IPM/P-2021/013

Entanglement Wedge Cross Section in Holographic Excited States

Mohammad Sahraei∗, Mohammad Javad Vasli∗,

M. Reza Mohammadi Mozaffar∗,† and Komeil Babaei Velni∗,†

∗ Department of Physics, University of Guilan, P.O. Box 41335-1914, Rasht,
Iran
† School of Physics, Institute for Research in Fundamental Sciences (IPM),
P.O. Box 19395-5531, Tehran, Iran

E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
We evaluate the entanglement wedge cross section (EWCS) in asymptotically AdS
geometries which are dual to boundary excited states. We carry out a
perturbative analysis for calculating EWCS between the vacuum and other states
for a symmetric configuration consisting of two disjoint strips and obtain
analytical results in the specific regimes of the parameter space. In
particular, when the states described by purely gravitational excitations in
the bulk we find that the leading correction to EWCS is negative and hence the
correlation between the boundary subregions decreases. We also study other
types of excitations upon adding the extra matter fields including current and
scalar condensate. Our study reveals some generic properties of boundary
information measures dual to EWCS, e.g., entanglement of purification,
logarithmic negativity and reflected entropy. Finally, we discuss how these
results are consistent with the behavior of other correlation measures
including the holographic mutual information.

###### Contents

  1. 1 Introduction
  2. 2 AdS Black Brane Geometries
    1. 2.1 Thermal Excitations
    2. 2.2 Charged Excitations
  3. 3 AdS Plane Wave Geometries
    1. 3.1 EWCS in Boosted AdS Black Brane
    2. 3.2 EWCS in AdS Plane Wave: Case A
    3. 3.3 EWCS in AdS Plane Wave: Case B
  4. 4 Momentum Relaxation Geometries
  5. 5 General Analysis
  6. 6 Conclusions and Discussions

##  1 Introduction

The gauge/gravity duality has stimulated a wide variety of recent efforts
investigating the entanglement properties of quantum field theories (QFTs) and
also the connection between entanglement and geometry, e.g., see [1, 2]. In
this context, certain entanglement measures of the boundary QFT can be related
to the geometric quantities that live in the bulk spacetime. The best studied
example is the Ryu-Takayanagi (RT) prescription, which provides a holographic
realization of the entanglement entropy corresponding to a spatial boundary
subregion $A$[3]. In this case, the holographic entanglement entropy (HEE) can
be computed as

$\displaystyle S_{A}={\rm min}\frac{{\rm area}(\Gamma_{A})}{4G_{N}},$ (1.1)

where $\Gamma_{A}$ is a bulk minimal hypersurface which ends on the boundary
of $A$. It is important to highlight that entanglement entropy (EE) is a
unique measure of quantum entanglement when the global state is pure. However,
for a mixed state EE measures both classical and quantum correlations and in
order to isolate the quantum correlations we should consider other measures,
e.g., entanglement of purification (EoP), logarithmic negativity and reflected
entropy. Much of our analysis in this paper will focus on studying EoP in
different holographic settings, so we proceed by reviewing its definition.

EoP is a measure of total (both quantum and classical) correlations which
coincides with EE for pure states. In general, it is a difficult quantity to
obtain and only recently numerical lattice calculations have been developed
for QFTs[4, 5]. In order to define this quantity, let us consider a bipartite
system with Hilbert space equal to the direct product of two factors, i.e.,
$\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$. Assuming that a mixed
state on $\mathcal{H}$ is described by a density matrix $\rho_{AB}$, we can
purify it into a pure state $|\psi\rangle$ by adding auxiliary degrees of
freedom to $\mathcal{H}$ such that $\rho_{AB}={\rm
tr}_{A^{\prime}B^{\prime}}|\psi\rangle\langle\psi|$. In this case the EoP is
defined as[6]

$\displaystyle E_{P}(A,B)\equiv\min_{|\psi\rangle}S(\rho_{A\cup A^{\prime}}),$
(1.2)

where $\rho_{A\cup A^{\prime}}={\rm
tr}_{BB^{\prime}}|\psi\rangle\langle\psi|$. There are several useful
inequalities which the EoP satisfies generally, e.g.,

$\displaystyle I(A,B)\leq 2E_{P}(A,B)\leq 2\;{\rm min}\\{S_{A},S_{B}\\},$
$\displaystyle I(A,B)+I(A,C)\leq 2E_{P}(A,B\cup C),$ (1.3)

where $I(A,B)$ is the mutual information given as follows

$\displaystyle I(A,B)=S_{A}+S_{B}-S_{A\cup B}.$ (1.4)

Recently, there are many attempts to construct a holographic prescription for
mixed state correlation measures which have led to a remarkably rich and
varied range of new insights, e.g., [7, 8, 9, 10]. A common feature of these
studies is that all of the aforementioned correlation measures have a unique
holographic counterpart which is the entanglement wedge cross section (EWCS).
Before we proceed, let us recall the definition of EWCS in a simple
holographic setup. Considering a spatial region A in the boundary field
theory, the entanglement wedge is the bulk region corresponding to the reduced
density matrix $\rho_{A}$ and whose boundary is $A\cup\Gamma_{A}$ (see the
left panel in figure 1).

Figure 1: Left: The shaded region is the entanglement wedge corresponding to
the boundary subregion $A$. Middle: The case where the entanglement wedge is
connected, i.e., $\Gamma_{A\cup B}=\Gamma_{A\cup B}^{(1)}\cup\Gamma_{A\cup
B}^{(2)}$. In this case EWCS is proportional to the area of $\Sigma_{A\cup
B}$. Right: The case where the entanglement wedge is disconnected, i.e.,
$\Gamma_{A\cup B}=\Gamma_{A}\cup\Gamma_{B}$ and the corresponding EWCS
vanishes.

Note that for static geometries the entire configuration lies on a constant
time slice. Similarly, this construction can be easily extended to more
general cases where the boundary region is composed of multiple subregions. In
particular, when the boundary region is union of two disjoint subregions $A$
and $B$ the boundary of the entanglement wedge is $A\cup B\cup\Gamma_{A\cup
B}$. In this case, keeping the geometry of $A$ and $B$ fixed while their
separation varies, the entanglement wedge has a phase transition due to the
competition between two different configurations for the corresponding RT
hypersurface $\Gamma_{A\cup B}$ (see the middle and right panels in figure 1).
For small separations the connected configuration is favored and we have
$\Gamma_{A\cup B}=\Gamma_{A\cup B}^{(1)}\cup\Gamma_{A\cup B}^{(2)}$. On the
other hand, for large separations the disconnected configuration is favored,
and hence $\Gamma_{A\cup B}=\Gamma_{A}\cup\Gamma_{B}$. Now for connected
configuration the EWCS is defined to be the minimal cross-sectional area of
the entanglement wedge, i.e.,

$\displaystyle E_{W}(A,B)={\rm min}\frac{{\rm area}(\Sigma_{A\cup
B})}{4G_{N}}.$ (1.5)

Note that for disconnected configuration $\Sigma_{A\cup B}$ becomes empty and
the corresponding $E_{W}$ vanishes. In [7, 8], a holographic proposal has been
developed to describe the EoP corresponding to a general state in the boundary
theory as follows

$\displaystyle E_{P}(A,B)=E_{W}(A,B).$ (1.6)

Based on this conjecture, it was shown that, the resultant quantity reproduces
all the desired properties of EoP, e.g., eqs. (1). As we have already
mentioned above there exist other boundary measures which seem to be dual to
EWCS. Other proposals that make a connection between $E_{W}$ and different
boundary correlation measures can be summarized as follows[9, 10]

$\displaystyle
E_{W}(A,B)=\frac{S_{R}(A,B)}{2}=\frac{\mathcal{E}(A,B)}{\chi_{d}},$ (1.7)

where $S_{R}$ and $\mathcal{E}$ are reflected entropy and logarithmic
negativity, respectively.111However, we must add that the connection between
EWCS and logarithmic negativity was recently called into question by [11]. In
particular, the derivations of [10] were shown to not apply for bulk solutions
that break the replica symmetry. Here $\chi_{d}$ is a constant which depends
on the dimension of the spacetime. Let us also mention that in [12] another
measure of correlations the so-called odd (entanglement) entropy introduced
which can be extracted from EWCS using

$\displaystyle S_{O}(A,B)=E_{W}(A,B)+S(A\cup B).$ (1.8)

The above proposals pass a variety of consistency checks which provide
important pieces of evidence for finding holographic duals of EWCS. See [13,
14, 15, 16, 17, 18, 19, 20] for various studies of general properties of EWCS
and its holographic counterparts in static geometries. Further, the
nonequilibrium evolution of EWCS for various quench protocols has been
considered in [21, 22, 23, 24, 25, 26]. Related investigations attempting to
better understand the corresponding measures from the perspective of the
boundary field theory have also appeared in [27, 28, 29, 30, 31].

This paper presents another step in this research program, in which we
investigate the behavior of EWCS for different bulk geometries dual to
boundary excited states. Indeed, developing a proper understanding of EWCS on
the gravity side is essential to properly test the various holographic
proposals noted above in eqs. (1.6), (1.7) and (1.8). In particular, we will
consider states described by purely gravitational excitations in the bulk
where the stress tensor is the only operator that has a nonvanishing
expectation value. In this case, we will determine the leading corrections to
$E_{W}$ and other correlation measures including HEE and HMI. We also consider
more generic perturbations away from the vacuum, when other operators acquire
nontrivial expectation values. Although for a generic excited state our
analysis requires numerical treatment, we present some analytic results for
the variation of EWCS in the specific regimes of the parameter space. Let us
recall that similar studies as we consider here were also considered in [32,
34, 33, 35, 36, 37, 38, 39] to obtain insights into the behavior of HEE for
excited states. We will elaborate more on these issues in the discussion
section.

The rest of the paper is organized as follows: In section 2, we briefly review
the related analysis on EWCS for AdS black brane geometries including both
thermal and charged excitations. In section 3, we consider a specific class of
anisotropic boundary excited states which are dual to AdS plane wave
geometries and study the properties of EWCS, where we present both numerical
and analytic results. Next, in section 4 we study this quantity for momentum
relaxation geometries where in the dual description additional scalar
operators are excited. In section 5, we extend our studies to general boundary
excited states which are dual to asymptotically AdS geometries. Finally, we
briefly discuss our results and indicate some possible future directions, in
section 6.

##  2 AdS Black Brane Geometries

In this section we review the holographic information measures for AdS black
brane geometries. In particular, we consider a $d$-dimensional boundary field
theory with nonzero temperature and density dual to a charged AdS black brane
geometry with a metric given by

$\displaystyle
ds^{2}=\frac{R^{2}}{r^{2}}\left(-f(r)dt^{2}+\sum_{i=1}^{d-1}dx_{i}^{2}+\frac{dr^{2}}{f(r)}\right),\hskip
42.67912ptf(r)=1-\left(1+q^{2}r_{h}^{2}\right)\frac{r^{d}}{r_{h}^{d}}+q^{2}\frac{r^{2(d-1)}}{r_{h}^{2(d-2)}},$
(2.1)

where $R$ and $r_{h}$ denote the AdS and horizon radius, respectively. Also
$q$ is proportional to the charge density and related to the boundary chemical
potential $\mu$, i.e., $q=\sqrt{\frac{d-2}{2d-2}}\mu$. From this metric, one
obtains that the temperature and energy density of the equilibrium state are
given by

$\displaystyle T=\frac{d}{4\pi
r_{h}}\left(1-\frac{d-2}{d}q^{2}r_{h}^{2}\right),\hskip
42.67912pt\mathcal{E}=\left(1+q^{2}r_{h}^{2}\right)\varepsilon,$ (2.2)

where $\varepsilon=\frac{(d-1)R^{d-1}}{16\pi G_{N}}\frac{1}{r_{h}^{d}}$.
Notice that the expectation value of the energy-momentum tensor dual to this
geometry takes the form of that for an ideal fluid, i.e.,

$\displaystyle\langle T_{\mu\nu}\rangle={\rm
diag}\left(1,\frac{1}{d-1},\cdots,\frac{1}{d-1}\right)\mathcal{E}.$ (2.3)

In the extremal limit where the temperature vanishes we have
$q^{2}r_{h}^{2}=\frac{d}{(d-2)}$ and the blackening factor has a double zero
at $r=r_{h}$.

