaking.
 A previous study of such a model was performed  numerically.
In this work, we find an approximate analytic solution for the winding-one monopole,
which includes the first-order backreaction on the metric.

Applying the change of variables introduced in   \cite{Horowitz:1999gf}, we map
the time-independent global AdS$_4$ solution
 to a falling monopole configuration in the Poincar\'e patch.
On the field theory side, this is dual to a
local quench in a perturbed CFT, induced by the injection
of a condensate which breaks some of the global  $SU(2)$ symmetries of the theory.
Outside the quench, the global symmetry of the CFT
remains unbroken.

To investigate the field theory dual of the falling monopole,
we  compute the expectation values of various local operators.
Depending on the choice of boundary conditions for the scalar,
 several interpretations are possible on the CFT side.
For Dirichlet or Neumann boundary conditions, the falling monopole
is dual to a CFT deformed by a time-dependent source, which performs a non-zero external
work on the system.
For  a particular choice of the multitrace deformation,
given in eq. (\ref{multitrace-speciale}),
 the monopole is dual to a theory with a time-independent Hamiltonian.
In this case, the expectation value of the energy-momentum
tensor has the same functional form as the one in the background of a  falling black hole  \cite{Nozaki:2013wia}.
In other words, the energy density of the quench is not sensitive to the presence of the condensate.

To further characterise the field theory dual of the falling monopole,
we  perturbatively compute the entanglement entropy for spherical regions.
Let us denote by $\Delta S$ the difference of entropy between the excited state and the vacuum.
We find a rather  different behaviour for $\Delta S$
compared to the case of the falling black hole:
for the monopole quench, $\Delta S$
for a region centered
 at the origin is always negative,
 while in the BH case $\Delta S$ is positive.
The negative sign of $\Delta S$  for the monopole quench
 is consistent with the expectation that the
formation of  bound states, which are responsible for the condensate at the core of the quench,
corresponds to a decrease of the number of degrees of freedom  \cite{Albash:2012pd}.

The paper is organised as follows. In section \ref{sect:monopole-global-AdS}
we consider a static monopole solution in global AdS and we  find an analytical
solution in the regime of small backreaction.
In section \ref{sect:falling-monopole} we apply the change of variables
introduced in \cite{Horowitz:1999gf} to the global AdS monopole.
This trick transforms the global AdS static solution
to a  falling monopole in the Poincar\'e patch, which
provides the holographic dual of the local quench. In section \ref{sect:boundary-interpretation}
 we  compute the expectation value of some local operators, including the energy-momentum tensor.
In section \ref{sect:entanglement-entropy} we study the entanglement entropy for various subsystem geometries.
We conclude in section \ref{sect:conclusions}. Some technical details
are discussed in  appendices.

\section{A static monopole in global AdS }
\label{sect:monopole-global-AdS}

We consider the same theoretical setting as in \cite{Esposito:2017qpj}
which, in global AdS, is dual to  a  boundary theory with a spontaneously-broken $SU(2)$ global symmetry.
The action of the model is:
\begin{equation}
S=\int d^4 x \sqrt{-g} \left[ \frac{1}{16 \pi \, G} \left( R -2 \Lambda  \right) +  \mathcal{L}_M  \right] \, ,
\end{equation}
where $ \mathcal{L}_M$ is the matter lagrangian
\begin{equation}
\mathcal{L}_M =
- \frac14 F_{\mu \nu}^a F^{a \, \mu \nu} - \frac12 D_\mu \phi^a D^\mu \phi^a - \frac{ m_\phi^2}{2}  (\phi^a \phi^a)   \, .
\label{Lmatter}
\end{equation}
We choose  the cosmological constant  and the scalar mass as follows
\begin{equation}
\Lambda=-\frac{3}{L^2} \, , \qquad m^2_\phi=-\frac{2}{L^2} \, ,
\label{Lambda-e-mphi}
\end{equation}
where $L$ is the AdS radius.
In eq. (\ref{Lmatter}),  $F_{\mu\nu}=F_{\mu\nu}^a \frac{\sigma_a}{2}$ denotes
the non-abelian field strength of the $SU(2)$ gauge field $A_\mu=A_\mu^a \frac{\sigma_a}{2}$, i.e.
\begin{equation}
F_{\mu\nu}^a=\partial_\mu  A^a_\nu -  \partial_\nu  A^a_\mu+ e \, \epsilon^{abc} A^b_\mu A^c_\nu \, ,
\end{equation}
with $e$ the Yang-Mills coupling.
The covariant derivative acting on the adjoint scalar is
\begin{equation}
D_\mu \phi_a = \partial_\mu \phi_a  +  e \, \epsilon_{abc} A_\mu^b \phi^c \, .
\end{equation}
The equations of motion  are:
\begin{eqnarray}
&& D^\mu   F^{a}_{\mu \nu}
- e \, \epsilon^{abc} \phi^b D_\nu \phi^c = 0 \, ,
\qquad
 g^{\mu \nu} D_\mu D_\nu \phi^a
- m_\phi^2 \phi^a=0 \, ,
\nonumber \\
&& R_{\mu \nu}- \frac12 R g_{\mu \nu} + \Lambda g_{\mu \nu} =8 \pi G \,  T_{\mu \nu} \, ,
\end{eqnarray}
where $D_\mu$ denotes the combination
of the gravitational and $SU(2)$ gauge covariant derivatives,
and $T_{\mu \nu}$ is the bulk energy-momentum tensor
\begin{equation}
 T_{\mu \nu} =
D_\mu \phi^a  D_\nu \phi^a +F_{a \mu \alpha} F_{a \nu}^{\,\,\,\,\,  \alpha} + g_{\mu \nu} \mathcal{L}_M \, .
\label{bulk_T}
\end{equation}
We first consider the monopole in a global AdS$_4$ background, with metric
\begin{equation}
ds^2= L^2 \left( -(1+r^2) d \tau^2 +\frac{dr^2}{1+r^2} +
 r^2 (d \theta^2+\sin^2 \theta d \varphi^2) \right) \, .
\label{global-AdS-1}
\end{equation}
At large $r$ the field $\phi^a$ has the following expansion
\begin{equation}
\phi^a = \alpha^a \frac{1}{r^{\Delta_1}}+ \beta^a \frac{1}{r^{\Delta_2}} + \dots \, ,
\end{equation}
where $\Delta_{1,2}$ are the dimensions of the sources and vacuum expectation values (VEV)
of the global $SU(2)$  triplet of
operators $\mathcal{O}^a$ which are dual to the scalar triplet $\phi^a$.
For our choice of mass, see eq. (\ref{Lambda-e-mphi}), the dimensions are
\begin{equation}
\Delta_1=1 \, , \qquad \Delta_2=2
\end{equation}
and both the $\alpha^a$ and the $\beta^a$ modes are normalisable.
For this reason, we can choose among different possible boundary interpretations
of the source and VEV\footnote{The subscript in the operator $\mathcal{O}^a$ refers to its dimension.}:
\begin{itemize}
\item the Dirichlet quantisation, where ${\alpha}^a$ corresponds to the source
and ${\beta}^a$ to the VEV
\begin{equation}
J_D^a = {\alpha}^a \, , \qquad   \langle \mathcal{O}_2^a \rangle =  \beta^a \, .
\end{equation}
\item the Neumann quantisation,
 where $-{\beta}^a$ corresponds to the source
and ${\alpha}^a$ to the VEV
\begin{equation}
J_N^a=-  {\beta}^a \, , \qquad     \langle \mathcal{O}_1^a \rangle = {\alpha}^a \, .
\end{equation}
\item
the  multitrace deformation \cite{Witten:2001ua,Berkooz:2002ug,Papadimitriou:2007sj,Faulkner:2010gj,Caldarelli:2016nni},
where  $\langle \mathcal{O}_1^a \rangle = {\alpha}^a$ and
 the boundary dual is deformed by the action term
\begin{equation}
S_{\mathcal{F}}= \int d^3 x \sqrt{-h} \, [ J_{\mathcal{F}}^a \, {\alpha}^a+\mathcal{F}({\alpha}^a) ]\, , \qquad
J_{\mathcal{F}}^a= -{{\beta}}^a - \frac{ \partial \mathcal{F}}{\partial {\alpha}^a}\, ,
\end{equation}
where $\mathcal{F}$ is an arbitrary function.
Imposing $J_{\mathcal{F}}^a=0$, in order to consider an isolated  system, we find the boundary condition
\begin{equation}
{\beta}^a=- \frac{\partial \mathcal{F}}{\partial {\alpha}^a} \, .
\label{zero-source}
\end{equation}
\end{itemize}
If we use either Dirichlet or Neumann quantisation,
there is no non-trivial monopole solution with zero boundary scalar sources.
Multitrace deformations, instead, allow  finding a monopole solution with
a zero boundary source (which satisfies eq. (\ref{zero-source}) for an opportune $\mathcal{F}$),
thus in a situation compatible
 with spontaneous symmetry breaking.

