aainstitutetext: Perimeter Institute for Theoretical Physics,
Waterloo, ON N2L 2Y5, Canadabbinstitutetext: Center for Gravitational Physics
and Quantum Information,
Yukawa Institute for Theoretical Physics, Kyoto University,
Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japanccinstitutetext:
Department of Physics, Rikkyo University, Toshima, Tokyo 171-8501, Japan

# Double Holography of Entangled Universes

Robert C. Myers b Shan-Ming Ruan b,c and Tomonori Ugajin
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We employ double holography to examine a system of two entangled gravitating
universes that live on two codimension-one branes in an asymptotically AdS3
spacetime with two disjoint conformal boundaries. There are distinct brane
configurations depending on the temperature of the thermoﬁeld double (TFD)
state between the left and right systems. The topology transition between two
branes is naturally identified with the emergence of an Einstein-Rosen bridge
connecting the two entangled universes. This doubly holographic construction
offers a holographic perspective on gravitational collapse and black hole
formation in brane universes. Through this holographic framework, we analyze
the quantum information structure of the two gravitating universes.
Specifically, we calculate the mutual information between defects present in
the boundary theories on the left and right sides. Furthermore, we investigate
the decoupling process in the Hayden-Preskill protocol applied to the two
copies of the defect field theory and discuss the interpretation of the
Yoshida-Kitaev decoding protocol.

††arxiv: 2403.XXXX

YITP-24-33

##  1 Introduction

Recent studies have shown that many quantum information-theoretic ideas
naturally appear in quantum gravity and its holographic realizations. The
basic example of this phenomenon is the Ryu-Takayanagi (RT) formula Ryu:2006bv
; Ryu:2006ef ; Hubeny:2007xt  in the AdS/CFT, which relates the entanglement
entropy of the boundary CFT to the area of an extremal surface in the bulk
geometry of the dual gravity theory. The island formula Almheiri:2019hni  for
the entropy of Hawking radiation is obtained by suitably generalizing the RT
formula to include quantum corrections Engelhardt:2014gca ; Faulkner:2013ana .
This enables a controlled calculation of the Page curve from the semi-
classical description of spacetime (plus quantum field theory on top)
Almheiri:2019psf ; Penington:2019npb . Therefore, we have gained surprising
new insights into why semi-classical gravity is consistent with the unitarity
of the underlying quantum theory. One of the surprising outcomes of the island
formula implies that the area of the black hole interior called ”island” is
encoded in the degrees of freedom of Hawking radiation as a quantum error-
correcting code, and thus is reconstructable from it in principle. The island
formula is proven by a replica calculation of the radiation entropy
Penington:2019kki ; Almheiri:2019qdq . The gravitational path integral for its
Rényi entropies contains saddles called replica wormholes, and they dominate
the path integral after the Page time, resulting in the formation of the
island region.

In the original setup of the island formula, the bath collecting Hawking
quanta is non-gravitational. Upon activating gravity within the bath system, a
geometric connection between the black hole and the gravitating bath system
shows up and modifies the formula for the entropy when the entanglement
between two systems is large, as shown in Balasubramanian:2021wgd ;
Miyata:2021qsm  (see also Anderson:2020vwi ; Geng:2020fxl ; Geng:2021iyq ;
Balasubramanian:2021xcm ; Liu:2022pan ; Balasubramanian:2023xyd  for related
studies). In these studies, the authors considered the following setup: First,
we start with two disjoint gravitating universes denoted by L and R, where
both are asymptotically AdS and each contains an eternal black hole. On these
gravitating universes, we define the bulk effective quantum field theory as
QFT${}_{\textrm{\tiny L}}$, QFT${}_{\textrm{\tiny R}}$, and take these to be
the same theory111For simplicity, the gravity sector was treated as
classical.. The Hilbert space of this sector is given by $\mathcal{H}_{\rm
QFT_{\textrm{\tiny L}}}\otimes\mathcal{H}_{\rm QFT_{\textrm{\tiny R}}}$. On
this Hilbert space, we consider the following thermofield double (TFD) state,

$|\Psi\rangle=\frac{1}{\sqrt{Z(\beta)}}\,\sum_{i=0}^{\infty}e^{-\beta
E_{i}/2}\,|\psi_{i}\rangle_{\textrm{\tiny{\tiny{\tiny
L}}}}\,|\psi_{i}\rangle_{\textrm{\tiny R}}\,,$ (1)

where the inverse temperature $\beta$ is now a parameter controlling the
entanglement between these two universes. This generalizes the setup of
Balasubramanian:2020coy ; Balasubramanian:2020xqf  for the island formula
where one of two universes is non-gravitating. The main statement in
Balasubramanian:2021wgd  was that in the high temperature limit
$\beta\rightarrow 0$ the entanglement entropy of the universe L

$S(\rho_{\textrm{\tiny{\tiny L}}})=-{\rm tr}\rho_{\textrm{\tiny
L}}\log\rho_{\textrm{\tiny L}},\quad\rho_{\textrm{\tiny L}}={\rm
tr}_{\textrm{\tiny R}}|\Psi\rangle\langle\Psi|\,,$ (2)

is given by the generalized entropy $S_{\rm gen}(\mathrm{L/R})$ on a geometry
denoted $\mathrm{L/R}$, which is constructed by suitably connecting the two
universes together through a wormhole,

$S(\rho_{\textrm{\tiny L}})=S_{\rm
gen}(\mathrm{L/R})=\min_{C}\left[\underset{C}{\text{ext}}\left(\frac{{\rm
Area}_{\textrm{\tiny L/R}}[\partial
C]}{4G_{N}}+S_{\rm{eff}}[C]\right)\right]\,,$ (3)

where $C$ is a codimension-two surface in the wormhole on $\mathrm{L/R}$,
${\rm Area}_{\textrm{\tiny L/R}}[\partial C]$ denotes the area of the boundary
of $C$ on the connected geometry $\mathrm{L/R}$, and $S_{\rm{eff}}[C]$ is the
entanglement entropy of the bulk effective QFT on $\mathrm{L/R}$. If each of
these universes is an asymptotic AdS with two boundaries, then $\mathrm{L/R}$
again has two boundaries, and its metric is obtained by solving Einstein
equations with the boundary condition for $\rm{L}$ on one of these boundaries
and for $\rm{R}$ on the other. Notably, this calculation is consistent with
the ER=EPR hypothesis Maldacena:2013xja , which suggests that strong quantum
entanglement can lead to the emergence of an Einstein-Rosen bridge in quantum
gravity.

Doubly holographic framework Almheiri:2019hni ; Chen:2020uac ; Chen:2020hmv ;
Hernandez:2020nem ; Grimaldi:2022suv  is an efficient tool to investigate
quantum extremal surfaces and the black hole information paradox for AdS black
holes, e.g., see Chen:2019uhq ; Rozali:2019day ; Balasubramanian:2020hfs ;
Chen:2020jvn ; Caceres:2020jcn ; Krishnan:2020fer ; Sully:2020pza ;
Omiya:2021olc ; Neuenfeld:2021bsb ; Neuenfeld:2021wbl ; Chu:2021gdb ;
Li:2021dmf ; Wang:2021xih ; Geng:2021wcq ; Ling:2021vxe ; Geng:2022slq ;
Suzuki:2022xwv ; Karch:2022rvr ; Deng:2022yll ; Anous:2022wqh ;
Neuenfeld:2023svs ; Chang:2023gkt ; Liu:2023ggg ; Basak:2023bnc ;
Jeong:2023lkc ; Kawamoto:2023wzj  for various studies about double holography.
The configuration involves a $d+1$ dimensional asymptotically anti-de Sitter
(AdS) space, acting as the dual to a holographic conformal field theory
governing Hawking radiation. Additionally, there is a codimension-one brane
housing a black hole and extending to the conformal boundary. The asymptotic
AdS space’s conformal boundary acts as a heat bath, collecting radiation
emitted by the black hole. This perspective is termed the bulk perspective, as
it realizes the original system of the evaporating black hole on the
codimension-one surface of the $d+1$ dimensional semi-classical spacetime. The
alternative description, known as the CFT perspective, is non-gravitational.
It involves a boundary CFT on the $d$-dimensional conformal boundary coupled
to defects that are dual to the black hole on the brane. In this framework,
the radiation’s entropy corresponds to the entanglement entropy of the
boundary CFT in the presence of these defects (dual to the brane). The
holographic calculation utilizes the standard RT formula, revealing the
dominance of the extremal surface ending on the brane at sufficiently late
times, leading to the formation of the island region in the black hole
interior.

This paper delves into the dynamics of two entangled gravitating universes in
two dimensions, from the three-dimensional bulk perspective. In contrast to
the discussion in Balasubramanian:2021wgd , which focused on two-dimensional
JT gravity with limited calculations due to multiple saddles in the
gravitational path integral, our approach, based on the three-dimensional bulk
perspective, allows for a comprehensive exploration of the entanglement
entropy.222The effective gravitational action on the brane takes a nonlocal
Polyakov form Chen:2020uac ; Chen:2020hmv ; Grimaldi:2022suv  in the present
paper (see the discussion in Appendix A.5), but the analysis is easily
extended to Jackiw-Teitelboim (JT) gravity gravity, as discussed in
Grimaldi:2022suv . The power of holographic duality enables us to follow the
entire temperature dependence. The analysis uncovers a diverse phase structure
dependent on $\beta$ and the distance between the brane endpoints. We model
the matter’s entangled state (1) using a BTZ black hole in three-dimensional
bulk spacetime, which is dual to the thermofield double state. The two-
dimensional universe $L$ corresponds to two defects, $D^{(\textrm{\tiny
L})}_{a}$ and $D^{(\textrm{\tiny L})}_{b}$, situated on the left boundary.
Similarly, the universe R is represented by the defects $D^{(\textrm{\tiny
R})}_{a}$ and $D^{(\textrm{\tiny R})}_{b}$ on the opposite boundary.

###  1.1 Double Holography

Motivated by the framework explored in Balasubramanian:2021wgd , we
investigate a doubly holographic setup involving two gravitating branes
entangled by matter degrees of freedom. This scenario offers three equivalent
perspectives for understanding the system Chen:2020uac .

The brane perspective: From this point of view, the system comprises two
disjoint asymptotically AdS2 spacetimes, denoted as the left universe
$U_{\textrm{\tiny L}}$ and the right universe $U_{\textrm{\tiny R}}$. Each
possesses two one-dimensional boundaries, represented by $D^{(\textrm{\tiny
L})}_{a}$, $D^{(\textrm{\tiny L})}_{b}$ for $U_{\textrm{\tiny L}}$, and
$D^{(\textrm{\tiny R})}_{a}$, $D^{(\textrm{\tiny R})}_{b}$ for
$U_{\textrm{\tiny R}}$. Additionally, a two-dimensional non-gravitational bath
(${\bf Bath}_{\textrm{\tiny L}}$ and ${\bf Bath}_{\textrm{\tiny R}}$) is
prepared for each spacetime. ${\bf Bath}_{\textrm{\tiny L}}$ is attached to
$U_{\textrm{\tiny L}}$ at its asymptotic boundaries $D^{(\textrm{\tiny
L})}_{a}$, $D^{(\textrm{\tiny L})}_{b}$, and similarly for the right universe.
The degrees of freedom of CFT matter can propagate in both the gravitational
and non-gravitational regions. We assume these two non-gravitational baths are
disjoint, ensuring no direct interactions between the total regions
$U_{\textrm{\tiny L}}\cup{\bf Bath}_{\textrm{\tiny L}}$ and $U_{\textrm{\tiny
R}}\cup{\bf Bath}_{\textrm{\tiny R}}$. The quantum state on the total system
is thus the TFD state (1).

The boundary perspective: Given that $U_{\textrm{\tiny L}}$ and
$U_{\textrm{\tiny R}}$ are asymptotically AdS2, it is natural to anticipate
the dual holographic description. In this perspective, $U_{\textrm{\tiny L}}$
is dual to “defect degrees of freedom” on its asymptotic boundaries
$D^{(\textrm{\tiny L})}_{a},D^{(\textrm{\tiny L})}_{b}$. The total system
involves ambient matter modes on ${\rm Bath}_{\textrm{\tiny L}}$ interacting
with modes on two defects $D^{(\textrm{\tiny L})}_{a},D^{(\textrm{\tiny
L})}_{b}$. A similar system is constructed for ${\rm Bath}_{\textrm{\tiny
R}}\cup D^{(\textrm{\tiny R})}_{a}\cup D^{(\textrm{\tiny R})}_{b}$, entangled
with the left. To distinguish from the standard CFT, we will refer to the
boundary field theory as the defect conformal field theory, denoted by DCFT.
The boundary setup as illustrated in figure 1 is characterized by the inverse
temperature $\beta$ as well as $L_{1}$ and $L_{2}$ measuring the distance
between two defects.

The bulk perspective: In this paper, we assume that the matter QFT is a
conformal field theory with a large central charge. Then the boundary
entangled state (1) has a dual gravity description in AdS3 bulk spacetime. The
dual geometry for the bulk spacetime is either the thermal AdS3 or BTZ black
hole, depending on the temperature. Two asymptotic boundaries of the AdS3 bulk
spacetime correspond to the bath regions ${\bf Bath}_{\textrm{\tiny L}}$,
${\bf Bath}_{\textrm{\tiny R}}$. Furthermore, in this setup the 2D gravitating
universe in the brane perspective is realized as a codimension-one brane in
AdS3 bulk spacetime, which anchors on the asymptotic boundary at the locations
of the defects. Because we require the brane to be sufficiently massive, it
will backreact to the 3D bulk geometry. The backreaction of the codimension-
one brane is treated by first preparing two AdS3 (the masses $m_{1},m_{2}$ of
two spacetimes can be different in general) and then gluing two spacetime
regions (say $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$) along the brane, as
depicted in figure 2. The brane profile is determined by solving the junction
conditions between the two sides, which are given by the so-called Israel
junction condition.

###  1.2 Boundary perspective: entangled DCFTs

Figure 1: Boundary perspective: The two defect field theories at the
temperature $T_{\textrm{\tiny DCFT}}$ are on living on the left and right
boundaries, respectively and are entangled together.

Here, we briefly discuss the information we need to fix the boundary setup. In
the main text, the dual bulk spacetime would be determined by using the
boundary data. For the left/right boundary, we consider a boundary defect
field theory living on a circle as shown in figure 1. The metric
characterizing the conformal boundary takes the form:

$ds^{2}\big{|}_{\rm{bdy}}=-dt^{2}+R_{\rm bdy}^{2}d\phi^{2}\,.$ (4)

The presence of two defects on the boundary circle partitions the boundary
theories into distinct regions, each with a fixed length $L_{1}$ and $L_{2}$.
In other words, the two-dimensional boundary theory is living on an infinitely
long cylinder whose circumference is fixed as $L_{\rm bdy}\equiv
L_{1}+L_{2}=2\pi R_{\rm bdy}$. To maintain simplicity, we posit that the
defect field theories residing within the two regions share an identical
central charge, namely $c_{1}=c=c_{2}$. Within this context, the set of
boundary data encompasses five independent parameters:
$(c,T_{\mathrm{DCFT}},L_{1},L_{2})$ along with the defect entropy $2\log g$,
which is determined by the boundary condition at the defects and measures the
ground state degeneracy of the defect – see eq. (7). However, because the
boundary theory is conformal, the physics will only depend on dimensionless
combinations of these parameters. Hence we can parametrize the boundary
physics by the four following combinations:

$\left\\{c\,,\quad\log g\,,\quad T_{\textrm{\tiny
DCFT}}\left(L_{1}+L_{2}\right)\,,\quad\frac{L_{1}}{L_{2}}\right\\}$ (5)

with each of these parameters retaining its scaling-invariant nature.

###  1.3 Bulk perspective: summary of three possible phases

Figure 2: Sketch illustrating a time slice in the bulk for the three possible
phases. We are only showing a single asymptotic boundary.

In the following section, we will discuss potential phases from the 3D bulk
perspective, focusing on plausible bulk brane profiles that connect four
defects on the boundary, given a specific set of boundary parameters (5).
Generally, three distinct phases characterize the bulk spacetime
Bachas:2021fqo , as illustrated in figure 2.

Cold phase: In this scenario, when the temperature is sufficiently low, the 3D
bulk geometry consists of two disconnected pieces, denoted as
$\mathcal{M}_{\textrm{\tiny L}}$ and $\mathcal{M}_{\textrm{\tiny R}}$. In this
phase, each brane connects two defects on the same boundary. This implies that
the bath CFT is in a confined phase, and the correlation primarily exists
between $D^{(\textrm{\tiny L})}_{a}$ and $D^{(\textrm{\tiny L})}_{b}$.
Consequently, the mutual information between the left and right defects
(denoted as $I_{\textrm{\tiny L:R}}$) is of the order of $O(c^{0})$,
indicating a low correlation between the left and right.

Warm phase: As the temperature increases, the bath CFT becomes deconfined. In
the dual gravity description, this corresponds to the Hawking-Page transition,
and the bulk geometry transforms into a black hole. At this middle stage, the
bulk BTZ black hole remains small and the branes connecting the two defects on
the left/right boundary stay outside the black hole horizon. Consequently, the
entanglement entropy between the two sides increases, but the correlation
between the left and right defects remains negligible.

Hot phase: As the system heats up further, the black hole expands, eventually
swallowing parts of the branes. In this case, each brane connects two defects
- one on the left boundary and the other on the right. Due to the topology
change of the brane profiles, the correlation between the left and right
defects becomes significant in this hot phase.

###  1.4 Entanglement structure of the holographic gravitating universes

We use this holographic construction to perform a detailed study of the
quantum information structure of the two gravitating universes. Our primary
focus is on assessing the entanglement between the two branes and how it
evolves through the phase transition. In the CFT perspective, this translates
into the entanglement between two groups of defects $D^{(\textrm{\tiny
L})}_{a,b}$, with the corresponding quantity being the mutual information
between them. We holographically compute this mutual information in each phase
by meticulously determining the dominant RT surface for the entanglement
entropy of each subregion. The summarized results are shown in figure 20. The
key observation is that the mutual information is non-trivial only in the hot
phase. This is consistent with expectations since each brane in the bulk can
be interpreted as a “bit thread” Freedman:2016zud , providing a visualization
of the entanglement structure of the DCFT state.

###  1.5 The Hayden-Preskill setup and the Yoshida-Kitaev protocol

The Hayden-Preskill Gedanken experiment provides a valuable framework for
investigating how information thrown into an evaporating black hole is
recovered from Hawking radiation, particularly after the Page time. The
crucial mechanism for achieving recovery is the phenomenon of decoupling,
where the remaining black hole $C$ and the reference system $R$, initially
maximally entangled with the diary, rapidly lose correlation as the size of
the late Hawking radiation becomes comparable to that of the diary. We
demonstrate that the setup of Hayden and Preskill finds a natural realization
in our two entangled DCFTs setup. This is accomplished by establishing an
explicit correspondence between subsystems in these two setups (see figure
21). We then use holography to study this decoupling in the DCFT setup by
computing the mutual information $I_{R:C}$ between the radiation and the black
hole. Our analysis shows that in the hot phase, the contribution of defects to
this mutual information vanishes, indicating a successful decoupling.