In order to find the HEE, in the following, we focus on the case of a strip
entangling region lies on a constant time slice, i.e.,

$\displaystyle-\frac{\ell}{2}\leq x_{1}\equiv x\leq\frac{\ell}{2}\hskip
56.9055pt0\leq x_{2},\cdots,x_{d-1}\leq L$ (2.4)

where $\ell\ll L$. In this case assuming that the minimal hypersurface has a
translation invariance, its profile will be determined by an embedding $x(r)$
and then using eqs. (1.1) and (2.1) the entropy functional is computed as

$\displaystyle
S=\frac{R^{d-1}L^{d-2}}{4G_{N}}\int\frac{dr}{r^{d-1}}\sqrt{{x^{\prime}}^{2}+\frac{1}{f(r)}},$
(2.5)

where the prime indicates derivative with respect to $r$. The profile for the
minimal hypersurface is then obtained by minimizing the above functional.
Since there is no explicit $x(r)$ dependence, the corresponding momentum is a
conserved quantity and hence the equation determining the profile simplifies
to

$\displaystyle
x^{\prime}=\frac{1}{\sqrt{\left(\left(\frac{r_{t}}{r}\right)^{2d-2}-1\right)f(r)}},$
(2.6)

where $r_{t}$ denotes the location of the turning point of $\Gamma_{A}$.
Further, using the above expression the width of the entangling region can be
written as follows

$\displaystyle\ell=2\int_{0}^{r_{t}}dr\frac{1}{\sqrt{\left(\frac{r_{t}^{2(d-1)}}{r^{2(d-1)}}-1\right)f(r)}}.$
(2.7)

Also plugging eq. (2.6) back into eq. (2.5) the HEE reads

$\displaystyle
S=\frac{R^{d-1}L^{d-2}}{2G_{N}}\int_{\epsilon}^{r_{t}}\frac{dr}{r^{d-1}}\frac{1}{\sqrt{\left(1-\frac{r^{2(d-1)}}{r_{t}^{2(d-1)}}\right)f(r)}}.$
(2.8)

On the other hand using holographic prescription we can find the HMI and EoP
corresponding to a certain combined boundary region $A\cup B$. In the
following, we consider a symmetric configuration consisting of two disjoint
strips with equal width $\ell$ separated by $h$. Note that these measures are
nontrivial only for connected configurations, i.e., $S_{A\cup
B}=S(2\ell+h)+S(h)$, and vanish for disconnected ones, i.e., $S_{A\cup
B}=2S(\ell)$, when $\Sigma_{A\cup B}$ becomes empty. Hence throughout the
following, we will assume that the connected configuration is favored. In this
case due to a reflection symmetry about $x=0$, $\Sigma_{A\cup B}$ runs along
the radial direction and connects the corresponding turning points of
$\Gamma_{h}$ and $\Gamma_{2\ell+h}$. Note that, keeping $\ell$ fixed while $h$
increases, the EWCS has a discontinuous phase transition such that
$\Sigma_{A\cup B}$ becomes empty when the two regions are distant enough [7].
This behavior is due to the competition between two different configurations
for the entanglement wedge. Assuming $\ell\gg h$, the connected configuration
has the minimal area and we obtain a non-zero EWCS. Using eq. (1.5) the
corresponding functional becomes

$\displaystyle
E_{W}=\frac{R^{d-1}L^{d-2}}{4G_{N}}\int_{r_{d}}^{r_{u}}\frac{dr}{r^{d-1}\sqrt{f}},$
(2.9)

where $r_{d}$ ($r_{u}$) denotes the corresponding turning point of
$\Gamma_{h}$ ($\Gamma_{2\ell+h}$). Finding the explicit dependence of HEE and
EWCS on boundary quantities such as temperature or chemical potential requires
computing the above integrals, which is a rather intricate task. So in the
following, we present two specific examples in which we evaluate the variation
of different measures due to the perturbations around the vacuum. We will
consider thermal and charged excitations separately for which the leading
order corrections can be obtained explicitly.

###  2.1 Thermal Excitations

In this case we consider $(q=0,T\neq 0)$ limit where the excited state is a
thermal state at zero charge density. Focusing on low temperature limit we
have $\ell T\ll 1$. From eq. (2.2) this constraint can be expressed in terms
of the bulk parameters as $r_{t}\ll r_{h}$. In this case $r_{t}$ is close to
the boundary and eq. (2.7) has the leading behavior

$\displaystyle\ell=r_{t}\left(c+\frac{\sqrt{\pi}}{(d+1)}\frac{\Gamma\left(\frac{d}{d-1}\right)}{\Gamma\left(\frac{d+1}{2d-2}\right)}\left(\frac{r_{t}}{r_{h}}\right)^{d}\right),$
(2.10)

where
$c=\frac{2\sqrt{\pi}\Gamma\left(\frac{d}{2d-2}\right)}{\Gamma\left(\frac{1}{2d-2}\right)}>0$.
Inverting this equation, we can represent the turning point as a function of
$\ell$

$\displaystyle
r_{t}=\frac{\ell}{c}\left(1-\frac{\sqrt{\pi}}{(d+1)c^{d+1}}\frac{\Gamma\left(\frac{d}{d-1}\right)}{\Gamma\left(\frac{d+1}{2d-2}\right)}\left(\frac{\ell}{r_{h}}\right)^{d}\right).$
(2.11)

That is, increasing the temperature, the turning point of the extremal
hypersurface decreases. In this limit, the leading order behavior of eq. (2.8)
reduces to

$\displaystyle
S=\frac{R^{d-1}L^{d-2}}{2(d-2)G_{N}}\left(\frac{1}{\epsilon^{d-2}}-\frac{c}{2r_{t}^{d-2}}+\frac{\sqrt{\pi}(d-2)}{4}\frac{\Gamma\left(\frac{d}{d-1}\right)}{\Gamma\left(\frac{d+1}{2d-2}\right)}\frac{r_{t}^{2}}{r_{h}^{d}}\right).$
(2.12)

In principle then, we can invert the above expression to write our result in
terms of the boundary quantities $\ell$ and $\varepsilon$. Combining the above
results, we obtain the first order correction to HEE as follows

$\displaystyle\Delta S\equiv S-S_{\rm
AdS}=\tilde{c}L^{d-2}\ell^{2}\varepsilon,$ (2.13)

where

$\displaystyle\tilde{c}=\frac{2\pi^{3/2}}{(d^{2}-1)c^{2}}\frac{\Gamma\left(\frac{1}{d-1}\right)}{\Gamma\left(\frac{d+1}{2d-2}\right)}.$
(2.14)

Notice that $S_{\rm AdS}$ is the vacuum contribution given by $S_{\rm
AdS}=\frac{R^{d-1}L^{d-2}}{2(d-2)G_{N}}\left(\frac{1}{\epsilon^{d-2}}-\frac{c^{d-1}}{2\ell^{d-2}}\right)$.
In what follows we will use this $\Delta S$ notation in several parts of this
paper. Note that $\tilde{c}>0$ and hence thermal excitations increase the HEE.
This behavior is due to the fact that the number of degrees of freedom grows
as we excite the system. It is worth to mention that the above relation
between the leading order variation of HEE and the expectation value of the
stress tensor, known as the first law of entanglement, was first noted in the
holographic calculations of [32] and further derived in [34] using the
positivity of the relative entropy (see also [35, 36] for related studies).
Indeed, in [34] it was shown that the first order variation in the
entanglement entropy for a spatial subregion equals with the first order
variation in the expectation value of the modular Hamiltonian which is a
complicated object that cannot be expressed as an integral of local operators.
Note that, in cases where the modular Hamiltonian is explicitly known, it is
given by an integral of the energy density over the interior of the entangling
region weighted by a shape dependent profile. In addition, as shown in [40]
the first law of entanglement captures the same information as the linearized
Einstein’s equations.

Further, these results allow us to find the variation of other correlation
measures. For example, considering a boundary configuration consisting of two
disjoint strips with equal width $\ell$ separated by $h$, we can obtain the
variation of HMI

$\displaystyle\Delta I\equiv I-I_{\rm
AdS}=-2\tilde{c}L^{d-2}\left(\ell+h\right)^{2}\varepsilon,$ (2.15)

where $I_{\rm AdS}$ is the vacuum contribution given by $I_{\rm
AdS}=\frac{R^{d-1}L^{d-2}c^{d-1}}{2(d-2)G_{N}}\left(\frac{1}{2\left(2\ell+h\right)^{d-2}}+\frac{1}{2h^{d-2}}-\frac{1}{\ell^{d-2}}\right)$.
The minus sign shows that the thermal excitations decrease the HMI and hence
reduce the total correlation between the subregions.

Next we examine the variation of EoP due to the increase in energy at low
temperature using eq. (2.9). At this limit, we can expand $E_{W}$ for large
$r_{h}$, which yields [14]

$\displaystyle
E_{W}=\frac{R^{d-1}L^{d-2}}{4(d-2)G_{N}}\left(\frac{1}{r_{d}^{d-2}}-\frac{1}{r_{u}^{d-2}}\right)+\frac{R^{d-1}L^{d-2}}{16G_{N}}\frac{r_{u}^{2}-r_{d}^{2}}{r_{h}^{d}}.$
(2.16)

Now, we use eq. (2.11) to express $r_{d}$ and $r_{u}$ in terms of $\ell$ and
$h$. As a result, the variation of EoP becomes

$\displaystyle\Delta E_{W}\equiv E_{W}-E_{W\rm
AdS}=\mathcal{C}L^{d-2}\ell(\ell+h)\varepsilon,$ (2.17)

where

$\displaystyle\mathcal{C}=\frac{4\pi}{(d-1)c^{2}}\left(1-\frac{4\sqrt{\pi}}{(d+1)c}\frac{\Gamma\left(\frac{d}{d-1}\right)}{\Gamma\left(\frac{d+1}{2d-2}\right)}\right),$
(2.18)

and $E_{W\rm AdS}$ is the vacuum contribution given by $E_{W\rm
AdS}=\frac{R^{d-1}L^{d-2}c^{d-2}}{4(d-2)G_{N}}\left(\frac{1}{h^{d-2}}-\frac{1}{\left(2\ell+h\right)^{d-2}}\right)$.
Notice that $\mathcal{C}<0$ and hence the finite temperature corrections
decrease the EoP. Regarding the EoP as a measure of total correlation between
the two subregions, we see that thermal excitations promote disentangling
between them. Further, comparing eq. (2.17) with eq. (2.13) we see that in the
case of two adjacent intervals, i.e., $h\ll\ell$, the variation of EoP is
proportional to the variation of HEE up to a negative constant. At this point,
let us recall that in $d=2$ the variation of these measures in the same limit
takes the following simple form [14]

$\displaystyle\Delta
S=\frac{\pi}{3}\ell^{2}\varepsilon+\mathcal{O}\left(\frac{h}{\ell}\right),\hskip
56.9055pt\Delta
E_{W}=-\frac{\pi}{3}\ell^{2}\varepsilon+\mathcal{O}\left(\frac{h}{\ell}\right),$
(2.19)

which shows that at leading order $\Delta E_{W}=-\Delta S$ and hence the
measures change with the same rate (in this case we have
$|\mathcal{C}|=\tilde{c}=\frac{\pi}{3}$). On the other hand, in higher
dimensions $|\mathcal{C}|<\tilde{c}$ and thus the variation of HEE becomes
more pronounced.