\subsection{Monopole solution in the probe limit}

Let us first consider the zero backreaction limit $G \rightarrow 0$.
The monopole solution can be built by a generalisation of  't Hooft-Polyakov  ansatz in global AdS$_4$
 (see e.g. \cite{Lugo:1999fm,Lugo:1999ai,Esposito:2017qpj}):
\begin{equation}
\phi^a=\frac{1}{L} H(r) n^a \, , \qquad
A^a_l=   F(r) r \,  \epsilon^{a i k} n^k \partial_l \, n^i  \, ,
\label{monopole-ansatz}
\end{equation}
where $x^l=(r,\theta,\varphi)$ and $n^k$ is the unit vector on the sphere $S^2$
\begin{equation}
n^k=(\sin \theta \cos \varphi, \sin \theta \sin \varphi, \cos \theta) \, .
\label{direzione-enne}
\end{equation}
The resulting equations of motion are shown in appendix \ref{Appe-eqs-erre}.
The regularity of the solution at small $r$ requires that
both $H(r)$ and $F(r)$ approach zero linearly in $r$.
On the other hand, at $r \rightarrow \infty$, the choice of boundary conditions
depends on the physics we want to describe on the boundary.
Such a choice is determined in terms of the coefficients $(\alpha_H,\beta_H,\alpha_F,\beta_F)$
specified in the expansion of the scalar and gauge fields nearby the boundary:
\begin{equation}
H(r) = \frac{\alpha_H}{r}+\frac{\beta_H}{r^2} + \dots \, ,
\qquad
F(r)  = \frac{\alpha_F}{r}+\frac{\beta_F}{r^2} + \dots \, .
\label{boundarty-condition-F-H}
\end{equation}
We choose $\alpha_F=0$ in order to describe a theory
with spontaneous breaking of the $SU(2)$ global symmetry, as in  \cite{Esposito:2017qpj}.
Instead, in order to get a background which is different from empty AdS,
we have to look for solutions where
$\alpha_H$ and $\beta_H$ are generically non-vanishing.
Once that $\alpha_H$ is fixed, $\beta_H$ is determined by the requirement
that the solution is smooth.
In appendix \ref{flux-appendix} we compute the monopole magnetic flux,
which is independent of the boundary conditions expressed in eq. (\ref{boundarty-condition-F-H}).

An exact solution to eqs. (\ref{sistema-H-F-senza-backreaction}) can be found at the leading order in $\alpha_H$:
\begin{eqnarray}
H(r) &=& \frac{\alpha_H}{r}  \,  \left[ 1-\frac{\tan^{-1} r}{r}  \right] \, ,
\nonumber \\
F(r) &=& \frac{ e \, \alpha _H^2}{16 r^3}  \,  \left[  \pi ^2 r^2-4 \left(r^2+1\right) (\tan ^{-1} r )^2-\left(\pi
   ^2-4\right) r \tan ^{-1} r  \right] \, .
\label{H-F-analitico}
\end{eqnarray}
Such a solution entails the following coefficients
\begin{equation}
\label{beta-H-analitico}
 \beta_H=-\frac{\pi}{2} \alpha_H \, , \qquad
 \beta_F=e \, \alpha_H^2 \, \frac{12 \pi -\pi^3}{32} \, .
\end{equation}
At higher order in $\alpha_H$, eqs. (\ref{sistema-H-F-senza-backreaction})
can be solved numerically. To this purpose, it is convenient
to use the compact variable $\psi$ defined by
\begin{equation}
r=\tan \psi \, .
\end{equation}
The equations of motion in the variable $\psi$ can be found in appendix \ref{Appe-eqs-psi}.
An example of numerical solution is shown in figure \ref{monopole-profiles-ads}.
A plot of $(\alpha_H,\beta_H)$ for various numerical solutions is shown in figure \ref{alfa-beta}.

\begin{figure}
\begin{center}
\includegraphics[scale=0.52]{monop-ads-H.pdf} \qquad
\includegraphics[scale=0.52]{monop-ads-F.pdf}
\caption{
Numerical solutions for $H(\psi)$ and $F(\psi)$ are shown in black
(the values $e=1$,  $\alpha_H=1$ have been used).
As a comparison, the analytical approximations (\ref{H-F-analitico})
are shown in blue.
}
\label{monopole-profiles-ads}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[scale=0.7]{alfa-beta.pdf}
\caption{ Plot of the relation between the coefficients $\alpha_H$ and $\beta_H$
for different values of $e$. At small $\alpha_H$,  eq. (\ref{beta-H-analitico}) is satisfied.
}
\label{alfa-beta}
\end{center}
\end{figure}

\subsection{Monopole backreaction}

We now introduce the monopole backreaction, modelled by the metric
\begin{equation}
ds^2= L^2 \left( -(1+r^2)  h(r) g(r) d\tau^2 +\frac{h(r)}{g(r)} \frac{dr^2}{1+r^2}
+ r^2 (d \theta^2+\sin^2 \theta d \varphi^2) \right) \, .
\label{monopole-backreaction}
\end{equation}
In order to recover asymptotically global AdS, we impose the following boundary conditions at large $r$
\begin{equation}
\lim_{r \rightarrow \infty} h=\lim_{r \rightarrow \infty} g=1 \, .
\end{equation}
The full set of equations of motion in this background is given in appendix \ref{Appe-eqs-erre}. \\
The asymptotic form of the equations of motion fixes the following large $r$ expansions
\begin{equation}
h(r) = 1 + \frac{h_2}{r^2}+\frac{h_3}{r^3} + O(1/r^4) \, ,
\qquad
g(r) = 1 + \frac{g_2}{r^2}+\frac{g_3}{r^3} + O(1/r^4) \, ,
\label{hg-expansion}
\end{equation}
with
\begin{equation}
g_2=-h_2=\frac{2 \pi G \alpha_H^2  }{L^2} \, , \qquad h_3= -\frac{16 \pi G}{3 L^2}  \alpha_H \beta_H  \, .
\end{equation}
The unfixed parameter  $g_3$ can be found by requiring that the solution is smooth.

At the leading order in $\alpha_H$, the $H(r)$ and $F(r)$ solutions
are still given by eq. (\ref{H-F-analitico}). The leading-order backreaction on the metric
can be solved analytically too, giving:
\begin{equation}
h(r)=1 + \epsilon \, h_\epsilon + O(\epsilon^2) \, , \qquad
g(r)=1 + \epsilon \, g_\epsilon + O(\epsilon^2) \, , \qquad
\epsilon=\frac{\pi \, G \, \alpha_H^2 }{L^2}  \, ,
\label{epsilon-definition}
\end{equation}
where
\begin{eqnarray}
    \label{h-g-analitico}
h_\epsilon &=&   \pi ^2     -\frac{4}{r^2}  - \frac{2}{r^2+1}
-4 \frac{2 \left(r^2-1\right) r \tan^{-1} r +  \left(r^4+1\right) (\tan^{-1} r)^2 }{r^4}  \, , \nonumber \\
\nonumber
g_\epsilon &=&  \pi ^2 + \frac{1}{r^2} -
\frac{ 2 r \tan^{-1} r+3  }{ r^2} \left(1-\frac{2}{1+r^2} \right)   \nonumber \\
&&
 -2  ( \tan^{-1} r  ) \frac{  2 \left(r^4-1\right) \tan^{-1} r +r \left(3 r^2+4\right) }{ r^4}
  \, .
\end{eqnarray}
These solutions set
\begin{equation}
g_3=-\frac{10 \,  \pi^2 \, G \,  \alpha_H^2 }{3 \, L^2}  \, .
\label{g3-analitico}
\end{equation}
The profile functions at higher order in $\alpha_H$ are again accessible by numerically solving the equations of motion. As in the probe limit, it is convenient to introduce the variable $\psi =\tan^{-1} r$, getting the equations of motion shown in appendix \ref{Appe-eqs-psi}.
A comparison between the numerical  and the analytical solutions at small $\alpha_H$
is shown in figure \ref{monopole-profiles-ads-back}.

\begin{figure}
\begin{center}
\includegraphics[scale=0.52]{monop-ads-back-h.pdf} \qquad
\includegraphics[scale=0.52]{monop-ads-back-g.pdf}
\caption{
Numerical solutions for the metric functions $h(\psi)$ and $g(\psi)$
are shown in black (the values $e=1$,  $\alpha_H=1$, $L=1$ and $G=0.1$ have been used).
As a comparison, the analytical approximations (\ref{h-g-analitico})
are shown in blue.
}
\label{monopole-profiles-ads-back}
\end{center}
\end{figure}

\section{A falling monopole in Poincar\'e patch}
\label{sect:falling-monopole}

The gravity dual of local quench in a CFT can be realised by
considering a falling particle in AdS \cite{Nozaki:2013wia}.
To this purpose, a nice trick was introduced in  \cite{Horowitz:1999gf}.
The idea is to start from a spherically symmetric geometry
in global AdS, and to transform it to a time-dependent
Poincar\'e AdS geometry by performing a change of variables.

The Poincar\'e AdS$_4$ metric with coordinates $(t,z,x,{\varphi})$ is
\begin{equation}
ds^2=L^2 \left( \frac{dz^2 -dt^2 +d x^2 +x^2 d {\varphi}^2}{z^2} \right) \, .
\label{poincare-AdS}
\end{equation}
The  metric in eq. (\ref{poincare-AdS}) and the global AdS metric
in eq. (\ref{global-AdS-1}) can be mapped into each other
 via the coordinate transformations
\begin{eqnarray}
\label{A-changing}
 \sqrt{1+ r^2} \cos \tau &=& \frac{A^2+ z^2+ x^2-t^2 }{2  A \,  z} \, , \qquad
 \sqrt{1+ r^2} \sin \tau = \frac{ t}{z} \, ,
\nonumber \\
r \, \sin \theta &=& \frac{ x}{z} \, , \qquad  r \, \cos \theta = \frac{   z^2+x^2-t^2 - A^2 }{2 A \, z} \, ,
\end{eqnarray}
leaving the angular coordinate ${\varphi}$ unchanged. \\
These transformations  can be inverted as follows:
\begin{eqnarray}
\label{A-changing-inverse}
r &=& \frac{\sqrt{A^4  + 2 A^2 (t^2+ x^2-z^2) + (z^2 + x^2 -t^2)^2 } }{2 A \,  z} \, ,
\nonumber \\
\tau &=& \tan^{-1} \left(  \frac{2 A \, t}{ z^2+x^2-t^2 + A^2}  \right) \, ,
\nonumber \\
\theta &=& \tan^{-1} \left( \frac{2 A  \, x}{ z^2+x^2-t^2 - A^2 }  \right) \, .
\end{eqnarray}
The change of variables in eq. (\ref{A-changing}) maps
a configuration with a static particle in the center of global AdS
to a falling particle in the Poincar\'e patch, that can be used to model a local quench.
We will apply this method to the monopole solution we discussed in section
\ref{sect:monopole-global-AdS}.

