aainstitutetext: Department of Electrophysics, National Yang Ming Chiao Tung
University, Hsinchu, ROCbbinstitutetext: Center for Theoretical and
Computational Physics, National Yang Ming Chiao Tung University, Hsinchu,
ROCccinstitutetext: National Center for Theoretical Physics, ROC

# Page Curve of Effective Hawking Radiation

Chia-Jui Choua, Hans B. Laoa, Yi Yanga,b,c

###### Abstract

We study the generalized entanglement entropy in the higher dimensional two-
sided eternal black hole by double holography. By introducing a moving ETW
brane, which defines the effective Hawking radiation region, we find a new
type of Ryu-Takayanagi surface besides the Hartman-Maldacena surface and the
island Ryu-Takayanagi surface known previously. We study the phase transition
among the three Ryu-Takayanagi surfaces at different temperatures and obtain
the phase diagram as well as the Page curve.

##  1 Introduction

The calculations of Hawking showed that black holes have thermodynamic
properties like temperature and entropy. The idea is that a pair production
that occurs near the horizon could lead to one particle to fall into the black
hole and the other to escape to infinity, this leads to the well known Hawking
radiation BF02345020 ; PhysRevD.13.191 . Since a black hole is formed from a
pure state, this state should evolve in a way that it ends up in a final state
that is also pure if the evolution were to obey unitarity. However, as the
black hole evaporates completely, all that is left is thermal radiation so
that the final state is in a mixed state. Apparently, the information that
fell into the black hole vanishes. This leads to the black hole information
paradox and violates unitarity which is one of the fundamental principle of
quantum mechanics.

Don Page showed that if a black hole is formed from a pure state and
evaporates unitarily then the von Neumann entropy of Hawking radiation should
initially rise until the so-called Page time when it starts to drop down to
zero as the black hole completely vanishes. This corresponds to the process
where information can leak out from the black hole and is encoded in the
Hawking radiation. However, this can only happen at late times past the Page
time when the black hole has evaporated around half of its original state.
This trend followed by the Hawking radiation is called the Page curve 9306083
. Over the years, there were many proposals in addressing the black hole
information paradox 1207.3123 ; 1211.6767 ; 1301.4504 ; 1304.6483 ; 1306.0533
; 1308.4209 ; 1310.6334 ; 1607.05141 ; 1712.04955 . It was believed that the
Page curve can only be obtained in the quantum gravity theory.

It has been shown by Bekenstein and Hawking that the entropy of a black hole
is proportional to its horizon area which is the so-called Bekenstein-Hawking
(BH) formula BF02757029 ; PhysRevD.7.2333 . This is a clear demonstration of
the connection between a quantum-mechanical quantity and a geometric quantity.
This connection was later generalized by Ryu and Takayanagi through the
AdS/CFT correspondence, where they proposed a prescription connecting the
entanglement entropy of a region $\mathcal{A}$ in a field theory to an area of
a codimension two minimal surface $\mathcal{E}_{\mathcal{A}}$, called the RT
surface, in its dual bulk spacetime 0603001 ; 0605073 ; 0705.0016 . The
entanglement entropy calculated in this way is called the holographic
entanglement entropy.

To include the quantum correction, the generalized entropy (fine-grained or
von Neumann entropy) in a CFT was proposed in 1307.2892 . Soon after, the
holographic prescription for the generalized entropy was proposed by including
the bulk entropy and finding a quantum extremal surface (QES) 1408.3203 ;
1904.08423 . Remarkably, based on the entanglement wedge reconstruction
1601.05416 , the authors in 1905.08255 ; 1905.08762  showed that the Page
curve can be obtained in the semiclassical calculation of the generalized
entropy by holographic correspondence.

The QES in the context of a two-dimensional Jackiw-Teitelboim (JT) gravity
theory coupled to non-gravitational conformal matter, i.e. the so called
thermal bath, has been studied in 1908.10996 ; 1911.03402 . The bulk entropy
was calculated by the double holography scheme. It was found that in order for
the Hawking radiation to follow the Page curve for an evaporating black hole,
an island should emerge along with a QES inside the horizon at late times to
account for unitarity. While for the eternal black hole, it was showed that
the QES could be outside the horizon 1910.11077 ; 2004.00598 ; 2004.01601 ;
2107.14746 . The island formula has soon been extended to higher dimensional
spacetime 1910.12836 ; 1911.09666 ; 2004.05863 ; 2005.08715 ; 2006.02438 ;
2006.04851 ; 2006.10754 ; 2006.11717 ; 2006.16289 ; 2007.06551 ; 2007.15999 ;
2010.00018 ; 2010.00037 ; 2010.04134 ; 2010.16398 ; 2012.03983 ; 2012.05425 ;
2012.07612 ; 2101.05879 ; 2101.10031 ; 2101.12529 ; 2102.01810 ; 2102.02425 ;
2103.15852 ; 2103.16163 ; 2104.00006 ; 2104.02801 ; 2104.07039 ; 2104.13383 ;
2105.00008 ; 2105.01130 ; 2105.08396 ; 2105.09106 ; 2106.07845 ; 2106.10287 ;
2106.11179 ; 2106.12593 ; 2107.00022 ; 2107.01219 ; 2107.03390 .

Later on, coupling the JT gravity theory to a thermal bath at a finite
temperature has been studied in 2007.11658 , where the doubly holographic bulk
spacetime is a black hole instead of a pure AdS spacetime. However, coupling a
higher dimensional gravity theory to a thermal bath at a finite temperature is
still not clear.

On the other hand, boundary conformal field theory (BCFT) is a conformal field
theory defined on a manifold with boundaries where suitable boundary
conditions are imposed 9505127 . Early studies of holographic dual of defect
or interface CFT can be found in 0011156 ; 0111135 . The holographic dual of
BCFT by including extra boundaries in the gravity dual was proposed in
1105.5165 ; 1108.5152 ; 1205.1573 . In other words, a holographic construction
of conformal field theories with boundaries can be established with some
appropriate boundary conditions. Using holographic BCFT to study the Page
curve has been addressed in 1910.12836 ; 1911.09666 ; 2006.16289 .

Figure 1: The three-dimensional illustration of the BCFT model where the
different RT surfaces are displayed. The green inner cylinder is the Hartman-
Maldacena surface, the red hat-like surfaces are the boundary RT surfaces, and
the blue hemispheres are the island RT surfaces. The boundary/ETW brane of the
CFT is represented by the outer cylinder, while the Planck brane is the axis
in the middle of the cylinder.

In this paper, we study the holographic entanglement entropy of a
($d+1$)-dimensional BCFT at finite temperatures, whose dual bulk spacetime is
an asymptotically $\rm{AdS}_{d+2}$ black hole. We introduce a moving end-of-
the-world (ETW) brane which is the hypersurface that the earliest Hawking
radiation can reach. This ETW brane defines a time-dependent effective
radiation region, which supports a new type of RT surface instead of the two
known ones, see the sketch in Fig.1. The two black circles on the two ends
represent the entanglement surfaces. There are three possible RT surfaces that
may appear. One is the two disconnected blue hemispheres that anchors on the
entanglement surfaces. Another is the green inner cylinder which connects the
entanglement surfaces directly. In addition, there is a third one, the two red
hat-like surfaces, if there exists an ETW brane represented by the outer
cylinder. This RT surface anchors on both the entanglement surfaces and the
ETW brane. We examine the competition among the three RT surfaces that may
appear in the course of the evolution of the Page curve by studying the phase
transitions among them and the phase diagram.

This paper is organized as follows. In section 2, we review the basics of
holographic entanglement entropy and discuss our setup including the
purification, double holography, and BCFT that will be used in our work. In
section 3, we calculate the holographic entanglement entropy for the three RT
surfaces. In section 4, we examine the phase transitions among the RT surfaces
then discuss the phase diagram and the Page curve at different temperatures.
We conclude our results in section 5.

##  2 Background and Setup

###  2.1 Generalized Entanglement Entropy

Consider a Hilbert space $\mathcal{H}$ of a quantum field theory on a Cauchy
time slice and divide $\mathcal{H}$ into two regions $\mathcal{A}$ and its
complement $\mathcal{A}^{c}$. We denote the density matrix of the states in
$\mathcal{H}$ by $\rho$. The entanglement entropy between the states in
$\mathcal{A}$ and $\mathcal{A}^{c}$ is defined as,

$\mathcal{S}_{\mathcal{\mathcal{A}}}=-\rm{Tr}\left[\rho_{\mathcal{\mathcal{A}}}\ln\rho_{\mathcal{A}}\right],$
(1)

where

$\rho_{\mathcal{\mathcal{A}}}=\rm{Tr}_{\mathcal{A}^{c}}\rho,$ (2)

is the reduced density matrix of the region $\mathcal{A}$.

Assuming the quantum field theory has a dual gravity theory living in a higher
dimensional asymptotic AdS bulk spacetime $\mathcal{M}$ through the
holographic correspondence 9711200 ; 9802150 , the entanglement entropy
between $\mathcal{A}$ and $\mathcal{A}^{c}$ can be calculated by the Ryu-
Takayanagi (RT) formula 0603001 ; 0705.0016 ,

$S^{\rm{class}}_{\mathcal{A}}=\underset{\mathcal{E}_{\mathcal{A}}}{\rm{min}}\frac{\text{Area}(\mathcal{E}_{\mathcal{A}})}{4G_{N}},$
(3)

where $G_{N}$ is the Newton constant and $\mathcal{E}_{\mathcal{A}}$ is a
codimension two RT surface in the bulk spacetime $\mathcal{M}$ which is
anchored on the boundary $\partial\mathcal{A}$ of the entangling region
$\mathcal{A}$. It is important that the RT surface $\mathcal{E}_{\mathcal{A}}$
is homologous to $\mathcal{A}$. The presence of the boundaries $\mathcal{Q}$
in the BCFT spacetime allows the RT surface $\mathcal{E}_{\mathcal{A}}$ to
reach the boundary $\mathcal{Q}$ as shown in Fig.2.