Before closing this section, let us comment further on the computation of the
first order corrections to the holographic correlation measures, e.g., HEE and
EoP, under the metric perturbation. In the absence of perturbations, we have a
pure AdS geometry and the HEE for a spatial subregion $A$ is given in terms of
the area of a minimal hypersurface $\Gamma_{A}$. Now, consider an arbitrary
small perturbation to the AdS geometry which is dual to small deviation of the
boundary vacuum state. We expect that both the area functional and the shape
of the extremal hypersurface are modified and the corresponding HEE changes.
More explicitly, we have $\Gamma_{A}\rightarrow\Gamma_{A}+\delta\Gamma_{A}$
and $S_{A}\rightarrow S_{A}+\delta S_{A}$. However, considering the first
order corrections to HEE, one can show that any change in the profile of the
minimal hypersurface will not contribute to $\delta S_{A}$. Hence we can
evaluate the linear change in the HEE by evaluating the area functional on
$\Gamma_{A}$ in the perturbed background [40]. Note that in this case the
boundary subregion $A$ is fixed and thus the boundary condition for the
minimal hypersurface does not change.

Turning now to the computation of the first order correction to EoP, we should
find the minimal cross-sectional area of the entanglement wedge, i.e.,
$\Sigma_{A\cup B}$, in the perturbed geometry. Recall that by definition, the
entanglement wedge is bounded by $A\cup B\cup\Gamma_{A\cup B}$ and hence any
change in the position of the RT hypersurface, modifies the profile of
$\Sigma_{A\cup B}$. This means that the boundary condition for $\Sigma_{A\cup
B}$ changes under metric perturbation, while keeping $A$ and $B$ fixed. Hence
we expect that the variation of $\Gamma_{A\cup B}$ plays a central role in
evaluating the correction to EoP even at leading order. As an example,
consider the symmetric configuration consisting of two disjoint strips with
equal width. In this case due to the reflection symmetry $\Sigma_{A\cup B}$
lies entirely on $x=0$ slice both for pure AdS geometry and for any
homogeneous excitations around it.222Here by homogeneous excitations we mean
excitations being independent of $x$, so that the profile of EWCS does not
change. On the other hand, because the turning points of the corresponding RT
hypersurfaces, i.e., $\Gamma_{h}$ and $\Gamma_{2\ell+h}$, are modified the
limits of integration in eq. (2.9) change which give a nontrivial contribution
to $\Delta E_{W}$. Further, we note that in the present example, turning on
the temperature, the RT hypersurfaces move away from the horizon to smaller
values of $r$. In the following, we will see that similar results apply for
other types of excitations.

###  2.2 Charged Excitations

In this case we consider a charged black brane which is dual to a finite
temperature boundary state with a nonzero chemical potential for some
conserved charge. We assume that the chemical potential is of the same order
as temperature, i.e, $\mu\sim T$, and of course, as in the previous case, we
assume $\ell T\ll 1$. Again, these constraints can be expressed in terms of
the bulk parameters as $qr_{h}\sim\mathcal{O}(1)$ and $r_{t}\ll r_{h}$. Thus,
$r_{t}$ is close to the boundary and eq. (2.7) yields

$\displaystyle
r_{t}=\frac{\ell}{c}\left(1-\frac{\sqrt{\pi}}{(d+1)c^{d+1}}\frac{\Gamma\left(\frac{d}{d-1}\right)}{\Gamma\left(\frac{d+1}{2d-2}\right)}\left(\frac{\ell}{r_{h}}\right)^{d}\left(1+q^{2}r_{h}^{2}\right)+\frac{\sqrt{\pi}d}{(2d-1)c^{2d-1}}\frac{\Gamma\left(\frac{d}{2d-2}\right)}{\Gamma\left(\frac{1}{2d-2}\right)}\left(\frac{\ell}{r_{h}}\right)^{2d-2}q^{2}r_{h}^{2}\right).$
(2.20)

We see once again (as in the neutral case) that at leading order the turning
point is a decreasing function of the excitation parameter. Substituting the
above expression into eq. (2.8), we obtain

$\displaystyle\Delta S(q)=\left(1+q^{2}r_{h}^{2}\right)\Delta
S-\frac{R^{d-1}L^{d-2}}{4G_{N}}\frac{\sqrt{\pi}(d-1)}{(2d-1)c^{d}}\frac{\Gamma\left(\frac{d}{2d-2}\right)}{\Gamma\left(\frac{1}{2d-2}\right)}q^{2}r_{h}^{2}\frac{\ell^{d}}{r_{h}^{2d-2}},$
(2.21)

where $\Delta S$ is defined in eq. (2.13). Note that at leading order the
charged excitations increase the HEE, while higher order correction is
negative. A similar calculation for EoP functional eq. (2.9) shows that

$\displaystyle
E_{W}=\frac{R^{d-1}L^{d-2}}{4G_{N}}\left(\frac{1}{(d-2)r_{d}^{d-2}}-\frac{1}{(d-2)r_{u}^{d-2}}+(1+q^{2}r_{h}^{2})\frac{r_{u}^{2}-r_{d}^{2}}{4r_{h}^{d}}-\frac{q^{2}r_{h}^{2}}{2d}\frac{r_{u}^{d}-r_{d}^{d}}{r_{h}^{2d-2}}\right).$
(2.22)

Hence using eq. (2.20) the variation of EoP becomes

$\displaystyle\Delta E_{W}(q)=\left(1+q^{2}r_{h}^{2}\right)\Delta
E_{W}+\frac{R^{d-1}L^{d-2}}{4G_{N}}\mathcal{C}_{q}q^{2}r_{h}^{2}\frac{(2\ell+h)^{d}-h^{d}}{r_{h}^{2d-2}},$
(2.23)

where $\Delta E_{W}$ is given by eq. (2.17) and $\mathcal{C}_{q}$ is a
constant factor which can be written as

$\displaystyle\mathcal{C}_{q}=\frac{\sqrt{\pi}d}{(2d-1)c^{d+1}}\frac{\Gamma\left(\frac{d}{2d-2}\right)}{\Gamma\left(\frac{1}{2d-2}\right)}-\frac{1}{2dc^{d}}.$
(2.24)

Therefore, the charged excitations decrease the EoP at leading order and hence
reduce the total correlation between the two subregions. Note that
$\mathcal{C}_{q}>0$ and thus the subleading term is positive. Further, using
eq. (2.21) we see that the corresponding HMI decreases at leading order. This
is perfectly consistent with eq. (2.23) since both HMI and EoP are measures of
total correlation between the subregions.

##  3 AdS Plane Wave Geometries

In this section, we would like to generalize the previous analysis to
determine corrections to the EoP in a class of boundary excited states which
are dual to the AdS plane wave geometries. These gravitational backgrounds are
dual to anisotropically excited states in a QFT with a constant energy flux.
The bulk geometry will be an asymptotically AdSd+1 black brane with a metric
given by [41]

$ds^{2}=\frac{R^{2}}{r^{2}}\left(-2dx^{+}dx^{-}+\sum_{i=1}^{d-2}dx_{i}^{2}+dr^{2}\right)+R^{2}\mathcal{T}r^{d-2}(dx^{+})^{2}.$
(3.1)

Here $\mathcal{T}$ is proportional to the boundary energy density, i.e.,
$\mathcal{T}\propto T_{++}$ and based on the null energy condition, the range
$\mathcal{T}>0$ is allowed. The $x^{+}$ and $x^{-}$ coordinates are light-like
defined as

$x^{\pm}=\frac{t\pm x_{d-1}}{\sqrt{2}}.$ (3.2)

As shown in [42, 43] the metric (3.1) arises from the boosted AdS black brane
in the limit of infinite boost. Let us recall that the boosted AdS black brane
geometries, also known as the regularized AdS plane waves, are dual to a
uniformly boosted strongly coupled large $N$ thermal state. The corresponding
metric is given by [34]

$\displaystyle
ds^{2}=\frac{R^{2}}{r^{2}}\left(-dt^{2}+dx_{d-1}^{2}+\gamma^{2}\frac{r^{d}}{r_{h}^{d}}\left(dt+v\,dx_{d-1}\right)^{2}+\sum_{i=1}^{d-2}dx_{i}^{2}+\frac{dr^{2}}{f(r)}\right),\hskip
28.45274ptf(r)=1-\frac{r^{d}}{r_{h}^{d}},$ (3.3)

where $v$ is the velocity and $\gamma^{-2}=1-v^{2}$. This geometry is dual to
a thermal plasma which is boosted along the $x_{d-1}$ direction with the
following expressions for temperature and energy density

$\displaystyle T=\frac{d}{4\pi\gamma r_{h}},\hskip
56.9055pt\mathcal{E}=\left(1+\frac{d}{d-1}\gamma^{2}v^{2}\right)\varepsilon.$
(3.4)

Note that defining $\mathcal{T}=\frac{(1+v)^{2}}{2(1-v^{2})r_{h}^{d}}$ and
considering $v\rightarrow 1$ while $r_{h}\rightarrow\infty$ one can show that
eq. (3.3) reduces to eq. (3.1). It is important to mention that the dual
states described by purely gravitational excitations in the bulk because the
stress tensor is the only operator that has a nonvanishing expectation value.

In the following, we first study the behavior of the correlation measures in a
boosted AdS black brane and then we examine their variation in the limit of
infinite boost. Note that, due to the boosting along a certain direction the
rotational symmetry in the boundary theory is broken and the corresponding HEE
shows a number of interesting features [44, 45]. It is worth to mention that
the HEE in AdS plane wave geometry has been studied in [46]. Also, the
behavior and phase transitions of HMI in this background has been analyzed in
[47].

###  3.1 EWCS in Boosted AdS Black Brane

In this section, we apply holographic prescription to find the EoP using eq.
(1.5) for configurations consisting of thin long strips. In order to
investigate the behavior of EoP, we consider two different types of boundary
subregions: (a) the width of the entangling region is orthogonal to the
direction of boost, i.e., $x_{d-1}$, called case A, or (b) the width of the
entangling region is along the $x_{d-1}$-direction, called case B.