The holographic quench is symmetric under time reversal $t \to -t$:
for $t<0$ the monopole is approaching the boundary, while for $t>0$
it moves in the direction of the bulk interior.
Physically, we can think of the initial condition at $t=0$ as
the initial out-of-equilibrium state, which can be prepared
in the dual conformal field theory by some
appropriate operator insertion.

The position of the monopole center, corresponding to $r=0$ in global AdS,
in the Poincar\'e patch is time-dependent and follows the curve
\begin{equation}
x=0 \, , \qquad z = \sqrt{t^2 +A^2} \, .
\label{traiettoria-monopolo}
\end{equation}
In the approximation in which the monopole is a pointlike particle,
eq. (\ref{traiettoria-monopolo}) can be interpreted as the trajectory of the monopole.
From the gravity side, the parameter $A$ can be interpreted as the initial position along the $z$-direction
of the free-falling monopole.
From the CFT perspective, the parameter $A$ fixes the size of the local quench.

\subsection{Bulk energy density of the falling monopole}
\label{subsec-energy density and flux}

One may be tempted to imagine the monopole as a pointlike particle
which is falling along the trajectory in eq. (\ref{traiettoria-monopolo}).
To check this intuition, it is natural to consider
the bulk energy-momentum tensor (\ref{bulk_T}).

Working in the limit of negligible monopole backreaction,
we perform the coordinate change in eq. (\ref{A-changing-inverse})
\begin{equation}
x^{\mu} = \left( \tau \, , r \, , \theta \, , \varphi \right) \rightarrow x'^{\mu} = \left( t \, , z \, , x \, , \varphi \right)
\end{equation}
The energy-momentum tensor in Poincar\'e patch is given by
\begin{equation}
T_{\alpha \beta}' \left( x' \right) = \frac{\partial x^{\mu}}{\partial x'^{\alpha}}
 \frac{\partial x^{\nu}}{\partial x'^{\beta}} \, T_{\mu \nu}\left( x \right) \, .
\end{equation}
To properly normalise the energy-momentum tensor, we introduce the vierbein $e^{\mu}_m$ such that
\begin{equation}
T'_{mn} \left( x' \right) = e^\mu_m e^\nu_n \, T'_{\mu \nu} \left( x' \right)  \, ,
\qquad
e^\mu_m \, e^\nu_n  \,  g'_{\mu \nu} = \eta_{mn} \, ,
\end{equation}
where $g'_{\mu \nu}$  and  $\eta_{mn}$  are  the Poincar\'e AdS and the Minkowski
 metric tensors, respectively. In particular, we choose\footnote{In this section,
 the Minkowski indices $m,n$ take the values $0,1,2,3$,
while the curved spacetime indices are $t,z,x,{\varphi}$.}
\begin{equation}
e^\mu_0 = \left( \frac{z}{L} \, , 0 \, , 0 \, , 0 \right) \, , \qquad e^\mu_1 = \left( 0 \, , \frac{z}{L} \, , 0 \, , 0 \right) \, , \qquad e^\mu_2 = \left( 0 \, , 0 \, , \frac{z}{L} \, , 0  \right) \, .
\end{equation}
The energy density as measured in such a orthonormal frame is
\begin{equation}
\rho = T'^{00} = \frac{z^2}{L^2} \, T'_{tt} \, ,
\end{equation}
and the components of the Poynting vector $\vec{s}=(s_z, s_x,s_{\varphi})$ are
\begin{equation}
s_z = T'^{01} = - \frac{z^2}{L^2} \, T'_{tz} \, , \qquad s_x = T'^{02} = - \frac{z^2}{L^2} \, T'_{tx} \, , \qquad s_{\varphi}= 0 	\, .
\end{equation}
In figure \ref{EM00_monopole-plot} and \ref{EMpointing_monopole-plot}
 we show the numerical results for the energy density and the energy flux into the bulk at fixed time.

 \begin{figure}
\begin{center}
\includegraphics[scale=0.45]{en_density_t0}
\qquad
\includegraphics[scale=0.45]{en_density_t3}
\qquad
\includegraphics[scale=0.45]{en_density_t5}
\qquad
\includegraphics[scale=0.45]{en_density_t7}
\caption{Contour lines of constant energy density for fixed time.
The monopole center is represented by the black spot.
The numerical values $A=1$ and $L=1$ have been chosen. }
\label{EM00_monopole-plot}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[scale=0.45]{pointing_TOT}
\caption{Direction of the bulk Poynting vector for fixed time.
The  numerical values $A=1$ and $L=1$ have been chosen.}
\label{EMpointing_monopole-plot}
\end{center}
\end{figure}

  The pictures clearly illustrate how the energy density,
  initially localised near the AdS boundary, spreads into the bulk.
The energy distribution resembles a pointlike particle only at early times,
while at late times the energy is spread  along a spherical wavefront.
At $t=0$ all the components of the Poynting vector vanish,
implying that there is no energy flux at initial time.

\section{Expectation values of local operators}
\label{sect:boundary-interpretation}

To understand the physical interpretation on the boundary CFT,
it is useful to study the expectation values of some important local
operators. In particular, in this section we will focus on expectation values
of scalar operators, global $SU(2)$ currents and energy-momentum tensor.
The details depend on the boundary conditions chosen
for the scalar field $\phi^a$. In general, the boundary energy-momentum tensor $T_{mn}$
is not conserved because the external sources perform work on the system.
With a particular choice of multitrace deformation, see eq. (\ref{multitrace-speciale}),
the system is isolated and $T_{mn}$ is conserved.

\subsection{Boundary conditions for the scalar}

In the AdS/CFT correspondence, the asymptotic behaviour of the bulk fields
is dual to the source and expectation values of operators in the CFT.
For this reason, we will focus on asymptotics of the scalar field $\phi^a$
nearby the boundary.
In global AdS, the direction $n^a$ of $\phi^a$ in the internal $SU(2)$ space is
given by eq. (\ref{direzione-enne}).
In Poincar\'e patch, by performing the coordinate transformation in eq. (\ref{A-changing-inverse})
we find that, nearby the boundary at $z=0$,
\begin{equation}
n^a=\frac{1}{\omega^{1/2}} \, \left(-2 A x \cos (\varphi ),-2 A x \sin (\varphi ),A^2+t^2-x^2\right)
+O(z^2)  \, ,
\label{enne-poincare}
\end{equation}
where, for convenience, we introduce the  quantity $\omega(x,t)$
that appears in many subsequent expressions
\begin{equation}
\omega(x,t)=A^4+2 A^2 \left(t^2+x^2\right)+\left(t^2-x^2\right)^2 \, .
\label{omega}
\end{equation}

The core of the quench can be thought of as localised at
\begin{equation}
x=\sqrt{t^2 + A^2} \, ,
\label{cono-luce-largo}
\end{equation}
which, at large $t$, coincides with good approximation with the lightcone of the origin $x=t$.
For the value in eq. (\ref{cono-luce-largo}),
 the adjoint scalar field points in the direction $n=n^a \sigma^a$ given by
\begin{equation}
n=- (\sigma_1 \cos {\varphi} + \sigma_2 \sin {\varphi}) \, .
\end{equation}
The scalar points along the $\sigma_3$ direction inside the lightcone,
and along the $-\sigma_3$ outside the lightcone, see figure \ref{direction-fig}.
As we will see later, at large $t$, the absolute value of the
scalar field is peaked on $x$ given by eq. (\ref{cono-luce-largo}),
and is almost zero both inside and outside the lightcone.

 The configuration at $t=0$ resembles a baby skyrmion,
 with a field pointing along $\sigma_3$ in the core and along $-\sigma_3$ far away
 (actually, it is not a skyrmion because the VEV tends to zero at infinity).
 As time increases, this configuration expands along the lightcone.
 At large time we end up with two region of vacuum
 (inside and outside lightcone) separated by an expanding
 shell of energy.

\begin{figure}
\begin{center}
\includegraphics[scale=0.52]{scalar-direction.pdf}
\caption{ Value of $n^3$ as a function of $(t,x)$ for $A=1$.
Negative values of the radial cylindrical coordinate $x$ correspond to $\varphi \to - \varphi$.
}
\label{direction-fig}
\end{center}
\end{figure}

In order to extract the sources and the expectation values of the local operator
triggering the quench, it is useful to expand the change of variables in eq. (\ref{A-changing-inverse})
nearby the boundary.
The global AdS radial coordinate reads
\begin{equation}
r=\frac{a}{z} +O(z)\, , \qquad
a= \frac{ \omega^{1/2 }}{2 A} \, .
   \label{erre-vs-zeta}
\end{equation}
By means of eq. (\ref{erre-vs-zeta}),
we obtain the boundary expansion of $H(r)$
\begin{equation}
H=\frac{\alpha_H}{r} + \frac{\beta_H}{r^2}+O(r^{-3})  =
\tilde{\alpha}_H z + \tilde{\beta}_H z^2 + O(z^3)  \, ,
\label{H-poincare}
\end{equation}
where
\begin{equation}
   \tilde{\alpha}_H =   \frac{\alpha_H}{a} =    \alpha_H \,   \frac{2 A}{\omega^{1/2}} \, ,
     \qquad
\tilde{\beta}_H =  \frac{\beta_H}{a^2}    = \beta_H \,  \frac{4 A^2}{ \omega} \, .
   \label{alfa-e-beta-tilde}
\end{equation}
A plot of $ \tilde{\alpha}_H $  and $ \tilde{\beta}_H $ is shown in figure \ref{alpha-beta-tilda}.
It is interesting to note that
\begin{equation}
\frac{\tilde{\beta}_H}{\tilde{\alpha}_H^2}=\frac{\beta_H}{\alpha_H^2} = \kappa \, ,
\label{alfa-e-beta-tilde-rel}
\end{equation}
where $\kappa$ is a constant. In the limit of small backreaction, from eq. (\ref{beta-H-analitico})
we find
\begin{equation}
\kappa=-\frac{\pi}{ 2 \alpha_H} \, .
\end{equation}

\begin{figure}
\begin{center}
\includegraphics[scale=0.52]{alpha-tilde-H.pdf}  \qquad
\includegraphics[scale=0.52]{beta-tilde-H.pdf}
\caption{ The quantities ${\tilde{\alpha}}_H$ (left) and  ${\tilde{\beta}}_H$ (right) as a function of $(t,x)$.
Here we set $A=1$, $\alpha_H=1$ and we use the relations in eq. (\ref{beta-H-analitico}),
 valid for small backreaction.
}
\label{alpha-beta-tilda}
\end{center}
\end{figure}