(a) Sunset

(b) Rainbow

(c) Sky

Figure 2: Three possible RT surfaces $\mathcal{E}_{\mathcal{A}}$ of an
entangling region $\mathcal{A}$ extending to the bulk spacetime $\mathcal{M}$
where the holographic dual gravity theory lives. The colored regions represent
the entanglement wedges in different configurations.

To include the quantum correction, we need to consider the generalized
holographic entanglement entropy 1307.2892 ,

$S^{\rm{gen}}_{\mathcal{A}}(\mathcal{E}_{\mathcal{A}})=\frac{\text{Area}(\mathcal{E}_{\mathcal{A}})}{4G_{N}}+\mathcal{S}_{\rm{bulk}}(\mathcal{E}_{\mathcal{A}}),$
(4)

where $\mathcal{S}_{\rm{bulk}}(\mathcal{E}_{\mathcal{A}})$ is the bulk entropy
between the entanglement wedge $\mathcal{R}_{\mathcal{A}}$ bounded by
$\mathcal{A}\cup\mathcal{E}_{\mathcal{A}}$, and its complementary region in
the bulk space.

The fine-grained entanglement entropy is defined as the minimal value of the
generalized entanglement entropy,

$\mathcal{S}_{\mathcal{A}}=\underset{X}{\rm{min}}\left[S^{\rm{gen}}_{\mathcal{A}}(X)\right]=\underset{X}{\rm{min}}\left[\frac{\text{Area}(X)}{4G_{N}}+\mathcal{S}_{\rm{bulk}}(X)\right],$
(5)

where $X$ is the QES that minimizes the generalized entanglement entropy
$S^{\rm{gen}}_{\mathcal{A}}(X)$.

###  2.2 Black Hole Coupled to Thermal Bath

In the papers of $\rm AEM^{4}Z$ 1905.08762 ; 1908.10996 , a $2\rm d$ Jackiw-
Teitelboim (JT) gravity on $\rm AdS_{2}$ is coupled to a matter $\rm CFT_{2}$.
This 2d system is holographically dual to a 1d quantum mechanical system. If
this $2\rm d$ theory containing a black hole is coupled to a bath which
consists of the same $\rm CFT_{2}$ but now living on a flat space, then the
black hole is permitted to evaporate into the bath.

To study the Hawking radiation for an evaporating black hole, one has to deal
with a time-dependent spacetime. Alternatively, one can consider a thermofield
double state (TFD) that is dual to a two-sided eternal black hole 1910.11077 .
The Penrose diagram of the eternal black hole system is shown in Fig.3. There
are two copies of the CFT, $\rm{CFT}_{L}$ and $\rm{CFT}_{R}$, living on the
boundaries of two copies of the exterior region (shaded in lighter blue
color), which are connected by a wormhole (shaded in darker blue color). The
bifurcation dashed lines are the event horizons of the eternal black hole.

Figure 3: The Penrose diagram of the two-sided eternal black hole. The violet
dashed lines represent the event horizon of the black hole ($z=z_{h}$), which
separates the black hole into interior (darker blue) and exterior (lighter
blue) regions. Two copies of the CFT live on the two boundaries of the eternal
black hole. Two copies of the bath (light yellow) are coupled to the
boundaries of the black hole. The vertical dashed curves are the cutoff at
$x=b_{L,R}$. The vertical solid curves represent the ETW brane at different
times. The horizontal curves are Cauchy slices at different boundary times
with the red, green, and blue fragments representing the entanglement regions
of the Hawking radiation. The 45 degree lines with arrows are the outgoing
null surfaces on which the Hawking radiation travels.

The system is time-invariant as time evolves forward on one side while
backwards on the other side. However, one can consider time evolving forward
on both sides to introduce the time dependence in this system 1303.1080 .

In addition, in order to allow the eternal black hole to radiate, we couple
two copies of an auxiliary thermal bath (shaded in light yellow color) at the
boundaries of the two exterior regions. The two copies of the thermal bath are
in flat spacetime and have the same finite temperature $T$ with the black
hole. The eternal black hole together with the two copies of the thermal bath
comprises a pure system.

We are going to calculate the entanglement entropy of the CFTs on the
boundaries of the exterior regions using the generalized entanglement entropy
in eq.(5). To do so, we introduce a cutoff at $x=b_{L,R}$ which is slightly
inside the bath region. The cutoff separates the whole system into two
regions: the gravitational region that includes the black hole with its
boundaries, and the radiation region where the Hawking radiation escapes. The
entanglement region is thus the surface at the cutoff $x=b_{L,R}$.

Through the holographic correspondence, the entanglement entropy of the CFTs
can be obtained by finding a minimal area surface in the bulk gravity
spacetime that is homologous to the entanglement region. Classically, the
minimal surface is the union of the two horizons, i.e. the classical RT
surface, which is a time independent constant. The classical RT surface gives
the coarse-grained entropy that sets the upper bound of the entanglement
entropy of the Hawking radiation.

Including quantum corrections, the QES is the surface that minimizes the
generalized entanglement entropy eq.(5). The location of the QES on a Cauchy
slice is generically different at different slices, so the fine-grained
entropy of the eternal black hole is time-dependent. Furthermore, there could
be more than one QES that minimize the entanglement entropy locally. The
dominant QES should be the one which globally minimizes the entanglement
entropy. This implies the possible phase transitions between different QES.

Having the basic picture in mind, let us now consider the radiation region in
more detail. We couple the bath to the eternal black hole at a certain time
$t=t_{h}$. We assign the bath to have the same temperature as that in the
eternal black hole. As the Hawking radiation from the black hole escapes to
the thermal bath, there is the same amount of energy that goes back to the
black hole from the bath so that the whole system is in thermodynamic
equilibrium.

To decode the information inside the black hole, we need to measure the
Hawking radiation escaping to the thermal bath. Once the Hawking radiation has
entered the bath, it then travels along the outgoing null surface with the
speed of light as shown in Fig.3. At time $t=t_{b}$, the earliest Hawking
radiation reaches the cutoff and enters into the radiation region. We then
start to measure the Hawking radiation from the time $t_{b}$ and set it as our
initial time.

At a later time $t>t_{b}$, the earliest radiation will reach a surface of
distance $L=b+c(t-t_{b})$ which behaves as an end-of-the-world (ETW) brane.
Since there is no radiation beyond the ETW brane, the effective radiation
region is just the space-like region between the cutoff and the ETW brane,
namely $[b_{L},L]\cup[b_{R},L]$, which increases with time. The ETW brane at
different times are plotted as vertical curves in Fig.3.

As we have mentioned, the whole system including both the eternal black hole
and the thermal bath is a pure system, so that the bulk entanglement entropy
in eq.(5) equals the entanglement entropy of the Hawking radiation in the
effective radiation region $\mathcal{R}$ as shown in the Fig.3. The
entanglement entropy of the radiation region $\mathcal{R}$ can be calculated
using the island formula 1908.10996 ; 2006.04851 ,

$\mathcal{S}_{EE}(\mathcal{R})=\min\left\\{\text{ext}\left(S(\mathcal{R}\cup\mathcal{I})+\frac{\text{Area}(\partial\mathcal{I})}{4G_{N}^{(d+1)}}\right)\right\\},$
(6)

It can be shown that, at early times, the entanglement entropy of the
effective radiation region is always dominated by the vanishing island
$\mathcal{I}=\emptyset$ with $\text{Area}(\partial\mathcal{I})=0$. The bulk
entropy $S(\mathcal{R}\cup\emptyset)$ corresponding to the vanishing island,
monotonically increases from zero and will eventually exceed the coarse
grained entropy implying the information paradox.

Remarkably, there exists a non-trivial QES outside of the horizon for the
eternal black hole 1303.1080  that implies a non-vanishing island
$\mathcal{I}$ between the two non-trivial QES. The bulk entropy corresponding
to this non-vanishing island equals to the entanglement entropy of the union
of the radiation region with the island and will dominate at late times as
shown in the Fig.3.

Since the QES is outside of the horizon, the generalized entanglement entropy
for the non-vanishing island is a constant. At a later time, e.g. $t=t_{I}$ in
Fig.3, the non-vanishing island case will dominate the system. The time at
which the phase transition between the vanishing and non-vanishing island
takes place is called the Page time. The phase transition of the generalized
entanglement entropy leads to the well known Page curve for the eternal black
hole.

The eternal black hole in 2-dimensions has been studied in 1910.11077 .
Extension to higher dimensions was previously considered in 2006.04851 . The
main difficulty in obtaining the generalized entanglement entropy is
calculating the bulk entropy in curved spacetime. In 2-dimensions, the replica
trick is used to compute the entanglement entropy through the path integral
approach. However, this trick is difficult to be generalized to higher
dimensions.

In section 2.3, we will use the idea of the double holography to calculate the
bulk entanglement entropy 1908.10996 . It has been shown that combining double
holography with BCFT is a powerful method to calculate the bulk entanglement
entropy and can be generalized to higher dimensions 2010.00018 .

###  2.3 Double Holography

To calculate the bulk entanglement entropy in JT gravity, the authors of
1908.10996 ; 1911.03402  introduced a $3\rm d$ spacetime that is
holographically dual to the $2\rm d$ matter CFTs and the baths, i.e. the
double holographic duality. The $3\rm d$ double holographic bulk metric is
locally an $\rm AdS_{3}$ spacetime with a boundary on which the $2\rm d$
theory is living. This is similar to the Randall-Sundrum model, where the
boundary is called the Planck brane 9905221 ; 9906064 ; 0011156 .

Extending this idea to a higher dimensional thermofield double state as
discussed in 2006.04851 , there are three perspectives to describe the system.
In the boundary perspective, a pair of $(d+1)$-dimensional CFTs live on a
region $[0,L]$ with a $d$-dimension CFT living on its boundary at $x=0$
indicated by the green square dots as shown in Fig.4(a). where $x=b_{L,R}$ is
the cutoff and the other boundary is at $x=L$. The effective Hawking radiation
region is $[b_{L},L]\cup[b_{R},L]$.