Yoshida and Kitaev Yoshida:2017non  proposed an actual recovery protocol for
the diary from early Hawking radiation in the Hayden-Preskill setup. In this
protocol, the postselection to the maximally entangled state of the late
Hawking radiation and its copy plays a key role, inducing the quantum
teleportation of the diary to early Hawking radiation. We show that this
postselection procedure has a natural realization in the DCFT setup, achieved
by changing the distance between two defects.

###  1.6 Organization of this paper

The rest of the paper is organized as follows: Section 2 reviews the
holographic construction of the system of interest, i.e., the thermofield
double state entangling two copies of a two-dimensional defect CFT. Further,
we describe the various phases that the system goes through as the temperature
increases. This review is primarily based on Bachas:2021fqo . While the
underlying structure is not simple enough that it can be described
analytically, the final expressions for, e.g., the phase diagram can be
determined numerically. However, there are two domains where the system can be
understood fully analytically: $L_{1}=L_{2}$ and $L_{1}\gg L_{2}$, which are
described in sections 2.2 and 2.4 respectively. In section 3, we then turn to
examine the quantum entanglements between various subsystems, e.g., the
quantum correlations between the various defects. We use mutual information as
our measure of the strength of these correlations and examine how the mutual
information varies in various phases. In the cold and warm phases, the mutual
information between the various subsystems is independent of time. In the hot
phase, we examine the time dependence of the entanglement and the appearance
of quantum extremal islands. We again provide analytical results for
$L_{1}=L_{2}$ and $L_{1}\gg L_{2}$ in sections 3.1 and 3.2, respectively. In
section 4, we demonstrate that this doubly holographic setup can be used to
realize the Hayden-Preskill protocol. We conclude in section 5 with a
discussion of our results and possible future directions. Appendix A provides
studies on various aspects of the brane perspective, such as phase transitions
on the brane, renormalized action and so on. In appendix B, we describe some
technical details of the UV regulator for the bulk branes. Appendix C provides
an alternative analysis of the thermodynamics of the defect CFT in the large
tension limit.

##  2 Bulk Perspective: Three Phases

In this section, we apply the bulk perspective to examine the holographic dual
of entangled defect CFTs on the boundary. As discussed above, the bulk
gravitational description involves two codimension-one branes extending
through the AdS3 geometry and anchored to the asymptotic boundaries at the
location of the boundary defects. In the following discussion, it will be
useful to introduce $\mathcal{S}_{1,2}$ to denote the bulk regions bounded by
the branes and the asymptotic boundary regions of length $L_{1,2}$ (as
measured by the boundary metric (4)) – see figures 1 and 2. In principle, we
could consider ‘interfaces’ where the CFTs on either side of the defects are
distinct. However, for simplicity, we will assume that the two CFTs are the
same, which implies that in the bulk, the cosmological curvature scale $\ell$
is constant throughout the AdS3 geometry.333The first half of this section is
largely a review based on Bachas:2021fqo . However, we note this reference
considers a more general set-up where the AdS scale differs on either side of
the brane. Hence the central charges of the two CFTs between the defects are
identical, i.e., $c_{1}=c_{2}=c$, with

$c=\frac{3\ell}{2G_{\textrm{\tiny N}}}\,.$ (6)

While the bulk spacetime is locally equivalent to AdS3 everywhere (and in all
of the different phases), the backreaction of the two-dimensional branes
enlarges the geometry in the vicinity of the branes, e.g., as illustrated in
figure 3.

The boundary theory is also characterized by Affleck-Ludwig entropy
Affleck:1991tk  associated with the defects. This constant can be interpreted
as the central charge of the conformal degrees of freedom living on the
defects. As we demonstrate below (see eq. (50)), it is determined by the brane
tension Grimaldi:2022suv 444Following Grimaldi:2022suv , we include a factor
of 2 here because we are considering a defect (with two sides) rather than a
boundary. With this normalization, our result matches the holographic
calculation of $\log g$ in the usual AdS/BCFT correspondence Takayanagi:2011zk
; Fujita:2011fp .

$S_{\rm def}\equiv 2\log
g=\frac{c}{6}\,\log\left(\frac{1+T_{o}\ell}{1-T_{o}\ell}\right)=\frac{c}{3}\,\text{arctanh}\left(T_{o}\ell\right)\,.$
(7)

The bulk gravitational action has three components

$I_{\rm tot}=I_{\rm{bulk}}+I_{\rm{brane}}+I_{\rm{boundary}}\,.$ (8)

We describe these terms in Euclidean signature since this choice is relevant
for evaluating the action and determining the phase diagram in the subsequent
sections. Then the bulk action is, of course, simply given by the Einstein
action with a negative cosmological constant,

$I_{\rm{bulk}}=-\frac{1}{16\pi G_{\textrm{\tiny N}}}\int
d^{3}x\,\sqrt{g}\left(\mathcal{R}(g)+\frac{2}{\ell^{2}}\right)\,.$ (9)

Note that the overall minus sign is associated with the Euclidean signature.
We write the brane contribution as

$I_{\rm{brane}}=-\frac{1}{8\pi G_{\textrm{\tiny N}}}\int
d^{2}\sigma\,\sqrt{h}\left(\Delta K-2T_{o}\right)\,.$ (10)

where $\Delta K_{ab}\equiv K_{ab}^{(1)}+K_{ab}^{(2)}$ denotes the
discontinuity of the extrinsic curvature across the brane. One can think of
this Gibbons-Hawking-York contribution as representing the curvature
contribution that would be created in the bulk Einstein term by the
$\delta$-function source of the brane Israel:1966rt . Of course, $h_{ab}$
denotes the induced metric on the brane surface, and the extrinsic curvatures
are given by $K_{ab}^{(i)}=h^{\mu}_{a}h^{\nu}_{b}\nabla_{\mu}n_{\nu}^{(i)}$
with the normal vector $n_{\mu}^{(i)}$ pointing outward from the region
$\mathcal{S}_{i}$. Eq. (10) also contains the usual worldvolume action of the
brane proportional to the tension $T_{o}$.555Note the present normalization
differs from that in Chen:2020uac ; Chen:2020hmv ; Hernandez:2020nem ;
Grimaldi:2022suv . That is, $T_{o}^{\textrm{\tiny(present)}}=4\pi
G_{\textrm{\tiny N}}\,T_{o}^{\textrm{\tiny(previous)}}$. Finally, we have the
boundary terms on the asymptotic boundary,

$I_{\rm{boundary}}=-\frac{1}{8\pi G_{\textrm{\tiny
N}}}\int_{\partial(\mathcal{S}_{1}\cup\mathcal{S}_{2})}\sqrt{h}\left(K-\frac{1}{\ell}\right)\,.$
(11)

with the usual Gibbons-Hawking-York term and the boundary counterterm for
$d=2$ Emparan:1999pm . As discussed in appendix B, the boundary action may
also contain two additional terms associated with the branes, depending on the
choice of the cut-off surface near the defects. The first is a Hayward term
Hayward:1993my ; Brill:1994mb  for the joint where the asymptotic boundaries
of $\mathcal{S}_{1,2}$ meet at the brane. The second is a boundary counterterm
for the gravitational theory on the brane. However, these two contributions
precisely cancel (after taking $\epsilon\to 0$) and so for simplicity, we have
not included them above in eq. (11). Appendix B provides an explicit
discussion of the joint terms and their cancellation.

The dual bulk spacetime is ultimately obtained by gluing the bulk regions
$\mathcal{S}_{i}$ along the two branes $\mathcal{B}_{i}$. Since we are in
three dimensions, the bulk regions are simply AdS3 geometries locally because
the bulk action (8) is nothing but the Einstein-Hilbert term involving a
negative cosmological constant. Hence we must now determine to profile of the
branes. (Note that we are employing Lorentzian signature for this discussion.)

Referring to the bulk coordinates in bulk region $\mathcal{S}_{i}$ as
$(t_{i},r_{i},\phi_{i})$, we can express the metric of each region as:

$ds^{2}=-\left(\frac{r^{2}_{i}}{\ell^{2}}-m_{i}\right)dt_{i}^{2}+\frac{dr^{2}_{i}}{\frac{r^{2}}{\ell^{2}}-m_{i}}+r_{i}^{2}d\phi_{i}^{2}\,,$
(12)

where the dimensionless parameter $m_{i}$ is related to the ADM mass of the
gravitational region by $M_{i}=\frac{m_{i}}{8G}$. In the following discussion,
we consistently consider the cut-off surface positioned at

$r_{i}\approx\frac{\ell^{2}}{\epsilon}+A_{i}+B_{i}\epsilon+\mathcal{O}(\epsilon^{2})\to\infty\,.$
(13)

where we introduce some higher-order terms in order to ensure that the cut-off
surface is smooth when it crosses the brane – see Appendix B for more
discussion about this point. For physical quantities that are independent of
the choice of the regulator, any extra terms involving the coefficients
$A_{i},B_{i}$ will automatically vanish after taking $\epsilon\to 0$. The
boundary metric for each interval is fixed as

$ds^{2}\big{|}_{\rm bdy}=-dt^{2}_{i}+\ell^{2}d\phi^{2}_{i}\,.$ (14)

Given the thermal state on the conformal boundary, we note that the Euclidean
time will be periodic.

It is important to highlight that the two mass parameters, $m_{1}$ and
$m_{2}$, are typically distinct since the boundary sizes, $L_{1}$ and $L_{2}$,
will generally be different. Moreover, the sign of $m_{i}$ is not
predetermined either. In cases where $m_{i}$ is negative, the bulk region
$\mathcal{S}_{i}$ essentially corresponds to a portion of global AdS3. In such
instances, it is always possible to reformulate the bulk metric using global
coordinates $(T,R,\Omega)$ with

$ds^{2}=-\left(\frac{R^{2}}{\ell^{2}}+1\right)dT^{2}+\left(\frac{R^{2}}{\ell^{2}}+1\right)^{-1}dR^{2}+R^{2}d\Phi^{2}\,,$
(15)

which is accomplished by rescaling the coordinates as follows

$T=\sqrt{-m_{i}}\,t_{i}\,,\quad
R=\frac{r_{i}}{\sqrt{-m_{i}}}\,,\quad\Phi=\sqrt{-m_{i}}\,\phi_{i}\,.$ (16)

Note that in this case, we always adopt the ‘periodicity’ of $\Phi$ to be
$2\pi$ and hence of $\phi_{i}$ to be $2\pi/\sqrt{-m_{i}}$. Hence there is no
conical singularity at the ‘center’ of the geometry, i.e., $r_{i}=0$.

With positive $m_{i}$, the bulk metric corresponds to the BTZ black hole,
i.e.,

$ds^{2}=-\frac{r^{2}-r_{h}^{2}}{\ell^{2}}\,dt^{2}+\frac{\ell^{2}}{r^{2}-r_{h}^{2}}\,dr^{2}+r^{2}d\phi^{2}\,,\quad\text{with}\quad
m_{i}>0\,,$ (17)

where the black hole horizon is located at $r=r_{h}=\sqrt{m_{i}}\ell$. The
temperature of the black hole reads

$T_{\textrm{\tiny
BH}}\equiv\frac{r_{h}}{2\pi\ell^{2}}=\frac{\sqrt{m_{i}}}{2\pi\ell}=T_{\textrm{\tiny
DCFT}}\,,$ (18)

where the latter equality stands as a consequence of the holographic duality.

Now, the position of the branes in the bulk $\mathcal{S}_{i}$ is derived by
solving the the so-called Israel junction conditions Israel:1966rt :

$\Delta K_{ab}-\Delta K\,h_{ab}-2T_{o}\,h_{ab}=0\,.$ (19)

Recall that $\Delta K_{ab}$ was introduced below eq. (10) and denotes the
discontinuity of the extrinsic curvature across the brane. Implicitly above,
we have been using that the induced metric on both sides of the brane must
match, i.e., $h_{ab}^{(1)}=h_{ab}^{(2)}=h_{ab}$, to successfully glue the two
regions $\mathcal{S}_{1,2}$ together.

The static solutions for the brane have been extensively discussed in
Bachas:2021fqo , and we will offer a concise overview of the pertinent results
in the following discussion. First, let us denote the brane coordinates as
$(t,\sigma)$. Then given our focus on static brane profiles, we can make the
gauge choice, that the timelike brane coordinate coincides with that in the
bulk, i.e., setting $t=t_{1}=t_{2}$. In this context, we can thus write the
brane metric as

$ds^{2}\big{|}_{\mathcal{B}}=-f(\sigma)dt^{2}+g(\sigma)d\sigma^{2}\,.$ (20)

The profile of the brane within the bulk spacetime is then determined by four
functions, namely $\\{\phi_{i}(\sigma),r_{i}(\sigma)\\}$ with $i=1,2$.

Considering the pullback of the metric to the brane from both sides then
yields the following equations:

$f(\sigma)=\frac{r^{2}_{1}}{\ell^{2}}-m_{1}=\frac{r^{2}_{2}}{\ell^{2}}-m_{2}$
(21)

and

$g(\sigma)=\frac{r_{1}^{\prime 2}}{f}+r_{1}^{2}\phi^{\prime
2}_{1}=\frac{r_{2}^{\prime 2}}{f}+r_{2}^{2}\phi^{\prime 2}_{2}\,.$ (22)

Moreover, an additional independent constraint arises from the Israel junction
conditions. For instance, considering the $(tt)$-component of eq. (19) yields

$\sqrt{\frac{f(\sigma)}{g(\sigma)}}\frac{r_{1}^{2}\phi^{\prime}_{1}+r_{2}^{2}\phi^{\prime}_{2}}{\ell^{2}}=-2T_{o}f(\sigma)\,.$
(23)

These are three independent equations for four variables. The undetermined
degree of freedom corresponds to reparametrization of the spacelike
worldvolume coordinate $\sigma$. It is convenient to fix it by imposing a
gauge condition $f(\sigma)=\sigma$. In this gauge, we simply obtain

$\frac{r_{i}(\sigma)}{\ell}=\sqrt{\sigma+m_{i}}\,,$ (24)

where the dimensionless coordinate $\sigma$ plays the role of the ‘radius’ on
the brane. The asymptotic boundary is evidently defined as $\sigma\to\infty$.
For the BTZ black hole with positive $m_{i}$, its black hole horizon is
situated at $\sigma=0$ with $r=\sqrt{m_{i}}\ell\equiv r_{h}$.

The induced metric of the brane can now be reformulated as

$ds^{2}\big{|}_{\mathcal{B}}=-\sigma
dt^{2}+\frac{\ell^{2}}{4(1-T_{o}^{2}\ell^{2})(\sigma-\sigma_{+})(\sigma-\sigma_{-})}d\sigma^{2}\,,$
(25)

with the two poles in the ($\sigma,\sigma$)-component are given by666As we
will see only the pole at $\sigma_{+}$ will be physical and it represents a
mere coordinate singularity since the $\sigma$ coordinate cannot fully
parameterize a complete brane. In appendix A, we introduce an alternative
global coordinate system on the brane to avoid this issue.

$\sigma_{\pm}=\frac{1}{4(1-T_{o}^{2}\ell^{2})}\left(-(m_{1}+m_{2})\pm\sqrt{\left(\frac{m_{1}-m_{2}}{T_{o}\ell}\right)^{2}+4m_{1}m_{2}}\right)\,.$
(26)

The physical solutions can only be obtained under the condition

$|T_{o}|\leq\frac{1}{\ell}\,,$ (27)

which ensures that the brane geometry is asymptotically AdS2, as we will
demonstrate later. It is also straightforward to verify that $\sigma_{+}\geq
0$ and $\sigma_{-}<0$. Finally, the profiles of the branes are parameterized
by $\phi_{i}(\sigma)$ and can be obtained by integrating

$\begin{split}\phi^{\prime}_{i}&=-\frac{4(T_{o}\ell)^{2}\sigma+m_{i}-m_{j}}{8T_{o}\ell(\sigma+m_{i})\sqrt{(1-T_{o}^{2}\ell^{2})\sigma(\sigma-\sigma_{+})(\sigma-\sigma_{-})}}\,.\\\
\end{split}$ (28)

An explicit expression can be found for $\phi_{i}(\sigma)$ involving elliptic
functions. We note that

$\lim\limits_{\sigma\to\sigma_{+}}\frac{d\sigma}{d\phi_{i}}=0\,,$ (29)

due to the pole in eq. (28) at $\sigma=\sigma_{+}$. This implies that the
point at $\sigma=\sigma_{+}$ can be interpreted as a turning point positioned
at the middle of the brane $\mathcal{B}$. Consequently, as the worldvolume
radius ranges over $\sigma\in[\sigma_{+},+\infty)$, one-half of the brane is
covered by the brane extending from this middle point to the boundary defect
on the asymptotic boundary.

###  2.1 Three phases of bulk spacetime

In the preceding analysis, the bulk regions $\mathcal{S}_{1}$ and
$\mathcal{S}_{2}$ are not fully determined yet, because we have not specified
the dimensionless parameters $m_{1}$ and $m_{2}$. Depending on the sign of
these mass parameters, we can define three distinct phases for the system: the
cold phase with $m_{1}<0$ and $m_{2}<0$, the warm phase with $m_{1}>0$ and
$m_{2}<0$, and the hot phase with $m_{1}>0$ and $m_{2}>0$. The significance of
these phase names will become evident in the phase diagram shown in figure 6.
Each of these bulk geometries is characterized by two dimensionless
parameters, namely the masses $m_{1}$ and $m_{2}$ of the two sides of the
brane. In cases where the brane crosses the horizon, as occurs for $m_{i}>0$,
we have an additional dimensionless parameter $\Delta\phi_{{\rm Hor}}$, which
signifies the area of the black hole horizon in the bulk region
$\mathcal{S}_{i}$.

From the perspective of the boundary, the boundary DCFT setup is described by
four independent and dimensionless parameters, as described at eq. (5).
Serving as a holographic duality, the geometric characteristics of the dual
bulk spacetime can be deduced from these boundary parameters. For example, the
central charge $c$ is determined by the dimensionless ratio
$\ell/G_{\textrm{\tiny N}}$, as shown in eq. (6), while the boundary entropy
$\log g$ is determined by the dimensionless bulk parameter $T_{o}\ell$. In the
following, we will delve into the explicit relations between bulk information
and the CFT data ${T_{\textrm{\tiny DCFT}}(L_{1}+L_{2}),L_{1}/L_{2}}$ for each
bulk phase. The bulk phase diagram will be determined by evaluating the
gravitational path integral using the total action (8) through the saddle
point approximation in the semi-classical limit as $G_{\textrm{\tiny
N}}\rightarrow 0$, with the three bulk phases represented by competing saddle
points.