Combining eqs. (\ref{enne-poincare})
and (\ref{H-poincare}),
 the  expansion of $\phi^a$ nearby the Poincar\'e patch boundary  is
\begin{equation}
\phi^a= \frac{H(z)}{L} n^a =\frac{1}{L} \left( {\tilde{\alpha}}^a \, z + {\tilde{\beta}}^a \, z^2 + O(z^3) \right) \, , \qquad
{\tilde{\alpha}}^a=n^a \tilde{\alpha}_H \, , \qquad
{\tilde{\beta}}^a=n^a \tilde{\beta}_H \, .
\end{equation}
 As for the global AdS case, we can consider several quantisations for the scalar field $\phi^a$:
\begin{itemize}
\item the Dirichlet condition, where $\tilde{\alpha}_H$ corresponds to the source
and $\tilde{\beta}_H$ to the VEV
\begin{equation}
J_D^a  =  \tilde{\alpha}^a \, , \qquad   \langle \mathcal{O}_2^a  \rangle = \tilde{\beta}^a  \, .
\end{equation}
\item the Neumann condition,  where $-\tilde{\beta}_H$ corresponds to the source
and $\tilde{\alpha}_H$ to the VEV
\begin{equation}
J_N^a= -  \tilde{\beta}^a \,  , \qquad     \langle \mathcal{O}_1^a  \rangle = \tilde{\alpha}^a  \, .
\end{equation}
\item
the  multitrace deformation, where
 the boundary dual is deformed by the action term
\begin{equation}
S_{\mathcal{F}}= \int d^3 x \sqrt{-h}  \, [ J_{\mathcal{F}}^a \, \tilde{\alpha}^a+\mathcal{F}(\tilde{\alpha}^a) ]\, , \qquad
J_{\mathcal{F}}^a= -{\tilde{\beta}}^a - \frac{ \partial \mathcal{F}}{\partial \tilde{\alpha}^a}\, ,
\end{equation}
and  $\langle \mathcal{O}_1^a \rangle = \tilde{\alpha}^a$.
\end{itemize}

All these boundary conditions correspond in general
to a monopole in presence of external time-dependent sources.
Among such possible choices of boundary conditions, it is
interesting to consider the multitrace deformation with
\begin{equation}
\mathcal{F}_\kappa(\tilde{\alpha}^a) = -\frac{\kappa}{3} \left( \tilde{\alpha}^a \tilde{\alpha}^a \right)^{3/2}
=-\frac{\kappa}{3}  \tilde{\alpha}_H^3 \, .
\label{multitrace-speciale}
\end{equation}
In this case, the monopole is a solution with a vanishing source, because it satisfies
\begin{equation}
\tilde{\beta}^a=- \frac{\partial \mathcal{F}}{\partial \tilde{\alpha}^a} \, ,
\label{zero-source-poincare}
\end{equation}
as can be checked from eq. (\ref{alfa-e-beta-tilde-rel}).

\subsection{The boundary global currents}

Our monopole ansatz in global AdS is given by eq.
(\ref{monopole-ansatz}), with boundary conditions in
eq. (\ref{boundarty-condition-F-H}) and with $\alpha_F=0$.
As a consequence, we deduce that in Poincar\'e patch the gauge field
$A^a_\mu$ vanishes at the boundary $z=0$.
In other words, if the sources for the global symmetries are set to zero in global AdS,
they also vanish after the change of coordinates leading to the Poincar\'e patch.

From the order $z$ terms in the boundary expansion of $A^a_\mu$  we can extract the expectation value of the three currents $J^a_l$
\begin{eqnarray}
\langle J^1_l \rangle&=& \frac{8 A^2 \beta _F}{\omega^{3/2}}
 \left( t x \sin (\varphi ),-\frac{1}{2} \sin (\varphi )
   \left(A^2+t^2+x^2\right),-\frac{1}{2} x \cos (\varphi )
   \left(A^2+t^2-x^2\right) \right) \, , \nonumber \\
   %
\langle J^2_l \rangle &=& \frac{8 A^2 \beta _F}{\omega^{3/2}}
\left( -t x \cos (\varphi ),\frac{1}{2} \cos (\varphi )
   \left(A^2+t^2+x^2\right),-\frac{1}{2} x \sin (\varphi )
   \left(A^2+t^2-x^2\right)\right) \,  ,\nonumber \\
\langle J^3_l \rangle &=& \frac{8 A^2 \beta _F}{\omega^{3/2}}
\left( 0,0,-A x^2 \right) \, ,
\end{eqnarray}
where $x^l=(t,x,\varphi)$ are the boundary spacetime coordinates,
and $a=1,2,3$ is the $SU(2)$ global index.
Plots of the charge density $J^2_t$ is shown in figure \ref{density-fig}.

\begin{figure}
\begin{center}
\includegraphics[scale=0.52]{density.pdf}
\caption{Charge density of the second component of isospin  $J^2_t$ as a function of $(t,x)$
for ${\varphi}=0$ (positive $x$) and ${\varphi}= \pi$ (negative $x$).
 We set $A=1$, $\alpha_H=1$ and $e=1$ and we use the relations in eq. (\ref{beta-H-analitico}),
 valid for small backreaction.
The peak and the pit correspond to positive and negative sign global charges, which are taken apart
from each other by the quench.
}
\label{density-fig}
\end{center}
\end{figure}

It is interesting to compare the direction in the $SU(2)$ space
of the current expectation value with the direction of the scalar expectation value $n^a$.
We find that the expectation value of the global current is always orthogonal to the direction of the scalar
expectation value  in the isospin space
\begin{equation}
\langle J^a_m \rangle  \, n^a = 0 \, .
\end{equation}
 So the quench breaks
  all the three global symmetry group generators.
 This is true just on top of the "lightcone" in eq. (\ref{cono-luce-largo});
inside and  outside this surface
both the scalar and the current expectation values tend to zero
and  the $SU(2)$ global symmetry of the boundary theory is unbroken.

 \subsection{The boundary energy-momentum tensor}

  For illustrative purposes, we will compare the result with the one obtained for
 a quench modelled by a falling black hole, studied in \cite{Nozaki:2013wia}.
 In this case the metric is
\begin{equation}
ds^2= L^2 \left( -\left( 1+r^2-\frac{M}{r} \right)  d\tau^2 +\frac{dr^2}{1+r^2-\frac{M}{r}}
+ r^2 (d \theta^2+\sin^2 \theta d \varphi^2) \right) \, ,
\label{metric-BH}
\end{equation}
where $M$ is a dimensionless parameter proportional to the black hole mass.
In order to find the quench background, we apply the change of variables in eq. (\ref{A-changing}).
 Then, to extract the energy-momentum tensor with the method in  \cite{Balasubramanian:1999re},
  it is convenient to pass to Fefferman-Graham (FG) coordinates.  Details of the calculation are
   in appendix \ref{appe-energy-momentum-tensor:BH}, where
 expressions for all the components of the energy-momentum tensor can also be found, see eq. (\ref{T-cono-luce-BH}).
 In particular, the energy density is
 \begin{equation}
 T_{tt}^{(BH)}=\frac{A^3 L^2  M }{\pi  \, G } \,\,
 \frac{ \omega + 6 t^2 x^2 } { \omega^{5/2}} \, .
 \end{equation}
The energy-momentum tensor is conserved and traceless, and
the total energy is
\begin{equation}
\mathcal{E}^{(BH)}= \frac{ M \, L^2 }{  2 \,  G   A } \, .
\end{equation}

 In the falling monopole case a more accurate discussion is needed, since
  the boundary  energy-momentum tensor depends on the details of the boundary conditions of bulk fields.
 We first focus on  Dirichlet boundary conditions  \cite{deHaro:2000vlm}.
  Starting  from the backreacted metric in  eq. (\ref{monopole-backreaction}),
we apply the change of variables in eq. (\ref{A-changing}).
The intermediate metric expression is a bit cumbersome, so in appendix \ref{appe-energy-momentum-tensor:D} we just specify the coordinates expansion that puts it in the FG form. All the non-vanishing components of the energy-momentum tensor $T_{mn}^{(D)}$ obtained from such a metric are given in eq. (\ref{T-cono-luce-D}).
The energy density is:
\begin{eqnarray}
T_{tt}^{(D)}&=&\frac{A^3 }{3 \pi  G   \, \omega^{5/2}}
   \left[ 48 \pi  G \alpha _H \beta _H  \, x^2 t^2
+  (8 \pi  G \alpha _H \beta _H-3    g_3 L^2  )  (\omega + 6 t^2 x^2)
\right]  \nonumber \\
&=&
 2 \pi  \alpha_H^2   A^3  \, \frac{   \omega + 2 t^2 x^2    }{       \omega^{5/2}}  \, ,
\end{eqnarray}
where in the second line we have inserted the analytic approximations for small backreaction  in eqs. (\ref{beta-H-analitico})
and (\ref{g3-analitico}).
In this limit, the total energy is
\begin{equation}
\mathcal{E}^{(D)}=   \frac{ \pi^2  \alpha_H^2   }{A} \left( 1- \frac{ 2}{3 }  \frac{t^2}{A^2 +t^2}\right) \, .
\end{equation}
The energy is a decreasing function of time, meaning that the Dirichlet boundary
conditions absorb energy from the bulk.
The non-conservation of energy motivates the investigation of a different quantisation.