### Case A

Considering the strip entangling region to be along $x_{1}$-direction, we can
parameterize the minimal hypersurface by $x_{1}=x(r)$. Then the HEE is
computed as

$\displaystyle
S=\frac{R^{d-1}L^{d-2}}{4G_{N}}\int\frac{dr}{r^{d-1}}\sqrt{1+v^{2}\gamma^{2}\frac{r^{d}}{r_{h}^{d}}}\sqrt{{x^{\prime}}^{2}+\frac{1}{f}}.$
(3.5)

By minimizing this entropy functional, we can determine the profile of $x(r)$.
Once again, we have a conserved quantity and hence finding the minimal
hypersurface is a straightforward exercise. Plugging the resultant profile in
the above functional, we are left with

$\displaystyle
S=\frac{R^{d-1}L^{d-2}r_{t}^{d-1}}{2G_{N}}\int_{0}^{r_{t}}\frac{dr}{r^{2d-2}}\frac{1+v^{2}\gamma^{2}\frac{r^{d}}{r_{h}^{d}}}{\sqrt{f}}\frac{1}{\sqrt{\left(\frac{r_{t}}{r}\right)^{2(d-1)}+v^{2}\gamma^{2}\frac{r_{t}^{d}}{r_{h}^{d}}\left(\left(\frac{r_{t}}{r}\right)^{d-2}-1\right)-1}},$
(3.6)

where the relation between $\ell$ and $r_{t}$ is given by

$\displaystyle\ell=2\int_{0}^{r_{t}}\frac{dr}{\sqrt{f}\sqrt{\left(\frac{r_{t}}{r}\right)^{2(d-1)}\frac{r_{h}^{d}+v^{2}\gamma^{2}r^{d}}{r_{h}^{d}+v^{2}\gamma^{2}r_{t}^{d}}-1}}.$
(3.7)

Next, we can also compute EoP in this symmetric configuration for a boundary
state dual to eq. (3.3). In this case the corresponding expression can be
written as

$\displaystyle
E_{W}=\frac{R^{d-1}L^{d-2}}{4G_{N}}\int_{r_{d}}^{r_{u}}\frac{dr}{r^{d-1}\sqrt{f}}\sqrt{1+v^{2}\gamma^{2}\frac{r^{d}}{r_{h}^{d}}}.$
(3.8)

Now we are equipped with all we need to study the desired correlation measures
in this background. Before examining the full dependence of different
correlation measures on boost parameters, we would like to study the leading
order variation of them. First, we expand eq. (3.7) in the limit $r_{t}\ll
r_{h}$ to find the leading corrections to $r_{t}$ compared to its static value

$r_{t}=r_{t}^{(0)}+\frac{\sqrt{\pi}}{(d-1)c^{d+2}}\left(\frac{\Gamma(\frac{d}{2d-2})}{\Gamma(\frac{1}{2d-2})}-\frac{\Gamma(\frac{d}{d-1})}{\Gamma(\frac{d+1}{2d-2})}\right)v^{2}\gamma^{2}\frac{\ell^{d+1}}{r_{h}^{d}},$
(3.9)

where $r_{t}^{(0)}$ is the turning point for $v=0$ given by eq. (2.11). Using
the above expression and eq. (3.6), the HEE can be written in terms of the
following expansion

$S=S^{(0)}+\frac{2\pi^{3/2}L^{d-2}\Gamma\left(\frac{d}{d-1}\right)}{c^{2}(d-1)\Gamma\left(\frac{d+1}{2d-2}\right)}v^{2}\gamma^{2}\ell^{2}\varepsilon,$
(3.10)

where again $S^{(0)}$ is the HEE for $v=0$, i.e., eq. (2.12). In comparing the
above expression with eq. (2.13), we see that

$\displaystyle\Delta S^{\prime}\equiv
S-S^{(0)}=\left(1+\frac{d+1}{d-1}v^{2}\gamma^{2}\right)\Delta S.$ (3.11)

Of course, this reproduces the result first found in [34]. Clearly, the HEE
increases after the boost transformation. Using this result the variation of
HMI becomes

$\displaystyle\Delta
I^{\prime}=\left(1+\frac{d+1}{d-1}v^{2}\gamma^{2}\right)\Delta I.$ (3.12)

Now we proceed to examine the behavior of EoP in low temperature limit. First,
we expand eq. (3.8) in the limit $r_{d}\ll r_{u}\ll r_{h}$ to find the leading
corrections to $E_{W}$. Next, we use eq. (3.9) for $r_{d}$ and $r_{u}$
separately to express the variation of EoP in terms of the boundary quantities
as follows

$\displaystyle\Delta
E_{W}^{\prime}\equiv{E_{W}}-E_{W}^{(0)}=\frac{d+1}{d-1}L^{d-2}v^{2}\gamma^{2}\ell(h+\ell)\varepsilon.$
(3.13)

We note again that using eq. (2.17), the above result can be written as
follows

$\displaystyle\Delta
E_{W}^{\prime}=\left(1+\frac{d+1}{d-1}v^{2}\gamma^{2}\right)\Delta E_{W}.$
(3.14)

Comparing the above result with eqs. (3.11) and (3.12) shows that the boost
transformation has the same effect on the correlation measures as expected.

Now we turn our attention to numerically evaluating the correlation measures
for different values of the parameters. We will mainly focus on three
dimensional boundary theory, because the interesting qualitative features of
the measures are independent of the dimensionality of the field theory. In
figure 2 we show the subtracted HEE, the corresponding HMI and EoP for
specific values of $h$ and $\ell$. For simplicity, we have rescaled the
holographic measures, i.e.,
$\\{S,I,E_{W}\\}\rightarrow\frac{4G_{N}}{R^{d-1}L^{d-2}}\\{S,I,E_{W}\\}$. The
left panel shows the dependence of HEE on the width of the entangling region
for different values of temperature and velocity. Note that the HEE is
regularized by subtracting the AdS contribution. The dashed violet curve
corresponds to AdS black brane geometry. The middle panel demonstrates the HMI
as a function of the dimensionless boundary quantity $h/\ell$. Here the dashed
purple curve corresponds to pure AdS geometry. In the right panel we show
$E_{W}$ for the same values of the parameters. Based on this plots, we observe
that although the HEE and EoP increase with the boost parameter, the HMI
decreases with $v$. Also the phase transition of EoP happens at smaller
separations between the two subregions comparing to $v=0$ case.

Figure 2: Subtracted HEE (left), HMI (middle) and EWCS (right) as functions of
the width and separation between subregions for different values of $r_{h}$
and $v$ in $d=3$.

### Case B

In this case the width of the strip entangling region is along
$x_{d-1}$-direction and we can parameterize the minimal hypersurface by
$x_{d-1}\equiv x(r)$. Given the metric in eq. (3.3) the corresponding entropy
functional becomes

$\displaystyle
S=\frac{R^{d-1}L^{d-2}}{4G_{N}}\int\frac{dr}{r^{d-1}}\sqrt{\left(1+v^{2}\gamma^{2}\frac{r^{d}}{r_{h}^{d}}\right){x^{\prime}}^{2}+\frac{1}{f}}.$
(3.15)

Applying the same steps as in our previous calculations, one can evaluate the
leading corrections to the HEE at small temperature as

$\Delta^{\prime\prime}S=\left(1+\frac{2}{d-1}v^{2}\gamma^{2}\right)\Delta S,$
(3.16)

where $\Delta S$ was defined in eq. (2.13). We again use the symmetry of the
configuration to evaluate the EoP in this case. The corresponding area
functional can be obtained by setting $x^{\prime}=0$ in eq. (3.15). Expanding
the integrand and evaluating the resultant expression we have

$E_{W}=\frac{R^{d-1}L^{d-2}}{4G(d-2)}\left(\frac{1}{r_{d}^{d-2}}-\frac{1}{r_{u}^{d-2}}\right)+\frac{R^{d-1}L^{d-2}}{16Gr_{h}^{d}}\left(r_{u}^{2}-r_{d}^{2}\right),$
(3.17)

which can be re-express as follows

$\Delta^{\prime\prime}E_{W}=\left(1+\frac{2}{d-1}v^{2}\gamma^{2}\right)\Delta
E_{W},$ (3.18)

where $\Delta E_{W}$ was defined in eq. (2.17). In obtaining the above result,
we have used that at leading order

$r_{t}=r_{t}^{(0)}-\frac{\sqrt{\pi}\gamma^{2}v^{2}\ell^{d+1}}{(d-1)c^{d+2}r_{h}^{d}}\left(\frac{2\Gamma\left(\frac{d}{d-1}\right)}{(d+1)\Gamma\left(\frac{d+1}{2d-2}\right)}-\frac{\Gamma\left(\frac{d}{2d-2}\right)}{\Gamma\left(\frac{1}{2d-2}\right)}\right).$
(3.19)

To close this section, we present plots of EoP in various dimensions in fig. 3
for several values of $v$. Clearly the EoP is an increasing function of the
boost parameter which is consistent with our previous results. Note that the
derivative of EoP for $v=0.9$ in the left panel has a discontinuity which is
due to the appearance of a new disconnected configuration for the
corresponding RT hypersurface. As discussed in [46], there is a disconnected
hypersurface simply given by $x^{\prime}=0$ such that for small regions the
connected surface has the minimal area, while for large regions the RT surface
changes topology and the disconnected surface is favored.

Figure 3: EoP for $d=3$ (left) and $d=4$ (right) as functions of the width and
separation between subregions with $r_{h}=1$ for different values of $v$.

###  3.2 EWCS in AdS Plane Wave: Case A

In this section, our goal is to calculate the variation of EoP in an AdS plane
wave geometry when the width of the strip is along the $x_{1}$-direction.
Recall that in this case the energy flow is parallel with the entangling
region. Upon substituting the profile of the minimal hypersurface, i.e.,
$x_{1}\equiv x(r)$ into the metric (3.1), the HEE functional simplifies to

$S=\frac{R^{d-1}L^{d-2}}{4G_{N}}\int\frac{dr}{r^{d-1}}\sqrt{(1+{x^{\prime}}^{2})(1+\frac{\mathcal{T}r^{d}}{2})}.$
(3.20)

Minimizing the above expression yields the equation of motion for $x(r)$ which
reads

${x^{\prime}}^{2}=\frac{2+\mathcal{T}r_{t}^{d}}{(2+\mathcal{T}r^{d})r_{t}^{2d-2}-(2+\mathcal{T}r_{t}^{d})r^{2d-2}}r^{2d-2}.$
(3.21)

We note again that $r_{t}$ is the turning point of the minimal hypersurface
which is fixed by setting $x^{\prime}(r_{t})=\infty$. Further, using the above
result the width of the entangling region and the HEE can be written in terms
of the following expressions

$\displaystyle\ell$ $\displaystyle=$ $\displaystyle
2\sqrt{2+\mathcal{T}r_{t}^{d}}\int_{0}^{r_{t}}dr\;r^{d-1}\;\frac{1}{\sqrt{(2+\mathcal{T}r^{d})r_{t}^{2d-2}-(2+\mathcal{T}r_{t}^{d})r^{2d-2}}},$
(3.22) $\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{L^{d-2}R^{d-1}}{2G_{N}}\int_{\epsilon}^{r_{t}}\frac{dr}{r^{d-1}}\frac{(2+\mathcal{T}r^{d})r_{t}^{d-1}}{\sqrt{2(2+\mathcal{T}r^{d})r_{t}^{2d-2}-2(2+\mathcal{T}r_{t}^{d})r^{2d-2}}}.$
(3.23)

Note that using the above expressions one can easily show that the divergent
part of HEE is the standard area law which does not depend on $\mathcal{T}$ as
we expected, since the metric (3.1) is asymptotically AdS. Also we can
evaluate the EWCS for the same configuration of entangling regions that we
considered before. Once again, since there is a reflection symmetry, we expect
that $\Sigma$ lies entirely on $x=0$ slice. Thus the EoP can be written as
follows

$E_{W}=\frac{R^{d-1}L^{d-2}}{4G_{N}}\int_{r_{d}}^{r_{u}}\frac{dr}{r^{d-1}}\sqrt{1+\frac{\mathcal{T}}{2}r^{d}}.$
(3.24)

In figure 4 we show the HEE, the corresponding HMI and EoP for specific values
of $h,\ell$ and $\mathcal{T}$. The left panel shows the subtracted HEE as a
function of $\ell$ for different values of $\mathcal{T}$. The middle panel
demonstrates the HMI as a function of $h/\ell$ which has a continuous
transition due to the competition between connected and disconnected
configurations. Again, the dashed purple curve corresponds to AdS black brane
geometry. In the right panel we show EoP for the same values of the
parameters. As is clear from the graphs, all three correlation measures are
monotonically increasing function of the energy density.

Figure 4: Subtracted HEE (left), HMI (middle) and EoP (right) as functions of
the width and separation between subregions for different values of
$\mathcal{T}$ in $d=3$.