####  2.1.1 Cold phase ($m_{i}<0$)

Figure 3: Bulk perspective in the cold phase: the bulk dual spacetime for both
the left and right sides is created by joining two portions of AdS3 vacuum
geometry with $m_{i}<0$, along the two-dimensional brane. The codimension-one
branes, labeled as $\mathcal{B}_{i}$, possess a constant tension $T_{o}$ and
are represented by the purple curve. The shaded regions, colored orange and
red, correspond to the bulk regions $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$,
respectively. This configuration characterizes the bulk spacetime in the cold
phase. The middle gray dot (not a conical defect) indicates the center of the
geometry at $r_{i}=0$.

We begin with the cold phase by taking $m_{1}<0,m_{2}<0$. The construction of
the bulk geometry then involves gluing two AdS3 vacua, described by the metric
(12) with $m_{i}<0$, along the constant-tension branes, as illustrated in
figure 3. In accordance with the analysis in eq. (16), the periodicity of the
angular coordinate is set as

$\phi_{i}\sim\phi_{i}+\frac{2\pi}{\sqrt{-m_{i}}}\,,$ (30)

to avoid a conical singularity in the bulk $\mathcal{S}_{i}$. The profile of
the brane, determining the interior boundary of $\mathcal{S}_{i}$, is derived
from eq. (28). The crucial feature among various bulk geometries for
$\mathcal{S}_{i}$ is whether the center of the original AdS3, situated at
$r=0$, is encompassed by the brane $\mathcal{B}_{i}$. If the center of the
bulk geometry $\mathcal{S}_{i}$ is included by the brane $\mathcal{B}_{i}$,
the spatial cycle contracts, leading to $\phi^{\prime}_{i}(\sigma_{+})<0$.
Utilizing eq. (28), it is straightforward to show that the sign of $\phi_{1}$
is proportional to

$\phi^{\prime}_{1}(\sigma_{+})\propto
m_{2}+m_{1}\left(2T_{o}^{2}\ell^{2}-1\right)+T_{o}\ell\sqrt{(m_{1}-m_{2})^{2}+4m_{1}m_{2}T_{o}^{2}\ell^{2}}\leq
m_{1}\left(2T_{o}-1\right)\left(T_{o}\ell+1\right)\,,$

where the middle expression is a monotonically increasing function of $m_{2}$.
Thus, it can be concluded that777For the regime with $T_{o}\ell<\frac{1}{2}$,
we have $\phi^{\prime}_{1}(\sigma_{+})<0$ for
$m_{2}<m_{1}\left(1-4T_{o}^{2}\ell^{2}\right)$ and
$\phi^{\prime}_{2}(\sigma_{+})<0$ for
$m_{1}<m_{2}\left(1-4T_{o}^{2}\ell^{2}\right)$

$\phi^{\prime}_{1}(\sigma_{+})\leq 0\,,\qquad\text{when}\qquad
T_{o}\ell\geq\frac{1}{2}\,.$ (31)

In a similar fashion, for the $\mathcal{S}_{2}$ region to comply, it must also
include the center $r_{2}=0$ if $T_{o}\ell\geq\frac{1}{2}$. This type of
configuration, illustrated in figure 3, is denoted as $[E_{1},E_{1}]$ in
Bachas:2021fqo .

Given that both bulk regions $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ encompass
the center, the dictionary between CFT data $L_{1},L_{2}$ and the bulk
parameters $m_{1},m_{2}$ is given by

$\begin{split}\frac{2\pi}{\sqrt{-m_{1}}}&=\frac{L_{1}}{\ell}+2\int_{\infty}^{\sigma_{+}}\phi_{1}^{\prime}(\sigma)d\sigma\,,\\\
\frac{2\pi}{\sqrt{-m_{2}}}&=\frac{L_{2}}{\ell}+2\int_{\infty}^{\sigma_{+}}\phi_{2}^{\prime}(\sigma)d\sigma\,,\\\
\end{split}$ (32)

where both $\sigma_{+}$ and $\phi^{\prime}_{i}(\sigma)$ depend on mass
parameters $m_{i}$, as defined in eqs. (26) and (28), respectively. For
example, the explicit expression of the boundary size $L_{i}$ in terms of bulk
information $m_{i}$ and $m_{j}$ (with $i=1$, $j=2$ or $i=2$, $j=1$) can be
written as follows:

$\begin{split}L_{i}&=\frac{2\pi\ell}{\sqrt{-m_{i}}}-2\Delta\phi_{i}\ell\,,\end{split}$
(33)

with

$\Delta\phi_{i}\equiv\int_{\infty}^{\sigma_{+}}\phi_{i}^{\prime}(\sigma)d\sigma=\frac{\left[4T_{o}^{2}\ell^{2}\,\mathbf{\Pi}\left(-\frac{m_{i}}{\sigma_{+}},\frac{\sigma_{-}}{\sigma_{+}}\right)+\frac{m_{i}-m_{j}}{m_{i}}\left(\mathbf{K}\left(\frac{\sigma_{-}}{\sigma_{+}}\right)-\mathbf{\Pi}\left(-\frac{m_{i}}{\sigma_{+}},\frac{\sigma_{-}}{\sigma_{+}}\right)\right)\right]}{4T_{o}\ell\sqrt{(1-T_{o}^{2}\ell^{2})\
\sigma_{+}}}\,,\\\ $ (34)

in which $\Delta\phi_{i}$ represents the angular integral from the conformal
boundary to the turning point on the brane, $\mathbf{\Pi}(z|m)$ refers to the
incomplete elliptic integral of the third kind, and $\mathbf{K}(m)$ represents
the complete elliptic integral of the first kind.

As suggested by the term “cold phase,” this configuration prevails at low
temperatures with respect to fixed boundaries. To demonstrate this, a
comparison of the Euclidean action $I_{\textrm{\tiny E}}$ for different phases
is required. Notably, the periodicity of the Euclidean time $t_{\textrm{\tiny
E}}$ is given by the inverse temperature $\frac{1}{T_{\textrm{\tiny DCFT}}}$,
just as in the boundary theory. Upon performing the Euclidean integral in eq.
(8), which includes the appropriate boundary counterterms, it is
straightforward to show that the renormalized Euclidean action in the cold
phase is given by

$\begin{split}I^{\rm cold}_{\textrm{\tiny E}}&=\frac{1}{16\pi G_{\textrm{\tiny
N}}T_{\textrm{\tiny DCFT}}\ell}\left(L_{1}m_{1}+L_{2}m_{2}\right)\\\
&=-\frac{c}{6\pi}\left(\frac{(\pi-\sqrt{-m_{1}}\,\Delta\phi_{1})^{2}}{T_{\textrm{\tiny
DCFT}}L_{1}}+\frac{(\pi-\sqrt{-m_{2}}\,\Delta\phi_{2})^{2}}{T_{\textrm{\tiny
DCFT}}L_{2}}\right)\,.\end{split}$ (35)

where the mass parameters $m_{i}$, as well as $\Delta\phi_{i}$, are understood
to be functions of boundary sizes $L_{i}$ by solving eq. (33). The expression
is then recast in terms of the boundary data in the second line. It should be
noted that obtaining the $m_{i}$ in terms of the boundary data is generally
challenging. Nevertheless, we will delve into two analytical limits in the
later subsections – see also appendix C.

####  2.1.2 Warm phase ($m_{1}>0,m_{2}<0$)

Figure 4: Bulk perspective within the warm phase. The bulk dual spacetime for
left/right side is constructed by gluing a portions of BTZ black hole with
$m_{1}>0$ and AdS3 vacuum with $m_{2}<0$ along a two dimensional branes.

As described by the standard Hawking-Page transition, the AdS3 bulk geometry
is dominated by a BTZ black hole solution rather than a vacuum AdS3 when the
temperature surpasses a critical value $T_{\textrm{\tiny HP}}$. Applying this
concept to the construction of bulk spacetime corresponding to the TFD state
of DCFT at temperature $T_{\textrm{\tiny DCFT}}$, it is reasonable to
anticipate the potential inclusion of a black hole geometry when the thermal
temperature is sufficiently high. Consider the scenario where only one of the
bulk regions, say $\mathcal{S}_{1}$, contains a black hole. This configuration
corresponds to selecting $m_{1}>0$ and $m_{2}<0$ and is termed the warm phase.
The dual bulk spacetime is thus given by gluing part of the BTZ spacetime
$\mathcal{S}_{1}$ containing a black hole and a portion of AdS3 vacuum
$\mathcal{S}_{2}$, as sketched in figure 4.

Given the configuration $m_{1}>0$ and $m_{2}<0$, we need to establish a
relationship between these bulk parameters and the corresponding boundary
data, similar to what was done for the cold phase. In the context of the BTZ
black hole as defined in (17), it is apparent that the temperature of the
black hole $T_{\textrm{\tiny BH}}$ matches that of the boundary DCFT, as
indicated by eq. (18). This implies that we can set the first mass parameter
$m_{1}$ as:

$m_{1}=\left(2\pi\ell\,T_{\textrm{\tiny
BH}}\right)^{2}=\left(2\pi\ell\,T_{\textrm{\tiny DCFT}}\right)^{2}\,.$ (36)

Furthermore, it is also worth noting that, in the warm phase, the turning
point $\sigma=\sigma_{+}$ satisfies the condition:

$\sigma_{+}>\frac{|m_{1}+m_{2}|-(m_{1}+m_{2})}{4\left(1-T_{o}^{2}\ell^{2}\right)}\geq
0\,.$ (37)

This indicates that the brane $\mathcal{B}_{1}$ remains a certain distance
away from the horizon, as illustrated in figure 4.

On the other hand, the size of the initial planar BTZ black hole, as defined
by the metric (17), is arbitrary. We can specify the area of the black hole
horizon included in the bulk region $\mathcal{S}_{1}$ as $\Delta\phi_{1}^{\rm
Hor}r_{h}$. Since the topology of the black hole horizon in $\mathcal{S}_{1}$
forms a closed circle, it is possible to obtain a relation between the
boundary size and the horizon area, i.e.,

$\frac{L_{1}}{\ell}=\Delta\phi_{1}^{\rm
Hor}-2\int_{\infty}^{\sigma_{+}}\phi_{1}^{\prime}(\sigma)d\sigma\equiv\Delta\phi_{1}^{\rm
Hor}-2\Delta\phi_{1}\,,\\\ $ (38)

where the explicit form of the integral $\Delta\phi_{1}$ along $\phi_{1}$
direction has been provided in (34). This equation establishes the connection
between the bulk parameter $\Delta\phi_{1}^{\rm Hor}$ and the boundary data
$L_{1}$.

Considering another bulk region $\mathcal{S}_{2}$ with $m_{2}\leq 0$, it can
be verified that the turning point on the brane satisfies:

$\phi^{\prime}_{2}(\sigma_{+})<0\,,\qquad\text{when}\qquad
m_{2}<\frac{m_{1}}{1-4T_{o}^{2}\ell^{2}}<0\,,$ (39)

by employing eq. (28). Similar to the situation in the cold phase, this
indicates that the corresponding bulk region $\mathcal{S}_{2}$ also includes
the center at $r_{2}=0$. As a result, the boundary size $L_{2}$ can be
expressed as

$L_{2}=\frac{2\pi\ell}{\sqrt{-m_{2}}}-2\Delta\phi_{2}\ell\,,$ (40)

where $\Delta\phi_{2}$ is shown explicitly in eq. (33). On the contrary, the
condition $\phi^{\prime}_{2}(\sigma+)>0$ arises for the following parameter
ranges888These conditions are derived for arbitrary configurations of
$m_{1},m_{2}$. And of course the condition $m_{2}<m_{1}$ is trivial in the
warm phase considered here.,

$\begin{split}\frac{m_{1}}{1-4T_{o}^{2}\ell^{2}}<&m_{2}<m_{1}\,,\qquad\text{for}\qquad
T_{o}>\frac{1}{2\ell}\,,\\\ &m_{2}<m_{1}\,,\qquad\text{for}\qquad
T_{o}<\frac{1}{2\ell}\,.\\\ \end{split}$ (41)

It implies that $\mathcal{S}_{2}$ does not contain the center point and the
boundary-bulk dictionary is instead established as

$L_{2}^{(+)}=+2\ell\int^{\infty}_{\sigma_{+}}\phi_{2}^{\prime}(\sigma)d\sigma\equiv-2\Delta\phi_{2}\ell\,.$
(42)

These equations outline how the boundary sizes $L_{2}$ and $L_{2}^{(+)}$ are
connected with the bulk parameter $\Delta\phi_{2}$ for the warm phase.

Since the mass parameter $m_{1}$ in the warm phase is fixed by the
temperature, as in eq. (36) above, the boundary size $L_{2}$ will be
determined by $m_{2}$ according to eq. (40) or (42). It is important to note
that this relationship between $L_{2}$ and $m_{2}$ may not be a one-to-one map
in general. Finally, let us note that we can focus on the case where the
center is included in $\mathcal{S}_{2}$ in the large tension limit
$T_{o}\sim\frac{1}{\ell}$. This is because the boundary-bulk map (42) is ruled
out by the constraint that $L_{2}^{(+)}>0$ in the large tension limit. The
critical tension could be derived by solving

$0=\lim\limits_{m_{2}\to\frac{m_{1}}{1-4T_{o}^{2}\ell^{2}}}L_{2}^{(+)}=\frac{\ell}{\sqrt{-m_{2}}}\left(\pi-\frac{2T_{o}\ell\,\mathbf{K}\left(\frac{T_{o}\ell}{T_{o}\ell-1}\right)}{\sqrt{1-T_{o}^{2}\ell^{2}}}\right)\,,$
(43)

which gives an approximate solution of $T_{o}\ell\approx 0.7927$. For brane
tensions larger than this critical value, the bulk integral in eq. (42) would
always lead to $L_{2}^{(+)}<0$, which is unphysical. Because we are interested
in the large tension limit for doubly holographic construction, this paper
will focus on the scenario where $\mathcal{S}_{2}$ in warm phase also contains
the center and uses the relation $L_{2}(m_{2})$ defined in eq. (40) or its
inverse function $m_{2}(L_{2})$.

To determine the saddle point among the competing phases, the gravitational
action of the warm phase needs to be evaluated. The final result can be
expressed as follows Bachas:2021fqo :

$\begin{split}I^{\rm warm}_{\textrm{\tiny E}}&=\frac{1}{16\pi G_{\textrm{\tiny
N}}T_{\textrm{\tiny BH}}\ell}\left((L_{1}-2\ell\Delta\phi_{1}^{\rm
Hor})m_{1}+L_{2}m_{2}\right)\\\ &=-\frac{c}{6}\left(\pi T_{\textrm{\tiny
DCFT}}\left(L_{1}+4\ell\Delta\phi_{1}\right)+\frac{(\pi-\sqrt{-m_{2}}\,\Delta\phi_{2})^{2}}{\pi
T_{\textrm{\tiny DCFT}}L_{2}}\right)\,.\end{split}$ (44)

In this equation, $\Delta\phi_{1}$ and $\Delta\phi_{2}$ can be interpreted as
functions of $T_{\textrm{\tiny DCFT}}L_{2}$ by solving the inverse function
$m_{2}(L_{2})$ from equation (40). This expression provides a way to compare
the warm phase with other phases, which is essential for determining the
dominant saddle point in the semi-classical limit.

####  2.1.3 Hot phase ($m_{i}>0$)

Figure 5: Bulk perspective within hot phase. The bulk dual spacetime for
left/right side is constructed by gluing a portion of BTZ black hole and a
portion of another BTZ black hole with the same mass. The horizon area
encompassed within each part is denoted as $\Delta\phi_{i}^{\rm Hor}$,
respectively.

We proceed to the third phase, referred to as the hot phase, characterized by
$m_{1}>0$ and $m_{2}>0$. This phase emerges as the dominant classical geometry
in the regime of elevated temperatures. Within this phase, our construction
involves the initiation of two BTZ black hole geometries. By fusing the two
designated portions, denoted as $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$, along
the brane, we establish the corresponding holographic dual spacetime. This
configuration is visually represented in figure 5.

Recalling our earlier analyses for the $\mathcal{S}_{1}$ region for the warm
phase, it becomes evident that the temperature of the black hole should match
that of the thermal DCFT on the boundary due to the identical Euclidean time
periodicity. Consequently, the two mass parameters are constrained to be
equal, viz,

$m_{1}=\left(2\pi\ell T_{\textrm{\tiny BH}}\right)^{2}=\left(2\pi\ell
T_{\textrm{\tiny DCFT}}\right)^{2}=m_{2}\,.$ (45)

This matching is inherently expected, given our consideration of static
profiles that correspond to a state of thermal equilibrium. Importantly, this
equivalence leads to a straightforward outcome: the turning point on the brane
is positioned at

$\sigma_{+}\big{|}_{m_{1}=m_{2}}=0\,,$ (46)

which coincides with the location of the black hole horizon at $r=r_{h}$ – see
comments below eq. (24). This implies that, at higher temperatures, the black
hole horizon expands, exerting a stronger gravitational pull that attracts the
branes towards this enlarged horizon. Drawing a parallel with the warm phase
scenario, we label the area of the black hole horizon encompassed within the
$\mathcal{S}_{i}$ region as $\Delta\phi_{i}^{\rm Hor}r_{h}$. These two
parameters capture the crucial bulk information that characterizes the
holographic spacetime.

Furthermore, upon closer examination, it becomes evident that the angular
integral in the hot phase (with $m_{i}>0$) can be simplified into a more
explicit expression. Specifically, we have

$\Delta\phi_{i}\equiv\int_{\infty}^{0}\phi_{i}^{\prime}(\sigma)d\sigma=\int^{\infty}_{0}\frac{T_{o}\ell}{2(m_{i}+\sigma)\sqrt{m_{i}+(1-T_{o}^{2}\ell^{2})\sigma}}d\sigma=\frac{\text{arctanh}\left(T_{o}\ell\right)}{\sqrt{m_{i}}}\,.$
(47)

Analogous to the relation in eq. (38), the connection between the bulk
parameters $\Delta\phi_{i}^{\rm Hor}$ and the boundary data can be rendered in
a simpler form:

$L_{i}=\Delta\phi_{i}^{\rm Hor}\ell-\frac{1}{\pi T_{\textrm{\tiny
DCFT}}}\text{arctanh}\left(T_{o}\ell\right)\,.$ (48)

where $\Delta\phi_{i}^{\rm Hor}$ represents the size of the horizon within the
bulk region $\mathcal{S}_{i}$, as illustrated in figure 5.