(a) boundary perspective

(b) brane perspective

(c) bulk perspective

Figure 4: Three different perspectives of the gravity system, Hawking
radiation system and the holographic dual of these systems.

The brane perspective can be realized by using holographic correspondence to
replace the $\rm CFT_{d}$ by a Planck brane on which an asymptotically $\rm
AdS_{d+1}$ eternal black hole lives, as shown in Fig.4(b). We assign the
radial direction of the eternal black hole as $z$ with the $\rm CFT_{d}$
sitting at $z=0$. The horizons of the eternal black hole are indicated by the
two purple square dots. As argued in 0011156 ; 0105132 ; 0109017 ; 0106261 ;
0112166 ; 0201261 ; 0207003 ; 0608089 ; 2006.02438 , the transparent boundary
condition between the gravitational and the radiation regions imply that the
stress tensor in $\rm CFT_{d}$ is not conserved and will generate an anomalous
dimension, so that the dual graviton gets a mass. However, if the central
charge of the $\rm CFT_{d}$ is large, the effect from the mass can be
negligible.

In the bulk perspective, we use the holographic correspondence again to
replace the $\rm CFT_{d+1}$ by an asymptotically $\rm AdS_{d+2}$ black hole
bulk spacetime. This geometry can be properly described by a holographic BCFT
setup, as shown in Fig.4(c). The $(d+1)$-dimensional BCFT is dual to an $\rm
AdS_{d+2}$ black hole bulk spacetime. The boundary $x=0$ extends to the bulk
spacetime as a Plank brane and the other boundary $x=L$ extends to the bulk
spacetime as an ETW brane. The embedding of the branes into the bulk spacetime
is determined by solving the holographic BCFT system. The purple dashed lines
represent the two horizons in the bulk black hole. In the bulk perspective,
the bulk entanglement entropy in eq.(4) can be calculated by a classical RT
surface in the bulk spacetime anchored on the entangling surface at
$x=b_{L,R}$.

In summary, in our setup, the boundary on the left corresponds to the
gravitational region containing the Planck brane and a small region to the
left of the cutoff at $x=b_{L,R}$. The other boundary on the right at $x=L$
corresponds to an ETW brane which moves at the speed of light to the right.
The effective radiation region is $[b_{L},L]\cup[b_{R},L]$.

###  2.4 Brief Review of Black Hole Solution in BCFT

In this subsection, we will briefly review a black hole solution in the
holographic BCFT that we will use in this work.

The $(d+2)$-dimensional bulk spacetime $\mathcal{M}$ is holographically dual
to a $(d+1)$-dimensional CFT defined on the conformal boundary
$\partial\mathcal{M}$. $\partial\mathcal{M}$ has a $d$-dimensional boundary
$\mathcal{P}$, which has a $(d+1)$-dimensional hypersurface dual $\mathcal{Q}$
in $\mathcal{M}$ anchored at $\mathcal{P}$, see Fig.2.

The action of the gravitational theory in the bulk is,

$S=\mathcal{S}_{\mathcal{M}}+S_{GHY}+S_{\mathcal{Q}}+S_{\partial\mathcal{M}}+S_{\mathcal{P}},$
(7)

where

$\displaystyle S_{\mathcal{M}}$
$\displaystyle=\int_{\mathcal{M}}\sqrt{-g}(R-2\Lambda_{\mathcal{M}}),$ (8)
$\displaystyle S_{\mathcal{Q}}$
$\displaystyle=\int_{\mathcal{Q}}\sqrt{-h}(R_{\mathcal{Q}}-2\Lambda_{\mathcal{Q}}+2K),$
(9) $\displaystyle S_{\partial\mathcal{M}}$
$\displaystyle=2\int_{\partial\mathcal{M}}\sqrt{-\gamma}K^{\prime},$ (10)
$\displaystyle S_{\mathcal{P}}$
$\displaystyle=2\int_{\mathcal{P}}\sqrt{-\sigma}\theta.$ (11)

$S_{\mathcal{M}}$ is the Einstein–Hilbert action of $\mathcal{M}$ with $R$ and
$\Lambda_{\mathcal{M}}$ being the intrinsic Ricci curvature and the
cosmological constant of $\mathcal{M}$ respectively. $S_{GHY}$ is the Gibbons-
Hawking-York boundary term. $S_{\mathcal{Q}}$ is the action of the
$(d+1)$-dimensional hypersurface $\mathcal{Q}$ with the induced metric
$h_{ab}=g_{ab}-n_{a}^{\mathcal{Q}}n_{b}^{\mathcal{Q}}$, where
$n^{\mathcal{Q}}$ is the unit normal vector of $\mathcal{Q}$, and
$R_{\mathcal{Q}}$, $\Lambda_{\mathcal{Q}}$, $K$ being the intrinsic Ricci
curvature, the cosmological constant and the extrinsic curvatures of
$\mathcal{Q}$ in $\mathcal{M}$ respectively. $S_{\partial\mathcal{M}}$ is the
action of the conformal boundary $\partial\mathcal{M}$ with the induced metric
$\gamma_{ab}=g_{ab}-n_{a}^{\partial\mathcal{M}}n_{b}^{\partial\mathcal{M}}$,
where $n^{\partial\mathcal{M}}$ is the unit normal vector of
$\partial\mathcal{M}$, and $K^{\prime}$ is the extrinsic curvature of
$\partial\mathcal{M}$. $S_{\mathcal{P}}$ is the boundary term of $\mathcal{Q}$
and $\partial\mathcal{M}$ with $\sigma_{ab}$ being the metric of $\mathcal{P}$
and $\theta=\cos^{-1}\left(n^{\mathcal{Q}}\cdot
n^{\partial\mathcal{M}}\right)$ being the supplementary angle between
$\mathcal{Q}$ and $\partial\mathcal{M}$, which makes a well-defined
variational principle on $\mathcal{P}$.

Comparing the above BCFT system with the Penrose diagram in Fig.3,
$\mathcal{Q}$ represents the Planck brane containing the eternal black hole
(blue region), $\partial\mathcal{M}$ represents the thermal baths (light
yellow region), and $\mathcal{P}$ is the boundary of the eternal black hole at
$x=0$ where the CFTs live.

A simple asymptotic AdS black hole solution of the above system has been
obtained in 1805.06117 ,

$ds_{\mathcal{M}}^{2}=\frac{l_{AdS}^{2}}{z^{2}}\Bigg{[}-f(z)dt^{2}+\frac{dz^{2}}{f(z)}+dx^{2}+\sum_{i=1}^{d-1}(dx_{i})^{2}\Bigg{]},$
(12)

where

$f(z)=1-\frac{z^{d+1}}{z_{h}^{d+1}},$ (13)

and $z_{h}$ corresponds to the horizon of the black hole. The
$(d+1)$-dimensional conformal boundary $\partial\mathcal{M}$ is at $z=0$. The
temperature of the BCFT is given by the Bekenstein-Hawking temperature of the
black hole,

$T=\frac{d+1}{4\pi z_{h}}.$ (14)

In the double holographic setup, there are two $(d+1)$-dimentional
hypersurfaces $\mathcal{Q}$s, the Planck brane and the ETW brane. By varing
$S_{\mathcal{Q}}$ with $h^{ab}$ we get the equations of motion of
$\mathcal{Q}$,

$R_{\mathcal{Q}ab}+2K_{ab}-\left(\frac{1}{2}R_{\mathcal{Q}}+K-\Lambda_{\mathcal{Q}}\right)h_{ab}=0,$
(15)

which is the Neumann boundary condition proposed by Takayanagi in 1105.5165 .
This condition is too strong and gives more constraint equations than the
degrees of freedom, so in 1701.04275 ; 1701.07202 , the following mixed
boundary condition is proposed by Chu et al.

$(d-1)(R_{\mathcal{Q}}+2K)-2(d+1)\Lambda_{\mathcal{Q}}=0.$ (16)

The induced metric $h_{ab}$ of the Planck brane and the ETW brane is,

$ds_{\mathcal{Q}}=\frac{l_{AdS}^{2}}{z^{2}}\Bigg{[}-f(z)dt^{2}+\frac{dz^{2}}{f(z)}+\sum_{i=1}^{d-1}(dx_{i})^{2}\Bigg{]},$
(17)

with a simple embedding $x=\rm{constant}$. The intrinsic curvature, extrinsic
curvature, and the cosmological constant on $\mathcal{Q}$ are,

$R_{\mathcal{Q}}=-\frac{d(d+1)}{l_{AdS}^{2}},~{}K_{ab}=0,~{}\Lambda_{\mathcal{Q}}=-\frac{d(d-1)}{2l_{AdS}^{2}},$
(18)

which satisfies the mixed boundary condition eq.(16). In all of the
calculations, we are going to denote the $(d-1)$-dimensional volume as,

$\int d^{d-1}\mathbf{x}=V_{d-1}.$ (19)

##  3 Holographic Entanglement Entropy of Hawking Radiation

As we have explained in section 2.3, we are going to calculate the
entanglement entropy with the effective radiation region
$[b_{L},L]\cup[b_{R},L]$ as the entanglement region. By holographic
correspondence, it is proportional to the area of the RT surface in the doubly
holographic bulk spacetime. The RT surface is anchored at the entanglement
surface $x=b_{L,R}$ and is homologous to the effective radiation region
$[b_{L},L]\cup[b_{R},L]$. At the leading order, we only need to consider the
classical RT surface,

$S=\int d^{d+1}\xi\mathcal{L}=\int d^{d+1}\xi\sqrt{\det G_{ab}},$ (20)

where the induced metric is defined by

$G_{ab}=g_{\mu\nu}\partial_{a}X^{\mu}\partial_{b}X^{\nu},$ (21)

and the bulk metric $g_{\mu\nu}$ is given in eq.(12).