To better understand these behaviors, let us explore the scaling of the
correlation measures in the specific regimes of the parameter space. A simple
analysis shows that in $\mathcal{T}\ell^{d}\gg 1$ limit, we have $\ell\sim
r_{t}$ and the finite part of the HEE takes the following form [46]

$S_{f}\sim\frac{R^{d-1}L^{d-2}}{G_{N}}\sqrt{\mathcal{T}}\begin{cases}\pm\ell^{2-\frac{d}{2}}&\qquad
d\neq 4\\\ \log(\ell\mathcal{T}^{1/4})&\qquad d=4\end{cases},$ (3.25)

where $+(-)$ sign is for $d<4(>4)$. Clearly, the finite part of HEE is always
smaller than the thermal entropy and is a monotonically increasing function of
$\mathcal{T}$ and $\ell$. The latter is due to the fact that the number of
degrees of freedom grows as we excite the system.

On the other hand, in $\mathcal{T}\ell^{d}\ll 1$ limit the metric (3.1) is a
small deformation of pure AdS, thus we can use a perturbative expansion to
compute the variation of HEE [47]. Again, in this case $r_{t}$ is close to the
boundary and from eq. (3.22) we have

$\ell=r_{t}\left(c+\frac{\sqrt{\pi}}{2(d-1)^{2}}\left(\frac{\Gamma(\frac{1}{d-1})}{\Gamma(\frac{d+1}{2d-2})}-(d-1)\frac{\Gamma(\frac{d}{2d-2})}{\Gamma(\frac{1}{2d-2})}\right)\mathcal{T}r_{t}^{d}\right).$
(3.26)

Further, in this limit, the leading order behavior of eq. (3.23) reduces to

$\Delta S=\frac{R^{d-1}L^{d-2}(d+1)\tilde{c}}{32\pi
G_{N}}\mathcal{T}\ell^{2},$ (3.27)

which again shows that HEE increases with $\mathcal{T}$. Combining the above
result with eq. (3.25) we can compute HMI in different scaling regimes. In
$1\ll h^{d}\mathcal{T}\ll\ell^{d}\mathcal{T}$ limit upon substituting eq.
(3.25) into eq. (1.4), we find

$I\sim\frac{R^{d-1}L^{d-2}}{G_{N}}\sqrt{\mathcal{T}}\begin{cases}\pm\left(2\ell^{2-\frac{d}{2}}-h^{2-\frac{d}{2}}-(2\ell+h)^{2-\frac{d}{2}}\right)&\qquad
d\neq 4\\\ \log\left(\frac{\ell^{2}}{h(2\ell+h)}\right)&\qquad
d=4\end{cases}.$ (3.28)

Subsequently, we can simply use eq. (3.27) in
$h^{d}\mathcal{T}\ll\ell^{d}\mathcal{T}\ll 1$ limit which yields

$\displaystyle\Delta I=-\frac{R^{d-1}L^{d-2}(d+1)\tilde{c}}{16\pi
G_{N}}\mathcal{T}(\ell+h)^{2}.$ (3.29)

This result indicates that HMI decreases when $\mathcal{T}$ is turned on.
Finally, we turn to the computation of EoP in this background which can be
determined by simply evaluating the integral in eq. (3.24) which gives an
exact result

$\displaystyle
E_{W}=\frac{R^{d-1}L^{d-2}}{4(d-2)G_{N}}r^{2-d}{{}_{2}}F_{1}\left(-\frac{1}{2},\frac{2}{d}-1,\frac{2}{d},-\frac{\mathcal{T}r^{d}}{2}\right)\bigg{|}_{r_{u}}^{r_{d}}.$
(3.30)

Now, in $1\ll h^{d}\mathcal{T}\ll\ell^{d}\mathcal{T}$ limit, the turning point
can be approximated by $r_{t}\sim\ell$ and we find

$E_{W}\sim\frac{R^{d-1}L^{d-2}}{8G_{N}}\sqrt{2\mathcal{T}}\begin{cases}\frac{2}{d-4}\left(h^{2-\frac{d}{2}}-(2\ell+h)^{2-\frac{d}{2}}\right)&\qquad
d\neq 4\\\ \log\left(1+\frac{2\ell}{h}\right)&\qquad d=4\end{cases}.$ (3.31)

Interestingly, we see that in this limit the EoP increases with $\mathcal{T}$
which is consistent with the results presented in Figure 4. On the other hand
in $h^{d}\mathcal{T}\ll\ell^{d}\mathcal{T}\ll 1$ limit we use eq. (3.26) for
$r_{u,d}$ to express the EoP as follows

$\Delta{E_{W}}=\frac{R^{d-1}L^{d-2}}{G_{N}}\frac{(d+1)\mathcal{C}}{32\pi}\mathcal{T}\ell(\ell+h),$
(3.32)

where $\mathcal{C}$ is defined in eq. (2.18). Recall that $\mathcal{C}$ is
negative and hence in this limit the EoP decreases as the energy injected to
the system increases. Based on this result and eq. (3.31), we conclude that
the EoP is not a monotonic function of $\mathcal{T}$. This is better shown in
figure 5 that reports the ratio $E_{W}/E_{W}(\mathcal{T}=0)$ versus
$\mathcal{T}$ for different values of the parameters in $d=3$. We do not
understand what is the reason for this behavior at present and leave this
issue for future study.333Similar results for the behavior of EoP in a
different setup has been reported in [48].

Figure 5: EoP as a function of the energy flux for different values of $h$ in
$d=3$ with $\ell=1$.

###  3.3 EWCS in AdS Plane Wave: Case B

In this section, we consider extending our analysis to the case where the
width of the strip is along the $x_{d-1}$-direction. Therefore, in terms of
the null coordinates we fix the boundary entangling region to be

$-\frac{\ell}{2\sqrt{2}}\leq x^{\pm}\leq\frac{\ell}{2\sqrt{2}},\qquad 0\leq
x_{1},\cdots,x_{d-2}\leq L.$ (3.33)

The corresponding HEE functional is given by

$S=\frac{R^{d-1}L^{d-2}}{4G_{N}}\int dr\frac{\mathcal{L}}{r^{d-1}},\hskip
56.9055pt\mathcal{L}=\sqrt{1-2(\partial_{r}x^{+})(\partial_{r}x^{-})+\mathcal{T}r^{d}(\partial_{r}x^{+})^{2}}.$
(3.34)

Minimizing the above expression yields the equations of motion for
$x^{\pm}(r)$. Since there is no explicit $x^{\pm}$ dependence, we have two
conserved momentums. After some work, the corresponding expression for $\ell$
and $S$ can be simplified as follows [46]

$\displaystyle\ell$ $\displaystyle=$ $\displaystyle
2\sqrt{2}\int_{0}^{r_{t}}dr\frac{r^{d-1}}{\sqrt{H(r)}}=-2\sqrt{2}\int_{0}^{r_{t}}dr\frac{r^{d-1}(\mathcal{T}r^{d}-B)}{\sqrt{H(r)}},$
(3.35) $\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{R^{d-1}L^{d-2}}{2G_{N}}\int_{\epsilon}^{r_{t}}\frac{\mathrm{d}r}{r^{d-1}}\frac{AB}{\sqrt{H(r)}},$
(3.36)

where $H(r)=\sqrt{A^{2}B^{2}-2Br^{2d-2}+\mathcal{T}r^{3d-2}}$ and $A$ and $B$
are integration constants which can be written as

$\displaystyle
A=\frac{r^{d-1}\mathcal{L}}{\mathcal{T}r^{d}\partial_{r}x^{+}-\partial_{r}x^{-}},\hskip
56.9055ptB=\frac{\mathcal{T}r^{d}\partial_{r}x^{+}-\partial_{r}x^{-}}{\partial_{r}x^{+}}.$
(3.37)

Further, the turning point is determined by

$\frac{A^{2}B^{2}}{r_{t}^{2(d-1)}}+\mathcal{T}r_{t}^{d}-2B=0.$ (3.38)

Next, we turn our attention to evaluating the EoP in this case. The expression
takes the same form as in pure AdS geometry and as a result we obtain

$E_{W}=\frac{R^{d-1}L^{d-2}}{4G_{N}}\int_{r_{d}}^{r_{u}}\frac{dr}{r^{d-1}}=\frac{R^{d-1}L^{d-2}}{4(d-2)G_{N}}\left(\frac{1}{r_{d}^{d-2}}-\frac{1}{r_{u}^{d-2}}\right).$
(3.39)

Before examining the behavior of the EoP for general values of the parameters
and dimension, it is instructive to analyze the particular case of two
dimensional boundary theory, since the correlation measures can be determined
analytically. In fact, this allows us to derive in detail some general
features of the EoP for the excited states. We will treat this case separately
in the following.

#### A case study: $d=2$

In this case, using eqs. (3.35) and (3.38) the relation between $r_{t}$ and
$\ell$ as a function of the energy flux can be expressed analytically in
closed form as follows [46]

$\displaystyle\ell\tanh\left(\sqrt{\frac{\mathcal{T}}{2}}\frac{\ell}{2}\right)=\sqrt{2\mathcal{T}}r_{t}^{2}.$
(3.40)

For $\mathcal{T}\ell^{2}\ll 1$, the turning point has the expansion

$\displaystyle
r_{t}=\frac{\ell}{2}\left(1-\frac{\mathcal{T}\ell^{2}}{48}+\cdots\right),$
(3.41)

which decreases monotonically from its pure AdS value of $\frac{\ell}{2}$ to
$0$, as the energy flux is increased from zero to infinity at fixed $\ell$.
Again, this result shows that turning on the perturbation the extremal
hypersurface moves to smaller values of $r$. Now using the above result and
eq. (3.36) for HEE, we obtain

$\Delta
S=\frac{R}{4G_{N}}\log\frac{\sinh(\sqrt{\frac{\mathcal{T}}{2}}\ell)}{\sqrt{\frac{\mathcal{T}}{2}}\ell}.$
(3.42)

Further, we can expand the variation of HEE for small and large
$\mathcal{T}\ell^{2}$, which yields

$\Delta
S=\frac{R}{4G_{N}}\begin{cases}\sqrt{\frac{\mathcal{T}}{2}}\ell-\log\left(\sqrt{2\mathcal{T}}\ell\right)+\cdots&\qquad\mathcal{T}\ell^{2}\gg
1\\\
\frac{\mathcal{T}\ell^{2}}{12}-\frac{1}{5}\left(\frac{\mathcal{T}\ell^{2}}{12}\right)^{2}+\cdots&\qquad\mathcal{T}\ell^{2}\ll
1\end{cases}.$ (3.43)

We note again that the excitations increase the HEE. With these expressions,
the variation of HMI can be evaluated as

$\Delta
I=\frac{R}{4G_{N}}\log\frac{h(2\ell+h)\sinh^{2}\left(\sqrt{\frac{\mathcal{T}}{2}}\ell\right)}{\ell^{2}\sinh(\sqrt{\frac{\mathcal{T}}{2}}h)\sinh(\sqrt{\frac{\mathcal{T}}{2}}(2\ell+h))}.$
(3.44)

As a result, in different scaling regimes we have

$\Delta
I=\frac{R}{4G_{N}}\begin{cases}-\frac{\mathcal{T}}{6}(\ell+h)^{2}+\cdots&\qquad\mathcal{T}h^{2}\ll\mathcal{T}\ell^{2}\ll
1\\\
-\log\sqrt{\frac{\mathcal{T}}{2}}\ell-\sqrt{\frac{\mathcal{T}}{2}}h+\cdots&\qquad\mathcal{T}h^{2}\ll
1\ll\mathcal{T}\ell^{2}\end{cases}.$ (3.45)

Again, we see that injecting a finite amount of energy into the system
decreases the HMI and hence reduce the mutual correlation between subregions.
Finally, using eqs. (3.39) and (3.40), we obtain the variation of EoP

$\Delta
E_{W}=\frac{R}{8G_{N}}\log\frac{h\tanh\left(\sqrt{\frac{\mathcal{T}}{2}}\frac{2\ell+h}{2}\right)}{(2\ell+h)\tanh\left(\sqrt{\frac{\mathcal{T}}{2}}\frac{h}{2}\right)}.$
(3.46)

Expanding in different scaling regimes, we thus have

$\Delta
E_{W}=\frac{R}{8G_{N}}\begin{cases}-\frac{\mathcal{T}}{6}\ell(\ell+h)+\cdots&\qquad\mathcal{T}h^{2}\ll\mathcal{T}\ell^{2}\ll
1\\\
-\log\sqrt{\frac{\mathcal{T}}{2}}\frac{\ell}{2}-\frac{h}{2\ell}+\cdots&\qquad\mathcal{T}h^{2}\ll
1\ll\mathcal{T}\ell^{2}\end{cases}.$ (3.47)

Here we see that the EoP is a decreasing function of the energy which is
consistent with our previous results. Further, comparing the above result with
eq. (3.43) we see that in the case of two adjacent intervals and small
perturbation we have $\Delta E_{W}=-\Delta S$ which shows that these measures
change with the same rate. This behavior resembles that found in eq. (2.19)
where the boundary excited state is an equilibrium state at finite
temperature.