Evaluating the renormalized Euclidean action in the hot phase then yields

$\begin{split}I_{\textrm{\tiny E}}^{\rm hot}&=\frac{1}{16\pi G_{\textrm{\tiny
N}}T_{\textrm{\tiny BH}}\ell}\left(\left(L_{1}-2\Delta\phi_{1}^{\rm
Hor}\ell\right)m_{1}+\left(\left(L_{2}-2\Delta\phi_{2}^{\rm
Hor}\ell\right)m_{2}\right)\right)\\\ &=-\frac{c}{6}\pi T_{\textrm{\tiny
BH}}\left(L_{1}+L_{2}\right)-\frac{2c}{3}\log\left(\sqrt{\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)\,,\\\
&=-\frac{c\pi}{6}T_{\textrm{\tiny DCFT}}\left(L_{1}+L_{2}\right)-2\,S_{\rm
def}\,,\\\ \end{split}$ (49)

where we have identified the boundary entropy associated with one defect as

$S_{\rm def}\equiv 2\log
g=\frac{c}{6}\log\left({\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)\,.\\\ $ (50)

as given in eq. (7). This relationship reproduces the same result in the
AdS/BCFT correspondence Takayanagi:2011zk ; Fujita:2011fp . The factor of $4$
that precedes the boundary entropy in eq. (49) accounts for the total number
of defects existing on both the left and right boundaries. In subsequent
subsections, we will further confirm that the aforementioned boundary entropy
(50) corresponds to the holographic entanglement entropy of a single defect.

###  2.2 Phase diagram

Figure 6: Phase diagram with respect to the dimensionless boundary data
$\\{T_{\textrm{\tiny DCFT}}\left(L_{1}+L_{2}\right),\frac{L_{1}}{L_{2}}\\}$
with choosing $T_{o}\ell=0.9$.

In the previous subsection, we reviewed the possible solutions for the bulk
spacetime and evaluated the gravitational action (8) for the corresponding
phases. These solutions encompass three distinct phases, namely the cold, warm
and hot phases. Upon specifying the boundary data, we now determine the
dominant saddle point by comparing the Euclidean actions of these different
phases. As discussed at eq. (5), the boundary configurations are specified by
four dimensionless parameters. The first two $c$ and $\log g$ characterize the
boundary theory and are fixed here. The last two $T_{\textrm{\tiny
DCFT}}\left(L_{1}+L_{2}\right)$ and ${L_{1}}/{L_{2}}$ describe the boundary
geometry and the temperature and will be varied to determine the phase
diagram. By using numerical solutions of the inverse functions inherent in the
boundary-bulk dictionary, it is straightforward to evaluate the gravitational
action for a particular choice of these parameters. Subsequently, we can draw
the phase diagram as shown in figure 6 with respect to these two dimensionless
parameters.

To provide insight into the shape of the phase diagram depicted in figure 6,
we will derive the approximate expressions for the boundaries between the
various phases. Beginning with the transition between the warm and hot phases,
we compare the renormalized actions in eqs. (44) and (49) and find that the
corresponding boundary in the phase diagram is determined by solving

$4\pi T_{\textrm{\tiny
DCFT}}\ell\Delta\phi_{1}+\frac{(\pi-\sqrt{-m_{2}}\,\Delta\phi_{2})^{2}}{\pi
T_{\textrm{\tiny DCFT}}L_{2}}=\pi T_{\textrm{\tiny
DCFT}}L_{2}+4\text{arctanh}\left(T_{o}\ell\right)\,,$ (51)

where $\Delta\phi_{i}$ are defined with taking $m_{1}=(2\pi T_{\textrm{\tiny
DCFT}}\ell)^{2}$, and $m_{2}$ is obtained by the inverse function presented in
eq. (40). Since $L_{1}$ has dropped out of this equation, the corresponding
phase boundary is precisely determined by

$T_{\textrm{\tiny DCFT}}L_{2}=C_{\textrm{\tiny WH}}(T_{o}\ell)\,,$ (52)

which represents a constant governed by the boundary entropy or the
combination $T_{o}\ell$. Consequently, the boundary between the warm and hot
phases becomes a straight line in figure 6, i.e., $T_{\textrm{\tiny
DCFT}}(L_{1}+L_{2})=C_{\textrm{\tiny WH}}\left(1+{L_{1}}/{L_{2}}\right)$.

Figure 7: The coefficients $k_{1},k_{2}$ as a function of the dimensionless
tension $T_{o}\ell$.

Although obtaining the analytical solution for the phase transitions is
challenging, a notable observation is that the phase transitions associated
with the warm phase simplify when $L_{1}\gg L_{2}$. This regime can be
approached by considering the limit $m_{2}\to-\infty$, corresponding to the
condition $T_{\text{DCFT}}L_{2}\ll 1$. In this context, the angular integrals
defined in eq. (34) can be approximated by

$\lim\limits_{m_{2}\to-\infty}\Delta\phi_{1}\approx\frac{k_{1}}{\sqrt{-m_{2}}}+\cdots\,,\qquad\lim\limits_{m_{2}\to-\infty}\Delta\phi_{2}\approx\frac{k_{2}}{\sqrt{-m_{2}}}+\frac{k_{1}m_{1}}{2(-m_{2})^{3/2}}+\cdots\,,$
(53)

with the constants defined by

$\begin{split}k_{1}&=2\sqrt{\frac{T_{o}\ell}{1+T_{o}\ell}}\left((1+T_{o}\ell)\mathbf{E}\left(\frac{T_{o}\ell-1}{T_{o}\ell+1}\right)-\mathbf{K}\left(\frac{T_{o}\ell-1}{T_{o}\ell+1}\right)\right)\,,\\\
k_{2}&=\frac{1}{2\sqrt{T_{o}\ell(1+T_{o}\ell)}}\left(\mathbf{K}\left(\frac{T_{o}\ell-1}{T_{o}\ell+1}\right)+(4T_{o}^{2}\ell^{2}-1)\mathbf{\Pi}\left(4T_{o}\ell(1-T_{o}\ell);\frac{T_{o}\ell-1}{T_{o}\ell+1}\right)\right)\,.\\\
\end{split}$ (54)

We show the numerical plot for $k_{1},k_{2}$ in figure 7. Further, we note
that in the large tension limit, i.e., $T_{o}\ell\to 1$ limit, one finds

$k_{1}\approx\frac{\pi}{\sqrt{2}}\left(1-7\,\frac{1-T_{o}\ell}{8}\right)+\mathcal{O}((1-T_{o}\ell)^{2})\,,\quad
k_{2}\approx\frac{\pi}{\sqrt{2}}\left(1+\frac{1-T_{o}\ell}{8}\right)+\mathcal{O}((1-T_{o}\ell)^{2})\,.$
(55)

Both of these approximations help to simplify the bulk-boundary dictionary. As
an example, let us focus on $m_{2}\to-\infty$ for which the bulk-boundary
dictionary in the warm phase simplifies

$L_{1}\simeq\left(\Delta\phi^{\rm
Hor}_{1}-\frac{2k_{1}}{\sqrt{-m_{2}}}\right)\ell\,,\quad
L_{2}\simeq\frac{(2\pi-2k_{2})\ell}{\sqrt{-m_{2}}}-\frac{(2\pi
T_{\textrm{\tiny DCFT}}\ell)^{2}k_{1}\ell}{(-m_{2})^{3/2}}\,.$ (56)

Hence disregarding higher-order terms in powers of $T_{\text{DCFT}}L_{2}$, we
can approximate the phase transition between the hot and warm phases as:

$\displaystyle T_{\textrm{\tiny DCFT}}L_{2}\big{|}_{\textrm{\tiny WH}}$
$\displaystyle=$ $\displaystyle C_{\textrm{\tiny WH}}\simeq\frac{(\pi-
k_{2})}{(\pi-k_{1}-k_{2})\pi}$ $\displaystyle\times\ \left(\sqrt{(\pi-
k_{2})(\pi-
k_{1}-k_{2})+4\text{arctanh}^{2}(T_{o}\ell)}-2\text{arctanh}(T_{o}\ell)\right)\,.$

It is useful for later to noting that $C_{\textrm{\tiny WH}}$ is a small
parameter in the large tension limit $T_{o}\ell\to 1$ because of

$C_{\textrm{\tiny WH}}\simeq\frac{(\pi-k_{2})^{2}}{2\pi}\
{\log^{-1}\\!\left({\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)}\,.$ (58)

The approximations with $|m_{2}|\gg 1$ can also be applied to explore the
phase transition between the cold and warm phases. By comparing the
renormalized actions in the cold and warm phases, as given by eqs. (35) and
(44), one can describe the phase transition as follows

$T_{\textrm{\tiny DCFT}}L_{1}\big{|}_{\textrm{\tiny CW}}=C_{\textrm{\tiny
CW}}\simeq 1-\frac{k_{1}}{\pi-
k_{2}}\frac{L_{2}}{L_{1}}+\left(\frac{k_{1}}{\pi-
k_{2}}\frac{L_{2}}{L_{1}}\right)^{2}+\mathcal{O}\left(\left(\frac{L_{2}}{L_{1}}\right)^{3}\right)\,.$
(59)

With $L_{2}/L_{1}\to 0$, the phase transition is located at $T_{\textrm{\tiny
DCFT}}L_{1}=T_{\textrm{\tiny DCFT}}L_{\rm bdy}=1$.

Combining the above results, we can immediately find that the triple point of
the phase diagram is situated at

$\text{triple point:}\qquad T_{\textrm{\tiny DCFT}}L_{1}\simeq
1-\frac{k_{1}}{\pi-k_{2}}C_{\textrm{\tiny WH}}\,,\quad T_{\textrm{\tiny
DCFT}}L_{2}=C_{\textrm{\tiny WH}}\,.$ (60)

Then the ratio of the boundary lengths at the triple point is approximated by
$\frac{L_{1}}{L_{2}}\big{|}_{\rm triple}\simeq\frac{C_{\textrm{\tiny
CW}}}{C_{\textrm{\tiny WH}}}$ with

$\frac{C_{\textrm{\tiny CW}}}{C_{\textrm{\tiny
WH}}}\simeq\frac{k_{1}(k_{2}-\pi)+2\pi\text{arctanh}(T_{o}\ell)+\pi\sqrt{(\pi-
k_{2})(\pi-k_{1}-k_{2})+4\text{arctanh}^{2}(T_{o}\ell)}}{(\pi-k_{2})^{2}}\,.$
(61)

Near the triple point, the phase transition between the cold and hot phases
can be approximated by $T_{\text{DCFT}}(L_{1}+L_{2})=C_{\textrm{\tiny CH}}$
where the latter constant is given by

$\sqrt{\left(\frac{L_{2}}{L_{1}}\frac{k_{1}}{(k_{2}-\pi)}+\left(\frac{L_{2}}{L_{1}}+1\right)\left(1+\frac{L_{1}}{L_{2}}\left(1-\frac{k_{2}}{\pi}\right)^{2}\right)\right)+\frac{4}{\pi^{2}}\text{arctanh}^{2}(T_{o}\ell)}-\frac{2}{\pi}\text{arctanh}(T_{o}\ell)\,,$
(62)

in terms of the dimensionless parameter $L_{1}/L_{2}$. However, it is
important to note that these approximations, derived from a series expansion
with respect to $1/\sqrt{-m_{2}}$, are not valid in the vicinity of the regime
$L_{1}\approx L_{2}$. Instead, we will demonstrate the full analytical
understanding of the symmetric setup where $L_{1}=L_{2}$ below in subsection
2.4.

###  2.3 Fusion limit

Figure 8: The fusion limit of boundary defect filed theory is defined by
taking $L_{2}\to 0$.

Despite the challenge of finding analytical results for generic values of the
parameters, there are several interesting limits where we can derive simple
results. Considering the fusion limit Bachas:2007td  with $L_{2}/L_{1}\to 0$,
the boundary system behaviors like a thermal CFT2 on a circle whose
circumference is fixed as $L_{1}$. As shown in the phase diagram 6, the fusion
limit leads us to the right boundary of the phase diagram. Moreover, there are
only two phases in this limit, i.e., the warm and cold phases.999We note that
the three phases could all exist at the limit $L_{1}/L_{2}\to\infty$. Whether
we can probe the hot phase depends on the limit at which we approach
$L_{1}/L_{2}\to\infty$. In the fusion limit with $T_{\textrm{\tiny
DCFT}}L_{2}\to 0$, which is below the black hole threshold given by eq. (2.2),
the hot phase is absent.  The thermal partition function of holographic CFT2
is universal. If the CFT2 resides on a circle of circumference $L_{\rm bdy}$
and the temperature is $T_{\textrm{\tiny CFT}}$, we have, e.g.,
Hartman:2014oaa

$\ln Z_{\rm thermal}^{\textrm{\tiny
CFT}}=\begin{cases}\frac{\pi\,c}{6}\,\frac{1}{T_{\textrm{\tiny CFT}}L_{\rm
bdy}}\,,\qquad T_{\textrm{\tiny CFT}}L_{\rm bdy}<1\,,\\\ \,\\\
\frac{\pi\,c}{6}\,T_{\textrm{\tiny CFT}}L_{\rm bdy}\,,\qquad T_{\textrm{\tiny
CFT}}L_{\rm bdy}>1\,,\\\ \end{cases}$ (63)

From the bulk perspective, the renormalized action at the fusion limit is
given by

$\begin{split}\lim\limits_{L_{2}/L_{1}\to 0}I_{\textrm{\tiny E}}^{\rm
cold}&=-\frac{c}{6}\left(\frac{\pi}{T_{\textrm{\tiny DCFT}}L_{1}}+\frac{(\pi-
k_{2})^{2}}{\pi T_{\textrm{\tiny DCFT}}L_{2}}\right)+\mathcal{O}(L_{2})\,,\\\
\lim\limits_{L_{2}/L_{1}\to 0}I_{\textrm{\tiny E}}^{\rm
warm}&=-\frac{c}{6}\left(\pi T_{\textrm{\tiny DCFT}}L_{1}+\frac{(\pi-
k_{2})^{2}}{\pi T_{\textrm{\tiny DCFT}}L_{2}}\right)+\mathcal{O}(L_{2})\,,\\\
\end{split}$ (64)

where the second term, which is common to both phases, originates from the
Casimir energy of the $\mathcal{S}_{2}$ region and is divergent in this limit
as the defects bounding the region approach one another. The phase transition
definitely happens at

$T_{\textrm{\tiny DCFT}}=\frac{1}{L_{1}}=\lim\limits_{L_{2}\to
0}\frac{1}{L_{\rm bdy}}\,,$ (65)

which agrees with the Hawking-Page temperature of AdS3. The phase transition
is universal in the sense that it is not affected by the properties of the
defects as it obviously does not depend on the boundary entropy $\log g$ or
the brane tension $T_{o}$.

Ignoring the corrections from the higher powers of $L_{2}$, one can also
derive the energy and entropy of the system (see Appendix C for more details)
in the fusion limit as follows:

$\begin{split}E^{\rm{cold}}&=-T_{\textrm{\tiny DCFT}}^{2}\frac{\partial
I^{\rm{cold}}_{\textrm{\tiny E}}}{\partial T_{\textrm{\tiny
DCFT}}}\approx-\frac{\pi\,c}{6\,L_{1}}-\frac{c}{6}\frac{(\pi-k_{2})^{2}}{\pi
L_{2}}+\mathcal{O}(L_{2}/L_{1}^{2})\,,\\\
S^{\rm{cold}}&=-\frac{\partial(T_{\textrm{\tiny
DCFT}}\,I^{\rm{cold}}_{\textrm{\tiny E}})}{\partial T_{\textrm{\tiny
DCFT}}}=0\,,\end{split}$ (66)

and

$\begin{split}E^{\rm{warm}}&=-T_{\textrm{\tiny DCFT}}^{2}\frac{\partial
I^{\rm{warm}}_{\textrm{\tiny E}}}{\partial T_{\textrm{\tiny
DCFT}}}\approx\frac{c}{6}\pi T^{2}_{\textrm{\tiny
DCFT}}L_{1}-\frac{c}{6}\frac{(\pi-k_{2})^{2}}{\pi
L_{2}}+\mathcal{O}(T_{\textrm{\tiny DCFT}}^{2}L_{2})\,,\\\
S^{\rm{warm}}&=-\frac{\partial(T_{\textrm{\tiny
DCFT}}\,I^{\rm{warm}}_{\textrm{\tiny E}})}{\partial T_{\textrm{\tiny
DCFT}}}\approx\frac{\pi c}{3}T_{\textrm{\tiny
DCFT}}L_{1}+\mathcal{O}(T_{\textrm{\tiny DCFT}}L_{2})\,.\end{split}$ (67)

Obviously, apart from the Casimir contributions appearing in both and whose
coefficient is determined by the brane tension, the results are the same as
that of a thermal CFT2 living on a circle with a circumference $L_{1}$.

###  2.4 Symmetric boundary

In the preceding analyses, two distinct parameters, $L_{1}$ and $L_{2}$, were
introduced to define the sizes of the two boundary intervals. However, the
analysis also simplifies for the specific scenario where
$L_{1}=L_{2}$.101010An extensive examination of this configuration can also be
found in appendix C of Grimaldi:2022suv . From the perspective of the boundary
theory, this choice produces a $Z_{2}$ symmetry between the two regions across
the defect. By the symmetry argument, we can expect that the dual bulk
spacetime should also exhibit a $Z_{2}$ symmetry around the branes
$\mathcal{B}_{i}$. From the viewpoint of bulk spacetime, it is then apparent
that only the cold and hot phases are compatible with this symmetry
requirement, while the warm phase is precluded due to the absence of symmetry.
The absence of the warm phase is also evident upon examining the left boundary
of the phase diagram 6. Further, the symmetry demands that the mass parameters
$m_{i}$ for the two bulk regions $\mathcal{S}_{i}$ are identical, i.e.,
$m_{1}=m=m_{2}$.

This configuration not only simplifies various expressions but also enhances
the clarity of the boundary-bulk dictionary. For instance, the angular
integral $\Delta\phi_{i}$, defined in eq. (34), reduces to

$\Delta\phi_{i}\big{|}_{m_{1}=m_{2}}=\begin{cases}\frac{\pi}{2\sqrt{-m}}\,,\qquad
m<0\,,\\\ \frac{\text{arctanh}(T_{o}\ell)}{\sqrt{m}}\,,\qquad m>0\,.\\\
\end{cases}$ (68)

Furthermore, the symmetry requires the brane profiles in $\mathcal{S}_{1,2}$
to be the same and so the corresponding extrinsic curvatures are equal, i.e.,
$K^{(1)}_{ab}=K^{(2)}_{ab}$. Hence, eq. (19) reduces to

$K^{(1)}_{ab}-K^{(1)}\,h_{ab}-T_{o}h_{ab}=0\,.$ (69)

The most general solutions for the brane profile satisfying eq. (69) can be
found in, e.g., ref. Kawamoto:2023wzj . In the subsequent discussion, we will
concentrate on the corresponding brane solutions within the cold and hot
phases, respectively.