A changing of the quantisation conditions causes a
shift of $T_{mn}^{(D)}$ by finite parts (see e.g.  \cite{Caldarelli:2016nni}).
Here we specialise to a class of multitrace deformations that do not break the $SU(2)$
global symmetry.
Assuming that
\begin{equation}
\tilde{\alpha}^a= n^a \tilde{\alpha}_H \, , \qquad
\tilde{\beta}^a= n^a \tilde{\beta}_H \, ,
\end{equation}
which is true for the monopole, we can
write the source as
\begin{equation}
J_{\mathcal{F}}^a = n^a \, J_{\mathcal{F}} \, .
\end{equation}
As a function $\mathcal{F}$ parameterising the multitrace deformation  we choose
\begin{equation}
\mathcal{F}(\tilde{\alpha}^a)=\mathcal{F}(\tilde{\alpha}^a \tilde{\alpha}^a)=\mathcal{F}(\alpha_H) \, .
\end{equation}
The current can be written in terms of $\tilde{\alpha}_H,\tilde{\beta}_H$ as follows
\begin{equation}
J_{\mathcal{F}}=-\tilde{\beta}_H - \mathcal{F}'(\tilde{\alpha}_H) \, .
\end{equation}
The energy-momentum tensor, (see appendix \ref{appe-energy-momentum-tensor:multitrace} for further details) is
\begin{equation}
T^{(\mathcal{F})}_{ij}= T_{ij}^{(D)}+\eta_{ij} [\mathcal{F}({\tilde{\alpha}}_H) -{\tilde{\alpha}}_H {\tilde{\beta}}_H - \mathcal{F}'({\tilde{\alpha}}_H) {\tilde{\alpha}}_H ] \, .
\end{equation}

Note that this result also applies to the Neumann conditions, that can be seen as a multitrace deformation with $\mathcal{F}=0$.
If we instead specialise to $\mathcal{F}=\mathcal{F}_\kappa$, see eq.  (\ref{multitrace-speciale}),
 the external source is zero and the energy-momentum tensor is conserved.
 Moreover, an explicit computation reveals that the energy-momentum tensor
has the same functional form as the one for the falling BH:
\begin{equation}
T^{(\kappa)}_{ij}=  \frac{16 \pi G \alpha_H \beta_H-3 L^2 g_3}{3 L^2 M}   \, T_{ij}^{(BH)}
\,  .
\label{Tkappa-monopole}
\end{equation}
Using the analytic values for small backreaction of $\beta_H$ and $g_3$ in
eqs. (\ref{beta-H-analitico}) and (\ref{g3-analitico}), we find
\begin{equation}
T^{(\kappa)}_{ij}=\frac{2  \pi^2  G \, \alpha_H^2}{3 L^2 M} \, T_{ij}^{(BH)}  \, , \qquad
T_{tt}^{(\kappa)} =
\frac{2  \pi   \alpha_H^2 A^3}{3  } \,
 \frac{ \omega + 6 t^2 x^2 } { \omega^{5/2}} \, .
\end{equation}
The total energy is
\begin{equation}
\mathcal{E}^{(\kappa)}=\frac{  \pi^2   \alpha_H^2 }{3 A  } \, .
\end{equation}
As apparent from eq. (\ref{Tkappa-monopole}), the energy-momentum tensor
is not a probe enough precise to distinguish between a falling monopole or a falling
black hole in the bulk. In the next section, we will see that the entanglement entropy
of these two falling objects behaves instead in a radically different way.

\section{Holographic entanglement entropy}
\label{sect:entanglement-entropy}

In this section we will study the effect of
the  leading-order backreaction on holographic entanglement entropy.
It is useful to consider $\epsilon$ as defined in eq. (\ref{epsilon-definition}) as
an expansion parameter. In the asymptotically global AdS case, the metric at the leading order in $\epsilon$ is
\begin{equation}
ds^2= L^2 \left( -(1+r^2) \left[ 1+ \epsilon \, (h_\epsilon +  g_\epsilon) \right] d\tau^2 + \left[ 1+ \epsilon \, (h_\epsilon - g_\epsilon )  \right]  \frac{dr^2}{1+r^2}
+ r^2 (d \theta^2+\sin^2 \theta d \varphi^2) \right) \, ,
\label{monopole-backreaction-epsilon}
\end{equation}
where $h_\epsilon$ and $g_\epsilon$ are given by eq. (\ref{h-g-analitico}).
We will be interested in the evolution of entanglement entropy for the quench
in Poincar\'e patch, so we  apply the change of variables in eq. (\ref{A-changing-inverse}),
obtaining a time-dependent background.

The metric tensor can be written as follows
\begin{equation}
g_{\mu \nu}=g_{\mu \nu}^{(0)} + \epsilon \, g_{\mu \nu}^{(1)} + O(\epsilon^2) \, , \qquad \epsilon=\frac{\pi G \, \alpha_H^2 }{L^2} \, .
\end{equation}
Given a codimension-two surface $x^{\mu} \left( y^\alpha \right)$ parameterised with coordinates $y^\alpha = \left( y^1, y^2 \right) $, the induced metric is
\begin{equation}
G_{\alpha \beta} = \frac{\partial x^\mu}{\partial y^\alpha}  \frac{\partial x^\nu}{\partial y^\beta} g_{\mu \nu} \,  .
\end{equation}
Such an induced metric
can also be expanded as a power series in $\epsilon$
\begin{equation}
\label{Gk}
G_{\alpha \beta}=G_{\alpha \beta}^{(0)} + \epsilon \, G_{\alpha \beta}^{(1)} + O(\epsilon^2) \, , \qquad
G_{\alpha \beta}^{(k)} = \frac{\partial x^\mu}{\partial y^\alpha}  \frac{\partial x^\nu}{\partial y^\beta} g_{\mu \nu}^{(k)} \, ,  \qquad k=0,1 \, .
\end{equation}
We can compute the change of area of the Ryu-Takayanagi (RT) surface at the leading order in $\epsilon$, as in  \cite{Nozaki:2013wia}.
To this purpose, we can expand the determinant of the metric in the area functional.
The first order term of this expansion is
\begin{equation}
\Delta \mathcal{A}=\frac{\epsilon }{2} \int d^2 y \,   \sqrt{G^{(0)}} \, {\rm Tr}  \left[ G^{(1)} (G^{(0)})^{-1} \right] \, .
\label{Delta-area}
\end{equation}
It is important to note that, at first order, it is enough to work with the unperturbed RT surface $x^{\mu} \left( y^\alpha \right)$, which simplifies the computation a lot.
The difference in entropy between the excited state and the vacuum at the leading order
 is proportional to eq. (\ref{Delta-area})
\begin{equation}
\Delta S = \frac{\Delta \mathcal{A}}{4 G} \, .
\end{equation}
We will apply this procedure to various examples of subregions.

\subsection{Disk centered at the origin}

We take as a boundary subregion a disk of radius $l$ centered at $x=0$ and
lying at constant time $t$. The RT surface in unperturbed Poincar\'e patch of AdS$_4$ is  the half sphere
\begin{equation}
z=\sqrt{l^2-x^2} \, .
\label{minimal-surface}
\end{equation}
From eqs. (\ref{Delta-area}) and (\ref{monopole-backreaction-epsilon}) we obtain
\begin{equation}
\label{entropy-centrata}
\Delta S(l,t) = \frac{\pi^2 \, \alpha_H^2}{4} \frac{1}{l} \int_0^l
 \frac{   \left(h_{\epsilon }- g_{\epsilon } \right)  x^3    }
   { \left(l^2-x^2\right)^{3/2}} \,
     \frac{\omega(l,t) }{    \left(A^2-l^2+t^2\right)^2+4 A^2 x^2 }
    \, dx  \, ,
\end{equation}
where $\omega$ is defined in eq. (\ref{omega}). The functions $h_{\varepsilon}$ and $g_{\varepsilon}$ depend on the variable $r$, which
on top of the RT surface  reads
\begin{equation}
\label{erre-RT-centrata}
r= \frac{ \sqrt{\left(A^2-l^2+t^2\right)^2+4 A^2 x^2 } }{2 A \sqrt{l^2-x^2} } \, .

\end{equation}

For the entropy, the $A$ dependence can be completely reabsorbed
by the following rescaling of the quantities $l$, $x$ and  $t$
\begin{equation}
l \to \frac{l}{A} \, , \qquad  x \to \frac{x}{A} \, , \qquad  t \to \frac{t}{A} \, .
\label{rescale-1}
\end{equation}
For this reason, the numerical analysis has been performed  for $A=1$ without loss of generality.
Numerical results are shown in figure \ref{entropy-plot}.
We find that  $\Delta S$ is always negative,
meaning that the perturbed entanglement entropy is always smaller than the vacuum value.
We can think of the quench  as a region of
spacetime where a condensate (which breaks a global symmetry on the boundary, as in holographic
superconductors \cite{Hartnoll:2008vx}) is localised. A lower entropy
fits with the intuition that some degrees of freedom have condensed \cite{Albash:2012pd}
and so there should be fewer of them compared to the vacuum
(which has zero scalar expectation value).

\begin{figure}
\begin{center}
\includegraphics[scale=0.45]{entropy-time-dep}
 \qquad
\includegraphics[scale=0.45]{entropy-l-dep}
\caption{
Left: time dependence of $\Delta S$ for a spherical subregion with fixed radius $l$  centered at the origin of the quench.
 Right: dependence of    $\Delta S$ at fixed $t$ as a function of $l$. The numerical values $\alpha_H=1$, $A=1$ are used.
 }
\label{entropy-plot}
\end{center}
\end{figure}