#### Correlation Measures in Higher Dimensions

The integrals in eqs. (3.35) and (3.36) cannot be carried out analytically for
general $d>2$, so we turn our attention to numerically evaluating the
correlation measures in $d=4$ noting that the qualitative features of the
measures are independent of the dimensionality of the field theory. Indeed, as
shown in [46] in $d\geq 3$ there are two candidates for the minimal
hypersurface with the same endpoints at the asymptotic boundary. The first one
is the union of two disconnected straight lines extending from $r=\epsilon$ to
$r=\infty$. In this case we have $\partial_{r}x^{\pm}=0$ and the corresponding
HEE is independent of $\ell$. The second is a curved surface which extends
from $r=\epsilon$ to $r=r_{t}$ and whose area depends on $\ell$. In general,
there is a critical length $\ell_{c}$ such that for $\ell<\ell_{c}$ the
connected configuration has the minimal area, while for $\ell>\ell_{c}$ the RT
hypersurface changes topology and the disconnected configuration is favored.
Note that in this case the finite part of the HEE is given by $S_{f}=S_{\rm
con.}-S_{\rm dis.}$. The behavior of correlation measures can be read off from
Fig. 6. The left panel shows the finite part of the HEE as a function of the
width of the entangling region for different values of $\mathcal{T}$. From
this figure one finds that the critical width when the transition happens is
approximately given by $\mathcal{T}^{1/4}\ell_{c}\sim 1.17$. Note that the
dashed curve corresponds to pure AdS geometry with $\mathcal{T}=0$ where the
connected configuration is always favored for any $\ell$. The middle panel
demonstrates the HMI as a function of $\frac{h}{\ell}$ for $\mathcal{T}=1$. In
the right panel we show $E_{W}$ for the same values of the parameters.

Figure 6: Finite part of the HEE (left), HMI (middle) and EWCS (right) as
functions of the width and separation between subregions for different values
of $\mathcal{T}$ and $h$ in $d=4$.

We can also examine the behavior of the EoP in $\mathcal{T}\ell^{d}\ll 1$
limit. The analysis follows identically to the previous cases, hence we only
present the results. In this limit we can expand eqs. (3.35) and (3.36) to
find

$\Delta S=\frac{R^{d-1}L^{d-2}\tilde{c}}{16\pi G_{N}}\mathcal{T}\ell^{2},$

where $\tilde{c}$ is defined in eq. (2.14). We see that the variation of HEE
is positive and monotonically increasing as a function of $\mathcal{T}$. Using
this result one can show that the corresponding variation of HMI is negative
as expected. Also expanding eq. (3.39) to leading order yields

$\Delta E_{W}=\frac{R^{d-1}L^{d-2}\mathcal{C}}{16\pi
G_{N}}\mathcal{T}\ell(\ell+h),$

where $\mathcal{C}$ is defined in eq. (2.18). Again, in this limit the EoP
decreases as the energy injected to the system increases. Note that similar to
the previous case our numerical results show that the EoP is not a monotonic
function of the energy flux, although we do not explicitly show the
corresponding figures here.

##  4 Momentum Relaxation Geometries

In this section, we consider a specific class of boundary excited states in
which a scalar operator acquiring an expectation value. Based on holographic
dictionary the dual description involves adding a scalar field to the bulk. In
particular, we focus on a model of translational symmetry breaking in which
inhomogeneous scalar profiles can be constructed in such a way that the
corresponding geometry remains isotropic and homogeneous. In this model the
gravitational theory consists of Einstein-Maxwell theory with minimally
coupled massless scalar fields. As shown in [49], the spatially-dependent
marginal scalar operators cause momentum relaxation in the deformed dual field
theory. The corresponding metric in $d>2$ reads

$\displaystyle
ds^{2}=\frac{R^{2}}{r^{2}}\left(-f(r)dt^{2}+\sum_{i=1}^{d-1}dx_{i}^{2}+\frac{dr^{2}}{f(r)}\right),\hskip
28.45274ptf(r)=1-mr^{d}-\frac{\alpha^{2}r^{2}}{2(d-2)},$ (4.1)

where $\alpha$ is a constant which is proportional to the strength of the
source for the boundary scalar operator. From (4.1), one obtains that the
Hawking temperature and the horizon radius can be specified by

$\displaystyle T=\frac{d}{4\pi
r_{h}}\left(1-\frac{\alpha^{2}r_{h}^{2}}{2d}\right),\hskip
56.9055ptm=\frac{1}{r_{h}^{d}}\left(1-\frac{\alpha^{2}r_{h}^{2}}{2(d-2)}\right).$
(4.2)

On the other hand for $d=2$ the blackening factor becomes logarithmic, i.e.,

$\displaystyle f(r)=1-mr^{2}+\frac{\alpha^{2}r^{2}}{2}\log r.$ (4.3)

Further, the extremal limit is obtained with $\alpha^{2}r_{h}^{2}=2d$ and also
by setting

$\displaystyle m=\Bigg{\\{}\begin{array}[]{rcl}\frac{2}{(2-d)r_{h}^{d}}&{\rm
for}&d>2\\\ \frac{1+2\log r_{h}}{r_{h}^{2}}&{\rm for}&d=2\end{array}.$ (4.6)

Before we proceed, let us recall that different aspects of non-local probes
including HEE and HMI in this model has been studied in [50]. Indeed, as shown
in this paper for $d>2$ due to momentum relaxation effects, new logarithmic
universal terms may appear in the HEE. On the other hand, in the case of a two
dimensional boundary theory, the momentum relaxation leads to a non-critical
correction and the universal term remains unchanged. Further, the correlation
length is a decreasing function of momentum relaxation parameter and hence the
phase transition of HMI happens at smaller separation between the spatial
subregions.

Regarding the above observations we would like to study the effect of momentum
relaxation on EoP. As before, we consider a symmetric configuration consisting
of two parallel strips. The analysis follows identically to the previous cases
discussed in section 2, with the replacement of the blackening factor.
Therefore, the corresponding expressions for the width of the entangling
region and HEE are given by eqs. (2.7) and (2.8) respectively. Further the EoP
functional still takes the form presented in eq. (2.9). We copy it here for
convenience,

$\displaystyle
E_{W}=\frac{R^{d-1}L^{d-2}}{4G_{N}}\int_{r_{d}}^{r_{u}}\frac{dr}{r^{d-1}\sqrt{f(r)}}.$
(4.7)

Before examining the full $\alpha$-dependence of EoP, we would like to study
the variation of this quantity in more detail in $m=0,h\ll\ell\ll\alpha^{-1}$
regime which corresponds to the small deformation of the boundary state with
scalar condensation at finite temperature. Of course, the overall conclusions
are independent of the specific value of $m$, but we will focus on this regime
for concreteness. In this case, evaluating the above integral yields

$\displaystyle
E_{W}=\frac{R^{d-1}L^{d-2}}{4(2-d)G_{N}}\frac{1}{r^{d-2}}\,_{2}F_{1}\left(\frac{1}{2},1-\frac{d}{2},2-\frac{d}{2},\frac{\alpha^{2}r^{2}}{2(d-2)}\right)\bigg{|}_{r_{d}}^{r_{u}}.$
(4.8)

Now, expanding the above result in $h\ll\ell\ll\alpha^{-1}$ limit which
corresponds to $r_{d}\ll r_{u}\ll\alpha^{-1}$ we have

$\displaystyle
E_{W}=\frac{R^{d-1}L^{d-2}}{4(d-2)G_{N}}\left(\frac{1}{r_{d}^{d-2}}-\frac{1}{r_{u}^{d-2}}+\frac{\alpha^{2}}{4(d-4)}\left(\frac{1}{r_{d}^{d-4}}-\frac{1}{r_{u}^{d-4}}\right)\right).$
(4.9)

It is straightforward to write the above expression in terms of the boundary
quantities $h$ and $\ell$ to produce the result

$\displaystyle\Delta
E_{W}=\frac{R^{d-1}L^{d-2}c^{d-4}(1+\mathcal{C}^{\prime})}{16(d-2)(d-4)G_{N}}\left(\frac{1}{h^{d-4}}-\frac{1}{(2\ell+h)^{d-4}}\right)\alpha^{2},$
(4.10)

with

$\displaystyle\mathcal{C}^{\prime}=\frac{2\sqrt{\pi}(d-4)}{(d+2)c}\frac{\Gamma\left(\frac{3d}{2d-2}\right)}{\Gamma\left(\frac{2d+1}{2d-2}\right)}.$
(4.11)

Note that the overall coefficient in eq. (4.10) is positive for $d>4$ and
hence the scalar condensation can increase the EoP. We also note that this
result does not hold in $d=2,4$. It is easy to show that in
$h\ll\ell\ll\alpha^{-1}$ limit we have

$\displaystyle\Delta
E_{W}=\frac{R^{d-1}L^{d-2}}{32G_{N}}\alpha^{2}\Bigg{\\{}\begin{array}[]{rcl}\frac{1}{2}\log\frac{\ell}{h}&{\rm
for}&d=4\\\ \frac{1}{3}\log\alpha\ell&{\rm for}&d=2\end{array}.$ (4.14)

Based on the above expression and eq. (4.10), we conclude that the variation
of EoP is positive for $d>3$. It is worth to mention that considering the
extremal limit which corresponds to a boundary state at zero temperature, the
qualitative features of the above results do not change.

To close this section, we present plots of HEE, HMI and EoP in fig. 3 for
several values of the momentum relaxation parameter in a three dimensional
boundary theory. In the left panel, we present the HEE as a function of the
width of the entangling region. Clearly it is an increasing function of the
momentum relaxation parameter in accordance with the previous semi-analytic
results [50]. The middle panel demonstrates HMI as a function of $h/\ell$
which undergoes a continuous phase transition beyond which it is identically
zero. In the right panel we show EoP for the same values of the parameters
which has a discontinuous phase transition. Our numerical results make it
clear that in this case the phase transition of HMI and EoP happens at smaller
separation between the spatial subregions compared to pure AdS case with
$\alpha=0$.

Figure 7: The HEE (left), HMI (middle) and EoP (right) as a function of $h$
and $\ell$ for different values of the momentum relaxation parameter. Here we
have set $d=3$ and $m=0$.