Generically, there could be more than one QES with each of them being a
locally minimum surface. In the following of this section, we will find that
there are three RT surfaces in our system. One is the Hartman-Maldacena
surface (HM), which penetrates the horizon and connects the cutoff at
$x=b_{L,R}$. Another one is the boundary RT surface (BRT), which is anchored
not only on the cutoffs at $x=b_{L,R}$ but also on the ETW brane at $x=L$. The
third one is the island RT surface (IRT) that starts from the cutoff
$x=b_{L,R}$ and is anchored on the Planck brane at the quantum extremal
surfaces at $z=z_{\rm QES}$. The region between the QES and its symmetric copy
is the island which includes the whole interior of the eternal black hole.

(a)

(b)

(c)

(d)

(e)

Figure 5: The time evolution of the RT surfaces in the BCFT system. The
red/green/blue line represents the BRT/HM/IRT surface. The color of the shaded
region represents the entanglement wedge whose entanglement entropy is
dominant at that time.

The RT surfaces at different times are shown in Fig.5. The red/green/blue line
represents the BRT/HM/IRT surface. At time $t=t_{h}$, as shown in Fig.5(a),
the Hawking radiation from the black hole reaches the boundary $z=0$ and
enters into the thermal bath at $x=0$. At the initial time $t=t_{b}$, as shown
in Fig.5(b), the Hawking radiation reaches the cutoff $x=b$ and enters into
the effective radiation region. Since the ETW brane is located at $x=b_{R}$ at
the initial time $t=t_{b}$, the BRT surface shrinks to zero, and only the HM
and the IRT surfaces are present as shown in Fig.5(b).

The BRT surface dominates the system at the early time with the red region
being the entanglement wedge as shown in Fig.5(c). As time evolves, the
stretching of the interior of the black hole causes the growth of the HM
surface. However, at the same time, the ETW brane moves to the right with the
speed of light $c$ and coordinate $L=b+c(t-t_{b})$, so that the BRT surface
increases faster than the HM surface. At a certain time, the BRT surface will
exceed the HM surface and the latter becomes dominant with the green region
being the entanglement wedge as shown in Fig.5(d).

Finally, after a critical time, the BRT and HM surfaces will exceed the IRT
surface so that it becomes dominant with the blue region being the
entanglement wedge as shown in Fig.5(e). When the IRT surface dominates, the
entanglement wedge includes the part between the QESs on the Planck brane,
i.e. the island. Since the QES is outside of the horizon in the eternal black
hole, the island contains the whole interior of the black hole. Based on the
entanglement wedge reconstruction, after the critical time, we are able to
reconstruct all the information inside the black hole from the data observed
in the effective radiation region. The critical time is called the Page time
$t_{P}$.

###  3.1 Hartman-Maldacena Surface

Since the bulk spacetime is symmetric, we calculate only one side. However,
the complete description of the entanglement entropy can be considered by
taking into account the two sides. The HM surface is a locally minimum surface
penetrating the horizon and connecting the two cutoffs at $x=b_{L,R}$. To
describe the interior region behind the horizon, we make the following
coordinate transformation,

$t=v+\int\frac{dz}{f(z)}~{}\Rightarrow dt=dv+\frac{dz}{f(z)}.$ (22)

The metric eq.(12) then becomes,

$ds^{2}=\frac{l^{2}_{AdS}}{z^{2}}\left[-f(z)dv^{2}-2dvdz+dx^{2}+\sum_{i=1}^{d-1}dx_{i}^{2}\right],$
(23)

where the HM surface is described by $x=b_{R}$ and $v=v(z)$.

The entanglement entropy eq.(20) on the $t-z$ plane (or the $v-z$ plane) after
the coordinate transformation eq.(22), can be calculated as,

$\mathcal{S}_{\rm HM}=\frac{l_{AdS}^{d}}{4G_{N}^{(d+2)}}\int
dzd^{d-1}\mathbf{x}\frac{1}{z^{d}}\sqrt{-\left(\frac{dv}{dz}\right)\left(f(z)\left(\frac{dv}{dz}\right)+2\right)}.$
(24)

The conjugate momentum of $v$ is,

$\frac{\partial\mathcal{L}}{\partial
v^{\prime}}=C_{v}=\frac{f(z)v^{\prime}+1}{z^{d}\sqrt{-v^{\prime}\left(f(z)v^{\prime}+2\right)}},$
(25)

where $v^{\prime}={dv}/{dz}$. From eq.(25) we can solve for $v^{\prime}$ as,

$v^{\prime}=\frac{1}{f(z)}\left[-1\pm\sqrt{\frac{C_{v}^{2}z^{2d}}{f(z)+C_{v}^{2}z^{2d}}}\right].$
(26)

We choose the solution with the negative square root and integrate it on $z$
to get,

$v=\int_{0}^{z_{M}}\frac{dz}{f(z)}\left[-1-\sqrt{\frac{C_{v}^{2}z^{2d}}{f(z)+C_{v}^{2}z^{2d}}}\right],$
(27)

where $z_{M}$ is the turning point of the HM surface with the boundary time
$t$ as shown in Fig.6. Substituting eq.(27) back into eq.(22), we obtain the
expression of the boundary time,

$t=-\int_{0}^{z_{M}}\frac{C_{v}z^{d}dz}{f(z)\sqrt{f(z)+C_{v}^{2}z^{2d}}}.$
(28)

The turning point of the HM surface can be found by imposing a boundary
condition at $z=z_{M}$,

$\left.\frac{dz}{dv}\right|_{z=z_{M}}=f(z_{M})\sqrt{\frac{f(z_{M})+C_{v}^{2}z_{M}^{2d}}{-f(z_{M})}}=0,$
(29)

which leads to the following expression of the conjugate momentum $C_{v}$ in
terms of $z_{M}$,

$C_{v}^{2}=-\frac{f(z_{M})}{z_{M}^{2d}}.$ (30)

As we have mentioned above, we will use

$t_{h}=-\int_{0}^{z_{h}}\frac{\sqrt{f(z_{h})}z^{d}dz}{f(z)\sqrt{f(z_{h})z^{2d}-f(z)z_{h}^{2d}}},$
(31)

as the reference time as shown in Fig.6.

Figure 6: The Hartman-Maldacena Surface at time $t$.

Plugging eq.(26) and eq.(30) into eq.(24), we finally obtain the entanglement
entropy corresponding to the HM surface,

$\mathcal{S}_{\rm HM}=\frac{l^{d}_{AdS}\cdot
V_{d-1}}{4G_{N}^{(d+2)}}\int_{0}^{z_{M}}\frac{z_{M}^{d}dz}{z^{d}\sqrt{f(z)z_{M}^{2d}-f(z_{M})z^{2d}}}.$
(32)

###  3.2 Boundary RT Surface

The effective Hawking radiation region at a fixed time is111Due to the
symmetry between the left and right sides, we will only consider the right
side as the example in the following calculations. The calculation for the
left side is exactly the same.

$\mathcal{R}=\\{x\in\mathbb{R},\textbf{x}_{d-1}\in\mathbb{R}^{d-1}\mid
x\in(b_{R},L),x_{i}\in\mathbb{R}\quad\text{for}\quad i=1,2,\cdots,d-1\\},$
(33)

which preserves $(d-1)$-dimensional translation invariance, so that we can
describe the BRT surface by $t=\text{constant}$ and $z=z(x)$, The entanglement
entropy eq.(20) on the $z-x$ plane can be calculated as,

$\mathcal{S}_{\rm BRT}=\frac{\ell_{AdS}^{d}}{4G^{(d+2)}_{N}}\int
dxd^{d-1}\textbf{x}\frac{1}{z^{d}}\sqrt{1+\frac{1}{f(z)}\cdot\left(\frac{dz}{dx}\right)^{2}}.$
(34)

The conjugate momentum of $z(x)$ is,

$p_{z}=\frac{\partial\mathcal{L}}{\partial
z^{\prime}}=\frac{z^{\prime}}{z^{d}\cdot f(z)\sqrt{1+\frac{z^{\prime
2}}{f(z)}}},$ (35)

where $z^{\prime}=dz/dx$. The Hamiltonian can be calculated as,

$\mathcal{H}=z^{\prime}p_{z}-\mathcal{L}=\frac{-1}{z^{d}\sqrt{1+\frac{z^{\prime
2}}{f(z)}}}.$ (36)

Since the Lagrangian $\mathcal{L}$ is not explicitly dependent on $x$, the
Hamiltonian is conserved. We thus define a constant,

$C_{z}=\frac{-1}{z^{d}\sqrt{1+\frac{z^{\prime 2}}{f(z)}}}.$ (37)

From eq.(37), we can solve for $z^{\prime}$ as,

$z^{\prime}=\pm\frac{\sqrt{f(z)\left(\frac{1}{C_{z}^{2}}-z^{2d}\right)}}{z^{d}}.$
(38)

It has been shown that the minimum surface must be perpendicular to the ETW
brane 1701.07202 , so that we have the following boundary conditions,

$z(x=b_{R})=0,\quad\left.\frac{dz}{dx}\right|_{x=L}=0.$ (39)

Applying eq.(37) at $x=L$, we can determine the constant,

$C_{z}=-\frac{1}{z_{L}},$ (40)

where $z_{L}=z(L)$ is the location that the BRT surface reaches on the ETW
brane at $x=x_{L}$. By integrating eq.(38) and using eq.(40), the length of
the effective radiation region can be calculated,

$L-b_{R}=\int_{0}^{z_{L}}\frac{z^{d}dz}{\sqrt{f(z)(z_{L}^{2d}-z^{2d})}}.$ (41)

Plugging eq.(38) and eq.(40) into eq.(34), we finally obtain the entanglement
entropy corresponding to the BRT surface,

$\mathcal{S}_{\rm BRT}=\frac{l_{AdS}^{d}\cdot
V_{d-1}}{4G_{N}^{(d+2)}}\int_{0}^{z_{L}}\frac{z_{L}^{d}dz}{z^{d}\sqrt{f(z)(z_{L}^{2d}-z^{2d})}}.$
(42)

The entanglement entropy defined in eq.(42) is divergent near $z=0$. For the
purpose of our numerical calculations, we regularize it as,

$\mathcal{S}_{\rm BRT-reg}=\frac{l_{AdS}^{d}\cdot
V_{d-1}}{4G_{N}^{(d+2)}}\left[\int_{0}^{z_{L}}\frac{z_{L}^{d}dz}{z^{d}\sqrt{f(z)(z_{L}^{2d}-z^{2d})}}-\int_{0}^{z_{L}}\frac{z_{L}^{d}dz}{z^{d}\sqrt{(z_{L}^{2d}-z^{2d})}}\right].$
(43)

We note that the entanglement entropies corresponding to the HM surface, which
we calculated in the last subsection, and the IRT surface, which we will
calculate in the next subsection, have the same divergence. However, since we
are only concerned about the difference between the entanglement entropies,
the divergences will cancel each other out, so that we do not need to
regularize the other ones. The regularization of the BRT in eq.(43) is only
for the purpose of numerical plotting.