Analytic results can be found in some regimes.
Nearby the boundary $r \rightarrow + \infty$, we can use the expansion
\begin{equation}
h_{\epsilon } = -\frac{2}{r^2} + \dots \, , \qquad
g_{\epsilon } = \frac{2}{r^2}  + \dots \, .
\label{expansion-h-and-g}
\end{equation}
Since the minimal $r$ on the RT surface is given by eq. (\ref{erre-RT-centrata})
with $x=0$, this expansion will be valid in the whole integration region in eq. (\ref{entropy-centrata}) in the regime
\begin{equation}
{| A^2 + t^2 -l^2 |}\gg 2  l  A\, .
\label{regime-approssimato}
\end{equation}
Equation  (\ref{entropy-centrata}) can then be evaluated explicitly
\begin{equation}
\Delta S =  -  \pi^2 \, \alpha_H^2 \,  \left[ \left( \frac{A l}{ \sqrt{\omega(l,t)}} +  \frac{ \sqrt{\omega(l,t)}}{4 A l}\right)
\tanh^{-1} \left( \frac{2 A l}{\sqrt{\omega(l,t)}} \right) -\frac12
\right] \, .
\end{equation}
We can specialise the approximation in eq. (\ref{regime-approssimato})
to the following situations:
\begin{itemize}
\item
small $l \ll A$
 \begin{equation}
  \Delta S  = - \frac{8}{3} \pi^2 \, \alpha_H^2 \,   \frac{A^2  \, l^2}{(A^2+t^2)^2} \, .
\end{equation}
\item large $t \gg A$  and  $t \gg l$
 \begin{equation}
  \Delta S  =  - \frac{8}{3} \pi^2 \, \alpha_H^2 \, \frac{ l^2 \, A^2}{t^4} \, .
  \end{equation}

  \item $t=0$ and $l \gg A$
  \begin{equation}
  \Delta S=  - \frac{8}{3} \pi^2 \, \alpha_H^2 \,  \frac{A^2}{l^2} \, .
  \end{equation}
  \end{itemize}

  It is useful to note that, for given $(l,t)$,  the minimal surfaces in eq. (\ref{minimal-surface}) in the Poincar\'e patch
are mapped by eq. (\ref{A-changing-inverse}) to constant $\tau$ surfaces in global AdS.
These surfaces are attached at $r \to \infty$ to a circle
with constant $\theta=\theta_0$, where
\begin{equation}
\theta_0(l,t)=\tan^{-1} \left( \frac{2 A \, l}{l^2 -t^2-A^2} \right) \, ,
\label{theta-zero}
\end{equation}
which corresponds to a parallel on the $S^2$ boundary.
This shows that $\Delta S(l,t)$ is a function just
of the combination in eq. (\ref{theta-zero}).

The RT surfaces with
 \begin{equation}
l= l_0 = \sqrt{t^2 + A^2}
 \end{equation}
corresponds to a parallel with $\theta_0 = \pi/2$.
These surfaces are special, because they lie on the equator of $S^2$.
Due to symmetry,  we conclude that the RT surface at $l=l_0$
 has either the maximal or the minimal $\Delta S$.
For the monopole case, we know that $\Delta S$ is  negative
and close to zero for $t \to 0$ and large $l$.
So we expect that $l=l_0$ is a minimum of $\Delta S$,
as also confirmed by the numerics in figure \ref{entropy-plot}.

 For $l=l_0$, $\Delta S$ can be computed exactly:
  \begin{equation}
  \Delta S_0 = \Delta S (l_0) = \frac{\pi^2 \, \alpha_H^2}{4}   \int_0^{\infty}
 \left(h_{\epsilon }- g_{\epsilon } \right)  \frac{r}{\sqrt{1+r^2}}
\, dr = - \Upsilon   \frac{\pi^2 \, \alpha_H^2}{4} \, ,
\label{Delta-S-zero}
  \end{equation}
where
\begin{equation}
\Upsilon=6 \pi -12 -8 \pi \, \beta(2) +14 \, \zeta(3) \approx 0.658 \, .
\end{equation}
In this expression, $\beta(2)\approx 0.916$ is the Catalan constant
and $\zeta$ is the Riemann zeta function.

Summarising, the entropy of the disk with radius $l_0=\sqrt{t^2+A^2}$
remains constant as a function of the time $t$ and equal to the minimum $\Delta S_0$. This can be heuristically justified as follows.
At large $t$, the bound from causality on the speed of entanglement propagation is saturated:
$\Delta S$, which originated at $t=0$ from a region nearby $x=0$, spreads at the speed of light.
At small $t$, the speed of propagation is smaller, because
at $t \to 0$ also the matter of the quench has zero velocity:
entanglement spreads with matter.

\subsection{Translated disk}

For convenience, we introduce
\begin{equation}
 \vec{x} = \left( x_1, x_2 \right)=  \left( x \cos {\varphi}, x \sin {\varphi} \right)\, .
\end{equation}
We now consider as subregion  a disk of radius $l$ centered at $\left( x_1, x_2 \right) = \left( \xi, 0 \right)$
  and lying at constant time $t$. The corresponding RT surface in unperturbed Poincar\'e AdS$_4$ is the translated half sphere
\begin{equation}
z = \sqrt{l^2 - \left( x_1 - \xi \right)^2 - x_2^2} \,  .
\end{equation}
In appendix \ref{appe:entanglement-entropy-trans-disk} we write the explicit integral for the
first-order correction to the
 holographic entanglement entropy.
It is convenient to rescale spatial and time coordinates as in eq. (\ref{rescale-1}), with $\xi \to \xi/A$ as well.
Numerical results can be obtained for arbitrary radius $l$, see figure \ref{trans_disk-plot}.

\begin{figure}
\begin{center}
\includegraphics[scale=0.45]{disk_down}
\qquad
\includegraphics[scale=0.45]{disk_up}
\qquad
\includegraphics[scale=0.45]{disk_trans_l_vari}
\caption{
Left and right:  Time dependence of  $\Delta S$ for a disk-shaped subregion of radius $l=5$ centered
at $\left( x_1, x_2 \right) = \left( \xi, 0 \right)$ for different values of $\xi$. For large $\xi$, the maximum is reached for $t \approx \xi$.
Bottom: Plot of $\Delta S$ as a function of $l$ for a translated disk-shaped subregion, for various values of $t=\xi$.
Numerical values $\alpha_H=1$, $A=1$ have been fixed.
}
\label{trans_disk-plot}
\end{center}
\end{figure}

In the regime of small $l$, the RT surface stays at large $r$, so we can use the expansion
(\ref{expansion-h-and-g})
and a compact expression can be found
\begin{equation}
  \Delta S(l,t,\xi)
    =  - \frac{8}{3} \pi^2 \, \alpha_H^2 \,   \frac{A^2  \, l^2}{A^4+2 A^2 \left(\xi ^2+t^2\right)+\left(t^2-\xi ^2\right)^2} \, .

  \label{Delta-S-small-disk}
\end{equation}
This shows that $\Delta S$ is always negative for disks with small radius $l$.
The entanglement entropy of small disks is then dominated by the negative contribution
due to the scalar condensation.

At large $l \gg \xi$, the subregion is, with good approximation, a
disk centered at $\xi \approx 0$, and so, from the results of the previous
section, we expect a negative $\Delta S$.
For intermediate $l$, the quantity $\Delta S$ can become positive,
see figure \ref{trans_disk-plot}.
In this regime we can interpret the positive contribution to $\Delta S$ as due to
quasiparticles.

\begin{figure}
\begin{center}
\begin{tikzpicture}
\draw [->] (-4,0)--(4,0) node [at end, right] {$x_1$};
\draw [->] (0,0)--(0,4) node [at end, right] {$t$};
\draw[<->] (-0.02,-0.25)--(0.48,-0.25);
\node at (0.2,-0.6) {$A$};
\draw[red, very thick] (-0.5,0)--(0.5,0);
\draw[blue, very thick] (1.5,2.5)--(3.5,2.5);
\draw[<->] (2.52,2.75)--(3.48,2.75);
\node at (3,3.1) {$l$};
\draw[dashed] (2.5,0)--(2.5,2.5);
\node at (2.5,-0.3) {$\xi$};
\draw[->,red,thick] (0,0)--(2.5,2.5);
\draw[->,red,thick] (0,0)--(-2.5,2.5);
\draw[->,red,thick] (0.45,0)--(2.95,2.5);
\draw[->,red,thick] (0.45,0)--(-1.95,2.5);
\draw[->,red,thick] (-0.45,0)--(-2.95,2.5);
\draw[->,red,thick] (-0.45,0)--(2.05,2.5);
\end{tikzpicture}
\end{center}
\caption{In the quasiparticle model, the quench creates EPR pairs of entangled quasiparticles which
subsequently propagate without interactions. When just one of the quasiparticles belonging to an EPR pair
is inside the blue region, the entanglement entropy of the region increase.}
\label{quasi-particles-figure}
\end{figure}

Free quasiparticles provide a simple model of
entanglement propagation  \cite{Calabrese:2005in}.
In this picture, the quench is assumed to create many copies of
Einstein-Podolsky-Rosen (EPR)
pairs, which hen  propagate without interactions,
see figure \ref{quasi-particles-figure}.
When just one of the entangled particles in an EPR pair
is inside a given region, there is a positive contribution to the entanglement entropy.
This model can reproduce several  aspects of the spread of entanglement
in global and local quenches. Models with interacting quasiparticles
 have also been studied  \cite{Casini:2015zua}.
In all these models, the contribution of the excitations
to the entanglement entropy is always positive.
In the monopole quench, there is also
a negative contribution to the entanglement entropy due to the scalar
condensate. In general, we expect that there is a competition
between the quasiparticle and the condensate contribution,
which is responsible for the change of sign of the entanglement
entropy of the translated disk region.

\subsection{Half-plane region}

We take as a boundary subregion the half-plane $x_1 \geq 0$ at constant time $t$.
The unperturbed RT is the bulk surface at $x_1=0$ and constant time $t$. A convenient choice of parameters is
\begin{equation}
y^{\alpha} = \left( z, x_2 \right) \, .
\end{equation}
Details of the calculations are in appendix \ref{appe:entanglement-entropy-half-plane}.
From the closed-form expression, we deduce that the entropy variation $\Delta S$ is a function of $t/A$.
Numerical result is shown in figure \ref{halfplane-plot}.

For $t=0$, the entropy is given by $\Delta S_0$ in eq. (\ref{Delta-S-zero}).
This is because, due to the change of variables in eq. (\ref{A-changing-inverse}),
the $t=0$ plane with $x_1=0$ (which corresponds to ${\varphi}=\pm \pi/2$)
is mapped in global AdS to a disk with $\tau=0$ and constant ${\varphi}=\pm \pi/2$.
Then, an explicit computation easily leads to the same entropy as in  (\ref{Delta-S-zero}).