##  5 General Analysis

In this section we extend our studies to general boundary excited states which
are dual to asymptotically AdS geometries. We start with the asymptotically
AdSd+1 metric in Fefferman-Graham (FG) coordinates444Note that in the
following, $z$ denotes the bulk radial coordinate in a generic asymptotically
AdS geometry, hence for the pure AdS spacetime $z=r$ as in previous sections.

$\displaystyle
ds^{2}=\frac{R^{2}}{z^{2}}\left(dz^{2}+g_{\mu\nu}(z,x^{\rho})dx^{\mu}dx^{\nu}\right)\hskip
56.9055ptg_{\mu\nu}(z,x^{\rho})=\eta_{\mu\nu}+\gamma_{\mu\nu}(z,x^{\rho}),$
(5.1)

where for small deviations from the vacuum we consider $|\gamma_{\mu\nu}|\ll
1$. In the following, we are interested in the first order correction of EoP
under a small perturbation. While this analysis can be done for a generic
entangling surface, in order to gain better insight into the properties of
EoP, let us illustrate the discussion with the example of a configuration
consisting of two parallel strips with arbitrary widths along, say, the
direction of $x_{1}\equiv x$. As in the previous sections, the symmetry of our
setup implies that the corresponding profile for the minimal hypersurface
$\Sigma$ parametrized as $x=x(z)$. Note that, for pure AdS geometry, i.e.,
$\gamma_{\mu\nu}=0$, the corresponding hypersurface lies entirely on a
constant time slice. Hence, we assume that in the perturbed geometry, the
profile for the minimal hypersurface can be given as follows

$\displaystyle x(z)=x_{0}(z)+x_{1}(z)+\cdots.$ (5.2)

In this case using eq. (5.1), the EoP functional can be written as

$\displaystyle
E_{W}=\frac{R^{d-1}L^{d-2}}{4G_{N}}\int_{\bar{z}_{d}}^{\bar{z}_{u}}\frac{dz}{z^{d-1}}\sqrt{g}\sqrt{g^{xx}+x^{\prime
2}},$ (5.3)

where $g\equiv\det(g_{ij})$. Here $\bar{z}_{u,d}$ denote the values of $z$ at
the intersection of $\Sigma$ and $\Gamma$s in the perturbed background which
coincide with $z_{u,d}$ for $\gamma_{\mu\nu}=0$. Thus to leading order

$\displaystyle\bar{z}_{u,d}={z}_{u,d}+{z}^{(1)}_{u,d}+\cdots.$ (5.4)

Using eq. (5.1) the first factor inside the integral in eq. (5.3) becomes

$\displaystyle\sqrt{g}=\sqrt{1+\gamma_{i}^{i}}=1+\frac{1}{2}\gamma_{i}^{i},$
(5.5)

where $\gamma_{i}^{i}=\delta_{ij}\gamma^{ij}$. We can also expand the final
factor in eq. (5.3) as follows

$\displaystyle\sqrt{g^{xx}+x^{\prime 2}}=\sqrt{1+x_{0}^{\prime
2}}\left(1-\frac{\gamma_{xx}-2x_{0}^{\prime}x_{1}^{\prime}}{2(1+x_{0}^{\prime
2})}\right).$ (5.6)

Combining the above results, we obtain the variation of EoP

$\displaystyle\Delta
E_{W}=\frac{R^{d-1}L^{d-2}}{4G_{N}}\left(\frac{z_{u}^{(1)}\sqrt{1+x_{0}^{\prime}(z_{u})^{2}}}{z_{u}^{d-1}}-\frac{z_{d}^{(1)}\sqrt{1+x_{0}^{\prime}(z_{d})^{2}}}{z_{d}^{d-1}}+\int_{z_{d}}^{z_{u}}\frac{dz\sqrt{1+x_{0}^{\prime
2}}}{2z^{d-1}}\left(\gamma^{i}_{i}-\frac{\gamma_{xx}-2x_{0}^{\prime}x_{1}^{\prime}}{1+x_{0}^{\prime
2}}\right)\right),$ (5.7)

where we subtracted the vacuum contribution which is in this case given by

$\displaystyle E_{W{\rm
AdS}}=\frac{R^{d-1}L^{d-2}}{4G_{N}}\int_{z_{d}}^{z_{u}}\frac{dz}{z^{d-1}}\sqrt{1+x_{0}^{\prime
2}}.$ (5.8)

Turning now to the symmetric configuration where the width of the subregions
are equal and $\Sigma$ lies entirely on $x_{0}=0$ slice, i.e.,
$x_{0}^{\prime}=0$, eq. (5.7) can be rewritten as follows

$\displaystyle\Delta
E_{W}=\frac{R^{d-1}L^{d-2}}{4G_{N}}\left(\frac{z_{u}^{(1)}}{z_{u}^{d-1}}-\frac{z_{d}^{(1)}}{z_{d}^{d-1}}\right)+\frac{R^{d-1}L^{d-2}}{8G_{N}}\int_{z_{d}}^{z_{u}}\frac{dz}{z^{d-1}}\left(\gamma^{i}_{i}-\gamma_{xx}\right).$
(5.9)

Notice that the first term in the above expression comes from the variation of
the RT hypersurfaces, i.e., the boundary condition for EWCS, while the second
term is the contribution coming from the variation of EoP functional.

The above calculations hold for a generic perturbation of the vacuum state at
leading order. Recall that the perturbation of the boundary vacuum state
described by different excitations in the dual geometry. Indeed, the deviation
of the bulk metric from pure AdS is given by [34]

$\displaystyle\gamma_{\mu\nu}=\frac{16\pi
G_{N}}{dR^{d-1}}z^{d}\sum_{n=0}z^{2n}\;T_{\mu\nu}^{(n)},$ (5.10)

where using the equations of motion one can show that $T^{(n)}_{\mu\nu}$ is
traceless and conserved, i.e.,

$\displaystyle T^{(n)\mu}_{\mu}=0,\hskip
56.9055pt\partial_{\mu}T^{(n)\mu}_{\nu}=0.$ (5.11)

Further, employing the Einstein equations, $T_{\mu\nu}^{(n>0)}$ can be
determined in terms of $T_{\mu\nu}^{(0)}$, i.e.,
$T^{(n)}_{\mu\nu}\propto\Box^{n}T^{(0)}_{\mu\nu}$. Throughout the following,
we will assume that the boundary excitations are uniform and hence
$T^{(n>0)\mu\nu}=0$. Equipped with eq. (5.9), we evaluate the variation of EoP
for three different kinds of excitations in more detail in the following. Let
us recall that similar analysis for the variation of HEE when the entangling
subregion is a sphere has been done in [34].

### Case i) $\langle T_{\mu\nu}\rangle\neq 0$

Let us first focus our attention on the boundary states described by purely
gravitational excitations in the dual geometry. In this case the stress tensor
is the only operator that has a nonvanishing expectation value. Assuming that
$\langle T_{\mu\nu}\rangle\sim\delta\ll 1$, the change in the EoP may receive
contributions at all orders in $\delta$.555Note that we introduce $\delta\ll
1$ to control the overall amplitude of the boundary excitations which allows
us to easily keep track of the perturbative expansion to all orders. Hence
after evaluating the corrections we set $\delta=1$. However, in the following,
our calculations will be to linear order in $\delta$.

Notice that, we would like to find the variation of the EoP in terms of
boundary quantities $\ell$ and $h$. We do so by first finding the leading
correction to the turning point of the RT hypersurface. It is relatively
straightforward to evaluate this correction to linear order in $\delta$, with
the result

$\displaystyle z_{t}=\frac{\ell}{c}+\frac{8\pi
G_{N}}{d(d-1)R^{d-1}}\left(T_{00}-\left(T_{00}-\frac{d-1}{d+1}T_{xx}\right)\frac{2\sqrt{\pi}}{c}\frac{\Gamma\left(\frac{d}{d-1}\right)}{\Gamma\left(\frac{d+1}{2d-2}\right)}\right)\left(\frac{\ell}{c}\right)^{d+1},$
(5.12)

where we have used $T^{i}_{i}=T_{00}$ due to the tracelessness of the stress
tensor. Upon substituting eqs. (5.10) and (5.12) into eq. (5.9) we may
evaluate the variation of EoP to find

$\displaystyle\Delta
E_{W}\left(T_{\mu\nu}\right)=\frac{\mathcal{C}}{d}L^{d-2}\bigg{(}(d+1)T_{00}-(d-1)T_{xx}\bigg{)}\ell(\ell+h),$
(5.13)

where $\mathcal{C}$ is defined in eq. (2.18). Clearly, the first term above is
completely determined by the energy density. Whereas the second term receives
contribution from the $xx$ component of the stress tensor. Precisely, the same
situation arose in [34] in investigating the structure of the first order
variation of the HEE. There it was shown that although for spherical
subregions only the energy density contributes to the variation of HEE, for
entangling surfaces with a less symmetric geometry, other components have
nontrivial contributions. For example, if we consider a strip entangling
region, the variation of HEE becomes [34]

$\displaystyle\Delta
S\left(T_{\mu\nu}\right)=\frac{\tilde{c}}{d}L^{d-2}\bigg{(}(d+1)T_{00}-(d-1)T_{xx}\bigg{)}\ell^{2},$
(5.14)

where $\tilde{c}$ is defined in eq. (2.14). For the case of $d=2$, comparing
the above result to eq. (5.13) shows that for adjacent intervals at leading
order $\Delta E_{W}=-\Delta S$. Further, in the case of an isotropic
background, i.e., $T_{xx}=T^{i}_{i}/(d-1)$, eq. (5.13) yields

$\displaystyle\Delta E_{W}=\mathcal{C}L^{d-2}\ell(\ell+h)T_{00}.$ (5.15)

Recall that $\mathcal{C}<0$ and hence the above result demonstrates that the
excitations decrease the EoP. One simple consistency check on our result is
that in the case of AdS black brane geometry, we recover the expected results
eqs. (2.17) and (2.23). We would also like to stress that for the present
case, the variation of HEE is always positive. More explicitly, using eq.
(5.14) the leading correction to HEE for isotropic backgrounds is given by

$\displaystyle\Delta S=\tilde{c}L^{d-2}\ell^{2}T_{00}.$ (5.16)

This result indicates that the corresponding HMI decreases, similar to what
was observed above for the EoP.

### Case ii) $\langle T_{\mu\nu}\rangle\neq 0,\;\langle J_{\mu}\rangle\neq 0$

Suppose the boundary theory has a conserved $U(1)$ current $J^{\mu}$ such that
$\langle J^{0}\rangle$ is the corresponding charge density. This means that
there should be a massless gauge field $A_{\mu}$ in the bulk whose boundary
value determines the chemical potential conjugate to $\langle J^{0}\rangle$.
Taking into account the leading backreaction of the gauge field on the
geometry, the relevant part of the metric perturbation in the FG expansion
will take the form [34]

$\displaystyle\gamma_{\mu\nu}=\frac{16\pi
G_{N}}{dR^{d-1}}z^{d}T_{\mu\nu}+\frac{z^{2(d-1)}}{4(d-1)^{2}(d-2)}\left(2(1-d)J_{\mu}J_{\nu}+\eta_{\mu\nu}J^{2}\right).$
(5.17)

Note that the leading order contribution to EoP coming from the stress tensor
is given by eq. (5.13) and so in the following, we focus on the contribution
due to the second term above. Assuming an isotropic current density, i.e.,
$J_{x}^{2}={\vec{J}}^{\,2}/(d-1)$, we may evaluate the variation in eq. (5.9)
to find

$\displaystyle\Delta
E_{W}\left(J^{\mu}\right)=\frac{R^{d-1}L^{d-2}}{16G_{N}c^{d}}\frac{d-1}{d(d-2)(2d-1)}\left((J^{0})^{2}+{\vec{J}}^{\,2}\right)\left((2\ell+h)^{d}-h^{d}\right).$
(5.18)

This result indicates that EoP increases when the current density is turned
on. Hence we find that the correlation between subsystems is enhanced by
adding the chemical potential. On the other hand one can evaluate the
variation of HEE due to the current as follows

$\displaystyle\Delta
S\left(J^{\mu}\right)=-\frac{R^{d-1}L^{d-2}}{16G_{N}c^{d-1}}\frac{1}{(d-2)(2d-1)}\left((J^{0})^{2}+{\vec{J}}^{\,2}\right)\ell^{d},$
(5.19)

which is always negative. Based on this result one can show that the current
perturbations produce a positive contribution to the change in the HMI which
is consistent with the behavior of EoP. Finally, note that the qualitative
features of the above results are in complete agreement with eqs. (2.21) and
(2.23).