In the cold phase, the bulk geometry is described by the AdS3 vacuum geometry,
parametrized as follows

$\begin{split}ds^{2}&=-\left(\frac{r^{2}}{\ell^{2}}-m\right)dt^{2}+\left(\frac{r^{2}}{\ell^{2}}-m\right)^{-1}dr^{2}+r^{2}d\phi^{2}\,.\\\
\end{split}$ (70)

The static brane profile in this geometry is derived as

$\frac{r}{\ell\sqrt{-m}}\sin\left(\sqrt{-m}|\phi\pm\phi_{\rm
bdy}|\right)=\frac{T_{o}\ell}{\sqrt{1-(T_{o}\ell)^{2}}}\,,$ (71)

where the brane intersects the conformal boundary at
$\phi=\pm\phi_{\text{bdy}}$. In our conventions, the boundary region with a
fixed length of $L_{i}$ is given by $\phi\in\left[-\phi_{\rm bdy},\phi_{\rm
bdy}\right]$ with the assumption that the midpoint of the boundary lies at
$\phi=0$. It is evident that the boundary size is
$L_{1}=L_{2}=2\phi_{\text{bdy}}\ell$. For a regular bulk spacetime without
defects, the periodicity condition, $\phi\sim\phi+\frac{2\pi}{\sqrt{-m}}$,
fixes the free parameter $\phi_{\text{bdy}}$ as
$\phi_{\text{bdy}}=\frac{\pi}{2\sqrt{-m}}$. Consequently, the boundary-bulk
map in the cold phase can be summarized as:

$L_{1}=L_{2}=\frac{\pi\ell}{\sqrt{-m}}\,.$ (72)

This result can also be derived from eq. (33) by employing $\Delta\phi_{i}$
from eq. (34). It can be understood as a consequence of the fact that the
brane profile in the global AdS3 configuration always connects two antipodal
points on the boundary circle. Finally, the total Euclidean action for the
cold phase reduces to

$\begin{split}I_{\textrm{\tiny E}}^{\rm
cold}&=\frac{c}{24\pi\ell^{2}T_{\textrm{\tiny
DCFT}}}\left(L_{1}m_{1}+L_{2}m_{2}\right)=-\frac{\pi\,c}{6}\frac{1}{T_{\textrm{\tiny
DCFT}}(L_{1}+L_{2})}\,.\end{split}$ (73)

In the hot phase, characterized by $m_{1}=m_{2}=m>0$, the bulk spacetime is
represented by a (static) BTZ black hole geometry whose line element is given
by

$ds^{2}=-\frac{r^{2}-r_{h}^{2}}{\ell^{2}}dt^{2}+\frac{\ell^{2}}{r^{2}-r_{h}^{2}}dr^{2}+r^{2}d\phi^{2}\,,$
(74)

where $r_{h}$ denotes the radius of the horizon and is related to the positive
mass parameter by $m=(r_{h}/\ell)^{2}$. Similar to eq. (71), the static brane
profile living in BTZ black hole spacetime (74) is given by

$\frac{r}{r_{h}}\sinh\left(\frac{r_{h}}{\ell}|\phi\pm\phi_{\rm
bdy}|\right)=\frac{T_{o}\ell}{\sqrt{1-(T_{o}\ell)^{2}}}\,.$ (75)

It is obvious that the brane always intersects the BTZ horizon at:

$\phi=\phi_{\rm hor}=\pm\left(\phi_{\rm
bdy}+\frac{\ell}{r_{h}}\text{arctanh}\left(T_{o}\ell\right)\right)\,,$ (76)

which results in the boundary-bulk map shown in (48) by utilizing
$L_{i}=2\phi_{\rm bdy}\ell$ $\Delta\phi_{i}^{\rm Hor}=2\phi_{\rm hor}$, and
$T_{\textrm{\tiny DCFT}}=T_{\textrm{\tiny BH}}=\frac{r_{h}}{2\pi\ell^{2}}$. By
explicitly employing the BTZ metric and the brane profiles from eq. (75), it
is straightforward to derive the renormalized action for the two-sided system:

$\begin{split}I_{\textrm{\tiny E}}^{\rm
hot}&=-\frac{\pi\,c}{6}\,T_{\textrm{\tiny
DCFT}}\left(L_{1}+L_{2}\right)-\frac{c}{3}\log\left({\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)\,.\\\
\end{split}$ (77)

The transition from the cold phase to the hot phase bears similarities to the
Hawking-Page transition. Upon comparing the effective actions for this
symmetric configuration (i.e., $L_{1}=L_{2}$), as given in eqs. (73) and (77),
one finds that the transition occurs at

$C_{\textrm{\tiny
CH}}\equiv-\frac{2\,\text{arctanh}\left(T_{o}\ell\right)}{\pi}+\sqrt{1+\frac{4\,\text{arctanh}^{2}\left(T_{o}\ell\right)}{\pi^{2}}}\in[0,1]\,.$
(78)

The same result has been derived before in e.g., Fujita:2011fp ;
Cooper:2018cmb ; Grimaldi:2022suv . In the limit of a large tension with
$T_{o}\ell\to 1$, the phase transition point is shifted to a very low-
temperature regime with $C_{\textrm{\tiny
CH}}\simeq\frac{\pi}{4\text{arctanh}(T_{o}\ell)}\ll 1$. Following
Grimaldi:2022suv , we can also express this transition point in terms of the
boundary data

$C_{\textrm{\tiny CH}}=\sqrt{1+\left(\frac{6}{\pi}\,\frac{S_{\textrm{\tiny
def}}}{c}\right)^{2}}-\frac{6}{\pi}\,\frac{S_{\textrm{\tiny def}}}{c}\,.$ (79)

using the defect entropy in eq. (7). As discussed above, the limit
$T_{o}\ell\to 1$ corresponds to the regime where $S_{\textrm{\tiny def}}/c\gg
1$ and we find $C_{\textrm{\tiny CH}}\simeq\frac{\pi\,c}{12\,S_{\textrm{\tiny
def}}}$.

##  3 Holographic Entanglement of the Defects

In this section, we holographically compute entanglement entropies of various
subsystems in our setup. From the CFT perspective, we are particularly
interested in the mutual information $I_{\textrm{\tiny L:R}}$ between the
defects $D^{(\textrm{\tiny L})}_{a,b}$ on the left boundary and
$D^{(\textrm{\tiny R})}_{a,b}$ on the right, i.e.,

$I_{\textrm{\tiny L:R}}:=S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny
R})}_{a,b})+S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny
L})}_{a,b})-S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny R,L})}_{a,b})\,.$ (80)

This mutual information is interesting because, in the brane perspective, this
corresponds to measuring the correlation between two branes representing “two
gravitating universes”. The entanglement entropy between two gravitating
disjoint spacetimes in two dimensions was considered in
Balasubramanian:2021wgd . It was found that even if these two are originally
disjoint, as we increase the entanglement of the matter degrees of freedom on
them, the two geometries become connected by an Einstein-Rosen bridge, which
is reminiscent of the ER=EPR conjecture Maldacena:2013xja . One of the
purposes of this paper is to see the dynamics of this spacetime connection
from the bulk perspective, and to provide independent support for the
arguments given in Balasubramanian:2021wgd .

To begin with, it is useful to recall that the entanglement entropy between
the gravitating region and the non-gravitating region from the brane
perspective which is computed with the conventional island formula and is
equal to the entanglement entropy between the degrees of freedom of the
boundary CFTs and those on the defects, from the boundary perspective.

In the current case, we are interested in the entanglement between the two
gravitating universes L and R communicating through exchanges of QFT degrees
of freedom. From the boundary CFT perspective, this corresponds to the
entanglement between two groups of defects $D^{(\textrm{\tiny
L})}_{a,b}=D^{(\textrm{\tiny L})}_{a}\cup D^{(\textrm{\tiny L})}_{b}$ and
$D^{(\textrm{\tiny R})}_{a,b}=D^{(\textrm{\tiny R})}_{a}\cup D^{(\textrm{\tiny
R})}_{b}$. More importantly, we need to regulate our holographic entanglement
entropy calculations for the defects by considering a small interval around
each defect, whose size is fixed as $\Delta L_{\textrm{\tiny D}}$, as was done
in Grimaldi:2022suv . This setup is depicted in figure 9. From the boundary
perspective, these small defect intervals are considered as the boundary dual
of the two-dimensional braneworld. We would like to further note that the
regulator length $\Delta L_{\textrm{\tiny D}}$ should be considered as much
smaller than the AdS radius or the boundary size, but it is still distinct
from (and larger than) the UV cut-off scale. As a summary, the defect length
satisfies

$\epsilon\ll\Delta L_{\textrm{\tiny D}}\ll\ell\,.$ (81)

Figure 9: The defect denoted as $\\{D^{(\textrm{\tiny
R})}_{a},D^{(\textrm{\tiny R})}_{b},D^{(\textrm{\tiny
L})}_{a},D^{(\textrm{\tiny L})}_{b}\\}$ is regulated by including a boundary
interval with a length $\Delta L_{\textrm{\tiny D}}\ll\ell$.

Furthermore, these defects are coupled through the CFT degrees of freedom
which correspond to degrees of freedom of Hawking quanta from the brane
perspective – see figure 11. Therefore, one way to quantify the entanglement
between them is to compute the mutual information $I_{\textrm{\tiny L:R}}$.
This holographic mutual information obviously depends on the bulk phases which
are determined by the dimensionless CFT parameters as shown in the phase
diagram 6. In order to allow the existence of all three phases, we will focus
on the regime with $L_{1}/L_{2}\gg 1$, i.e., the fusion limit discussed in
subsection 2.3, which also simplifies the analysis. But first, we will
consider the symmetric case with $L_{1}=L_{2}$ in the next subsection, where
we can analytically derive all expressions. This probes the regime where
$L_{1}\sim L_{2}$ and there are only cold and hot phases.

In the following, to compute the holographic entanglement entropy in different
phases, we need to evaluate the length of various geodesics or extremal
curves. This simplifies because the bulk spacetime is always locally an AdS3
geometry. It is useful to regard AdS3 as a hyperbola,

$X_{0}^{2}+X_{1}^{2}-X_{2}^{2}-X_{3}^{2}=\ell^{2},$ (82)

living in the embedding space with the metric

$ds^{2}=-(dX_{0})^{2}-(dX_{1})^{2}+(dX_{2})^{2}+(dX_{3})^{2}\,.$ (83)

It is then easy to find that the geodesic length $d_{\rm geodesic}$ between
two points in AdS3 is given by

$d_{\rm geodesic}\equiv\ell\,\text{arccosh}\left(D_{\rm
geodesic}\right)=\ell\,\text{arccosh}\left(\frac{X_{0}X_{0}^{\prime}+X_{1}X_{1}^{\prime}-X_{2}X_{2}^{\prime}-X_{3}X_{3}^{\prime}}{\ell^{2}}\right)\,.$
(84)

For example, in global coordinates (15), this geodesic distance becomes

$D_{\rm global}=\sqrt{1+\frac{R^{2}}{\ell^{2}}}\sqrt{1+\frac{R^{\prime
2}}{\ell^{2}}}\cos\left(\frac{T-T^{\prime}}{\ell}\right)-\frac{RR^{\prime}}{\ell^{2}}\cos\left(\Phi-\Phi^{\prime}\right)\,.$
(85)

Considering the BTZ black hole metric defined in (17), the geodesic distance
becomes

$D_{\mathrm{BTZ}}=\frac{rr^{\prime}\cdot\cosh\left(\frac{r_{h}\Delta\phi}{\ell}\right)-\sqrt{\left(r^{2}-r_{h}^{2}\right)\left(r^{\prime
2}-r_{h}^{2}\right)}\cosh\left(\frac{r_{h}\Delta
t}{\ell^{2}}\right)}{r_{h}^{2}}\,,$ (86)

with $\Delta\phi\equiv\phi-\phi^{\prime}$ and $\Delta t=t-t^{\prime}$. One can
also define the two-sided BTZ geometry by using the Kruskal coordinates.
Correspondingly, we can obtain the length of a geodesic connecting the left
and right regions as

$D_{\mathrm{BTZ}}=\frac{r_{\textrm{\tiny R}}r_{\textrm{\tiny
L}}\cdot\cosh\left(\frac{r_{h}(\phi_{\textrm{\tiny L}}-\phi_{\textrm{\tiny
R}})}{\ell}\right)+\sqrt{\left(r_{\textrm{\tiny
R}}^{2}-r_{h}^{2}\right)\left(r_{\textrm{\tiny
L}}^{2}-r_{h}^{2}\right)}\cosh\left(\frac{r_{h}}{\ell^{2}}\left(t_{\textrm{\tiny
R}}-t_{\textrm{\tiny L}}\right)\right)}{r_{h}^{2}}\,.$ (87)

Obviously, the latter is related to the one-sided geodesic (86) by taking the
analytic continuation $(r,r^{\prime},t,t^{\prime})\to(r_{\textrm{\tiny
L}},r_{\textrm{\tiny R}},t_{\textrm{\tiny L}},t_{\textrm{\tiny
R}}+i\frac{\pi\ell^{2}}{r_{h}})$. Note here that our convention for the time
arrows on the two asymptotic boundaries is that $t_{\textrm{\tiny L}}$ runs
down on the left boundary of the Penrose diagram and $t_{\textrm{\tiny R}}$
runs up on the right boundary.

###  3.1 Entanglement with the symmetric boundary

In this subsection, our focus lies on the investigation of holographic
entanglement entropy and mutual information in the symmetric setup with
$L_{1}=L_{2}$. We carry out these calculations analytically utilizing the
explicit geodesic distances given above for any two points in AdS3 spacetime.
Notably, we will also need to find the extremal curves when the geodesics
intersect the branes. This entails a process of extremization over the
intersection points along the brane. For the sake of conciseness, we shall
omit the intricate details of these calculations and provide the salient
results for the holographic entanglement entropy. Given multiple extremal
surfaces with respect to fixed boundary subregions, the final result of the
holographic entanglement entropy is governed by the smallest distance among
the choices of these extremal curves. Since there are only cold or hot phases
for the symmetric setup, we need to evaluate the entanglement entropy and
mutual information in both phases. To distinguish the two phases, we denote
the transition temperature as $T_{\textrm{\tiny CH}}$, i.e.,

$T_{\textrm{\tiny CH}}\equiv\frac{C_{\textrm{\tiny
CH}}}{L_{1}+L_{2}}=\frac{1}{L_{1}+L_{2}}\left(\sqrt{1+\frac{4}{\pi^{2}}\,\text{arctanh}^{2}\left(T_{o}\ell\right)}-\frac{2}{\pi}\,\text{arctanh}\left(T_{o}\ell\right)\right)\,,$
(88)

which was derived in eq. (78). In the following, we consider the left and
right boundaries on the time slices $t_{\textrm{\tiny L}}$ and
$t_{\textrm{\tiny R}}$, respectively. We start with the holographic
entanglement entropy of different defect regions and then proceed to the
evaluation of the mutual information.

####  3.1.1 Holographic entanglement entropy of defects

We begin by examining the holographic entanglement entropy associated with
different defects, which will subsequently be used to evaluate the mutual
information. As explained above eq. (81), we begin by regularizing each defect
with a small interval of fixed length $\Delta L_{\textrm{\tiny D}}$, followed
by taking the limit as $\Delta L_{\textrm{\tiny D}}\to 0$ to derive the final
results.

For the case of a single defect, such as $D^{(\textrm{\tiny R})}_{a}$ or
$D^{(\textrm{\tiny R})}_{b}$, the corresponding holographic entanglement
entropy is obtained from the extremal geodesic connecting the two endpoints of
the defect interval. We will refer to this type of geodesic as “connected”
geodesics (as shown in the left column of figure 20). For both phases, the
results are summarized as

$S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny
R})}_{a})=\begin{cases}\frac{c}{3}\left[\log\left(\frac{L_{\rm
bdy}}{\pi\epsilon}\sin\left(\frac{\pi\Delta L_{\textrm{\tiny D}}}{L_{\rm
bdy}}\right)\right)+\log\left(\sqrt{\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)\right]\,,\quad
T_{\textrm{\tiny DCFT}}\leq T_{\textrm{\tiny CH}}\,\\\ \,\\\
\frac{c}{3}\left[\log\left(\frac{\beta}{\pi\epsilon}\sinh\left(\frac{\pi}{\beta}\Delta
L_{\textrm{\tiny
D}}\right)\right)+\log\left(\sqrt{\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)\right]\,,\quad
T_{\textrm{\tiny DCFT}}\geq T_{\textrm{\tiny CH}}\,.\\\ \end{cases}$ (89)

Here, the notation for the inverse temperature $\beta=1/T_{\textrm{\tiny
DCFT}}$ is used to maintain consistency with the established convention in
other literature. Importantly, it is worth noting that the two expressions,
pertaining to the cold and hot phases, can be interchanged using the following
replacement

$L_{\rm bdy}\equiv L_{1}+L_{2}\quad\longleftrightarrow\quad
i\beta=\frac{i}{T_{\textrm{\tiny DCFT}}}\,.$ (90)

However, it is obvious that the length of connected extremal geodesic becomes
ill-defined if we take the limit as $\Delta L_{\textrm{\tiny D}}$ approaches
0. However, as in eq. (81), while we wish $\Delta L_{\textrm{\tiny D}}$ to be
smaller than any macroscopic scales, we keep $\Delta L_{\textrm{\tiny
D}}\gg\epsilon$. Hence in evaluating the above expressions, we have

$\frac{\Delta L_{\textrm{\tiny D}}}{\epsilon}=\lim\limits_{\Delta
L_{\textrm{\tiny D}}/L_{\rm bdy}\to 0}\left(\frac{L_{\rm
bdy}}{\pi\epsilon}\sin\left(\frac{\pi\Delta L_{\textrm{\tiny
D}}}{L_{\rm{bdy}}}\right)\right)=\lim\limits_{\Delta L_{\textrm{\tiny
D}}/\beta\to
0}\left(\frac{\beta}{\pi\epsilon}\sinh\left(\frac{\pi}{\beta}\Delta
L_{\textrm{\tiny D}}\right)\right)\,.$ (91)

It is worth noting that keeping ${\Delta L_{\textrm{\tiny D}}}/{\epsilon}$ as
fixed in both phases ensures the entanglement entropy $S_{\textrm{\tiny
EE}}(D^{(\textrm{\tiny R})}_{a})$ is continuous at the transition point. This
choice effectively regulates the entanglement entropy of each defect, yielding

$S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny
R})}_{a})=\frac{c}{3}\left(\log\left(\frac{\Delta L_{\textrm{\tiny
D}}}{\epsilon}\right)+\frac{1}{2}\,\log\left(\frac{1+T_{o}\ell}{1-T_{o}\ell}\right)\right)\equiv
S^{\textrm{\tiny(reg)}}_{\rm{def}}\,,$ (92)

which remains the same for both the cold and hot phases. Note that the second
term is just the defect entropy $S_{\rm{def}}$ defined in eq. (7), while the
first term is simply the entanglement entropy that would be found for the
small interval $\Delta L_{\textrm{\tiny D}}$ in the CFT vacuum. Notably, this
regulated defect entropy is independent of the bulk phases and the
temperature, as it primarily characterizes the intrinsic information related
to the boundary conditions on the defects.