At large $t$, from the analysis in appendix \ref{appe:entanglement-entropy-half-plane}, we find
that $\Delta S$ scales in a linear way with time, i.e.
\begin{equation}
\Delta S = K \, \alpha_H^2 \,  \frac{ t}{A}  \,  , \qquad K \approx 0.636 \, .
\label{large-time-half-plane}
\end{equation}
An exact expression for $K$ is given in eq. (\ref{K-expression}).
This is consistent
with numerical results, shown in figure \ref{halfplane-plot}.
We emphasise that this result is valid only in the regime
where we can trust our perturbative calculation in the parameter $\epsilon$.
At very large $t$, we expect that the large backreaction effects spoil
the results in eq. (\ref{large-time-half-plane}).
A linear behaviour at large $t$ is also realised for the perturbative entropy
 in the case of a falling black hole in AdS$_4$ \cite{Jahn:2017xsg}.

The numerical plot in figure \ref{halfplane-plot} is consistent
with both the $t=0$ and large $t$ analytical calculations.
At $t=0$, $\Delta S$ is negative, in agreement with the expectation
that the condensate decreases the entanglement entropy.
Immediately  after $t=0$, the quantity $\Delta S$  enters in a linear growth regime
and becomes positive around $t \approx 2 A$.
This linear behaviour is similar to the one of the black hole quench
and we expects that it is due to the contribution of quasiparticles.

\begin{figure}
\begin{center}
\includegraphics[scale=0.45]{halfplane_2}
\caption{
Time dependence of  $\Delta S$ for the half-plane subregion.
 The numerical values $\alpha_H=1$, $A=1$ have been chosen.
}
\label{halfplane-plot}
\end{center}
\end{figure}

\subsection{Comparison with the BH quench}

In the case of a falling black hole in AdS$_4$, the perturbative entropy
for a centered disk is
\begin{equation}
\Delta S^{(BH)}=\frac{\pi M L^2}{4 G \,  A l}\left(\frac{l^{4}-2 l^{2} t^{2}
+\left(A^{2}+t^{2}\right)^{2}}{\sqrt{l^{4}+2 l^{2}\left(A^{2}
-t^{2}\right)+\left(A^{2}+t^{2}\right)^{2}}}-\left|t^{2}+A^{2}-l^{2}\right|\right) \, ,
\end{equation}
where $M$ is the mass parameter, as defined in eq. (\ref{metric-BH}).
In this case, $\Delta S^{(BH)}$ is always positive
and has a maximum for $l=l_0=\sqrt{t^2+A^2}$.
 The First Law of Entanglement Entropy (FLEE)  \cite{Bhattacharya:2012mi}
is valid in the regime of small $l$:
\begin{equation}
\Delta S= \frac{\Delta E}{ T_E} \, , \qquad T_E=\frac{2}{ \pi l} \, ,
\label{FLEE}
\end{equation}
where $T_E$ is the entanglement temperature and $\Delta E$ is the energy.

FLEE is generically invalidated in the case
of backgrounds with scalars, because the Fefferman-Graham expansion
of the metric does not start at order $z^d$ \cite{Blanco:2013joa,OBannon:2016exv}, where $d$
is the dimension of the spacetime boundary.
Indeed, as can be checked from eq. (\ref{Delta-S-small-disk}),  FLEE
is not satisfied for the monopole quench background.
The behaviour of $\Delta S$ for small $l$ in eq. (\ref{Delta-S-small-disk}) is rather different from the FLEE regime. In particular:
\begin{itemize}
\item  $\Delta S$ is negative;
\item the quantity $\Delta S$ scales as $l^2$, and not as $l^3$ as predicted by FLEE;
\item there is no  choice of boundary conditions for which
 the energy $\Delta E$ is proportional to $\Delta S$.
\end{itemize}

The  FLEE can be derived from
the notion of relative entropy  \cite{Blanco:2013joa,Rangamani:2016dms},
which is a quantity that measures how far is a density matrix $\rho$ from a reference density matrix $\sigma$:
\begin{equation}
S(\rho || \sigma) = {\rm Tr} (\rho \log \rho) -{\rm Tr} (\rho \log \sigma) \, .
\end{equation}
As a general property, $S(\rho || \sigma) $ is positive definite and it
vanishes if and only if $\rho= \sigma$.
The relative entropy can be written as
\begin{equation}
S(\rho || \sigma) =  \Delta \langle \mathcal{K}_\sigma \rangle - \Delta S \, ,
\end{equation}
where $\mathcal{K}_\sigma$ is the modular Hamiltonian of the density matrix $\sigma$
\begin{equation}
\mathcal{K}_\sigma = - \log \sigma \, .
\end{equation}
From positivity of relative entropy we get the relation
\begin{equation}
\Delta \langle \mathcal{K}_\sigma \rangle  \geq \Delta S \, .
\label{relativeE-ineq}
\end{equation}
The modular Hamiltonian operator $\mathcal{K}_\sigma$ for a spherical domain with radius $l$ in the vacuum state of a $d$-dimensional CFT
can be expressed \cite{Casini:2011kv}  in terms of the energy momentum operator as follows
\begin{equation}
\mathcal{K}_\sigma= 2 \pi \int_{\rm sphere } d^{d-1} \vec{x} \, \,\, \frac{l^2 -x^2}{2 l} T_{tt}(\vec{x}) \,  ,
\end{equation}
where $x=|\vec{x}|$.
In the limit of small spherical subregion and in $d=3$, we find
\begin{equation}
\Delta \langle \mathcal{K}_\sigma  \rangle = \frac{\pi l}{2}\Delta  E \, ,
\end{equation}
so the FLEE in eq. (\ref{FLEE}) follows from
 the saturation of the identity (\ref{relativeE-ineq}), i.e.
 \begin{equation}
 \Delta \langle \mathcal{K}_\sigma \rangle  = \Delta S \, .
 \end{equation}

If we consider two nearby density matrices, i.e.
\begin{equation}
\sigma=\rho_0 \, , \qquad \rho=\rho_0 + \varepsilon \rho_1+ \mathcal{O}(\varepsilon^2) \, ,
\end{equation}
where $\varepsilon$ is an expansion parameter,
the relative entropy scales with $\varepsilon$ as follows
\begin{equation}
 S(\rho || \sigma) = \mathcal{O}(\varepsilon^2) \, .
 \end{equation}
Since the $ \mathcal{O}(\varepsilon)$ contribution to relative entropy vanishes,
there is a general expectation \cite{Blanco:2013joa} that
for small deformations eq. (\ref{relativeE-ineq}) is saturated.
However, the question if FLEE is satisfied in quantum field theory is subtle:
there the density matrix $\rho$ is infinite dimensional
and so it is not clear, in principle, when a perturbation might be considered small.

\section{Conclusions}
 \label{sect:conclusions}

 In this paper, we  studied a magnetic monopole solution in the
static global AdS$_4$ setup introduced in   \cite{Esposito:2017qpj}.
The boundary conditions of the monopole are
specified by the parameter $\alpha_H$, which in the multitrace quantisation
is proportional to the VEV of the scalar operator in the boundary CFT.
We found an approximate analytic solution for the monopole
in the regime of small $\alpha_H$  which includes the leading-order backreaction on the metric.
By using the map introduced in \cite{Horowitz:1999gf},
the static monopole in global AdS$_4$ is mapped into a falling monopole in the Poincar\'e patch.
This bulk configuration is dual to a local quench on the CFT side.
The expectation values of local operators depend on the choice of the boundary conditions.
With Dirichlet or Neumann conditions, the falling monopole is dual to a field theory
with a time-dependent source. With the special choice of multitrace deformation
in eq. (\ref{multitrace-speciale}), the monopole is dual to a field theory with zero sources.
In this case, there is no energy injection and the form of the energy-momentum
tensor is the same as the one of a falling black hole  \cite{Nozaki:2013wia}.

The behaviour of entanglement entropy is instead rather
different compared to the case of the falling black hole.
For spherical regions centered on the local quench,
the perturbed entanglement entropy is always less than the vacuum value, i.e. $\Delta S \leq 0$.
This is consistent with the presence of a condensate
at the core of the local quench \cite{Albash:2012pd}.

In the case of a spherical region not centered at the origin,
there is a competition in $\Delta S$ between
the negative
contribution from the condensate and the
positive one due to quasiparticles \cite{Calabrese:2005in}.
Depending on the radius $l$ and on the distance $\xi$ from the origin of the spherical region,
 $\Delta S$ can be positive or negative, see figure \ref{trans_disk-plot}.
In the case of half-plane region,
the negative contribution to $\Delta S$ due to the condensate wins
at early times, while the positive contribution due to quasiparticles
dominates at late times (see figure \ref{halfplane-plot}).

For a quench dual to a falling black hole,
the First Law of Entanglement Entropy (FLEE)  \cite{Bhattacharya:2012mi}
is satisfied for small subregions.
For the quench dual to the monopole, we find that the FLEE is not satisfied.
This is a feature shared with other AdS backgrounds which involve the backreaction
of scalar bulk fields, see \cite{Blanco:2013joa,OBannon:2016exv}.
On the field theory side, the violation of the FLEE
comes from the non-saturation of the inequality in eq. (\ref{relativeE-ineq}).
It is still an open question whether a given deformation
obeys  the FLEE \cite{OBannon:2016exv}
in a quantum field theory. It would be interesting to further
investigate the FLEE in non-equilibrium systems, in order to
understand its general validity conditions.

Analytical  soliton solutions which include  backreaction
 are quite rare in AdS spacetime.
The monopole solution found in this paper
can be the starting point for several further investigations.
In particular, it would be interesting to study
more general solitonic objects in AdS.
For instance, vortex strings in AdS were considered by many authors
\cite{Albash:2009iq,Montull:2009fe,Keranen:2009re,Domenech:2010nf,Iqbal:2011bf,Dias:2013bwa,Maeda:2009vf,Tallarita:2019czh,Tallarita:2019amp}.
A static configuration in Poincar\'e patch with a monopole attached to a vortex
string should also be possible, as proposed in \cite{Dias:2013bwa}:
the vortex string pulls the vortex and it can counterbalance
the gravitational force that makes it fall towards the center of AdS.
It would be interesting to find explicit solutions for these objects  and to investigate
their field theory duals.