###  3.3 Island RT Surface

The entanglement entropy of the IRT surface on the $z-x$ plane reads,

$\mathcal{S}_{\rm IRT}=\frac{l^{d}_{AdS}}{4G_{N}^{(d+2)}}\left[\int
dxd^{d-1}\textbf{x}\frac{1}{z^{d}}\sqrt{1+\frac{1}{f(z)}\cdot\left(\frac{dz}{dx}\right)^{2}}+\frac{1}{z_{I}^{d-1}}\right],$
(44)

which is the same as $\mathcal{S}_{\rm BRT}$ in eq.(34) with an extra term
representing the contribution from the QES. Also, $z_{\rm QES}=z_{I}$ is the
location of the QES on the Planck brane as shown in Fig.3. In a similar way to
section 3.2, we can define a constant $C_{\sigma}$,

$C_{\sigma}=\frac{-1}{z^{d}\sqrt{1+\frac{z^{\prime 2}}{f(z)}}},$ (45)

and solve for $z^{\prime}$ as,

$z^{\prime}=\pm\frac{\sqrt{f(z)\left(\frac{1}{C_{\sigma}^{2}}-z^{2d}\right)}}{z^{d}}.$
(46)

However, in the presence of the QES, the IRT surface does not need to be
perpendicular to the Planck brane. The boundary conditions now are given by,

$z(x=0)=z_{I},\quad z(x=b_{R})=0,$ (47)

With the above boundary conditions, we get,

$\frac{dz}{dx}=\pm\frac{\sqrt{f(z)\left(z_{I}^{2d}-z^{2d}+\frac{z_{I}^{2d}\sigma^{2}}{f(z_{I})}\right)}}{z^{d}},$
(48)

which can be integrated to give,

$b_{R}=\int_{0}^{z_{I}}\frac{z^{d}dz}{\sqrt{f(z)\left(z_{I}^{2d}-z^{2d}+\frac{z_{I}^{2d}\sigma^{2}}{f(z_{I})}\right)}},$
(49)

where $\left.\sigma=dz/dx\right|_{x=0}$ is the slope as the IRT surface
reaches the Planck brane at $x=0$. The slope $\sigma$ can be determined by
finding the location $z_{I}$ of the QES on the Planck brane that minimizes
$\mathcal{S}_{\rm IRT}$ for a fixed cutoff $b_{R}$. Without the QES
contribution, the minimum condition implies $\sigma=0$, i.e. the RT surface is
perpendicular to the brane as we have known.

Finally, the entanglement entropy corresponding to the IRT surface eq.(44) can
be expressed as,

$\mathcal{S}_{\rm IRT}=\frac{l^{d}_{AdS}\cdot
V_{d-1}}{4G_{N}^{(d+2)}}\left[\int_{0}^{z_{I}}\frac{z_{I}^{d}}{z^{d}}\sqrt{\frac{\left(1+\frac{\sigma^{2}}{f(z_{I})}\right)}{f(z)\left(z_{I}^{2d}-z^{2d}+\frac{z_{I}^{2d}\sigma^{2}}{f(z_{I})}\right)}}dz+\frac{1}{z_{I}^{d-1}}\right].$
(50)

Now, the condition for the location $z_{I}$ of the QES on the Planck brane
that minimize $\mathcal{S}_{\rm IRT}$ is,

$\frac{d\mathcal{S}_{\rm IRT}}{dz_{I}}=0.$ (51)

Combining eq.(49) and eq.(51), we can solve for $z_{I}$ as well as $\sigma$
for a fixed $b_{R}$.

##  4 Phase Transitions and Page Curve

As we have seen in the last section, there are three RT surfaces in the
eternal black hole system, i.e. the HM, BRT, and IRT surfaces, whose
corresponding entanglement entropies have been calculated in eq.(32), eq.(42),
and eq.(50). The dominant one is the minimum among them,

$\mathcal{S}=\min(\mathcal{S}_{\rm HM},\mathcal{S}_{\rm BRT},\mathcal{S}_{\rm
IRT}).$ (52)

Among the three entanglement entropies, $\mathcal{S}_{\rm IRT}$ is a constant
at a fixed temperature, while the other two increase with time. At the initial
time $t=t_{b}$, $\mathcal{S}_{\rm BRT}=0$ dominates. At a later time, there
could be phase transitions between each pair of the three entanglement
entropies. In this section, we will investigate the phase transitions in
detail and obtain the Page curve for the entanglement entropy of the eternal
black hole.

###  4.1 Phase Transitions

To be concrete in this subsection, we set the cutoff $b_{L}=b_{R}=b=0.1$, and
the black hole horizon $z_{h}=10$ which corresponds to the temperature
$T=0.032$.

Since the entanglement entropy corresponding to the IRT surface is a time
independent constant, we will first consider the phase transitions between the
IRT surface and the other two surfaces. The phase transition between
$\mathcal{S}_{\rm BRT}$ and $\mathcal{S}_{\rm IRT}$ can be obtained from

$\Delta\mathcal{S}_{\rm BRT-IRT}=\mathcal{S}_{\rm BRT}-\mathcal{S}_{\rm
IRT}=0,$ (53)

where $\mathcal{S}_{\rm BRT}$ and $\mathcal{S}_{\rm IRT}$ are given in eq.(42)
and eq.(50).

The regularized222We regularize the entanglement entropy using eq.(43) to plot
it numerically. However, we should note that the exact phase transition time
is determined from eq.(53) without regularization. entanglement entropy
corresponding to the BRT and IRT surfaces are plotted in Fig.7(a).
$\mathcal{S}_{\rm BRT}$ (red line) starts from zero at $t=t_{b}$ and increases
almost linearly. It dominates until $t-t_{b}=t_{\rm BRT-IRT}\simeq 12.01$ when
the phase transition between $\mathcal{S}_{\rm BRT}$ and $\mathcal{S}_{\rm
IRT}$ (blue line) takes place, then $\mathcal{S}_{\rm IRT}$ will dominate for
$t-t_{b}>t_{\rm BRT-IRT}$.

(a) BRT-IRT

(b) HM-IRT

(c) BRT-HM

Figure 7: The phase transitions between BRT (red), HM (green), and IRT (blue)
surfaces for $z_{h}=10$. (a) $\mathcal{S}_{\rm BRT}$ and $\mathcal{S}_{\rm
IRT}$. (b) $\mathcal{S}_{\rm HM}$ and $\mathcal{S}_{\rm IRT}$. (c)
$\mathcal{S}_{\rm BRT}$ and $\mathcal{S}_{\rm HM}$.

Similarly, the phase transition between $\mathcal{S}_{\rm HM}$ and
$\mathcal{S}_{\rm IRT}$ can be obtained from

$\Delta\mathcal{S}_{\rm HM-IRT}=\mathcal{S}_{\rm HM}-\mathcal{S}_{\rm IRT}=0,$
(54)

where $\mathcal{S}_{\rm HM}$ and $\mathcal{S}_{\rm IRT}$ are given in eq.(32)
and eq.(50).

The regularized entanglement entropy corresponding to the HM and IRT surfaces
are plotted in Fig.7(b). $\mathcal{S}_{\rm HM}$ (green line) starts from a
finite value at $t=t_{b}$ and increases. It dominates until $t-t_{b}=t_{\rm
HM-IRT}\simeq 15.95$ when the phase transition between $\mathcal{S}_{\rm HM}$
and $\mathcal{S}_{\rm IRT}$ takes place, then $\mathcal{S}_{\rm IRT}$ will
dominate for $t-t_{b}>t_{\rm HM-IRT}$.

Finally, there is also a phase transition between $\mathcal{S}_{\rm BRT}$ and
$\mathcal{S}_{\rm HM}$ that can be obtained from

$\Delta\mathcal{S}_{\rm BRT-HM}=\mathcal{S}_{\rm BRT}-\mathcal{S}_{\rm HM}=0,$
(55)

where $\mathcal{S}_{\rm BRT}$ and $\mathcal{S}_{\rm HM}$ are given in eq.(42)
and eq.(32).

The regularized entanglement entropy corresponding to the BRT and HM surfaces
are plotted in Fig.7(c). Both $\mathcal{S}_{\rm BRT}$ and $\mathcal{S}_{\rm
HM}$ increases with time. Among the two, $\mathcal{S}_{\rm BRT}$ dominates
until $t-t_{b}=t_{\rm BRT-HM}\simeq 6.47$ when the phase transition between
$\mathcal{S}_{\rm BRT}$ and $\mathcal{S}_{\rm HM}$ takes place, then
$\mathcal{S}_{\rm HM}$ will dominate for $t-t_{b}>t_{\rm BRT-HM}$.