Furthermore, let us consider the holographic entanglement entropy associated
with two defects on the same side, e.g., $D^{(\textrm{\tiny
R})}_{a,b}=D^{(\textrm{\tiny R})}_{a}\cup D^{(\textrm{\tiny R})}_{b}$. Apart
from the connected geodesics introduced above, there are disconnected
geodesics that can connect $D^{(\textrm{\tiny R})}_{a}$ and $D^{(\textrm{\tiny
R})}_{b}$. For the cold phase, the holographic entanglement entropy is given
by the minimal area of the connected and disconnected extremal surfaces, i.e.,

$S_{\textrm{\tiny EE}}^{(\textrm{\tiny C})}(D^{(\textrm{\tiny
R})}_{a,b})=\min\begin{cases}\frac{c}{3}\log\left(\frac{L_{\rm
bdy}}{\pi\epsilon}\sin\left(\frac{\pi(L_{1}-\Delta L_{\textrm{\tiny
D}})}{L_{\rm bdy}}\right)\right)+\frac{c}{3}\log\left(\frac{L_{\rm
bdy}}{\pi\epsilon}\sin\left(\frac{\pi(L_{2}-\Delta L_{\textrm{\tiny
D}})}{L_{\rm bdy}}\right)\right)\,,\\\ \,\\\
\frac{2c}{3}\left[\log\left(\frac{L_{\rm
bdy}}{\pi\epsilon}\sin\left(\frac{\pi\Delta L_{\textrm{\tiny D}}}{L_{\rm
bdy}}\right)\right)+\log\left(\sqrt{\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)\right]\,,\\\
\end{cases}$ (93)

which does not depend on the temperature $T_{\textrm{\tiny DCFT}}$. Taking the
defect limit with $\Delta L_{\textrm{\tiny D}}\to 0$ yields

$\lim\limits_{\Delta L_{\textrm{\tiny D}}\to 0}S_{\textrm{\tiny
EE}}^{(\textrm{\tiny C})}(D^{(\textrm{\tiny
R})}_{a,b})=\min\,\left\\{\frac{2c}{3}\log\left(\frac{L_{1}+L_{2}}{\pi\epsilon}\right)\,,\,2\,S^{\textrm{\tiny(reg)}}_{\rm{def}}\right\\}\,.$
(94)

In this limit, the first expression (coming from the connected geodesics) is
minimal when

$\sqrt{\frac{1+T_{o}\ell}{1-T_{o}\ell}}\geq\frac{L_{1}+L_{2}}{\pi\Delta
L_{\textrm{\tiny D}}}\,,\qquad\text{condition-}\alpha$ (95)

is satisfied.

At high temperatures, the bulk spacetime for each side is given by a BTZ black
hole at a temperature $T_{\textrm{\tiny DCFT}}=T_{\textrm{\tiny
BH}}=\frac{r_{h}}{2\pi\ell^{2}}$. The corresponding holographic entanglement
entropy for two defects on one side, e.g., $D^{(\textrm{\tiny R})}_{a,b}$, is
derived as

$S_{\textrm{\tiny EE}}^{(\textrm{\tiny H})}(D^{(\textrm{\tiny
R})}_{a,b})=\min\begin{cases}\frac{c}{3}\log\left(\frac{\beta}{\pi\epsilon}\sinh\left(\frac{\pi\left(L_{1}-\Delta
L_{\textrm{\tiny
D}}\right)}{\beta}\right)\right)+\frac{c}{3}\log\left(\frac{\beta}{\pi\epsilon}\sinh\left(\frac{\pi\left(L_{2}-\Delta
L_{\textrm{\tiny D}}\right)}{\beta}\right)\right)+S_{\rm{horizon}}\,,\\\ \,\\\
\frac{2c}{3}\left[\log\left(\frac{\beta}{\pi\epsilon}\sinh\left(\frac{\pi}{\beta}\Delta
L_{\textrm{\tiny
D}}\right)\right)+\log\left(\sqrt{\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)\right]\,.\\\
\end{cases}$ (96)

Here, we also include the area of black hole horizon $S_{\rm horizon}$ in the
contribution to the entanglement entropy because the disconnected-disconnected
phase for the RT surface also encompasses the black hole sitting at the middle
of the glued spacetime. The corresponding RT surfaces for $S_{\textrm{\tiny
EE}}^{(\textrm{\tiny H})}(D^{(\textrm{\tiny R})}_{a,b})$ are represented by
the two plots to the far left in figure 17 after taking
$L_{1}=L_{2}$.111111Obviously, the RT surfaces shown in the right two plots in
figure 17 can not dominate here since they are not symmetric. The Bekenstein-
Hawking entropy $S_{\rm horizon}$ is thus defined by the area of horizon,
i.e.,

$\begin{split}S_{\rm{horizon}}&\equiv\frac{r_{h}\Delta\phi^{\rm{Hor}}_{1}+r_{h}\Delta\phi^{\rm{Hor}}_{2}}{4G_{\textrm{\tiny
N}}}=\frac{c}{3}\pi T_{\textrm{\tiny
BH}}\left(L_{1}+L_{2}\right)+\frac{2c}{3}\log\left(\sqrt{\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)\,.\\\
\end{split}$ (97)

where $\Delta\phi^{\rm{Hor}}_{i}$ is given in eq. (48). The latter expression
corresponds to the entropy of the thermal CFT on the boundary of length
$L_{1}+L_{2}$ plus the entropy $S_{\rm def}$ of the two defects. Hence we note
that $S_{\rm{horizon}}$ is not the black hole entropy $S_{\textrm{\tiny BH}}$
of the original BTZ black hole when the tension of the brane is nonzero.
Similar to the observation for $S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny
R})}_{a})$, it is obvious that the holographic entanglement entropy
$S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny R})}_{a,b})$ in the cold phase
(without $S_{\rm{horizon}}$ part) and hot phase are related to each other by
the replacement in eq. (90). Although there are two types of extremal surface
in the hot phase, we can find that the minimal one is always given by the
connected geodesics in the limit $\Delta L_{\textrm{\tiny D}}\to 0$. As a
result, the holographic entanglement entropy of the left/right defects in hot
phases are nothing but twice the defect entropy $S_{\rm Defect}$, namely

$\lim\limits_{\Delta L_{\textrm{\tiny D}}\to 0}S_{\textrm{\tiny
EE}}^{(\textrm{\tiny H})}(D^{(\textrm{\tiny
R})}_{a,b})=\frac{2c}{3}\left(\log\left(\frac{\Delta L_{\textrm{\tiny
D}}}{\epsilon}\right)+\frac{1}{2}\,\log\left(\frac{1+T_{o}\ell}{1-T_{o}\ell}\right)\right)=2\,S^{\textrm{\tiny(reg)}}_{\rm{def}}\,.$
(98)

We also note that there are some other types of disconnected extremal surfaces
which can cross the brane twice. However, we can confirm that their areas
cannot yield the minimal value in the limit $\Delta L_{\textrm{\tiny D}}\to
0$.

Finally, we investigate the most symmetric case by including four defects on
both the left and right boundary. Since the left and right bulk spacetime are
disconnected in the cold phase, the holographic entanglement entropy of four
defects is simply given by

$S_{\textrm{\tiny EE}}^{(\textrm{\tiny C})}\left(D^{(\textrm{\tiny
R,L})}_{a,b}\right)=S^{(\textrm{\tiny C})}_{\textrm{\tiny
EE}}(D^{(\textrm{\tiny R})}_{a,b})+S^{(\textrm{\tiny C})}_{\textrm{\tiny
EE}}(D^{(\textrm{\tiny L})}_{a,b})\,.$ (99)

However, we have several types of geodesics which can cross the horizon and
connect the defects on the left and right sides. The candidates of RT surfaces
are shown by the first two plots in figure 18 and the left plot in figure 19
(after taking $L_{1}=L_{2}$). The corresponding entanglement entropy is
provided by the minimum among the three candidates, i.e.,

$S_{\textrm{\tiny EE}}^{(\textrm{\tiny H})}(D^{(\textrm{\tiny
R,L})}_{a,b})=\min\begin{cases}\frac{4c}{3}\log\left(\frac{\beta}{\pi\epsilon}\sinh\left(\frac{\pi\left(L_{1}-\Delta
L_{\textrm{\tiny D}}\right)}{\beta}\right)\right)\,,\\\ \,\\\
\frac{4c}{3}\left[\log\left(\frac{\beta}{\pi\epsilon}\sinh\left(\frac{\pi}{\beta}\Delta
L_{\textrm{\tiny
D}}\right)\right)+\log\left(\sqrt{\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)\right]\,,\\\
\,\\\
\frac{4c}{3}\log\left(\frac{\beta}{\pi\epsilon}\cosh\left(\frac{\pi}{\beta}(t_{\textrm{\tiny
R}}-t_{\textrm{\tiny L}})\right)\right)\,.\\\ \end{cases}$ (100)

where the factor $\cosh\left(\frac{2\pi}{\beta}(t_{\textrm{\tiny
R}}-t_{\textrm{\tiny L}})\right)$ originates from the (almost) linear growth
of extremal surface in the wormhole region. Similar to the simplifications
shown before, our regularization for the defect limit $\Delta L_{\textrm{\tiny
D}}\to 0$ leads to

$S_{\textrm{\tiny EE}}^{(\textrm{\tiny H})}(D^{(\textrm{\tiny
R,L})}_{a,b})=\min\left\\{\frac{4c}{3}\log\left(\frac{\beta}{\pi\epsilon}\sinh\left(\frac{\pi
L_{\rm
bdy}}{2\beta}\right)\right),4\,S^{\textrm{\tiny(reg)}}_{\rm{def}}\,,\frac{4c}{3}\log\left(\frac{\beta}{\pi\epsilon}\cosh\left(\frac{\pi}{\beta}(t_{\textrm{\tiny
R}}-t_{\textrm{\tiny L}})\right)\right)\right\\}\,,$ (101)

which is bounded from above by $4S^{\textrm{\tiny(reg)}}_{\rm{def}}$. In the
regime with a large defect entropy, i.e., in the large tension limit, one find
that the entanglement entropy jumps to a lower value after the phase
transition to the hot phase, signifying the formation of a wormhole connecting
the left and right regions.

####  3.1.2 Holographic mutual information

With the results of entanglement entropy of different defect regions, it is
straightforward to evaluate the mutual information between different
subregions.

Figure 10: The mutual information between the left and right defects in the
symmetric set-up. Left: the temperature dependence with fixing
$t_{\textrm{\tiny L}}=t_{\textrm{\tiny R}}$ but varying the tension of the
brane. Right: the time dependence with $\Delta t\equiv|t_{\textrm{\tiny
L}}-t_{\textrm{\tiny R}}|$.

Mutual Information $I_{\textrm{\tiny R}_{a}:\textrm{\tiny R}_{b}}$: The mutual
information between the two defects on one side is defined by

$I_{\textrm{\tiny R}_{a}:\textrm{\tiny R}_{b}}=S_{\textrm{\tiny
EE}}(D^{(\textrm{\tiny R})}_{a})+S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny
R})}_{b})-S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny R})}_{a,b})\,.$ (102)

In the cold phase, this reduces to

$I_{\textrm{\tiny R}_{a}:\textrm{\tiny R}_{b}}^{(\textrm{\tiny
C})}=\max\,\left\\{0\,,\frac{2c}{3}\log\left(\frac{\pi\Delta L_{\textrm{\tiny
D}}}{L_{1}+L_{2}}\sqrt{\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)\right\\}\,,$
(103)

which is non-zero if the condition-$\alpha$ in eq. (95) is satisfied. Recall
to be in the cold phase, the temperature must also be lower than the critical
one, i.e., $T_{\textrm{\tiny DCFT}}\leq T_{\textrm{\tiny CH}}$. On the other
hand, in the hot phase (with $T_{\textrm{\tiny DCFT}}\geq T_{\textrm{\tiny
CH}}$), the mutual information $I_{\textrm{\tiny R}_{a}:\textrm{\tiny R}_{b}}$
simply vanishes, viz,

$I_{\textrm{\tiny R}_{a}:\textrm{\tiny R}_{b}}^{(\textrm{\tiny H})}=0\,.$
(104)

This is expected because, in the hot phase, the branes $\mathcal{B}_{i}$
extend from a defect on the right to another on the left, rather than between
the two defects on the right side, i.e., from $D^{(\textrm{\tiny R})}_{a}$ to
$D^{(\textrm{\tiny L})}_{a}$ rather than $D^{(\textrm{\tiny R})}_{b}$.

Mutual Information $I_{\textrm{\tiny R}_{a}:\textrm{\tiny L}_{a}}$: The mutual
information of two defects on the left and right side is defined by

$I_{\textrm{\tiny R}_{a}:\textrm{\tiny L}_{a}}:=S_{\textrm{\tiny
EE}}(D^{(\textrm{\tiny R})}_{a})+S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny
L})}_{a})-S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny R})}_{a}\cup
D^{(\textrm{\tiny L})}_{a})\,.$ (105)

After taking account of the regularization with $\Delta L_{\textrm{\tiny
D}}\to 0$, it reduces to

$\begin{cases}I_{\textrm{\tiny R}_{a}:\textrm{\tiny L}_{a}}^{(\textrm{\tiny
C})}=0\,,\\\ \,\\\ I_{\textrm{\tiny R}_{a}:\textrm{\tiny
L}_{a}}^{(\textrm{\tiny
H})}=\max\left(0\,,\frac{2c}{3}\log\left(\frac{\sinh\left(\frac{\pi}{\beta}\Delta
L_{\textrm{\tiny D}}\right)}{\cosh\left(\frac{\pi}{\beta}(t_{\textrm{\tiny
R}}-t_{\textrm{\tiny
L}})\right)}\sqrt{\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)\right)\,.\end{cases}$
(106)

Mutual Information $I_{\textrm{\tiny L:R}}$: Finally, we are interested in the
mutual information between the left and right defects, i.e.,

$I_{\textrm{\tiny L:R}}:=S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny
R})}_{a,b})+S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny
L})}_{a,b})-S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny R,L})}_{a,b})\,.$ (107)

It is straightforward to obtain

$\begin{cases}I_{\textrm{\tiny L:R}}^{(\textrm{\tiny C})}=0\,,\\\ \,\\\
I_{\textrm{\tiny L:R}}^{(\textrm{\tiny
H})}=\max\left\\{\frac{4c}{3}\log\left(\frac{\sinh\left(\frac{\pi}{\beta}\Delta
L_{\textrm{\tiny
D}}\right)}{\sinh\left(\left(\frac{\pi}{\beta}\left(L_{1}-\Delta
L_{\textrm{\tiny
D}}\right)\right)\right)}\sqrt{\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)\,,0\,,\frac{4c}{3}\log\left(\frac{\sinh\left(\frac{\pi}{\beta}\Delta
L_{\textrm{\tiny D}}\right)}{\cosh\left(\frac{\pi}{\beta}(t_{\textrm{\tiny
R}}-t_{\textrm{\tiny
L}})\right)}\sqrt{\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)\right\\}\,.\end{cases}$
(108)

Taking the symmetric time slice with $t_{\textrm{\tiny L}}=t_{\textrm{\tiny
R}}$, we find that the mutual information $I_{\textrm{\tiny L:R}}$ in the hot
phase is dominated by the defect contributions

$I_{\textrm{\tiny L:R}}^{(\textrm{\tiny H})}\simeq
4\left(\frac{\pi\,c}{3}\,T_{\textrm{\tiny DCFT}}\,\Delta L_{\textrm{\tiny
D}}+S_{\rm def}\right)+\cdots\,.$ (109)

That is, we have $S_{\rm def}$ in eq. (7) for each defect plus the thermal
entropy of the CFT in the interval around these. This comes from the extremal
surfaces connecting the left and right boundaries in the hot phase. At the
late time with $|t_{\textrm{\tiny L}}-t_{\textrm{\tiny R}}|\gg L_{1}$, the
mutual information $I_{\textrm{\tiny L:R}}^{(\textrm{\tiny H})}$ between two
sides always reduces to a constant. These behaviours are also illustrated in
the figure 10.

###  3.2 Holographic mutual information

Figure 11: A time slice ($t_{\textrm{\tiny L}}=0=t_{\textrm{\tiny R}}$) of the
bulk spacetime in the warm (left figure) and hot (right figure) phases. The
gray curves denote the geodesics identified to make a quotient of the global
AdS3 with left and right boundaries. The black line in the middle indicates
the position of the horizon of BTZ black hole, i.e., $r=r_{h}$.

In contrast to the regime with $L_{1}\sim L_{2}$ like the symmetric case
above, the bulk spacetime allows three distinct phases when $L_{1}\gg L_{2}$.
To illustrate the corresponding entanglement, we consider the holographic
calculation of the mutual information while assuming $L_{1}\gg L_{2}$, which
helps to simplify the calculations because the relevant extremal surfaces are
limited.

In the holographic calculations for the symmetric setup, we regulated the
defects by including a small boundary interval around each defect, with a
length $\Delta L_{\textrm{\tiny D}}$ which is a small length measure compared
with the AdS radius $\ell$ – see eq. (81). This is motivated by our goal that
the defect region is considered as the boundary dual of the braneworld in our
doubly holographic construction. However, in order to make it true, one can
expect that the defect region cannot be too small otherwise all information is
localized inside the defects. Taking the symmetric case as an example, one can
find that all mutual information between these defects vanishes if one simply
takes $\Delta L_{\textrm{\tiny D}}$ to be too small. For example, examing eq.
(103)), we find that a critical size given by $\frac{\Delta L_{\textrm{\tiny
D}}}{L_{\rm bdy}}=\frac{1}{\pi}\,\sqrt{\frac{1-T_{o}\ell}{1+T_{o}\ell}}$ where
$I_{\textrm{\tiny R}_{a}:\textrm{\tiny R}_{b}}^{(\textrm{\tiny C})}=0$.
Inspired by this result, we extend eq. (81) to restrict the following
holographic analysis to the following regime

$\epsilon\ll\ell\sqrt{\frac{1-T_{o}\ell}{1+T_{o}\ell}}\ll{\Delta
L_{\textrm{\tiny D}}}\ll\ell\,.$ (110)

The first condition guarantees that the brane still resides below the UV
cutoff surface in the bulk AdS3 spacetime and the last two conditions will
ensure that the defect is well approximated by the holographic gravity theory
on the brane.121212The AdS radius $\ell_{\textrm{\tiny B}}$ of braneworld is
fixed by the brane tension $T_{o}$ , as we discuss in appendix A. From eq.
(193), we have $\ell_{\textrm{\tiny B}}=\ell/\sqrt{1-(T_{o}\ell)^{2}}\gg\ell$
in the large tension limit with $T_{o}\ell\sim 1$. From the brane perspective,
this regime suppresses the contributions from the higher curvature terms in
the effective gravitational action (257) on the branes, e.g., see
Grimaldi:2022suv .