### Case iii) $\langle T_{\mu\nu}\rangle\neq 0,\;\langle\mathcal{O}\rangle\neq
0$

Now consider perturbing the boundary theory by adding a scalar operator with
scaling dimension $\Delta$. This boundary operator is dual to a massive bulk
scalar field whose mass should satisfy $m=\Delta(d-\Delta)$. Note that the
unitarity implies a lower bound on $\Delta$, i.e., $\Delta>\Delta_{\rm
min}=\frac{d}{2}-1$. In the absence of any sources the leading backreaction of
the scalar field on the geometry in the FG expansion will take the form [34]

$\displaystyle\gamma_{\mu\nu}=\frac{16\pi
G_{N}}{dR^{d-1}}z^{d}T_{\mu\nu}-\frac{\mathcal{N}^{2}}{4(d-1)}z^{2\Delta}\eta_{\mu\nu}\mathcal{O}^{2},$
(5.20)

where $\mathcal{N}$ is a normalization constant and $\mathcal{O}$ denotes the
scalar condensate. Again, the leading order contribution coming from the
stress tensor is given by eq. (5.13) and so in the following, we focus on the
contribution due to the second term. It is relatively straightforward to
evaluate the leading correction to EoP for a uniform scalar condensate with
the result

$\displaystyle\Delta
E_{W}\left(\mathcal{O}\right)=\frac{R^{d-1}L^{d-2}\widetilde{\mathcal{C}}}{16G_{N}}\left((2\ell+h)^{2\Delta-d+2}-h^{2\Delta-d+2}\right)\mathcal{N}^{2}\mathcal{O}^{2},$
(5.21)

where

$\displaystyle\widetilde{\mathcal{C}}=\frac{-\Delta}{(d-1)(2\Delta-d+2)c^{2\Delta-d+2}}\left(1-\sqrt{\pi}\frac{2\Delta-d+2}{(d-1)c}\frac{\Gamma\left(\frac{2\Delta+d}{2d-2}\right)}{\Gamma\left(\frac{2\Delta+2d-1}{2d-2}\right)}\right).$
(5.22)

Using the unitarity bound, one can show that $\widetilde{\mathcal{C}}$ flips
its sign at a critical point which can be approximated by $\Delta-\Delta_{\rm
min}\sim\frac{\pi}{4}$ (see figure 8). Thus for a sufficiently relevant scalar
operator $\Delta E_{W}\left(\mathcal{O}\right)$ is negative which means that
the scalar condensate decreases the EoP. Note that the total variation of the
EoP for an excited state with nontrivial expectation values for stress tensor
and scalar operator can be obtained by combining eqs. (5.13) and (5.21).
Assuming that the magnitude of $T_{\mu\nu}$ and $\mathcal{O}$ is controlled by
a single scale $M$ we have

$\displaystyle\frac{\Delta E_{W}\left(\mathcal{O}\right)}{\Delta
E_{W}\left(T_{\mu\nu}\right)}\sim\left(M\mathcal{L}\right)^{2\Delta-d}.$
(5.23)

where $\mathcal{L}$ is a length scale depending on $h$ and $\ell$. Since we
are working in a regime where $M\mathcal{L}\ll 1$, for $\Delta<\frac{d}{2}$
the contribution due to the scalar condensate is dominant.

Figure 8: $\widetilde{\mathcal{C}}$ as a function of scaling dimension for
different values of $d$. The vertical dashed line at $\Delta-\Delta_{\rm
min}\sim\frac{\pi}{4}$ indicates the critical point where $\Delta
E_{W}\left(\mathcal{O}\right)$ flips its sign.

Further, one can compute the variation of HEE in the present case as follows

$\displaystyle\Delta
S\left(\mathcal{O}\right)=-\frac{R^{d-1}L^{d-2}}{16G_{N}}\frac{\Delta}{(d-1)(2\Delta-d+2)c^{2\Delta-d+2}}\frac{\sqrt{\pi}\Gamma\left(\frac{2\Delta+d}{2d-2}\right)}{\Gamma\left(\frac{2\Delta+2d-1}{2d-2}\right)}\ell^{2\Delta-d+2}\mathcal{N}^{2}\mathcal{O}^{2}.$
(5.24)

Again, considering the unitarity bound, the overall factor in the above
expression is negative and hence turning on the scalar operator decreases the
HEE. Also note that based on the above result the variation of HMI is
positive. This behavior contrast with the corresponding result for the
variation of EoP when $\mathcal{O}$ is sufficiently relevant. Interestingly,
while both HMI and EoP are measures of total correlation between subregions,
they do not behave in the same manner as the vacuum state perturbed by a
scalar operator. We do not fully understand what is the reason for this
behavior and leave it for future study. Note that based on figure 8 for
sufficiently irrelevant operators, the EoP and HMI behave in the same manner.

##  6 Conclusions and Discussions

In this paper, we explored the behavior of entanglement wedge cross section
(EWCS) in different asymptotically AdS geometries dual to boundary excited
states. We used the holographic proposal established in [7] for computing this
quantity which gives $E_{W}$ in terms of the minimal cross-sectional area of
the entanglement wedge. In particular, we computed the variation of EWCS under
the small perturbations away from the vacuum, when generic operators acquire
nontrivial expectation values. To get a better understanding of the results,
we also compared the variation of $E_{W}$ to other correlation measures
including HEE and HMI. Our study was mainly for a symmetric configuration
consisting of two disjoint strips with equal width, which is the simplest case
to utilize the holographic proposal to compute the mixed state correlation
measures. However, we expect that the qualitative features of our results are
independent of the specific configuration. Although, for finite excitations we
did a numerical analysis, considering small perturbations around the vacuum,
we evaluated the leading order variation of holographic correlation measures
analytically.

A key observation was that the variation of RT hypersurfaces plays a central
role in evaluating the correction to EWCS even at leading order. This is
completely different from what happens for the HEE (or HMI) in the same setup,
where any change in the profile of the RT hypersurface does not contribute to
the first order correction.

Our analysis in this paper focused mainly on boundary excited states described
by purely gravitational excitations in the bulk where the stress tensor is the
only operator that has a nonvanishing expectation value. Generally, the
leading correction to $E_{W}$ is negative which means that the excitations
decrease the correlation between the subregions and hence promote
disentangling between them. Of course, this is in agreement with the previous
results for the variation of HMI in the same setup [51]. Regarding the EWCS
and HMI as measures of total correlation between the subregions, this behavior
seems reasonable. However, this result is different from what happens for the
HEE which an increasing function of the excitation parameter due to the fact
that the number of degrees of freedom grows as we excite the system. These
results hold in different excited states dual to perturbed bulk geometries
including AdS black brane and AdS plane wave backgrounds. Further, correction
to EWCS in a general asymptotically AdS geometry with purely gravitational
excitations is given in eq. (5.13). In this case, we found that based on the
symmetry of the entangling regions a combination of the components of the
stress tensor contribute to the variation of $E_{W}$.

In extending these calculations to states in which additional matter fields
are excited in the dual geometry, we considered two different types of
excitations: perturbing with a current and perturbing with a scalar
condensate. Upon adding the extra matter fields, a number of interesting
features arose. First, the EWCS increases when the current density is turned
on and hence the total correlation between the subregions increased. This is
consistent with the variation of HMI being positive in this case. Second, the
scalar condensate produces a negative contribution to the change in the EWCS,
when the scalar operator is sufficiently relevant and its dimension respects
the unitarity bound. Further, comparing the contributions to $\Delta E_{W}$
due to the stress tensor and scalar operator, we see that when the scaling
dimension of the scalar operator satisfy $\Delta<d/2$, the contribution due to
the scalar condensate is dominant.

At this point, let us recall that there exist different correlation measures
which can be regarded as the holographic dual to EWCS, e.g., entanglement of
purification (EoP), reflected entropy and odd entropy. Given the holographic
prescriptions in eqs. (1.6) and (1.7), it is clear that all the above results,
up to overall multiplicative constant, hold for the EoP and reflected entropy.
However, the situation is different for odd entropy where the corresponding
expression is given by eq. (1.8). Remember that we always consider the regime
where the connected configuration is favored, i.e., $S_{A\cup
B}=S(2\ell+h)+S(h)$, it is straightforward to compute the odd entropy using
our previous results for different excited states. In general, the explicit
expression for $S_{O}$ is somewhat complicated, so we will not write it out
here. However, when $A$ and $B$ are two adjacent intervals in an excited state
of a 2-dimensional CFT, the resultant odd entropy has a rather simple
expression. In particular, when $T_{\mu\nu}$ is the only operator that has a
nonvanishing expectation value, using eqs. (5.15) and (5.16) we have

$\displaystyle\Delta S_{O}=\pi T_{00}\ell^{2},$ (6.1)

which is positive. Based on this result we see that the odd entropy increases
under the thermal excitations (see section 2). Of course, this behavior is in
qualitative agreement with the earlier QFT results obtained in [28]. There the
authors considered a two dimensional scalar field theory and found that the
odd entropy is an increasing function of the temperature.

We can extend this study to different interesting directions. An interesting
question is if either of these behaviors is reflected in the QFT calculations
of mixed state correlation measures dual to EWCS. Indeed, free field theories
provide a particularly well-controlled setup for studying various aspects of
the corresponding measures, e.g., EoP [4, 5] and reflected entropy [29, 30].
Let us emphasize that to the best of our knowledge, these studies mainly
focused on vacuum (Gaussian) states without any excitations or Ising model at
the critical point which has conformal symmetry. Notice that the analysis in
[28] includes the finite temperature corrections to odd entropy in free scalar
theories in which mixed states of interest, e.g., thermal states, are again
Gaussian. To better clarify the behavior of these measures in generic excited
states, it would be interesting to extend these studies to more general
setups, e.g., [52, 53, 54]. This analysis can help us to examine our results
and also choose a unique holographic counterpart for EWCS among different
proposals.

In this paper we restricted our discussion to the symmetric configuration for
the boundary entangling regions which significantly simplifies the calculation
of the EWCS. It is interesting to consider more general configurations where
the widths of the strips are different, using the techniques of [55]. Another
topic to explore would include extending our results to field theories dual to
higher curvature or massive gravity theories as in [56, 57]. Finally it would
be interesting to study EWCS in nonrelativistic holographic setups with
nontrivial excitations. We plan to explore some of these directions in the
near future.

### Acknowledgements

We would like to thank Mohammad Hasan Vahidinia for ongoing collaboration on
related ideas. We would also like to thank Ali Mollabashi for clarifications
about aspects of ref. [28]. KBV thank the School of Particles and
Accelerators, Institute for Research in Fundamental Sciences (IPM) for
hospitality where part of this work was carried out.

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