We have obtained the phase transitions of every pair of the three entanglement
entropies corresponding to the BRT, HM, and IRT surfaces for the black hole
horizon $z_{h}=10$ or $T=0.032$. Now we put them together in Fig.8(a) with the
red/green/blue line representing $\mathcal{S}_{\rm BRT}$/$\mathcal{S}_{\rm
HM}$/$\mathcal{S}_{\rm IRT}$. The three critical points are labeled by black,
orange, and violet dots.

(a) $\rm z_{h}=10$

(b) $\rm z_{h}=0.5$

(c) $\rm z_{h}=0.1355$

(d) $\rm z_{h}=0.1$

Figure 8: The behavior of the entanglement entropies, $\mathcal{S}_{\rm BRT}$
(red), $\mathcal{S}_{\rm HM}$ (green), and $\mathcal{S}_{\rm IRT}$ (blue), at
different temperatures or black hole horizons. The black, orange, and violet
dots indicate the phase transitions of $\mathcal{S}_{\rm
BRT}\sim\mathcal{S}_{\rm HM}$, $\mathcal{S}_{\rm BRT}\sim\mathcal{S}_{\rm
IRT}$, and $\mathcal{S}_{\rm HM}\sim\mathcal{S}_{\rm IRT}$, respectively.

At early times, $\mathcal{S}_{\rm BRT}$ dominates the system. Later on, at
$t-t_{b}\simeq 6.47$, the phase transition between $\mathcal{S}_{\rm BRT}$ and
$\mathcal{S}_{\rm HM}$ takes place, and $\mathcal{S}_{\rm HM}$ becomes
dominant. Then, at $t-t_{b}\simeq 15.95$, the phase transition between
$\mathcal{S}_{\rm HM}$ and $\mathcal{S}_{\rm IRT}$ occurs, and
$\mathcal{S}_{\rm IRT}$ becomes dominant. In addition, there is another phase
transition between $\mathcal{S}_{\rm BRT}$ and $\mathcal{S}_{\rm IRT}$ that
occurs between the above two phase transitions at $t-t_{b}\simeq 12.01$.
However, this phase transition is irrelevant since it will not occur in the
real physical process as shown in Fig.8(a).

Having obtained the phase diagram of the entanglement entropy at $z_{h}=10$ or
$T=0.032$, let us consider the phase diagrams at other temperatures. Fig.8(b)
shows the entanglement entropies for the black hole horizon $z_{h}=0.5$, which
corresponds to a higher temperature $T=0.637$. As in the case of $z_{h}=10$,
there are three phase transitions in this case. The phase transitions of
$\mathcal{S}_{\rm BRT}\sim\mathcal{S}_{\rm HM}$, $\mathcal{S}_{\rm
BRT}\sim\mathcal{S}_{\rm IRT}$, and $\mathcal{S}_{\rm HM}\sim\mathcal{S}_{\rm
IRT}$ takes place at $t-t_{b}\simeq 0.444$, $t-t_{b}\simeq 0.636$ and
$t-t_{b}\simeq 0.759$, respectively.

The phase structure for $z_{h}=0.5$ is similar to that for $z_{h}=10$.
Nevertheless, we have an important observation that the three critical points
become closer for a smaller horizon or a higher temperature, so that the
duration of $\mathcal{S}_{\rm HM}$ domination shrinks. Therefore, an
interesting question is whether or not the duration of $\mathcal{S}_{\rm HM}$
would shrink to zero if the temperature continuously increases? We found that
the answer is yes.

Fig.8(c) shows the entanglement entropies for the black hole horizon
$z_{h}=0.1355$, which corresponds to the critical temperature $T_{c}=2.349$.
At this temperature, the three phase transitions coincide at the same point at
$t-t_{b}\simeq 0.236$ when the time duration of $\mathcal{S}_{\rm HM}$ shrinks
to zero, and $\mathcal{S}_{\rm BRT}$ transit to $\mathcal{S}_{\rm IRT}$
directly at this critical point.

For $T\geq T_{c}$, for example, in the case of the black hole horizon
$z_{h}=0.1$, which corresponds to the temperature $T=3.183\geq T_{c}$, the
entanglement entropies are shown in Fig.8(d). The phase transitions of
$\mathcal{S}_{\rm BRT}\sim\mathcal{S}_{\rm HM}$, $\mathcal{S}_{\rm
BRT}\sim\mathcal{S}_{\rm IRT}$, and $\mathcal{S}_{\rm HM}\sim\mathcal{S}_{\rm
IRT}$ take place at $t-t_{b}\simeq 0.217$, $t-t_{b}\simeq 0.200$ and
$t-t_{b}\simeq 0.190$, respectively. We notice that the order of the three
phase transitions reverses. Because the phase transition between
$\mathcal{S}_{\rm BRT}$ and $\mathcal{S}_{\rm HM}$ occurs after that between
$\mathcal{S}_{\rm BRT}$ and $\mathcal{S}_{\rm IRT}$, the dominant entanglement
entropy $\mathcal{S}_{\rm BRT}$ will transit directly to $\mathcal{S}_{\rm
IRT}$.

###  4.2 Page Curve

Now we are ready to plot the Page curve from the phase transitions among the
three RT surfaces obtained in section 4.1. The Page curves for different
temperatures are plotted in Fig.9.

(a) $T<T_{c}$

(b) $T=T_{c}$

(c) $T>T_{c}$

Figure 9: The Page curve for different temperatures. (a) For the case of
$z_{h}=0.5$, $\mathcal{S}_{\rm BRT}$ transitions to $\mathcal{S}_{\rm HM}$ at
$t-t_{b}\simeq 0.444$, then to $\mathcal{S}_{\rm IRT}$ at $t-t_{b}\simeq
0.759$. (b) For the case of $z_{h}=0.1355$, $\mathcal{S}_{\rm BRT}$
transitions directly to $\mathcal{S}_{\rm IRT}$ at $t-t_{b}\simeq 0.236$. (c)
For the case of $z_{h}=0.1$, $\mathcal{S}_{\rm BRT}$ transitions directly to
$\mathcal{S}_{\rm IRT}$ at $t-t_{b}\simeq 0.200$.

(a)

(b)

Figure 10: Phase diagram of the Hawking radiation entanglement entropy. The
red/green/blue region represents the region where $\mathcal{S}_{\rm
BRT}$/$\mathcal{S}_{\rm HM}$/$\mathcal{S}_{\rm IRT}$ dominates. The black,
orange, and violet lines represent the phase transitions of $\mathcal{S}_{\rm
BRT}\sim\mathcal{S}_{\rm HM}$, $\mathcal{S}_{\rm BRT}\sim\mathcal{S}_{\rm
IRT}$, and $\mathcal{S}_{\rm HM}\sim\mathcal{S}_{\rm IRT}$. The dashed parts
do not occur in the real physical process. The black dot labels the triple
point where the three phases meet together. (a) Phase diagram in the
temperature $T$ and time $t-t_{b}$ plane. (b) Phase diagram in the black hole
horizon $z_{h}$ and time $t-t_{b}$ plane.

At low temperature $T<T_{c}$, e.g. $z_{h}=0.5$ or $T=0.637$, the Page curve is
plotted in Fig.9(a). At early times, the entanglement entropy is dominated by
$\mathcal{S}_{\rm BRT}$ which increases from zero at $t=t_{b}$. After the
first phase transition between $\mathcal{S}_{\rm BRT}$ and $\mathcal{S}_{\rm
HM}$ takes place at $t-t_{b}\simeq 0.444$, $\mathcal{S}_{\rm HM}$ becomes
dominant until the second phase transition between $\mathcal{S}_{\rm HM}$ and
$\mathcal{S}_{\rm IRT}$ takes place at $t-t_{b}\simeq 0.759$ when
$\mathcal{S}_{\rm IRT}$ becomes dominant. In this case, the Page time is
determined by the phase transition between $\mathcal{S}_{\rm HM}$ and
$\mathcal{S}_{\rm IRT}$ labeled by the violet dot.

As the temperature grows, the duration of $\mathcal{S}_{\rm HM}$ shrinks. At
the critical temperature $T_{c}=2.349$ or $z_{h}=0.1355$, the Page curve is
plotted in Fig.9(b). The contribution from the HM surface completely vanishes,
and $\mathcal{S}_{\rm BRT}$ transits to $\mathcal{S}_{\rm IRT}$ directly. The
Page time now is determined by the phase transition between $\mathcal{S}_{\rm
BRT}$ and $\mathcal{S}_{\rm IRT}$ labeled by the orange dot.

As we go beyond the critical temperature as shown in Fig.9(c) for the case of
$T=3.183$ or $z_{h}=0.1$, $\mathcal{S}_{\rm BRT}$ always transits to
$\mathcal{S}_{\rm IRT}$ directly. The other two phase transitions are
irrelevant. The Page time is determined by the phase transition between
$\mathcal{S}_{\rm BRT}$ and $\mathcal{S}_{\rm IRT}$ labeled by the orange dot.

We conclude that the Page time decreases as the temperature grows. Among the
three RT surfaces, only the IRT surface intersects with the Planck brane, and
its entanglement wedge includes the interior of the eternal black hole.
According to the entanglement wedge reconstruction, we can reconstruct the
interior of the black hole when $\mathcal{S}_{\rm IRT}$ is dominant. The
decreasing of the Page time as the temperature is growing implies that one can
reconstruct the interior of the black hole earlier for a higher temperature
black hole. This is consistent with the intuitive expectations since higher
temperature black holes evaporate faster.

###  4.3 Phase Diagram

Figure 11: Different slices of the phase diagram as the cutoff $b$ is varied.