####  3.2.1 Mutual information in the cold phase

Figure 12: The extremal surfaces in the cold phase. Left: The connected
extremal surface (green curves) for a single defect $D^{(\textrm{\tiny
R})}_{a}$. Right: Two types of extremal surfaces (green curves) with respect
to one-sided defects, i.e., $D^{(\textrm{\tiny R})}_{a,b}$. The left and right
one are referred to as the disconnected and connected phases, respectively.

We begin with the cold phase which is the dominant solution at low
temperatures. More precisely, it is controlled by the dimensionless boundary
size, i.e., $T_{\textrm{\tiny DCFT}}L_{1}\leq C_{\textrm{\tiny CW}}$ as
discussed around eq. (59). Noting that we are interested in the regime with
$L_{1}\gg L_{2}$, one concludes that the cold phase arises with

$T_{\textrm{\tiny DCFT}}L_{2}\leq C_{\textrm{\tiny CW}}\
\frac{L_{2}}{L_{1}}\approx\frac{L_{2}}{L_{1}}\ll 1\,.$ (111)

We first need to evaluate the holographic entanglement entropy for the
individual defects, as we did in the symmetric setup. However, this
calculation requires more work here since the two boundary intervals have
different sizes. The extremal surface associated with a single defect, e.g.,
$D^{(\textrm{\tiny R})}_{a}$ is depicted in the left plot of figure 12. As
before, since this surface crosses the brane and connects the two endpoints of
the defect interval, we need to minimize over the intersection point where it
meets the brane to find the minimal surface. However, since the defect region
is small with $\Delta L_{\textrm{\tiny D}}\ll\ell$, we expect to find that the
leading contribution in the limits $\Delta L_{\textrm{\tiny D}}\to 0$ and
$T_{o}\ell\to 1$ is

$S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny
R})}_{a})=\frac{c}{3}\left(\log\left(\frac{\Delta L_{\textrm{\tiny
D}}}{\epsilon}\right)+\log\left(\sqrt{\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)+\mathcal{O}(1)\right)\simeq
S^{\textrm{\tiny(reg)}}_{\rm{def}}\,,$ (112)

where we have ignored subleading corrections and in the last expression,
recast the formula as the regulated defect entropy defined in eq. (92). This
leading contribution is universal because near the conformal boundary at
$r\to\infty$, the brane geometry is asymptotically given by AdS2 and the
tension fixes the angle between the AdS2 brane and conformal boundary. We will
also use the same formula for $S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny
R})}_{a})$ in both warm and hot phases in the later subsections.

Considering the two defects on a single asymptotic boundary, the extremal
surfaces associated with, e.g., $S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny
R})}_{a,b})$, come in two distinct classes, as shown in the right plot of
figure 12. The corresponding holographic entanglement entropy is determined by
the minimal length among the two pairs of candidate surfaces, i.e.,

$S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny
R})}_{a,b})=\min\begin{cases}2S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny
R})}_{a})\simeq 2S^{\textrm{\tiny(reg)}}_{\rm{def}}\,,\\\ \,\\\
\frac{c}{3}\log\left(\frac{L_{\mathcal{S}_{1}}}{\pi\epsilon}\sin\frac{\pi
L_{1}}{L_{\mathcal{S}_{1}}}\right)+\frac{c}{3}\log\left(\frac{L_{\mathcal{S}_{2}}}{\pi\epsilon}\sin\frac{\pi
L_{2}}{L_{\mathcal{S}_{2}}}\right)\,,\\\ \end{cases}$ (113)

where $L_{\mathcal{S}_{1}}=\frac{2\pi\ell}{\sqrt{-m_{1}}}$ and
$L_{\mathcal{S}_{1}}=\frac{2\pi\ell}{\sqrt{-m_{2}}}$ are the boundary
perimeter which the full bulk spacetimes associated with $\mathcal{S}_{1}$ and
$\mathcal{S}_{2}$ would have had if they had not been cutoff by the brane in
constructing the holographic dual of the defect CFT. Generally, the
minimization defined in eq. (113) is nontrivial and depends on all of the
boundary parameters. However, it is easy to realize that the extremal surface
crossing the brane (as shown in figure 12) is not preferred in the regime of
present interest (110) because of the large contribution coming from the
defect entropy. As a result, we always have non-vanishing mutual information
between the two defects regions $D^{(\textrm{\tiny R})}_{a},D^{(\textrm{\tiny
R})}_{b}$. More explicitly, this mutual information can be expressed as

$I_{\textrm{\tiny R}_{a}:\textrm{\tiny R}_{b}}^{(\textrm{\tiny C})}\simeq
4\log g-\frac{c}{3}\log\left(\frac{L_{\mathcal{S}_{1}}}{\pi\Delta
L_{\textrm{\tiny D}}}\sin\frac{\pi
L_{1}}{L_{\mathcal{S}_{1}}}\right)-\frac{c}{3}\log\left(\frac{L_{\mathcal{S}_{2}}}{\pi\Delta
L_{\textrm{\tiny D}}}\sin\frac{\pi L_{2}}{L_{\mathcal{S}_{2}}}\right)\,.\\\ $
(114)

The two defects are obviously connected by the brane living in the bulk
spacetime. The non-vanishing mutual information here is then a manifestation
of the “ER=EPR” proposal Maldacena:2013xja . Although it is hard to obtain a
generic analytic answer, we can consider the limit $L_{2}/L_{1}\to 0$ which is
realized by taking $m_{2}\to-\infty$. From the results derived in eqs. (53)
and (54), it is straightforward to find the dominant contributions given the
boundary perimeters, i.e.,

$\begin{split}L_{\mathcal{S}_{1}}&\equiv\frac{2\pi\ell}{\sqrt{-m_{1}}}\simeq
L_{1}+\frac{k_{1}L_{2}}{\pi-k_{2}}+\mathcal{O}(L_{2}^{\,2}/L_{1})\,,\\\
L_{\mathcal{S}_{2}}&\equiv\frac{2\pi\ell}{\sqrt{-m_{2}}}\simeq\frac{\pi
L_{2}}{\pi-k_{2}}+\mathcal{O}(L_{2}^{\,2}/L_{1})\,.\\\ \end{split}$ (115)

Correspondingly, the mutual information $I_{\textrm{\tiny R}_{a}:\textrm{\tiny
R}_{b}}^{(\textrm{\tiny C})}$ reduces to

$I_{\textrm{\tiny R}_{a}:\textrm{\tiny R}_{b}}^{(\textrm{\tiny
C})}\simeq\frac{2c}{3}\log\left(\frac{\pi-k_{2}}{\sqrt{k_{1}\sin(\pi-
k_{2})}}\frac{\Delta L_{\textrm{\tiny
D}}}{L_{2}}\sqrt{\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)+\mathcal{O}\left(\left(\frac{L_{2}}{L_{1}}\right)^{2}\right)\,,$
(116)

where we have substituted the simplified entanglement entropy

$S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny
R})}_{a,b})\simeq\frac{c}{3}\log\left(\frac{\sin(\pi-k_{2})}{\pi-
k_{2}}\,\frac{L_{2}}{\epsilon}\right)+\frac{c}{3}\log\left(\frac{k_{1}}{\pi-
k_{2}}\frac{L_{2}}{\epsilon}\right)+\mathcal{O}\left(\left(\frac{L_{2}}{L_{1}}\right)^{2}\right)\,.\\\
$ (117)

We would like to comment on some features of the above approximations. First,
the leading contributions do not depend on the boundary size $L_{1}$ because
the extremal surfaces are far from the center of bulk spacetime when
$L_{2}/L_{1}\to 0$. Second, the square of the ratio $\frac{L_{2}}{L_{1}}$
suppresses the subleading corrections. Furthermore, note that the three phases
only coexist in the regime where $\frac{L_{2}}{L_{1}}\ll 1$, which implies
that the leading contributions remain good approximations. It is interesting
to note that even in the fusion limit $L_{2}\ll L_{1}$, the entanglement
entropy like $S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny R})}_{a,b})$ does not
reduce to the vacuum value. This nontrivial value of entropy exists because in
this limit, the Casimir energy between two defects becomes large and negative.
Hence, this can even affect the physics of the ultraviolet, resulting in the
detectable difference between this state and the vacuum even in this short
distance limit.

Finally, let us mention that the mutual information $I_{\textrm{\tiny L:R}}$
between the left and right defects simply vanishes in the cold phase because
the corresponding bulk spacetimes are disconnected.

####  3.2.2 Mutual information in the warm phase

Next, we turn our attention to the warm phase. As a reminder, the warm phase
arises in the regime

$\frac{L_{2}}{L_{1}}\approx\frac{L_{2}}{L_{1}}C_{\textrm{\tiny CW}}\leq
T_{\textrm{\tiny DCFT}}L_{2}\leq C_{\textrm{\tiny WH}}\,.$ (118)

Here the second inequality signifies the transition between the warm and hot
phases, which is parametrized by

$C_{\textrm{\tiny WH}}\simeq\frac{(\pi-k_{2})\left(\sqrt{(\pi-k_{2})(\pi-
k_{1}-k_{2})+4\text{arctanh}^{2}(T_{o}\ell)}-2\text{arctanh}(T_{o}\ell)\right)}{(\pi-
k_{1}-k_{2})\pi}\,,$ (119)

which was given above in eq. (2.2). The first inequality is equivalent to

$T_{\textrm{\tiny DCFT}}L_{1}\geq C_{\textrm{\tiny CW}}\approx
1-\frac{k_{1}}{\pi-
k_{2}}\frac{L_{2}}{L_{1}}+\mathcal{O}\left((\frac{L_{1}}{L_{2}})^{2}\right)\,,$
(120)

which marks the transition between the warm and cold phases. We have assumed
that the boundary interval with a length $L_{1}$ is the bigger side. The
boundary temperature is associated with the mass of the bulk black hole in the
bulk region $\mathcal{S}_{1}$ by $m_{1}=(2\pi\ell T_{\textrm{\tiny
DCFT}})^{2}$, as in eq. (45). The other bulk region $\mathcal{S}_{2}$ is a
portion of the AdS3 spacetime with a mass denoted by $m_{2}<0$.

#### Holographic entanglement entropy $S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny
R})}_{a,b})$

First of all, let us consider the entropy of the two defect regions
$D^{(\textrm{\tiny R})}_{a,b}$ on the right boundary. It is obvious that there
are two families of extremal surfaces. The first two classes of geodesics stay
away from the brane as shown in two plots of figure 13. In contrast, in the
second two classes, the geodesics cross the brane as depicted in figure 14.

Figure 13: Relevant extremal surfaces in the warm phase (Green lines) for
$S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny R})}_{a,b})$.

Beginning with those shown in the left plot of figure 13, one can easily
obtain the corresponding geodesic length because each bulk portion
$\mathcal{S}_{i}$ is still a locally AdS3 spacetime. As a result, adding the
two lengths yields the candidate result of the entanglement entropy

$S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny
R})}_{a,b})\overset{?}{=}\frac{c}{3}\log\left[\frac{\beta}{\pi\epsilon}\;\sinh\frac{\pi(L^{\prime}_{\mathcal{S}_{1}}-L_{1})}{\beta}\right]+\frac{c}{3}\log\left[\frac{L_{\mathcal{S}_{2}}}{\pi\epsilon}\sin\frac{\pi
L_{2}}{L_{\mathcal{S}_{2}}}\right]\,,$ (121)

where the first term corresponds to a geodesic in the BTZ black hole and the
second, in global AdS3. The two boundary perimeters in the warm phase are
defined as

$L^{\prime}_{\mathcal{S}_{1}}\equiv\Delta\phi_{1}^{\rm
Hor}\ell=L_{1}+2\Delta\phi_{1}\ell\,,\quad
L_{\mathcal{S}_{2}}\equiv\frac{2\pi\ell}{\sqrt{-m_{2}}}\,.$ (122)

Similar to the previous calculations, we can evaluate the above formula by
taking the series expansion with $m_{2}\to-\infty$, which gives rise to

$\begin{split}S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny
R})}_{a,b})&\overset{?}{=}\frac{c}{3}\log\left(\frac{\beta}{\pi\epsilon}\sinh\left(\frac{\pi
L_{2}}{\beta}\frac{k_{1}}{\pi-
k_{2}}\right)\right)+\frac{c}{3}\log\left(\frac{\sin(\pi-k_{2})}{\pi-
k_{2}}\frac{L_{2}}{\epsilon}\right)+\mathcal{O}\left(\frac{L_{2}^{2}}{L_{1}^{2}}\right)\\\
&\simeq\frac{c}{3}\log\left(\frac{k_{1}}{\pi-
k_{2}}\frac{L_{2}}{\epsilon}\right)+\frac{c}{3}\log\left(\frac{\sin(\pi-
k_{2})}{\pi-
k_{2}}\frac{L_{2}}{\epsilon}\right)+\mathcal{O}\left(\frac{L_{2}^{2}}{\beta^{2}}\right)+\mathcal{O}\left(\frac{L_{2}^{2}}{L_{1}^{2}}\right)\,.\\\
\end{split}$ (123)

At the leading order, this result for the entanglement entropy
$S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny R})}_{a,b})$ in the warm phase is
equal to the result (117) in the cold phase. Nevertheless, it should be noted
that the subleading terms differ because the corrections in the warm phase
involve additional terms which are not suppressed by powers of $L_{2}/L_{1}$.

Now we turn to the geodesics shown in the right plot of figure 13. In this
case, the geodesic in $\mathcal{S}_{2}$ remains the same as above, while the
geodesic in $\mathcal{S}_{1}$ bulk now surrounds the black hole. The
corresponding geodesic lengths yield

$S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny
R})}_{a,b})\overset{?}{=}\frac{c}{3}\log\left(\frac{\beta}{\pi\epsilon}\;\sinh\frac{\pi
L_{1}}{\beta}\right)+S_{\rm
horizon}+\frac{c}{3}\log\left[\frac{L_{\mathcal{S}_{2}}}{\pi\epsilon}\sin\frac{\pi
L_{2}}{L_{\mathcal{S}_{2}}}\right]\,,$ (124)

where $S_{\rm horizon}$ denotes the Bekenstein-Hawking entropy in the warm
phase, i.e.,

$\begin{split}S_{\rm{horizon}}=\frac{r_{h}\Delta\phi^{\rm{Hor}}_{1}}{4G_{\textrm{\tiny
N}}}=\frac{c}{3}\pi T_{\textrm{\tiny
BH}}\left(L_{1}+2\Delta\phi_{1}\ell\right)\simeq\frac{c}{3}\pi
T_{\textrm{\tiny BH}}\left(L_{1}+\frac{k_{1}L_{2}}{\pi-k_{2}}\right)\,.\\\
\end{split}$ (125)

where we have used the leading expansion for $\Delta\phi_{1}$ from eq. (53) to
obtain the final approximation. It is obvious that we always have the result
in eq. (121) is always smaller than that in eq. (125) thanks to the inequality
$\log(\sinh X)<X$ where here we take $X={2\pi\Delta\phi_{1}\ell}/{\beta}$.

Figure 14: Irrelevant extremal surfaces in the warm phase (denoted by green
curves) for $S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny R})}_{a,b})$.

On the other hand, we also have the possible geodesics crossing the brane, as
shown in figure 14. Considering the simplest example on the left plot of the
figure. The minimal geodesic requires the minimization over the crossing point
on the brane. As we argued before, the extremal geodesic length would be
universally given by eq. (112) in the limit $\Delta L_{\textrm{\tiny D}}\to
0$, which is not sensitive to the details of the bulk spacetime. Explicitly,
these geodesics can be approximated by

$2S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny
R})}_{a})\simeq\frac{2c}{3}\log\left(\frac{\Delta L_{\textrm{\tiny
D}}}{\epsilon}\right)+4\log g+\mathcal{O}(1)\,,$ (126)

while those in the right plot yield

$\frac{c}{3}\log\left(\frac{\beta}{\pi\epsilon}\;\sinh\frac{\pi
L_{1}}{\beta}\right)+\frac{c}{3}\log\left(\frac{\beta}{\pi\epsilon}\;\sinh\frac{\pi
L_{1}}{\beta}\right)+4\log g+\mathcal{O}(1)\,.$ (127)

However, these candidates for the entanglement entropy do not compete with the
previous configurations which do not cross the brane because of the large
$\log g$ contributions (in the regime of our interest).

As a summary, we conclude that the holographic entanglement entropy
$S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny R})}_{a,b})$ in the warm phase is
approximated by

$S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny
R})}_{a,b})\simeq\frac{c}{3}\log\left(\frac{\beta}{\pi\epsilon}\sinh\left(\frac{\pi
L_{2}}{\beta}\frac{k_{1}}{\pi-
k_{2}}\right)\right)+\frac{c}{3}\log\left(\frac{\sin(\pi-k_{2})}{\pi-
k_{2}}\frac{L_{2}}{\epsilon}\right)\,,$ (128)

from eq. (123). It is evident that this expression for $S_{\textrm{\tiny
EE}}(D^{(\textrm{\tiny R})}_{a,b})$ increases with temperature. Conversely,
$2S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny R})}_{a})$ remains constant. Thus,
it might be inferred that a critical temperature exists above which the latter
becomes the dominant. Roughly, this temperature would be given by

$L_{2}\,T_{\textrm{\tiny DCFT}}\simeq\frac{\pi-
k_{2}}{k_{1}}\log\left(\sqrt{\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)\,.$ (129)

However, from eq. (118), we recall that the warm phase only exists up to

$L_{2}\,T_{\textrm{\tiny DCFT}}\leq C_{\textrm{\tiny WH}}\simeq\frac{(\pi-
k_{2})^{2}}{4\pi}{\log^{-1}\\!\left(\sqrt{\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)}\,.$
(130)

Therefore, this potential critical temperature is significantly higher than
the temperature for the transition from the warm to hot phase, and the
dominant holographic entanglement entropy in the warm phase is always
determined by eq. (128). Hence using eq. (128) for the entanglement entropy
$S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny R})}_{a,b})$ and eq. (112) for
$S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny R})}_{a})$, we obtain the mutual
information in the warm phase as

$I_{\textrm{\tiny R}_{a}:\textrm{\tiny R}_{b}}^{(\textrm{\tiny W})}\simeq
4\log g-\frac{2c}{3}\log\left(\frac{L_{2}}{\Delta L_{\textrm{\tiny
D}}}\,\frac{\sqrt{k_{1}\sin(\pi-k_{2})}}{\pi-k_{2}}\right)\,,$ (131)

where we have only kept the leading contributions.