Another possible direction is the study of holographic complexity \cite{Susskind:2014rva,Stanford:2014jda,Brown:2015bva}.
Quantum computational complexity is a recent quantum information entry in the holographic dictionary,
which was motivated by the desire of understanding the growth of the Einstein-Rosen bridge
inside the event horizon of black holes.
Complexity in several examples of global and local quenches
has been studied by several authors, e.g. \cite{Moosa:2017yvt,Chapman:2018dem,Chapman:2018lsv,Chen:2018mcc,Auzzi:2019mah,DiGiulio:2021oal,Ageev:2018nye,Ageev:2019fxn,DiGiulio:2021noo}.
 It would be interesting to investigate complexity for quenches dual to a falling monopole.
 This analysis may give us useful  insights to understand the impact of condensates
 on quantum complexity.

\section*{Acknowledgments}

We are grateful to Stefano Bolognesi for useful discussions.
N.Z. acknowledges the Ermenegildo Zegna's Group for the financial support.

 \section*{Appendix}
\addtocontents{toc}{\protect\setcounter{tocdepth}{1}}


\section{Introduction}
\label{sec:intro} Narrow-Line Seyfert 1 (NLS1) Galaxies are a relatively peculiar subclass of active galactic
nuclei (AGNs), which are characterized by their optical spectra with narrow permitted lines (FWHM (H$\beta$) $<$
2000 km s$^{-1}$), the ratio of [O III]$\lambda$5007$/$H$\beta$ $<$ 3, and the bump of Fe II (e.g., Pogge 2000).
NLS1s are also interesting for their low masses of central black holes and high accretion rate. NLS1s are
generally radio quiet; only a small percentage of them is radio-loud ($<7\%$; Komossa et al. 2006). So far, four
NLS1s are confirmed to be GeV emission sources by Fermi/LAT, which should have relativistic jets as predicted by
Yuan et al. (2008).

PMN J0948+0022 (z = 0.5846) is the first NLS1 detected in GeV band by Fermi/LAT. The inverted spectrum in radio
band indicates the presence of a relativistic jet viewed at small angles (Zhou et al. 2003), which is similar to
the properties of blazars. The significant variabilities of PMN J0948+0022 are detected, especially in GeV band
(Abdo et al. 2009; Foschini et al. 2012). It is found that the observed luminosity variations are usually
accompanied with the shift of peak frequencies of the SEDs, similar to some GeV-TeV BL Lacs (Zhang et al. 2012).
The abundant broadband observational data provide an opportunity to study the physical mechanism of the spectral
variations of PMN J0948+0022.

\begin{figure}
\centering
\includegraphics[angle=0,scale=0.35]{f1.eps}
\caption{{\em Panel a}---Observed SEDs ({\em scattered data points}) with model fitting ({\em lines}) for PMN
J0948+0022. The four SEDs are marked as states ``a" ({\em blue points and thick solid line}), ``b" ({\em magenta
points and dashed line}), ``c" ({\em red points and thin solid line}), and ``d" ({\em green points and dotted
line}), respectively. {\em Panels b, c}---$L_{\rm c}$ as a function of $\nu_{\rm c}$ in both the observer ({\em
Panel b}) and co-moving ({\em Panel c}) frames.}
\end{figure}

\section{SED Selection and Modelling }
\label{sec:using} We compile the observed broadband SEDs of PMN J0948+0022 from literature. Four available SEDs
as shown in Figure 1(a), are defined as SEDs ``a", ``b", ``c", and ``d" according to their peak luminosity of the IC
bump of the SEDs, respectively. The data of SEDs ``a" and ``b" are from Foschini et al. (2012) and are obtained
with the observations in 2010 July 8th and 2011 October 9th-12th, respectively. The SEDs ``c" and ``d" of this
source are taken from the observations in 2009 June 14th and 2009 May 5th (Abdo et al. 2009).

The broadband SEDs of PMN J0948+0022 are similar to the typical FSRQs and thus the $\gamma$-ray emission should
be dominated by jet emission. We use the syn+SSC+EC model to fit its SEDs because the contributions of the
external field photons from the broad line region (BLR) need to be considered. The total luminosity of the BLR
is calculated using the luminosity of its emission lines (Zhou et al. 2003) with equation (1) given in Celotti
et al. (1997). The size of the BLR is calculated using the BLR luminosity with equation (23) in Liu \& Bai
(2006). The energy density of the BLR measured in the co-moving frame is $U^{'}_{\rm
BLR}=6.76\times10^{-3}\Gamma^{2}$ erg cm$^{-3}$, where we take $\Gamma\sim\delta$. The minimum variability
timescale is taken as $\Delta t=12$ hr. The details of the model and the strategy for constraining its
parameters constraints can be found in Zhang et al. (2012).

\begin{figure}
\centering
\includegraphics[angle=0,scale=0.35]{f3.eps} \caption{$L_{\rm c}$ as functions of $B$, $\delta$, and $\gamma_{\rm b}$ in both the observer ({\em
three top panels}) and co-moving ({\em three bottom panels}) frames.}
\end{figure}

The SEDs of PMN J0948+0022 are well explained with the syn+SSC+EC model, as shown in Figure 1(a). The EC component of the SEDs for PMN J0948+0022 presents a
further constraint on $\delta$, and thus makes a tighter constraint on $\delta$ and $B$ than that for BL Lacs in
Zhang et al. (2012). The fitting parameters of SEDs for PMN J0948+0022 are also more tightly clustered; the
magnetic field strength $B$ is from $2.24\pm0.11$ G to $3.34\pm0.50$ G, the Doppler factor is from $12.4\pm0.8$
to $23.0\pm0.8$, and the break Lorenz factor of electrons is from $91\pm3$ to $107\pm9$.

\section{Spectral Variation of IC Bump}
\label{sec:frontmatter}

The broadband SEDs of PMN J0948+0022 are dominated by the EC process, and there are significant variabilities in
GeV band. The peak luminosity ($L_{\rm c}$) as a function of peak frequency ($\nu_{\rm c}$) of the IC bump in
both the observer and co-moving frames are given in Figures 1(b),(c). A tentative correlation between the peak
luminosity and peak frequency in both the observer and co-moving frames is found, i,e., $L_{\rm c} \propto
\nu_{\rm c}^{(2.12\pm0.83)}$ with $r = 0.87$ (Pearson correlation coefficient) and $p=0.13$ (chance
probability), and $L^{'}_{\rm c}\propto \nu^{'(1.05\pm0.21)}_{\rm c}$ with $r = 0.96$ and $p=0.04$,
respectively, indicating that the luminosity variations of the IC bump are accompanied with a spectral shift.

To investigate the possible physical reason of this phenomenon, we show the IC peak luminosity as functions of
$B$, $\delta$, and $\gamma_{\rm b}$ in both the observer and co-moving frames in Figure 2. It can be found that:
(1) Both $L_{\rm c}$ and $L^{'}_{\rm c}$ are anti-correlated with $B$. The Pearson correlation analysis and the
best liner fits yield $L_{\rm c} \propto B^{(-1.09\pm0.28)}$ with $r=-0.94$, $p=0.06$ and $L^{'}_{\rm c}\propto
B^{(-0.37\pm0.09)}$ with $r=-0.94$, $p=0.06$. (2) $L_{\rm c}$ seems to be correlated with $\delta$ with $r=0.93$
and $p=0.07$. (3) No correlation between $\gamma_{\rm b}$ with $L_{\rm c}$ and $L^{'}_{\rm c}$ is found, which
may be due to the uncertainties of the synchrotron radiation peak for the SEDs. These facts indicate that the
spectral variations of the IC peak for PMN J0948+0022 may be attributed to the variations of $\delta$ and $B$, similar to the results of a typical FSRQ 3C 279 (Zhang et al. 2013).

The significant variations of the IC peak for PMN J0948+0022 in GeV band are dominated by
the EC process. The energy density of the external photon field would be magnified by $\Gamma^2$ and the energy
of the seed photons would be magnified by $\Gamma$ due to the motion of the emitting regions, hence a small
variation of $\delta$ would result in significant variations of $\nu_{\rm c}$ and $L_{\rm c}$. As mentioned
above, $B$ is also anti-correlated with $L_{\rm c}$ in both the observer and co-moving frames, indicating the
variations of $B$ for this source between different states, which are also accompanied with the variations of
$\delta$, might be linked to the variations of some intrinsic physical parameters of the center black hole, such
as the disk accretion rate or the corona (Zhang et al. 2013). The instabilities of corona or disk accretion rate may
result in the variations of the jet physical condition and the variations of jets emission.

\section{Conclusion}
The SEDs observed at four epochs for PMN J0948+0022, which can be explained well with the syn+SSC+EC model, are
compiled from literature to investigate its spectral variation. A tentative correlation between the peak
luminosity and peak frequency of its IC bumps is found, indicating that a higher GeV luminosity corresponds to a
harder spectrum for the emission in the GeV band, similar to the properties of some blazars. The SEDs of PMN
J0948+0022 are dominated by the EC bumps and thus the magnification of the external photon field by the bulk
motion of the radiation regions is an essential reason for the spectral variation. The variations of $B$ and
$\delta$ for PMN J0948+0022 between different states may be produced by the instabilities of the corona or the
disk accretion rate.

\section{Acknowledgments}
This work is supported by the National Basic Research Program (973 Programme) of China (Grant 2009CB824800), the
National Natural Science Foundation of China (Grants 11078008, 11025313, 11133002, 10725313), Guangxi Science Foundation (2011GXNSFB018063, 2010GXNSFC013011), and Key Laboratory for the Structure and Evolution of Celestial Objects of Chinese Academy of Sciences.