The phase diagram in the temperature vs. time plane is plotted in Fig.10(a).
The red region in the lower left part is dominated by $\mathcal{S}_{\rm BRT}$,
the blue region in the upper right part is dominated by $\mathcal{S}_{\rm
IRT}$, and the green region in between is dominated by $\mathcal{S}_{\rm HM}$.
The black, orange, and violent lines are the phase boundaries between
different phases. The solid parts represent the phase boundaries in the real
physical process. There is a triple point at $T_{c}=2.349$ and $t-t_{b}=0.236$
labeled by a black dot, where the three phases meet together. Below the
critical temperature $T_{c}$, the system undergoes three phases,
$\mathcal{S}_{\rm BRT}$, $\mathcal{S}_{\rm HM}$ and $\mathcal{S}_{\rm IRT}$ as
time evolves, with the violet line indicating the Page time. While above the
critical temperature, the system undergoes only two phases, $\mathcal{S}_{\rm
BRT}$ and $\mathcal{S}_{\rm IRT}$, with the orange line indicating the Page
time.

This behavior of the phase diagram is consistent with our intuitive
expectation: as the temperature grows, it becomes "easier" to decode the
information inside the black hole due to the earlier Page time since the black
hole radiates faster at higher temperatures.

So far we have only considered the system with a fixed cutoff $b=0.1$. In the
following, we will show how the phase diagram is affected by the variation of
the cutoff. By using the simple relation in eq.(14), the phase diagram can be
plotted in the black hole horizon vs. time plane in Fig.10(b), which will be
used to illustrate the effect from varying the cutoff $b$.

The phase diagrams in the black hole horizon vs. time plane for three
different cutoffs are plotted in Fig.11. We can see that, as the cutoff $b$
increases, the triple point moves to a new location with a larger black hole
horizon (or lower temperature) and a later Page time. Remarkably, numerical
calculations show that the path of the triple point is a straight line shown
as the black line in Fig.11. However, the physical interpretation of this
linear behavior of the triple point is not obvious to us and we will
investigate it in the future.

##  5 Conclusion

In this work, we studied the entanglement entropy for a $(d+1)$-dimensional
two-sided eternal black hole. We calculated the bulk entropy in the
generalized entanglement entropy eq.(5) by doubly holographic correspondence.
Our concrete setup is given by a black hole solution in a holographic BCFT as
discussed in section 2.3.

In our setup, we introduced an ETW brane which is the hypersurface of the
earliest Hawking radiation. The ETW brane moves away at the speed of light
$L=b+c(t-t_{b})$, and it defines a time-dependent effective radiation region
$[b_{L},L]\cup[b_{R},L]$.

There are three RT surfaces associated with the bulk entropy. One is the HM
surface which penetrates the horizon and connects the cutoffs $b_{L}$ and
$b_{R}$ directly. The HM surface increases with time due to the stretching of
the interior of the black hole. The second one is the BRT surface which
intersects the ETW brane. The BRT surface increases with time due to the
movement of the ETW brane. The last one is the IRT surface which intersects
the Planck brane and supports an island. The IRT surface is a time-independent
constant.

We investigated the phase transitions between every pair among the three RT
surfaces in section 4.1. The phase transition time for each pair of the RT
surfaces was determined by comparing their corresponding entanglement
entropies. Putting all phase transitions together, we obtained the phase
diagram of the entanglement entropy as shown in Fig.10.

We found a critical temperature $T_{c}\simeq 2.394$ for the Page curve with
$b=0.1$. When the temperature is lower than the critical temperature, i.e.
$T<T_{c}$, there are three durations for the entanglement entropy as shown in
Fig.9(a). $\mathcal{S}_{\rm BRT}$ starts from zero at the initial time
$t=t_{b}$ and increases until the first phase transition, when
$\mathcal{S}_{\rm HM}$ takes over and continuously increases. After the second
phase transition, $\mathcal{S}_{\rm IRT}$ becomes dominant and keeps as a
constant. On the other hand, when the temperature is higher than the critical
temperature, i.e. $T\geq T_{c}$, the duration of $\mathcal{S}_{\rm HM}$
disappears, and $\mathcal{S}_{\rm BRT}$ transits to $\mathcal{S}_{\rm IRT}$
directly as shown in Fig.9(b,c).

### Acknowledgements

This work is supported by the Ministry of Science and Technology (MOST
109-2112-M-009-005) and National Center for Theoretical Science, Taiwan.