#### Holographic entanglement entropy $S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny
L,R})}_{a,b})$

Figure 15: Relevant extremal surfaces in the warm phase (denoted by green
curves) for $S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny L,R})}_{a,b})$. Figure
16: Irrelevant extremal surfaces in the warm phase (denoted by green curves)
for $S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny L,R})}_{a,b})$.

Let us move now to the holographic calculation of $S_{\textrm{\tiny
EE}}(D^{(\textrm{\tiny L,R})}_{a,b})$ in the warm phase where we include all
four defects on two sides. Similar to the previous analysis in the symmetric
case, we will assume the left and right boundary times are $t_{\textrm{\tiny
L}},t_{\textrm{\tiny R}}$, respectively. Because of the time translation
invariance of the TFD state, the final result can only depend on the time
difference $(t_{\textrm{\tiny R}}-t_{\textrm{\tiny L}})$.

Since we are interested in the regime where the boundary entropy $2\log g$ is
large, we split all extremal surfaces into two families according to whether
they cross the brane or not, as shown in figures 15 and 16. From the previous
results, we know the ones crossing the brane acquire contributions
proportional to $\log g$ and will not be the minimal extremal surface. So we
examine the relevant extremal surfaces depicted in figure 15. Most of the
geodesics appearing there have already appeared in the previous calculations
of $S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny R})}_{a,b})$. The new type is the
geodesic that crosses the BTZ black hole horizon and connects the left and
right boundaries. Its geodesic length is also known and has been discussed for
eq. (100) for the symmetric case.

Collecting all previous results, the holographic entanglement entropy
$S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny L,R})}_{a,b})$ for all four defects
is given by the minimization as follows:

$S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny
L,R})}_{a,b})\simeq\frac{2c}{3}\log\left(\frac{\sin(\pi-k_{2})}{\pi-
k_{2}}\frac{L_{2}}{\epsilon}\right)+\min\begin{cases}\frac{2c}{3}\log\left(\frac{\beta}{\pi\epsilon}\sinh\left(\frac{\pi
L_{2}}{\beta}\frac{k_{1}}{\pi-k_{2}}\right)\right)\,,\\\ \,\\\
\frac{2c}{3}\log\left(\frac{\beta}{\pi\epsilon}\;\sinh\frac{\pi
L_{1}}{\beta}\right)\,,\\\ \,\\\
\frac{2c}{3}\log\left(\frac{\beta}{\pi\epsilon}\cosh\left(\frac{\pi}{\beta}(t_{\textrm{\tiny
R}}-t_{\textrm{\tiny L}})\right)\right)\,.\\\ \end{cases}$ (132)

The time growth in the third expression is related to the growth of the
wormhole connecting the two sides of the boundary. These three results from
top to bottom above correspond to the three candidates for the geodesics from
left to right illustrated in figure 15. The first term which is common to all
three expressions arises from the geodesics in the $\mathcal{S}_{2}$ region
which is common to all three configurations in the figure. As a result, the
geodesics in the $\mathcal{S}_{1}$ region determine the minimal entropy and
the minimization in eq. (132) is determined by

$\min\,\left\\{\sinh\left(\frac{\pi L_{2}}{\beta}\frac{k_{1}}{\pi-
k_{2}}\right)\,,\quad\sinh\left(\frac{\pi
L_{1}}{\beta}\right)\,,\quad\cosh\left(\frac{\pi}{\beta}(t_{\textrm{\tiny
R}}-t_{\textrm{\tiny L}})\right)\right\\}\,.$ (133)

First of all, it is easy to show that $\sinh\left(\pi L_{1}/\beta\right)$
cannot be the minimal quantity in the regime of interest. Noting that the warm
phase is preferred when $T_{\textrm{\tiny DCFT}}L_{1}\geq C_{\textrm{\tiny
CW}}$ and $T_{\textrm{\tiny DCFT}}L_{2}\leq C_{\textrm{\tiny WH}}$, we always
have

$L_{1}\geq\frac{C_{\textrm{\tiny CW}}}{C_{\textrm{\tiny
WH}}}L_{2}\simeq\frac{4\pi L_{2}}{(\pi-
k_{2})^{2}}\,\log\left(\sqrt{\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)\,.$ (134)

in the warm phase. Due to the appearance of the boundary entropy in this
inequality, the second term in eq. (133) will always be large compared to the
first. More explicitly, the second condition for the warm phase implies

$\sinh\left(\frac{\pi L_{2}}{\beta}\frac{k_{1}}{\pi-
k_{2}}\right)\leq\sinh\left(\frac{k_{1}\pi C_{\textrm{\tiny WH}}}{\pi-
k_{2}}\right)\simeq\frac{(\pi-
k_{2})k_{1}}{4}\,{\log^{-1}\\!\left(\sqrt{\frac{1+T_{o}\ell}{1-T_{o}\ell}}\right)}\ll
1\,.$ (135)

While the last inequality above is shown by taking the limit $T_{o}\ell\to 1$,
we can also numerically show that $\sinh\left(\frac{\pi
L_{2}}{\beta}\frac{k_{1}}{\pi-k_{2}}\right)<1$ for any
$T_{o}\ell\in(\frac{1}{2},1]$. These results also ensure that the first term
is also always less than $\cosh\left(\frac{\pi}{\beta}(t_{\textrm{\tiny
R}}-t_{\textrm{\tiny L}})\right)$, the third term in eq. (133).

Following the above analysis, we can find that the minimization in eq. (132)
always selects the first expression and hence the holographic entanglement
entropy $S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny L,R})}_{a,b})$ reads

$\begin{split}S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny
L,R})}_{a,b})&=S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny
L})}_{a,b})+S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny R})}_{a,b})\\\
&\simeq\frac{2c}{3}\log\left(\frac{\beta}{\pi\epsilon}\sinh\left(\frac{\pi
L_{2}}{\beta}\frac{k_{1}}{\pi-
k_{2}}\right)\right)+\frac{2c}{3}\log\left(\frac{\sin(\pi-k_{2})}{\pi-
k_{2}}\frac{L_{2}}{\epsilon}\right)\,.\\\ \end{split}$ (136)

This tells us that the mutual information $I_{\textrm{\tiny L:R}}$ in warm
phase is always vanishing, i.e.,

$I_{\textrm{\tiny L:R}}^{(\textrm{\tiny W})}=0\,.$ (137)

In the warm phase, two defects on the same boundary are connected by the
brane. The brane appears as a thread-like structure that geometrically
manifests the entanglement structure among the defects. As a result, any
defects situated on two distinct boundaries are not entangled in the warm
phase, which leads to the vanishing of mutual information (to leading order in
the large $c$ expansion).

####  3.2.3 Mutual information in the hot phase

With the increase in temperature, the bulk spacetime finally locates at the
hot phase with a black hole sitting in the middle, as shown in the global
picture 11. More precisely, the hot phase is dominant when

$T_{\textrm{\tiny DCFT}}L_{2}\geq C_{\textrm{\tiny WH}}\,,\qquad
T_{\textrm{\tiny DCFT}}(L_{1}+L_{2})\geq C_{\textrm{\tiny CH}}\,,$ (138)

where the former applies for $L1/L2>1$ and the latter, for $L1/L2\lesssim 1$.
The transition parameters $C_{\textrm{\tiny WH}},C_{\textrm{\tiny CH}}$ are
approximately given by eqs. (2.2) and (62), respectively. The advantage of the
hot phase is that the brane profile can be analytically derived as eq. (75)
which is the same as that in the symmetric setup. This simplification is
traced back to the fact that the temperatures of bulk black hole
$T_{\textrm{\tiny BH}}$ in $\mathcal{S}_{1},\mathcal{S}_{2}$ are the same. The
asymmetric feature of two boundary sizes $L_{1},L_{2}$ is decoded in the area
of the black hole horizon in $\mathcal{S}_{1},\mathcal{S}_{2}$. From the
boundary-bulk dictionary derived in eq. (48), the horizon size of each portion
is precisely shown as

$\Delta\phi_{i}^{\rm Hor}\ell=L_{i}+\frac{1}{\pi T_{\textrm{\tiny
BH}}}\,\text{arctanh}\left(T_{o}\ell\right)\,,$ (139)

which would be a significant contribution in the limit $T_{o}\ell\to 0$. For
example, the Bekenstein-Hawking entropy $S_{\rm horizon}$ associated with the
black hole in the hot phase is given by

$\begin{split}S_{\rm{horizon}}&\equiv\frac{r_{h}\Delta\phi^{\rm{Hor}}_{1}+r_{h}\Delta\phi^{\rm{Hor}}_{2}}{4G_{\textrm{\tiny
N}}}=\frac{c}{3}\pi T_{\textrm{\tiny BH}}\left(L_{1}+L_{2}\right)+4\log
g\,.\\\ \end{split}$ (140)

In the following analysis, we will find that this fact can help us to simplify
the discussion on the relevant extremal surfaces.

#### Holographic entanglement entropy $S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny
R})}_{a,b})$

Figure 17: Candidates of extremal surfaces for $S_{\textrm{\tiny
EE}}(D^{(\textrm{\tiny R})}_{a,b})$ in the hot phase.

As in the other phases, for a single defect, we have $S_{\textrm{\tiny
EE}}(D^{(\textrm{\tiny R})}_{a})=S^{\textrm{\tiny(reg)}}_{\rm{def}}$ where the
latter was defined in eq. (98). Hence we begin here with the extremal surfaces
associated with $S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny R})}_{a,b})$ for two
defects on the same asymptotic boundary. In contrast to the cases in either
the cold phase or warm phase, the extremal surfaces here always cross the
brane, as depicted in figure 17. The two configurations on the far left of the
figure were examined in the discussion of the symmetric case – see eq. (96).
Obviously, the length of the geodesics in the first configuration is smaller
than that of the second, i.e.,

$\frac{c}{3}\log\left(\frac{\beta}{\pi\epsilon}\sinh\left(\frac{\pi
L_{1}}{\beta}\right)\right)+\frac{c}{3}\log\left(\frac{\beta}{\pi\epsilon}\sinh\left(\frac{\pi
L_{2}}{\beta}\right)\right)+S_{\rm{horizon}}>2\,S^{\textrm{\tiny(reg)}}_{\rm{def}}=2S_{\textrm{\tiny
EE}}(D^{(\textrm{\tiny R})}_{a})\,,$ (141)

where we are considering small defect intervals with $\Delta L_{\textrm{\tiny
D}}\ll\ell$.

In contrast to the symmetric case, there are two other candidates of minimal
configuration which were ruled out by symmetry argument before. One can also
derive analytical results for these, the third and fourth candidates in figure
17, by using the explicit brane profile (75) and performing the minimization
along the joint point on the brane. However, there is a simpler way to show
that the last two candidates will not be the minimal extremal surface in the
present case either. The key point is that the geodesic connecting the two
branes in either $\mathcal{S}_{1}$ or $\mathcal{S}_{2}$ is bounded by the size
of the horizon. For example, let us consider the geodesic in $\mathcal{S}_{2}$
which connects two points $(r_{2},\phi_{2})$ and $(r_{2},-\phi_{2})$ on the
brane (i.e., the third plot in figure 17). Using the generic formula for
geodesic length in eq. (86), one finds

$D_{\textrm{\tiny
BTZ}}=1+\frac{2T_{o}^{2}\ell^{2}}{1-T_{o}^{2}\ell^{2}}\frac{\sinh^{2}\left(\frac{r_{h}\phi_{2}}{\ell}\right)}{\sinh^{2}\left(\frac{r_{h}(\phi_{2}-\phi_{\rm
bdy})}{\ell}\right)}\,,$ (142)

where we have substituted $r_{2}(\phi_{2})$ by using the brane profile in eq.
(75). It is then straightforward to show that
$\partial_{\phi_{2}}D_{\textrm{\tiny BTZ}}<0$. In order words, it means that
the minimal geodesic connecting the two branes is located at the intersection
between the brane and horizon, i.e., the minimal geodesic is the horizon
itself. On the other hand, we also know that the area of the minimal geodesic
starting from the brane to the defect living in $\mathcal{S}_{1}$ is given by
$\frac{1}{2}S^{\textrm{\tiny(reg)}}_{\rm{def}}$. As a summary, we can conclude
that the geodesic length of the third candidate is bounded from below by

$\begin{split}&\quad\frac{c}{3}\log\left(\frac{\beta}{\pi\epsilon}\sinh\left(\frac{\pi
L_{2}}{\beta}\right)\right)+S^{\textrm{\tiny(reg)}}_{\rm{def}}+\frac{r_{h}\Delta\phi^{\rm{Hor}}_{2}}{4G_{\textrm{\tiny
N}}}\\\ &=\frac{c}{3}\log\left(\frac{\beta}{\pi\epsilon}\sinh\left(\frac{\pi
L_{2}}{\beta}\right)\right)+\frac{c}{3}\frac{\pi
L_{2}}{\beta}+\frac{c}{3}\log\left(\frac{\Delta L_{\textrm{\tiny
D}}}{\epsilon}\right)+4\log g\\\ &>2S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny
R})}_{a})\,,\end{split}$ (143)

where the last inequality simply holds since we consider the limit $\Delta
L_{\textrm{\tiny D}}\ll\ell$. The contribution of the fourth candidate in
figure 17 is similarly bounded from below by

$\quad\frac{c}{3}\log\left(\frac{\beta}{\pi\epsilon}\sinh\left(\frac{\pi
L_{1}}{\beta}\right)\right)+S^{\textrm{\tiny(reg)}}_{\rm{def}}+\frac{r_{h}\Delta\phi^{\rm{Hor}}_{1}}{4G_{\textrm{\tiny
N}}}>2S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny R})}_{a})\,.$ (144)

In conclusion, the holographic entanglement entropy $S_{\textrm{\tiny
EE}}(D^{(\textrm{\tiny R})}_{a,b})$ in the hot phase is fixed as

$S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny
R})}_{a,b})=2\,S^{\textrm{\tiny(reg)}}_{\rm{def}}=2\,S_{\textrm{\tiny
EE}}(D^{(\textrm{\tiny R})}_{a})\,,$ (145)

which implies the vanishing of mutual information, i.e., $I_{\textrm{\tiny
R}_{a}:\textrm{\tiny R}_{b}}^{(\textrm{\tiny H})}=0$.

#### Holographic entanglement entropy $S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny
L, R})}_{a,b})$

Figure 18: Relevant extremal surfaces for $S_{\textrm{\tiny
EE}}(D^{(\textrm{\tiny L, R})}_{a,b})$ in the hot phase. Figure 19: Irrelevant
extremal surfaces for $S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny L, R})}_{a,b})$
in the hot phase.

There are two classes of extremal surfaces for $S_{\textrm{\tiny
EE}}(D^{(\textrm{\tiny L, R})}_{a,b})$ as depicted in figures 18 and 19. One
observes that the geodesics in the second figure are exactly two copies of
those considered for $S_{\textrm{\tiny EE}}(D^{(\textrm{\tiny R})}_{a,b})$
above. It is straightforward to argue that none of these can provide the
minimal length because they cross the brane and we are interested in the
regime with a large defect entropy. Hence, we focus on the relevant geodesics
in figure 18. The holographic entanglement entropy $S_{\textrm{\tiny
EE}}(D^{(\textrm{\tiny L, R})}_{a,b})$ then comes by picking the minimum area
amongst the four relevant extremal surfaces, i.e.,

$S_{\textrm{\tiny EE}}^{(\textrm{\tiny H})}(D^{(\textrm{\tiny
R,L})}_{a,b})=\min\begin{cases}S_{1}^{(\textrm{\tiny
H})}=\frac{2c}{3}\log\left(\frac{\beta}{\pi\epsilon}\sinh\left(\frac{\pi
L_{1}}{\beta}\right)\right)+\frac{2c}{3}\log\left(\frac{\beta}{\pi\epsilon}\sinh\left(\frac{\pi
L_{2}}{\beta}\right)\right)\,,\\\ \,\\\ S_{2}^{(\textrm{\tiny
H})}=\frac{4c}{3}\log\left(\frac{\beta}{\pi\epsilon}\cosh\left(\frac{\pi}{\beta}(t_{\textrm{\tiny
R}}-t_{\textrm{\tiny L}})\right)\right)\,,\\\ \,\\\ S_{3}^{(\textrm{\tiny
H})}=\frac{2c}{3}\log\left(\frac{\beta}{\pi\epsilon}\cosh\left(\frac{\pi}{\beta}(t_{\textrm{\tiny
R}}-t_{\textrm{\tiny
L}})\right)\right)+\frac{2c}{3}\log\left(\frac{\beta}{\pi\epsilon}\sinh\left(\frac{\pi
L_{2}}{\beta}\right)\right)\,,\\\ \,\\\ S_{4}^{(\textrm{\tiny
H})}=\frac{2c}{3}\log\left(\frac{\beta}{\pi\epsilon}\cosh\left(\frac{\pi}{\beta}(t_{\textrm{\tiny
R}}-t_{\textrm{\tiny
L}})\right)\right)+\frac{2c}{3}\log\left(\frac{\beta}{\pi\epsilon}\sinh\left(\frac{\pi
L_{1}}{\beta}\right)\right)\,,\\\ \end{cases}$ (146)

where the four expressions above are denoted by $\\{S_{1}^{(\textrm{\tiny
H})},S_{2}^{(\textrm{\tiny H})},S_{3}^{(\textrm{\tiny
H})},S_{4}^{(\textrm{\tiny H})}\\}$ from top to bottom correspond to the four
configurations from left to right shown in figure 18. Further, note that we
have ignored the corrections involving $\Delta L_{\textrm{\tiny D}}$ by
assuming this regulator scale is much smaller than any of the macroscopic
scales, e.g., $\beta$, $L_{1}$ and $L^{2}$.

First of all, let us note that the entanglement entropy at late times always
reduces to the constant, i.e.,

$S_{\textrm{\tiny EE}}^{(\textrm{\tiny H})}(D^{(\textrm{\tiny
R,L})}_{a,b})\big{|}_{|t_{\textrm{\tiny R}}-t_{\textrm{\tiny L}}|\geq t_{\rm
crt1}}=S_{1}^{(\textrm{\tiny