## References

  * [1] S. W. Hawking.  Particle Creation by Black Holes.  Commun. Math. Phys., 43:199–220, 1975.  [Erratum: Commun.Math.Phys. 46, 206 (1976)].
  * [2] S. W. Hawking.  Black Holes and Thermodynamics.  Phys. Rev. D, 13:191–197, 1976.
  * [3] Don N. Page.  Information in black hole radiation.  Phys. Rev. Lett., 71:3743–3746, 1993.
  * [4] Ahmed Almheiri, Donald Marolf, Joseph Polchinski, and James Sully.  Black Holes: Complementarity or Firewalls?  JHEP, 02:062, 2013.
  * [5] Kyriakos Papadodimas and Suvrat Raju.  An Infalling Observer in AdS/CFT.  JHEP, 10:212, 2013.
  * [6] Daniel Harlow and Patrick Hayden.  Quantum Computation vs. Firewalls.  JHEP, 06:085, 2013.
  * [7] Ahmed Almheiri, Donald Marolf, Joseph Polchinski, Douglas Stanford, and James Sully.  An Apologia for Firewalls.  JHEP, 09:018, 2013.
  * [8] Juan Maldacena and Leonard Susskind.  Cool horizons for entangled black holes.  Fortsch. Phys., 61:781–811, 2013.
  * [9] Seth Lloyd and John Preskill.  Unitarity of black hole evaporation in final-state projection models.  JHEP, 08:126, 2014.
  * [10] Kyriakos Papadodimas and Suvrat Raju.  Black Hole Interior in the Holographic Correspondence and the Information Paradox.  Phys. Rev. Lett., 112(5):051301, 2014.
  * [11] Ning Bao, Adam Bouland, Aidan Chatwin-Davies, Jason Pollack, and Henry Yuen.  Rescuing Complementarity With Little Drama.  JHEP, 12:026, 2016.
  * [12] Ning Bao, Sean M. Carroll, Aidan Chatwin-Davies, Jason Pollack, and Grant N. Remmen.  Branches of the Black Hole Wave Function Need Not Contain Firewalls.  Phys. Rev. D, 97(12):126014, 2018.
  * [13] J. D. Bekenstein.  Black holes and the second law.  Lett. Nuovo Cim., 4:737–740, 1972.
  * [14] Jacob D. Bekenstein.  Black holes and entropy.  Phys. Rev. D, 7:2333–2346, 1973.
  * [15] Shinsei Ryu and Tadashi Takayanagi.  Holographic derivation of entanglement entropy from AdS/CFT.  Phys. Rev. Lett., 96:181602, 2006.
  * [16] Shinsei Ryu and Tadashi Takayanagi.  Aspects of Holographic Entanglement Entropy.  JHEP, 08:045, 2006.
  * [17] Veronika E. Hubeny, Mukund Rangamani, and Tadashi Takayanagi.  A Covariant holographic entanglement entropy proposal.  JHEP, 07:062, 2007.
  * [18] Thomas Faulkner, Aitor Lewkowycz, and Juan Maldacena.  Quantum corrections to holographic entanglement entropy.  JHEP, 11:074, 2013.
  * [19] Netta Engelhardt and Aron C. Wall.  Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime.  JHEP, 01:073, 2015.
  * [20] Netta Engelhardt and Sebastian Fischetti.  Surface Theory: the Classical, the Quantum, and the Holographic.  Class. Quant. Grav., 36(20):205002, 2019.
  * [21] Xi Dong, Daniel Harlow, and Aron C. Wall.  Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality.  Phys. Rev. Lett., 117(2):021601, 2016.
  * [22] Geoffrey Penington.  Entanglement Wedge Reconstruction and the Information Paradox.  JHEP, 09:002, 2020.
  * [23] Ahmed Almheiri, Netta Engelhardt, Donald Marolf, and Henry Maxfield.  The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole.  JHEP, 12:063, 2019.
  * [24] Ahmed Almheiri, Raghu Mahajan, Juan Maldacena, and Ying Zhao.  The Page curve of Hawking radiation from semiclassical geometry.  JHEP, 03:149, 2020.
  * [25] Hong Zhe Chen, Zachary Fisher, Juan Hernandez, Robert C. Myers, and Shan-Ming Ruan.  Information Flow in Black Hole Evaporation.  JHEP, 03:152, 2020.
  * [26] Ahmed Almheiri, Raghu Mahajan, and Juan Maldacena.  Islands outside the horizon.  10 2019.
  * [27] Friðrik Freyr Gautason, Lukas Schneiderbauer, Watse Sybesma, and Lárus Thorlacius.  Page Curve for an Evaporating Black Hole.  JHEP, 05:091, 2020.
  * [28] Takanori Anegawa and Norihiro Iizuka.  Notes on islands in asymptotically flat 2d dilaton black holes.  JHEP, 07:036, 2020.
  * [29] Vijay Balasubramanian, Ben Craps, Mikhail Khramtsov, and Edgar Shaghoulian.  Submerging islands through thermalization.  JHEP, 10:048, 2021.
  * [30] Moshe Rozali, James Sully, Mark Van Raamsdonk, Christopher Waddell, and David Wakeham.  Information radiation in BCFT models of black holes.  JHEP, 05:004, 2020.
  * [31] Ahmed Almheiri, Raghu Mahajan, and Jorge E. Santos.  Entanglement islands in higher dimensions.  SciPost Phys., 9(1):001, 2020.
  * [32] Koji Hashimoto, Norihiro Iizuka, and Yoshinori Matsuo.  Islands in Schwarzschild black holes.  JHEP, 06:085, 2020.
  * [33] Mohsen Alishahiha, Amin Faraji Astaneh, and Ali Naseh.  Island in the presence of higher derivative terms.  JHEP, 02:035, 2021.
  * [34] Hao Geng and Andreas Karch.  Massive islands.  JHEP, 09:121, 2020.
  * [35] Hong Zhe Chen, Robert C. Myers, Dominik Neuenfeld, Ignacio A. Reyes, and Joshua Sandor.  Quantum Extremal Islands Made Easy, Part I: Entanglement on the Brane.  JHEP, 10:166, 2020.
  * [36] Venkatesa Chandrasekaran, Masamichi Miyaji, and Pratik Rath.  Including contributions from entanglement islands to the reflected entropy.  Phys. Rev. D, 102(8):086009, 2020.
  * [37] Dongsu Bak, Chanju Kim, Sang-Heon Yi, and Junggi Yoon.  Unitarity of entanglement and islands in two-sided Janus black holes.  JHEP, 01:155, 2021.
  * [38] Raphael Bousso and Elizabeth Wildenhain.  Gravity/ensemble duality.  Phys. Rev. D, 102(6):066005, 2020.
  * [39] Chethan Krishnan.  Critical Islands.  JHEP, 01:179, 2021.
  * [40] Roberto Emparan, Antonia Micol Frassino, and Benson Way.  Quantum BTZ black hole.  JHEP, 11:137, 2020.
  * [41] Hong Zhe Chen, Robert C. Myers, Dominik Neuenfeld, Ignacio A. Reyes, and Joshua Sandor.  Quantum Extremal Islands Made Easy, Part II: Black Holes on the Brane.  JHEP, 12:025, 2020.
  * [42] Yi Ling, Yuxuan Liu, and Zhuo-Yu Xian.  Island in Charged Black Holes.  JHEP, 03:251, 2021.
  * [43] Aranya Bhattacharya, Anindya Chanda, Sabyasachi Maulik, Christian Northe, and Shibaji Roy.  Topological shadows and complexity of islands in multiboundary wormholes.  JHEP, 02:152, 2021.
  * [44] Juan Hernandez, Robert C. Myers, and Shan-Ming Ruan.  Quantum extremal islands made easy. Part III. Complexity on the brane.  JHEP, 02:173, 2021.
  * [45] Jaydeep Kumar Basak, Debarshi Basu, Vinay Malvimat, Himanshu Parihar, and Gautam Sengupta.  Islands for Entanglement Negativity.  12 2020.
  * [46] Elena Caceres, Arnab Kundu, Ayan K. Patra, and Sanjit Shashi.  Warped Information and Entanglement Islands in AdS/WCFT.  JHEP, 07:004, 2021.
  * [47] Feiyu Deng, Jinwei Chu, and Yang Zhou.  Defect extremal surface as the holographic counterpart of Island formula.  JHEP, 03:008, 2021.
  * [48] Anna Karlsson.  Concerns about the replica wormhole derivation of the island conjecture.  1 2021.
  * [49] Rong-Xin Miao.  Codimension-n holography for cones.  Phys. Rev. D, 104(8):086031, 2021.
  * [50] Constantin Bachas and Vassilis Papadopoulos.  Phases of Holographic Interfaces.  JHEP, 04:262, 2021.
  * [51] Alex May and David Wakeham.  Quantum tasks require islands on the brane.  Class. Quant. Grav., 38(14):144001, 2021.
  * [52] Kohki Kawabata, Tatsuma Nishioka, Yoshitaka Okuyama, and Kento Watanabe.  Probing Hawking radiation through capacity of entanglement.  JHEP, 05:062, 2021.
  * [53] Aranya Bhattacharya, Arpan Bhattacharyya, Pratik Nandy, and Ayan K. Patra.  Islands and complexity of eternal black hole and radiation subsystems for a doubly holographic model.  JHEP, 05:135, 2021.
  * [54] Wontae Kim and Mungon Nam.  Entanglement entropy of asymptotically flat non-extremal and extremal black holes with an island.  Eur. Phys. J. C, 81(10):869, 2021.
  * [55] Lars Aalsma and Watse Sybesma.  The Price of Curiosity: Information Recovery in de Sitter Space.  JHEP, 05:291, 2021.
  * [56] Dominik Neuenfeld.  The Dictionary for Double Holography and Graviton Masses in d Dimensions.  4 2021.
  * [57] Hao Geng, Severin Lüst, Rashmish K. Mishra, and David Wakeham.  Holographic BCFTs and Communicating Black Holes.  jhep, 08:003, 2021.
  * [58] Vijay Balasubramanian, Arjun Kar, and Tomonori Ugajin.  Entanglement between two gravitating universes.  4 2021.
  * [59] Christoph F. Uhlemann.  Islands and Page curves in 4d from Type IIB.  JHEP, 08:104, 2021.
  * [60] Dominik Neuenfeld.  Double Holography as a Model for Black Hole Complementarity.  5 2021.
  * [61] Kohki Kawabata, Tatsuma Nishioka, Yoshitaka Okuyama, and Kento Watanabe.  Replica wormholes and capacity of entanglement.  JHEP, 10:227, 2021.
  * [62] Jinwei Chu, Feiyu Deng, and Yang Zhou.  Page curve from defect extremal surface and island in higher dimensions.  JHEP, 10:149, 2021.
  * [63] Yizhou Lu and Jiong Lin.  Islands in Kaluza-Klein black holes.  6 2021.
  * [64] Jorrit Kruthoff, Raghu Mahajan, and Chitraang Murdia.  Free fermion entanglement with a semitransparent interface: the effect of graybody factors on entanglement islands.  SciPost Phys., 11:063, 2021.
  * [65] Ibrahim Akal, Yuya Kusuki, Noburo Shiba, Tadashi Takayanagi, and Zixia Wei.  Holographic moving mirrors.  Class. Quant. Grav., 38(22):224001, 2021.
  * [66] Jaydeep Kumar Basak, Debarshi Basu, Vinay Malvimat, Himanshu Parihar, and Gautam Sengupta.  Page Curve for Entanglement Negativity through Geometric Evaporation.  6 2021.
  * [67] Elena Caceres, Arnab Kundu, Ayan K. Patra, and Sanjit Shashi.  Page Curves and Bath Deformations.  6 2021.
  * [68] Hidetoshi Omiya and Zixia Wei.  Causal Structures and Nonlocality in Double Holography.  7 2021.
  * [69] Hao Geng, Andreas Karch, Carlos Perez-Pardavila, Suvrat Raju, Lisa Randall, Marcos Riojas, and Sanjit Shashi.  Inconsistency of Islands in Theories with Long-Range Gravity.  7 2021.
  * [70] Hong Zhe Chen, Zachary Fisher, Juan Hernandez, Robert C. Myers, and Shan-Ming Ruan.  Evaporating Black Holes Coupled to a Thermal Bath.  JHEP, 01:065, 2021.
  * [71] D. M. McAvity and H. Osborn.  Conformal field theories near a boundary in general dimensions.  Nucl. Phys. B, 455:522–576, 1995.
  * [72] Andreas Karch and Lisa Randall.  Locally localized gravity.  JHEP, 05:008, 2001.
  * [73] Oliver DeWolfe, Daniel Z. Freedman, and Hirosi Ooguri.  Holography and defect conformal field theories.  Phys. Rev. D, 66:025009, 2002.
  * [74] Tadashi Takayanagi.  Holographic Dual of BCFT.  Phys. Rev. Lett., 107:101602, 2011.
  * [75] Mitsutoshi Fujita, Tadashi Takayanagi, and Erik Tonni.  Aspects of AdS/BCFT.  JHEP, 11:043, 2011.
  * [76] Masahiro Nozaki, Tadashi Takayanagi, and Tomonori Ugajin.  Central Charges for BCFTs and Holography.  JHEP, 06:066, 2012.
  * [77] Juan Martin Maldacena.  The Large N limit of superconformal field theories and supergravity.  Adv. Theor. Math. Phys., 2:231–252, 1998.
  * [78] Edward Witten.  Anti-de Sitter space and holography.  Adv. Theor. Math. Phys., 2:253–291, 1998.
  * [79] Thomas Hartman and Juan Maldacena.  Time Evolution of Entanglement Entropy from Black Hole Interiors.  JHEP, 05:014, 2013.
  * [80] Lisa Randall and Raman Sundrum.  A Large mass hierarchy from a small extra dimension.  Phys. Rev. Lett., 83:3370–3373, 1999.
  * [81] Lisa Randall and Raman Sundrum.  An Alternative to compactification.  Phys. Rev. Lett., 83:4690–4693, 1999.
  * [82] Andreas Karch and Lisa Randall.  Open and closed string interpretation of SUSY CFT’s on branes with boundaries.  JHEP, 06:063, 2001.
  * [83] Massimo Porrati.  Mass and gauge invariance 4. Holography for the Karch-Randall model.  Phys. Rev. D, 65:044015, 2002.
  * [84] Andreas Karch, Emanuel Katz, and Lisa Randall.  Absence of a VVDZ discontinuity in AdS(AdS).  JHEP, 12:016, 2001.
  * [85] M. Porrati.  Higgs phenomenon for 4-D gravity in anti-de Sitter space.  JHEP, 04:058, 2002.
  * [86] Massimo Porrati and Andrei Starinets.  On the graviton selfenergy in AdS(4).  Phys. Lett. B, 532:48–54, 2002.
  * [87] M. J. Duff, James T. Liu, and H. Sati.  Complementarity of the Maldacena and Karch-Randall pictures.  Phys. Rev. D, 69:085012, 2004.
  * [88] Ofer Aharony, Adam B. Clark, and Andreas Karch.  The CFT/AdS correspondence, massive gravitons and a connectivity index conjecture.  Phys. Rev. D, 74:086006, 2006.
  * [89] En-Jui Chang, Chia-Jui Chou, and Yi Yang.  Holographic entanglement entropy in boundary conformal field theory.  Phys. Rev. D, 98(10):106016, 2018.
  * [90] Rong-Xin Miao, Chong-Sun Chu, and Wu-Zhong Guo.  New proposal for a holographic boundary conformal field theory.  Phys. Rev. D, 96(4):046005, 2017.
  * [91] Chong-Sun Chu, Rong-Xin Miao, and Wu-Zhong Guo.  On New Proposal for Holographic BCFT.  JHEP, 04:089, 2017.